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Appendix A.
VOL. I.
Census of the Commonwealth of
Australia.
The Mathematical Theory of Population,
of its Character and Fluctuations,
and of the Factors which influence them,
BEING AN
Examination of the general scheme of Statistical Representation,
with deductions of necessary formulae; the whole being applied to
the data of the Australian Census of 191 1, and to the elucidation of
Australian Population Statistics generally.
BY
G. H. KNIBBS, C.M.G.. F.S.S., F.R.A.S., etc., ^
Member of the International Institute of Statistics,
Honorary Member of the Soci^t^ de Statistique of Paris, and of the
American Statistical Association, etc., etc.
COMMONWEALTH STATISTICIAN.
Published under Instructions from the
MINISTER OF STATE FOR HOME AND TERRITORIES,
Melbourne. ■«
By Authority:
McCARRON, BIRD & CO., Printers, 479 Collins Street, Melbourne
[C.S.— No. 312.]
FOREWORD.
The following monograpH on the Mathematical Theory of Population,
in form an appendix to the Report on the AustraHan Census of 1911, is
intended to serve a double purpose. It aims on the one hand at supply-
ing the elements of a mathematical technique, such as are needed for
the analysis of the various aspects of vital phenomena that come under
statistical review, and, on the other, at interpreting material made
available by the first Census of AustraHa which has been carried out upon
uniform lines and' by a central authority. The earUer portion of the
appendix has consequently been almost wholly devoted to the creation
of the requisite technique. Later technical solutions are introduced only
when required by way of application to any statistical analysis under
immediate review.
In the realm of official statistics there is an enormous amount of
accumulated material, which, decade after decade, remains unanalysed
and uninterpreted.- This is due to several things', viz., to the fact that
routine tabulations largely occupy the energies of the staffs of statistical
bureaux; to the fact that much of the mass of material itself is defective
and its correction involves more time than is available ; and perhaps
still more to the fact that appropriate schemes of mathematical analysis
have as yet either not been developed, or are regarded as inapphcable.
The present analyses and interpretations have yielded many results
' which, it is believed, mil be seen to be of value. They have brought into
clearer rehef the necessity for recognising that the variation of any one
statistical element affects all other statistical elements, so that the satis-
factory reduction of " crude data" to a common system is by no means
an easy undertaking, and the comparability of the statistic of two com-
munities can never be rigorously exact in all particulars. It is fortunate,
however, that practically exactitude means merely " a precision sufficient
for any particular purpose in view."
In substance this monograph consists of two elements, viz., (i.) a
technical one, and (ii.) an interpretative one. Formulse essential for the
purposes of interpretation have been deduced, and their use has been
illustrated by appHcation to the data of the AustraHan Censuses, or to
intercensal statistical data which, subject otherwise to considerable
uncertainty, could be adjusted only by means of information derived
from the Census. Thus results of immediate value are obtained simul-
taneously with an exposition of the theory and technique of the subject.
FOBEWORD.
The various formulae developed have been carefully checked through-
out, but it is too much to hope that among so many results error has
been completely avoided. The author will, therefore, be grateful if any
discoverer of errors or misprints will communicate with him.
As a rule corrections to data have been pushed as far as seemed to be
desirable ; theoretically it is often possible to push them even still farther.
It is doubted, however, whether the precision of the data'would justify
this. An example will illustrate the point. In determining the ratios
which reveal the age of maximum fecundity, if the number of women at
risk be taken as the total of the same age-group, the denominator wiU be
too large and the derived ratio too small. Hence allowances must be
made for the diminution of risk for prior cases of child-birth. But there
is no well-defined time-limit at which these allowances should stop.
In general, however, their apphcabihty becomes more questionable as
they become smaller.
A synopsis shews the general treatment of the subject, and an index,
at the end of this appendix, makes reference thereto easy. Where it
has been deemed necessary to coin technical expressions their derivation
has been indicated.
Finally it may be mentioned that many of the formulae developed
will be found serviceable in other investigations in which statistical
methods are called into requisition.
G. H. KNIBBS.
Commonwealth Bureau of Census and Statistics.
Melbourne, March 1917.
CORRIGENDA
Page 3. — Under figures in footnote : after " small figures" read " in brackets."
Page 4. — ^§ 4, line 8 : for " an" read " on."
Page 7. — Line 3 ; for " acurately" read " accurately."
Line 3, footnote, for " Gesellsohaftsehre" read " Gesellschaftalehre."
Page 8. — Sub-heading (iv.). For " interpolation" read " interpolations."
Line 7, last paragraph, insert " the" after " given."
Page 40. — ^Line H from bottom, after log x, insert " and k being log k."
Line 9 from bottom, for " fc," " 21c," " 3k," read " k," " 2k," " 3(c."
Page 55.— ^Line 1, for " of a curve" read " of the curve."
Page 68.— Formula (197(i), for (" 1 — " read " (i — ."
Line 13, after " above" add " the numerical coefficients remaining, of
course, the same."
Page 72. — In formula (211), the y should follow the sign of integration.
Page 81. — Line 4 from bottom, for (" n" read (" h."
Page 104. — ^Line 5, for" difference" read" the differences."
Line 27, for " the comparison of" read " comparisons among."
Page 144. — Lines 10 and 11, for" section" and "sections," read"Part" and"Parts,"
and for XII., read XI.
Page 163.— Line 4, for " M " read " M."
Line 7, for " 2Mr + " read " 2Mr,."
Page 213. — Line 3 from bottom, for " occupying " read " occurring."
Page 233. — Line 4 of paragraph, for " in part of the" read " in part the."
Page 240. — ^Line 4, § 8, add after " maternity," " each birth being regarded a case
of maternity."
Page, 242. — ^Throughout table read " births" for " maternity "
Page 277.— Table LXXXVIII., in " Duration," for 251-160, read "251-260,"
and for" 251-170" read" 261-270."
Page 306. — Add to end of paragraph : — " Twins produced from one ovum have
been called ' univiteUins ' and those from two ova ' bivitelUns *."
Page 307. — ^Line 3 from bottom, for " uniovulate" read " uniovular."
Note. FormulsB 374, and 396 are omitted.
SYNOPSIS.
THE MATHEMATICAL THEORY OF POPULATION, OF ITS CHARACTERS AND
FLUCTUATIONS, AND THE FACTORS WHICH INFLUENCE THEM.
I. Introductory.
1. General
2. SignificEuioe of analysis
3. The nature of the problem
4. Necessity for the mathematical ex
pression of the conditions of the problem
5. Conception applies equally to a popula-
tion de/octo or a population de/iwe
6. Nature of population fluctuations
7. Changes in the constitutions of popula-
tions
8. Organic adjustments of populations
9. Continuous and finite fluctuations
10. Curves required to represent various
fluctuations and the solution of the same
II. Various Types of Population Fluctuations.
1. Mathematical conception of rate of in-
crease
2. Determination of a population for any
instant when the rate is constant
3. Kelation of instantaneous rate to the ratio
of increase for various periods
4. Determination of the mean population for
any period ; rate constant
5. Error of the arithmetical mean ; rate
constant . .
6. Empirical expression for any population
fluctuation
7. Mean population for any period ; rate not
constant . .
8. Change, with change of epoch, of the
coefficients expressing rate
9. Error of the arithmetical mean ; rate not
constant . .
10. Expression of the coefficients in the em-
pirical formula for rate in terms of the
constant rate
11. Investigation of rate is complete only
when its variations are ascertained
12. Rate is a function of elements that varies
with time
13. Factors which secularly influence the
rate of increase . .
14. Variations which depend on natm-al re-
sources, irrespective of human interven-
tion
15. Variations of rate of long periods
16. Representation of periodic elements in
non-periodic form
17. Influence of natural resoittces disclosed by
advancing knowledge . .
18. Influences of resources dependent upon
huinan intervention
19. Effects of migration
20. Simple variation of rate, returning
asymptotically to original value
Formulse.
(l)..(la)
(2).. (4)
(5)
(6).. (7)
(8).. (86)
(9)..(9o)
(10).. (10a)
(11). .(12)
(13).. (13a)
(14)
(15)
(16).. (17)
(18). .(19)
(20)
Tables.
Fig.
Page.
1
2
3
4
5
10
10
11
11
12
12
12
13
13
13
14
14
14
16
16
17
17
17
18
18
APPENDIX A.
n. Various Types etc. — continued.
21. Examination of exponential curves ex-
pressing variation of rate
22. Determination of constants of such expon-
ential curves
23. Case of total non-periodic migration re-
presented by an exponential curve
24. Simple variation of rate, returning
asymptotically to a particular value . .
25. Examination of the preceding curve
' 26. Determination of the constants of the
curve
27. Total non-periodic migration resulting in
permanent increase but returning to
original rate
28. The utility of the exponential curve of
migration
29. Fluctuation of annual periodicity
30. Discontinuous periodic variations of rate
31. Empirical expression for secular fluctua-
tion of rate
32. Growth of- various populations . .
33. Rate of increase of variotis populations . .
34. The population of the world and the rate
of its. increase
m. Oeteiminatiou of Cuive-constants and of in-
termediate Values when the Data are
Instantaneous Values.
1. General
2. Determination of constants where a
fluctuation is represented by an integral
function of one variable
3. Evaluation of the differences from the
coefficients
4. Subdivision of intervals
5. Evaluation of constants of periodic
fluctuations . .
6. Constants of exponential curves
7. Evaluation of the constants of various
curves representing types of fluctuations
8. Polymorpluc and other fluctuations
9. Projective anamorphosis
IV. Special Types of Curves and their Character-
istics.
1. General
2. Curves of generalised probability
3. The method of evaluating the constants of
the curves of generalised probability . .
4. Flexible Curves
5. Determination of the constants of a flex-
ible curve . .
6. Generalised probability-curves derived
from projections of normal curves
7. Development of type-curves
8. Evaluation of the constants of the pre-
ceding type-curves
9. To determine the surface on which the pro-
jeotion of a normal probability-curve
wiU result in a given skew-curve
10.- Reciprocals of curves of the probability-
type
11. Dissection of multimodal fluctuations
into a series of unimodal elements
Formulse.
(20a).. (24)
(25).. (30)
(31). .(316)
(32)
(32a ..(36)
(37).. (38)
(39).. (39a)
(40).. (42)
(43).. (436)
(44) . . (45a)
(46).. (69)
(70)
(71)
(72).. (101)
(102). .(104)
(105).. (122)
(123).. (133)
134
(135.. 145)
146
(147).. (166)
(167).. (176)
(177).. (181)
(182).. (183)
(184)
(185)
Tables.
II.
III.
IV., V.
Fig.
Page.
—
19
—
21
—
22
2
22
23
6, 7
8
9-20
21-27
28-33
24
24
25
25
25
26
26
28
30
34
34
37
37
38
40
40
42
45
47
49
52
52
53
57
61
62
62
63
63
SYNOPSIS.
V. Group Values, their Adjustment and
Analysis.
1. Group-values and their limitations
2. Adjustment of group-values
3. Representation of group-values by equa-
tions with integral indices
4. Formulae depending on successive differ-
ences of group-heights . .
5. Formulae depending on the group-heights
themselves
6. Formulae depending upon the leading
differences in the groups or in group-
heights
7. Determination of differences for the con-
struction of curves
8. Cases where position of curve on axis of
ordinates has a fixed value
9. Determination of group-values when con-
stants are known
10. Curves of group-totals for equal intervals
of the variable expressed as an integral
function of the central value of the
interval
11. Average values of groups
VI. Summation and Integration for Statistical
Aggregates.
1. General
2. Areal and volumetric summation formulae
3 . The value of groups in terms of ordinates
4. The value of group-subdivisions in terms
of groups
5. Approximate computation of various
moments
6. Statistical integrations
7 . The Eulerian integrals or Beta and Gamma
functions
8. Table of indefinite and definite integrals
and limits . . . ,
Vn. The Place o£ Graphics and Smoothing in
the Analysis of Population-Statistics.
1. General
2. The theory of smoothing statistical data
3. Object of smoothing
4. Justification for smoothing process
5. Mode of application of smoothing processes
6. On smootlung by differencing •
7. Effect of changing the magnitude of the
differences
8. Smoothing, by operations on factors . .
9. Logarithmic smoothing . .
On the diHerenoe between instantaneous
and grouped results
Determination of the exact position and
height of the mode
12. The testing of smoothed or graphic results
10
11.
Formulse.
(186)
(187).. (189)
(190)..(194d!)
(195)..(197d)]
(198)..(198d)]
(199).. (200c)
(201)..(209d)
(210)..(210e)
(211).. (216)
(217).. (224)
(225).. (252)
(253).. (268)
(269).. (274)
(275).. (281)
(282).. (288)
Tables.
Fig.
34
VI.
VII.
VIII.
(289)
(290)
(291)
(292).. (298)
IX.
Page.
64
64
65
66
67
69
69
72
72
73
75
75
80
81
82
84
84
35, 36
85
86
87
87
88
89
90
91
91
91
92
94
APPENDIX A.
Vni. Conspectus of Fopulation-chaiacters.
1. General
2. Characters directly given or derivative . .
3. Characters in their instantaneoTis and
progressive relations
4. Conspectus of population-characters
5. The range of the wider theory of population
6. The creation of norms
7. Homogeneity as regards populations
8. Population norms . .
9. Variation of norms
10. Norms representing constitution of popula-
tion according to age . .
11. Mean age of a population
12. Population norm as a fvmction of age
IX. Population in the Aggregate, and its Distribu-
tion according to Sex and Age.
1. A census and its results
2. Causes of misstatement of age
3. Theory of error of statement of age . .
4. Characteristics of accidental misstatements
and their fluctuations
5. Characteristics of systematic misstate-
ment
6. Distribution of misstatement according
to amount and age of persons . .
7. The smoothing of enumerated populations
in age-groups
8. The error of linear grouping
9. Graphic process of eliminating systematic
error
■^ 10. Summation methods
' 11. Advantages of graphic smoothing over
summation and other methods
12. Graphs of Australian population distri-
buted according to age and sex for
various censuses
1 3. Growth of population when rate is identical
for all ages
14. Growth of population where migration
element is known
15. Growth of population rate of increase
varying from age to age
16. The prediction of future population and
its distrib\ition . .
X. The Masculinity of Population.
1. General
2. Norms of masculinity and femininity . .
3. Various defifiitions of masculinity and
femininity . . . . ....
4. Use of norms for persons and masculinity
only
5. Relation between masculinity at birth
and general masculinity of population
6. Masculinity of still and live nuptial and
ex-nuptial births
7. Coefficients of ex-nuptial and still-birth
masculinity
8. Masculinity of first-bom
9. Masculinity of populations according to
age, and its secular fluctuations
10. Theories of masculinity . .
Formulse.
(299) to (306)
(307)
(308).. (309)
(310)
(311).. (323)
(324).. (325)
(326).. (330)
(331)
(332)
(333).. (335)
(336)
(337).. (339)
Tables.
X.
XI.
XII.
XIII.
XIV.
XV., XVI.
XVII., XVIII
XIX., XX.
XXI.
XXII.
XXIII., XXIV,
XXV., XXVI.
XXVII.
XXVIII,
XXIX.
XXX.
XXXI.
Fig. Page.
96
96
97
98
102
103
103
104
104
37 & 38
39
40 &41
42
43 &44
125
—
127
—
128
—
128
—
129
—
130
131
—
131
—
132
45,46
133
—
136
—
137
138
47
139
140
105
106
107
108
109
109
111
112
114
116
117
119
120
124
SYNOPSIS.
XI. Natality.
1. General
2. Crude birth-rates . .
3. Influence of the births upon the birth-
rate itself
4. Influence of infantile mortality on birth-
rate
5. World-relation between infantile mortal-
ity and birth-rate
6. Eesidual birth-rates
7. Determination of proportion of infantile
deaths arising from births in the year of
record, number of births constant
8. Equivalent year of birth in cases of infan-
tile mortality
9. Proportion of infantile deaths arising from
births in year of record, number of
births increasing
10. Secular fluctuation in birth-rates
11. The Malthusian law ,
12. Malthusian equivalent interval . .
13. The Malthusian coefficient and Malthusian
gradient
14. Reaction of the marriage-rate upon the
birth-rate
15. Annual periodic fluctuation of births
16. The subdivision of results for equaUsed
quarters into values corresponding to
equalised months
17. Equalisation of periods of irregular length
18. Determination of a purely physiological-
annual fluctuation of birth-rate
19. Periodicities due to Easter
Xn. Nuptiality.
1. General
2. The nuptial-ratio . .
3. The crude marriage-rate . .
4. Secular fluctuation of marriage-rates
5. Fluctuation of annual period in the fre-
quency of marriage
6. General. — Conjugal constitution of the
population . . • . .
7. Relative conjugal numbers at each age
8. The curves of the conjugal ratios
9. The norms of the conjugal ratios
10. Divorce and its secular increase . .
11. The abnormality of the divorce curve
12. Desirable form of divorce statistics
13. Frequency of marriages according to pairs
of ages . . . . ....
14. Numbers corresponding to given differ-
ences of age
15. Errors in the ages at marriage
16. Adjustment nvmibers for ages 18 to 21 in
elusive
17. Probability of marriage of bride or bride
groom of a given age to a bridegroom or
bride of any unspecified age
18. Tabvdation in 5-year groups
Formula.
(340).. (341)
(342).. (342a)
(3426)
(343) to (348)
(349) (350),
(351)
(352),(353),(354)
(355) to 362)
(363).. (364)
(365).. (366)
(367).. (368)
(369)
(370), (371),
(371a)
(372).. (373)
Tables.
XXXII.
XXXIII.
XXXIV.
XXXV.
(375) to (395)
(397), (398)
(399)
(400)
XLV.
XL VI.
XL VII.
—
XL VIII.
—
XLIX.
L.
LI.
(401).. (402)
LII.
LIII.
(403)
(404), (405)
(406), (407)
XXXVI,
XXXVII.
XXXVIII,
XXXIX.
XL.
Fig.
48
49 to 52
XLI., XLII.,
XLIII.
XLIV.
LIV.
LV.
LVI., LVII.
LVIIL, LIX.
LX.
53
54
55
66
57, 58
59
60, 60o
Page.
142
143
144
145
147
150
152
155
158
160
162
163
164
166
166
169
171
172
173
175
175
176
179
180
180
182
185
186
186
188
189
189
192
193
195
198
198
APPENDIX A.
Xn. Nuptiality — coniimied.
19. Frequency'of marriage according to age
representable by a system of cvirved
lines . . . . . . '■ ■
20. The error of adopting a middle value of a
range
21. General theory of protogamio andgamio
sm'f aces . .
22. Orthogonal trajectories . .
23. Critical characters on the protogamio sur-
face
24. Apparent peculiarities of the protogamio
frequency
25. The contours of the protogamic surface
27. Relative marriage frequency in various
age-groups
28. The numbers of the unmarried and their
masculinity . . '. .
29. The theory of the probability of marriages
in age-groups
30. Masculmity of the unmarried in various
age-groups
31. The probabihty of marriage according to
pairs of ages
32. The relative numbers of married persons
in age-groups
33. Conjugal age-relationships
34. Non-homogeneous groupings of data . .
35. Average differences in age of husbands
and wives according to census
36. Average diSerences ofage at marriage
37. The gamic surface
38. Smoothing of surfaces
39. Solution for the constants of a surface re-
presenting nine contiguous groups . .
40. Nuptiality and conjugality norms
41. The marriage-ratios of the unmarried
Xni. Fertility and Fecundity and Reproductive
Efficiency.
1.
2.
3.
4.
5.
6.
7.
8.
9.
General
Definitions . . . . ■ •
The measurement of reproductive efficiency
Natality tables
Norm of population for estimating repro-
ductive efficiency and the genetic index
The natality-index
Age of beginning and of end of fertility . .
The maternity frequency, nuptial and ex-
nuptial, according to age, and the
female and male nuptial-ratios
Nuptialand ex-nuptial maternity and their
frequency -relations
Maximum probabilities of marriage and
maternity, etc.
11. Probabihty of a first-birth occurring with-
in a series of years after marriage . .
12. Maximum probabiUties of a first-birth . .
13. Determination of the co-ordinates of the
vertices . .
14. Average age of a group . .
15. Curves of probability for different inter-
vals derived by projection . .
Number of first-births according to age
and duration of marriage
10,
16,
FormulsB,
(408), (409),
(410), (411)
(412) to (416)
(417), (418),
(419)
(420) to (424)
(425) to (435)
(436)
(437)
(438), (439)
(440), (441)
(442) to (452)
Tables.
LXI., LXII.
LXIII. .
LXIV.
LXV., LXVa
LXVI., LXVII
LXVIII.
LXIX.
LXX.
(453)
(454)
(455), (*6)
(457 to (461)
(462) to (465)
(466)
(467)
(468), (469)
(470), (471)
(472), (473)
Fig. Page,
199
200
201
203
203
61, 62
63
64
65
LXXI., LXXII.
LXXIII.
LXXV.
LXXVI.
66 to 70
71
I LXXVII.
208
208
211
212
214
218
223
223
224
224
225
226
228
229
230
232
232
233
233
235
236
237
237
238
240
243
245
245
248
249
250
250
251
SYNOPSIS.
Xm. Fertility and Fecundity — continiied.
17. The nuptial protogenesic boundary and
agenesic surface
18. Curve of nuptial protogenesic maxima . .
19. Bx-nuptial protogenesis . ,
20. Average age for quinquennial age-groups
of primiparae . .
21. Average interval between marriage and a
first-birth, a function of age
22. The protogenesic indices
23. Exact evaluation of the average interval
from a limited series of age-groups
Evaluation of group intervals for an ex-
tended number of groups
Average interval for curves of the expon-
ential type
Positions of average intervals for groups
of all first-births
27. The unprejudiced protogenesic interval . .
24,
25,
26
28.
29.
30.
Protogenesic index based on age at and
duration of marriage . .
Protogenesic quadratic indices and quad-
ratic intervals . .
Correction of the protogenesic interval for
H. population whose characters are not
constant . .
31. Proportion of births occurring up to any
point of time after marriage
32. Range of gestation period
33. Proportion of births attributable to pre-
nuptial insemination . .
34. Issue according to age and duration of
marriage
35. Initial and terminal non-linear character
of the average issue according to dura-
tion of marriage . .
36. The polygenesio, fecundity, and gamo-
genesic distributions
37. Diminution of average issue by recent
maternity
38. Crude fertility, according to age, corrected
for preceding cases of maternity
39. Age of greatest fertility
40. Feoundity-oorrection for infantile mortality
41. Secular trend of reproductivity
42. Crude and corrected reproductivity
43. Progressive changes in the survival co
efficients
XIV. Complex Elements of Fertility and Fecundity.
1. General
2. Correspondence and correlation
3. Corrections necessary in statistics involv-
ing the element of duration . .
4. Distribution of partially and wholly speci-
fied quantities in tables of double entry
5. Unspecified cases follow a regular law . .
6. Number of children at a confinement — a
function of age
7. Relative frequency of multiple births . .
8. Uniovular and diovular multiple births
Formulse.
(474)
(475) to (4786)
(479) to (490)
(491) to (495o)
(496) to (510)
(511) to (517)
(518) to (521)
(522), (523.)
(524)
(525)
(526), (527)
(528) to (533)
(534)
(535), (536)
(537) to (540)
(541)
(542)
(543) to (547)
(548),(549),(550)
(o51),(552),(553)
Tables.
Fig. Page.
72,73
LXXVIII.
LXXIX.,
LXXX.
LXXXI.
LXXXII.
LXXXIII.,
LXXXIV.
LXXXV.-
LXXXVI.
LXXXVII.
LXXX VIII.,
LXXXIX.
XC, XCI.,
XCII.
XCIII.
XCIV.
xcv.
XCVI.
74,75
76 to 79
80
81
XCVII.
XCVIII.
XCIX.
C, CI.
cii., cm.,
CIV.
82
255
256
257
■257
257
260
261
262
264
267
268
271
272
274
276
276
278
279
282
285
286
289
290
291
292
293
295
297
297
298
300
302
303
305
306
APPENDIX A.
StV. Complex Elements of Fertility — continued.
9. Small frequency of triovulation . .
10. Nuptial and ex-nuptial probability of
twins, according to age
11. Probability of triplets according to age ..
12. Probability of twins, according to dura-
tion of marriage
1 3. Probability of triplets according to duration
of marriage
14. Remarkable initial fluctuation in the fre-
quency of twins according to interval
after marriage . .
15. Frequency of twins according to order of
confinement
16. Secular fluctuations in multiple births . .
17. Comparison of nuptial and ex-nuptial
fertility
18. Theory of fertility, sterility and fecundity
1 9. Past fecundity of an existing population . .
20. Fecundity during a given year
21. Nvimber of married women without child-
ren, all durations of marriage
22. SteriEty-ratios according to age and
duration of marriage . .
23. Curves of sterility according to duration
of marriage
24. Fecundity according to age and duration
of marriage
25. The age-genesic distribution
26. The durational genesic distribution
27. The age-fecundity distribution . .
28. The durational fecundity distribution . .
29. The age-polyphorous distribution
30. The durational polyphorous distribution
31. Fecimdity distribution according to age,
duration of marriage and number of
children borne
32. The duration and age-fecundity distri-
butions . .
33. The duration and age-polyphorous dis-
tributions
34. The age and durational fecundity distri-
butions . .
35. The age and durational polyphorous
distributions
36. Fecundity-distributions according to age
at marriage
37. Complete tables of fecimdity
38. Digenesic surfaces and diisogenic contours
39. Diisogenic graphs and their significance . .
40. Diisogens, their trajectories and tangents
41. Digenesic age-equivalence in two popula-
tions
42. Birth-rate equivalences for given age-
differences
43. Diisogeny in Australia
44. Diisogeny generally
45. Multiple diisogeny
46. Twin and triplet frequency according to
ages
47. Apparent increase of frequency of twins
with age of husbands
48. Triplet diisogeny . .
49. Frequency according to age and according
to order of confinerhent
50. Unexplored elements of fecundity
Foimulie.
(554)
(555)
(556)
(557)
(558),(559),(560)
(561)
(562)
(563)
(564) to (569)
(570)
(571) to (575)
(576)
(577),(578),(579)
(580) to (586)
(587)
(588) to (591)
(592)
(593)
(594)
Tables.
CV.
CVI.
CVII. CVIII.
cix., ex.
CXI., CXII.
CXIII., CXIV.
CXV., CXVI.
CXVII,
CXVIII.
CXIX., CXX.
CXXI.
CXXII.
CXXIII.
CXXIV.,
CXXV.
CXXVI.,
CXXVII.,
CXXVIII.
Fig. {Page.
CXXIX.
cxxx.
CXXXI.
CXXXII.,
CXXXIII.
83
84
85
86
87
88
89, 90
91
92
93
94
95
96, 97
309
309
310
311
311
312
314
316
317
319
321
324
326
327
331
331
333
333
334
335
335
336
337
340
340
340
340
345
349
349
350
352
353
354
366
361
363
364
367
367
368
368
SYNOPSIS.
XV. Mortality.
1. General
2. Secular changes in crude death-rates .
3. Secular changes in mortality according to age
4. The changes in the ratio of female to male
mortality according to time and age . .
5. Secular changes in mortality vary with age
6. Fluent life-tables
7. Determination of the general trend of the
secular changes in mortality
8. Modification of the general trend by age
9. Significance of the variations in the mor-
tality improvement ratio
10. The plasticity curve
1 1. Rate of mortality at the beginning of life
12. Composite character of aggregate mortal-
ity according to age . .
1 3. The curve of organic increase or decrease
14. Exact value of abscissa corresponding to
the quotient of two groups . .
15. Absence of climacterics in mortality
16. Fluctuations of the ratio of female to male
death-rates according to age
17. Rates of mortality as related to conjugal
condition
18. Exact ages of least mortality
19. General theory of the variation of mor-
taUty with age . .
20. The Gompertz-Makeham-Lazarus theory
of mortality
21. Theory of an actuarial population
22. The relation between the mortaUty curve
and the probabiUty of death
23. Limitations of the Gompertz theory and
its developments
24. Senile element in the force of mortaUty . .
25. The force of mortality in earlier childhood
26. Genesic and gestate elements in mortaUty
27. Norm of mortality-rates . .
28. Number of deaths from particular causes
29. Relative frequency of deaths from par-
ticular diseases according to age & sex
30. Death-rates from particular diseases ac-
, cording to age and sex
. 31. Rates of mortality during the first twelve
months of life . .
32. Annual fiuctuation of death-rates
33. Studies of particular causes of death,
voluntary death
XVI. Migration.
1. Migration
2. Proportion bom in a country . .
3. Correlation, owing to migration between
age and length of residence . .
4. The theory of migration
5. Migration-ratios for Australia
6. Periodic fluctuations in migration
7. Migration and age
8. Defects in migration records and the closure
of results
Formulae.
(595) to (600)
(601)
(602) to (604)
(605),(606),(607)
(608),(609),(610)
(611),(612),(613)
(614), (615) ■
(616) to (627)
(628)
(629) to (629/)
(630) to (638)
(639) to (644)
(645), (646)
(647) to (649)
(650), (651)
(652) to (654)
(655) to (660)
(661), (662)
Tables.
CXXXIV.
CXXXIVa
CXXXV.
CXXXVI.,
CXXXVII.
CXXXVIII.
CXXXIX..
CXL.
CXLI.
CXLII.
CXLIII.
CXLIV.
CXLV.
CXL VI.
Fig.
98
99
100
101
CXL VII.
CXL VIII.
102
103
CXLIX
CLIIL, CLIV.
CLV., CLVE.
CLVII.
CLVIII. to
CLX.
CLXI. to
CLXIII.
104
105
106
Page.
370
373
374
375
378
380
382
382
387
389
389
392
394
395
399
399
400
401
402
405
407
408
410
411
412
413
413
414
414
415
415
424
426
429
429
431
431
433
435
439
439
APPPENDIX A.
XVn. Miscellaneous.
Formulae.
Tables.
Fig.
Page-
1. General
_
_
,
440
2. Subdivision of population and other
groups
(663) to (667)
—
—
440
3. The measure of precision in statistical
results
(668)
—
—
441
4. Indirect relations
—
—
107
442
5. Limits of uncertainty
—
CLXIV.
—
443
6. The theory of happenings or " occurrence
frequencies"
(669) to (686)
—
—
444
7. Actual statistical curves do not coincide
with elementary type-forms . .
. —
—
. —
448
8. International norm-graphs and type-
curves
—
—
—
449
9. Tables for facilitating statistical com-
putations
—
—
—
450
10. Statistical integrations and general for-
mulae
—
—
—
450
Table of Integrals' and Limits . .
—
—
—
451
XVm. Conclusion.
1 . The larger aim of population statistic
453
2. The impossibUity of any long-continued
increase of population at the present
rate
—
—
—
454
3. Need for analysis of existing statistical
material . .
—
_
—
455
4. The trend of destiny
—
—
—
456
APPENDIX A.
THE MATHEMATICAL THEORY OF POPULATION, OF ITS
CHARACTER AND FLUCTUATIONS, AND OF THE
FACTORS WHICH INFLUENCE THEM.
L— INTRODUCTORY.
1. General.^The fundamental elements of social statistics are the
fluctuations of the numbers and constitution of the population and of
its various characteristics. These fluctuations are profoundly affected
by many factors, only some of which are susceptible of physical ex-
pression. For example, the extraordinary development, characteristic
in the last few decades, of every branch of science and technology, and the
skill with which acquired knowledge has been applied to the exploitation
of Nature's resources, have probably created the possibility of develop-
ing a considerably larger population than the world has yet carried, at
least in historic times. On the other hand, the social standards have
been so profoundly altered as to strongly counteract the effect indicated.
Thus the raising of the standard of living, and an increased complexity in
social organisation have held in check, more or less, that increase of
population which might otherwise have been possible.
The opposition of tendency involved by the coexistence of these
two factors necessarily reinforces the interest, while it increeises the
difficulty of the problems which depend for solution on an evaluation
of the degree of influence exerted by particular factors. The interest
of any theory is evident when we ask : " What, on the whole, is indicated
by past statistical history as to the future populations of the various
races of the world ? " This is a question, the correct answer to which is
a necessary guide for national policy, and one which involves not only
the accumulation of statistical facts that have now become available,
but also a theory by means of which a forecast can be made as to what the
immediate future has in store for each community.
An interesting illustration of this may be drawn from the history
of the United States. In the year 1815, Elkanah Watson predicted with
extraordinary accuracy the population of the United States up to the
year 1860, by some method which, though not absolutely doing so, was
sensibly equivalent to simply assuming a constant rate of increase.
As a matter of fact, had Watson actually assumed that the rate of in-
crease from 1790 to 1800 would remain constant till 1860, he would have
predicted the population with still greater accuracy than he actually
did. This will be made apparent hereinafter ; see also Figs. 3 and 4,
APPENDIX A.
The more complex conditions of the world to-day and the rapidity
of the development of the arts and sciences, make the accuracy of pre-
diction for so lengthy a period extremely doubtful ; nevertheless an
attempt to forecast the affairs of any country, to be well founded, must
be based upon the results of a review, among other things, of aU the
facts of its population development, and upon a study of this develop-
ment in aU other parts of the world.
Of no less interest is the constitution of a population in respect of
age, sex and race, and the influence of birth-rates and death-rates there-
upon. The effect of age at marriage, the reproductivity as measured
by frequency of childbirth, and the age at which it occurs, the pro-
bability of living at every age, and the variation of this probability
with increasing scientific, hygienic and economic knowledge, are problems
of the first order of importance.
The attempt is here made to give a rough outhne of the theory of
the subject, elucidating that theory where it seemed desirable by quanti-
tative examples.
2. Significaiice of analysis. — ^The fluctuations in the number and
constitution and other characters of populations present, ia general,
complex and dissimilar changes, and depend upon elements which will
not readily lend themselves to prediction. They would thus appear
at first sight not to be amenable to mathematical analysis. Never-
theless, when the fluctuations are analysed and expressed in mathe-
matical form, their trend often becomes much more definite, and their true
significance is more clearly revealed. ^
^ An example will illustrate what is meant. The populations in the United
States in 1790 and 1820 were respectively 3.93 and 9.64 milUons of people. If the
number were supposed to iaorease at each instance at a uniform rate so as to give
these numbers in the years mentioned, the deduced populations would be very nearly
the actual ones, not only for the iutermediate decades, but even up to the year 1860,
as is evident from the following table, viz. : —
Year ..
1790
1800
1810
1820
1830
1840
1850
1860
Population supposed
to increase at uni-
form rate (millions)
Actual population
(millions)
Difference (millions)
3.93
3.93
.00
5.30
5.31
.01
7.15
7.24
.09
9.64
9.64
.00
13.00
12.87
.13
17.53
17.07
.46
23.65
23.19
.46
31.89
31.44
.45
A remarkable prediction by Elkanah Watson is referred to later : see Figs.
3 and 4.
This fact, viz., that the supposition made is approximately true, throws light
on the other facts. Thus, that to accord with this supposition the figures for 1800
and 1810 are very slightly too small, while those for 1830 to 1860 are somewhat in
excess ; and the excess is constant for 1840, 1850, and 1860 ; illustrate the value of
the scheme of analysis by means of which the fundamental idea is ascertained. The
deviations of the actual values from those computed on the assumption of uniform
rate of increase may thus, indeed, become in turn the starting point of a further
analysis undertaken with a view to the interpretation of the departure from the law
of imiform increase, arbitrarily adopted as the norm of the phenonxena.
INTRODUCTORY.
For this reason it is proposed to develop the mathematical con-
ceptions which may serve as the foundation of definite analyses of the
fluctuation of any population ; to express these conceptions by formulae ;
to so develop and resolve the formulae that they may be readily applied ;
and, where necessary, to illustrate their application.
3. The nature of the problem. — ^An ideal theory of population is
one which would enable the statistician not only to determine definitely
the influences thereupon of the various elements of human development,
and of the phenomena of Nature, but also to examine all facts of interest
to mankind, as they stand in relation to population. And however
hopeless may be the expectations of establishing such a theory with
meticulous precision and in all detail, it nevertheless remains true that
fluctuations of population can often be adequately understood only
when they are analysed by means of definite mathematical conceptions.
Moreover, since all important facts concerning population are susceptible
of numerical expression, analjrtical conceptions formulated for the pur-
pose of giving exactitude to a knowledge of its variations, should be
ultimately cast, if possible, in a mathematical mould. *
The total population-aggregates of some countries have been found
to increase almost exactly at a uniform rate ; in general, however, the rate
fluctuates. " Can the characteristics of such fluctuations be subsumed
under any conception ? " is a question which naturally presents itself.
* To revert to a previous illustration, for example, if we ask : " What uniform
rate of increase would cause a population of 3.93 millions to become 9.64 millions
in 30 years ? " the answer is that it would be necessary that each million persons
should receive at each instant an addition at the rate of 29,910 persons per annum,
that is to say, the rate of continuous increase would have to be 0.02991 per annum.
More exactly, this would give the following figures, viz. : —
3,930,000; (+ 1,370,173) = 5,300,173; (+ 1,847,877) = 7,148,050;
(-1- 2,492,128) = 9,640,178^^^^^^^
The differences, shewn by the small figures^o not in themselves disclose the
fact that the increase is at a uniform rate, but on dividing each by the preceding popiUa-
tion figures it is seen to be equivalent to adding 348,644 persons per million per
decennium. Hence, obviously, the rate of increase was constant. This rate will
be found to be equal to an increase of 30,361.8 annually per million of the population
at the beginning of each year.
The facts just indicated, viz., that starting with a population of 3,930,000, and
uniform increases at the rate of 0.02991 per anniim, gives a population of 5,300,173
in ten years, etc. ; that an equivalent figure is given for the population if, at the end
of each year, there is added to it an absolute increment of the amount of 0.0303618
of the population at its beginning ; that the figvu^es at the end of a decennium are
given by adding an increment of 0.348644 of the population at the beginning of the
decennium— can be elucidated only by formulating a definite conception of rate,
and studying the consequences that flow therefrom. It is, for example, by no means
immediately obvious that, used with the limitations above indicated, the three
sets of figures will give identical results. The last will accurately give only decennial
results ; the middle value only annual ; the rate of continuous increase is the only
one which is appropriate to furnish correct results for any moment during the whole
period under review : see Fig. 4.
APPENDIX A.
Such answer as may be given must, if it is to be explicit, obviously be
in the form of a mathematical theory of the subject. Such a theory will
be found to involve two elements, viz. : —
{a) The appropriation of suitable conceptions of a mathematical
character, and
(6) The development of a scheme of using them.
The propriety of the apphcation of such conceptions is to be
measured by the extent to which they are capable of illuminating the
actual facts, and of reducing them to system.
What has been said regarding total population, appUes equally to
each constituent part, viz., to the totals for each sex, to the number of
both sexes or of either sex at birth or at a particular age, to the ratio of
the sexes, to the fluctuations in the rates of birth or death, and to all
the circumstances of migration.
In other words, any fact, either of the condition or constitution of
population at any moment, or of the relation of these at different moments
can be readily subsumed under appropriate mathematical conceptions
with suiScient precision for practical purposes.
Again, in deaUng with the co-ordination of population with other
related facts susceptible of statistical statement, the question often
arises : " How can the nature of the relation be best defined or best
disclosed ? " The selection of appropriate mathematical conceptions,
and the means of bringing the facts under them, also constitute phases
of the theory to be considered.
4. Necessity foi the mathematical expression of the conditions
of the problem. — ^Although, in the nature of the case, the population of
any territory necessarily changes through births and deaths by whole
units, and in instances of immigration and emigration sometimes by
relatively large groups of units, no appreciable error will ordinarily be
committed, at least where the aggregate population is large, if all its
fluctuations be supposed to take place continuously and by iuiinitesimal
increments. This supposition, which might appear . ai^^nsuffioient
consideration to be physically invahd, very fairly represents, after all,
the, actual facts, in their totality.*
1 For, when all tjie oircufnstanoes are tak;en into account, it is obviojis that the
extent or degree to whicli the individuals of a community participate in its economic
and general life, pr in territorial occupation, passes through a wide range of values.
These considerations have application even to, the circumstances of birth, and death,
and even moreover to those of immigration and emigration. The ordinary involve-
ment of a community by each individual through the circun;stances preceding
birth and following upon death, s,hew clearly that in ijnany important respects the
introduction and disappearance of a imit of the population is, virtually, not quite
instantaneous.
It is obvious, too, that this consideration would apply even if registration, or
rather the statistical recognition of that fact, were contemporaneous with birth and
death, which, however, it is not, since ordinarily it follows these events by a period
of varying length. In cases of birth it also stretches over a longer period. It
INTRODUCTORY.
Thus the fluctuations of population therein may at least in ordinary
cases, be represented with precision by an imaginary or fictitious popula-
tion, the ideal fluctuations of which, varying with time, conform to all the
laws of infinitesimal increment or decrement, in this way rendering
those fluctuations amenable to a rigorous analysis by the methods of the
infinitesimal calculus. Such an imaginary population, changing con-
tinually by infinitesimal amounts, not only accurately represents the
totaUty of facts, but is amenable to mathematical treatment.
It is nevertheless important to bear in mind that actual pojJulation-
changes may be oscillatory, as will later be shewn.
5. Conception applies equally to a population " de facto " or a
population "de jure." — ^Population may be related to territory in two
ways, viz., by actual presence, and by legal relationship therewith ; that
is to say, the relationship may be " de facto " or " de jure " ; and official
statements regarding population are of each kind. In some countries,
as where the floating population is large, or where citizens are under
special obHgations {e.g., military service, etc.), the main concern may be
to ascertain the population which may be said to belong to, or to be
domiciled in the place, the foreign migratory element, whatever its
magnitude, being regarded as of relatively little moment. Again,
where communal rights are exphcit and of an important character, the
general reasons for deciding to adopt the " de jure " relationship for the
official enumeration of population may be very cogent. ^
The association of a human being, however, with any particular
territory, defimited by frontiers of any tj^e whatever, is, after aU, only
one of degree, so that any criterion {e.g., nationality, domicile, etc.),
other than that of mere presence in the territory, however necessary for
certain purposes, is more or less indeterminate for others, particularly
in countries where the freedom of movement of the individual is practic-
ally unrestricted. The actual presence of an individual in any territory
involves, in varying degree, ^ the whole scheme of general relationship
which every unit has to the general community in which he finds himself,
and which that community has to the territory it is occupying. He is
is considerably influenced by legal prescriptions in regard thereto, as well as by the
traditions and cireiuustaiioes of the community. Thus the registration of death
must perforce quickly follow on its occurrence ; not so the registration of birth.
In a sparsely -populated district, the registration of birth may be very late as com-
pared with registration in a densely -populated area.
We may remark in passing, that official estimates of population, at least when
based upon accurate vital and migration records, as ordinarily kept and reported,
are usually slightly in error as regards actual populations, viz., to an extent cor-
responding to the want of balance between inclusions at the beginning of a period of
record, really belonging to a previous record, and exclusions at the end of the period
owing to complete information not being to hand. In an increasing population
the error tends on the whole to be one of defect.
^ As, for example, in some of the Cantons in Switzerland.
^ The economics and general relationship of individual with a community
passes through a wide range of values, and in each individual the value varies with
his age.
APPENDIX A.
subject to the laws and to the same extent also the general civic and other
responsibilities of the place, while the community, on the other hand, is
concerned with his protection and well-being. Hence the " de facto "
population may often be statistical desideratum. For other purposes
obviously the " de jure " population is a necessity.
For the general purposes of economics there are features character-
istic of population which may be considered either in the " de jure " or
the " de facto " relationship, which may call for specialisation in any
mathematical treatment. For mere enumeration, however, the mathe-
matical conception as above defined will apply with equal rigour to
either.
6. Nature of population fluctuations.— The fluctuations of the en-
tire population of the earth, if available for long periods, would probably
disclose in their most general aspect the secular characteristics of its
increase, which must have greatly varied. Merely local effects would
to a large extent disappear in the total ; opposite periodicities, dependent
on seasons, would be balanced by the inclusion of results from both
hemispheres ; by taking quinquennial, decennial, or longer means or
averages, the effect of minor fluctuations would be correspondingly
eliminated ; and the broad outhnes of the facts of the growth of the
world's population would be brought into reUef. Were the curve of
secular increase of population for the entire earth available, it would
obviously constitute the most suitable norm for general comparative
purposes. Statistic unfortunately, has, however, not yet attained to
this. All we can assert with certainity is that the present rate of in-
crease can have existed for a relatively short time only.
Limiting the consideration to particular countries, changes will be
found exhibiting the following features, viz. : —
(i.) The rate of appearance of individuals by birth, and disappear-
ance by death is not, in general, uniform throughout the year,
but shews more or less definitely an annual period.
(ii.) The movement of floating population is also non-uniform,
disclosing, in many instances, definite annual periodicity.
(iii.) Improvements of natural conditions are in general followed by
changed rate of increment to the population, which may
have a period of a considerable number of years, or may be
brief.
(iv.) Variations of social and economic traditions profoundly affect
the rate of increase of population.
For the larger purposes of statistic, elements of the type (i.) and (ii.)
are ordinarily negUgible ; while those of the type (iii.) and (iv.) are of
the first order of importance. For minor purposes the converse may be
true. Hence, the scheme of any investigation must be adapted to the
element under consideration.
INTRODUCTORY.
In general, secular and long-period changes must be eliminated in
order to accurately study minor and short-period changes ; and con-
versely, minor periodic changes must be eliminated in order to acurately
ascertain the characteristics of the secular changes.
7. Changes in the constitution of populations. — ^The ratio of the total
numbers of each sex, the proportion of the sexes at each age, the relative
birth, marriage, and death rates, the circumstances affecting fecundity,
the consequences upon all of these of migration, of disease, of war, and of
economic and social traditions and developments, as well as their fluctua-
tions with the lapse of time, are necessarily matters of statistical concern.
Such changes may be called " constitutive changes," or perhaps " organic
changes," and their analysis and subsumption under mathematical ex-
pressions are often of importance and are essential in various statistical
analyses.
8. Organic adjustments of populations. — ^In reviewing the constitu-
tion of population as a whole, it is obvious that organic adjustments
occur.^
The nature and drift of such adjustment as has been indicated, or
of the deviations of the actual constitution of a population at any moment
from some norm adopted for comparison, and the changes in such devia-
tions, can be effectively studied only by the estabhshment of a system
of suitable mathematical relations. For such deviations to be made the
subject of prediction, the law of their fluctuation with time, must, of
course, be ascertained. The principles guiding the constitution of a
norm will be illustrated hereinafter.
9. Continuous and finite fluctuations. — ^The scope of the mathe-
matical theory of the fluctuation of population reveals its fundamental
importance. Every form of fluctuation, whether of total population,
or of its constitutive elements, of its characters, or of the influences
to which these are subject, may ordinarily be regarded as changing
continuously by infinitesimal increments or decrements within the
period during which it is assumed to vary. In special cases the fluctua-
tions may even be discontinuous.
1 In Europe, for example, of those bom living, there are about 105 male births to
every 100 female births : of those still-bom the proportion is about 133 (see
" Pie Geborenen nach dem Geschleoht," in " Statistik und Gesellsohaftsehre," by
Prof. Dr. Georg von Mayr. Bd. II., § 56, p. 189), and the deviation from these
figures for different countries is, in general, small. Nevertheless, in the total popula-
tion of Europe there is a ratio of only about 97.6 males to 100 females. To war and
unhealthy occupation, and accident, the death of a considerable number of males is
directly attributed. Thus there are no less than about 108 deaths of males to 100
deaths of females, for a number of countries. Nevertheless, because of the larger
number of male births, the percentage does not materially change.
APPENDIX A.
The aim of any definitive consideration of the subject is to express
the fluctuations of population or of its constituent elements, and of its
characters, in forms which will serve —
(i.) To render intelligible the characteristics of such fluctuations.
(ii.) To assist attempts at tracing the cause and effect of fluctuations.
(iii.) To determine means and averages, etc.
(iv.) To make aU required interpolation of values.
(v,) To make prediction by extrapolation possible, or to make it
possible by the result of a general analysis.
(vi.) To bring into clear reUef the various characters of a population.
10. Curves required to represent vaxious fluctuations and the solution
of the same. — ^When a curve or " graph " representing a series of
statistical results can be defined with sufficient accuracy by some form
which is susceptible of geometrical or algebraical representation, such
definition constitutes an advance as regards the understanding of the
essential nature of the facts : a clearer conception of the statistical
results is attained. For example, if the rate at which a population is
growing be constant, then the curve passing through the terminals of the
ordinates (whose length represents the successive values of the population)
plotted against distances along an axis representing time, is a curve which
is concave upward. This curve is of character such that, it, instead of
plotting the ordinates on the natural scale, their logarithms be plotted,
the terminals will be found to lie upon a straight hne. Thus, if when the
logarithms of the numbers of any population at different dates are plotted
as ordinates, and the times as abscissse, the points are found to lie on a
straight Une, we know that the rate of increase is constant.
To thoroughly represent and to analyse the nature of the changes
in the size of any population or the changes in its constituent elements
or characters, a considerable command of schemes of curve-representation
is a desideratum. For the mathematical representation of fluctuation,
therefore, it is, in general, necessary to know the geometrical form or
graph of various algebraic or other mathematical expressions ; in order
that, given geometrical form or graph of a series of results, the mathe-
matical expression appropriate to represent it wiU be "recognised. For
this reason a considerable number of type-curves and a knowledge of
their graphs must be at the disposal of the statistical analyst, so that
the appropriate expression may be selected. As soon as it is decided
upon, the mode of solving for the constants of the representative ex-
pression becomes of importance. With this ia view, it has been found
desirable to give a considerable number of formulae, and to indicate the
methods by means of which the constants that make the expression
definitive can be found.
INTRODUCTORY.
This has been the more necessary, because, after all, the scheme
of statistical representation, or the "fitting of curves," is an art of
much difficulty, and one which is only in its infancy.
The fluctuations of the numbers representing population and its
various characters make considerable demands in regard to knowledge of
this kind, and consequently not only are formulae given herein from time
to time, but their " graphs " are also drawn. These exhibit the character
of the curves represented. It will be seen that the interpretation of
statistical results therefore make considerable demands of what is called
curve-tracing.^
1 The "Spezielle algebraische iind transzendente ebene Kurven, Theorie und
Geschiohte," of Dr. Gino Loria, 2 vols., Teubuer, Leipzig, 1910-1911 ; the "Samm-
luug von Formehi der reinen und angewandten Mathematik," by Dr. W. Laska,
Ft. Vieweg und Sohn, Braunschweig, 1888-1894; and Frost's well-known "Curve
Tracing," give much valuable information in regard to the possibility of representing
certain important forms. These works, however, are neither adequate nor exhaus-
tive. The work of Felix Auerbach on " Physik in graphischen Darstellungen,"
Teubner, Leipzig, 1912, has also a large number of forms of importance to statis-
ticians.
n— VARIOUS TYPES OF POPULATION FLUCTUATIONS.
1 . Mathematical conception of rate of increase. — Whether diminish-
ing or gaining, any actual population may be replaced by a " representa-
tive population," assumed to change at every moment by infinitesimal
amounts at some rate (p say) per unit of time. That is to say, p will
denote the fraction of a unit which, at the instant under consideration,
measures the rate of change of the population for a unit of time.
Hence, if Pj be the population at the time t, and Pt + dt that at the time
t -\- 8t, then where 8tis small we shall have
(1) P.^st = Pt{l + pSt)=P^e''''
as the fundamental expression for its fluctuation. In other words-
Pt pSt is the absolute change in the time St. If p be positive, the change
is an increase ; if negative, it is a decrease.
The rate p may be either constant, in which case we shall denote
it by r, or it may on the other hand vary in some determinate way
with time, in which case we shall retaia the Greek letter. If the fate
be regarded as a function of time, then we should have
(la) P^^^^ = Pt\p{l+<l>t)dt]
We shall consider initially the case where it is constant.
2. Determination of a population for any instant when the rate is
constant.^ If increments of population be supposed to be added at N
uniform intervals of time, extending over the period t, at the uniform
rate r per unit of population per unit of time, then, putting P^ for the
initial population and Pt for that at the end of the time t, we shall have,
(2) Pt = Po ( 1 + ^)'"; = Poe'
when N becomes finite.
As usual e denotes the base of Napierian logarithms, viz. : —
2.7182818284590, etc.
It is sometimes convenient to put this expression in the form of a
series : thus, by the exponential theorem, we have
(2a) Pt = Po(l +>t+~ + ~ + etc.)
Taking logarithms of both sides of (2), we notice that
1 AVhen p is constant the investigation is analogous to that for determining
the increase in a sum of money when interest is supposed to accrue at every instant of
time. For a development of the theory of continuous interest and a kindred investi-
gation of population, see a paper by J. M. Allen, Joiim. Inst. Actuaries, Vol XLI
p. 305.
TYPES OF POPULATION FLUCTUATIONS. 1 1
(3) log Pt =logPo + (»■ log e) t
hence, if r be constant, the graph obtained by passing a line through the
points formed by plotting as ordinates the logarithms of the population
for successive years, quinquenniums or decenniums, opposite the cor-
responding values of t as abscissae, wUI be a straight line, the tangent of
whose angle with the axis of abscissae is r log e. We shall call this graph
the partial^ logarithmic homologue of the graph of equation (2).
The value of log^„ e is 0.4342944819032, etc., and of logj„ (log^^e)
is 9.6377843113005, etc.^ Both are required in practical calculation,
to, however, only few places of decimals.
To find the constant rate of increase, we have
(4) r= (log Pt -^ log Po )/{t\oge)
3. Relation of instantaneous rate to the ratio of increase for various
periods. — We may call the constant r the constant rate of continuous
increase, and similarly the variable p the instantaneous rate of con-
tinuous increase. It is often necessary, however, to substitute for r
the equivalent rate for a year, or for five or ten years, that is to say, to
measure the ratio at which the population at the beginning of the period
must be increased in order to give it its proper value at the end thereof.
Calling this rt, we have
(5).- r, ={Pt -P„ )/P„ = /'-I; ore''' =1 -f r^
4. Determination of the mean population for any period : rate
constant. — ^Let Pq denote the population at the beginning of any period
and Pt the population after the time t : then, since /e""' dt = e'V^, the
mean population P,„ is obviously
I/-'. .. Po rt.,... PqK'-D ^ I , rtrH
2
(6)....-J^Ptdt=^'lertdt^ r^ =Po{ l + 2r+^+etc. )
a formula which is suitable for determining the mean from the initial
population. This expression may be put also in the form, see (5)
(7) (Pt — Po)/rt; or P^rt / rt
by means of which, when the rate is constant and known, the mean
population can be calculated, either from the absolute increase for a
given period, or from the ratio of the increase for a given period to the
initial population for that period.
^ Partial, because the values of t and not of the logarithms of t are not used as
the abscissee.
' 9 is used instead of I.
12 APPENDIX A.
5. Error of the arithmetical mean : rate constant. — ^The arith-
metical mean of the population at the beginning and end of any finite
period differs, of course, from the true mean. The magnitude of this
difference is sometimes required. From (2a) and (6) we obtain —
„ ffH^ 2rH^ 3rH^ , n
(8) -Pm = 4 (Po + -P. ) -n (2X! + 2ir + 2:5r + ^^- )
which may also be written —
(8a) Pm=i{Po+ Pt)-Pt{2M-'2A-\+ ^X! " ^**'- >*
When expressed in terms of the arithmetical mean itseH, the odd powers
of r and t disappear, thus
(8b) -Pm = i (^0 + -Pt) ( 1 -2:3^ + ^X! - -JVV.-^^ )
This last is the most convenient formula. The values of the coefficients
are "I'j, Y^jj, 55rTC' ^^''•
Remembering that the maximum value of r is about 0.03, all these
series converge with sufficient rapidity.
6. Empirical expression for any population-fluctuation.— If the
population of a country be determined at « + 1 different dates, then a
curve of the w** degree can be arbitrarily drawn, passing through the
graph of the coordinates. In the absence of any information as to the
magnitude of the population between the given dates, the ordinate to
the curve drawn from the terminal of the abscissa corresponding to the
date may be assumed to be a probable value for the population at that
date.
The curve in question may be written ^ —
(9) Pt = Pq {I +at+bt^ + ct^ + etc.)
which, for purposes of practical calculations or computational check,
may be found convenient in the form : —
(9a). ...Pt=Po {1 +t [a + t{b+tc+ etc.)] [
7. Mean population for any period : rate not constant. — ^Using the
same notation as in II., 4, equation (6), we have —
(io)....P. =,-^/;=p,* =
Po{^ + I (h+h)+ I («,H<2«i+<i'')+ I {t^Hh^k+hhHti^)+ etc.}
Since, in the majority of calculations, <i is 0, in which case t^ becomes
simply t, we may write the result thus : —
(10a)....Pm = Po(l +ia< + i6(2 + Jct^+etc.); or
Po{l+«[|+«(|+<|+etc.)]}
the latter form being sometimes the more convenient for practical cal-
culation.
' See equation (45a) hereinafter. The fitting-efficiency of equation (9) is not
equal to that indicated later, but it is more convenient to use.
TYPES OF POPULATION FLUCTUATIONS. 13
t ■ ^ . „__ ^
8. Change, with change of epoch, of the coefficients expressing
rate. — ^If the coefficients a, b, etc., have been determined for Pg at a par-
ticular date, and it be desired to make the population, Pj the origin P'^
for new computations, so that —
(11) P; = P;,(1 +aT+i3T« + yT»+etc.)
in which t denotes the interval of time after the new epoch ; that is to say,
P^ = Pt + r and P'„ = Pt
Gn putting Pt / P^ = 1 4- r', we shall then have
(a = {a+2bt+Bct^ +4: dt^ + etc.)/(l + r')
(12). ..... ■ ^ = {b +3ct + 6dt^ + etc.)/(l + r')
^y = {c+4:dt + etc.)/(l + r')
which is perhaps the best form for computation. If the quantity en-
closed in brackets in equation (9) be denoted hj y = (f>t, the several
quantities in the brackets in (12) are dy/dt ; {d^y/dt^)/2 ! ; (d^y/dt^)/3 ! ;
etc., and the coefficients can be written out by a reference to Pascal's
triangle. They are, of course, simple " figurate numbers " of the second,
third, fourth, etc., orders.
That the coefficients must be altered when a new origin for t is
selected, exposes one of the inherent limitations of the empirical equation.
9. Error of the arithmetical mean : rate not constant. — ^The arith-
metical mean will always be in excess with either a uniform or a growing
rate of increase. From (9) and (10a) we obtain —
bt^ 2ct^ 3dt*
(13) Pm =UPo + Pt)-Poi 273+ 2A + 2:5 + ^^- )
which may also be readUy expressed in terms of the mean itself, as in
(8b), thus—
,,^ ^>,, 6 , 2a6-6r-, 5a^b-10b^-l5ac-36d,^ , ,
(13a)..P,„=i(Po+P,)-|l-3-,<^+-4T-«=' gi t*-eto.}
This, however, is more tedious to use than (13).
10. Expression of the coefficients in the empirical formula for rate
in terms of the constant rate. — ^If in equation (9), viz. : —
P( = p^ (1 4- a< + 6<2 _^ c(3 ^ etc.)
a=r; b=r^/2\; c = r^/3 I ; etc.
the equation would express a constant rate, that is to say, it would be
simply another form of equation (2a) ; and if a, b, c, etc., have not these
values, the rate of increase is variable.
14 APPENDIX A.
By substituting the corresponding values of r in (13a), it may eadily
be seen to be identical with (8b) ; and similarly as regards (13) and (8).
11. Investigation of rate is complete only when its variations are
ascertained. — Heverting to II., 1, equation (1) may be written —
(14) 8P = Pplt = P.^(<)S«
which may be regarded as the fundamental differential form for increase
of population, the final form being required, since the rate p is rarely
if ever, constant, even for short periods of time. Hence in its theoretical
form, an investigation of the fluctuations of population cannot be com-
plete tiU all variations of its rate of growth are definitively ascertained,
in other words, <^(<) must be ascertained.
12. Bate is a function of elements that vary with time. — ^The rate
at which population increases is dependent upon elements external to
and beyond the control of man, as well as upon elements within him,
more or less under control. Both change with the lapse of time. In
Fig. 3, § 32, hereinafter, examples are given shewing the curve of popula-
tion of different countries, and in Fig. 4, of the same section, the cor-
responding logarithmic homologues of the populations. As already
pointed out, the latter would be straight lines, if the rates of increase
were constant. Hence, in the sense that it is dependent upon elements
that vary with time, and may thus be directly related to the latter, the
rate p =</>(<) may be investigated as a function of the elapsed time.
13. Factors which secularly influence the rate of increase. — ^Where
not otherwise expressed, the rate of increase will be assumed to refer to
total population. Let us consider primarily a community which grows
by natural increase alone. This increase will be profoundly affected
by four types of things, viz. : —
(i.) The material natural resources of the occupied territory,
(ii.) The various cosmic energies which facilitate man's development.
(iii.) Knowledge which increases the power of utihsing natural
resources,
(iv.) Sociological and other analogous standards, which react upon
human activities, particularly upon man's productiveness,
and the magnitude and character of his consumption of
what he has produced.
Regarding (i.), it may be said that the natural resources of the ter-
ritory occupied may be either actual or potential. Even without human
intervention, a territory' may be prodigal of those forms of animal and
vegetable life, for example, which provide immediately for human wants.
Its climate and meteorology may be propitious. It may possess large
stores of readily available wealth, or of energy convertible into wealth.
TYPES OF POPULATION FLUCTUATIONS. 15
Or yet, again, though in the state of Nature infertile, it may respond to
weJl-directed efforts to make it so. It may have large hidden resources
which can be recognised, and can become available only through a con-
siderable development of scientific and technical knowledge, and through
practical abiUty in applying the same. Lastly, it may contain types of
wealth^l as for example mineral wealth generally, which, though valueless
per se to sustain life, may be made contributory to the growth of popula-
tion through the part they play in the world-economy.
All these may be summed up under two headings, viz. : —
(i.) Natural fertility and resources of the territory independent of
human action,
(ii.) Wealth or resources dependent on human action.
Both, however, are potentialities rather than actuaUties in regard
to population : how they eventuate in respect thereto depends upon
other and very subtle factors inhering in that order of things which
concerns the general sociological and economic beliefs and in the traditions
and activities of the people. For example, the general attitude of a
people in respect to the question of fecundity and the prevaUing view as
to what should constitute a reasonable standard of living, profoundly
affect the rapidity of the increase of the population, and the reaching of
the' time when natural limitations of fecundity operate severely.
There is still another factor of an analogous nature that plays a part,
the significance of which is each year becoming more manifest, viz. : —
The attitude of a people toward the development of the intellectual
powers of man, and toward the application of such powers to the avail-
ment of the resources of Nature. Indeed, in general, the great advantages
of the human being over the larger mammals is due to the efficiency in
this direction of his intellectual endowment, and his power by systematis-
ing to store and apply acquired knowledge.
If we denote natural fertihty or wealth of resources of the territory,
say, by w ; what may be called its geographical and climatic advantages
by g ; its other available resources when better scientific knowledge is
applied, or even when new wants are created by advancing civiUsation,
by u ; the factors expressing themselves in the matter of fecundity by / ;
through standard of Uving, including hygiene, by I ; through intellectual
knowledge and its range, energy, and wisdom of appUcation by i ; then
we must regard the increased population as really a function of all these,
that is to say —
(15) P = P„ ^ iw,g, u,f, I, i,. . . A)
The influences of these elements are, in general, secular in character,
i.e., they produce slow changes, some being manifest in the years of a
decade, others only in many decades. They are all determining factors
of the possibilities of population, but do not necessarily express its
actuality.
16 APPENDIX A.
Their specific character is such that ordinarily they produce gradual
and more or less remote effects, rather than effects which are instantaneous
and immediately of great magnitude. Such effect may tend towards a
constant value, may increase, or diminish, but in all cases the consequent
changes will be gradual. It is to be noted, however, that some of the
factors may acquire for a short time an importance which, locally at any
rate, may lead to rapid changes.
Factors of the kind considered are probably either non-periodic, or
if periodic their period is secular.
A general solution, if it were possible, would presuppose that the
way in which w, g, u, f, I and i, varied with elapsed time was determinable.
This variation, however, is not susceptible of exact definition : never-
theless, the form of the functions expressing their effect on the rate of
increase p is not always wholly indeterminable.
14. Variations which depend on natnial resources, irrespective of
human intervention. — ^This may include both periodic and non-periodic
elements. The natural wealth of a territory, as unaffected by the inter-
vention of man, is, in general though not invariably, a maximum
initially, 1 though its values may oscillate between very wide Umits,
owing to variations of meteorological or climatological factors. Where
natural wealth is of a type that is subject to steady decUne, its effect on
the rate of increase may be represented for all practical cases probably
by a very simple function of the elasped time.
15. Variations of rate of long periods. — ^Any periodicity in meteoro-
logical and other factors, affecting the natural wealth of a territory,
however much their influence may be masked by other factors, wiU in
most cases cause a collateral periodicity in rate of increase. This can
be represented by such a formula as the following, viz. : —
(16)..pt/po = 1+ [tto+ai sin (ai+ jr ) +a^sia{a^+Y )+etc.J+ Q
in which Ti, T^, etc., will represent the lengths of the various periods
to which the elapsed time t is related ; ai, a^, etc., are intervals deter-
mining the epochs of Ti, T^, etc. ; and finally o^, a^, etc., are the
amplitudes of the variation from the mean value. Thus necessarily —
(17) Ug = — {tti sin ai -f a^ sin a^ + etc.)
and Q wiU of course represent the effect of the other elements influencing
the rate of increase to which reference will be made later. Equation
(16) is specially suitable for representing fluctuations of long period,
which are expressible in terms of a sine series.
' Examples could be drawn in recent times from America or Australia. It
may, however, even in regions which nevertheless can be made habitable, be actually
zero, as for example, in the Sahara, in Arizona, and in some parts of Australia.
TYPES OF POPULATION FLUCTUATIONS. 17
(18)
16. Representation of periodic elements in non-periodic form. —
Where T is exceedingly long as compared with t, the numerator of the
expression (16) may take a much more simple form, available probably
for all practical cases. For putting —
^A,=2 [(ai cos ai )/Ti ]; A, ^ —1 2 [(«< sin a,- ) / T/ ] ;
1 ^3= -^^ 2 [(ai cos ai )/T/]; ^, = + 1 i;[( («; sin a^ )/r/].
etc., etc., ; etc., etc.
the limits of the summation being from i = 1 to i = w, and n being the
number of periodic terms. Then remembering that
Ug + 2 (ai sin tti ) =
with the same limits, we can express (16) in the form
(19)../3j/p=l+ao+aisin(ai+ ^ )+ etc.=l+.4i< + A,t^+. . + etc.
which, with (18), connects the coefficients with the amplitude and epoch
of the periodic fluctuations.
The values of Ai, A^, etc., may be either positive, negative, or
zero.
17. Influence of natural resources disclosed by advancing know-
ledge. — ^Turning now to the question of the various terms in Q, viz.,
those representing in equation (15) the effect of m, /, I, and i on the rate
of increase, we remark first of all that increased scientific knowledge,
especially in physics and chemistry, suggests that possibly the available
resources of Nature are practically without Umit, (that is m = oo ). This
being so, the rate of increase may be regarded as dependent, not so much
upon Nature's Umitations as upon the extent and character of our know-
ledge, and of our energy and wisdom in applying it; that is, in the formula,
it depends upon i, not upon %. We shaU find, however, that Nature's
limitations are very real, for rates of increase of population which
characterise many countries at the present time cannot be maintained
for several thousand years.
18. Influences of resources dependent upon human intervention. —
There is a narrower sense, however, in which % may represent specific
and finite quantities, which can be sufficiently indicated by two or three
illustrations. Territories hke portions of the Sahara in Africa, and of
Arizona in America, apparently hopeless waste, may in response to the
appUcation of artesian water, become fertile and habitable. In ordinary
agriculture, land, practically valueless in the state of Nature, may become
valuable by the appHcation of suitable fertilisers. The infertility of
land which is due to the absence of the necessary micro-organisms, may,
when once such organisms are introduced, quite disappear, and the
potential wealth in the territory existing may have been quite undreamt
of. Or yet again, the value to man of a natural product, utilisable in
18 APPENDIX A.
the natural state, or after being treated technically, may be wholly
unknown ; the discovery of its real value may so change the economic
conditions of a territory as to greatly facilitate increase of population.
In these and many other similar ways, natural resources reacting to man's
operation may be found to be very great, though at first apparently
non-existent. It would obviously therefore be very difficult to assign
a form to the function which is in any way to represent the effect of natural
resources.
19. Effects of migration. — ^Migration operates in several ways on
the rate of increase of population, viz. : (i.) By the actual addition or
withdrawal of the migrants ; (ii.) by the change of the constitution of
the population, thus affecting its rate of fecundity ; (iii.) by consequential
economic changes which favour or impair the rate of increase. A com-
plete expression for its effects would therefore be elaborate in form.
Since, however, the community changed by migration tends to adjust
itself to the economic condition of the country, the real elaboration into
each component element is unnecessary, and the resultant of all the
elements operating may take a relatively simple form.
Migration itself is of two forms — ^periodic and non-periodic. The
population of countries, for example, which at certain seasons are visited
by large numbers of tourists, or from which large numbers depart, may be
taken as affording illustrations of periodic migration. The rate of influx
or ef&ax is usually slow initially ; it then increases, becoming a maximum ;
when it dechnes much in the same way. In form, the curve of absolute
increase or decrease is somewhat similar to the probability curve, but the
curve is probably rarely symmetrical with respect to the maximum
ordinate.
Non-periodic migration may, in addition to the effect of its absolute
amount, change the final rate of increase or leave it as it was originally.
Although both periodic and non-periodic migration may be actually
discontinuous, no material error wiU ordinarily be committed if it be
assumed to be continuous, provided that in amount it be negligibly small
for the part of the year when it has actually ceased. So that there is no
serious objection to the use of an essentially continuous function.
20. Simple variation of rate, returning asymptotically to ordinal
value. — ^Non-periodic migration of population, frequent in new countries,
may produce a simple variation of rate which ultimately disappears.
Owing to the reputation the territory acquires in respect of some real or
supposed advantage, immigration sets in, increasing in rate till a maximum
is reached, and declining again till the original rate is restored. For
the territory or territories from which the emigration takes place, the
converse effect may be true. If the rate can be ascertained at several
periods, the total effect on the population can then be deduced with fair
accuracy.
TYPES OF POPULATION FLUCTUATIONS.
19
The simplest variation of this type, and one which will probably
represent most instances with sufficient precision, may be expressed in
the form —
(20).
■Pi/P^ = I +T7<«
■q being positive for cases of immigration, and negative for those of
emigration. This form would be suitable for deductions as to population
based on the determination of rate of increase at various times.
By suitably selecting the unit of {, the parameter rj and the index-
numbers m and n, equation (22) may be made to represent the very
different circumstances which may obtain at the commencement, and
during the development and passing away of the effect of migration on
the original rate of increase. For example, it wiU express that type of
migration in which the increments per unit of time to the rate of increase,
though initially slow, grow and decrease with continually changing
velocity, tiU the original rate is restored ; or, on the other hand, it will
express that type where the migration effect on the rate is sudden.
This is illustrated by the curves in Fig. 1., viz. : —
Curve Tj^™-"'
Curves graphed
,,=1
a wi = A w =
h 1
c 1
6, 2
e 4
4
^
/
\
^
/
\
\
2
\
/ /
^<
k
1
/
^
//
s
A
/ /'
'~^.
■*^.c
\
■•V
16^
y
K
:^
"■■-
v
-~.
^_
«„
= 1/e:
1— m/nt
4 5
Fig. 1.
7 Values of t
Curves y = tj*™ + "*
in which the parameter ij is unity throughout. The possible varieties
of change of rate of increase are obvious from the figure^ when it is re-
membered also that the horizontal proportions can be maintained, and
the vertical changed at pleasure by simply altering the value of ij.
21. Examination of exponential curves expressing variation of
rate. — ^The curve of equation (20) demands special consideration. For
brevity put E for {pt — p^ / p^, then we can re-express (20) in the
form^
1 An expression of still greater fitting power is 1/ = At'"'' e"* • See a paper on the
curve by G. H. Knibbs. Journ. Roy. Soc. N.S. Wales, Vol. XLIV., pp. 341-367.
20
APPENDIX A.
therefore
(20a) R =rit'
„4m-nt
{2l)....dR/dt= J-^- ;w— w<(l +logc<)}
(21a) ^ = 7,«-+««(« \ogt + ^ +«)
(21b).
dt
dhj , , , , »* >o wi — m ,
dt
and hence the value of t, which gives the maximum value for R, is found
by solving the equation —
(22) m/n = <max (1 + loge <max)
For the maximum to correspond to a value of t less than unity and
greater than '^/e, the equation will be of the form i™ - "* (w and n being
positive) ; or less than Ye the equation will be of the form i- ("»+««)
This equation can be solved by inspection, by means of the following
table : —
TABLE I.
Argument t. Values of t log, t,t{\ + logs t), and t ( — 1 + logg *)
t
t log, t
«(l+log,«)
e(-l+log,«)
t
( log, t
«(l+log,«)
0.1
0.2303n
0.1303n
— ,
1
0.0000
1.0000
0.2
0.3219m
0.1219n
—
2
1.3863
3.3863
0.3
0.3612re
0.0612n
—
3
3.2958
6.2958
0.4
0.3665re
0.0335
—
4
6.5452
9.5452
0.5
0.3466n
0.1534
— .
5
8.0472
13.0472
0.6
0.3065n
0.2935
—
6
10.7506
16.7506
0.7
0.2497n
0.4503
—
7
13.6214
20.6214
0.8
0.1785?i
0.6215
—
8
16.6355
24.6355
0.9
0.0948n
0.8052
—
9
19.7750
28.7750
1.0
0.0000
1.0000
9.0000
10
23.0259
33.0259
1.1
0.1048
1.2048
9.0048
11
26.3768
37.3768
1.2
0.2188
1.4188
9.0188
12
29.8189
41.8189
1.3
0.3411
1.6411
9.0411
13
33.3443
46.3443
1.4
0.4711
1.8711
9.0711
14
36.9444
50.9444
1.5
0.6082
2.1082
9.1082
15
40.6208
55.6208
1.6
0.7520
2.3520
9.1520
16
44.3614
60.3614
1.7
0.9021
2.6021
9.2021
17
48.1646
65.1646
1.8
1.0580
2.8580
9.2580
18
52.0267
70.0267
1.9
1.2195
3.1195
9.3195
19
55.9443
74.9443
2.0
1.3863
3.3863
9.3863
20
59.9146
79.9146
2.1
1.5581
3.6581
9.4581
21
63.9350
84.9350
2.2
1.7346
3.9346
9.5346
—
. —
2.3
1.9157
4.2157
9.6157
2.4
2.1011
4.5011
9.7011
—
2.5
2.2907
4.7907
9.7907
2.6
2.4843
5.0843
9.8843
,
2.7
2.6818
5.3818
9.9818
" —
2.8
2.8829
5.6829
10.0829
— ,
2.9
3.0877
5.9877
10.1877
,
3.0
3.2958
6.2958
10.2958
—
—
•
Note. — ^The n denotes that the quantity is negative. In the column for t log, t
and t ( — I+logj t), the whole number 9 has been used in preference to the more awk-
ward form r : in these cases the values given therefore exceed the true values by 10.
TYPES OF POPULATION FLUCTUATIONS. 21
The suitability of the assumption of a curve of the type in question
may be fairly well ascertained in the following way. For t = I, B = rj
in (20a) (the unit of t may be 1 week, 1 month, 1 quarter, or 1 year, say,
according to the character of the migration under review).
Taking the logarithm of both sides of (20a) we have —
(23) log R = log rj +{m—nt) log t
From the observed values of B, the values of Jl
(24) 31 =(logiJ — log ri)/logt =m-~nt
may be formed. These are plotted as ordinat-es, with the corresponding
values of t as abscissae ; then if the points ^ lie on a straight Hne, m
will be the intercept on the axis of ordinates, and n will be the tangent
of the negative angle which the line of points makes with the axis of
abscissae.. If they do not lie on a straight line, the assumption is invalid.
If, moreover, we have the epoch at which the rate was a maximum,
we have also from (22) the ratio of m/n, and obviously the two should be
in agreement. This is a further test of the vahdity of the assumption.
22. Determination of constants of such exponential cuives. — ^The
constants ij, m and n in equation (20) may be found from three observa-
tions at any suitable intervals, say at the times <,, t^, and t^, the
commencements of the fluctuation being therefore also known. If the
value of -B for i = 1 is not known, put —
(25)
^,.,= log B—log Bi ; /S3., = log i?3 —log B,
Wj.i= log <2 — log h ; v,.j = «2 log (2 — h log «i
M,_, =logf3— log*2 ; «3., =t^logt^ —t^logt
2
then we shall have —
(26) m = ('Sf3.,u,.^ —S^.ti'sJ / K-.'^^.i — -a.i-a.s/
and
(27) W = (-Ss.A.i— 'S,.,M3., )/(«,.,«,.! —U^.^■V,J
The values of m and n being found, ij is best found from.
(28) logTj =log B — (m — nt)logt.
the suffixes of B and t being identical.
If, however, the rate for < = 1 be known, then tj is B^^^•, and, see
(24), the suffixes of |l , iJ and t being identical, we shall have —
(29) m = (3l2«3- ^3«2)/(*3 - h )
and
(30) n = (M^~ Ws)/(h-h)
a solution much less tedious than the former. The values of t^ and t^
must be well selected.
22 APPENDIX A.
Obviously, if more than three values of B are taken, the application
of the test indicated by equation (24) is necessary.
23. Case of total non-periodic nidation represented by an ex-
ponential curve. — ^Where migration adds or subtracts its quota to the
population only temporarily, as in cases of temporary migrations to or
from a country, the exponential curve of equation (20) will often represent
with exactitude, not merely the variations of the rate of increase, but of
the absolute population. The complete expression put into non-periodic
form would thus be, for the case in question —
(31) Pt = P^{ ept + 7, (qtr - '^
the factor q depending upon the value of the unit of t used in (20) {i.e.,
if 7j, m and t be determined for months, then q = 12, t in ept being in
years). This formula would represent a single migration effect, vanishing
asymptotically. If the migration be itself proportioned to the magnitude
of the population at each instant, as may often be the case, then the
preceding equation (31) will become —
(31a) Pt = P„ept |1 +7](qt)
m — nt\
In using either (31) or (31a), it is of course necessary that Pq be the
population at the commencement of the migration effect ; i.e., the
origin of t must be identical in both parts of the complete expression.
Yet again, if the expression represents only the variation of the rate,
we shall have —
(31b) P4 = P„eP«(l +vt"'-"')
which, however, wiU be considered in a more general form hereinafter.
24. Simple variation of rate, returning asymptotically to a par-
ticular value. — ^A variation of rate may tend to return to some new value,
greater say than the original. Such a variation can be expressed in the
following way, viz. : —
(32) pt/po=l+r,t^^+'^
m and n being positive.
Since this curve becomes asymptotic to a line parallel to the t axis
at the distance rj, and has the ordinate value rj for t = 1, the unit by
which t is measured must give an abscissa of unity for the first value ri of
the ordinate. This somewhat limits the convenience of its application.
Some of the forms of the curve are illustrated by Fig. 2.
TYPES OF POPULATION FLUCTUATIONS.
23
18
^
-N
1
\
Ifi
\,
\
14
\
\
N,
1»
,
\
/
V
^?.
/
\
/
\,
11
/
\,
\
s
10
\
\
R
\
c
R
7
1
fl
A
■—
—
—
-J,
f
/■
"■^
■-.
^
4
/
■^
^
.
/
" ""
--.
i.
n
/
^
1
^^
—
■ —
— ,
f>
('
h
a
1
K^
''
_j_
— -.
—
—
--
--
—
--
—
r
'?/
n
^
'^J
Cuives r^t
m + f\i
Curves graphed
a m = 4 ; w = 4
b 1
c i i
. d i i
*max — ^
1 +
8 9
Fig. 2.
25. Examination of the preceding curve.— As in section (21), put
1
then
(32a) iJ'=Tj<±'» + '
(33).... dW /dt = "nL
n loge t
m + nt ^ t ±m+nt\
and consequently the value of t which gives the maximum value for B'
is found by solving the equation ^ = £^^t '^^''^ l^ads to—
(34).
± m
= t (log« t - 1)
24 APPENDIX A.
For a maximum to correspond to a value of t greater than e, the
1
equation will be of the form «»> + «« (m and n being positive) ; or less
than e, equation will be of the form (-»»+"«
This may be solved for the series of values already given in Table I. for
t (loge t — 1) : see section 21 hereinbefore.
Similarly to the preceding case we take the logarithm of both sides of
(32a), we have —
(35) log R'= log 7] + log < / (± m + nt)
Hence as before, finding ^' from observed values of R' we have —
(36) '§, = log t / (log B' —log 7]) = ± m + nt
which enables us to examine the vaHdity of the assumption, since it is
the equation of a straight line of which the values of "g{' and t are
respectively ordinates and abscissse.
For the point of inflexion the second differential will be required :
the sign of m being positive, it is —
26. Deteimination of the constants of the curve. — ^In this case the
rate for < = 1 is known, and r/ = Ri = i', thus formula (29) holds when
^ is changed for ^', and similarly in regard to (30) changing the sign,
that is —
(37) m = (il',*3 - W,t,) I (<3 - t,)
(38) n = {W^-3\)/{h-h)
The test of (36) is necessary if there be more than three values of ^'.
For the case of immigration tj is positive, for emigration negative.
27. Total non-periodic migration resultii^ in permanent increase
but returning to original rate. — ^Where the migration effect on total
population adds or subtracts its quota, but leaves the original rate
practically undisturbed, the result may be expressed similarly to (31), i.e.,
(39) Pt = Po ieP« + y) {qt)^^^^^\
and if as supposed in section 23 the migration be itself influenced at
every moment by the magnitude of the population, (39) will become —
(39a) Pt =Po ep< {1 + •ij(g<)± »»+»»;
TYPES OF POPULATION FLUCTUATIONS. 25
28. The utility of the exponential curve of migration. — ^Formulae
(20) to (31b) are serviceable, when the population has to be determined
by taking into account the rate of migration determined only at several
suitable occasions, the intermediate migration being supposed to conform
to the exponential curve assumed to represent all values intermediate
to those determining it, and all future values so long as it is apphed.
29. Fluctuation of annual periodicity. — ^The instantaneous rate of
increase of the population of any country, at least where the population
is at aU numerous, must, during the course of the year, indicate a yearly
period, since both the migration rate and the birth and death rates have,
in general, a characteristic annual fluctuation. There is sometimes a
difference, however, between the migration fluctuation, and that due to
births and deaths, for the former, owing to local circumstances, is some-
times conflned to a part of the year only, while the two latter extend over
the entire year. The scheme of expressing long periodic fluctuations has
already been indicated, viz., in equations (16) to (19). Continuous
fluctuations of short periods may with advantage be put in the form —
(40). . . .Pf/p = 1 + Oq + ai sin (ai + fiit) + a.^ (sin a^ + fx^t) + etc.
where jx\ and /Aj are whole numbers or proper or improper fractions,
deflning definitely ascertained periods, and where, as before, we must
necessarily have —
(41) ttj = —Z asma;
see section (17) ; or yet again, if the true period is not known and a curve
known by experience is to be empirically reproduced, then we may put
2tt 2tt
(42). .pt/p^ = 1 + tto + «! sm (ai + — + a^ sm 2{a^ + -t) +
ttg ain 3(a3-| t) -\- etc.,
the unit of t being the period (e.g., one year) embracing aU the fluctuations
to be reproduced in the period following.
30. Discontinuous periodic variations of rate. — We may assume
that the continuous rate is any function of t, i.e., pt = (f>{t) say.
Suppose that superimposed on this curve, there is a migration effect
existing for parts of the year only, reappearing at the corresponding times
in each following year. Let us suppose further that in the intervals,
there is no variation of rate through migration, the fluctuation being
fully expressed by <f>{t) above. Then, provided that suitable values
are given to the constant oSq to the amphtudes ai, a^, etc., and to the
epochal angles ai, a.. 2, etc., the fluctuation of rate may be represented
by such an expression as —
277
(43) |0j//>„ = ^(«) ± V [Wo + ai sin (ai + — - «) + etc.J.
26 APPENDIX A.
the + sign denoting immigration effects, and the — sign emigration
effects. For the final term will have no real values when the quantity
under the radical sign becomes negative : a^ must of course satisfy the
conditions expressed by equation (17) hereinbefore.
Similarly, fluctuations of other character may be represented by —
(43a). . . . p^/p^ = cf>(t) ±V{ao + «! sin (ai + t/Ti) + etc. }
or again by —
(43b). . . .p^/p^ = <f>(t) ± V{at +bt^ +ct' + etc.)
Since only real values can have any meaning the expressions under the
radical sign in (43), and (43a) and (43b) are discontinuous, the discon-
tinuity extending from each value of t where the value of the expression
changes from + to — , to where it changes from — to + again.
31. Empiiical expiessiou for secular fluctuations of rate. — ^For the
purpose of prediction it is usual to deal either with mean population or
the population at a particular date, say the end of the year. The
fluctuations of rate may be empirically determined from past records
and put in the exponential form, viz.,
(44) Pf/p = 1 + Tjt* + "»« + «t'+ etc.
7], k, m, n, etc., being integral- or fractional, positive or negative. Or
again, it may be expressed in the form —
(45) p^/p^ = 1 + ai + j8<2 + yt^ + etc.
or yet again in the form —
(45a) p^/p^ = \ -\- atv + ^ta + ytr + etc.
in which p, q, r, etc., are in ascending order of magnitude, but not re-
stricted to integral values. The fitting efficiency of this latter form is
much greater than where the indices are restricted to integral values, ^
but the determination of the constants a, j8, y, etc., andp, q, r, etc., are
not so convenient.
32. Growth of various populations. — ^Populations increase when the
additions by birth and immigration together exceed the deductions
through death and emigration together. The rate of increase differs
greatly as between country and country, and differs from decade to decade,
so that it cannot be regarded as in any sense uniform even for short periods
of time. This is evident from Fig. 3, in which the growth of the popula-
tions of a larger number of countries is shewn by their progression every
decade, and is still more obvious in Fig. 4 (shewing their logarithmic
homologues) by the changes in the slope of the Unes. In the following
table, the populations, given in millions and decimals of a miUion, are
those shewn on Fig. 3.
1 Obviously, since both the coefficients and indices are at our disposal, it is easy
to see that attempts to apply (45) to the curve y=atP, where p is a proper or im-
proper fraction, are invalid. It is also invalid for the curve y=atP + btP+a + etc.
TYPES OF POPULATION FLUCTUATIONS.
27
The Populations of Various Countries from 1790 to 1910.
The scale for the lower part of the figure denotes ten times the
numbers of the scale for the higher part. The predicted population for
the United States was based on the assumption that the rate for 1790 to
1800 would be maintained constant. On the scale of the figure this
curve substantially agrees with the prediction by Elkanah Watson in
1815.
28
APPENDIX A.
Table n. — Populations in Millions, of Various Countries.
Years.
COl IHTRT.
1790-9.
1800-9.
1810-9.
1820-9.
1830-9.
1840-9.
1850-9.
Commonwealth
.002
.005
.01
.03
.07
.19
.41
United Kingdom
15.90
1
17.91
1
20.89
1
24.03
1
26.71
1
27.37
Scotland
1.61
1
1.81
1
2.09
1
2.36
1
2.62
1
2.89
Ireland . . • •
..
5.40
1
5.94
1
6.80
1
7.77
1
8.18
1
6.55
Austria
15.59
16.58
17.53
Belgium
6
4.34
6
4.53
Denmark
.93
4
1.22
1.28
1.41
France
26.93
1
29.87
1
31.89
1
33.40
1
34.71
Germany
23.18
6
24.83
2
27.04
1
29.77
32.79
2
35.96
Hungary
7
13.77
Italy
6
18.38
5
19.73
H
21.98
«
23.62
8
24.86
Norway
.88
5
1.05
5
1.19
5
1.33
5
1.49
Portugal
8
3.92
Spain
,
, ,
7
16.46
Sweden
2.19
2.35
2.40
1)
2.58
2.88
3.14
3.48
Finland
U
.71
.83
.86
1.18
1.37
1.45
1.64
Servia
.40
U.S. America
U
3.93
5.31
7.24
U
9.64
12.87
17.07
23.19
Years.
CODNTRT.
18
60-9.
1870-9.
18
80-9.
1890-9.
1900-9.
1910-9.
Commonwealth
1.15
1.65
2.23
3.65
3.77
4.43
United Kingdom
1
28.93
1
31.49
1
34.88
1
37.73
1
41.46
1
45.22
Scotland
1
3.06
1
3.36
1
3.74
1
4.03
1
4.47
1
4.76
Ireland
i
5.80
1
5.41
1
5.17
1
4.70
1
4.46
1
4.39
Austria
9
20.39
22.14
23.90
26.15
28.57
Belgium
6
4.83
5.52
6.07
6.69
7.42
Denmark .
1.60
1.78
1.97
2.17
1
2.45
1
2.78
France
1
35.84
2
36.10
1
37.41
1
38.13
1
38.45
1
39.60
Germany .
1
38.14
1
41.06
45.23
49.43
56.37
64.93
Hungary .
V
1.22
15.51
15.74
17.46
19.25
20.89
Italy
2
25.00
1
25.96
1
28.46
1
30.46
1
32.48
34.67
Japan
2
36.70
40.45
44.83
50.50
Norway .
1.70
b
1.82
1
1.99
(1
2.22
2.39
Portugal .
8
4.00
V
4.16
1
4.31
4.66
5.02
1
5.55
Spain
V
16.43
7
17.55
7
18.32
18.61
19.59
Sweden
3.86
4.17
4.57
4.78
5.14
5.52
Finland
1.75
1.77
2.06
u
2.38
2.71
3.12
Servia
u
1.00
4
1.35
4
1.90
2.16
2.49
2.91
U.S. America
31.41
38.56
50.16
62.62
76.21
93.35
33. Bate of increase of various populations. — ^Fig. 3 and the accom-
panying table reveal directly only the relative magnitude of the popula-
tions, but not their exact rate of growth. The latter is displayed on Kg. 4,
in which (the scale being constant) the steepneas of slope of the line repre-
sents the rapidity of the rate of increase. As before mentioned, this rate
is very irregular from decade to decade, as would be revealed by dividing
the population at the end of each decade by that at the beginning thereof
and comparing the numbers ; i.e., by finding and comparing, for example,
the values of P„/Po giving those of 1+ r. The rates tabulated here-
under are the anrnud rates which, if maintained constant, would produce
the populations at the end of the decades ; that is, they are the values
of r found from log (1+ r) = (log P„— log Po)/n, where n is the inter-
vening number of years.
TYPES OF POPULATION FLUCTUATIONS.
29
Bates of Increase of Various Populations, 1790 to 1910.
90 1800 10 30 30 40 1850 60 70 80 90 1900 10
F denotes Finland ; N, Norway ; S, Servia.
* The logarithms for Australia, Denmark, Finland, Ireland, Norway, Scotland,
and Servia are shewn on the right of the figure ; for the others, on the left.
Fig. 4.
30
APPENDIX A.
Table in. — Annual Rate of Increase per 10,000 of Population of Various Countries.
Approximate Decade.
C'OLXTRY.
1790
to
1799
1800
to
1809
1810 1820
to to
1819 , 1829
1830
to
1839
1840
to
1849
1850
to
1859
1860
to
1869
1870
to
1879
1880
to
1889
1890
to
1899
1900
to
1909
C'wealth . .
V. K'dom*
Scotland*
Ireland* . .
Austria
Belgiumt ••
Denmark*
France* . .
Germany . .
Hungary . .
Italy
Norway* ..
Portugal . .
Spain
Sweden
Finland . .
Servia
Japan*
U. States . .
976
71
157
306
829
120
118
96
52t
43(a)
74(d)
21
36
315
1124 1 764
155 141
145 1 122
136 ' 134
52t ' 66
143 114
— 79(W
74(d)| 126
73 1 111
321 150
291 293
1052
106
105
52
62
80 §
46
108
83(c)
112
87
57
2i«\
785
24
99
-225
56
43
97
39
77
114
103
124
311
1095
56
57
-122
152
64
127
32
66
41**
133
104
65
309
370
85
94
—70
41t
96**
107
7
74
86**
42
69
20
sot
77
12
190**
206
308
103
108
—45
41t
96**
102
36
108
17
92
56(a)
58(c)
30 1
92
153
348
267
351
79
75
—96
77
96
97
19
89
104
68
56(a)
66
45
145
216
116
224
180
95
104
—53
90
98
122
8
132
64
64
122
75
33((!)
73
131
143
110
198
163
87
63
—16
89
104
127
30
142
51
58
74
92
67
71
142
157
118
205
* Add 1 year to date for proper decade, t Add 6 years up to 1860 inclusive, t Kate
for 20 years. ** Bate for 14 years. § Rate for 6 years, (a) Rate lor 16 years. (6) Bate
for 9 years, (c) Rate for 13 years, (rf) Bate lor 24 years.
34. The population oi the world and the rate of its increase. — ^In
(iealing with the magnitude of the population of any country and the
rate of its growth, the most general comparison is that made with the
entire population of the world and its rate of growth. This, however,
is not well siscertaiaed. Recently, for example, the estimate for China's
population has been reduced over 100 millions. The following table
gives results of different estimates : —
Table IV.— Estimates of World's Population.*
Year.
Authority.
Estimate
(Millions).
Year.
Authority.
Estimate
(MiUions).
1660
Riccioli
1,000
1813
Graberg v. Hemso
686
1685
Isaak Vossius
500
1816
A. Balbi
704
1740
Nio. Struyok
500
1822
Reichard
732
1672
Riccioli
1,000'
1824
G. Hassel
938
1742
J. P. Sussmilch
9S0 to 1,000
1828
G. Hassel
850
1753
Voltaire
1,600
1828
I. Bergius
893
1761
J. P. Sussmilch
1,080
1828
A. Balbi
737
1789
W. Black
800 to 1,000
1828
Balbi*
847
1804
Malte-Brvm*
640
1833
Stein
872
1804
Volney
437
1838
Franzl
9S0
1805
Pinkerton
700
1838
V. Rougemont
850
1805
Fabri
700
1840
OmaUus d'Halloy
750
1809
G. Hassel
682
1840
Bernoulli
764
1810
Abnanach de Gotha*
682
1840
V. Roon
864
1812
Morse
766
1843
Balbi
739
TYPES OF POPULATION FLUCTUATIONS.
31
Table IV. — Estimates ot World's Population*— oonSinited.
Year.
Authority.
H. BerghauB
Estimate
(Millions).
Year:
Authority.
Estimate
(Millions).
1843
1,272
1880
Behm & Wagner
1,456
1845
Miohelot*
1,009
1882
Behm & Wagner
1,434
1854
V. Reden
1,135
1883
Behm & Wagner*
1,433
1889
Dieterioi
1.288
1886
Levasseur*
1,483
1866
E. Behm
1,360
1891
Ravens tein*
1,467
1868
Kolb
1,270
1896
Statesman's* Year
1.493
1868
E. Behm
1,375
1903
Jurasohek* [Book
1,512
1870
E. Behm
1,359
1906
Jurasohek*
1,538
1872
Behm & Wagner
1,377
1910
Annuaire Statistique
1873
Behm & Wagner
1,391
d. I. Rep. Fran9ai8e*
1874
Behm & Wagner*
1,391
Jurasohek*
1,610
1878
Levasseur*
1,439
1913
Knibbs*
1,632
1878
Levasseur
1,439
1914
Knibbs
1,649
* These will be found on the graph, Fig. 5.
This table shews, for the period 1804 to 1914, rates of annual increase
ranging between 0.0015 and 0.0121^ and averaging about 0.00864.
We may obtain some idea of the present rate of growth by taking the
weighted mean of the rate for the known countries ; that is, each rate of
increase is weighted according to the population. In this way, it is found
for the quinquennium 1906 to 1911, and for the group of countries
in the Table V. hereinafter, that the general result is a rate of increase of
0.01159 per annum, or 1.159 per cent, of the population.
Table V.— Annual Increase per 10,000 Population for the quinquennium 1906-1911.
Country.
Rate
Years t
Country.
Rate.
Yearst
Ireland
— 6
Switzerland
+ 121
57.6
France
+ 16
436
Netherlands
122
57.2
Jamaica
28
248
Denmark
126
66.4
Scotland
65
126
Grerman Empire
136
61.3
Norway
66
105
Finland
143
48.8
Belgium
69
101
Rilmania
148
47.2
Italy
80
87
Servia . .
155
45.1
Sweden
84
82.9
Chile
156
44.8
Hungary
84
82.9
United States . .
182
38 4
Austria
86
80.9
Commonwealth
203
34.6
Spain
87
80.0
New Zealand . .
256
27.4
England and Wales
104
67.0
Canada . .
298
23.6
Japan
108
64.6
Ceylon
120
58.1
Weighted Average* .
+ 115.9
60.1
* Weighted average according to population. f Years necessary for the
population to be doubled in value at the rate indicated.
The number of years n in which a population, increasing at the rate r, is doubled,
may be very readily computed thus : —
(1+r)" = 2 ; therefore n log, (1 + r) = log, 2 = 0.693147
consequently n == 0-69316 0.69315
log, (1+r)
r(l-
+ 3-.)
but when r is very small we may neglect powers higher than the second (that is ^ in
the brackets) ; hence
0.69315 ,, , , , ... 0.693 , „ , .,
n = ; — ( 1 + i»-), sensibly, = + 0.347-
' On taking the mean of Levasseur and Behm & Wagner, and again of Levasseur
and Ravenstein.
32 APPENDIX A.
Either this rate of increase must be enormously greater than has
existed in the past history of the world or enormous numbers of human
beings must have been blotted out by catastrophes of various kinds from
time to time. For, putting the present population at 1,649,000,000,
at the average rate of increase this number would be produced from a
single pair of human beings in about 1782 years,* that is to say, since
A.D. 132, or since Salvius JuKanus revised under Hadrian the Edicts of
the Prstors. Even the rate given by the world -populations 1804 and
1914, viz. (0.0086) gives only 2397 years, carrying us back only to B.C. 483,
or since the days of Darius I. of Persia.
The profound significance of this fact, accentuated also by the
extraordinary increase in the length of life (expectation of life at age 0),
which has revealed itself of recent years, is obvious when the correlative
food requirements are taken into account. The resources of Nature will
have to be exploited in the future more successfully than in the past to
maintain this rate of increase (0.01159), which doubles the population
every 60.15 years, and would give for 10,000 years the colossal number
22,184. with 46 noughts (lO*^) after it.
This number is so colossal that it is difficult to appreciate its
magnitude. Assuming the earth to be a globe of 3960 miles radius, of a
density 5.527 compared with water, that water weighs about 62J lbs.
per cubic foot, and that a human being weighs on the average, say,
100 lbs. (7 st. 2 lbs.), the actual mass of the earth would be equivalent
only to, say, 132,265 x 10^* persons; that is, it would require 16,771
X 10*' times as much " matter " as there is in the earth. Or, to consider
it as a question of surface, allowing 1 J square feet per person, the earth's
entire surface area would provide standing room for only 36,625 x 10"
persons. That is, the population would be 60,570 x 10** times as
great as there would be standing room if the whole earth's surface were
available. It is evident from this that the rate of increase of human
beings must have been more approximate to the rate for France at the
present time, if the earth has been peopled for 10,000 years : the French
rate, 0.0016, would require 12,842 years to give the present population
from a single pair. This rate, however, would give a population of only
17.55 millions in 10,000 years.
The foregoing analysis of the effect of the rate of increase, with
which we are familiar, establishes the fact that the rate must have passed
through great changes, and could not have been maintained for any long
period, either at its present average, or that characteristic of the last
century. (See II. § 12, 13, 14 and 15.) It is not improbable that the
rate of the last quinquennium will not be long maintained ; and it is
* Thus dividing by 2, we have 824,500,000 = (1.01159)" where n is the
immber of years, that is, n = 1.782.
TYPES OF POPULATION FLUCTUATIONS.
33
certain that however great human genius or effort may be, in enlarging
the world's food supplies, that rate cannot possibly be maintained for
many centuries. The contention of Malthus is thus placed beyond
question, from a different point of view.
The analysis also suggests that there are probably great oscilla-
tioMs of the rate of increase, but since accurate records date back
for so comparatively short a time, no general indication of their character
can be given.
THE WORLD'S POPULATION, 1806-1914.
17
16
15
14
o 13
S
12
,(!'>-
y
'\
[o/
7
/
■ /
/
'/
,^
/
/
'/
>'
/
»'
V
/
./'
f^
J
V
/
/
^
"
.S 11
a 10
H
9
8
7
10 20 30 40 50 60 70 SO 90 1900 10
Fig. 5.
In Fig. 5 some of the estimates are shewn by black dots. The
firm line drawn among these dots is intended to represent the probable
development of the world's population. The thin broken line among the
dots, though adhering more closely to the various estimates, is, however,
of doubtful probability. The lower broken line represents a population
increasing at a uniform rate from 640 millions in 1804 to 1649 milhons
in 1914 ; i.e., 110 years. From the figure it is evident that the rate of
increase in the early part of last century has fallen off, and the world's
population increase will continue at a less rapid rate. Thus it is beyond
question that there have been oscillations of rate, but their period has
not yet been determined, and is perhaps not determinable, owing to lack
of data. One thing is assured, viz., that the present rate of increase
cannot be maintained for any lengthy period.
in.— DETERlffllNATION OF CURVE-CONSTANTS AND OF INTER-
MEDIATE VALUES WHEN THE DATA ARE INSTANTANEOUS
VALUES.
1. Creneial. — The data of statistics are usually to hand in two
essentially different forms, viz., [a) instantaneous values or numbers
which are true at a given moment ; as, for example, the population of a
country at a given instant ; and (b) group values or numbers belonging
to some particular interval of time, as the number of births per month,
or per annum, for a population of given magnitude. Some indications
have already been given of suitable formulEe for instantaneous values,
and in one or two instances the mode of deducing their constants was also
furnished. We proceed to consider the solution for the .constants of
equations which are appropriate for representing instantaneous values.
In mathematical language, if «/ = / {x), then having chosen the form of
the function, it remains to determine its constants from the data. In the
case of group values, the equations must denote the value of the integral
of the function between given limits, and the problem has special features,
the study of which will be undertaken later (IV.) There are a considerable
number of cases of importance, some of which are aperiodic, and others
periodic.
2. Determination of constants wheie a fluctuation is represented
by an integral function of one variable. — ^When, as is ordinarily the case ,
the data consist of values corresponding to equal intervals of time, as,
for example, the population at the end of each quarter, at the end of each
year, or at the end of each ten years, etc., the fluctuation may be empiric-
ally represented by the equation.
(46) y (or — )=a +hx + cx^ + dx^ + etc.,
in which, in the above illustration, x represents time. In this case the
number of constants to be determined wiU depend upon the number of
instants for which we have data. Two classes of cases arise, viz., (i.)
oases in which the data furnish the initial value ; (?/„), that is, a in the
equation above, and (ii.) cases in which the initial value is not furnished,
but is for a unit interval of time before the first result available. In
other words, in the equation above, we require a series of solutions for the
cases where a has a fixed value, including zero, and when it is undeter-
mined ; or what is the same thing, when we have either y, or y^ as the
«
CURVE- CONSTANTS AND INTERMEDIATE VALUES.
35
initial datum. If we have the value of y„, then n subsequent points will
require an integral equation of the rath degree. If not, n points, including
2/1, necessitate an integral equation of the (w-l)th degree.^
If h denote the common interval of time (represented by distance
between the ordinates), the values of y in the preceding expression are : —
(47). .2/o=a ; 2/i=a -\- bk -{- ck -\- etc. ; y^=a + 2bk + 4 cfc^ + etc., etc.
If a be kmyum, then by subtracting a from the values of y we have a
series of equations identical with the above in which —
(48) 2/0= ; 2/1 =bk +ch -\- etc. : 2/3 = 2bk -{- ^cJc"^ + etc.
We deal first with the cases where a is known and assume that the
ordinates 2/1, 2/2> ^te., are the values computed from the axis X, so taken
that a = 0. Then the following formulae, in which yi is denoted by i,
2/2 by ii., etc., may be readily deduced : —
Formulae when j/g = = a.
For-
mula.
Data.
Value of 6.
Value of c.
(49)
Vi
^(1.)
(50)
j/iandj/s,
-^ (4i.-u.)
2^ (—21. + 11.)
(51)
»i to Vs
-gjr- (18i.— 9ii.+2iii.)
25r(— 51.+411.— 111.)
52)
!/ito»^
^ (48i.— 36ii. + 16iu.— 3iv.)
24P (—1041. + 1141i.— 56111. + lllv.)
(53)
Vi to Vb
-^ (3001.— 30011. H-200iii.—75iv. + 12v.)
24j;«(-154i.+214il.-156111. + 611v.-10».)
For-
mula,
Data.
Value o{ d.
Value of e.
(51a)
»i to 2/3
^ (3i.— 3il.-t-Ui.)
(52a)
!/ito Vt
iW' (18i— 2«i.+l«u.— 31v.)
2^. (-«.+«".— 41U.+1V.)
(53a)
Vi to Vs
2Sk' ( + 71i.-118ii. + 98111 .-411v. + 7v.)
24p(— 141. + 2611.— 24iii. + lliT.—2v.)
(53b)
Vitove
Value of / = f25jfc5 (+51.-1011. + lOUl. — 51v. -I- Iv.)
Instead of using the value of the ordinates it is often convenient to
form the successive differences, and then the coefficients b to f can be
expressed very briefly in terms of the leading differences of the ordinates,
corresponding to the values 0, k, 2k, etc., of the abscissa. In the follow-
ing results, Z)i, D^, etc., represent the successive leading differences,
that is, remembering that y^ = 0; Di = 2/1 ; D^ =y^ — 2yi ; Dg= y.^
- 3^2+3 2/1; etc.; etc.
1 See II., § 6, formulas (9) to (13a).
36
APPEISTDIX A.
For-
mula. Data, i
Value of b.
Value of «.
(54)
(55)
(56)
(57)
!/i &y,
i/i to I/s
!/ito Vt
-2J- C2Ji — i),)
6il;
(6i)i — 3Z), + 2D3)
(58) j/i to j/s
J2J (12iJi— 6i)a+4D3— 3Z).,)
gQj (60 2)i—30iJj + 20i),—15D4+ 121)5)
2J2 i'^
2jr (fl,— jD.,)
2^. (12D,— 12Da + lli).)
2^, (12D,— 127)3 + 112).— lOB,).
For-
mula.
Data.
Value of (f
Value of e
Value of /
(56a)
VltOJ/3
W »>
(57a)
Wi to v^
J2P (2D3— 3D,)
24F' •"*
(58a)
Vx to ;/5
24P (4Z)3— 62), + 72»5)
2jj. (X>4— 2 D.)
120*' ^=
Secondly, when a is not known, and the ordinates yi, y^, etc., are
distant Ic, 2k, etc., from the Y-asds, we may readily extrapolate a by
means of the differences. For the coefficients are simply the numbers
of Pascal's triangle (the binomial coefficients) with the first omitted.
Thus, the small Roman figures denoting suffixes only, we have —
(59). . . .a =2i. — ii. ; or 3i. — 3ii. + iii. ; or 4i. — 6ii. + 4Jii. — iv. ;
or 5i. — lOii. + lOiii. - 5iv. + v. ; or 6i. - 15ii. + 20iii.
— 15iv. + 6v, — vi.
for two, three, etc., ordinates given. When a is found, the problem
resolves itself into that for which solutions have already been given, or
it may be directly solved. For five ordinates given, not including a, we
have, for example : —
Formulee.
(60) a= 5i. — lOii. + lOiii. - 5iv. + v.
(61) ^= life (- '^'^i- + 214"- - 234iii. + 122iv. - 25v.)
(62) c = 24P (71i. - 236ii. + 294iii. - 164iv. + 35v.)
1
(63) d= J2P (-71+ 26u. - 36iii. + 22iv. - 5v.)
(64) e= ^Ji. - 4ii. + 6iii. - 4iv. + v.)
CURVE-CONSTANTS AND INTERMEDIATE VALUES. 37
The values of the coefficients in terms of the leading differences (D) are : —
(65) a = 2/1 - Z»i + D, - 2), + 2),
(66) ^ = 1^ (12^1 - 18^2 + 22i), - 25Z)J
(67) « = 2^-2 (^^^^ - ^^^^ + ^^^^^
(68) ^=1^-3(2^3 -5i)J
(69) e = _L D,
3. Evaluation of the differences from the coefficients. — ^When the
coefficients of an integral function, viz., one of the form (46), are known,
and it is desired to ascertain the values of the ordinates y^, j/i, y^, etc.,
the common interval between which is k, they may be rapidly computed
from differences, viz., from x=Q and y=a, together with the following
leading differences :—
Factor into numerical coefficient below —
Differences, bk + ck^ ^ dk^' -\- ek'^ + fk ^.
Di
1
1
1
1
1
D.
2
6
14
30
D.
6
36
150
D.
24
240
D.
120
(70)
For equations of less degree than the fifth the table still serves since
/, e, etc., may be put equal to 0.
4. Subdivision of intervals. — When the ordinates are to hand for
a series of intervals, those for a subdivision of these into m parts may
readily be determined by computing a new series of lesser leading differ-
ences, d say, using those, D say, of the original intervals, as a basis, as
follows : —
Differ- D, D^ D, D4 D,,
ence. m m' m^ m* m°
I- , m--l 2m^-3m+l 6m'-llm'' + 6m^l 2im*~50m^ +35m^-lQm+l
d^ = l_ +
(71)
2 6 24 120
llm^ — 18m + 7 10m^ — 21m' + Um—3
d^= 1 - (m - 1) +
12 12
3to — 3 7m' — l2m + 5
dj = . . 1 - 2 + 4
d^= ■■ ^ ■■ 1 - 2(m- 1)
d.= .." .. 1
38 APPENDIX A.
That is, we divide the wth difference by m", and this factor is multiplied
into the expression opposite d with the proper suffix. The sum of the
terms gives the leading difference in the corresponding d in the first
column.'^
When an interval is divided into 2, 4, 8 or 16, etc., parts, the ordinatea
may be found by successive " interpolations into the middle."^
5. Evaluation of constants of periodic fluctuations. — ^The general
empirical formula for a periodic curve which may be made to fit given
data is —
(72).. 2/ (or — )=a+6sin(j8+a;)+csin2 (y +a;)+rf sin 3 (S+a;)+ etc.
in which the number of terms to be taken depends upon the given data,
and is sufficiently illustrated hereunder.
When the values of y are given only for the beginning of the recurring
period of the total fluctuation and at the end of the first half period, we
have —
(73) y =a +bsm(p + x)
(74) a = i (2/0 + yi) ; 6 sin j8 = 1(2/0 - Vi)
Hence if any definite value be assigned to 6, j8 becomes determinate ;
or if to j8, 6 becomes determinate.
When there are values of y for the beginning of the total period, and
for the instantjS one-third and two-thirds of the period, then we have,
writing —
y^ - « = rj, ; 2/1 — a = ri ; etc.
(75) a = H,y^ + y, -f 2/,) ; tan ;8 = ^^ll-i
ri — r^
a and ^ being found, we have —
(76) b = r„ cosec ^
Using r„ throughout to denote 2/n — a, where n is 0, 1, 2, etc., we
have for four values, viz., at the beginning of a period and at one -fourth,
two-fourths, and three -fourths of the period, from the beginning —
(77)....a = i(2/„+2/i +2/2 +2/3); tan j8 = ^^
and in the expression for tan ^, we may write r for y.
These quantities being found, we then have —
(78) b = Tg cosec ^ = rx sec ^.
For fifth periods, that is, for equidistant ordinates to 4, the formulae
for the constants are : —
(79) y=a+b sin {^-\-x)-\-c sin 2 (y+x)
1 See Text Book Iiistitute of Actuaries, Pt. II., Ed. 1902, p. 443.
CURVE-CONSTANTS AND INTERMEDIATE VALUES.
39
and the solution gives —
(80) a = ts:y.
2 sin 360 [^
(81).. tan /3 =
(82)..
•2 cos 36«(r., + r,)}
r, — r., + 2 cos SB" (r, — rj
I _ cosec j8 ! r„ — 2 cos 36° (r^-\- r^ ) )
3 + 2 cos 72«-
(83) tan 2 y = ^ sin 36'']r„(2 + 2 cos 72")+ 2 cos 36" (r,+ r,) !
r,— r^— 2 cos 36" (r^— r J
(84).
c =cosec 2y.
r J — r^ — 2 cos 36*' (r^ — r^
r„(2 + 2 cos 72") + 2 cos 36° (r
3 + 2 cos 72°.
The values of sin 36°, cos 36°, sin 72°, and cos 72° are respectively : —
i V'(10— 2 V5) = 0.5877853 ; J (V5+1) = 0.8090170 ; J ^7(10+2 V5) =
0.9510565 ; and J (^5 — 1) = 0.3090170.
For sixth periods, that is, for equidistant ordinates to 5 the formulae
for the constants are : —
(85).
(86).
(87).
(88).
(89).
. tan j8 =
V3 {r, -
Ia+IJ
n + '•2 - 1^4,
.tan 2 y
6 = 1 cosec ^ {r^ — r
V3 (r, + r,)
sec 2 y
■'■4 +'■5)
c =
(n— J-g +>'4— *■.-,)
2V3
The solution for twelfth periods is sometimes required as, for example,
when values are to hand for the beginning of each month. Denoting as
before the remainders y„ — a by >•„ we have —
(90).
•« = 1-2 ^"
y-
Then making the following additions for brevity of working, viz. —
in =»-o +r3 +rg +r9 ; No = r^ ->r r^—r^—Tg
'•i + '•4 +'■7 +»'io ; -^1 = '•1 + »"7 — '•t — »"io
(91).,
■'0
-i^2 = »"2 + '•s + »■« + '•ii ; N^=r^+r^~-r^—r^
^0 = '■0 + '■4 + '"s ; -Bo =»'o + ''s— »"6 — '"8
Jlfi = n + r- H- rg ; i^i = ri + t-j — r, — r^
-M'2=r2 + re +r,„; iJ^ = r^ + r^ — r^ — r^„
■^3 ='■3 +^7 +'■11; -'^s ='"3 + '"5— »"9— '■11
(92).
(93).
(94).
R.
VS-Rg
2iZo + V3i?i
6 = tS cosec ^ (2i2o + -/S-R, — -Rj
2^1+ 21^0 --AT,
V3i?j)
40 APPENDIX A.
(95) " = 173 sec2y(i\ri+ 7\^,)
M M
(96)....tan3S=fP-|j
(97) rf = i cosec 38 (Jl/o — M .)
(98) taii4e= y^\
111 — I12
(99) ^ = i cosec 4eip
(101) /= J, cosec 5? (2i?(, — V3i?] — iJa + VS-Rs)
6. Constants of exponential curves. — The case of a curve of the
type
(102) y = 1 ± ^^i"**"'
see equation (20), has already been sufficiently considered : its constants
can be found as shewn by formulfe (23) to (30) ; and also that of the
type, see equation (32)
1
(103) y =1 + Tji^™±««;
see formulEe (35) to (38). In general, curves of this type may be solved
by forming the equations y' = y — 1 and taking logarithms when we get
such forms as —
(104) M = e + log i (± m + ni) and u — e -, ^ — ~
° ± m ± Mi
solutions for which have already been sufficiently indicated. As this
process of taking logarithms is the key to many solutions, we now refer
more fully to the matter. The essence of this method of solving is that
if a series of values on the axis of abscissae be taken in geometrical pro-
gression, their logarithms are in arithmetical progression. Thus, ^ being
log X, we have —
Quantities = x ; kx ; k^x ; k^x ; etc.;
Logarithms of same = x '> X+^! X+^fc; ;)^+3fc; etc.
Hence the problems of solution are reduced to those of the examples
illustrated by formulas (46) to (71).
7. Evaluation of the constants of various curves representing types
of fluctuation. — ^The evaluation of the constants of various curves can
often be effected by taking suitable ordinates to the curve and solving
from their logarithms. This is illustrated in the following series of
equations : — ■
(105).... 3J = (Te^^'" = QTe^'i^' = €S^^"' = QTJ^
M'
CURVE-CONSTANTS AND INTERMEDIATE VALUES. U
We have on taking napierian logarithms —
(106) Y = AX"" + O = AM"" + G
in which log ^ = F ; log y[ = C ; log Jl. = A ; log Z = a; ; and
log M = m.
The second curve may be called the first logarithmic homologue of
the first, and the first the first anti-logarithmic generatrix of the second.
Subsequent curves may be similarly defined as the second logarithmic
homologue, etc.
Yet again, if G be zero, we have on taking the logarithm of this last
expression —
(107) y = a + mx,
in which log Y =^ y ; and log A= a.
This will sufficiently illustrate the matter. Several examples of
solution wiU be given of important curves for representing fluctuation.
In the curve
(108) y = 4 + Bx'^
"If ^ = ; then the solution is found at once from any two values of y
and of X. For we have —
(109) log y = log B -\- m log x.
On Fig. 21 hereinafter, these curves are shewn by thick lines for
positive values of m, and by thin lines for negative values.
If, however, A be not zero, then we must take three values of y
for abscissas of the value x, xk, xk ^, when it may easily be shewn that —
(110) . . , . y^-y^ = k-; ovn = ^°g ^y^ - y4^-l°^-^^=:yj-^
2/2 — yi log k
The curve
(111) y = B + Ge'">
can be solved by taking the values of y for x, x + k, x -\- 2k, for
,/ . ,/, n [p a(x + 2Jc) g a{x + m
^'-^'^> y^ — y^- c [e "(^ + *) — e«^]
Consequently putting 7.,.^^ for the left-hand expression, and writing
2.3025851 for the modulus for changing common into Napierian
logarithms
(113) a = ^log,o F,,,
When a is found the solutions for B and G are obvious. Curves of the
equation e^ are shewn by thick Unes on Fig. 22 hereinafter, and those
of equation 1 / e^ by thin lines.
The exponential curve —
(114) y = ^ + .Be"^'
can be solved if A be zero, or if A be known, and a new series of y' =y— A
be formed. Thus A being zero,
(114a) log 2/ = log 5 + nxP log e.
42 APPENDIX A.
Hence, as before, taking three values of y for x, xk, xk ^, the solution is —
(115) P = A • l°g /[^-^-^ -^^}
log A; ^ \log2/2 — logyii
(116) ^ = ig^-i"gyi
' xv{ki>—l) log c
(117) log B = log 2/1 — wxP log e.
These curves are shewn for Fig. 23 hereinafter, for various values of
n and p.
The curve —
(118) 2/ = 4a;™e"^''
is solved by taking four ordinates, viz., for x, xk, xk^, xk^, when the
solution becomes 1 —
(119) ^ = 1 . log i iogy.-2iogy 3 +iogy4 |
log *: " t log 2/1-2 log 2/2 + log 2/3 )
using common logarithms. Then M denoting log e, we have also —
(1201 n = (^Qgyi ~ 2 log 2/2 + log 2/3) _ (log 2/2 - 2 log 2/3 +Iogy4 )
^ '" MxP(kP-l)^ ~ MxPkP{kP - l)^
(121) m = (log ^2 - log yi) - Mn xP (kv - 1).
log k
There are obviously two other possible formulae for m.
(122). . log A = log 2/1 — m logaii + Mnxv
the value of M being 0.4342945. Three other formulae are also possible
for A. For further formulae see (150) to (153) later ; see also Figs. 21
to 27, hereinafter, for the forms of the curve.
8. Polymoiphic and other fluctuations. — Monomorphic or rather
unimodal curves disclose a single maximum (or minimum) value. But
there are fluctuations which are polymorphic or multimodal. These may
be regarded as compounded of monomorphic curves. PracticaUy their
dissection is best effected by the graphic methods of analysis. In general
any curve can be represented with great accuracy by either
(123) y =a + bxP+ cxi + dx" + etc., or by
(224) Y=: ga + bxP + exv + etc.
where p, q, r, etc., are not restricted to integral values.
The latter curve is reduced to the former by taking the logarithm ;
thus, 2/ = log« Y. To solve for the constants we must have six points
besides the origin. If the value of a be known, the curve can be reduced
to one passing through the origin by subtracting a. Then we take values
of y for x, xk, xk ^, xk *, etc. For the case for terms in p and q only,
we can proceed as follows : —
1 For a more complete study of the curve, see "Studies in StatriBtical Repre-
sentation. On the Nature of the Curve," above given, viz. (118), by G. H. Knibbs
Joum. Roy. Soc, Vol. XLIV., pp. 34] -367, 1910.
CURVE-CONSTANTS AND INTERMEDIATE VALUES.
43
By writing L for bx'P and M for cx^ , and a for k^ and ^ for k^ ; we
have —
(125).. yi = L + M; y^ = La+M^; y^ = La^+M^^; y^^La^+M^^
Hence by eliminating L and M from the first three and from the last
three equations, we have respectively—
(126a).
1
2 ^2
2/1
2/2
= 0;
1 1 2/2
=0.
. (1266)
a ^ 2/8
a^ ;8'' 2/,
Consequently a and ^ are the roots of-
(127).
1 2/1 2/2=0
i 2/2 2/3
P 2/3 2/4
Thus the two values of f in the equation —
{128)..^^^i{a + p)+ap = ^^yiy^-yl)+i{y,y,-y,y^)+(y2y^-lfl)=0
are the values of kP and /fc«. And since k is known, the solution is to
hand by taking logarithms. •
The solution for three indices is similar. The six equations can be
written —
(129) 2/^ + 1 = La™ + ilf;8'» + iVy'»
and a, j8, and y ; that is ki>, M, and k^, are the roots of the equation.
(130).
1
2/1
2/2
2/3
2/4
2/2 2/3
2/3 2/4
2/4 2/.-.
2/0 2/e
=
^ 2, ^ and
which may be expanded in the form —
(130a) 4iP — 3^2^ + 3^3^ — ^^ =
where Ai, ZA^,ZA^ and A^ are the minors respectively of | ^
1 in the determinant.
If the constant a is included in (123) or (124), the solution is more
tedious. We must then have seven values of y. Thus —
(131) y„ + 1 = o + -Z^a™ + M^"' + By^
(131a) 2/™ + 2 - 2/m + 1 = i'a™ + M'^^ + R'y^
the accented values being L'— L [a — \); M' = M (^ — 1) ; etc.
Thus a, j8, and y are the roots of —
1 ^2 - 2/1 2/3 - 2/2
I 2/3-2/2 2/4 - 2/3
P 2/4 - 2/3 2/5 - Vi
P 2/.5 - 2/4 2/6 - Vs 2/7-2/6
Writing Fj, Fg' -^i' ^o for the minors of P, ^*, ^, and 1 in the
determinant, the equation becomes
(133) i'Y,^i^T,+^Yi- y„ = 0.
(132).
2/4
- 2/3
2/6
-2/4
2/e
- y..
=
44
APPENDIX A.
It will be seen from the preceding examples that when, a-s regards
their indices, the equations are not restricted to integral values, the
Acting power of the curve is enormously increased. To fit seven points
with integral indices we should have to have an equation of the sixth
degree. 1
Pigures 6 and 7 furnish graphs for simple cases with two indices
only. From these graphs, which also are for integral values of the index
only, it is immediately evident that the loci of curves with fractional
values must he between the curves drawn. The forms of the curves may,
of course, be modified also by varying the coefficients : hence the fitting
power of expressions of the type considered obviously becomes very great
when the limitation imposed by restricting the indices solely to integral
values is abandoned.^
24
'«~
/
/
ix
-1
X
*1
-x^
^+
ex"
/
/
V
/
g
^
<■
y
\
/
/
^^
y
^
^
t
/
/
t-
^
s^>
^--.
-^
/
/
ts-
[^
—
$A
<5
^
^
■^
''
'V
¥,
^
y
?^
^
Vs
-^
^
^
\
\^
^r*
1
1___^
—
/7
\,\
sc-
^
y
\,
^
^
"^
l_
—
^
'^
V
\
\
\
•
\
-bx
-^+
x*
i
■x"
I_
CZ
*1^
\
'vj
\,
\
-«4
3 4 1 2 S 4
Fig. 6.
1 See " Studies in Statistical Representation, III., Curves, their logarithmic
Homologues," etc., by G. H. Knibbs and F.W. Barford, Joum. Roy. Soc. K.S.Walee,
Vol. XL VIII., pp. 473-496
2 The limitations of Jthe fitting power of the curve are discussed in the paper
referred to in the preceding footnote. These limitations, in general, are of no
moment in statistical results.
CURVE -CONSTANTS AND INTERMEDIATE VALUES.
45
A.a.
X
»
X
"-J
.-t
/
7
^
«
/
«?
«
<e
i
r
/
/
/
/
/
A--
/
/
^y
/
/
/
<^;'
^
, ^
1
/
r
\
Li
/
/
.^
u
/
/
i>
^
\\\
/
/
/.
^
.4r
/
/
/•
/
>-
m\
L
^
^
-
-
L
6-
vVW
L^
.1'
1
— .
~^M
^
7/
V
r--
A
w)
^
:zz
^-
4"
J
//
\
\
-ii
— -
J
^^
■^
^
=?
^
1
^^"
hr"
T
~a.
^<?
■^
-CH
O"
1
j
Tjo.
"'*j
♦i
r'^c
r+^
J
,
^
c
/
^./
■"/
/
t/
/
/
/
'
/
/
/
\
/
1
/
\
/
7
\
\
»->
/
1
/
c
.D
5-
4-
3-
2
I
0-
A
N
/
y
/ 1
--
"\
-
^
/
/
/
y
1
\
^
XI
:>
L^
,-
/
^l
t'/
x:
<
^
^
J
>-
=i-i
?<
^
«^
\
s
/f
"^
. i
^
__£
^
i-;
1
^^
\i
^
\<^j
*r
/
S
;
'
»/
\
\s
T^
4
1 i
/
\
\ -
N<^
>^
I
/
{b\
+
j:'
^>
'
x-'^
cx
..l
^^
'
-
1
T
Fig. 7.
Some special cases of fluctuation will now be treated in dealing with
problems treating of fluctuating elements that directly or indirectly
influence the aggregate or constitution of the population.
9. Projective anamorphosis.— A symmetrical curve of frequency
(or symmetrical distribution) may become asymmetrical by the elements
being projectively varied by means of different types of projection (plane
or other). This change may be called projective anamorphosis. Any
character of a population may be regarded as subject to influence acting
46 APPENDIX A.
progressively (or retrogressively) with increase of the measiire of the
character in question, as for example, if the influence tending to increase
weight (or height) acted more or less powerfully with increase of that
character. This would lead to an asymmetric or skew frequency. Thus
if a normal frequency be denoted by y =(f) {x) ; a specialised frequency
conceived to originate therefrom would he given by y' = f («) <^ (a;).
This expression may also be skew, dimorphic, polymorphic, or in fact,
what we please, according to the character of/ (x). If/ {x)=mx or m/x, a
symmetrical curve is converted into a skew curve. If / (x) have a mode
such that it is not identical with that of ^ (a;) the latter will be dimorphic .
From this it is seen that the ordinates to a dimorphic curve may be the
sum or product of the ordinates to two monomorphic curves. It is not
proposed to elaborate just here, however, the general theory of anamor-
phosis by plane or other projection. It may be easily seen, however,
that a skew curve may be readily derived from a symmetrical one, while
retaining the general algebraical properties of the latter, by a projection,
from a hne parallel to the axis of the given symmetrical curve, through the
curve and on to a plane passing through the axis but inclined to the
plane of the given curve. This will be more fuUy considered hereinafter.
IV.— SPECIAL TYPES OF CURVES AND THEIR CHARACTERISTICS.
1. General. — ^When the characters of a population have a tendency
to deviate in either direction equally, and the number of the population
is P, the characters wiU be distributed as the coefficients {^ + ^)'»
i.e., as the numbers in Pascal's triangle, which, when m is infinite, becomes
the curve
(134) y = Pe * ; or say Pe *
the first form (viz., that when the power w = 2) being the ordinary
probability curve, in which k is the modulus. This type of distribution
is but one case of the more general expression which, interpreted in a
certain way,i has a cusp for the vertex for values of n equal to or less than
unity, and a curve convex upwards for aU values greater than unity,
the vertex however becoming more flat as n is increased.
The curves graphed are
a = axis
-J
-X
r~
'/^\\'
f
Vf\
_^
///(
\\1\
^yv\
///
i
1/ ^
^' 1
'^4
.^
'■
n
a.
■^
^
^. ■
-■
3:
c^
' a
/
\
s.
■^
..
—
=-
^^
cT
-^
.
/>
_
bJ
;^
--.
—
:i_J 1_|. X
Asymptote.
Fig. 8.
Asymptote.
The curve y = e~*" is coincident with a from the point Y to a point
y = 0.3678781 ; it is then parallel to the X axis. All the curves intersect
at this point.
Such a distribution is symmetrical, and takes the form in the figure
hereunder, Pig. 7, in which curve 'a' shews its form for to = ; 'b' for
TO = I ; ' c' for w = 1 ; ' d' for to = 2 ; ' e' for to = 4.
When the probabilities of distribution are not equal for possible
alternatives, and the probabilities of these alternatives are as p and q,
the sum of p and q being unity, then the distribution will be the coefficients
oi {p H-g)"*. Ifg'and^ are not equal the curve is not symmetrical,
but is of a form Uke Fig. 9 hereinafter. Whether results can be made to
conform to a particular tjrpe or not depends on the form of the curve, and
1 That is, BO that e-«" and e-«("+*'i are in the same spatial region, or on
the same side of an axis, and are not allocated to different regions according to
whether the number (n+Sn) is even or odd.
48 APPENDIX A.
in particular on the position of its vertex ; on whether its sides meet the
axis of the variable more or less sharply or asymptotically, on whether it
is monomorphic or polymorphic, or has one " mode" (is unimodal) or more
modes than one {is multimodal). Various types of unimodal fluctuations,
commencing and ending with zero values or otherwise, have been given
by Prof. Pearson. These are intended to reproduce the group-values of
statistical data, under appropriate forms of curves, by a method which
has been called the method of moments, the forms of the curves being
derived from the normal curve of probability. We shall later refer to
these, but remark first of all that the critical elements of the curves
representing distributions or fluctuations are as follows, viz. : —
(a) the value of the ordinate when the variable is zero ;
(6) the values of the variable for which the ordinates become zero ;*
(c) or, if they do not become zero, the value of the ordinate when
the variable is infinity ;
(d) the abscissa of the m,ode, or greatest ordinate, and the value of
that ordinate ;
(e) the abscissa of the ordinate which equally divides the curve
area (as, for example, the abscissa which corresponds to the
average value, or the centroid vertical) ;
(/) the distance between these two ordinates {d) and (e) (the numerator
of the quantity defining the skewness) ;
(g) the m£an-deviation of the curve (or denominator of the skewness) ;
{h) the abscissa of the point where the curvature changes its sign,
(point of inflexion) ;
[i) the abscissa of the point of most rapid change of direction of
the curve.
(a)......2/=/(0); (6) f (x) = ; (c) /(^)=A-orO;
id) x„ when df (x) / dx = ; and y„ = / (x„) ;
(e) Xa when the value of ^ xf (x) dx -^ jf (x) dx for the range of
the variable up to x^ is equal to that for x^ onward ;
(f) i^a-^m)
{g) Wj = ■\/[xJ (x) dx -^J/(x)^*]' in which x is measured
from the position of the mean (x^)-
(h) Xi whend^f (x) / dx^ = ;
(i) Xj, when d^f (x) /dx^ = 0.
* The approach of statistical curves to the axis of abscissae or to the axis of
ordinates is, in general, not determined by mathematical considerations, but by
a knowledge of the nature of the data itself. For example, the terminals of the curve
of fertility (discussed hereinafter) deduced from ex-nuptial births, shews a diminution
which may be represented very closely by the niunbers 1078, 154, 22, 3J^, for the
ages 16, 15, 14 and 13 respectively, i.e., each number is one-seventh of the number
preceding it. Merely mathematically, therefore, it is more probable that these
should continue for the ages 12, 11, 10, 9, etc., as 0.45 ; 0.064 ; 0.009 ; 0.0013, etc.
Even at age there would, of coiu'se, be still a positive value though small. But
physiological knowledge indicates that the earliest arrival of puberty is probably
over 10 years, hence 11 would be the earliest age for birth, and the ordinate must
be zero.
SPECIAI, TYPES OF CURVES AND CHARACTERISTICS.
49
2. Curves of generalised probability.— Prof. Pearson proposes to
reduce forms of distribution of statistical facts under a series of seven
type-forms of curves, representing what may be called curves of generalised
probability,! and much work has been reduced on this system.
Fig. 9
Type I. (i.).
Pig. 10.
Type I. (ii.).
Pig. 11.
Type I. (ii.a).
Pig. 12.
Type I. (iii.).
Pig. 13.
Type II. (i.).
Pig. 14.
Type n. (ii.)
His first type (Type I.) is :-
(135).
■Vo =2/(1 +
«i
(1 - „- )
which may take two other fundamental sub-forms, viz.,
(136).
y = 2/0 ( - - 1) (1
) ' ; and
(137).
2/ = 2/0 (1 - „- ) ( 1 + ;f )
which are represented respectively by the forms in Pigs. 9 to 12.^ When
V, Oi and 02 are positive the curve meets the X axis at the distances Oj
and a^, see the figures. The abscissa of the mode is and the curve is
skew.
1 See his " Contributions to the Mathematical Theory of Evolution." Phil.
Trans., Vol. 185 (1894) A, pp. 71-110; Vol. 186 (1895) A, pp. 343-414; Vol. 187
(1896) A, pp. 253-318; Vol. 191 (1898) A, pp. 229-311; Vol. 192 (1898) A, pp.
169-244; Vol. 192 (1899) A, pp. 257-330; Vol. 195 (1900) A, pp. 1-47; Vol. 195
(1900) A, pp. 79-150 ; Vol. 197 (1901) A, pp. 285-379 ; Vol. 197 (1901) A, pp. 443-459.
See Phil. Trans., Vol. 186 A, pp. 364-5.
50
APPENDIX A.
If, in the formula for Tj^e I., oj be made equal to a^, then the
formula becomes that of Type IT./ shewn by Figs. 13 and 14, viz. —
(138).
3/ = 2/o (1
;)
the basic form of which, when y^, is unity, is an elUpse with semiaxes a
and 1. The figure becomes a circle when i^ is ^ and a is 1. In general,
any form can be deduced from the basic form which, when va is unity,
is a parabola (the quantity within the brackets) in (138). If this quantity
be infinite and positive the figure becomes X' P' Y P X : see Pig. 13.
If positive and greater than unity, it is the curve r'r ; if unity it is the
parabola s's ; if less than unity, the curve t't in Fig. 13. The abscissa
of the mode is 0, and the curve is of course symmetrical.
If v be made negative in (138) the formula becomes
1
(139) y = 2/0
X^\ a,
and is shewn by Fig. 14. The abscissa of the mode (of mediocrity) is
at the origin.
If in the second sub-form of Type I. we make a^ infinity, then
(140).
■y = Voi- - 1)'
the form of which is shewn in Fig. 11 ; that is, the curve is asymptotic
to the ordinate whose abscissa is distant + a from the origin, and asymp-
totic also to the axis OX.
Fig. 15.
Type in.
Fig. 16.
Type IV.
Fig. 17.
TypeV.
?. -?i
^^y^V'^ola Asffmpit,te Jlsympcaee
?' > 7;
Asi/mpCo/^
Fig. 18.
Type VI.
Fig. 19.
Type VII.
Fig. 20.
Various.
1 Op. cit., pp. 364.5,
SPECIAL TYPES OF CURVES AND CHARACTERISTICS. 51
When in formula (135) a^ is infinity, then its form becomes Type III.,
viz.,
(141) y = y, (1 +^re-"'
and is of the form shewn in Mg. 15. The abscissa of the mode is at the
origin, and the curve is skew.
Type IV. is of the form shewn in Fig. 16 ; its equation being : —
1 + -A e-" to«-^*/<» ; or = 2/o cos*"' Q.t~'^
being the angle the tangent of which is x/a. The curve is asymptotic
to the X axis on both branches ; its mode is at the distance —va/2m
from the origin, and it is skew : see Fig. 16.
Tjrpe v., is of the form shewn in Fig. 17, and its equation is : —
(143) y = 2/0 ^^e~ "
The curve is limited on one side at the axis X, i.e., for a; = 0, and is
asymptotic thereto at the other ; its mode being at the distance y/f.
The curve is skew. The mean is at the distance y / {p —2) from the
origin.
Type VI. is of the form shewn in Fig. 18. Its equation is : —
(144) y = y^{x — a)«^ a;~«i
The curve is limited on one side only, viz., when a; = a. The mode
is at 0^1/(9-1 -g-z)-
Type VII. is the ordinary probability curve: see Fig. 19, viz. : —
?!
(145) y = j/o e-"
the mode being at the origin and the curve unhmited in either direction,
and of course symmetrical.
Curves a to e. Fig. 20, are typical forms of the following character-
istics in a population, viz. : —
(a) Infantile mortality, income, probates, value of houses, etc. ;
(6) MortaUty from scarlet fever, diphtheria, etc. ;
(c) Pauper frequency, divorce frequency with respect to duration of
marriage, frequency of scarlet fever with age, of typhus, etc. ;
(d) Senile mortality, mortality from enteric at different ages, fre-
quency of marriage of wives corresponding to age of husbands
at marriage, etc. ;
(e) Height, weight, strength frequency, anthropometric measure-
ments, etc,
52 APPENDIX A.
3. The method of evaluating the constants of the curves of generalised
piobabihty. — Two things are requisite in. using the Pearson curves,
viz. (i.) to select the appropriate type of curve ; and (ii.) to
evaluate the constants of the selected curve. The selection of a curve
which can be made to fit the given group-data depends upon relationships
among the moments calculated about the mean. These relationships
determine three criteria, which, after the necessary computations have
been made, indicate the appropriate selection.^
Solutions can also be effected by means of a combination of graphical
and numerical methods. The numerical solutions can be effected by
taking logarithms, that is,
(146) log 2/ = log 2/o + log / (x).
The process in detail can readily be followed from the examples in III.
(See in particular § 7). In general the solution must be tentative, and
it is important to notice that the type-curve selected is not valid if the
data have to be altered larger amounts than they are probably in error.
The principle which should be employed is the following : — ^The adoption
of a type-curve can be regarded as satisfactory only when it represents
the data within the limits of their probable errors. In other words the
geometric form and the algebraic processes should be subordinated to the
data and not vice versa.
4. Flexible curves. — Although the type-curves just considered fulfil
their general purpose fairly well, experience shews that their "fitting
pmoer" is somewhat limited. To overcome this, other types are necessary,
the " fitting power" of which is greater. In order to embrace as many
forms as possible under cover of a single formula a curve may be so taken
that its limiting forms shall include all parabolas, all hyperbolas (or
parabolas with negative indices), all exponentials with positive or negative
indices, and all curves of the normal probabihty type. Such a curve
wUl necessarily include all intermediate forms. I have called this type of
curve a, flexible curve.
Formula (149) in the next section is a curve of the type in question.
Its graph depends fundamentally upon the values of the indices m, n,
and p, and its vertical scale depends upon the constant A . The mode of
solving to determine its constants depends upon taking a series of values
of the abscissa in geometrical ratio, and is indicated in the next section.
1 See the article by Professor Pearson already referred to, also " Frequency
Curves and Correlation," by W. Palin Elderton (C and E. Layton, London, 1906) ;
and " Statistical Methods with special reference to Biological Variation," by C. B.
Davenport (Chapman and Hall Ltd., London, 1904).
The curves indicated on p. 57 and p. 81 of Mr. W. Palin Elderton'a work do
not satisfactorily represent the data, forasmuch as the curves chosen were in-
sufficiently flexible.
SPECIAL TYPES OF CURVES ANDXCHARACTERISTICS. 53
5. Determination of the constants of a flexible curve.— The probability
curve, see (134) hereinbefore, viz.,
(147) 2/ = Ce *" or,
~k^ + <'
in which c = log« G, may be put in a more general form, viz.-
(148) y = e /W
+ Fix) +
that is, its modulus k and constant G may be assumed to be functions of x.
If we suppose that
F(x) = a +^ log {±x) ; / (a;) = ya:« ; c = ;
and write p = 2 — ,s ; w = -1/y ; log ^4 = a ; m = j8, the expression
(148) can be written
(149) y = Ax'^e"^
see (118) in III. 7. This curve can fit a great variety of forms, viz.,
such as are shewn on Figs. 21 to 27, referred to later.
In practice it is not quite satisfactory to depend on four points.
A better fit can be secured by taking several, say r, series of ordinates
for values of the abscissa x^, k^x^ kl Xg, x„ k^x^
¥r Xf. Each set will give a value for p, say p^, p^, etc., and a mean
(geometric, arithmetic, or other) can be taken, p say. Or writing 7^,,, for
log Vp- ^ log y« + log yr, we have
(150) i) log {k,.k, k,) = log {n[{ Y,j r,,3)};
ill *■ denoting the product of r different sets of the quantities in the
brackets.
The use of this mean value of p, being inconsistent with each set of
four ordinates, gives for each set two solutions for n, three for m, and
four for A, that is in aU 2r, 3r and 4r solutions respectively for these
constants. Having found the mean value for p we use it, in solving for a
mean value of n, thus :— ^
, ni (log yi - log y2 - log ys + lo g y^}
n[{MxP{k^P - 1) (k^ -1)
(151).. r log n= log rTiii^^v^i.2v WTTv TVl ' °'
(151a) r ^ri»g iyiy^/y2ys)
Zl\MxV{kv +1) (kP - 1)2|
1 By comparing this with (120) it will be seen that the mean is taken of two
quantities each of which give n, on the principle that if a/b=c/d approximately
{a+c)/{b+d) is sensibly the arithmetic mean, or having two equations which give
n, we assign an equal weight to each. The geometric mean, however, is taken in
obtaining a mean result from the difierent sets. Of course {n-^ + nf)/r would
also be a satisfactory value, n here denoting the value obtained by using the mean
value of p. Although the two formulae are not identical, practically there is no
cogent reason for preferring one to the other.
54 APPENDIX A.
Adopting the mean values, thus found, for p and n, we have three
different values for m given by each set. Reverting to formula (121),
if we give double weight to the value found from the intermediate term
\ye get^
(152)
,„_ -S[(}ogy^+ logy2)+iyfiogys+logyi)~Mni:J,xP{k^P -IW+I)]
4i7^ log k.
Mean values for p, n and m being to hand, we have for A four values from
each set of ordinates, see (122) hereinbefore, the general formula being
(153) log ^ = log y^ - log (p-i x) - Mn (k"-^ xf .
hence for a mean of 4r values of A we have
1 3
( 153a) . . log ^ = j^ ; 27;; log (2/1 2/2 y%y^) - 4m2^ (log a; + g log ^)
- Mn i:[ [xv (F*" + k^f 4- AP 4- 1)])
M denotes throughout 0.434. .etc., if common logarithms are employed,
or imity if Napierian.
Ignoring the coefficient A the first and second derivatives of the
curve (149) are respectively
(154) dy / dx = x-^-'^ e^' (m + npx'); and
(155). .dhf/dx^ = x'^-^e^ {m{m—\)+npx'i' {2m-\-p—\)+ n^^x^*\
hence the mode (maximum or minimum value) is given by
J.
(156) X = {-m/npY
which becomes, for p = 1, simply— m/w. The point of inflection is
given by solving the equation
(157) P2 + p (2m +^ - 1) +m (m - 1) =
in which F denotes npx'^ ; this gives :
a58^ ^. _ ! 2m + y- 1 ± V[4mff + (p - 1)"] ^
which, when m = 1 gives the value
/I -1- \ - -
(159) ^i = - (-^j ^ and also x^ -(1 / np) "
for the abscissa of the mode.
1 The principle indicated in the preceding note applies, viz., if (a+6+c)/d
equals (a -\- ? -^ y)/S approximately, then (« + n -f 6 + j3 + c + 7) / (d + S) is
sensibly the aritlimetic mean.
t$
SPECIAL TYPES OF CURVES AND CHARACTERISTICS. 56
The integral of a curve can take a number of forms as follows, viz : —
(160). .fydx =\x'^e^ dx =
x^ I nxP (m + 1) ■nTal^v (m + 1) 1
m+l 1 "^(m+jj+l)!! +" ' + (m+r^+l) r ! +• -etc J; or
(161) 5!^e»4l ^P^^ (^^^)^
'••m+l ( m+^ + 1 "f"(m+39+l)(m+2i)+l) ■"••
, (-1)^ [np xoy , , I
+ "7 i 1 1 / 7 — 7 r-TT- ± etc. \ ; or
^ (m+p + l)....(m + r^ +1) =^ j'
(162) '^"'"''"^' e«^/l - ^-P+^ + (m-j)+l)(m-2j>+l) _
■«:P [ 7vpx!P {npx^)^
+{-lY ("^-y + l)----(m-r3> + l) ^ ^tc. 1
Between the limits and oo the integral may be put into the forms of
the second Eulerian integral, and is
(163). ...... f>">e -»»="«««= — !^ ^ ^
{pn P )
which, when m = o, gives
(164) j:^-'"^dx = r(^)/(pr^)
The abscissa, Xc say, of the centroid vertical, or mean of the distribution.
IS
jx«^+ie-'^dx \ p J
^^®®^ ^'^ Ja:'»e-"*'rfa; "" ^/m + l\ i
It is sometimes necessary to make the definite integral (163) when multi-
pHed by the coefficient A, equal to unity. In such a case we must have
the value of this constant the reciprocal of that given in the value of the
integral mentioned, viz. (163) ; that is
(166) A =pn P /^(^)
Simplioations of these general formulae are often possible. ^
1 For a fuUer study of this curve, see " Studies in Statistical Representation,"
by G. H. Knibbs. Jour. Roy. Soo. N.S.W., Vol. XLIV., pp. 341-367 ; 1910.
56
APPENDIX A.
The forms of the curves are as shewn on the Figs. 21 to 27. If w
in e"^ be zero, the curve degrades to Ax^, and we have the forms
in Fig. 21, in which the capital letters shew the curves when m is positive,
and the small letters when m is negative.
Fig. 21. Fig. 22.
If m be zero,
x^ will be unity,
and if f also be
unity, the curves be-
come e"^, the forms
of which are shewn
onFig.22,the upper
hues denoting the
values when w. is posi -
tive and the lower
when n is negative.
If p, however,
be not unity, and p
and n be positive,
we shall have such
forms as A, B, and
G on Fig. 23. If p
be negative and n
positive, the forms
become those shewn
by the curves D, E,
and F in the same
figure.
\\
b
/
/
M
/
/
1
c\
sM
[/.
c_^
^-
,/-
/7
v^
~. —
^.
/
^
■-^
ar--
t
/
/
/
c/
/^
/
/
y
,b
^
t^
(
)
z
Fig. 23.
Fig. 24.
If n be negative and p be positive, the forms become a, b, and c,
the reciprocals respectively of curves A, B, and C ; and if both n and p
be negative, the curves are such as d, e, and f, viz., the reciprocals re-
spectively of curves D, E and Fin the same figure, viz., Fig. 23.
Figs. 24 to 27 give values of the curves when both m, w, and p have
values other than zero, the Ught fines denoting the reciprocals of the
curves shewn by the heavy fines, and the curves being the following, viz.:
Fig. 24
Fig. 25
Values of—
VAIiUES OF —
m
n
V
m n p
.. A
=
\
-\
\
Fig. 26 .
. A
= 1 -i 6
.. B
=
\
-\
3
)j
. B
= _6 -\ -6
.. C
=
\
-1
\
3J
. C
= 6 -i 6
.. A
=
\
-2
5J
. D
= 6 -i 1
.. B
=
\
-1
Fig. 27 .
. A
= -1 i 6
.. C
=
\
-\
j»
. B
= -6 J 6
.. D
=
2
-1
'J
. C
= 1-11
.. E
=
2
-2
)J
. D
= 1-21
SPECIAL TYPES OF CURVES AND CHARACTERISTICS.
57
In the reciprocal curves, viz., a, b, c, d, etc., the signs of m and n
are changed, but not that of p?- These wUl sufficiently illustrate the
possible forms of the curve.
Fig. 26.
Fig. 25.
Fig. 27.
6. Generalised probability curves derived from projections of normal
curve.=ln Fig. 28 let'bYa denote a normal "error" (or probability)
curve, the ordinates of b and a being denoted by corresponding suffixes.
If a Une be drawn the distance I above OY and parallel thereto (and
parallel therefore also to the plane of the curve), it may be represented
by the point 0' in any plane at right angles to the plane of the curve.
1 It may be mentioned that H. P61abou, in dealing with the influence of
temperature on ohemioal reactions, developed a relation in the form
log y = a-'rb/x + c log x ;
which, ofjoourse, may be written in the form y=ah~'' x", which is merely a simple
case of formula (149). See M6m. d.l. Soc. des Sciences physiques et naturelles de
Bordeaux [5]. 3, pp. 141.257; 1898: Compt. Rend. 124, pp. 35, 360, 686 ; 1897.
58
APPENDIX A.
Let a line be drawn from any point, on the curve, viz., a, at right angles
to 0'. This will be the Une O'Q, which, when produced to q on a line
VOqU, making the angle 6 with the line PO, gives the point corres-
ponding to a. The abscissa then may be taken either as Oq or as its
orthogonal projection on OP. The latter is more simple. If it be pro-
duced to q' on a plane making the angle 6' with the axis OY, it will
give a result of greater skewness, see the points a^ and aa^ in the figure.
The scheme of projection will be obvious from the figure, and need not be
described in detail.
Let ^ denote any abscissa on the curve derived by projection,
and X the corresponding abscissa on the original curve. Then by similar
triangles we have at once the relation of x and £ in terms of I and 9,
inasmuch as
(167) x/l = ii ~x} / ita.n9.
This gives, on writing m for (tan 9) / I,
(168) i = x/{l -mx); x = $ / {1 + m^)
from which it is at once evident that the same result may be obtained
by any values of I and 9 whatsoever, which give the same value of m.
Thus, the point S
with the projecting
height 00" = T gives
the point U, the ortho-
gonal projection of
which Q" is identical
with the result with
the projecting height
00' = I, viz., R, as
is evident from the
figure.
Fig. 29 shews by
a heavy fine the curve
derived from the curve
in Fig. 28 by projection
on to the plane VOU,
and by a thin line the
curve similarly derived
by projection on to the
plane WOR in that
figure.
Fig. 29.
SPECIAL TYPES OF CURVES AND CHARACTERISTICS. 59
Hence, if for x in the probability-curve equation, we substitute its
numerical equivalent, we obtain
(169) y = i/g-' + ^omi+cf =i/e„(.+^)2.
K in the second expression being cm^, and jj, being the reciprocal of m.
The curve is asymmetric, since the denominator differs in value according
as ^ is negative or positive. Incidentally we notice that if I be relatively
large or d relatively small, m is small, and the asymmetry is not marked ;
and when I is infinite or d zero, the asymmetry vanishes, as is seen by the
projection. In this last expression when ^ is negative and equal to /x,
2/ = 0, so that there is a terminal of the curve on the negative side cor-
responding to a; = — 00 . When tan 6 = l/x, then mx = 1, and | is
infinity ; that is to say, the projecting line is parallel to the plane
through the axis. When — | is one-half of —x, then the point with the
same ordinate on the positive side is at infinity. This can also be seen
on the figure.^
This indicates the limitation of the method of projecting onto a plane,
namely, that if there is to be a corresponding point at a finite distance on
one side of the axis, the abscissae on the other side cannot be reduced to
a greater amount than one-half. This, however, can be overcome by
projection on a curved surface. Thus, if projected from the intersections
with an equilateral hyperbola orthogonally on to the X axis, from a line
parallel to and distant the height I from the Y axis, the Y axis 0"H of
the hyperbola being the distance p on the negative side, and the X axes,
being identical (see Fig. 30) we have
(I'O) ^ = i-A^ + ^) ;°^^ = ^-rTii
A denoting •p'^/l- Hence , substituting the former expression in the ordinary
[)robability curve equation, we obtain
(171) y = l/e-'Ci- xfK + f)"-!
This gives a terminal to the curve on one side, and an asymptotic relation
to the axis on the other, and may be made as skew as we please, as is
evident from Fig. 30 and from Fig. 32 giving a projection so derived.
A similar scheme of projection using a surface whose right section is a
parabola, the abscissa of whose vertex is p (from the origin), and whose
equation is tj = gf (f — i>)^ gives the result
in which y denotes g/l : see Fig. 31. The value of ^, therefore, is
(173) I = p +2^1^± ^^^y^" {p-x) + l]\
1 That is, when OQ' is one-half of OP', the corresponding point on the positive
side is at infinity.
60 APPENDIX A.
This gives terminals for both branches of the curve, viz. : —
Since both p and y may be arbitrarily determined, the position of the
terminals of the curve, in relation to the mode, may be made whatsoever
we please. Although this leads to a somewhat complicated expression
for I, it discloses the character of the curve obtained by projection. Its
equation is
(175) y = l/e'ti + Tfr
p)«]«
the asymmetry of which is evident. Fig. 30 illustrates the projection
on to a surface whose right section is an equilateral hyperbola, and the
type of resultant curve with one as5maptote is shewn on Fig. 32 : see
curve %, bi,. ..a^, bj.. ., thereon in a thin firm Hne, the thick curve
A, B,...A', B'... etc., being the probabihty curve from which it is
derived. Mg. 31 is similarly an example of a projection on to a surface
whose right section is a parabola, and is shewn on Fig. 32 by a broken
line: see curve a2, h^.-.a,'^, b^... etc. The scheme of projection is
sufficiently evident from the figures.
Reverting to projection on a plane, it may be noted also the pro-
jections may be varied by making I a function of y instead of a constant,
as, for example, I = ky'^, which, writing k for (tan d)/k, would give
(176) y = l/e<'0- + Ki/yy
This does not lead, however, to any simple expression for y in terms
of ^ only. We may notice that since Z = for y =0, both branches are
unlimited (that is to say, the asymptotic relation of the basic curve
remains) and the curve is more distant from the X axis than is the basic
curve ; the curve most closely approaches the type of that with I constant
if w be less than unity. If n be negative and numerically greater than
unity, we shaU have ^ sensibly equal to x for very small values of y, or
X = ^{\ — K-2/") approximately, and the branches are unlimited.
These projections shew that though initially a frequency may be
distributed according to the ordinary probabihty curve, yet the final
circumstances may be such that the " frequency is altered in several of
its characters," viz., its symmetry, asymptotic relations, etc.
SPECIAL TYPES OF CURVES AND CHARACTERISTICS.
61
Kg. 30.
Fig. 31.
j&if,- i^-
Pig. 32.
I, J^* qa 1, qt (5f„ H, "P, P. -^; ft Pi
Mg. 33.
7. Development of type-curves. — ^A consideration of the form of the
equations derived from projections shews that if we put as the funda-
mental form
(177).
.y, or l/y = ^oA *'^ + */*""" '^ - */«"
we may include all cases by variations of p, m, k, a and 6. When a; = 0,
the value is «/o> 't^t is to say, the mode is at the axis. If a and 6, each
supposed to be positive, are finite, then for a negative value of x equal to
a, or a positive value equal to 6, we have y = 0, that is the branches of
the curve terminate .at the axis of abscissa for the negative value of a; = a ;
and for the positive value of a; = 6. If 6 be infinite, the curve, which is
skew, becomes
(178).
■y = yo/e
* (1 + x/a)"" «
.ttta M—tnx
and if a be infinite and b finite, the curve is skew, and its equation is
(179) y = yo/e**"''^-*/*)'"
62 APPENDIX A.
If both a and b are infinite, then the preceding curves (178) (179) become
(180) y=yo/e *
and is symmetric, but if b (or a) be negative, then the curve is
(181) y =yo/c
he'"
This curve is asymmetric^ and both branches are asymptotic to the
axis. The reciprocals of these curves give the other forms required.
8. Evaluation of the constants of the preceding type-curves. — ^The
value of 2/q is assumed to be derived from the data. When all the
quantities are divided by the ordinate of the mode, viz., by yQ, we have a
series of redMced values of the ordinates, r) say. Then, as a rule, by taking
the logarithm twice we can obtain the necessary solution. Thus —
(182). .Tj =e-^'*); hence log rj ='?'=/(a;); andlogT}'=log/(a;)
which gives a linear equation. Thus, with the necessary number of
values of the ordinate and the corresponding values of the abscissa, a
solution of the constants is to hand. If more than the necessary number
are given, the least-square method of forming normal equations may be
employed. This method wiU not solve, however (177), (178), or (179),
where
(183)..logrj'=j3loga; -jlogA; + malog(l+^\ + mb\og{l + ^\
These, however, are very readily solved by expanding the logarithms,
and sometimes a and b can be estimated from the graph of the curve.
9. To determine the surface on which the projection of a normal
probability-curve will result in a given skew-curve. — ^From what has
preceded, and from I^s. 28 to 32, it is evident that the form and equation
of the curved surface, on which the projection of a normal probability-
curve will furnish any given skew curve, may readily be determined.
The problem more generally stated is : — Given two curves to find the
surface on which the projection of one will furnish the other. On.Eig. 33
let Y. .Pd and Y. .Qa be the branches of a normal probability curve,
and YQa Q^, and YPa P^ be the branches of a skew-curve,
the axis OY being identical for each. Draw radial lines from Y to the
orthogonal projections on to the X axis of various points on the normal
probabihty curve, viz., to the points qa, qb, etc., and Pa, Pb, etc., and
from the points Qa, etc.. Pa, etc., whose ordinates to the skew-curve
are identical with those of the corresponding points on the normal curve ;
and draw lines parallel to the axis OY. Then the intersections a, b, etc.,
a', b', etc., are points on the projection surface. Reference to the figure
^ p is to be understood merely as on operator raising the number in numerioaj
value, but not afieoting its sign.
SPECIAL TYPES OF CURVES AND CHARACTERISTICS. 6S
makes the proposition obvious. Thus, the equation to the normal curve
being known, that of the skew-curve can be found in the form y = l/ef(^\
as soon as the equation of the curve of the projection surface is ascertained.
In finding an equation to fit any series of groups the skew-curve
may, in practice, be drawn freehand : a suitable normal probability-
curve may then be drawn with the same mode and vertical height : the
points on the surface found by the method indicated. In general, this
wiU give a somewhat irregular projection-surface, which, however, may
ordinarily be so modified as to conform to some geometrical form easily
expressible algebraically, from which the requisite formula may then be
found. From Figs. 30, 31, and 33 it will be evident how the equation
may be ascertained.
10. Reciprocals of curves of the probability-type. — ^The curve -q = l/y,
also of type of practical importance, may similarly be derived by pro-
jection from the normal probabihty-curve : thus
(184) 7] = l/y = e " , or more generally, rj =e''
that is, its logarithmic homologue is the parabola 17 ' = yua;'', in which
rj '= log rj, and /x = l/k^. Thus in Fig. 33 the reciprocal of the normal
probability-curve (curve 1) is shewn by the curve marked 1', 1', while
the curve 2', 2', is the reciprocal of the curve marked 2, 2. The lateral
scale in the figure, however, for curve 2, is four times greater than for
curve 1. It wiU be seen that the type is somewhat similar to the curve
of instantaneous rate of mortality according to age.
11. Dissection of multimodal fluctuations into a series of miimodal
elements. — ^It is obvious that any multimodal fluctuation may be analysed
into a series of unimodal elements ; for example, a series of the form
(185) y= ^0 + ^6*''=-''^'/'' + ...4,6^'^-''^'/'=' + ..
may, with a sufficient number of terms, be made to fit any continuous
curve whatsoever to any assigned degree of accuracy.^ There is no
complete general solution of the problem, however, of dissection. We
have already shewn that a dimorphic curve may be the sum or product
of two monomorphic curves (see III., § 9, Projective anamorphosis).
The difficulties of dissection, however, are not unduly great with graphic
methods.
^ See " Contributions to the Mathematical Theory of Evolution" (on th«
dissection of Asjmunetrical and Symmetrical frequency curves, etc.). Prof. Karl
Pearson, Phil. Trans., Vol. 185-A, pp. 71-110; 1894.
" Sui massime delle curve dimorfiohe," Dr. F. de Helguero, Biometrika, Vol.
III., pp. 84-98, 1904 ; and also his " Per la risoluzione delle curve dimorfiche,"
Biometrika, Vol. IV., pp. 230, 231 ; 1905-6.
" Sulla statura degli Italiani," R. Livi, Firenze, 1883. " Die natarliohe Auslese
beim Menschen," O. Ammon, Jena, 1893,
v.— GROUP-VALUES, THEIR ADJUSTMENT AND ANALYSIS.
1. Group-values and their limitations. — ^The data of population
statistic are ordinarily given in the form of group-vcdues. For example,
in the age-distribution of a population the data are ordinarily in the form
of the numbers of persons between the ages x and x-\-k, x-\-k and x-\-2k,
and so on, where k may be a month, a year, 5 years, 10 years, etc. Hence,
when the number for any group of smaller limits is required, some curve
must be assumed which will give the same group-values if the latter are
to be regarded as correct.
In other words, if we suppose the numbers between the ages x and
X -\- dxto be P<f>{x) dx, then the number in the group between the ages
X and a; -f ^ is
(186) ^N,+,^Pj^-^''<f>ix)dx
in which, if P denote the total population of aU ages, the value of the
integral between the limits and the end of Ufe, say 105 (or c») is neces-
sarily unity. This is the fundamental conception of the use of group-
values. Thus, omitting the coefficient P, the value of the integral between
any Umits, when its total value is unity, is the proportion of the whole
population which lies between the limits in question.
When grov/p-values are known to be subject to error, each group can
be modified in amount so as to conform to some distribution regarded as
more probable than that furnished by the crude data. Thus, if in the
numbers according to age a census return gave for " ages last birthday "
29, 30 and 31, the numbers 20,000 ; 24,000 ; 18,000 ; we should know
ordinarily that the number 24,000 was in excess, since the numbers must
fall off as the ages increase unless immigration prevent. We deal
primarily with the case where the groups are assumed to be correct ;
having either been corrected, or having been taken accurately.
2. Adjustment of group-values. — ^In cases where group-values are
properly regarded as subject to appreciable error, they should either be
first adjusted before the constants of mathematical formulae representing
them are determined, or the computation should be so effected as to
automatically make the adjustment a minimum.
The Hmitations under which group-results are obtained are of two
kinds. The results furnished may be either —
(a) actually subject to large errors ; or
(6) insufficient in number to furnish a truly representative example.
For example, misapprehensions as to one's exact age must necessarily
have the effect of causing numbers of persons to be attributed to the
wrong age-group, thus diminishing some groups and increasing others.
GROUP VALUES : ADJUSTMENT AND ANALYSIS. 65
A certain tendency to misstatement is confirmed by census-results,
which reveal the fact that ages ending in are characterised by excessively
large numbers, and that the numbers for ages ending in 5 are also some-
what excessive, while the numbers for the adjoining years are in defect.
In the other case, hmitations in the numbers available prevent one
knowing exactly what would have been given had the numbers been
indefinitely large. In these latter cases, however, it is often possible
to surmise what the curve would have been had the numbers been large,
and the actual data may be redistributed so as to conform therewith.
In both instances the principle to be followed is that some groups should
be so increased, while others should be so diminished as to conform to the
most probable distribution which may, for convenience, be called the
" ideal distribution."
In effecting these changes in the numbers furnished by the data for
individual groups, the alterations should not only be as small as possible,
but also the accumulation of the alterations (that is, their algebraic sum)
should be alternately plus and minus, and should never become large in
amount.
Various considerations may serve as a guide in effecting the altera-
tion : for example, excluding the consideration of dehberate misstate-
ment of age and tendency to uniform error in one direction, the number
of cases in which the misstatement of age is one year only is, in general,
larger than the number in which the misstatement is two years ; and
so on. Experience shews also that large positive errors are Ukely to be
made for ages ending in ; for example, 30, 40, 50, etc. ; and lesser
positive errors are Ukely to be made for ages ending in 5 ; for example,
35, 45, 55, etc., while errors of defect are to be expected in ages 29 and 31,
etc., and 34 and 36, etc.
Adjustments are, as a rule, preferably made in the light of a full
consideration of aU the circumstances affecting the case, and not merely
by piirely mechanical or merely arithmetical methods.
A redistribution of values may be regarded as excellent when the
curve giving the values of the groups is, in the nature of the case, probable,
and when at all points it deviates from the successive values of the groups
in such wise that the deviation is always relatively small, and the aggre-
gate alternately plus and minus.
3. Representation of group-values by equations with integral indices. —
Any curve representing a series of statistical data may be represented by
the following expression, viz., —
(187) y = a -\- bx^ -\- cx^ + dx"" + etc.
and, if p, q, r, etc., be not necessarily integral, with a small number of
terms. Integrating this we shall have
(188) jydx =x{A + BxP+ Gx' + Dx' + etc.)
6 „ c d
in which A = a; B= — ^ ; = ^-py ; L> = ^-^;p^ ; etc.
66
APPENDIX A.
When p, q, r, etc., are the successive integers 1, 2, 3, we have for x=0,
k, 2k, 3k, etc.
Q Range of Factors into numbers below
^' the Abscissae, a ^bk ^ck''
(189).
I.
II.
III.
IV.
V.
k
2k
3k
4:k
k = k(l
2k = k{l
3k = k(l
U = k(l
5k = k(l
1
7
19
37
61
Idk^
1
15
65
175
369
1)
31)
211)
781)
2101)
It is easily seen that with integral indices, the above expression of w+l
groups can be fitted by an arbitrary equation of the nth degree. Denot-
ing the heights of the groups by the small Roman letters i. to v., the heights
being found by dividing the group-values by the base k, and the successive
differences of height by h^, hi, etc., the simplest scheme of solution is
to hand in the following series of equations, which are readily obtained by
differencing and substitution.
4. Formulae depending on successive differences of gioup-heights. —
We give first formulae depending merely on the difference of heights,
viz., the differences i.— ; ii. — i. ; iii. — ii. ; etc. ; that is, if we denote
the successive heights of the groups by
(190) ^0 ; ^0 + ^1 ; ^0 + ^1 + ^ ; etc.,
the successive differences of height will be
(191) hg ; hi; h^ ; h^ ; etc.
hg = i., denoting the height of the first group from the X axis, see Pig. 36.^
•
+
\
1
+
^
M
a
1
Curve
begins
at "a'
from
A
o ■
■a .^
A
cQ
A
V \
/ \
1
V
>
V
X
k
i
V.
k
*
Fig. 34.
1 These (191), are the first column of differences if the groups be divided by
their base-values viz., by fc.
GROUP VALUES : ADJUSTMENT AND ANALYSIS. 67
The following, as convenient formulae for the coefi&cients a, b, etc.. in
equation (187), can be deduced, viz. : —
For three groups :
For four groups : —
(192).. a = feo -^(5^1 -2/^2); b = \(2hi-h2); c=^^{-h^+h2]
(193) a =K-^ {Uhi-lOh+^h)
(193a) .... 6 = j^ (35A.1 - 34^2 +11^3)
(193b).... c= -^A-5hi + 8h2-3hs)
(193c).... d= A-^ih-^h+h)-
6k
For five groups :-
/
(194) ....a=Ao-^^i + ^(- ilh + 86^ - 51^3 + 12^*)
(194a) .... b = n ^1 + i^ (^^^1 - ^*^2 + 4:1^3 - 10^4 )
(194b). ... c= ^, (- 17^ + 37/^2 - 27^3 + 'Jh)
(194c).... d= ^^(3hi-8hz+'7hs-2hi)
(194d)....e= 25^^" ^+^^2 -3^3+^4)-
If instead of heights we use group-values, the quantities found, say
a', b', c', etc., will be k times those above given, and must be reduced
-accordingly
5. Formulse depending on the group-heights themselves. — ^Instead
of using the difference of the group-heights, the coefficients of the equation
may be expressed in terms of the successive group-heights themselves,
found by dividing the group-numbers by the value of the common interval
along the abscissa ; that is, by dividing the integrals between the successive
Hmits having a common interval k, by that quantity. It will be sufficient
to give the results for from three to five groups. These results are : —
For three groups : —
(195) a = -g- (Hi. - 7ii. -f 2iii.)
(195a) b = -j^ ( - 2i. + 3ii. - iii.)
(195b) c =p2(J- - 2ii- +in-)
68
APPENDIX A
For four groups
(196)
(196a).
(196b).
(196c).
a = Jo (25i- - 23ii. + 13iii. - 3iv.)
.6 =
12fc
(_ 35i. + 69ii. - 45iii. + lliv.)
c = jp (5i. - 13ii. + lliii. - 3iv.)
d =gi-3(-i. +3ii. -Siii. +iv.)
For five groups : —
(197) a --
(197a) b --
(197c).
(197d).
60
J_
12k
(1371. - 163ii. + 137m. - 63iv. + 12v,)
(_ 45i. 4- I09ii. - 105iii. +51iv. - lOv.)
(197b) c = gp (17i. - 54ii. + 64iii. - 34iv. + 7v.)
6P
(_3i. 4- iiii. _ I5iii. + 9iv. - 2v.)
^ =24F(^ - 4ii. + 6iii. - 4iv. + v.)
If the aggregate numbers or group^alv£s are used, instead of the heights,
the denominators will be 1/fc, 1/k'^ 1/k^ instead of those above.
6. Formulae depending upon the leading differences in the groups
or in group-hefehts. — It is often convenient in practice to work with
differences instead of the group-values or of heights. In the latter case
the coefficients are similarly given by the following equations : —
The coefficients of equation (187) expressed in t«rms of successive
leading difEerences of the group-heights are : —
(198)..
(198a).
(198b).
(198c)
(198d).
.a = \ (i.
Di
i(A
+ 6 A
- D.
-IDs\+1 D,j
11
12
+ ^oi>3
C =
d =
e =
Ds
f(+ j^3
->^
+
F(2i ^*
GROUP VALUES : ADJUSTMENT AND ANALYSIS. 69
In the above Di, D^, -D3 , and D^ are the leading differences of the
heights only, viz., of i., ii v. As before, if the group values are
subtracted, without first dividing by k, the denominators shovild be
l/k, l/fc^ l/k^, instead of those above given. Formulae (198)
to (198d) are correct for any number of groups up to five, the division
lines on the right hand side shewing the results for two, three, four and
five groups.
7. Determination of differences for the construction of curves. —
When the equation of the curve is to hand, it is often required to find
values of the ordinates corresponding to a series of values of the abscissa.
This is most conveniently effected by obtaining the successive leading
differences : from these the required values can be obtained. These
are : —
(199) f (x) = a + bx + cx^ + dx^ + ex^
(199a).... Di/(0) = .. b +c +d +e
(199b). . . . Z)2 / (0) = 2c +M + Me
(199c). . . . D3 / (0) = 6d + 36e
(199d). . . . Z>4 /(O) = Me
It may be remarked that when k=l these difference values become
(200) A /(O) = Ai. - I Ai. + I Dsi. -Id,L
(200a) Dzfm = Dzi. - | I>3i- + I D^i.
(200b) -D3 /(O) = Ai- - 2 -Oii-
(200c) i>4/(0) = i>4i-
in which the symbol D\i., D^i-, etc., denotes the leading differences
derived from the series from i., ii., iii., etc.
8. Cases where position of curve on axis of ordinates has a fixed value.
In the equation (187) it may happen that the curve is required to pass
through the intersection of the axes OX, OY; or at a fixed distance
therefrom on the Y-axis. In this instance the solutions given are
invalid, inasmuch as a is initially given, not determined from the group-
values. The most convenient procedure is to subtract this value a
from the heights i., ii., iii., etc., of the ordinates, or the value ka from the
APPENDIX A.
group-values (or areas) I., II., III., etc. This procedure gives new values,
viz., y' = y~a, and the solution required is then of the successive in-
tegrals (group values) divided by Ic.
(201) ^ fy'dx = ^ f{bx + cx^ + etc.) dx
that is, oi\bx -\- \cx^ -{■ etc.
It is obvious that in this instance n groups will require an equation
of the wth degree, instead of, as before, of the [n — l)th, the imposed
condition of a fixed value for a involving this limitation.
The following formulae give the value of the constants in terms of the
heights.
For two groups, curve passing through origin,
(202) 6=4 (7i - ii) ; c = jj-, ( - 91 + 3ii)
For three groups, curve passing through origin,
J_
9F
(203) ^ = iP (85i - 23ii + 4iu)
(203a) c = —^ (- lOi + 5ii - iii)
(203b) d = ^3 (Hi - 7ii + 2iii)
For four groups, curve passing through origin,
(204) b = ^^ (415i - leiii + 55iii - 9iv)
(204a) c = ggp ( - 755i + 493ii - 191iii + 33iv)
(204b) d = ^ (119i - 97ii + 47iii - 9iv)
(204c) e =^^{- 125i + 115ii - 65iii + 15iv)
For five groups, curve passing through origin,
(205).. . Jb = jg^^ (120191- 598111 + 3019111 - 981iv + 144v)
(205a). .c = g^ (- 343i + 273ii - 155iii + 53iv - 8v)
(205b). .d = g^3 (2149i - 211111 + 1429iii - 531iv + 84v)
(205c). ..e = -ggp- (- 133i + 147ii - llSiii + 47iv - 8v)
(205d). . ./ = J200P (13'^i - 163ii + 137iii - 63iv + 12v.)
i
GROUP VALUES : ADJUSTME2SrT AJSTD ANALYSIS. 71
The constants in the terms of the leading differences of the heights
are : — ^
For two groups, curve passing through origin,
(206) 6 = ^(3i- ^D,i)
(206a) ....c=^(-|-i+ Ad^I)
For three groups, curve passing through origin,
(207a)....c = ^ (-3i+|-Dii- ^ D^i^
(207b).... rf = ^ (li ~ ~D,i+ |-^2i)
For four groups, curve passing through origin,
i/2'i 1^ 7 1 \
(208) 6=^(-gi_ j2Ai+i8i>2i--8i)Bi)
(208a) ....c = J,(-f i + gi),i - l^i + |-^Z)3i)
(208b). ...(?= y ( |i- |i)ii+|i)2i - l^i)
(2080 . . . .e =^ (-|i + I Ai - 4i),i + |z)3i )
For five groups, curve passing through origin,
1 / 1S7 77 47 9 2
(209)....6=-^(^i- ggDii+ggD,i-^i)3i+ 2gi>4i
(209a) ••c = -p-( -"81+ jgAi- -g -021+ 32-^31- ^Dii
1/17 17 17 13 7
(209b) ..<«= p(^ 6" i- 12^1+ i8^2i- 24-^31+ -30-041
(209c) ••e = ^(-|-i+i^Ai-2^Ai+3l^3i- j^Ai
(209d) ../=^( ^ i_ ^Ai+ g^Ai- g^i)3i+ 4i)4i
^ i denotes the height of the first group-result ; D^i = ii — i ; D^i = iii —
2ii — i ; D^i = iv — 3iii + 3iii — i ; etc.; that is, they are the leading differences.
72 APPENDIX A.
9. Determination of group-values when constants are knovm. — When
the equation is in the form (187), jp, q, r, etc., being 1, 2, 3, etc., the most
ready way to compute a series of values of groups to k,h to 2k, 2k to Sk,
etc., is to form the leading differences, and from these the successive
values of the groups can be readily formed. The following formulae give
the required result : — ^
(210).... 1. =ak +^bk^ +^ck^ + ^^dki + ^^ek^ + ^fk^
(210a) . . Dil. = bk^ +2ck^ + 3idk* + 6ek^ + 10J/A;«
(210b) . . Dgl- = 2c/fc3 + Mjfc* + 30ek^ + 90//fc«
(210c) . .Dgl. = 6dk^ + 48ek^ + 260fk^
(210d) ..Dil. = 24e^s _^ 200fk«
(210e) . .D5I. = 120//fc«
When the equation is of a less degree than the fifth, zeros can be
substituted for the coefficients ; thus for a fourth degree, /=0 ; for a
third degree /=0 and e =0 ; and so on ; and the formulae stiU hold
good.
10. Curve of group-totals for equal intervals of the variable expressed
as an integral function of the central value of the interval.^If we have a
series of group-totals for equal intervals of the abscissa, as, for example,
for to k,k to 2k, etc., and if those values divided by the common interval
are represented by the ordinates at ^k, l^k, 2\k, etc., to a curve the
equation of which is an integral function of the type of formula (187),
then, whatever be the value of x in this equation, the ordinate for the
point X wiU give very approximately the group-total for x — \kto x-\-\k.
That is to say, denoting the ordinate to the curve representing the groups
a; ± JA; by Y, and that to the curve representing the original function by
y, if
(211) Y = F{x +\k) =yf^+''dx=f^+''f{x)dx
for the values x=0, 1, 2, etc., then it follows that very approximately
(212) Fix+lk+q) = f^' + 'flx) dx
provided that the forms of F and/ are the same, that is, that they are both
integral functions of a single variable. This result is important, and may
be estabhshed by the following consideration.
If we compute F {x) = /^f{x)dx so that the two are in agreement
for x=\k, \\k and 2|i, in the first function, with the limits to ifc. A; to 2k,
and 2k to 3fc in the second, then it is easy to establish that if the original
• D^l., D.2I; etc., denote the series of leadinc/ differences, viz. (II.— I)-
(III. - 211. + I.) ; (IV. - 3III. + 3III. - 1.) ; etc. '
GROUP VALUES : ADJUSTMENT AND ANALYSIS. 73
equation be a-\-hx-\-cx^, and if the equation for the group-total, divided
by the common interval, be A-\-Bx-\-Cx^, when x is the value of the
abscissa for the middle of the interval, then
(213) A=a + ^ck^; B = b ; C = c.
If we extend the solution to the third power of x, that is, extend the
limits to S^k and 3fc to 4fc respectively, we have
(214) A =a + ^ck^; B=b +-^dk^; G =c; D = d.
If we further extend the solution to the fourth power of x, and the hmits
to 4JA; and 4A to 5k respectively, we obtain
{215)..A=a+^ck^+^^ek*; B=b+^dk^; C=G+-^ek; D=d; E=e.
If the fifth power of the variable be included, that is, the hmits be 5\k
and 5k to Qk respectively, then
(216)..^=a+^cfc2+^eA*; B=b + \dk^+ ^^fk*; C=c+\ek^;
D=d+~fk^; E = e; F=f.
It will be observed that up to the second power of the variable, the
effect is that A differs from a only by a constant, consequently the
function F gives rigorously the correct result, viz., that given by integrat-
ing the function /. For powers higher than the second, the result is true
only for k=^, 1^, etc., in F, and for any other values is more or less in
error. This error cannot, in general, however, attain appreciable magni-
tude, because it is repeatedly reduced to zero at intervals of k, viz., at the
values of the abscissa, ^k, l^k, etc.
In practical statistical examples the coefficients b, c, d, e, f, etc.,
are generally in diminishing order of magnitude, and we see from the
equations (213) to (216) that the corresponding numerical factors also
rapidly diminish ; hence the difference between the rigorous value
\f{x)dx and the approximate value F {x) must generally be very smaU,
and, by the formulae given, can be readily tested in any numerical
examples.
11. Average values of groups. — An average value y^ of a group is
the quantity
(217) yr = -^j::ydx
in which y denotes the value of the ordinate, and Xx to X2, the range
of the variable. Reverting-to formulae (187) and (188), and retaining
the same meaning for the constants, the mean value of the range x to
X -\- kis
(218)..?/, =A + \_[B{{x+k)J> + -^-xP + '>']+ (7{(a;+A!)«+i-a;«+i}H-etc.J
APPENDIX A.
which takes a simpler form if j)> q> >". etc., are 1, 2, 3, etc. Where x has a
series of values 0, k, 2k, etc., as in (189) the averages are given by omitting
the factor k in the formulae. More generally, that is, for any value of
X and k we have
(219) yr = a+b (x+l^ +c(x^ +xk + jk^) +
d {x^ + lix% ^xk" + ^ k") +
e{x* + 2x^k + 2x^k^ + xk^ +"5 **)
For groups bounded by curves of the exponential type we may
note that
(220) a' = e-r log a == e"*^
Thus, the rate of change at any point of the curve y = we™^ is
'-) t'
and the mean rate y^ is
(221) / = <^ (we'»^)/da; = mwe"
(222) y^ = mne^''.
mk
that is, this is the mean ordinate to the curve.
If the ordinates for the beginning, middle, and end of any range of
values of the abscissa, that is, if the ordinates corresponding to the
values X, x-\-\k, and x-\-k, are to hand, and the group-values are the
integral of an equation of the type (199), then the value of y, is
1 1 fl 1 2^
(223). .2/,=g(y.+42/^+2/:»+*)-24**|5e+/(x+2A)+3? {x^+xk+^'^)+Q\^.
The negative term (in braces) is absolutely negative, x being positive,
if e, / and g are positive, and it is usually so small as to be neghgible.
When a;=0 and k=\, the value of (223) takes the very simple form
1 1 / 1 1 23 \
(224) yr = -Qiyo + ^ym + yk) ~ 2i\'5^ '^ J f ^ 28 ^)
2/ot denotes, of course, the middle ordinate.
This result is important, because it shews that group-values can be
calculated with considerable precision by the " prismoidal formula" if
we have middle as weU as terminal instantaneous values of each group.
VI.— SUMMATION AND INTEGRATION FOR STATISTICAL
AGGREGATES.
1 . General. — ^In effecting statistical summations, regard is to be
had to two elements, viz.: —
(i.) Order of accuracy significant in the case in point ;
(ii.) Arithmetical consistency of results.
Curves drawn freehand among data, that represent either groups or
instantaneous results, and which shew visible variations, can, for some
purposes, be integrated with sufficient precision by careful graphing
and the use of a planimeter.'- When arithmetical smoothing has followed
graphic, in order to enhance the accuracy, numerical calculations are
virtually required as being of corresponding precision. As a rule group
values (or the total area between any ordinates, the curve, and the axis of
abscissae) can, if the ordinates are relatively near each other, be computed
by means of the prismoidal, Simpson's, Weddle's and similar rules.
Finally, for work of the highest precision, actual integrations by the
method of the infinitesimal calculus are required. In general, however,
the precision then far transcends that of the data.
The extension of implied precision far beyond that of the data is
seen in all actuarial tables : this matter is referred to later, since the year
change in probabihty of life is a quantity of a much larger order than that
to which results are expressed.
2. Area! and volumetric summation formulae. — Statistics relating
to population involve both areal and volumetric summMions. The latter
can, however, always be represented by an areal graph. If the curve
represent instantaneous and not group-values^ about a particular value of
the variable, then the areal value can be computed without computing
the equation of the curve and integrating it.
It has been shewn* that if an axis be equally divided, that is, if
x=0, k, 2k. . . .nk, and the curve passing through the terminals of the
ordinates (y) from these points is assumed to be represented by an
integral function of x, then suitable multipliers or weights may be deter-
mined, which, appMed to the ordinates, will give the area. If there be an
* Amsler's Integrator will cover a considerable area, and gives in the one
operation (on four cylinders and discs) the values of following integrals, viz.: —
fydx; ^JyHx; ify'dx; ^fy'dx
that is, the area, the statical moment, the moment of inertia, and the cubic moment
about the axis x. No mechanical integrator, however, can possibly approximate
to the precision attainable by arithmetic.
" That is, represents the frequency y, for a given value x of the variable and
not the group-mean for x—^k to x + ^k. See V., 10.— -Curves representing group-
totals, formulae (211) to (216).
* See " Voliuues of solids as related to transverse sections," by G. H. Knibba,
Joum. Roy. Soc. N.S.W., Vol. XXXIV., pp. 36-71, 1900. See Prop. (O), p. 70.
70 APPE2SrDIX A.
odd. number of equidistant ordinates the curve may be of the same degree
as the number of ordinates, viz., (w+1) ; if the number of ordinates be
even, the degree of the curve must be one less than that number {n).
It has been shewn ako that if the curve bounding the area is of a
less degree than that satisfied by the number of ordinates, then there is
one-fold, two-fold, .... k-iolA infinity of multipliers which will exactly
give the area, according as the degree of the curve is 1, 2, .... fc less than
the number of ordinates .^
The formulae can be readily constructed, and are exhibited in the
table hereunder.^ The significance of this table may be indicated as
follows : —
When n-\-\ equidistant ordinates are given for a curve of the wth
degree, there is only one system of weights that will give the integral
correctly between the limits and n. In the table this system is in-
dicated in each case above by an asterisk (*). Further, when n is even,
the unique series of weights, applicable to n-\-\ equidistant ordinates,
is also applicable to a curve of the (w4-l)th degree, but this is not true
when n is odd.
When M+2 equidistant ordinates are given for a curve of the wth
degree, any value whatever may be assigned to one of the weights (say
Wg ), and the corresponding values of the other weights may be expressed
in terms of Wq. In this case there is evidently an infinite number of
possible systems of weights, each of which wiU give the integral accurately
for a curve of the nth. degree. In the foregoing table the systems of this
nature are indicated by a dagger (f), the coefficient (i.e., 1) of the arbit-
rarily selected weight being shewn in heavy type. As an example, there
may be taken the case in which seven equidistant ordinates of a fifth
degree curve are given. Here the weightings shewn by the table are
ti)g = Wq-, Wx = 3.3 — 6wo; w^ = — 4.2 + ISw,,; w^ = 7.8 — 2Qwg;
Wi= — 4.2 + 15wq; w^ = 3.3 — Gw^; w^ = Wq.
If Wq be given the value 0.3 this series becomes —(1, 5, 1, 6, 1, 5, Ij ,
which wiU be recognised as Weddle's rule .
Similarly, when w+3 equidistant ordinates are given for a curve of
wth degree, two weights may be arbitrarily selected and the remaining
n-{-\ may be computed in terms thereof, thus admitting of a two-fold
infinity of systems of weighting. In the foregoing table systems of this
nature are indicated by a double dagger {%).
Similarly, when w+4 ordinates are given for a curve of the wth
degree there is a three-fold infinity of systems, when n-\-5 ordinates are
given, a four-fold infinity, or, in general, when r ordinates are given for a
curve of the wth degree there is an (r— w— l)-fold infinity of systenjs of
weighting.
' Ibid, § 16, pp. 60-71. Examples of the development of fc-fold infinity of
multipliers are given on pp. 64-67.
2 Prepared by Mr. C. H. Wiokens, A.I.A.
SUMMATION AND INTEGRATION.
77
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■ ^ Tj< O -* i-H
^- 1 1
rf CO
+4-
1 1
-IS
1 1
p4 O lO to O '
IM ■* CO —
1 1
1
1
1
%
X
-f -* rt
X
r-i CO « •-'
X
00 Til 00
1
X
A
/ ^
(N (M t>. lO
(M CO 0<»
X
T* lO to o»
oq ^ CO
1 1
1 1
P4 CO CO ^
1 1
r^ CO 00 CO
O
1
»H CO CO -1
1 1
o
o
»H CO CO rt
IH CO CO rt
1 1
^ O 00 CO
1 1
r4 ;o ooco
1 1
r* O >0 CO
^^ 1
p4 O >0 to
7-1
SS2
*
-H |co
H- 1 1
++ 1 1
-Ico
1
1 1
X
X
X
X
X
rt ■* rt
t — — \
OS • CO
OtO 00
t- O 00
1
1 — * — \
1> to lO
IMOO -H
1
iP
1
O H «
^ & ^
S
g & & ^ ^
1
^ ^ ^ & *
CQ
1
= H d :i -T< W
* ^ ^ & ^ ^
1
■8(5
J
O
-a
O
o
to
I>
78 APPEXDIX A.
3. The value of groups in terms of ordinates. — ^It is often convenient
to ascertain the value of groups between certain limits of a variable.
If the ordinates be supposed to conform to the equation a+bz ; or
a-\-bx-\-cx^, etc., etc., we can construct a series of equations which are
rigorously true under the particular supposition, and may be regarded as
approximations in the general case. By comparing the expression for
the integral between assigned hmits ^^'ith the values of the ordinates, we
deduce the following expressions for the heights of the groups in terms of
the ordinates.
TABLE Vn.
Values of Group Heights for Difierent Ranges of the Variable in Terms of
the Ordinates to the Curve.
1st Approximation. Formulae (225) to (228).
Ranges of Integral 0-\ \-\ l-\\ l\-2
Semi-group-heights ^(82/0 + 2/1); ^(^o+Sj/i); |(-yo+%i); ^(-Syo + Tyi)
2nd Approximation. Pormulse (229) to (232).
Ranges of Integral 0- J \-\
Semi- group-heights ^ (Sy, -\-5y^-y^); — {2y^ ^Wy^^y^)
Ranges of Integral 1-1 J l|-2
Semi-group-heights -jg ( - 2/o + 1 l2/i + 22/2 ) ; ^ ( - 2/o + Sj/i + 82/2 )
3rd Approximation. Formulae (233) to (236).
Ranges of Integral 0-^ ^-1
^Td£tr^"l§2(^^^^«+^^'^2/i -431/2+92/3); ~(252/„+1972/i -373/2-1-7^3)
Ranges of Integral 1-1^ 1^-2
^^^g?tr^"l^(-^2/o+1552/i+532/2-7y3);j^(-72/„+532/i+155y2-92/3)
4th Approximation. Formulae (237) to (240).
Ranges of Integral. Semi-group-heights.
(237) 0-1 = 28^ (16942/0 + 1969yi - II9I2/2 -f 499i/3 - 91^^ )
(238) i-1 = 28^ (3142/, + 31992/1- 9212/2 -F 3492/3 -6I2/4)
(239) l-H = 2^ ( -9I2/0 + 21492/1 + 10592/2 - 281^3 + 44^/^ )
(240) lJ-2 = ~ (-612/0+6192/1 + 25892/2 - 3II2/3 + ^y^)
SUMMATION AND INTEGRATION. 79
« —
1st Approximation. Pormulse (241) to (243).
Ranges of Integral 0-1 h~^i 1~^
Group-heights -^ (Vo + yi) Vi ji -Vo +3^/1 )
2nd Approximation. Formulse (244) to (246).
Ranges of Integral 0-1 |-1| 1-2
Group-heights j^ (5t/o +82/1 -2/2 ); ^ (Vo +222/i +y2 Y, 12 (-^^o +§2/1 +^yz )
3rd Approximation. Formulse (247) to (249).
Ranges of Integral. Group-heights.
(247) 0-1= ^ (%o + 19J/1 - 52/2 + 2/3 ) •
(248) i-l|= ^ (2/0 + 222/1 + 2/2 + O2/3 )
(249) 1-2= 2^" 2/0+132/1+132/2-2/3)-
4th Approximation. Formulse (250) to (252).
Ranges of Integral. Group -heights.
(250) ^-1 = W ^^^^^° + ^^^^^ ~ ^^^^^ + ^*^^^' ~ ^^^' ^
(251) ^1*= 5^ (2232/0 + 53482/1 + I382/2 + 682/3 - 172/4 )
(252) li-2 = ^ ( - 192/0 + 3462/1 +456t/2 - 74ys + II2/4 )
In applying these formulse the actual common-range of the interval
on the axis of abscissse is immaterial ; that is, we may read throughout
to P ; ^kto k etc.; instead of to J ; ^ to 1 ; etc.; the ordinates
2/0, 2/1 > etc., being taken of course 0, k, 2k, 3k, etc. By these formulse,
therefore, we may halve groups.
It wiU be noticed that the coefficients are always symmetrically
opposed for semi-groups standing in the same relation to the ordinates ;
for example, with two ordinates, to | is the same form as | to 1 ; with
three, to ^ agrees with 1| to 2, and ^ to 1 with 1 to 1^ ; with four
ordinates, the only symmetrically opposed pair are 1 to 1^ and 1^ to 2.
From this it is evident that, for the third and fourth approximations the
formula for the remaining group-heights within the limits of the ordinates
80 APPENDIX A.
used can be written down by inspection. Thus for the 3rd and 4th
approximations the group-heights of the various semi -groups are as
follow : —
3rd Approximation.
The ordinates for the semi-group 2| to 3 are the inverse of those for to |
2 to2J „ „ , ^tol
Hto2 „ „ ., ltol|
(as already given).
4th Approximation.
The ordinates for the semi-group 3| to 4 are the inverse of those for to ^
3 to 3^ „ „ „ |tol
2^ to 3 „ „ „ 1 to 1^
„ ' „ 2 to 2^ „ „ „ lJto2
4. The value of group-subdivisions in terms of groups. — ^It is often
required to divide a group. Practically we may always halve a group and
halve again it necessary. If we divide groups with a common interval
(Jc) on the axis of abscissas we may, with advantage, use the growp-Jieight
(g) instead of the group number G ; that is, we may use g=0/k. Then we
obtain the following series of formulae, which, like the last, are rigorously
accurate if the groups are given by the integrals of the equation a-\-hx ;
a-\-bx-\-cx^ ; etc.; etc. They may therefore be regarded, as in the
previous instance, either as a series of approximations, or as rigorously
accurate, according as they represent exactly or approximately the sub-
divisions of groups given by the integral equations referred to.
TABLE Vm.
Values of Gteoup-heights foi different half-ranges of the variable in terms
of the heights of successive whole groups.
1st Approximation. Formulae (253) to (256).
Ranges of integral 0-J ^1 1-1^ 1^2
Semi-group-heights -J {5gi -g^); -^ (3gi +g2); -^ {gi -f 3^2 ); -^ ( -^i +5gz )
2nd Approximation. Formulae (257) to (260).
Ranges of integral 0-^ ^1
Semi-group-heights -g (ll?i — 4^2 + fl^s ) ; g- (5^i + 4g'2 — ffs )
Ranges of integral 1-1| 1^2
Semi-group-heights -^ (gi + ^2 - ffs) ' ^ i- ffi + ^9z + ff?)-
SUMMATION AND INTEGRATION. 81
3rd Approximation. Formulae (261) to (264).
Ranges of integral 0-J ^ to 1
Ranges of integral 1-1^ lJ-2
4tli Approximation. Formulae (265) to (268).
Ranges of integral. Semi-group-heights.
(265) 0-i = j|g (193gri - I22g2 + 88^3 - 88^4 + Ig^ )
(266) i-1 = j^ (689-1 + 122^2 - 889-3 + 389-4 - 7g^ )
(267) 1-^=118 ^'^^' + ^^^^' " ^^^' + ^^^* " ^^' ^
(268) l|-2 = jig ( - 79ri + gSgr^ +52^3 - 189r4 + 8^5 )
The opposite symmetry of the coefficients for semi-groups in S3mi-
metrically opposed positions, having regard to the total number of groups
in question, is obvious, as in the case for ordinates. The same remarks
apply, mutatis mutandis, as those made regarding the coefficients of the
ordinates.
5. Approximate computation of various moments.^In connection
with the application of the method of moments in statistical investigations
of distribution (population and other) it is often necessary to compute
moments from available data. This can also be done from the available
ordinates in the following manner : —
It. is obvious that the curved boundary of any group, covering a
limited range of the variable, can be represented with considerable pre-
cision by a curve of the second degree : see V., § 11, formulae (217) to
(224). Let the group-height be denoted by g, that is, let 9- denote the
group-area divided by k, that is, the group-range on the axis of abscissae.
If y' and y" are the ordinates to the curve for a—^k and a-\-^k respec-
tively, and «/(, be the central ordinate, viz., at the distance a from the
intersection of the axes, and if h be the distance of the mean of the
terminals y' and y" from the terminal of this central ordinate, that is,
if M=2/o— i iy'+y")' then the group-height is given by the equation
(269) 9=^{y'+y") +j^=-Qiy' +^ya + y")
and the equation to the curve is —
(270) .... y=t/a + ^-^-^ (^ -«)+ p (*-«)' = ^«+ ^ ('^-'*) +"<*- *)*
APPENDIX A.
the origin being at the distance a from the ordinate y^. This curve is
regarded as vaUd only for the group to which it applies, and not for
adjoining groups. From this last equation we can compute the successive
moments, Jf q denoting the area, M^ the statical moment, M2, the moment
of inertia, and M^ the moment of the fourth order.
It is important to attend to the signs of h and c. If y"— «/' is
positive, that is, if the ordinate is increasing in the direction of a-\-\k.,
then b is plus ; and c is plus if the curve is convex upward : that is, if
h is positive.
Thus the several moments are : —
(271) M^=lc{ya+Y2 "^'^ = ^ ^^/^ + \ ^)
(272) Ml -aMo = ~bk^ = ^^k^ (y" - y')
(273) M^ - 2aM, + a^, =^ *' (%« + 3^)
(274) Ms- iaM^ + 3aWi - a^^ = §^ ** (y"-y')
and may be very readily computed from these formulae, which are
rigorously exact on the supposition made, and will be sensibly correct
generally.
6. Statistical integrations. — Ordinarily, statistical data are subject
to considerable error and uncertainty, and meticulous precision in regard
thereto is, therefore, usually unmeaning. The approximations of statis-
tical technique itself, should, however, aim at a somewhat higher order of
accuracy than that characteristic of the data, in order that the error
should not prejudicially accumulate through mere computational vitia-
tion. The great majority of cases of integration occurring in ordinary
statistical practice will be found to have been solved. Valuable tables of
integrals are available.'
1 (i.) Sammlung von Formeln der reinen und angewondten Mathematik
W. LAska, Braunschweig, 1888-1894, pp. 1-1071.
(ii.) Tafeln unbestimmter Integrale. G. Petit-Bois, Leipzig, 1906.
(iii.) Een Aanhangsel tot de Tafels van onbepaalde Integraleu. D. Bierens de
Haan.
(iv.) BxpoB6 de la th^orie des propri6t^, des formules de transformation, et
des m6thodes d'6valuation des Int^grales dSfinies, partie 1, pp. 1-82 •
partie 2, pp. 83-181 ; partie 3, pp. 183-698. Bierens de Haan. Amster-
dam, 1860.
(v.) Nouvelles Tables d'int6grales d6finies. Bierens de Haan, parties
pp. 1-733, Engels, Leide, 1867, '
i #
SUMMATION AND INTEGRATION. 83
The integrals of curves of the type of (20), II., § 19, are sometimes
required : that is, —
(275) /a(6a;)±"'±"''-^ dx = '^f y±"'±''y dy = Ajy'^e^v^^svdy
= Afe^^™^'^y^^"svdy
in which A= a/b ; and y= bx. This last form may be expressed by
an exponential series. Or
{2'7G).. fyo+nvdy =/2/™ Jl + ny log y + | (««/ log y) " +....]dy
which may be integrated term by term. Again
(277)../x-c^x=.|l-2,+-3^-^+..;+-^|2-y^+-43----|
w ^a; 3 (log x) 2 Jl nx n^x^
2 ! 3 42 ^ 53
■■■}
+ ..Ketc.
Similarly, forms of the type of formula (32a), see II., § 23
1 log X
(278) ./a;<±'"±"""'da; =/e<±"'±™)''da;
can, if m and n be regarded as positive, be put in the form
^ '■ ■■•Jy ^ {m^nxy ^ 2! {m^nxfv ^'-^ h\ (m+nxfv ^' ' j*
which can be integrated term by term. The integrals, however, are
tedious. For example : —
/- log x , log X 1 f 1
1 1 1 1 1
(p—^)m4>v-^ "'"(p-4)m20J'-4 '^■■"'"2wP-Y^ ^ wP-^(f>]
1 J «
"'"(^-l)mJ'-i» °^ .^
^ denoting (m+wa;)''.
If ^ = 1, and n is positive, this takes the simpler form —
(281) . . . .ya;^^+^* = - log a; log {m+nx) - ^ (log nx)^ ^
■m.a m* , ^
Owing to the very great elaboration of the terms of many of the
integrals, practically it is preferable to compute a sufficient number of
ordinates, and integrate by any suitable summation-formula (given
hereinbefore).
84 APPENDIX A
7. The Eulerian integrals or Beta and Gamma functions. — ^The Beta
and Gamma functions are of special importance in statistical integrations.
They are : —
(282)
J ^x ^i X) ax~/^z ^i z) — /^, (1 + ^)'+"' - » (l+y)«+'«
that is, in the more brief notation —
(283) B (I, m) = B (m, I) = ^p^~^~^^
Further —
(284:)..C e'-'x^-^dx = (^ 0°^-)" dy=(n-l)r e'"^ x"-^dx = r{n)
respectively, from which it is evident that : — ^
(285) r(l) = l; r(w + l)=w! =nr(n)
Thus, in order to calculate F (n) we have, if it be an integer, it is
equal to (n—1) ! , if not an integer, it can be readily found, since its
logarithms have been tabulated for the range 1 to 2 to two places of
decimals and to 9 places of figures.^ Thus —
(286) n {n + 1) {n+2)....{n + k- 1) T {n) =r{n + k)
which, logarithmically, is perfectly convenient to use.
By putting kz = x, in (284), it becomes obvious that
(287) ./J'^e -"^ a;»-i dx = ^-^
(288) ^-^ W = ^'(^) =/o'"e-^a;"-i log a: dx.
Examples of the application of these formulae have already been
given: see IV., § 5, formulge (150) to (166).
8. Table of indefinite and definite integrals and limits. — ^In an
addendum small tables are given, for convenience, of indefinite integrals ;
of definite integrals, for example, between hmits such as zero and unity ;
zero and infinity ; etc., and of limiting values. These embrace those
which more frequently occur in statistical investigations.
' r(i) = ^/n.
' Traits des Fonctions EUiptiques, Legendre, Paris 1825-8 (logarithms to
12 places). Sammlung von Formeki, W. LAska, pp. 290-1. Braunschweig
(logarithms to 9 places). Biometrika, J. H. Duffell, Vol. VII., 1909-10, pp. 43-7
(logarithms to 7 places).
Vn.— THE PLACE OF GRAPHICS AND SMOOTHING, IN THE
ANALYSIS OF POPULATION-STATISTICS.
1. General. — Graphs of the data are necessary in any analysis of
population-statistics purporting to aim at thoroughness. A graph
indicates not only the general trend of the data, but also whether the
individual items conform with great exactitude to that trend, or whether
they deviate considerably therefrom. The criticism of deviations
ordinarily depends upon whether numbers or ratios are being analysed.
Where figures are of the nature of ratios, if, on the working-graphs the
numbers be written, it is possible to see at a glance whether changes in
any part of the graph of the crude data are significant or otherwise. Thus
a ratio resulting from 30,000 divided by 10,000 would be materially
changed so far as the numerical data are concerned by an alteration,
say, of one- thousandth. To change the ratio say from 3 to 2.997 would
mean an alteration of 30 in the numerator or of 10 in the denominator ;
whereas, if the original data were the numbers 3 and 1 , an alteration of a
single unit would greatly disturb the ratio.
In general, we are concerned with two kinds of alteration ; one
may be called the " redistribution of the data without alteration of their
aggregate ;" and the other may be called the " alteration of data to coincide
with what is deemed the most probable result," having regard to all the
facts. It is, for example, sometimes desirable to keep the aggregate of the
smoothed results identical with that of the data. In other cases this
is less essential, and it may be said that probably m^uch time is often
wasted in making re-distributed data agree with the original as to the
aggregate of units represented.
As to general method it may be noted that when the original facts
have been plotted, a curve may be drawn freehand by anyone familiar
with the characteristics of the various type-curves, and especially those
of probabiUty-curves. By means of sets of curves, French curves, and
sphnes of various kinds,* the freehand curves may then be improved so
as to be really smooth and conform to what might be called the probable
indication of the data. When the numbers represented are large,
limitations of scale may operate to Hmit the smoothness as deduced by
scaled values, from the graphs, but a little simple differencing wiU suggest
necessary adjustments, or the differences may be graphed. The adjust-
ments having been made, the aggregate can be formed by adding together
the scaled or properly differenced ordinates thus adjusted.
If this operation has been weU done the total will be so nearly in
agreement with the original data that a common factor of correction can
be used throughout, that is, all the ordinates may be increased or dimin-
ished in the same ratio, and the finally deduced ordinates will then agree
* Splines of transparent celluloid are most convenient.
86 APPENDIX A.
with the data, and at the same time form a smooth curve. If the data
when plotted are visibly irregular, meticulous precision in adjustment
is obviously but a waste of time. For this reason one of the great merits
of the graphic method is that, not only can the analyst see at a glance the
conformity or otherwise of the data to a particular type of curve, but he
can also judge whether the data yield results of a high order of precision.
It has already been mentioned (see IV., § 1) that the initial and
terminal characters of the curve and its mode (maximum and minimum)
are important. It may be added, that if the curve is not drawn as uni-
modal in type, the reason for the adoption ctf a particular form must
really depend on the character of the data, and may not be decided merely
upon mathematical considerations.
2. The theory of smoothing statistical data. — ^It may often be
known a priori that phenomena should exhibit a regular progression, and
that data, when graphed, shewing as zig-zag hnes, do not really represent
the ideal fact, owing either to the paucity of the data, or to unavoidable
error therein.
In a series of group^alues, i.e., totals or aggregates between a series
of limits of a variable, it is important to bear in mind that — ^assuming
the counts on which they depend to be correct — ^what is known is merely
the series of aggregates themselves : the probable distribution yielding
these aggregates has to be conjectured. When the totals or aggregates
are themselves regarded as subject to error, then the distribution may be
modified within the Umits of probable uncertainty, some groups being
diminished and others, particularly adjoining ones, increased.
There are four principal classes of data to which the process of curve-
smoothing is appUcable. These may be indicated as foUows : —
(i.) Frequencies of a phenomenon at successive epochs or during
successive periods of time ; as, for example, population
estimates at given dates and numbers of deaths occurring
during successive years,
(ii.) Rates of occurrence of a phenomenon per unit of reference
during successive periods ; as, for example, birth-rates per
thousand of population per annum for successive years,
(iii.) Frequencies in respect of successive values of characters capable
of continuous variation ; as, for example, the number of
persons at each age recorded at a given census,
(iv.) Rates of occurrence of a phenomenon per unit of reference in
respect of successive values of characters susceptible of con-
tinuous variation ; as, for example, rates of mortahty per unit
per annum during a given decennium in respect of each age.
In all these cases the characteristic of continuous variation^ is
assumed to exist either actually or virtually. Where statistical results
are discontinuoiia such a process is, strictly speaking, inapphcable ; as for
1 See I., § 9. ~
GRAPHICS AND SMOOTHING IN POPULATION STATISTICS. 87
example, in the tabulation of census ]Dopulation according to birthplace,
occupation, or reUgion. In some cases, however, although the data
are strictly speaking discontinuous, the principle may be applied partially ;
for example, in the case of a tabulation of dweUings according to number
of rooms or according to number of inmates. In such cases the character
possessed is progressive without being continuous ; nevertheless, with proper
qualifications, the smoothing principle may be applied even to these.
Another example, more nearly approaching but not attaining con-
tinuous variation, is the representation of dwellings according to rental
value.
3. Object of smoothing. — ^From the foregoing it wiU be seen that the
data to which the smoothing process is strictly appUcable are those which
may be regarded as functions of a continuous variable. But whether
such functions are readily expressible by means of algebraic formulae or
not, is, of course, reaUy immaterial. The essence of the matter is that in
any instance the data are in the main such as admit of representation by
means of a continuous hne, or a continuous surface or sohd in relation to
continuous units of reference. When such representation has been made
of the crude results of observation, it is ordinarily found that the line
surface or solid exhibits evidences of marked irregularities as between
adjacent points or series of points, their general trend, however, suggesting
an underlying basis of orderly progression. This progression is, of
course, afEected by minor influences operating at individual points, and
is more or less masked by the paucity of the data on which the repre-
sentation has been based ; thus, suggesting further that were it possible
to obtain data of unlimited extent, these irregularities would become
negligible. For this reason the object of the smoothing process may be
said to be that of removing these apparently accidental irregularities, and
of thus disclosing the basic or ideal uniformity which may be presumed to
represent the facts in aU their generahty.
4. Justification for smoothing process. — ^The justifications for the
smoothing process may thus be said to be : —
(a) That the irregularity does not represent the phenomenon in its
generality, since much of the observed irregularity is known
a priori to be due only to paucity of data ;
(6) or that it is known that the phenomenon subject to observation
is reaUy regular ;
(c) or, again, that the observed data suggest that regularity of trend
wiU not efficiently represent them.
It has been objected that any system of smoothing is, strictly speak-
ing, unwarrantable, since such a process virtually attempts to make the
facts accord with more or less questionable preconceptions regarding
them. To this view it may be rejoined that if the process were such as to
produce results which, though smooth, differed systematically and materi-
ally in their distribution from the original observations, the objection
would be valid. Where, however, due consideration is given to the
88 APPENDIX A.
relative magnitudes of the original data, and the smoothed results accord
therewith as closely as the data will allow when these exhibit a general
trend, then the only preconception that can be regarded as operative is the
justifiable one that ordinarily natural phenomena do not progress per
saltum. In this connection it must be noted that where there is distinct
evidence at any stage of a cataclysmic disturbance of results, the smoothing
process for such points or periods will usually be invaUd or not properly
applicable. Examples of such cataclysmic disturbances of statistical
data are war, famine, pestilence, earthquake, etc. Even in these cases,
however, it appears admissible under certain circumstances to apply a
smoothing process ; as, for example, in cases where the disturbances
referred to are of more or less frequent occurrence, and are not merely
isolated instances.
One of the most cogent justifications for the smoothing process has
its warrant in the fact that the recorded results of any statistical observa-
tions are necessarily approximative, and hence that the value of the
function recorded for any given value of the variable is probably not
usually more accurate than an estimate based on the recorded values in
respect of preceding and succeeding values of the variable. This con-
sideration suggests the idea of weighting successive observations to obtain
most probable values, which idea forms the basis of one of the leading
methods of adjustment. Again, where the results of the observations
are to be employed as guides to future action, it is clear that these results
should, as far as practicable, be freed from all fluctuations which may be
considered merely accidental, and thus unlikely to be reproduced in
future experience. This is of considerable importance in connection with
the construction of mortality and sickness, superannuation, and similar
tables to be used in the computation of rates of premium, and for the
conduct of valuations.
5. Mode of application of smoothing processes. — ^It has already been
indicated that one of the main objects of the smoothing process is the
discovery of a smooth series which presumably underKes the irregular
data furnished by a limited number of observations, and it has been
implied that a process to be justifiable must, in addition to smoothness,
be characterised also by what has been called " goodness of fit"; that
is, within reasonable limits it must reproduce the characteristic features
of the original data.
The methods of applying the smoothing process vhich have up to
the present been employed, may conveniently be grouped in three classes,
viz. : — (a) Graphic Methods ; (6) Summation Methods ; and (c) Methods
of Functional Conformity.
These methods have been employed in connection with observations
in many fields of research ; as, for example, general statistics, actuarial
science, physics and chemistry, astronomy, tidal theory, biology, etc.
In the actuarial field, an extensive and systematic use of the process has
been made, and a most detailed examination of the underlying principles
has been carried out.
GRAPHICS AND SMOOTHING IN POPULATION STATISTICS. 89
(a) Graphic method. — As its name indicates, this method is based
on the attainment of the desired smoothness by means of a graphical
representation and adjustment of the observed data. For example, the
subject of observation being the infantile mortality e;x:perienced in a
community during a given period, and the periods of observation being
calendar years, a base line is taken and divided into equal parts, each of
which represents a year. On these parts as bases a series of rectangles is
constructed, the area of each rectangle being proportional to the rate of
infantile mortality averaged for the corresponding year. The upper
parts of these rectangles will present in the case supposed the appearance
of flights of steps with uniform treads and unequal rises. The necessary
smoothing may be effected by drawing a continuous free-hand curve
through the upper portions of these rectangles in such a manner as to
include between certain limits the same area approximately as is contained
in the rectangles covering the same range.^ The area enclosed by the
part of the base hne relating to any year, the ordinates drawn from the
extremities of this part, and the portion of the curve between these
ordinates will represent the smoothed result for the year under review.
Whether, as in the example just given, the data should be represented by
areas, or, as is sometimes more suitable, by ordinates, is a matter which is
determined agreeably to the appropriate interpretation of the result
to be attained. It may be noted that the method of representation by
rectangular areas is specially applicable to cases where the data are
functions not of single values of the variable, but of ranges of such values.
For instance, in the above example, the rate of infantile mortaUty stated
for any year is a function not of any one point of time in that year, but
of the range of values representing the whole of the year. In most cases,
however, the system of representation by means of ordinates would be
equally valid, and sometimes more convenient.^ Referring again to the
above example, from a point on the base hne representing the end of each
year an ordinate could be drawn representing the rate of infantile mor-
tality for that year, and a free-hand curve being drawn amongst the upper
points of these ordinates, the ordinate to any point on the curve would
represent the rate of infantile mortahty for the year ending on the date
corresponding to the foot of the ordinate. Similarly, the ordinate for
smoothing" might be drawn from the beginning or the middle of the line
for each year, or, indeed, from any point uniformly selected in each, and a
corresponding interpretation of any point taken on the curves drawn
amongst the upper points of such ordinates would be apphcable.
6. On smoothing by differencing. — ^A curve continually convex (or
continually concave) upward might possibly be drawn with a single
difference. We have, by the theory of differentiation —
(289). .dy/dx=d(a+bxP+cx^+eto.)/dx=pbxP-^+qcxi-^ + etc.;
^ In practical examples it is rarely possible to make the curve such that the
adjusted areas are continually identical with the rectangles on the same base.
« See, however, V., § 10, formulae (212) to (216).
90
APPENDIX A.
hence, Up, org, etc., should happen to be integers, at some stage of differ-
entiation, this particular term of the expression wiU be x''=l, and hence
that difference wOl vanish. Probably in no case are population-statistical
results actually representable by integral values oip, q, etc., hence, strictly,
there is no limit to the series of differences. These, however, ultimately
become high negative powers of x, and consequently when x is large their
value is small : they must ultimately become of negUgible amount.
Again, statistical data often involve exponential forms, particularly
those of the type ae~"*, the differential of which is — nae~'^'', from which
it is evident the successive differences are interminable. Since, however,
de ~^/dx =l/e*, the higher differences for large values of x become insensible.
Hence, we shall always be justified in taking differences only to the stage
where they are appreciable. Thus if at any stage of smoothing we make
the second difference a constant, we are making the curve one which the
equation y^a-\-bx-\-cx^ wUl reproduce ; if we go on then with a constant
third difference, we add a stretch of a new curve, viz., y'=a'-f-6'a;+c'a;^
-{-d'x^; and so on. Such methods are unobjectionable when the
tangents to the curve at the point of junction may be regarded as sensibly
identical.
7. Effect of changing the magnitude of the differences. — ^It is often
useful to be able to recognise instantly the consequence of changing the
magnitude of a difference. This can be indicated at once by a table.
Table IX. — Efiect on the value of a function of a change of a unit in a
leading difference.
Difference in which
the change takes place.
Effect on the value of y where its suffix is-
1 2 ! 3
6
8 9 10
1st difference
2nd difference
3rd difference
4th difference
5th difference
1
2
1
1
!
I j
! '
5
10
10
5
1
6 7 8 9
15 21 28 36
20
15
35 I 56
35 1 70
21 i 56
84
126
126
10
45
120
210
252
It will be recognised that these are the figures of Pascal's triangle
taken diagonally, or the diagonal series in this are the figures of Pascal's
triangle taken vertically. By means of such a table one can see at a
glance the effect on any value of the function of changing a leading
difference.
GRAPHICS .\ND SMOOTHING IN POPULATION STATISTICS. 91
8. Smoothing, by operations on factors. — ^The smoothing of a suc-
cession of ordinates or of group-values may often advantageously be
effected not by operating upon these numbers themselves, but upon their
ratios to each other. This may be called factorial smoothing. Let
A, B, G, D, etc., be the series of quantities to be smoothed. The ratios
B/ A, C/B, D/0, etc., are formed, and denoted by b, c, d, etc. These
are graphed and smoothed by any process. '^ The smoothed values,
denoted hyb',c',d', etc., are then used to form a new series of quantities ;
thus A = A, Ab' = B"; B"0' = G", etc. The sum of these is then
made equal to the sum of the original series of quantities by a common
factor k, thus —
(.
(290).. /fc
A + B + C+D+ etc. ^ ]l+b'\l+c\l+d(l + ..)\\f-
' A+ Ab+ Abc+ Abcd+eto. ^ |i^^/ J^^' ; l-Fd(l-f .
I
then the smoothed values A', B', etc., are A'=kA; B'= kAb' ;
C'=kAb'c'; D'= kAb'c'd' ; etc.
Sometimes, on taking out the ratios, it becomes evident that they
should have a common value, since they shew no systematic progression.
In such a case, let m denote the mean value, then the denominator
A-\~Ab + Abe + etc. in (290) becomes A + Am + Am^ -\- etc.
Smoothing of this kind is serviceable for initial and terminal values.
9. Logarithmic smoothing. — ^In a similar manner quantities may
sometimes be advantageously smoothed by smoothing their logarithms.
In this connection we bear in mind that it a series of numbers are in
geometrical progression their logarithms are in arithmetical progression.
Let log A, log B, etc., be denoted by a, ^, etc., which are graphed, and
when smoothed denoted by a', j8', etc. If the sum of A", B", etc.,
corresponding to the smoothed values, do not agree with that of the
original values, k will be the factor of correction, and may be found as
before, that is, by (290). This process may be called logarithmic smoothing,
and like factorial smoothing, is often useful for initial and terminal
values.
10. On the difference between instantaneous and grouped results.^—
When instantaneous results are smoothed the resulting smooth curve
represents the equation which reproduces the values of y corresponding
to given values of the abscissas. When, however, group-results are
smoothed by differencing, the resultant curve strictly represents the
value of a group of the same base (supposed, of course, constant) with any
central value throughout the range smoothed: see V., § 10. When,
however, group results are few in number (that is, have relatively large
bases) the graph must be drawn upon a different principle, viz., it must,
as far as the probabOities of the case wiU admit, make the areas between
bounded by the curve, the abscissae, and the ordinates identical with the
' Arithmetically, i.e., by difference, or mechanically, by splines, etc.
92 .-UTENDIX A.
area of the group, or, in other words, the mean height of all the ordinates
to the curve in any given range of the abscissa must be equal to the height
of the group. That is, if /i is the height of the group, then : —
(291) h = ~-\_ r'f(x)dx.
X2 — Xi Jxi
/ (.r) denoting the smoothed curve drawn.
This method may be called " the method of equivalent grov/p-values,"
and it will, in general, either not depend on differencing at all, or depend
thereon to a less extent than when the bases are relatively smooth and
the groups numerous.
11. Determination of the exact position and height of the mode. —
It is often desirable to ascertain with such precision as is possible the
abscissa and height of the mode. Two approximate solutions are de-
sirable, viz. : — (a) when the graph shews that three groups should be taken
into consideration; and (6) when/owr groups. In the former case (a) the
formulse are extremely simple ; in the latter (6) they are much less so.
If more than four groups are to be taken into consideration it is better to
determine the general equation of the curve and solve to obtain that
value of X which makes dy/dx=0. As an approximate solution will be
available from the graph, there is usually very httle difficulty in obtaining
an exact value of x. Then the corresponding value of y can be found
from the equation: see V., §§ 3 to 7.
Case (a). In Fig. 35 let K denote the mean of the heights of the
groups on either side of the maximum group and the height of this last,
and let k be half the difference of the height of the groups on either side.
Let also a denote the difference of the height of one group and the
greatest group, and |3 similarly the difference of the height of the other
group and the greatest group. Then
(292) K =~ (a+/8); and A = 1 (a . ^).
Then a second degree curve, giving the same group values, gives the
abscissa of the mode: —
(293) M = -^5 ; aici ^,' = -J-^
and the height A, of the mode, above the maximum group is
(294) ^=r2^ + lT
If /, g, and h denote the heights of the rectangles we should have for
the constants of the curve —
(295) « =~{nf + 2h-lg)
(295a) . . . .b = 3g - 2f - h
(295b) ....c =^{h+f)-g
the base of the curve being considered unity.
GRAPHICS AND SMOOTHING IN POPULATION STATISTICS.
93
In the case (6), differences of height being as shewn in Fig. 36
the constants of the curve which must now contain dx^ will be
{29Q)..a=-^-^{y+y'); 6=^(15/3-8'); c
12
(y + y')'
a being reckoned from the point K, half-way between A and B to the
point L, that is, to the curve.
The value of the abscissa of the mode is given by
(297).. x„
y +y'
2(3^8
' 1 I ./fl I 2 (15|8-8')(3^-§') -
The sign of the term under the radical can readily be determined in a
practical example. The general expression for y,^ is lengthy. In cases
practically occurring we may compute it from x„ when that value is
found : that is, it is
(298). .y„ = ±^^y + y'(l -3a;;)+ (15^ - 8'):«„-2(3^-§')<|
the ordinate being reckoned from the line parallel to the axis of abscissae
and half-way between the points A and B in Fig. 36, i.e., the line MJ in
the figure.
IP 'a T
PG 0C+/3 »
Fig. 35
Fig 36.
The formulae (293) and (294) and (297) and (298) are not quite
satisfactory, and in general it is better to compute the coefficients of the
equation which fits a considerable stretch of the curve, and find the
position of the maximum by dy/dx = 0, if very great precision be
required.
94 APPENDIX A.
12. The testing of smoothed or graphic results. — ^When smoothed
graphed resxdts are obtained they will, in general, need, as already indi-
cated, to be arithmetically tested. The fundamentals of arithmetical
testing are the foUowing : —
(i.) The sum of the graphed results should be sensibly (or exactly)
equal to the sum of the original data ;
(ii.) The deviations, positive and negative, between the aggregate
of the smoothed results and the data up to each given value
of the argument should, consistently with the type of curve
adopted, be a minimum ;
(iii.) The position and ordinate of the mode should be carefully
fixed, and as well as the data will permit ;
(iv.) The position of the terminals should conform to the probabiUties
of the type of data so far as that can be determined.^
■^ In general, they cannot be determined mathematically. For example, the
frequency of births of given ages, so far as mathematical relations are concerned,
might be continued to start at the age 0, but in view of physiological considerations
we shoTild not be justified in starting at 0, but at, say, the age 11 ; similarly in regard
to the terminal, which may be made to meet the axis of abscissae for age 60 (or such
later age as may be indicated as occurring, should satisfactory information be to
hand).
Vm.— CONSPECTUS OF POPULATION-CHARACTERS.
1 . GeneraL — Thus far the consideration of the theory of population
has been concerned only with its numerical aspect, and with the mathe-
matical form of expressions under which it may be necessary to subsume
the facts. These constitute an essential preKminary only. It remains
now to consider in detail some of the various characters of importance.
Not only are population-statistics, in the narrower sense, signi-
ficant both (i.) in themselves, and (ii.) in comparison, but so also are all
facts that may properly be regarded as expressions of the various char-
acteristics of a population. Following the nomenclature of biology, these
may be called more briefly its characters. Such characters may relate to —
(a) Vital phenomena, that is, to birth, life and death, to repro-
duction in all its aspects, to disease and all the modes of its
incidence ;
(6) Anthropometry, that is, may relate merely to the human form
and its variations, or to its growth and decrepitude ,
(c) Anthropology, that is, they may refer to man's general evolu-
tion, both physical and psychical ;
(d) Sociology, that is, they may concern man in respect of his
social life, an important element in which is his economic
evolution, and they may concern also the reaction of this
upon his numbers and the density of his aggregation.
(e) Migration, aggregation, segregation, or wide dispersion, colonis-
ation, etc., that is, the direction and velocity of movement of
populations, the tendency to Uve in more or less dense
groups (large cities or villages) or to spread over the earth,
etc.
All these have significance in regard to the rate of development of
the world's people. It is well to bear in mind, also, that population-
characters may be in two forms, viz., either actual or potential.
The importance of the subject is seen in the impossibility of maintain-
ing the present rate of increase for any great length of time (see II.,
§ 34) ; and its range of subjects is best seen through a conspectus.
Characters may be simple or complex, their manifestation may be instan-
taneous or durational ; and the evidence of their nature direct or de-
rivative. The greatness of the range of population-characters, and the
number of significant relations subsisting among them is so vast that
no statistical presentation of them can be exhaustive. Thus important
questions are continually arising involving demands for new statistical
compilation, for human affairs can be properly analysed only with the
aid of a well-founded and technically satisfactory statistic. The simplest
population-characters are expressible in regard to units, as, for example,
the numbers in a population ; the wealth possessed, etc. The complex
96
APPENDIX A.
are those which involve multiple fields of comparison, for example, the
number of one sex, who, being between given limits of age, and belonging
to a given occupation, die of a particular disease.
That the number of comparisons possible is very great is obvious
from the fact that n things considered in their mutual instantaneous
relations, that is, n things considered each in relation to 1 n—l
other things, are 2"— 1. The following table will shew the number
possible up to w=10.
TABLE X.
No. of Elements
in Combination
Elements of Original Statistical Data.
1
2
3
4
5
6
7
8
9
10
4 , 5
6 10
4
1
10
5
1
6
15
20
15
6
1
7
21
35
35
21
7
1
8
9
10
28
36
45
56
84
120
70
126
210
56
126
252
28
84
210
8
36
120
1
9
45
1
10
1
Total possible
combinations
of elements
1
15 31 63 127 255 511
I
1023
The total possible for 12 is 4095, for 20 is 1,048,575.
There were, for example, 17 main questions to be answered in the
Australian Census ; thus there would be 2^' — 1 (viz., 131,071) possible
tables by combinations of these results, and a considerable proportion of
these would be of real significance.
2. Characters directly given or derivative. — ^Important characters
are not always immediately yielded by the data : they are often to be
ascertained only by analysis. Thus, as in the case of statistics generally,
population statistics may be either
A. Direct, viz.:-
A (i.) Instantane-
ous (numbers at a
given moment).
(Examples) : No.
of persons living ;
wealth possessed by
them at a particular
instant ; etc., or
A (ii.) Durational
(or number of
occurrences dur-
ing a unit of time)
(Examples): Num-
ber of persons
bom, married, or
deceased diu'ing a
day, month, or
year ; etc.
or) B
B (i.) Instantane-
ous (nvimbers de-
duced represent-
ing a state of
things for a given
moment or epoch).
(Examples) : Mas-
cuUnity at birth,
or at a census ;
wealth possessed,
per individual ;
expectation of
life ; etc., or
Derivative, viz.: —
B (ii.) Diu-ational
(numbers deduced
of occurrences dur-
ing a imit of time).
(Examples) : Birth,
marriage, or death-
rates per day,
month, or year ;
average wealth de-
duced from probate
returns ; etc.
The above indication of the nature of population statistic reveals
the reason of its extent, which is much greater than is implied in the
CONSPECTUS OF POPULATION CHARACTERS. 97
number of mere combinations of different fields of statistic considered
in their instantaneous relations alone.
3. Characters in their instantaneous and progressive relations. —
The characters of a population are fully studied only when examined both
in their instantaneous relations, and in the progression of these with time.
Suppose, for example, that characters A and B both vary with time, and
that such variation can be expressed by rational integral functions
thereof ; then the constant relation of the characters is given by
f2q9^ - = l^ = a2(l + b^t + c^t^ + etc.)
^ ' ' A ~ Fi{t) ^ ai(l + bit + Cit^ + etc.)
= ^ [1 +(62-6i)i-.|6i(62-6i)-{C2-Ci);-<Hetc.]
approximately ; or including the term in t^, and writing
a result of greater precision is given by : —
(301).. ^=^[l+i3<|l-(6i-y)*+[6i(6i-y)-Ci+S]«2+etc.lJ
The successive coefficients, in nearly all practical examples, converge
with sufficient rapidity to admit of the employment of the formula for
even large values of t.^
Derived characters involve, as a rule, a greater complexity of change
of relation with the lapse of time. Thus, for example, suppose a rate for
persons is to be deduced from the rates for males and females, and suppose
also that these do not change identically with the lapse of time (as, for
example, a death or morbidity-rate for a particular disease). Suppose
then that these rates for males and females are respectively : —
(302) mt = mo(l + b^t + c^t^ + etc.) ; and
(303) /t = /o (1 + 6/ « + c/ i^ + etc.)
Let us suppose that the ratio of the difference of the sexes to their
sum, that is (M- F)/(M + F) = ja ; or (F - M)/(F + M) = ; and that
the variation of this function with time is expressed by
(304) jLit = )tio(l + i3'< + y'f + etc.) ; and j>, = - ix.t\
then it will follow that the ratio f for persons will be
(305). .v = (I + |)m+ (-i - |)/= ^ N+/) + 4 /^ (»*-/)
^ 8av, ordinarily at least to « = 100.
98 APPENDIX A.
that is, it will be the mean weighted according to the relative numbers
of males and females. The result may at once be written out from (302)
and (303), and re-expressed is
{30Q)..pt = '^j[w(, + /o+/^o("*o-/o)]+'»Wo i6m(l + A^o) + /^o^'I
+ /o {cf (1 - /Lto) - f^oibf ^'+ Y')\]t'+ etcj
From this it is obviously impossible to secure consistency among
formulae for persons, males, and females, where the variation with time
of those for the two last is not identical, without complexity of expression.
Moreover, when variations with time have to be considered, as well as
many fields of comparison, not only do general formulcB become too
involved to be of practical value, but also the number of relations neces-
sary to exhaust the statistic becomes hopelessly large. For this reason
it is often desirable to compute the coefiBcients for males, females and
persons independently : if this be done with care the involved incon-
sistency may be regarded as negligible.
4. Conspectus of the population-characters with which the ordinary
census is concerned. — ^In Section 1 of Chapter II. of the general Census
Report, a classified statement and a brief review of the objects and uses
of a census are given. These present, however, only one aspect of some
of the leading characters of population. In the following conspectus a
somewhat different and more extensive sketch of such of these characters
as are capable of statistical measurement, and which constitute normal
bases for comparisons, is furnished : —
A. — ^Numerical constitution of population at a given epoch in regard to
(i.) Sex, and (ii.) age ;
(iii.) birthplace ; and (iv.) length of residence in country of
enumeration ;
(v.) nationaUty ; and (vi.) race ;
(vii.) conjugal condition ; (vui.) duration of marriage ; and
(ix.) size of family ;
(x.) infirmity ;
(xi.) degree of education ; and (xii.) school attendance ;
(xiii.) rehgion ;
(xiv.) occupation — {a) designation ; and (6) grade ;
(XV.) dwellings — (a) material ; and (6) number of rooms ;
(c) mode of occupancy ; and (d) rental ;
(xvi.) localisation.
In each case the statistical data initially represent the number of
persons possessing the character or group of characters specified, as, for
example, the number of persons having a family of a given size, the
number of persons having a given duration of marriage,
CONSPECTUS OF POPULATION CHARACTERS. 99
In the case of dwellings the enumeration is twofold, and comprises,
for example, the number of dwellings of a given material, as well as the
number of persons Hving in dwellings of a given material.
B. — ^Relative constitution of population in respect of characters
enumerated in A.
In this section are comprised the ratios of the numbers possessing a
given character or group of characters to the numbers possessing a wider
range of such characters, as, for example, the ratio of males under 21 years
of age to the total population of all ages and of both sexes.
C. — Variations of population at different epochs.
This may involve merely variations in aggregate population, or may
comprise variations in the numbers possessing any combination of the
characters enumerated in A, or in the relative constitutions deduced
under B.
D. — Mean population at a given period.
As in the case of C, this may involve merely the aggregate population
or may comprise the mean population possessing any combination of the
characters enumerated in A. The mean population for any unit of time
represents the number of such units of human life hved by the population
or section thereof under observation.
E. — ^Fluctuations of population during a given period.
These arise from : — (i.) Births (see F) ; (ii.) deaths (see G) ; (iii.)
migration (see H).
F.— Births,
(a) The statistical data initially represent the number of births
classed according to the following categories, taken either singly or in
combination.
(i.) Whether live or still birth ; (ii.) sex of child ;
(iii.) whether born in wedlock or not ;
(iv.) age of father ; and (v.) age of mother ;
(vi.) birthplace of father ; and (vii.) birthplace of mother ;
(viii.) occupation of father ;
(ix.) duration of parents' marriage (see I.) ;
(x.) locality ; and (xi.) date of birth ;
(xii.) date of registration ; and (xiii.) position of child in
family, i.e., whether first, second, etc.
(xiv.) single or multiple birth.
(6) The derivative statistical results comprise, inter alia, particulars
concerning the relations between
(i.) Live and stUI births ; and (ii.) nuptial and ex-nuptial
births ;
(iii.) male and female births ;
(iv.) number of births and population from which derived,
UBRARY
SEP 17 1945
DEer. OF
AGRIC, ECON.
100 APPENDIX A.
These may involve merely the relation between total births and
total population, or the relation between the number of births possessing
any character or group of characters enumerated in F (a) and the appro-
priate subdivision of population from which derived. In the one case
the result would be the crude birth-rate, or ratio of total births to total
population, in the other it would comprise such results as, say, the nuptial
birth-rate in a given area amongst fathers of a given age, birthplace, and
occupation, who had been married for a given period. Similarly (i.),
(ii.) and (iii.) may involve merely totals possessing the characters specified,
or may relate to subdivisions possessing any character or group of
characters enumerated in E : as, for example, the relation between
live and still births amongst the nuptial male births of women of a given
age and birthplace, who had been married for a given period.
G. — ^Deaths.
(a) The statistical data initially represent the number of deaths
classed according to the following categories, taken either singly or in
combination : —
(i.) Sex of deceased ; (ii.) age ; and (iii.) birthplace ;
(iv.) cause of death, (a) primary, and (6) secondary ;
(v.) occupation; (vi.) length of residence; and (vii.) locahty;
(viii.) age at marriage and re-marriage ;
(ix.) number of issue, according to sex, and whether hving or
dead ;
(x.) date of registration.
(6) The derivative statistical results consist mainly of particulars
concerning the relations between the number of deaths possessing any
character or group of characters enumerated in G (a) and the appropriate
subdivision of population from which derived, such, for example, as the
death rate from a specified cause in a given locality amongst males of a
given age, birthplace and occupation.
(c) As derivative results of the second degree may be classed such
particulars as
(i.) Index of mortahty ; and (ii.) corrected death-rates ;
(iii.) expectation of fife ; and (iv.) detailed mortahty tables.
H.— Migration.
Complete statistical data would initially represent an enumeration
of migrants classed according to the characters specified in A, with the
exception of (xi.) length of residence ; and (xv.) dwellings. Such detail
is quite impracticable, and the main characters available in Australia
are : —
(a) For traf&c by sea : —
(i.) Sex ; and (ii.) whether adult or child, or preferably exact
age;
(iii.) port of embarkation ; and (iv.) port of disembarkation ;
(v.) nationaUty or race ; and (vi.) date of migratioi),
CONSPECTUS OF POPULATION CHARACTERS. 101
(6) For land-trafific by rail : —
(i.) Sex ;
(ii.) state in which arrived ; and (iii.) from which departed ;
■ (iv.) date of migration,
(c) For land-traffic by road : —
Similar details as in (6).i
I. — ^IVIamage.
(a) The statistical data initially represent the number of marriages
granted in a given period classed according to the following categories
taken either singly or in combination : —
(i.) Age of bridegroom ; and (ii.) of bride ;
(iii.) conjugal condition of bridegroom ; and (iv.) of bride ;
(v.) birthplace of bridegroom ; and (vi.) of bride ;
(vii.) occupation of bridegroom ; (viii.) locaUty ; and (ix.) date
of registration ;
(x.) by whom celebrated ;
(xi.) ability of bridegroom to sign register ; and (xii.) of bride.
(6) The principal derivative statistical results are those concerning
the relations between the number of persons married during a given
period and possessing any character or group of characters enumerated
in I (a) and the appropriate subdivision of the population from which
derived, such, for example, as the marriage rate amongst bachelors of a
given age, birthplace and occupation.
J. — ^Divorce. ^
(a) Satisfactory statistical data would initially represent the number
of divorces granted in a given period classed according to the following
categories taken either singly or in combination : —
(i.) Age of husband ; and (ii.) of wife ;
(iii.) duration ; and (iv.) issue of marriage (a) males'; (b) females ;
(v.) locality ; and (vi.) birthplace of husband ; and (vii.) of
wife ;
(viii.) occupation of husband ;
(ix.) sex of petitioner ; and (x.) cause of petition ;
(xi.) date of rule nisi ; (xii.) and of making rule absolute ;
(xiii.) by whom marriage was celebrated.
(b) The principal statistical results derivative from the foregoing
would be relations between the numbers of persons divorced during a
given period and possessing any character or group of characters enumer-
ated in J (a), and the appropriate subdivision of the population from which
derived, as, for example, the proportion of husbands of a given age,
birthplace and occupation, who had been petitioners in granted divorce
cases.
' In Australia thia last information ia not available.
^ Complete statistics not available in Australia,
102 APPENDIX A.
K. — Sickness and Accident.^
(a) Satisfactory statistical data initially represent the cases of dis-
ablement by sickness or accident occurring in a given period classed
according to the following categories taken singly or in combination : —
(i.) Sex ; (ii.) age ; and (ui.) birthplace of person disabled ;
( iv.) cause of disablement ;
(v,) occupation ; and (vi.) locality ;
(vii.) date ; and (viii.) duration of disablement ;
(ix.) conjugal condition of person disabled ; and (x.) number of
issue :
(xi.) whether or not disablement terminated by death.
(6) Derivative statistical results would consist mainly of relations
between : —
(i.) cases and appropriate population ;
(ii.) cases of deaths,
(c) Derivative results of a second degree consist of sickness tables
constructed from initial data.
5. The range of the wider theory of population. — ^The conspectus
just given has obviously been hmited to matters with which the census
and ordinary vital statistics are more directly concerned. In a wide
consideration of population, however, the characters of importance
include a much larger range, embracing what has already been indicated
in § 1, hereinbefore, viz., the anthropometric, anthropological, and
sociological, including the economic. This has already been referred to :
see I., § 6, iii. and iv., and II., §§ 13-18. Because of this fact, a complete
theory of population must take account of (a) the reactions of eugenic
facts and arrangements upon the numbers and mode of growth of the
population of the entire world and of its constituent peoples, and (b) even
of the reactions thereupon of all economic and social conditions,
including those arising from mobility. This is seen when one contemplates
the part played by modern facUities in transport and communication.
Nor are the physical and psychical characters of the population less
foreign to a complete theory. For the same reason there are aspects
of subjects not directly enumerable as population facts, which have
immediate touch therewith ; such, for example, as national, munici-
pal and private wealth and their fluctuation, concentration and dis-
persion ; the productivity of such wealth, the economics of national and
municipal revenues, expenditures, and administrations ; the productivity
of private wealth, and, indeed, of wealth of all kinds ; the correlations
between population- fluctuations and such financial characters as national
UabiUties ; the quantity and velocity of the circulation of currency ; the
relations between primary and secondary production and population
development ; the growth of institutions expressive of a deepening
recognition of social solidarity in co-operative effort, and in the national-
isation of the greater pubUc services, etc. And finally, it may be said
^ Complete statistics are not available in Australia.
CONSPECTUS OF POPULATION CHARACTERS. 103
that all facts which throw any light whatever on the possibility of world-
production of food supplies and the fluctuations of population with
abundance or want belong to tlie wider theory of population, and demand
appropriate mathematical investigation.
These wider facts are, of course, beyond the range of the narrow
limits of ordinary official statistic, but no comprehensive view of the
significance of a study of population is possible, which excludes the study
of the reaction of material, psychical, or social conditions upon its growth
and fluctuation.
6. The creation of norms. — The significance of statistical results
is fully recognised only by comparisons with the similar results for other
populations. Such comparisons are effected in the most general way by
the creation of norms for each population-character. The principle
which governs the constitution of a norm is that it shall represent the
character selected on the widest possible basis. Thus, if statistical data
existed for every population in the world, world-norms would be possible
for every character statistically recorded. Western civiHsation is fairly
homogeneous and statistical data are available for many characters.
Thus it should be practicable in the near future to create a series of norms
for the greater part of the western world. These might be regarded as
the normal or usual value of any character in question, with which the
same character in any particular population may be compared. It is
obvious in order to compare a series of populations the best basis is
the average value of any character : furthermore, if a compared character
is affected by the deviation of any other from the average the value of
the norm and of the deviation therefrom furnish the best basis for
necessary corrections.
The essential nature of a norm is perhaps best seen by regarding
it as representing the characters of aU the populations included, considered
as a single population. Thus the deviation of the characters and
any particular population about the secular changes therein of this
great aggregate gives the most informative presentment of the position of
the population in question, that can possibly be had : in short, it makes
the scheme of comparison as broad as is possible.
7. Homogeneity as regards populations. — ^Two communities may
be said to be homogeneous with regard to any series of characters, when
those characters are identical. In comparisons between communities
in regard to any one character, it is necessary, in order that the com-
parison should be a just one, that aU other characters which have any
influence thereupon should be identical ; or, to put this more generally,
the comparisons of any selected characters in a community are legitimate
only when these communities are homogeneous with respect to all other
characters which may have any influence on the comparison. For ex-
ample, the birth-rates of two communities are immediately comparable
if the relative numbers of married and single at each age are the same,
because the birth-rate then (presumably) reveals the fertility under
identical physiological conditions.
104 APPENDIX A.
Since, however, different communities are more or less heterogeneous,
appropriate schemes must be developed through which rigorous com-
parisons can be effected. Thus, for example, corrections may be applied
in such a way that any character compared or contrasted wiU not be
affected by difference of other characters.
The most convenient way of securing such a result is to adopt, as
a basis for aU comparisons, a population so characterised as to represent
all others to be compared as nearly as possible. Such a population may
be caUed a " normal " or a " standard" population, and any character
in regard to which it has been standardised may be called a " norm."
8. Population norms. — In order that any character of a number of
populations or communities may be conveniently compared, it will be
necessary that whatever population be adopted as basis, it shall represent
each as nearly as possible. It is easy to see that, in regard to any character
under review, such a basis must be a weighted mean, so that the character
adopted as basic shall be the character of the population formed by
aggregating all populations which may have to enter into comparison.
Thus if P, Q, B, etc., be populations, and p, q, r, etc., be the values of
some one character in each, then the best basis of comparison is : —
Pp+ Qq+ Rr+etc . _ Ss _
^'^^^' P+ Q+ B + etc. - S - "
S being the sum of P -|- Q + -R + etc., and s the norm.
It is immediately obvious that, in general, the secular changes of
norms will be less marked than the secular changes in respect of the same
character of the individual populations from which the norm is determined.
For this reason it will be necessary for the progress of exact statistic to
estabhsh a series of norms for all elements the comparison of which are
important. That is, we must adopt a standard or normal population
of definite characters, or, in other words, create a series of population
norms to serve as a basis for comparisons. The scheme then of com
parison is to apply the ascertained attributes of each existing population
to the standard population. This process will reveal what would have
been manifested had each population been similarly constituted to the
standard population.
9. Variations of norms. — ^Inasmuch as, in the present development
of statistics, norms have not been created, except perhaps as regards the
constitution of population of each sex according to age, it will suffice to
indicate the outhnes of a general method of studying the variation of
norms. Since necessarily they can vary only slowly, a decennial determin-
ation will be probably always sufficient, and when a number of decennial
changes are to hand, the investigation of their variation will become
possible. Whether such variation wiU reveal any sign of periodicity or
not it is at present impossible to say. It is not unKkely that periodic
elements of variation will be found superimposed upon slow secular changes.
This, however, must be left for the future to determine, and the appropriate
method of analysis will depend upon the character of the data.
CONSPECTUS OF POPULATION CHARACTERS.
105
10. Norm representing constitution of population according to age. —
A norm for males and one for females of European race is of importance
for properly comparing death, marriage, birth and other rates. The use
of such a norm was proposed by Dr. Ogle at the meeting of the
" Institut International de Statistique, " in Vienna, 1891, and the index
of mortality at present used is based upon such a norm, though not a
properly constituted one. The aggregation of the populations of a con-
siderable group of countries between which also there is migration,
removes the speciahsing influence of this latter element, and secures the
general advantages of large numbers. The following results were obtained
from combining the populations of England and Wales, Scotland,
Ireland, the United States, the German Empire, Norway, Sweden, Italy,
Canada, Australia, and Newfoundland generally for the censuses of 1900
or 1901.1 The numbers are given in each age-group, and above a given
age :—
TABLE XI.
Population Norms for 1900.
European (1900).
India (1901).
European (1900).
Numbers in
Age-
Numbers in
Age-
Numbers at and
Group in total of
Group in total of
above age indie-
Age.
10,000
10,000
Age.
oated
Fe-
Per-
Fe-
Per-
Fe-
Per-
Males.
males.
sons.
Males.
males.
sons.
Males.
males.
sons.
270
263
266
266
276
271
10,000
10,000
10,000
1-4
971
953
962
988
1,063
1,025
1
9,730
9,737
9,734
5-9
1,139
1,119
1,129
1,394
1,382
1,388
6
8,759
8,784
8,772
10-14
1,057
1,038
1,047
1,264
1,081
1,174
10
7,620
7,665
7,643
15-19
975
980
977
866
835
861
16
6,563
6,627
6,596
20-24
915
931
923
787
892
838
20
5,588
5,647
5,619
23-29
808
813
810
879
894
887
25
4,673
4,716
4,696
30-34
715
705
710
848
851
850
30
3,865
3,903
3,886
35-39
640
624
632
609
657
583
35
3,150
3,198
3,176
40-44
563
550
557
648
652
650
40
2,610
2,574
2,544
45^49
470
463
467
370
339
356
46
1,947
2,024
1,987
50-54
413
417
415
437
452
445
50
1,477
1,661
1,520
65-59
331
344
337
177
169
173
55
1,064
1,144
1,105
60-64
272
290
281
254
303
278
60
733
800
768
65-69
197
212
205
66
79
72
65
461
510
487
70-74
136
150
143
76
91
84
70
264
298
282
75-79
79
88
84
27
32
29
76
128
148
139
80-84
36
43
39
30
35
33
80
49
60
55
85-89
10
13
12
5
6
5
85
13
17
16
90-94
3
3
3
6
7
6
90
3
4
4
95-
1
1
1
3
4
3
95
1
1
1
Total . .
10,000
10,000
10,000
10,000
10,000
10,000
10,000
10,000
10,000
10,000
^ See "The determination and uses of population norms representing the oon-
Btitution of populations according to age and sex, and also according to age only."
By G. H. Knibbs, and C. H. Wickens, Trans. 15th, Int. Congr. Hygiene and Demo-
graphy, Washington. Vol. VI., pp. 352-378.
106 APPENDIX A.
11. Mean age of population. — ^The mean age, x^, of a population is
given by the formula
/"xl-dx 1 if xL
(308) x^ = -^—1- = 4 + %rr^ • approximately.
Zj, denoting the relative frequency at the age x, co the greatest age attained
or considered, and L^ the number of age x last birthday, it being assumed
that this number may, on the average, be regarded as of age « + J.
Omitting the J, this last expression really gives the correct mean age last
birthday. The mean age next birthday, x^ of a population under the age
n is
rQnQ\ ^ - ' ^^n-i+ (w - 1) L„_2 + + Lq
y'^'J'') •'■n — r I r I IT
^n-l -r -L'ji-2 + + -^0
From this formula it is evident that, with a table giving the number
at and above each integral age, aU that is requisite to obtain the mean age
next birthday is to divide the total population into the sum of the num-
bers from the youngest to the oldest ages. Deducting ^ gives the usual
approximation to the mean exact age, while a deduction of unity gives
the mean age last birthday.
The mean age in years of the normal or standard population is, for
1901 :—
Males. Females. Persons.
26.934 years. 27.341. years. 27.148. years.
This mean age is, of course, not what is known actuarially as the
expectation of life at age 0, but is the average age of aU persons hving at
a given moment, or, in other words, it is the average past Hfetime of the
population at a given moment. On the other hand, the expectation of
hfe at age is the average future hfetime of all persons born. In the
case of a stationary population, however, with rates of mortaUty varying
with age, but remaining constant for each age through a great length of
time, the average past hfetime of the population at a given moment is
equal to its average future hfetime, that is, the average age of the popula-
tion is equal to the average " expectation of life" of the population as a
whole.^ Thus for the population of Europe in 1901 persons had Uved on
^ The expectation of life e° of the Ix dx persons of the exact age x is the future
lifetime T ^ of these, divided by their number, that is —
"■l = jx^dx/lx = Tx/ h
and consequently the total future lifetime of these Ix dx persons is
ex Ix dx - Ix dx Tx / Ix = Tx dx
Hence the total fvitiire lifetime of the whole existing population between and
w is
r e°. Ix dx = r Tx dx
aud as a whole existing population is J Tx dx, the average future lifetime or expecta-
tation of life of the whole existing population is J Tx dx / j Ix dx, which may be
shewn to be equivalent to J xlx dx/ j^ Ix dx, or the mean age.
CONSPECTUS OF POPULATION CHARACTERS. 107
the average about 27 years. The expectation of hfe changes with the
lapse of time, and is appreciably lengthening. Thus the secular change
of the norm will be the weighted average of the changes of the constituent
populations.
12. Population norm as a function of age. — ^The number of persons,
Y, at and above the age x may be closely represented by
(310). . . . y=;fca«'|8»' = 52674 (0.99961 )i'"8o8' (O.lsggS)!"*"*
which is a development of the Gompertz-Makeham type of formula. The
constants indicated fit very closely the values of the norm given in pre-
ceding table.' This matter will be dealt with more fully hereinafter.
^ For solution, vide op. oit. pp. 364-7.
IX.— POPULATION m THE AGGREGATE, AND ITS DISTRIBUTION
ACCORDING TO SEX AND AGE.
1. A Census and its results. — ^A well-conducted Census furnishes
results which are substantially correct so far as the aggregate number of
persons and the aggregate number of each sex is concerned. That is, if
p, m and/ denote the errors of the numbers of persons, males and females
respectively, and P, M and F their respective aggregates, thenp/P, m/M
and f/F are all extremely small quantities, which can have no important
bearing upon the general theory, or upon any deductions flowing from it.
Unfortunately this is not true regarding the numbers of either sex between
given age-limits.
In Chapter X of the Census Report, it has been shewn that for Aus-
tralia the Census results bear intrinsic evidence of great improvement
in regard to accuracy of statement respecting age ; see §§3 and 5. The
nature of this is shewn in the tables given of numbers and
percentages for the ages 28, 29 . . 32, and 48, 49 ... . 52. The exces
sive statements, for example, for the ages 30 and 50, became markedly
less. The results were as follow : —
Census
Age.
1891.
1901.
1911. 1911(adjusted).
Percentage of age- \
,
quinquennium in- [
30
23.35
22.98
20.90 19.96
eluding two years
50
29.06
25.77
21.75 20.16
on either side
A glance at Figs. 37 and 38 hereinafter will shew that the curves of
numbers according to age for ages 30 and 50 do not depart very much
from a straight line. For the former age the curves are concave upward ;
for the latter, convex upward. Hence at 30 the mean should be somewhat
less, and at 50 somewhat more, than 20 per cent. The ratio determined
from the smoothed results are shewn in the final column. We shall
consider the question of smoothing the results later.
For each it is seen that the numbers for the ages in question were
excessive, enormously so for 50 years of age, in the 1891 Census. The
error, however, was diminished for the Census of 1911, probably largely in
POPULATION AGGREGATES AND SEX DISTRIBUTION. 109
consequence of a special attempt to ensure the population appreciating
the necessity for accuracy.^ It may he said, however, that statements of
age leave much to be desired.
2. Causes of misstatement of age. — ^Many people are so indifferent
as regards their age that they are really unaware what it is, and for this
reason tend to assign round numbers (viz., ages ending with the figure
or the figure 5), as roughly expressing about their ages. In the case of
persons approaching 21 years of age, what may be called " matrimonial
reasons" exist for an overstatement, and this may continue to operate for
a year or two. In the case of females the tendency to overstate the age
is, on the whole, negative for a considerable period of life.^ For the older
ages, however, there is probably a distinct tendency in the opposite
direction.*
3. Theory of error of statement of age. — ^Assuming both a tendency
to express in round numbers ending in and 5, an age not accurately
known, and also particularly in the case of females some tendency to under-
state age, except for ages above, say, 60, we ought in general to find the
following characters in the crude results of a Census, viz. : —
(i.) In smoothing the crude results so as to conform to the general
trend, the results for ages ending in have to be considerably reduced ;
while those ending in 5 have to be reduced a somewhat smaller amount.
(ii.) The amounts of the corrections for ages above and below the
round numbers on the whole shew some asymmetry, though at the same
time, owing to the masking effect operating in ages so close as a; + and
X -\- 5, this character is not definite.
(ui.) The curves for males and females exhibit systematic differ-
ences of form due to systematic misstatement.
Figs. 37 and 38 shew the graphs of the numbers for each year from
to 100, for the Australian Commonwealth. It will be seen from these
that, for a population profoundly affected by migration, no systematic
difference of form actually exists of sufficient magnitude to unmistak-
ably indicate systematic misstatement of age. The marked tendency
to give ages ending with the figure is, however, very evident, so also
that to give ages ending with the figure 5 is also fairly clear.
1 Where the official admimstration of a commnnity is sufficiently systematic to
reqtiire every one to keep a card of identification, it is easy to get correct answers
to this and similar questions. The public appreciation of the importance of correct
answers is regrettably deficient.
' For matrimonial and economic reasons, and even reasons not entirely dis-
associated with personal vanity ; the two latter reasons also operate in the case of
males, but to an appreciably lesser extent.
' Certain investigations shew that vanity concerning longevity is not whoUy
absent in either sex.
110
APPENDIX A.
AUSTRALIA, 1911.
Fig. 37.
Coxumencing points of age-groups of one year at i
AUSTRALIA, 1911.
i indicated.
50
.0
30
20
10
^
FEM
AT.F.S
i
■s
\
N
V
\
\
1
V
\
^
\
O
v
tn
\
S
s,
\
_
B
a so i(
Fig. 38.
Commencing points of age-groups of one year at age indicated.
POPULATION AGGREGATES AND SEX DISTRIBUTION.
Ill
The curves in Figs. 37 and 38 are interpreted in the following way,
viz. : — ^The ordinate or vertical distance to the curve at any point repre-
sents in thousands the number of males (or females) in the age-group of
one year, commencing at the age in question. The zig-zag line denotes
the results furnished immediately by the Census, and the curve the
smoothed (and more probably correct) results.
4. Characteristics of accidental misstatements, and their fluctua-
tions. — ^The Censuses of the various States of Australia never having
been combined, it was desirable to compile the three preceding Censuses,
viz., those for 1881, 1891, 1901, in order to deal thoroughly with that of
1911. The results were not in age-groups for single years for 1881, but
were for the later Censuses. In doing this it was found on inspecting
the graphs for 1891, 1901 and 1911, of the numbers enumerated for each
age, that in the statements of age there were tendencies to concentrate
on certain ages, and to avoid, certain others. In order to definitely
examine these tendencies a tabulation was made of the data in respect of
the unit figure in the year of age stated in Australia at the Censuses of
1891, 1901 and 1911. To enable an estimate to be made of the degree of
error involved in these statements of age, the smoothed results were
similarly tabulated according to the unit figure in the year of age, and the
ratio of the former set of results to the latter was obtained for each sex
and each unit figure. The results should, of course, be unity if the
errors balanced, or had no tendency in any direction.
«
The ratios so obtained are as follows : —
Table XII. — ^Ratio of Number Recorded to Adjusted Number, Censuses 1891,
1901, 1911, Australia.
Year
Unit Figure in Age Last Birthday —
OF
[Census
1
2
3 4
5 6
i
7
8 ; 9
1891
1901
1911
MALES.
1.1388
.9167
1.0088
.9545
.9969
1.0366
1.0207
.9513
1.1044
.9369
1.0072
.9677
.9809
1.0343
1.0134
.9636
1.0485
.9956
.9944
.9787
.9990
1.0085
1.0097
.9691
1.0055; .9532
1.0144J .9667
1.01911 .9695
FEMALES.
1891
1901
1911
1.1251
1.0926
1.0367
.9288
.9270
.9895
.9978
1.0039
.9935
.9848
.9861
.9895
.9943
.9979
1.0056
1.0077
1.0106
1.0050
1.0117
1.0128
1.0066
.9640
.9708
.9770
1.0125
1.0165
1.0148
.9558
.9738
.9760
112
APPENDIX A.
The outstanding indications furnished by this table are for both
sexes
(i.) A marked tendency to concentrate on ages ending in 0.
(ii.) A less marked but persistent tendency to concentrate on ages
ending in 5, 6 and 8.
(iii.) A marked tendency to avoid ages ending in 1, 3, 7 and 9.
(iv.) A tendency to state ages ending in 2 and 4 with fair accuracy,
concentrations and avoidances being in evidence, but relatively
small in respect of these ages.
The table also furnishes an indication of the increasing accuracy
of statement of age at successive Censuses, the excess at ages ending in
having fallen from 13.88 per cent, in 1891, to 4.85 per cent, in 1911, iu
the case of males, and from 12.51 per cent, in 1891, to 3.67 per cent, in
1911, in the case of females.
Another interesting feature of the results is the evidence furnished
that inaccuracy of statement is more marked amongst mules than amongst
females. Thus, for the Census of
1891 the mean deviation from unity
(irrespective of sign) of the above
ratios was .0438 for males, as against
.0332 for females. The correspond-
ing figures in 1901 were .0358 for
males, as against .0281 for females,
and in 1911 they were .0181 for males,
as against .0143 for females.
ENGLAND AND WALES, 1911.
1
7
6
5
N
-
s
">«
\
K
3
2
1
(
\
\
\
L
\
»i
\
^^
u
3 23 30 4
) 5
a CO 7
8
90
for " persons"
at ages 50 and
; see Fig. 39.
Another remarkable feature, worthy
of attention, in the population-graphs
for Australia, as compared with those
of England and Wales, is the similarity
Fig. 39. of the features for ages 37, 38, 39, and
40, viz., in the graphs for " males" and
for " females" of the former, with that
for the latter country. There is also some similarity
60, due to excessive numbers for the ages ending in
5. Characteristics of systematic misstatement. — ^It having been
ascertained that in some cases the ages given in the Census cards were not
correct, notwithstanding the exphcit directions, persons who made mis-
statements were invited to send in corrections. Out of over 7000 re-
ceived, 1660, containing definite information as to the age given and the
amount of misstatement of age in the case of females, were tabulated in
age-groups, and according to the number of years the age had been mis-
stated. Of these, one-half (830) were for the State of Victoria, and the
balance of 830 for the State of New South Wales. The tabidated results
were as follows : —
POPULATION AGGREGATES AND SEX DISTRIBUTION
113
Table XIII. — Analysis of 1660 Cases of Misstatements of Age at Census of
1911, Australia.
CoKUECT Age.
No. PER 1000.
cokrbctios is
Years.
Un-
der
20
21
to
2-,
26
to
30
31
to
35
36
to
40
41
t.1
45
46
to
50
51
tio
55
56
to
60
61
to
70
Ov-
er
70
Total.
/o
Crude.
Smooth-
ed.
ca
a
Over
5 .
4 .
3 .
2 .
I 1 .
5
2
1
3
5
4
5
I
3
6
1
2
1
8
1
1
1
4
5
1
1
1
3
2
1
2
1
5
1
1
1
1
4
2
2
1
1
1
2
1
— ■
7
4
7
17
14
40
79
45
79
191
157
449
19
64
96
146
226
449
Total
Smootlied
20
5
10
17
12
18
1!
6
11
11
8
8
6
f.
4
3
3
1 ' —
2 ; 1
89
89
1000
-53.6
1000
S
■a
B
, 1 .
f 2 .
3 .
4 .
5 .
6 .
I:
11-15
Over
15 '.'.
10
2
1
55
21
4
80
62
36
18
8
5
1
1-
56
62
48
26
30
13
7
6
2
6
1
72
87
45
49
23
21
10
7
4
13
2
49
48
37
23
26
20
8
8
4
16
2
2
41
54
27
19
23
13
9
8
1
9
5
02
22
17
11
9
11
4
3
5
10
3
9
5
11
9
3
6
1
4
2
3
1
1
6
9
5
9
4
3
1
2
6
1
1
1
3
1
1
401
372
231
165
126
95
42
37
20
63
14
5
255
237
147
105
80
60
27
24
13
40
9
3
255
193 + 37*
145
107
79
58
41
27
18
11 + 25*
3
1
Total
Smoothed
13
13
80
77
211
168
257
284
333
337
243
268
209
189
117
120
55
64
46
44
7
7
1571
1571
1000
= 946.4
1000
Grand Total
Smoothed
33
18
90
94
223
186
269
298
339
348
254
276
217
195
123
124
58
67
47
46
7
8
1660
1660
= 1000
* The abnormality is about 37 in the one case, and 25 in the other. The 193 and 11 \(Ould
be the normal values in a total of 1000 — 37 — 25 = 938.
35 40
True Age of Females.
Belative frequency of Overstatements (A) and Understatements (B) of age with females
according to true age.
Kg. 40.
In the above table, the results of which are shewn in Figs. 40.. 41 and
42, the " smoothed" figures for the aggregate number of overstatements
according to age probably very closely represent the tendency in general :
the results, however, for under 20 years of age appear to be unduly large.
The smoothed results for the aggregate of understatements according to
age indicate the probable tendency in general. The smoothed result for
the total number of misstatements (over and under) according to age are
merely the sum of the preceding. The crosses, squares and circles
114
APPENDIX A.
represent the age-group aggregates for overstatements, understatements
and total misstatements, respectively. These results are shewn re-
spectively by curves A, B and C in Fig. 40.
The smoothed results of the aggregate number of overstatements according
to the amount of overstatement (see the vertical column at the right hand
side of the table) probably represent the distribution, but the aggregate
89 is so small that it can be regarded only as a rough indication. The
graph of this is curve A of Mg. 41.
AUSTRALIAN CENSUS, 1911.
S c3
So
la
\
ei
\
\
A
B
\
\
s
\
\
■^v
\
N
(
)
a?
si <
i« >
^ a a
ho 3
o c- O
80
3 4 6 6 7 a 9 10 11 13 13 14 16 16 17
Misstatement of age in years.
Curve A denotes overstatement ; cui.'ve B denotes understatement.
Fig. 41.
The smoothed result of the number of understatements according to
the amount of understatement, is probably represented by the final column
in the table. In this, however, the abnormality of understatements of
2 and 10 years is very striking. The graph is curve B of Fig. 41, and the
abnormal position for 2 and 10 year understatements is shewn by the
small squares with circles surrounding. This abnormality is probably on
the whole real ; that is to say, misstatements of 2 and 10 years had a real
predominance over the number which might have been expected according
to a probable law of frequency based upon misstatements of other amounts
(say, a frequency varying inversely as some power of the magnitude of the
misstatement).'^ At the same time it is also possible that in part it repre-
sents defects in the allegation as the amount of misstatement.
6. Distribution of misstatement according to amount and age of
persons. — ^By forming a series of 10-year groups from Table XIII., with
the central ages 20, 25, 30, etc. (completed years), and plotting these as
ordinates, some idea is obtained of the form of the function representing
the relative frequency of misstatement according to both age and magni-
tude of misstatement. Curves are then drawn among these positions,
the results shewn on Fig. 42 being thus obtained. The families of curves
are obviously fairly regular, and are skew. The positions of the ordinate -
terminals, obtained as described, are shewn in the following way. The
1 In a Census the frequency is for integral amounts of misstatement only.
POPULATION AGGREGATES AND SEX DISTRIBUTION.
115
^-^^
character of the mark denoting the terminal of the ordinate for a mis-
statement of 1 year is a dot ; for 2 years a vertical cross ; 3, a square ;
4, a slanting cross ; 5, a circle and vertical line ; 6, a lozenge ; 7, a circle
and horizontal line ; and 8, a slanting cross. After the age 55 the results
are rather irregular.
The broken lines for understatements of 2 years and 10 years shew
what may be regarded as the "normal" positions. That is, had there
been no peculiar predominance in the adoption of ages differing by these
amounts from the true age, the frequency curve would have been found
in about the position of these broken lines. They are numbered with
light-faced figures.
The frequency of misstatement according to age, as indicated
in Table XIII. and Fig. 42, refers to the number actually existing in
the age-groups, for which Table 18 of Part I. of the Australian
Census may be consulted
(pp. 32-33). To ascertain
the frequency for equal
numbers of females a cor-
rection is necessary, viz.,
division of each result by
the number in the age-
group to which it refers.
Although over 7000
acknowledgments of mis-
statements of age were
received, mostly from
women, the proportion
these bore to the aggregate
number of misstatements
was not ascertainable, and
after a study of other
errors revealed by the zig-
zag character of the
enumerated age-groups, it
was decided to regard
the characteristic misstate-
ment as sensibly negligible.
The absolute scale of the
frequency is not known,
since the total number of
misstatements could not
be inferred. Neverthe-
less its form is important
as throwing light upon the relative frequency of misstatements of
different amounts by women of different ages. The result may be
summed up as follows : —
40 50
Correct Ages.
The figures on the curves denote the amount of misstate-
ments in years.
Fig. 42.
116 APPENDIX A.
The analysis of acknowledged misstatements shewn in the table
gives the following indications (of course for females only) : —
(i.) Understatement of age constitutes 94.64 per cent., and over-
statement 5.36 per cent, of the aggregate cases of misstate-
ment,
(ii.) Excepting in the case of understatements of 2 years and 10
years, which are evidently abnormal, the frequency of mis-
statement diminishes with the number of years misstated,
at first very rapidly and later more slowly.
(iii.) The greatest frequency of understatement of all amounts
corresponds to the age of about 37J years.
(iv.) The age corresponding to the greatest frequency of understate-
ment of a given number of years increases with the amount of
understatement approximately in the ratio of about 1 J years
for every year of understatement, except in the case of 2 and
10 years.
(v.) The frequency of understatements of 2 years is about 1.2 times
that which would accord with the general tendency to under-
statement ; and the maximum is for the age of about 35 years .
(vi.) The frequency of understatement of 10 years is about 3.3 times
that which would accord with the general tendency to under-
statement ; and its maximum is for the age of about 30 years .
While these indications, being based upon only 1660 investigated
cases, have limited validity, they are probably substantially correct.
An insufficient number of returns were received from males to draw
any deductions as to the frequency of misstatement according to age and
amount of misstatement.
For cmrection purposes misstatements regarding age are best tabu-
lated according to the age declared ; on the other hand, for the expression
of the measurement of misstatement they are better tabulated according
to the true age. Since probably by far the greater number of persons give
their age correctly, it is probably desirable to regard the curves for over-
statement and understatement as discontinuous at the value zero.
7. The smoothing of enumerated populations in age-groups. — ^The
generalities of smoothing have been partially dealt with in VII., herein-
before ; see particularly §§ 1-9. Figs. 37 and 38 shew the graphs of the
enumeration in age-groups of the Australian Census of 1911 ; obviously
these are not the true results. It is obvious that the " smoothed" curve
must be of higher accuracy than the zig-zag results, since there are strong
reasons for believing that the numbers are sufficiently large to give a
" smooth curve." The following principles may be taken as a guide in
smoothing : —
(i.) Any smoothed curve so drawn as to equalise the zig-zag results
(doubtless) better represents the facts than the original data.
POPULATION AGGREGATES AND SEX DISTRIBUTION. 117
(ii.) The drawing of the smoothed curve can be assisted by arith-
metical and algebraic devices.
(iii.) The adoption of a particular position for the smoothed curve
must be governed not only by mathematical considerations,
but by the probabilities of each particular case.
(iv.) If arithmetic or algebraic methods are employed, they should
be such as do not involve systematic error.
(v.) The accumulations of error at all ages should be as small as
possible, and therefore should frequently change in sign, and
the grand total should be approximately (or exactly) the
enumerated total. ^
The method of smoothing by drawing a curve fulfilling the con-
ditions indicated is known as the graphic method. Before considering
it further, we shall examine the essential character of smoothing by
grouping, and the limitations of smoothing by grouping methods. First,
we consider the error introduced by mere means of aggregates.
8. The error of linear grouping. — ^If a series of points lie on a curve
say, convex upwards, their mean, weighted or otherwise, will obviously
lie below the curve, that is, x'^, y'^, denoting the mean of the co-ordin-
ates, and w the weight assigned to any point, the point having these
co-ordinates, viz. : —
will, in the case supposed, be below the curve. If the original points lie
on a straight line, the point wUl, of course, be on that line. Graphically,
the point may be determined for equal weights thus : —
Let P, Q, R, S, etc., be any points : the point midway between
P and Q is the mean of P, Q ; the point one-third of the distance of this
mean from R, towards R, is the mean of P,Q,R ; and, similarly, that
one-fourth of the distance of this last toward S, is the mean of P,Q,R,S ;
and, in general, the mean of n points is 1/wth of the distance of the mean of
{n — 1) points towards the wth point.
It follows from this that when n values are taken of any quantities,
which, being'graphed, are found to lie, not upon a straight, but upon a
curved line, then the mean of the independent variable (or argument)
does not correspond to the mean of the dependent variable (or value of
the function) unless the points representing them are all symmetrically
situated about the middle point. Thus, if we have the numbers in a
population at, say, ages 50 to 55, the mean does not correspond to the age
52. We proceed to consider the magnitude of the systematic error
involved.
1 Exact correspondence is neither essential nor extremely desirable, but as it
is easy to secure, there is no reason why it should not be insisted upon. A simple
way of securing it is to multiply each group-result by a correcting factor, viz., in
VII., § 7, herein.
118 APPENDIX A.
If we suppose the results to be representable by the equation
y= A + Bx-{-Cx^-}- etc., a,nd take points on either side of the middle
so that the correct value of «/ is A, we readily derive the following ex-
pressions shewing the errors of ternary, quinary, and larger groupings: —
(312).... ^IJy=A+ f Gk^+ I Ek^ + etc. (ternary).
(313) -^ Sy =A-{- 2 Gk^+ &% Ek^ + etc. (quinary).
(314) ^ Zy =A+ i Gk^+ 28 Ek^ + etc. (septenary)
(315). ... ^2y=A + 6^Gk^ + 18^ Ek* + etc. (nonary).
(316). ... ^Zy =A + lOGk^ + 178 Ek* + etc. (undecenary).
If the number of terms in the groups be denoted by n, the law of
increase in the numerical coefficients, y say, of G and e of ^ is as shewn
hereunder : —
(317) yG = ^(n^-l)G.
(318)..e^=.[-l(n-l)+i(«-l)2+l(,.-l)3+i(«-l)4ii7
The latter may be put in the more concise form in (319) hereunder.
Hence the error of a simple mean is shewn in the most general form by the
following expression, viz. : —
(319).. -^^=^+1 (wa-l)(7F+ -L .[(w^- 1) (3^2 _ 71 .BA* + etc.
The values of Gk^, Ek*, etc., can be very readily expressed in terms
of the ordinates to say the roughly smoothed curve. Thus, using accents
to denote ordinates symmetrically situated on either side of the middle
(unaccented) ordinate, we have —
(320) J [7], -27] + 7j') = Gk^+ Ek*+ etc.
(321) -g {v„+V,-^+v'+i')=^Gf^^+^ I Ek*+ etc.
We may therefore from the above equations obtain the value of y, free
from the systematic error due to curvature. Thus
(322) 2/0= ^ {■Sy-iv, -^ + v')}
and from (313) and (321)
(323) yo = j {Ey -(,,„ + ^ - 4^, +,,' +n")]
for ternary and quinary groupings respectively. These correction-terms
in the inner brackets are, as a rule, very small.^
1 To reduce the arithmetical work any one number may be taken from each of
the values of -q.
POPULATION AGGREGATES AND SEX DISTRIBUTION.
119
The repeated application of any system of grouping leads to more
highly smoothed results, but is unobjectionable only it freed from syste-
matic error. It, however, even then, never wholly removes the vitiating
influence of a value which is seriously defective or excessive.
It is easy to build up from the preceding formulae a system of
coefficients by means of which the repeated groupings can be performed in
one operation. Thus, each ordinate being assumed to have equal weight,
we have for repetitions of ternary groupings —
Table XIV.— Coefficients for Repeated Grouping.
No. of
Repeti-
tions.
Factor.
Resiolting
Grouping.
Weights toibe Applied to Co-ordinates.
1
3
Ternary
1 1 1
1
1
9
Quinary
12 3 2 1
2
1
27
Septenary
13 6 7 6 3 1
3
1
81
Nonary
1 4 10 16 19 16 10 4 1
4
1
243
Undecenary
1 5 15 30 45 51 45 30 15 6 1
The scheme of deriving these is evident.^ In the same way it is
necessary to buUd up also the scheme of corrections from (314), (316), etc.
9. Graphic process of eliminating systematic error. — A simple
approximate method of graphically eliminating the systematic error
indicated in the preceding section is based on the fact that the distance k
between the mean of a series of n ordinates on a parabolic curve and the
vertex of the curve is given in Table XV hereunder.
Table XV. — ^Position of Mean of n Points.-
Number {n) of points on curve . . n = 3
Proportional distance of mean of j k =
the ordinates from centre of j
chord towards vertex of curve I , _
4
4
J^ -Q^ -2^ 15'
9
7
11
9'' T%^ T^
.ZZh Aih .50h .53h .55h .58h .60h
the height h being the distance from the middle of the chord to the vertex.
Thus, if a series of means of n ordinates are plotted, and a curve be drawn
through them, this series can be taken to give an approximate guide to
1 Thus, 1.2.3
1
120 APPENDIX A.
the shape of the true curve. A section of double the stretch being then
taken^ the interval between the chord and curve along the ordinate is
assumed to be four times the similar distance for the central ordinate
of the original stretch. Hence in this case the points defined by the
means should be moved the following amounts, viz., those in Table XVT.
Table XVI. — Distance oJ Vertex from Mean oJ n Points.
Number of ordinates for
which a mean is taken n = 3 4 5 6 7 9 11
Proportion of vertex-dis- lr._]_Tj^Tj^TT '17 ^w ^ jr ^ tj
tance of the doiMe I*" 6 36 8 60 36 48 lO
stretch to be taken as 1
a correction ( k = .167H .139H .125H .117H .111 H .104H .lOOH
H denoting the height of the vertex above the chord double stretch.
This correction will eliminate the greater part of the systematic error,
but not the whole, inasmuch as the curve has been flattened by taking
the series of means : hence the corrections having been applied to the
mean points a new curve may be drawn, and the process repeated if
necessary. A smooth curve is then drawn among the points ultimately
defined.
This process, however, yields resultswhich, after aU, are but little better
than a direct attempt to draw a smooth curve among the points given by
the ordinate- terminals ; it is tedious, and its probabihtyis but little greater
than that obtained by directly drawing the smoothed curve and correcting
it by arithmetical (or algebraic) methods (" hand polishing"). To avoid
its tedium of drawing and hand-polishing, what are called summation
methods have been used. In these a weighted mean is obtained, the weight
factors having opposite signs in order to eliminate the systematic error
indicated in formulae (312) to (316).
10. Summation methods. — Summation methods in so far as they
are rigorous, eliminate the systematic error involved in weighted means
where the weights have no change of sign. Rigorously devised algorithms,
applied to a series of ordinates strictly conforming to a curve of the wth
degree, will reconstitute the given ordinates, whereas mere means of
a series of ordinates wiU not only not do so, but wUl increase the error
with every repetition of the grouping. The taking of the means of a
series of ordinates is therefore vahd only where the general trend is either
linear, or so nearly linear as to make the corrections referred to negligible.
Suppose, then, we have a series of ordinates, the terminals of which
0, P, Q, R . . . . Z, are to be smoothed. Evidently we can draw an
1 That is, if w -|- 1 be the nvunber of ordinates, a curve defined by 2 w -f- 1
ordinates is taken ; thus, if 3 points are originally taken, the curve of double stretch
will be that defined by five points.
POPULATION AGGREGATES AND SEX DISTRIBUTION. 121
integral curve of the nth degree through any n-\-l such points. Geo-
metrically, the summation smoothing process is the following :^Draw a
curve of the nth degree through the points 0, O+i; 0+2i; 0+m :
a similar curve through the points P P+wi : a third through the
points Q, Q+ni ; and so on.^
This will give a series of curves of the nth degree, usually close to one
another, and sometimes intersecting. The mean position of their inter-
sections on the ordinates (or ordinates produced) is the smoothed curve
required. The flexibility, or fitting power, of the curve depends, other
things being equal (a) on the degree of the curve ; and (6) on the nearness
of the points 0, 0+i, etc.; and consequently of P, P-fi, etc., to each
other.
It may readily be demonstrated, graphically or otherwise, that as
the value of i is increased, minor fluctuations are more and more obliterated.
The whole range being limited, the larger the value of n the more points
on the curve are fitted by one stretch : hence the smaller i will be ; and
the fitting power will consequently be increased.
Since the mean position of the intersection of the curves and the
ordinates defines the position of their terminals at the smoothed curve ;
and since each point O, P, etc., is the start of one of the component curves,
any abnormality in its position (i.e., deviation from the general trend)
is reflected in the mean result ; that is, it produces a deviation of a smaller
amount in the direction of the abnormal point.
The defect of all summation methods is seen, from their geometrical
representation, to be the following : —
(i.) The degree of obliteration of minor fluctuations is quite arbitrary
and depends upon the character of the summation-system.
(ii.) The result is vitiated by all abnormalities: the method, in fact,
does not lead to real smoothing, but to the reduction of the
magnitude of the oscillations of the curve.
This may be shewn analytically in the following way. We observe
first that if there are q-\-\ points in the total range of q intervals of any
component curve taken, then in a complete'^ series there wiU be g+l
intersection-points on the ordinates. The mean of these is to be taken.
The first complete term arranged according to the powers of the common
distance (k) between the ordinates, and the second term wiU be re-
spectively : —
(324) y^ = -^ {K +«a-i + - •«o) + (^-i+268-2+36«-3 + - OA;
+ (c^-2+2%g-3+3^g-4 + -)^^+K-S+2"(^,-4+3^<^g-5 + -)fc»+-}
^ Where h is the common interval on the axis of abscissae between ordinates,
the comjmon interval i between the points wiU always be an integral multiple of
Ic greater than 1 ; that is i = 2k, or 3fc, or 4k, etc.
^ It is, of course, not essential that the series should be what has been called
here complete, and in Woolliouse's method it was not complete. A complete series
may be defined as one where, q + I being the number of points including the terminal
ones ranged over by any curve, the initial point of the {q + l)th curve is on the
same ordinate as the final point of the initial range, viz., the zero (or first) curve.
122 APPENDIX A.
(325) 2/,+i = ^ <(a,+i + ..a,)+(l>,+ --)k+(Ct-i + ---)k^+
Thus the coefficients of the powers of x are changing every term, and con-
sequently the equation of the smoothed curve of, say, s+ 1 points will be
of the degree s, that is, it has no relationship whatever to the degree of
the originating equations of the wth degree passed through the points
; +i; +2i, etc.
It is thus seen that results of a " smoothing" by " summation"
methods are in principle toto coelo different from those obtained by
methods which ensure conformity to some function adopted for considera-
tions of the nature of the case.^
Numerous papers on the summation method have appeared from
time to time in the Journal of the Institute of Actuaries by various
investigators, of whom the principal are the following : — J. A. Higham,
W. S. B. Woolhouse, G. F. Hardy, J. Spencer, T. G. Ackland, G. J.
Lidstone, G. King, R. Todhunter. Some of these have contributed
several papers on the subject. A specially valuable one, on " The
rationale of formula for graduation by summation," by G. J. Lidstone,
appeared in the Journal of the Institute of Actuaries, Vol. XLI., pp. 348
et seq., and XLII., pp. 106 et seq. An important paper on the subject
by Dr. J. Karup wUl also be found in the Transactions of the Second
Actuarial C!ongress, p. 31 et seq.
The subject of graduation of summation has also quite recently
been re-examined by Mr. C. H. Wickens,^ and formulse based on ranges of
three determined points (0, +», and +2i) and four determined points
{i.e., including also +3i) are discussed for the developments of quinary
formulse and formulse other than quinary, the adjective denoting the
number of spaces into which i is divided. That is, if i =rk then the
formula derived is an r-ary formula. It is shewn that there are great
advantages in making the series complete, and that in taking the mean
it is advantageous to allow only haK-weight to the terminal points of
intersection on any ordinate.*
The following weights (Table XVII.) have been deduced by Mr.
Wickens for the different ordinates about the middle ordinates, th
1 Prof. Karl Peajson's scheme, adopted by many biometrioians, is to resolve
the data under a suitable type-form derived from a generalised theory of probability,
certain criteria being used to decide which form should be preferred. A single
Pearsonian curve, however, will not apply to population-enumerations, although
the population-curve may be empirically considered to be a combination either of
Pearsonian or of other curves.
2 An extension of the principle underlying Woolhouse's method of graduation,
read 30th October, 1911, Trans. Act. Soc, N.S.W., Session 1912, pp. 243-7.
' There are many physical analogies for this process. For example, if a
physical property be measured at equidistant points along a line including the
terminals the mean value is {a + 2b + 2c -\- ... -}- 2y -\- z)/2N, where N is
the number of spaces into which the points divide the line.
POPULATION AGGREGATES AND SEX DISTRIBUTION.
123
marked (3) and (4) being deduced from curves passed through 3 points
and 4 points respectively. The similarity is obvious. Other formute
may be obtained from the paper in question.
Table XVn. — Summation-formnla-coefficients to be applied to a Series of rOrdinates
Deduced on the Basis ot (3), and on a Basis oi (4) Determined Points.
Binary
Ternary
Quarternary
Quinary
Senary
Ordinates
i=U
i=Zk
t=4J;
i=
= 5fc
i=6fc
9+1
(x/k)
(
3)
(
i) (3)
(4)
(3)
(4)
(3)
(4)
(3)
(4)
- 12
- 11
-5
-3f
- 10
. ,
6
6
-8
-7*
- 9
-2
— 1^
-9
-9
- 8
6
6
-3
— 2^
-8
-8f
- 7
-3
-2*
-3
-3^
-5
-6J
- 6
6
6
-4
-4
-2
-2t
— 6
-1
-f
-3
-U
29
25f
- 4
-1
-%
12
10§
56
53i
- 3 -
1
—
1
19
17i
23
22|
81
81
- 2
7
63
36
36
33
33?
104
106|
- 1
9
9 13
13^
51
52J
42
43^
125
128A
1
6
1
6 18
18
64
64
50
50
144
144
1
9
9 13
131,
51
m.
42
43*
125
128i
2
J 7
&i,
36
36
33
33f
104
106J
3 -
1
—
1
19
17*
23
22|
81
81
4
-1
-1*
12
lOf
56
53i
5
— 1
-t
-3
-'H
29
25f
6
—4
-4
-2
-2*
7
-3
-2*
-3
-3J
-5
-6#
8
-3
-2*
-8
-8f
9
-2
-1^
-9
-9
10
-8
-7*
11
-5
-3S
12
Sum of Co-
efficients 3
2
3
2 54
54
256
256
250
250
864
864
For the mode of obtaining the values given by these formulae by
processes of summation, reference should be made to the paper, in which
also the smoothing coefficient is given as follows : —
Table aviu. — Smoothing Coefficients.
Interval
No. of Terms or
Ordinates
Series (3)
•v/(7«» + l)/4«"
Series (4)
s
V7(««-l) {s^+5) + 36s/6s*
4s-l
2
7
.1683
.1683
3
11
.0741
.0615
4
15
.0415
.0316
5
19
.0265
.0193
6
23
.0184
.0130
7
27
.0135
.0094
8
31
.0103
.0071
9
35
.0082
.0056
10
39
.0066
.0045
124 APPENDIX A.
The smallness of the smoothing coefficient is a measure of the
efficiency in smoothing.^
11. Advantages of graphic smoothing over summation and other
methods. — ^This graphing of the group-results of an enumeration (numbers
according to years of age in the instance immediately under review)
yields a succession of rectangles, or, if we prefer, points denoting their
heights. Smoothing in such a case consists essentially in transferring
numbers of. those who alleged they were a given age to some other nearly
identical age, the reason for this transfer being that it is judged a priori
(and justly so) that the irregular distribution indicated by the data does
not accord with the real facts. To do this there is no better way than to
draw among the tops of the rectangles (or the points representing them) a
smoothed curve following every variation of their general trend, which,
in the judgment of the analyst,^ is regarded as probably conforming to the
facts. This can be done, and the result scaled and smoothed arith-
metically, that is, by differencing. The aggregates as by enumeration
and by the smoothed curve can be formed, and the accumulated differ-
ences examined to see that they are kept within probable limits ; that is,
are alternately positive and negative, and are never great (see VII., §12).
The initial curve can then be amended whenever improvement seems
possible ; thus in its final form the grand total can be made identical
with the enumeration, and the difference between the enumerated and
smoothed aggregates up to any value of the variable (age) can be made the
least possible for the form of curve deemed to he best on examining the graph
of the enumerated results.^
The logic of this process has been admirably expressed by Whewell,
and before him again by Sir John Herschel, in the following passages : —
' ' This curve once drawn must represent .... the law .... much
better than the individual raw observations can possibly .... do
The series of lines joining the consecutive points . . . cannot possibly repre-
sent reality If, however, we thus take the whole mass of the
facts .... by making the curve which expresses the supposed observations
regular and smooth .... we are put in possession .... of something
more true than any (one) fact by itself." — Sir J. Herschel, Trans. Astr. Soc,
Vol. v., pp. 1-4.
1 See G. F. Hardy, Journ. Inst. Act., Vol. xxxii., p. 376.
^ Any attempt to dispense with the element of judgment is really illusive.
The adoption, for example, of a summation method will yield appreciably different
results according to the range taken. Thus » real undulation in a population curve
may be virtually obliterated by the process.
' There is a tendency to forget that technical processes are but instruments
in the hands of the user, and formulae employed confer no validity to the elements
depending upon judgment.
POPULATION AGGREaATES AND SEX DISTRIBUTION. 125
" The peculiar efficacy of the Method of Curves depends upon this . . .
that order and regularity are more clearly recognised when thus exhibited to
the eye as a picture (and) not only enables us to obtain laws of Nature from
good observations, but .... from observations which are very imperfect,
.... We draw our main regular curve not through the points given by
.... observations, but among them." — ^WhewelU, Novum Organon Re-
novatum, Bk. III., Chap, vii., p. 204, 3rd Edit., 1858.
Finally, it may be remarked that by adopting the graphic method
of smoothing, minor and unmeaning fluctuations are avoided. The
invalidity of merely mechanically applying various summation formulae
has been shewn by G. J. Lidstone ; he has indicated how, by the summa-
tion method, unmeaning fluctuations are introduced into what may be
known a priori to be a straight line.^
12. Graphs of Australian population distributed according to age
and sex for various Censuses. — ^Adopting the principles indicated, the
graphs of the enumerated population of Australia for the Census of 1911
distributed according to age, shewed that, both for females and for males,
the adoption of any function to which the results should be conformed
was out of the question. It was evident also that a " summation method"
was quite unsuitable. In the results for 1911 there was a sharp increase
in the numbers for ages 13 to 18 ; then a zig-zag result up to age 22
before a decided decrease appeared. It was thus evident that results
must be examined, and the smoothing based upon considerations as to the
possibility of misstatement. The data therefore were simply graphically
smoothed by drawing first a freehand curve among them, the changes of
direction of this curve being made a minimum, so far as that was possible,
while following all fluctuations deemed to represent the actual facts.
This curve was then carefully drawn with the aid of splines, French
curves, etc., the ordinates' scaled off and adjusted arithmetically.* The
result of this smoothing is shewn on Figs. 37 and 38
As has been shewn in § 10 and formulae (324) and (325) hereinbefore,
this is obvious from either geometrical or analytical considerations.
For that reason the graphic process has been preferred to summation
processes, which latter are regarded as theoretically invalid for the reasons
indicated.*
1 See also T. B. Sprague, Journ. Inst. Act., Vol. XXX., pp. 161-3, 1892 ;
James Sorley, Journ. Inst. Act., Vol. XXII., pp. 309-340, in particular 3 : The
Graphical Method, pp. 321-8 ; T. B. Sprague's works on " The Graphic Method,
etc.," Journ. Inst. Act., Vol. XLI., p. 182.
' On the rationale of the Formulae for graduation by summation. Joiu-n.
Inst. Act., Vol. XLI., 1907, p. 360, and diagrams A, B and C.
^ Identical methods were also applied to the data of the earlier Censuses.
* In the summation methods, as we have seen, fluctuations are introduced
into curves in order to conform-to a convenient algorithm, rationally deduced. But
a little re flection will convince any mathematician that the minute oscillations in
the directions of the tangents, involved in the process, would be better eliminated,
when that can conveniently be done ; and in any case, in the presence of large
departures of individual results from the smoothed curve, these small fluctuations
have neither real significance nor validity.
126
APPENDIX A.
Graphs shewing the distribution according to sex and age have been
prepared for the Australian Census of 1881, that of 1891 and 1901, as well
as that of 1911. The results for 1881 were deduced from quinquennial
groups ; those for the latter Censuses from year-groups ; and they are
shewn on Figs. 43 and 44. It will be seen that intervals of ten years
cause considerable differences in the forms of the curves ; these differences
are due of course to migration and to fluctuations in the birth and
mortality rates.
Commencing points of age-groups of one year at age indicated.
Fig. 43.
The curves in Figs. 43 and 44 are interpreted in the following way,
viz. : — ^The ordinate or vertical distance to the curve at any point repre-
sents in thousands the number of males (or females) in the age-group of
one year commencing at the age in question.
POPULATION AGGREGATES AND SEX DISTRIBUTION.
127
.n
1
r
p*
\
FEM.
SiLES
iV
^
o^^
\/
§
\
A
■s
\
\
\
\
1
\
V
v
®
20
\
\
^
Pi
DD
\
k\
g
\
\
, \
\
10
^
XV
. \
N>
^
V.
~-^
^=^
5 iJo
Commencing points of age-groups of one year at age indicate>1.
Fig. 44.
13. Growth of population when rate is identical for all ages. A
population P(, increasing at the instantaneous rate p per unit of time
becomes, if that rate be constant, as we have seen, Pt=PQ e''* see II.,
§§ l-IO, formulae (1) to (14). Hence, it the numbers between the ages
X and x-\-dx for the epoch i = 0, are represented by P^f(x)dx, in which
case
(326) J"/ {x) dx=l
and the rate of increase be the same for all ages, then the numbers between
the ages x and a; + da; at any later date t, must be
(327) Ptdx = Po e"^ f (x) dx.
the aggregate being Pg e"' ; that is to say /(a;) remains constant. Hence,
if the age-groups be divided by the total population, the results will be
identical, i.e., the relative numbers will be seen to remain the same and
their graphs will be identical. If, however, the aggregate numbers,
denoted by F (x), are graphed, the graphs will not be identical. For we
have in the latter case
(328) Ft{x)dx = eftPo (x) dx ;
and by hypothesis p is not a function of x ; hence
(329) ^) =e.^^ ; or tan 9, = e^" tan 0„
128 APPENDIX A.
that is, the slopes of the tangents to the graph of the population are
increased in the proportion 1 : e''*. In the absence of all information of
" migration" and " natural increase" (increase by excess of births over
deaths) the rate of increase of the preceding period must be assumed to
continue not only for the population as a whole, but also for each age ;
which is expressed by
(330) Pe=Poe'"/;/(.T)^.r.
and (327) hereinbefore.
14. Growth of population where migration element is known. If
the ages and numbers of migrants be known, as well as the ages and
numbers of the dying, then it is possible to determine the numbers in
each age-group by remembering that survivors after t years have increased
their age by t years.
Except for very small communities, this method of estimating
populations according to age (and sex), is, however, perhaps impracticable.
We shall, however, later consider it. Here it may be noted that the
estimation may be most conveniently treated in single year age-groups,
i.e., not by infinitesimal methods. The value of the method is that it
would enable aU rates to be finally made up intercensally, whereas,
after a Census has rendered the intercensal adjustments possible, they
have always to be corrected.
15 . Growth of population when rate of increase varies from age to age.
Changes in the birth-rate ( = rate of immigration at age 0), in the death-
rate for various ages (= rate of emigration at age x), in the rapidity of
migration and age of migrants (= rate of immigration or of emigration
at age x) causes a change to take place in the form both of/ (a;) and F {x)
referred to in. the previous section. The graphs of / (a;), i.e., of relative
numbers, at different epochs all give an area of unity between the limits
and CO (= end of the longest life) ; hence the curves for different epochs
necessarily intersect ; those of F {x), i.e., of absolute numbers, give the
areas Ft, and may or may not intersect. We consider the consequence
of those variations which change the form oif{x) ; see Figs. 43 and 44.
Where we have to interpolate to obtain intercensal populations,
or to extrapolate to predict a population, we may assume that the tangents
to the curve foix) change uniformly with time ; that is, they become
those of fT(x) by a linear change with time, T denoting the intercensal
period. Thus
,oqiN dft (x) dfo jx) , t _ dfT(x) dfo(x)
that is, o- is the total change in the tangent in the intercensal period T.
Hence, given the total population at the time t, we can effect its dis-
tribution according to age by determining merely /j (x) on the supposition
indicated.
POPULATION AGGREGATES AND SEX DISTRIBUTION. 129
This supposition (i) is of a more general character than that of sup-
posing that the number at any age changes Hnearly : supposition (ii.).
Graphically, the difference between the two is that, according to supposi-
tion (i.), the intercept on any ordinate between the graphs of /^{x) and
/r (cb), divided in the ratio t/T, gives the position of /( (x), while according
to supposition (ii.) it is the intercept between Fg {x) and -Fjt(.t) which is
uniformly divided. The advantage of supposition (i.) is that only the
form of ft{x) is fixed ; the graph of Ft {x) can then be made to agree
with any intercensal estimate of population.^
16. The prediction of future population and its distribution. — ^The
graphs of population of various countries for the years 1790 to 1910,
Fig. 3 hereinbefore, discloses no general law. All shew what may be
called oscillatory development. The graph of the population of Aus-
tralia from 1788 to 1914 (see Official Year Book No. 8 of the Common-
wealth of Australia, p. 127) shews also this feature in a fairly well marked
degree, and those of the individual States exhibit more striking oscilla-
tions. Hence accurate predictions even of total population of any pre-
cision are not possible. Figs. 43 and 44 shew that accurate predictions
for age-groups are not only not possible, but may be even more misleading
than the assumption of an unchanged distribution according to sex and
age. It may be noted, however, that there is a general similarity, though
there is by no means identity, in the forms of the graphs for males and
females. The great fluctuation in the masculinity of the population
according to age is also evident from a comparison of the results shewn
on Figs. 43 and 44. This, however, wiU be discussed later.
^ See Census Report, Vol. I., Chap. IX., post-censal adjustment of population
estimates for the intercensal period 1901-11.
X.— THE MASCULINITY OF POPULATION.
1 . General. —The ratio between males and females in any population
has been called its masculinity, and the fluctuations of such a ratio are
obviously important. The following ratios of the aggregate number of
males to the aggregate number of females in various populations will
give an idea of how closely the number approximates to unity.
Table XIX. — Mascnlinity of Various Populations (about Yeai 1900).
Norway
1891
.932
Ireland
1901
.974
Australia . .
1901
1.101
Sweden
1895
.944
Italy
1901
.990
C. of G.Hope
1904
1.024
Scotland
1901
.946
United States
1900
1.044
India
1901
1.038
Eng. & Wales
1901
.954
Canada
1901
1.050
Ceylon
1901
1.140
Germiny . .
1900
.969
Newfoundl'd
1901
1.053
The results given hereinbefore, viz., in VIII., § 9, Table XI., shew
that even when the total numbers for all ages for males is made equal to
that for females, there are easily discerned differences between Eastern
and Western populations.
In the foUowing Table, viz., XX., the aggregate number of males in
the different age-groups in the first eleven countries are divided by the
aggregate number of females in the same age-groups, the results being
shewn on line W ; for the last three countries the similar quotients are
shewn on line E.
Table XX. — Change of Masculinity with Age ; Aggregate of Various Populations,
about 1900.
Countries.
1-4
5-9
10-14
15-19
20-24
25-29
30-34
35-39
40-44
45-49
W
E
1.024
1.003
1.016
.966
1.014
1.047
1.015
1.212
.992
1.073
.979
.919
.991
1.022
1.005
1.037
1.021
1.135
1.020
1.035
1.012
1.131
Countries.
50-54
55-59
60-64
65-69
70-74
75-79
1
80-84 85-89
90-34
95-100
All
Ages.
W
E
.988
1.005
.962
1.095
.934
.870
.927
.882
.906
.873
.895
.885
.847. .784
.873 .905
.674
.880
.588
.880
.9964
1.0390
MASCULINITY OP POPULATION. 131
The figures in the table shew the relatively large range of " mas-
culinity" for different age-groups, and indicate the desirableness of the
determination of a norm for purposes of comparison. We proceed to
consider this aspect of the question.
2. Norms of masculinity and femininity. — ^The variations with the
lapse of time, of the norm of distribution according to age for the male
population of any community, and the same norm for the female popula-
tion of the same community wUl not, in general, be identical. The pro-
gressive changes, which may have both periodic and aperiodic elements,
are best studied by observing the fluctuation of the masculinity or of the
femininity of the population. These characters as ordinarily defined
are the number of males to one female (or in practice usually to 100
females), and the number of females to one male, respectively. Thus if
m = the number of males, / the number of females, and p = m -f- / the
number of persons of any age, the masculinity fxi and femininity (f>i for
that age wiU be expressed by the formulse : —
(332) 1^1 = j; <^i =^
with suffixes to denote the age. When these quantities and their varia-
tions are known, the changes taking place in the relative numbers of the
sexes are determined as soon as the variations in the norm for the entire
population (persons) are ascertained ; see VIII., §§ 8 to 10. The curve
shewing the variations of the norms for both sexes at each age from epoch
to epoch is not an essential, for their fluctuation is determinable from the
fluctuation of the norm for persons, and the fluctuation of either the
masculinity or the femininity. For this purpose a somewhat different
definition of masculinity is desirable ; this we shall now consider.
3. Various definitions of masculinity and femininity. — ^For many
purposes definitions other than that mentioned above have advantages.
Both of the functions referred to for ordinary populations approximate
to unity. But other functions may be adopted which hover either
about I or about zero. For example, the ratio of males (or of females)
to" the whole population, is a quantity which ordinarily approximates to
I ; or yet again the ratio of the difference of the number of males and
females to the total population is a number which ordinarily approximates
to zero. Algebraically, the three methods and their interrelations are as
follows : —
Ist Method : —
771 1 1
(333).. Masculinity = /xi = 7 ; Femininity (^1= ^ = —
Possible range to -|- oo ; ordinary value about 1.
132
APPENDIX A.
2nd Method : =
(334). .Masculinity =11.2 =
m
m
7
Ml
Femininity ^^2
__A_-^
m , , l+Mi
m -\- f 1+01 1+Mi
Possible range to + 1 ; ordinary value about \.
3rd Method : =
(335) . . Masculinity = /is =
m
m-f _f
-1
.Ml
»»+ / w» , 1 Ml +1 '
/"^
, f — m m 01—1
Femininity = ^3 = j^:^ = ^-^ - ^^^f
Possible range — 1 to + 1 ; ordinary value about zero.
The mutual relations subsisting among these several quantities are
set out in the following table : —
Table XXI. — Relations subsisting between Masculinity and Femininity according
to Various Definitions.
Func-
Expressed in terms of —
tion.
Ml
M2
M3
01
02
03
Ml
Ml
Ml
M2
1 - M2
M2
2n2 -1
1 + M3
1 — Ms
Hl+Ms)
MS
1
01
1
1+01
1-01
1 + 01
92
1-02
1-202
1-03
1 +03
i(l-0s)
— 03
M2
M3
1 + Mi
Ml — 1
Ml +1
01
02
03
1
Ml
1
1 1
M2
1 — M2
1-2(12
1 — Ms
1 +M3
HI -Ms)
- Ms
01
01
1+01
01 -1
01 +1
02
1-02
02
202-1
1+03
1-03
i(l+03)
03
1 +Mi
1 — Ml
1 + Mi
4. Use of norms for persons and masculinity only.— Instead of
having three norms, viz., one each for males, females and persons, it will
often suffice to have one for persons, and one for masculinity. Thus in
the norm of population the masculinity, by method 3, viz. (wi— /)/(m+/)
is as follows for Europe (i.) and for India (ii.).
MASCULINITY OF POPULATION.
133
Table XXn. — Change of Masculinity with Age.
Age Group.
1
to
4
5
to
9
10
to
14
15
to
19
20
to
24
25
to
29
30
to
34
35
to
39
40
to
44
45
to
49
(i.) . .
(ii.) . .
+
.013
.018
+
.009
.037
+
.008
+
.004
+
.009
+
.078
.003
+
.018
.009
.063
.003
.008
+
.007
.002'
+
.013
+
.045
+
.012
.003
+
.008
+
.044
Age Grodp.
45
to
49
50
to
54
55
to
59
60
to
64
65
to
69
70
to
74
75
to
79
80
to
84
85
to
89
90
to
94
95
to
105
(i.) . .
(ii.) . .
+
.008
+
.044
.005
.017
.019
+
.023
.032
.088
.037
.090
.049
.090
.054
.085
.089
.077
.130
.090
.000
.077
.000
.143
5. Relation between masculinity at biith and general masculinity of
population. — It has been suggested that some tendency exists which,
while not very strongly expressing itself, is nevertheless sufficiently
evident to equate the numbers of the sexes in the population of any
country, or at least that the masculinity at birth is in some way affected
by the masculinity of the population. ^ Masculinity here denotes merely
the ratio of males to females, that is, M/F.
The population of Australia has enormously changed in its mas-
culinity in a few decades, and consequently affords an opportunity of
examining this supposition. The masculinity at birth is compared with
that of the population for the years 1829-1913, the latter passing through
a wide range of falling values. The results are shewn in the following
table : —
Table XXUI.-
-Average Masculinity of Population and of Births, New South Wales,
over Various Periods.
Average
for
Masoulimty
Period.
Average
for
Masculinity
Period.
Years.
of Popu-
of Live
Years.
of Popu-
of Live
lation.
Births.
lation.
Births.
1829-34
6
2.961
1.016
1840-49
10
1.625
1.034
1835-89
5
2.436
1.031
41-50
10
1.560
1.035
40-44 ..
5
1.752
1.026
42-51
10
1.510
1.036
45-49
5
1.498
1.038
43-52
10
1.412
1.036
50-54 . .
5
1.309
1.031
44^53
10
1.433
1.033
55-59
5
1.281
1.033
45-54
10
1.404
1.035
1830-39
10
2.680
1.026
46-55
10
1.375
1.032
31-40 . .
10
2.568
1.018
47-56
10
1.352
1.033
32-^1 ..
10
2.443
1.021
48-57
10
1.325
1.029
33-42 . .
10
2.314
1.020
49-58
10
1.308
1.032
34r-43
10
2.205
1.029
50-59
10
1.295
1.032
35-44 . .
10
2.094
1.028
60-69
10
1.233
1.058
36-45
10
1.979
1.028
70-79
10
1.196
1.045
37-46
10
1.877
1.026
80-89
10
1.209
1.050
38-47 . .
10
1.784
1.027
90-99
10
1.147
1.054
39-48
10
1.698
1.030
1900-13
13
1.186
1.058
1 Diising, Das Geschlechtverhaltniss inx Konigreich Preussen.
134
APPENDIX A.
This table seems to shew that, on the whole, the masculinity of
birth jLtj, can be expressed approximately by such an equation, for ex-
ample, as
(336). ...^„ = ^p = 1.06 - 0.0325 [fj.^ - 1) + 0.0333 (fj,^ - 1)^ ;
/Xp denoting the total number of males divided by the total number of
females in the population over the period considered. The tabulated
mean values of the masculinity of the population, and the position of the
curve which represents the formula, are shewn on Fig. 45. The result
may, of course, not be directly due to the masculinity of the population :
both may have varied through some condition itself varying with time.
Fig. 46 shews such a variation. This, too, implies an opposite pro-
gression ; that is, it indicates clearly that while the mascuUnity of the
population was, on the whole, diminishing, that of the birth was, on the
whole, increasing.
The results for Victoria point less decisively in the same direction.
They are as foUows : —
Table XXTV.
Period
1851-60
1861-70
1871-80
1881-90
1891-1900
Of Population
1.765
1.303
1.142
1.108
1.049
Of Births
1.046
1.047
1.044*
1.049
1.050
* In conflict with the general indication.
These shew that as the masculinity of the population was diminishing,
that of birth was increasing, with the exception of the decennium 1871-
1880.
For the Commonwealth of Australia the results for the masculinity
of the population at the beginning of a year compared with that of the
births in the same year, set out in the order of the masculinities of the
population, are : —
Table XXV. — Masculinity in Australia.
Masculinity . .
1909
1910
1911
1908
1907
1912
1913
Of Population
1.0764
1.0771
1.0787
1.0793
1.0824
1.0854
1.0885
Of Birth
1.0520
1.0638
1.0473
1.0493
1.0489
1.0454
1.0476
The trends are again in opposite directions, but not markedly.
MASCXJLINITY OF POPULATION.
136
Mi
;. 45.
106 , -~~.^^
V--'
10
20
^*"~ *
?0
The curve Is that given
by formula (386) above. The
dots are individual results.
Masculinity of Population.
V
\
'^
^~_
A
_,^
-.—
'
I
ITBO
S-f, The curve A denotes
.S3 masculinity of population,'
"a i B masculinity of live-births,
g.j; The dots are individual
^ h^ results.
1000
1830 40 1850 60 70 80 90 1900 10 Year.
Masculinity of Population and of Live-births,
New South Wales, 1820-1913.
Fig. 46.
In the following table is set out the masculinity of the births, and in
decreasing order of the population of a number of countries ; these give
no definite indication : —
Table XXVI.— Masculinity of Various Countries, Arranged in Order of Masculinity
of Population.
Year
Masculin-
Period for
Mascvilin-
Masculin-
Country.
of
ity of
which
ity of all
ity of Ex-
Estimation
Population
M ^ F
Determined.
Births.
nuptial
Births only
Greece
1889
1.1037
1881-85
1.118
1.059
Australia
1907
1.0793
1901-13
1.051
1.042
Servia
1890
1.0548
1885-89
1.047
1.035
Rumania
1889
1.0373
1886-90
1.077
1.034
Italy
1881
1.0050
1887-91
1.058
1.044
Belgium
1890
.9950
1887-91
1.045
1.022
France . .
1891
.9930
1887-91
1.046
1.029
Hungary
1890
.9852
1887-91
1.050
1.029 •
Netherlands
1889
.9766
1887-91
1.055
1.047
Ireland
1891
.9713
1887-91
1.055
1.048
Finland . .
1890
.9690
1886-90
1'.050
1.052*
German Empire
1890
.9615
1886-90
1.052
1.047
Spain
1887
.9615
1878-82
1.083
1.079
Austria . .
1890
.9578
1887-91
1.058
1.055
Denmark
1890
.9515
1885-89
1.048
1.050*
Switzerland
1888
.9461
1887-91
1.045
1.016
England & Wales
1891
.9399
1887-91
1.036
1.044*
Sweden . .
1890
.9389
1887-91
1.050
1.043
Scotland . .
1891
.9330
1887-91
1.055
1.059*
Norway . .
1891
.9157
1887-91
1.058
1.059*
Aver.(uaweighted)
—
.9838
—
1.0568
1.0446
* The masculinity of ex-nuptial births is greater in these instances than that
of aU births ; in the other instances it is less.
136
APPENDIX A.
6. Masculinity of still and live nuptial and ex-nuptial births. — J. N.
and C. J. Lewis^ studied the " variations of mascvdinity under different
conditions" in 1906. Omitting seven of their quoted cases, in which
the information is incomplete, they shew that stUl-births disclose a mas-
culinity of 2 to 4 per centum greater than that for Uve-births. The un-
weighted averages of their cases with the omission mentioned (see p. 162),
viz., 17, give for the mascuUnity of live-births {M/F), 1.0504, and for
that of stiU-births 1.3032 ; that is, a masculinity 1.2407 greater than
that of Uve-birtlis. Results have been tabulated for Western Australia
for the years 1897 to 1913 for live and still-births, and from 1908-1913 for
ex-nuptial and nuptial stiU and live-births. These give the same general
indication. The results are as follows : —
Table XXVII. — Masculinity-ratios ior Nuptial, Ex-nuptial and StUl-biiths,
Western Australia,* 1897 to 1913.
Masculinity.
1897-1902.
11902-1907.
1908-1913.
M.
F.
M-^F
M.
F.
Mh-F
M.
F.
Mh-F,
Nuptial still-
births . .
Ex-nuptial still-
births . .
AU StiU-births . .
Ex-nuptial live-
births . .
Nuptial live-
births . .
All live -births . .
All birthst
507
759
15457
16216
16723
373
687
14658
15345
15718
1.359
1.1048
1.0545
1.0508
1.0639
672
982
21226
22208
22880
528
884
20108
20992
21520
1.273
1.1109
1.0556
1.0579
1-0632
804
49
853
1116
23941
25057
25910
641
37
678
1037
22882
23919
24597
1.254
1.325
1.258
1.0762
1.0463
1.0476
1.0534
* See Statistical Register, Western Australia, 1906 ; p. 12, 1914, Pt. I., p. 14.
■j- 1902 has been included twice in order to have 3 six -year periods.
J Including, that is, stiU-births.
The experience in Australia from 1901 to 1913 gave an unweighted
average of the masculinities determined for each year, for all births, and
for ex-nuptial births, the following results, viz. : —
Australia
,» • ■
Various Countries
(See Table XXV.)
All live -births . .
Ex-nuptial births
All hve-births . .
Ex-nuptial births
Average
Masculinity.
1.0508
1.0417
1.0568
1.0446
Kange of
Masculinity.
1.0411 to 1.0638
1.0098 to 1.0621
1.036 to 1.118
1.016 to 1.079
The unweighted average ratio of the " ex-nuptial" to all live-births
was 5.954 per centum for Australia.
^ See Jo\irn. Inst. Act., Vol. xl., pp. 154-188, April, 1906.
total
= m +/
= b
nuptial
= »*o + /o
= 6„
ex-nuptial
= »»i + A
= 61
nuptial
= '»2 +/2
= 62
ex-nuptial
= '"H + fs
= 63
total
= m' +f'
= 6'
MASCULINITY OF POPULATION. 137
It was stated by R. Mayo -Smith in his " Statistics and Sociology, "^
that " among illegitimate" {i.e., ex-nuptial) " children the excess of boys
is less than among legitimate" (i.e., nuptial). William Farr, however,
pointed out in his " Vital Statistics,"^ that he beUeved that " it is assumed
in the French returns that foundling children are illegitimate," but
that such an assumption is probably invalid, and he considered the matter
to be in doubt. The Australian results, however, tend to confirm those
for Europe given in Table XXVI.
7. Coefficients of ex-nuptial and still-birth masculinity. — It is a
somewhat remarkable fact that ex-nuptial and still-births shew increased
masculinity, and that among stUl-biiths the ex-nuptial shew a somewhat
different masculinity to the nuptial. For the analysis of this the follow-
ing notation will be convenient : — -
Live male and female births.
StiU male
If we call the ratio of the masculinity in the one case (say the ex-
nuptial) to that in the other (say the nuptial) the masculinity intensifica-
tion-coefficient k, its significance wiU vary according as we use ^1 , W2 > Ms J
see Table XXI. It may easily be shewn that
(337) For ^1; fc„=^.^A ;
(338) For jt.2; ''n^'^^^:
(339) For,x3; k„ = ''^^-^^;
that is, in regard to any character in the first case it is the relative number
of males born divided by the relative number of females born ; in the
second case it is the relative number of males born divided by the relative
number of births ; in the third case it is the ratio of the differences of the
males and females, divided by the relative number of births. The
coefficient intended can be indicated by suffixes and accents ; thus the
intensification-coefficient of ex-nuptial stUl-births on total stiU-births
would be yk'g ; of ex-nuptial on nuptial live-births, Aj^„ ; and so on ; see
the preceding scheme of notation in the beginning of this section.
1 Maomillan, London, 1895, p. 77. ' E. Stanford, Loudon, 1885 p. 104.
138
APPENDIX A.
The coefficients for Western Australia are as in the following table :
Table XXVin. — Masculinity Intensification-Coefficients, Western Anstralia,
1897-1913.
Ratio of Masculinity of
To the Masculinity of
1897-1902
1902-1907
190&-1913
All stiU-births
Ex-nuptial still-births
Ex-nuptial live-births
All live -births
Ex-nuptial live-births
Nuptial live -births . .
1.293
1.049
1.203 1.201
— 1.057*
1.052 1.029
* Depends upon limited numbers ; see Table XXVII.
For Western Australia for 1897 to 1913 inclusive, the ratio of mas-
culinity of all still-births, 1.287, on all live-births, 1.054, is 1.221. This
agrees excellently with the result of a series of values for Europe shewn in
Table XXVIII., the mean of which is 1.2397.
Table XXIX.— Ratio of Masculinity of Still-Births to that of Live-Biiths, in
various Countries.
Years.
Ratio.
Years.
llatio.
Years
Katio
Paris
Paris
Livonia . .
Montpellier
Alsace- Ijorraine
Netlierlands
8
10
10
10
5
1.157
1.179
1.205
1.208
1.208
1.210
■
Germany
W. AnstraUa
Prussia
Hungary
Italy
Amsterdam
Mean
5
17
10
5
5
12
1.220
1.221
1.225
1.238
1.239
1.241
Austria
Belgium . .
Switzerland
S\peden + Finland
Sweden
France
Mean . .
5
5
5
9
5
1.249
1.264
1.292
1.299
1.300
1.360
Mean . .
1.195
1.231
1.294
8. Masculinity of First-bom. — ^It has been supposed that masculinity
has some relation to primogeniture. For the six years 1908 to 1913
inclusive, there were in Australia 111,545 births, of which 25,708 were
first births. The number of males and females gave the foUowing re-
sults, viz. : —
Masculinity of Australian
Period.
First-births.
Other births.
All births.
1908-1913 . .
1.05260
1.05001
1.05066
Tabulated according to ages between marriage and birth, the results
were : =
Period
Masculinity of Australian First-births, the Interval
after Marriage being — ■
Under 1 year
1 year
2-5 years
&-25 years
1908-13
Difference from Mascu-
linity of all live-
births for same per-
iod, viz., 1.0607
1.0534
+ .0027
1.0514
+ .0007
1.0578
+ .0071
1.0091
-.0416
MASCULINITY OF POPULATION.
139
The numbers, however, are relatively small for the last group, in which
there were only 3490 births. The difference between the different
groups and the masculinity of aU live-births for the whole period is not
more remarkable than the difference between the masculiaity of all live-
births between one year and another. BertUlon's result from 1,140,860
births in Austria was 1.086 for first, and 1.054 for subsequent births ;
while Gteissler's result for Saxony for 4,794,304 births was 1.054. Lewis
for Scotland obtained from 85,964 births, for first births, 1.054 ; for
subsequent births, 1.048 ; Streda for Alsace-Lorraine, from 47,198 births,
for first births, 1.058 ; for subsequent births, 1.059.^
9. Masculinity of populations according to age, and its secular
fluctuation. — ^In any country where migration has a large influence, and
especially where also the migration is of a somewhat specialised character,
the masculinity is likely to shew considerable changes. In the following
Table, viz., XXX., are given the mascuUnities (jus) in age-groups, for
four Censuses, viz., 1881 to 1911, the masculinities in this case being
{M — F)/(M-j-F). This character is strikingly different from that of
England. The significance of the fluctuations of the masculinity are
best seen in Fig. 47.
a
3
.9
9 ®
^ "3 ■
g uS
CO og
— ^ «> a
!» ».S
"S "3
is go
■a z
S o
ES *
(B CD
a §
A
20
i/
/
"^
^-^
/ 1
/■
/
#•
"^
10
/
f"
/
"\
/''
■\
\
/
'
f
K
V
,-^-'
V-
\
*«^^
/ , -^
y
\
\
\
\
1901
1
6- 56 3
40 60 6
Ages.
Variation of Masculinity of Australian population according to age.
Fig. 47.
1 See Joum. Inst. Act., vol. xl., 1906, p. 164.
140
APPENDIX A.
Table XXX.— Masculinity* in Age-groups at Censuses 1881, 1891, 1901, 1911,
Australian Commonwealth, and England, 1911. Computed from Smoothed
Results.
Age-
Australian Commonwealth.
England.
Gbotjp.
1881.
1891.
1901.
1911.
1911.
0-4 ..
5-9 ..
10-14 . .
16-19 . .
20-24 . .
25-29 . .
30-34 . .
36-39 . .
40-44 . .
45-49 . .
60-64 . .
65-59 . .
60-64 . .
65-69 . .
70-74 . .
76-79 . .
80-84 . .
85-99 . .
90-94 . .
96-99 . .
100
.01018
.00898
.00943
.01332
.03493
.12482
.12489
.15176
.17886
.20734
.24498
.25646
.23988
.22504
.22228
.20038
.26350
.28965
.03175
-.05263
+ .20000
.01374
.00975
.01195
.00389
.04192
.11802
.15534
.14833
.16100
.14761
.15267
.16233
.19446
.19310
.17717
.19886
.17799
.12313
.25424
.23967
.17647
.01227
.01105
.00981
.00223
.00157
.02183
.07807
.11272
.13292
.14744
.13833
.10217
.08809
.13194
.16770
.13247
.07707
.06902
.05306
.06215
.01588
.01064
.00869
.01485
.02472
.03155
.03485
.04356
.07038
.10160
.12294
.10885
.07725
.06274
.05417
.06685
.07253
-.02107
-.05164
-.04651
+ .05263
+ .00463
-.00060
-.00126
-.00804
-.05366
- .05440
-.04369
-.03459
-.03693
-.03811
-.04132
-.04883
-.06437
-.09299
-.14419
-.17745
-.21752
-.27160
-.36311
-.40237
-.43750
Mascxilinity
of total
Population
.07983
.07362
.04824
.03840
-.03269
* (Males — Females) -4- Persons.
An examination of these results shews that where there is a consider-
able migration element, predictions as to the future movement of the
masculinity, by extrapolation, are somewhat uncertain both for any
age -group and for aU ages. Moreover, interpolations will lead to results
which can be regarded only as fairly accurate.
10. Theories of Masculinity. — ^The results given shew that the
masculinity of stiU-births is considerably higher than that of live -births,
roughly in the proportion of about 1.15 to about 1.35 greater ; and that
masculinity at birth generally is about 1.05 or 1.06. These facts are
remarkable, and have given rise to various attempted explanations.
J. A. Thomson in his " Heredity"^ says that, according to Blumenbach,
Drelincourt in the 18th century brought together 262 groundless hypo-
theses as to the determination of sex, and that Blumenbach regarded
' Murray, London, 1908, p. 477.
MASCULINITY OF POPULATION.
141
Drelincourt's theory as being the 263rd. Blumenbach postulated a
" Bildungstrieb" (formative impulse), but this was regarded as equally
groundless. It has been suggested that war, cholera, epidemics, famine,
etc., are followed by increase in the masculinity. These will have to
form the subject of later investigations. At present it would seem that
the first necessity is a sufficiently large accumulation of accurate statistic,
as a basis for study. The one point which is clear is that death in utero
(at least in the later stages) is marked by much greater masculinity than
that which characterises live -births. This wUl be referred to later in
dealing with infantile mortality.
That the effect of war is not apparently discernible in existing statistics,
is evident from the following table, viz., Table XXXI, shewing the
experience of France from 1865 to 1876. It will be seen that the war-
years, 1870 and 1871, and subsequent years reveal no change in the
masculinity.
Table XXXI.— Experience o£ France, 1865 to 1876.
Deaths of
Excess of
Rates per
1000 of Mean
Population.
Children
Males over
Year.
under 1 year
of age per
Females in
each 1000
Marriage.
Birth.
Death.
1000 births.
births.
1865
7.85
26.5
24.3
191
2.5
1866
8.00
26.4
23.2
162
2.6
1867
7.85
26.4
22.7
170
2.1
1868
7.85
25.7
24.1
192
2.3
1869
8.25
25.7
23.5
176
2.4
1870
6.05
25.5
28.4
191
2.3
1871
7.25
22.9
35.1
240
2.4
1872
9.75
26.7
22.0
152
2.3
1873
8.85
26.0
23.3
180
2.4
1874
8.30
26.2
21.4
158
2.6
1875
8.20
25.9
23.0
170
2.4
1876
7.90
26.2
22.6
165
2.3
XI.— NATALITY.
1. General. — The phenomena of human reproduction, as affecting
population, and the whole system of relations involved therein, may-
be subsumed under the term " natality." In one aspect they measure
the reproductive effort of a population ; in another they disclose the rate at
which losses by death are made good ; in a third they focus attention upon
social phenomena of high importance (e.g., nuptial and ex-nuptial
nataUty) ; in yet another they bring to light the mode of the reproductive
effort (e.g., the varying of fecundity with age, the fluctuation of the
frequency of multiple-birth, etc.) In this section we shall deal with the
questions which relate more directly to birth-rate, and shall treat of those
which relate more directly to nuptiality in section XII, and to fecundity
in section XIII.
Birth-rates are not immediately comparable. The physical and
social development of two communities being identical, their birth-rates
become roughly comparable only when the relative numbers of married
and of single women at each age are identical. In regard to the initial
qualification, it may be pointed out that any of the races of Western
Europe, for example, may be immediately compared on the basis of
identical numbers at the same ages ; but a population of the natives of
India would not be comparable to one of Western Europe because of
earlier physical development and earlier marriage. Comparisons of this
special character, however, may sometimes be founded on principles
indicated by the theory of " corresponding states" in physical investiga-
tions. This matter will be referred to later.
Populations similarly characterised in respect of features, material
to any question at issue, may be called homogeneous in that respect.
In order to compare the birth-rates of populations, otherwise homogene-
ous, but differently constituted in regard to age, it is necessary to take
account at least of three things, viz., (i.) the numbers at each age ; (ii.) the
relative fecundity at each age ; and (iii.) the relative numbers of married
and single women. In other words, a convenient and strict comparison
can be made satisfactorily only on the basis of what may be called a
" standard" or " normal" female population. This normal population
should represent the mean of the whole series of populations proposed to
be compared (i.e., the relative numbers of married and of single females at
each age should be their ratio to the entire aggregate). Comparison is
then effected by attributing to this population-norm the nuptial and ex-
nuptial birth-rates actually existing in the populations to be compared
with one another. Such a comparison is free from the effect of accidental
differences in constitution as to age ; thus the relative magnitude
NATALITY. 143
of the birth-rates and populations compared are revealed. The principles
of developing norms of this type have already been considered ; see
VIII., §§ 8 to 12.
We consider first the ngiture of a birth-rate.
2. Crude birth-rates. — ^While the total number born in any population
during any period, divided by the average number of the population
during the period, i.e., the crude birth-rate, is one element of the rate
at which the population is reconstituted, its nature and Hmitations are
importa-nt from certain points of view. We propose to consider these.
Since both births and population vary with time, we may regard their
variations of rate as represented by the functions / {t) and F (t). Thus
if Bjn denote the number of births occurring in a unit period (say 1 year),
and P^ be the mean population during that period, the average period-
rate (annual rate in the case supposed), which may appropriately be re-
ferred to the middle of the period, is : —
R R 1
(340) ^
Pw. P t^^F{t)dt
the instantaneous value passing through the range of values which deter-
mine the form of the functions / and F. P is the population as at the
middle of the year, and B the rate per annum at which births are
occurring at that moment.
In general, no serious error wiU be introduced in the value of j8 if,
instead of P^, the population at the middle of the year is used, though
more accurate results wiU be to hand if population-determinations at the
end of each half-year, or each quarter, or better stiU each month, are
used to ascertain the mean. The necessary formulae would be respectively
(341). .P„=i(P„+Pi) ; or = -f (P„ + 4Pi + Pi) ; or
= ^ (Po+4Pi+2Pi+4Pi-FPi); or
(P„ + 2P,., + 2PaH- . . . .2Px. + Pi ; or
= i Kn + P.% + Pf. + -Pie + ^i) + 2Pa +
5 (P^-f Pa + Pa+ Ph) + 6 (P,3, + Pft)} ;i
or any of these indicated in VI., § 2, Table VI.
1 The question of the formulae to be preferred was discussed for quarterly
results in the Population and Vital Statistics Bulletin for Australia, No. 1, pp. 20, 21,
and the coefficients adopted were 1, 4, 2, 4, 1, though previously 1, 2, 2, 2, 1 had been
used. The use of formulae based upon integral functions supposes that the recorded
population at the moment of record is substantially free from large deviations from
the nvunber represented by the functional change. If the functional change is small,
and the " accidental" deviation is large, the use of the functional formxilae does not
yield the advantages expected, and has the disadvantage of multiplying the " acci-
dental" deviation possibly by a very large or a very small factor (as the case may be);
if the former, the result is not satisfactory.
24
144 APPENDIX A.
Such formulae are, of course, more than abundantly accurate for all
statistical purposes.
Birth-rate is influenced by —
(a) the sex and age constitution of the population ;
(b) all forces restricting the fecundity of a population (e.g., frequency
of, and the age of, marriage ; social tradition and habits ; etc.) ;
(c) the frequency of multiple-births ;
(d) infantile mortality (since mothers who lose their offspring are
again exposed to the risk of maternity), etc.
These influencing factors will be considered either in this section, viz.,
Xn., or in later sections.
3. Influence of the births upon the birth-rate itself. — ^Let it be sup-
posed that the population of two communities be initially P and that in
the same period B births occur in one and 2 B in the other, of which in
each case the proportion s survive ; the numbers being thus sB and 2s B
at the end of the period. If there were no migration, and no deaths,
other than those arising from the births, the deduced birth-rates would be
R o i> 9 7?
a larger quantity. Hence the effect of an increase of a birth-rate, when a
proportion of the births is incorporated in the population, is to somewhat
diminish that ratio of births to population, which really represents
the relative frequency of birth, unless at least the population is increasing
in some manner which counteracts this. The preceding result is more
obvious if put in the form —
(342„)..2^a = p|:^{l+M-H4)Vetc.} ; ^, = ^
More generally we have —
(3426).... iSi: ^2 ^' ^^
(r denoting the increase, supposed linear) ; shewing that the birth-
rates and births are in the same ratio only if the mean populations are
identical. Hence as measures of fecundity birth-rates need some sHght
correction, owing to their influence on the magnitude of the population.
They are strictly comparable in this respect only when two populations
are homogeneous, and differences of birth-rate themselves disturb the
homogeneity and thus involve the apphcation of some correction. ^
^There is an analogous case in connection with the computation of interest
earned on assurance and similar funds. Thus if I denote the interest earned in the
course of a year, A and B the funds at the beginning and end of the year respectively,
and i the effective rate of interest earned on the funds during the course of the year,
then the value of i is approximately given by the following formula, now generally
adopted in practice : —
i = I/{^{A + B) - ^I }
NATALITY. 145
4. Influence of infantile mortality on birth-rate. — Denoting the
number of births by B, and of infantile deaths by M, and the number of
women of child-bearing age by P, we shall have for the birth-rate /3,
attributed not to the whole population but to the P women, and for |i the
rate of infantile mortality —
(343) j8 = B/P ; ,j, = M/B ; ^^ = M/P.
Suppose [L to change to some other value fi' = M' /B' ; M' being the
number of deaths and B' the number of births under the changed state
of things, assumed to have become constant. Then, since mothers who
lose their children are exposed to an increased risk of maternity, the
ratio of which is only the proportion q (a proper fraction) of the full risk,
we shall have for the number at risk as originally, viz., N, and also after
a change in the prevailing rate of infantile mortahty, N'
(344) N = P — B +qM ■ and N' = P — B' +qM'.
If the reproductivity of these two groups is the same, then B/N =
B' /N' ; from which it follows that —
(345, ^L + J^_1.1 + i*_l
and consequently, discarding the unit from each side and writing in the
values of the quantities as by (343) above, we have — ■
(346) -J + qti =j^ + q^';
that is —
(347) i8' = P {I +qp' ifi' - t.)}
It will be found that this change is sensibly a Hnear one, or any
increment in the rates of mortality will cause a sensibly constant but small
proportional increase in the birth-rate. If we call the birth-rate, freed
from the influence of infantile mortality, the normal birth-rate j8g, then —
(348) j8o = ;8 (1 + k^i).
in which k may be regarded as a constant for a particular community,
and a particular epoch. The value of k was found on the average for
Europe to be about + 0.033 fx, or Po = P + 0-033 fi, the birth-rate j8
being expressed per 1000 of the population, and the infantile mortality
rate u, expressed per 1000 births. An examination of the data for differ-
ent countries gave the following results : —
146
APPENDIX A.
TABLE XXXn.— Influence of the Rate of MantUe MortaUty on
the Crude
Birth-rate for Various Countries, about Year 1900.
Period.
Value
of /Sq and k
COUNTBY.
in;8 =
^0 + fcM.t
Birth.
MortaUty.
New Zealand
1881-1905
1882-1906
13.2
k
-^ 0.191
CoTTiTTionwealth . .
1887-1905
1888-1906
16.8
+ 0.118
Sweden
1881-1904
1882-1905
17.1
-i 0.100
Norway
1881-1905
1882-1906
20.5
i 0.100
Prussia . .
1881-1905
1882-1906
19.1
+ 0.085
Various Countries*
1901
1902
19.4
1- 0.083
Netherlands
1881-1905
1882-1906
22.6
+ 0.063
France
1881-1905
1882-1906
12.7
+ 0.061
Denmark
1881-1905
1882-1906
22.4
+ 0.060
Japan . .
1 1881-1904
1882-1905
22.3
+ 0.053
Ceylon
1881-1905
1882-1906
26.4
+ 0.042
Jamaica . .
1881-1905
1882-1906
34.3
+ 0.022
Switzerland
1881-1904
1882-1905
25.3
+ 0.018
Ireland . .
1881-1905
1882-1906
25.8
— 0.026
England and Wales
1881-1905
1882-1906
38.6
— 0.058
Scotland
1881-1905
1882-1906
38.9
— 0.068
* For one year only. t The birth-rate being expressed per 1000 of the
population, and the infantile mortality per 1000 births.
The infantile mortaUty rate {n) in the table is expressed by the number of
infants dying per 1000 of infants bom.
The crude birth-rate (3) is the number of births per 1000 of the total population.
It will be seen that the magnitudes of k, and therefore of q, have no
general relation to the magnitude of the birth-rate ; that is, a particular
value of the risk-factor is characteristic of a particular country.
In an investigation made in 1908"^ it was shewn that the influence
of infantile mortality was very irregular in its operation, and the following
deductions were stated, viz. : — ^
(i.) When either all mothers of deceased infants, or any constant
proportion thereof, may be regarded as subject to equal risk
of fecundity (i.e., equally hkely to bear children) then equal
increases in the rate of infantile mortality tend to be followed
by equal though relatively small increases in the birth-rate.
(ii.) The influence of infantile mortahty on the birth-rate must
always be very small. (The contrary proposition is not, of
course, necessarily true).
This type of investigation aims rather at ascertaining the form of the
function expressing the correction, so that the form being determined, the
constants can then be ascertained from the data. It would appear that
yearly irregularities of birth-rate are so great as compared with the
influence of infantile mortahty that the latter is virtually masked by the
former. Probably in any rigorous investigation of a measure of the
fecundity of a population the birth-rate should be corrected in some such
way as has been indicated.
1 By the writer. See Journ. Roy. Soc, N.S.W., Vol. xlii., pp. 238-250, par-
ticularly Fig. 1 on p. 243 therein.
2 Loc. cit. pp. 241-2.
NATALITY.
147
5. World-relation between infantile mortality and birth-rate. — ^In
order to ascertain whether in a world-wide survey of infantile mortality
and birth-rates any correlation manifested itself we may extend the pur-
view of all countries where fairly accurate statistics are available, viz.,
the following : —
Australia, Austria, Belgium, Chili, Ceylon, Demnark, England and
Wales, France, Ireland, Italy, Jamaica, Japan, Netherlands,
New South Wales, New Zealand, Norway, Queensland, Russia,
Scotland, South Australia, Spain, Sweden, Switzerland, Tasmania,
Victoria, West Australia.
The populations are, of course, repeated with different rates, and are
equivalent to 8776 millions,^ the results forming groups of available
results ; according to the magnitude of the infantile mortality we get the
results shewn in Table XXXIII. hereunder, the ranges of infantile
mortality being shewn therein.^
In Pig. 48, graph A denotes the relative frequency of the given ranges
of infantile mortality.* It will be observed that the graph is dimorphic,
that is, that while the characteristic rate of infantile mortality is about
.0150 (150 as usually expressed), there is also a second mode for the rate of
about .0255. The corresponding crude birth-rates are about .029 and
.048 respectively (or residual birth-rates, see hereinafter, about .025 and
.035). It will be seen that there can be a very high rate of infantile
mortality with low birth-rate, but it would appear, only for very limited
populations.*
TABLE XXXni.
-Relations of Infantile Mortality and Birth-rate, various Countries,
about Year 1900.
Popula-
Ranges of
Infantile
Crude
tion Ee-
Infantile
Mean of
Mean of
Mortality
Birth-rate
12 Months
presented
Mortality
Infantile
Crude
of
of
Residual of
(millions)
for Individual
Populations.
.0688-.0959
Mortalities.
.0821
Birth-rates.
Aggregate.
Aggregate.
Birth-rate.
344
.0291
.0911
.02692
.02447
479
.1018-. 1232
.1120
.0291
.1119
.02889
.02566
2035
.1276-. 1474
.1371
.0288
.1387
.02865
.02468
2172
.1519-.1724
.1618
.0291
.1598
.02904
.02440
1116
.1762-. 1974
.1872
.0340
.1880
.03391
.02753
851
.2032-. 2 179
.2098
.0367
.2085
.03365
.02663
297
2213-.2372
.2286
.0380
.2279
.03808
.02940
696
.2406-.2559
.2490
.0480
.2491
.04757
.03572
668
.2601-.2771
.2688
.0479
.2710
.04763
.03472
189
.2800-. 2920
.2870
.0446
.2845
.04885
.03495
105
.3040-. 3290
.3133
.0385
.3075
.04549
.03150
147
.3325-. 3490
.3406
.0366
.3392
.03701
.02446
91
.3660-.4120
.3890
.0372
.3800
.03681
.02282
1 The method is, of course, not perfectly satisfactory ; for, as pointed out bv
the writer (on p. 245), loc. cit. the populations are not homogeneous, and doubtless
if more moderate-sized districts could be analysed the material would give a clearer
indication of the true nature of the relation.
'^ See also loc. cit., p. 246, and Fig. 2, p. 247, in the same paper.
' See page 150 hereinafter.
* Similar indications are given by the analysis before referred to. See loc. cit.
p. 248, Fig. 3.
148 APPENDIX A.
This more general result shews that propositions (i.) and (ii.) in the
preceding section can be regarded as true only for individual populations
and probably for very Umited periods of time ; the effects are readily
masked by more potent influences.
In the table hereunder (XXXIV.), of results in the present century,
the following countries have been included, viz., in column (i.) New Zea-
land, 1913 ; Norway, 1912 ; Australia, 1913 ; Sweden, 1911 ; France,
1912 ; Netherlands and Denmark, 1913 ; Switzerland, 1913 ; Ireland,
England and Wales, and United Kingdom, 1913 ; Finland, 1912 ; Scot-
land and Ontario, 1913 ; Belgium, Italy and Prussia, 1912 ; Serbia, 1911 ;
German Empire, 1912 ; Spain, 1907 ; Bulgaria and Japan, 1910 ;
Jamaica, 1913 ; Austria and Hungary, 1912 ; Ceylon and Roumania,
1913 ; Russia (European), 1909 ; Chile, 1911 ; and in column (iv.)
France and Belgium, 1912 ; Ireland, England and Wales, and Ontario,
1913 ; Sweden, 1911 ; United Kingdom, 1913 ; Switzerland, 1912 ;
Scotland and Denmark, 1913 ; Norway, 1912 ; New Zealand, Nether-
lands, and Australia, 1913 ; German Empire, Prussia, Finland, Austria
and Italy, 1912 ; Spain, 1907 ; Japan, 1910 ; Jamaica, 1913 ; Serbia,
1911 ; Hungary, 1912 ; Chile, 1911 ; Ceylon, 1913 ; Bulgaria, 1910 ;
Roumania, 1913 ; Russia (European), 1909. The results are the weighted
means (or what is the same thing, the values are for the population-
aggregates) of the populations, combined in successive groups of ten,
arranged (in ascending order) according to infantile mortality in the one
case, and according to birth-rate in the other.
These results shew unequivocally that there is, in general, a relation
between birth-rate and infantile mortality. The calculated results are
as follows ; jS denoting birth-rate per unit of population, and /i denoting
infantile mortaUty rate per birth : —
Determined from groupings in the order of infantile mortality :—
(349). .j8 = 0.00956 + 0.1405 fj. ; (which gives /x = 0.06804 -f-7.117 )3) ;
and determined from grouping in the order of birth-rate : —
(350). . . .^ =-0.03661 + 5.970/3 ; (which gives ^ =0.06132 +0.1675/x).
The mean of these result.s is expressed with sufficient precision by —
(351).. /3 =0.00785(1 +19.6^); fj, =0.0510(1—127^)
jS being the rate per unit of population, and jj, per birth.
NATALITY.
149
TABLE XXXIV.— General Relation between InJantUe Mortality and Birth-rate,
Aggregates of various Countries, 1907 to 1913.
InFANGILE MoRTAUTY and BtETH-BATi;.
Birth
-BATE AND INFANTILE MoBTALITY.
Popula-
Infan-
Re-
Popu-
Tnffl.n -
Re-
tion in
tile
Birth-
Calcul-
duced
tion in
Birth-
tile
Calcul-
duced
Mil.
Mor-
rate, t
ated, t
Birth-
MU-
rate, t
Mor-
ated.§
Birth-
lions.
tality.*
rate.!
lions.
taUty.*
rate. *
(i-)
(ii.)
(ii
i.)
(iv.)
(V.)
(vi.)
107.6
90
22.6
22.2
20.6
154.2
22.7
99
99
20.5
152.5
96
23.0
23.1
20.8
116.9
24.0
105
107
21.5
153.3
96
23.1
23.1
20.9
110.4
24.1
104
107
21.6
153.2
97
23.0
23.2
20.8
112.2
24.4
103
109
21.9
150.4
99
23.0
23.5
20.7
80.1
24.8
99
111
22.3
118.3
107
24.3
24.6
21.7
143.5
26.4
121
121
23.2
147.2
113
26.1
25.4
23.2
179.1
27.1
128
125
23.6
185.4
121
26.7
26.6
23.5
136.2
28.2
134
142
24.4
184.5
122
26.9
26.7
23.6
161.3
28.8
144
135
24.7
246.3
129
27.3
27.7
23.8
191.6
29.6
142
140
25.4
228.6
135
28.4
28.5
24.6
208.0
30.0
144
132
25.7
186.9
142
29.8
29.5
25.6
256.1
30.8
148
147
26.2
234.2
146
30.7
30.1
26.2
255.9
30.8
148
147
26.2
230.4
147
30.8
30.2
26.3
252.7
31.0
150
148
26.4
256.6
151
31.0
30.8
26.3
269.0
31.4
154
151
26.6
270.1
155
31.6
31.3
26.7
206.3
32.5
159
157
27.3
239.3
159
31.6
31.9
26.6
169.5
33.6
163
164
28.1
205.6
163
32.5
32.5
27.2
170.6
33.9
164
166
28.3
319.2
194
36.7
36.8
29.6
149.1
34.8
163
171
29.1
256.5
208
38.9
38.8
30.8
230.6
39.8
211
201
31.4
* Per 1000 births.
t Per 1000 population.
§ By formula (350).
t By formula (349).
From these the lines B and C respectively are plotted and the cal-
culated values in columns (iii.) and (vi.) are computed. The dotted Hues
shew the positions of the other "graph for the purpose of comparison, and
the line which represents formula (351) is between the two.
That these results, though not identical, are very similar, is seen from
the graphs B and C, shewing the two series of values. What they estab-
lish is that, on the whole, the birth-rate and infantile mortality increase
together. Moreover, when the birth-rate is reduced to its effective value
twelve months later (that is, for one year of age), it is much more uniform
on the whole. Since, as shewn, the. increase of risk of maternity is re-
latively small (348), it follows that, on the whole, the social conditions
which characterise a large birth-rate are those associated with a high rate
of infantile mortahty. This, of course, is not necessarily so, but expresses
the general fact. In short, a high birth-rate is usually associated with a
high rate of infantile mortality, but high infantile mortality wiU, per se,
not appreciably affect the birth-rate. The importance of this result is
obvious.
160
APPENDIX A.
GENERAL RELATION BETWEEN INFANTILE MORTALITY AND BIRTH-RATE.
a s
^
40
/I
/
//
/ /
/
f
30
f
/y
I
/
y
/
//
i
M
/
/
/
/ /
/ /
10
J
/ /
;
\
X
10
/
/
/
■20
\
_^^
^
w
100
200
100-
200
For curve A.— Infantile mortality rate per birth.
For curves B and C— Infantile mortality rate per 1000 births.
Pig. 48.
6. Residual birth-iates. — Owing to the \evy high death-rate of infant.s,
the crude birth-rate, taken alone, is not a satisfactorj' expression of the
effective recuperative force of a population against the ravages of death.
It is not practicable, however, to assign any particular age as specially
appropriate for estimating the virtvxil efficiency of birth-rate, and as we
have seen high birth-rates, however, are ordinarily associated with a high
rate of infantile mortality.
For example, New Zealand and Australia had birth-rates in 1912 of
26.5 and 28.7 per thousand population, and infantile death-rates {i.e., deaths
under 12 months per 1000 bom) of 51 and 72, while Ceylon and Chile, in
1911, had birth-rates of 37.9 and 38.5, and infantile death-rates of 218
and 332. This question will be referred to later.
Birth-rates corrected so as to represent the number hving after a
given period may be called residual birth-rates, and the quantity multi-
plied into a birth-rate to give its residual value may be called the survival
coefficient, or survival factor. We shall consider these. Owing to the
fact that of all the deaths which occur in 12 months, about 42 per cent,
occur in the first month, the infantile mortahty may be referred to the
same calendar year as the births without sensible error, or we may correct
NATALITY. 151
it as explained hereinafter. Let ^ be the birth-rate and y the rate of
infantile mortality, the first expressed per unit of the population, the
latter per birth. Then the residual birth-rate ^j is^ —
(352) ^, = p {I -y)
The quantity in brackets is the '• survival-factor" and jS, is the " residual
birth-rate." For a population in which the number of births was con-
stant and the rate of mortality for the first twelve months was constant,
the probability of persons of age living to age 1, viz., ^pi, would be
the same as the survival factor, since under these conditions it would
denote the ratio of those surviving one year, viz., li to the number born,
viz., If,. Consequently, subject to this limitation —
(353) (l-y)=^i=Zi/Z„.
For a population in which the number of births is increasing,
say, at the rate rt, and the rate of infantile mortahty diminishing,^ say,
at the rate r't, these quantities become functions of time and are affected
by the interval of time between the year for which the births are recorded
and the somewhat later year for which the infantile deaths ought to be
recorded, in order to properly refer to the birth-group. As, however, the
error arising is of a small order as compared with the accidental deviations
from year to year, it is questionable whether a correction is worth apply-
ing. It may be mentioned that in Australia it was found by an investiga-
tion for the years 1909 and 1910, that all children who die in the first year
of life live on the average 99.3 days, and children are registered on the
average 38.2 days after birth. ^ The difference, 61.1 days, or say two
months, is regarded as the difference between the years. Thus the in-
fantile mortality in the following table was calculated on the births
occurring one-sixth of a year earlier. Similarly the birth-rate given for
the equivalent year to n, say ^e, is —
(354) iSe = i i8„_, + t i3n .
It may also be noted that an investigation of the question shewed that of
the deaths in Australia under 1 year of age occurring in any calendar
year, 0.72 to 0.74 per cent. — average about 0.73 — arose from births which
occurred within that calendar year, and 0.27 from those which occurred
in the preceding year. This proportion is doubtless approximately true
also for other countries.
1 These rates are commonly expressed per 1000 of the population, and per 1000
born respectively, in which case the formulae will be /S/ = /3' ( 1 — -JL- ) ; j8' and y
being 1000 times greater than /3 and y.
^ Infantile mortality has for years past been steadily diminishing in many
countries.
' This has ceased to be true because of the " maternity bonus."
162
APPENDIX A
This would suggest that the coefficients in the above equation
(354), should be ^ and f instead of ^ and |, but, only if the average late-
ness of the registration of births and deaths were the same, which, how-
ever, was not the case. The practical result of the difference is not great.
It wiU appear from a rigorous investigation in the next two sections,
that with the rate of infantile mortality as it stood during the years
1909 to 1913, the proper proportion is about 0.731, a proportion which
wiU be modified only by the difference in the registration interval. This
interval, owing to the payment of the maternity bonus, resulting in
earlier registration of births, has now become smaller.
TABLE XXXV.— Residual Birth-rates, AustraUa, 1904-14.
Crude
InfEintile
Crude
Year.
Birth-rate,
Death-rate t
Birth-rate
Survival
Residual
for Calendar
Calendar
for Equival-
Factor.
Birth-rate
Year.*
Year.
ent Year.
1903
25.29
1904
26.41
81.77
26.073
.91823
23.94
1905
26.23
81.76
26.260
.91824
24.11
1906
26.57
83.26
26.497
.91674
24.29
1907
26.76
81.06
26.728
.91894
24j61
1908
26.59
77.78
26.618
.92222
24.54
1909
26.69
71.58
26.673
.92842
24.76
1910
26.73
74.81
26.723
.92519
24.72
1911
27.21
68.49
27.297
.93161
25.43
1912
28.66
71.74
28.410
.92826
26.37
1913
28.26
72.71
28.317
.92729
26.26
1914
28.05
71.47
28.083
.92853
26.08
• Per 1000 population. t Per 1000 births.
The final column is the efficient birth-rate, the end of the first year
of life being taken as an appropriate point of time for determining the
efficiency, since the larger death toll from infantile troubles may be
regarded as then past.
7. Determination of proportion of infantile deaths arising from
births in the year of record, number of births constant.^Births, and
infantile and other deaths, are recorded as occurring during successive
equal periods of time, usually calendar years, half-years, quarters,
months, etc. ; and the deaths during such periods are distributed accord-
ing to a series of age-Umits, for adults usually whole years, 0-1, 1-2,
etc. In the case of " infantile deaths" or deaths of children under one
year of age, they are distributed according to age-hmits of weeks, months,
quarters, etc. Consequently the infantile deaths occurring in any year
are drawn from the births [and immigrants] both in the year of record
< *
NATALITY. 163
and in the previous year. More generally deaths of persons between the
ages x^, and x^ recorded in any period of time, say — i^ to 0, are drawn
[where there is no immigration] from those born [in the country] during
the period — (a;2+*z ) to — (ajj+O).^ In the same way deaths recorded
in any period — f2z to —t^ would be drawn from those born [either in the
country or from migrants entering it] during the period — (a;2+*2z) to
- (^1 + h )■
If the frequency of births be denoted by k' Fi (4), the number of
survivors after any period of time, x, of persons born at the moment t,
wUl, so long as the death rates at each age remain constant, also be this
function multiphed by the. probability of surviving to the age x. Thus
if this probability be denoted by Xx, or that of dying be denoted by
Sa;, = I — \x, then the survivors of age x, say 8x, and those who have
not attained that age, say Dx, will be —
(355) 8x =XxFi(t); and Dx = 8xFi(t)
for we may make A;' = 1 it ratios only are needed.^
With births increasing, the successive records of the dying of any
given age wiU also shew a similar progressive increase, proportional to
that of the births, the death-rates at each age being constant. Thus the
aggregate of births between the times <i and t^, will be —
(356) t,Bt, = kf 'Fiit)dt.
which would give merely B = K [tz — h) if the frequency of births
were constant, K being the number per annum when t is expressed in
years. If the frequency be not constant, but of the form indicated
hereinafter, viz., that in equation (359), then it will be [see also II., §6,
(10)]-
(357). .t.Bt=K [h - k)[l + \a {h-h) +\b {tl + hh+t\)
+ \c{h+h) {K+tD + etc.}
which, when ti is 0, takes the simpler iorm.-
— hfZ _i_
(358) oBt =Kt {1 + 1 a* + 4- &*' + T «*' + etc.
^ The words in sqaaxe brackets may be omitted, if proper care be taken in the
practical oonxputations in regard to the influence of ndgration.
' That is Xz = Zx //o in an " actuarial population" ; or is Ix if l^ be made
unity. Similarly Sx = {l^ — Ix )/lo ■
154 APPENDIX A.
With respect to survivors to age x, it may be noted that, in the earliest
stages of life, Xx decreases with the greatest rapidity, hence of the deaths
occurring in any year the greatest number is contributed by those of
the smallest age ; at least in the case of all aged less than 10 or II years,'-
and therefore for ages less than 10 years the greater number is contributed
by what may be called the ordinary year of reference, viz., in the case of
infantile deaths, the year of observation [or year of record] itself.^ "We
may obtain an exact measure of this if we have the values of Xx or 8x
For perfect rigour we must put these quantities = F^ix, t) for at the
present time the value Xx is sensibly increasing every year, and thus S^:
decreasing every year, for nearly all ages. It will, however, simplify
the solution, and lead to no sensible error, if we omit the t and assume
that either quantity is simply a function of x, say Xx = V-2,{x) and
Before envisaging the pertinent questions in their practical form,
rather than in their more general and theoretical form, it may be pointed
out that both these probabilities may readily be expressed as the sum of a
series of exponential terms in the form (360) hereunder. Moreover,
as has already been shewn, see II., §§ 2 to 10, formula (2a), (6), (9) to
(13), the birth-frequency [and if desired this may include the migration
element], may be put in the form —
(359) 6( = F^{t) = Z (1 + af + hf^ + etc.)
in which a, b, c, etc., may of course be positive, negative, or zero. As
above-stated, either —
(360). . . . Aa;,or 8x,= F^ix) or Fs(x) = (i^i e-»i=»+ A2e-".=^+ etc.)
in which, for values of x not greater than 10 or 11, w is numerically a
diminishing quantity. On expanding the exponential terms we have —
(361) Xx or 8.r = A (1 — ax + ^x^ - etc.)
in which it may be easily seen that —
"■ The instantaneous rate of mortality, or the so-called "force of mortality"
in actuarial terminology, is, in Australia, a minimum between the ages 11 and 12
years of age for males, and betvreeri 10 and 1 1 for females.
" The "ordinary"' rate of infantile mortality is the ratio of the number of
infants dying txnder one year of age, in any year of observation, to the number of
infants born in the same year. This ratio is usually multiplied by 1,000, to avoid
decimals.
NATALITY 155
Incidentally, it may be repeated that the function bt may be made
to embody all complications arising from migration, forasmuch as birth
may be regarded mathematically, merely as a case of immigration at
age 0, and the constant can be so determined as to represent birth and
[net] immigration combined for a succession of ages.
8. Equivalent year of birth in cases of infantile mortality. — ^In order
to avoid circumlocution and to simplify the statement of the problem,
we shall assume the period of observation of infantile mortality
to be successive calendar years ; and the record of births to be also
according to calendar years. The necessary variation of this statement
for other equal periods is self-evident. From what has been indicated
in the preceding section it is clear that the infantile deaths in any year
can be referred to a birth-year, which precedes the calendar year by
some period less than a half-year. We proceed to evaluate this interval,
which obviously depends upon : — (a) the rate at which the cases come
under initial observation, that is upon the frequency of birth [or of birth
and migration] ; and (6) upon the decrease in the rapidity of death in the
first year of life. The birth-rate in the ordinary sense is, of course,
immaterial. For so limited a period as one year, we can, for the purpose
in view, assume that I -\- rt expresses the increase with time of the fre-
quency of births. If we make the origin of the variable, i.e., time, the
end of the year of observation, we have to consider the deaths of persons
betA\een the age-limits and 1, occurring during the period —1 to 0,
drawn obviously from births [and immigration] during the period — 2 to 0,
inasmuch as survivors born at the time —2-\-t' will be within the age-
limits up to the time —1 -\- t', which also is in the year of observation.
Consequently also deaths among these must be taken into account.
Thus at the time — t, the ages of persons, the deaths among whom wiU be
included in the category of infantile deaths, will be between and x =
1 -\- t. This connects the time-limits with the age-limits. The range of
relative frequency extends from 1 for <= 0, to 1 — 2r for ^ = — 2,
(coming under observation, however, only for < = — 1, whenever in-
fantile deaths are observed for the same year as births). The question
for resolution then is : —
Given the form and constants of the function expressing the variation
in the frequency of births, and the form and constants of the function
expressing the probabiKty of Uving to age x, where x is less than 1, what
proportion of the infantile deaths in any calendar year is drawn from the
year of observation, and what proportion is drawn from the year pre-
ceding that of observation. The nature of the problem is illustrated
by Figs. 49 to 52, p. 157, which illustrate either the case of deaths? o
166 APPENDIX A.
that of births and survivors. The deaths occurring in a small unit of
time, At say, arising from births in the same unit, is represented by the
height of the first parallelepiped HA. It embraces all persons of age
to age -\- Ax, the period of observation being to + -^ *; the number
being the height HA, and A t being equal to A x, these quantities being
thus dx and dt when indefinitely small. The number of deaths at the
end of a period, say a year, from the survivors of those born at its begin-
ning, is represented by the height BI, and after a second period, say two
years, by the height CM, Fig. 49. The succession of deaths are thus
represented by the parallelepipeds 1 .i , 1 .2 , 1 .3 , etc . They are followed by
deaths occurring among those born during the period At to 2 At, repre-
sented by 2.1 , 2.2 , etc., and so on, these, in an increasing population,
being somewhat larger than the former series, since the births from which
they are drawn are greater in number. The parallelepiped P E, or 12.i,
represents the deaths in the last period, viz., 2 — At to 2 ; 7.1 represents
those in the period 1 to 1 + At, the deaths among the survivors in the
successive elementary periods being 7.2 , 7.3 , etc.* Thus, from Pigs. 49
and 50, it is at once evident that A'^, the deeper shaded figure
BIKLDB, represents deaths from survivors from the previous period.
The medium shaded figure, Bq, represents deaths in the period under
consideration born in that period, since G to L is contemporaneous with
G to F. The broken shaded figure Aq, or AHGKIBA, represents the
deaths occurring in the preceding periods from births in that period ;
they are similar to Bg. The black shaded figure, Ai ,or BILDCB, repre-
sents deaths at ages outside the limit, that is, at ages greater than one
year (exactly). In short, Aq, Bq, Dq represent deaths within the
year of record of persons less than one year old ; AJ, to CJ,, represent
deaths of survivors from the preceding year. Similarly in regard to the
other figures, A^ to C^ represeilt deaths of persons of age 1 to 2, attaining
that age in the year previous to the year of record ; while Af to B^
represent deaths of persons of age 1 to 2 who attain that age during the
year of record. Similarly, mutatis mutandis, in regard to A2, B2, and A2,
etc.
The figures of the type Aq, Bq, . . . . Ai, Bi, etc., are represented by the
solid Fig. 52, those of the type Ai, Bx,. . . .A2, B2, etc., are represented
by the solid Fig. 51.
If the origin for a; + 0, « + 0, be the point A in Figs. 49, 50, 52
then the lengths of an element of volume in areas Aq, A^, and Bg, taken
parallel to A G Q V, Fig. 62, are respectively t — x = \ — x, x, and
2t — {t + x) = 1 — X.
NATALITY.
157
Kg^,
158 APPENDIX A.
If the number of births in successive units of time be constant, it is
obvious that A^j. = B^ = C^^, etc. ; A^ = Bj = C^, etc. ; k being any
suffix, and that we have also —
(363) Ao = Bo = . . . . //(I - X) F3(x)dx
(364) Ao = Bo =.... =f^"xFs(x)dx
Consequently the ratio Bo/A'^ is foutid by dividing the value of
(363) by that of (364).
9. Proportion of infantile deaths arising from births in year of
record, number of births increasing. — ^If the number of births be increasing,
and the increase be assumed to be at the rate of e'* = 1 -\- rt -{- ^ rH^ -\-
etc. ; the quantities wiU increase, that is, with any common suffix
A<B<C. Since the maximum value of r is about 0.03, the effect of
the omission of the term t^, where it is one year, is of the order of a two-
thousandth, and may be ignored. Consequently, the increase in the
number of births may, with abundant precision, be taken to vary as
1 + rt. We may take the origin for t as at the point G in Fig. 50, hence
an element of the volume of Bq, and of A^ will be respectively —
(1 — x) F^ [x) 8a; (1 + Y r — rx) ; and x F3 (x) 8x (1 —rx) ;
the element being taken parallel to H G F. The terms in brackets,
containing r, represent the mean heights of the volume-elements taken
along the Unes Y I and G M respectively in Figs. 51 and 52. Hence,
multiplying out, the relative values of the volumes representing Bq and
A'^ are respectively as follows : —
(365) . . Bo=y^' Ul + i *■) -^^3(3:) — (1 + |-r) x F^{x) + r x^ Faix) \ dx
(366). .a; = J^" \x Fs(x) - r x^ F^ix) \ dx
From an analysis of the deaths of infants during four years, viz.,
1909, 1910, 1912 and 1913, the relative values of the functions to be
integrated were found to be as in the following table, viz., XXXVI., in
which the figures in the first column represent the number of deaths out
of 1,000,000 births, occurring up to the time after birth indicated in the
first column, that is up to age x.
i •
NATALITY.
159
TABLE XXXVI.— Values of above Integrals for Various Periods, Basis 1,000,000
Births, Australia, Years 1909-10 ; 1911-12.
Period, or A
ge X. ^'"F.^{x)dx
H - ""■■
{x) dx.
P x'^ IP
, (x) dx.
Number.
Number.
Proportion.
Nimiber.
Proportion
1 day
9,360
11
.0012
.0000
2 „
14,032
27
.0019
.0000
i „
. . 17,257
57
.0033
.0000
5 .,
. . 18,500
73
.0039
1
.0001
10 ..
. . 22,573
154
.0068
2
.0001
15 „
. . 25,335
247
.0097
5
.0002
20 „
. . 27,497
360
.0127
10
.0004
25 „
. . 29,228
457
.0156
17
.0006
30 „
. . 30,645
563
.0184
25
.0008
30.437 „ or
. mth. 30,757
664
.0184
26
.0008
60.874 „ i
I „ 37,231
1,362
.0366
128
.0034
91.311 „ ;
! „ .. 42,796
2,617
.0588
370
.0086
121.747 „ ^
t „ .. 47,735
3,963
.0828
790
.0165
162.181 „
; „ ,. 52,165
5,612
.1076
1,413
.0271
182.621 „ (
5 ,. . . 56,128
7,424
.1323
2,246
.0400
213.058 ,.,
1 „ .. 59,688
9,349
.1666
3,289
.0551
243.496 „ I
i „ .. 62,920
11,367
.1807
4,551
.0723
273.932 „ t
) „ .. 65,871
13,455
.2043
6,030
.0915
304.368 „ 10
„ . . 68,563
15,585
.2273
7,716
.1126
334.806 „ 1
I „ . . 71,045
17,766
.2499
9,615
.1363
365.?42 „ IS
! „ .. 73,366
19,973
.2722
11,736
.1600
* Proportion of number in second column.
From these results, the values given in the next table, viz., XXXVII.,
have been computed, agreeably to formulae (365) and (366). The results
shew that with quarterly records, over 94 per cent, of the recorded deaths
are referable to births in that quarter ; with half-yearly about 87 per cent.,
and with yearly records about 73 per cent., provided the intervals be-
tween occurrence and record are the same in both cases.
TABLE XXXVn.—Froportion of Deaths under One Year of Age, born during the
Year in which the Death is Recorded. Australia, 1909, 1910, 1913, 1913.
Elate of Increase of Births.
Interval of Time
Considered.
1.00
(constant).
1.01
1.02
1.03
1.04
15 days
1 month .
2 „
3 „
4
5 „
6 „
7
8 „
9 „
10 „
11 „
12 „
.9903
.9816
.9.634
.9412
.9172
.8924
.8677
.8434
.8193
.7957
.7727
.7501
.7278
.9903
.9817
.9636
.9415
.9176
.8930
.8685
.8444
.8204
7970
.7742
7517
.7296
.9904
.9818
.9639
.9418
.9181
.8936
.8693
.8454
.8216
.7983
.7756
.7534
.7314
.9905
.9820
.9641
.9422
.9185
.8942
.8701
.8465
.8227
.7996
.7771
.7550
.7332
.9906
.9821
.9644
.9425
.9190
.8949
.8709
.8475
.8239
.8009
.7786
.7567
.7351
160
APPENDIX A.
It will be seen from the above table that, for any ordinary increase
of birth-rate the effect thereof upon the proportions given may be ignored
for ordinary purposes ; that is, with the infantile -death rates experienced
in Australia, it may be assumed that 0.73 of the deaths recorded in any
year occur in that year, whatever the increase of the birth-rate, and this
will be approximately true generally.
10. Secular fluctuation in birth-rates. — ^The birth-rates since 1860
for Australia are as shewn in the following table : —
TABLE XXXVm.— Crude Biith-rates, and Marriage-rates, Australia, 1860 to 1914.
Kates per 100,000 of the Population.
Year
of
Decade.
1860.
1870.
1880.
1890.
1900.
1910.
B
M
B
M
B
M
B M
B
M
B
M
4,256
842
3,866
712
3,525
717
3,498 764
2,733
724
2,673
837
1
4,228
864
3,800
694
3,526
760
3,447
747
2,716
732
2,721
879
2
4,327*
876
3,707
696
3,448
?10
3,365
674
2,671
726
2,866
907
3
4.166
837
3,744
740
3,482
836
3,279
619
2,629t
667
2,826
866
4
4,291 |861
3,679
720
3,560
828
3,083
608
2,641
702
2,805
880
6
4,210 834
3,588
728
3,669
816
3,038
623
2,623
726
6
3,979 762
3,592
714
3,537
788
2,843
666
2,667
749
7
4,042 '751
3,499
731
3,660
769
2,821
668
2,676
787
8
4,046 i755
3,539
737
3,550
798
2,715
672
2,659
776
9
3,865 733
3,577
716
3,465
767
2,727 703
2,669
790
* The highest value was in 1862. f The lowest value was in 1903.
These rates shew a fairly steady decrease till 1903, and then an in-
conspicuous rise. Their significance will be dealt with later, viz., in the
part treating of " fecundity."
Secular as well as any other fluctuations in the birth-rate are of
course influenced by the marriage-rate. This rate is also shewn in the
above table. The figures in the columns denoted by B are birth-rates,
and in those denoted by M are marriage -rates. Fig. 53 shews both results
by dots, the general trends — ignoring small oscillations — ^being indicated
by the broken lines among the dots. The scale of values for the marriage
curve is ten times as much enlarged as that for the birth-rate curve : see
p. 165.
The long-continued fall in the birth-rate, which has been character-
istic of Australia, is characteristic also of the countries of the western
world. The rates for as many years as are available for various countries
are as shewn in the table hereunder. These are also graphed in Mg. 53.
They afford unmistakable evidence of what may be called the Malthusian
NATALITY.
161
drift of the world during the last 50 years, which drift, however, is on the
whole contemporaneous with a conspicuous reduction of infantile mortal-
ity, so that the " residual birth-rates" would shew a much less marked
effect. It is after all the residual rate which is of greater importance.
TABLE XXXIX.
Crude Birth-rates for Various Countries-'1860-1914— per 10,000 of the Population.
<
England and
Wales.
Scotland.
1
s
1
1
IX
t
1
1
CO
1
i
s
1
3
<
si
I
1
g
I860..
426
343
356
262
386
348
319
306
379
S81
1861..
423
346
349
269
377
326
sis
354
308
372
344
1862..
433
350
346
266
372
334
310
332
301
379
242
1863..
417
353
350
269
395
336
311
364
318
403
352
1864..
429
354
366
240
266
397
379
336
303
357
315
403
345
1865..
421
354
355
257
266
393
385
328
314
361
314
378
344
1866. .
398
352
354
262
264
393
390
331
322
354
327
379
421
350
1867..
404
354
351
260
264
371
367
308
305
354
321
366
388
340
1868..
405
358
353
268
257
369
354
275
312
349
325
379
424
341
1869..
387
348
343
267
257
379
372
282
295
343
316
393
426
339
1870. .
387
352
346
277
255
383
369
298
288
305
361
323
396
417
339
1871..
380
350
345
281
229
338
370
291
292
304
302
354
310
389
430
331
1872..
371
356
349
278
267
397
379
300
297
300
303
360
323
391
410
339
1873..
374
354
348
271
260
396
363
299
299
308
308
362
325
399
422
339
1874..
368
360
356
266
262
401
349
305
307
309
309
364
326
397
427
334
1875..
359
354
352
261
269
407
377
320
312
312
319
366
325
399
450
346
1876..
360
363
356
284
262
407
392
330
318
308
326
371
332
400
463
350
1877..
350
360
363
262
255
399
370
323
318
311
324
366
323
387
436
343
1878..
354
356
349
261
252
387
362
316
311
298
317
361
315
386
431
337
1879..
358
347
343
252
261
390
378
308
320
305
320
367
315
392
468
340
1880..
352
342
336
247
246
378
339
298
307
294
318
355
311
380
428
323
1881..
353
339
337
245
249
370
380
300
300
291
323
350
314
377
429
351
1882..
345
338
335
240
248
367
371
291
309
294
324
363
312
391
438
331
1883. .
348
335
328
235
248
371
372
288
309
289
318
343
305
382
448
328
1884..
356
336
337
239
247
376
390
285
310
300
334
349
305
387
466
334
1885..
357
329
327
235
213
377
386
280
313
294
326
344
299
376
448
328
1886..
354
328
329
232
239
377
370
280
309
298
326
346
296
380
466
328
1887..
356
319
317
231
235
377
389
280
308
297
320
337
294
382
442
326
1888..
355
312
313
228
231
374
375
278
308
288
317
337
291
379
438
322
1889..
346
311
309
227
230
371
383
276
297
277
313
332
295
379
437
313
1890..
360
302
304
223
21«
366
358
264
303
280
306
329
287
367
403
311
1891. .
345
314
312
231
226
377
372
278
309
283
309
337
296
370
423
319
1892..
337
304
307
225
223
363
362
274
296
270
295
320
289
362
404
309
1893..
328
307
308
230
228
375
365
277
307
274
305
338
295
379
426
316
1894. .
308
296
299
230
223
366
356
273
298
271
301
327
290
367
416
307
1895..
304
303
300
233
217
369
349
273
306
276
300
328
285
381
418
310
1896..
284
296
304
237
226
369
348
281
304
272
306
327
290
380
405
309
1897..
282
296
300
235
222
365
347
283
300
267
298
325
290
375
403
306
1898..
271
293
301
233
218
367
335
285
303
271
302
319
286
363
377
302
1899..
273
291
298
231
219
363
339
290
309
264
297
321
288
373
393
303
1900..
273
287
296
227
214
361
330
286
301
270
297
316
289
373
393
301
1901..
272
285
295
227
220
362
326
290
296
270
297
323
294
366
378
300
1902..
267
285
293
230
217
366
334
286
289
265
292
318
284
371
389
298
1003..
253
285
294
231
211
344
317
274
288
267
287
316
275
353
369
290
1904. .
264
280
291
236
209
347
329
273
281
268
289
314
271
356
374
290
1905..
262
273
286
234
206
335
327
269
274
267
284
308
261
339
363
285
1906..
266
272
286
235
206
337
321
269
267
257
286
304
257
350
365
285
1907..
268
265
277
232
197
330
317
262
264
265
282
300
253
340
367
281
1908..
266
267
281
233
201
227
337
264
263
257
286
297
249
337
369
282
1909..
267
258
273
234
195
317
327
265
263
256
282
291
237
334
377
278
1910..
268
251
262
233
196
305
333
250
261
247
275
286
237
325
367
273
1911..
272
244
256
232
187
294
315
242
259
240
267
278
229
314
350
265
1912..
286
238
259
230
190
289
324
241
266
237
267
281
226
313
363
247
1913..
282
239
265
228
190
■•
252
231
256
281
••
••
246
Mean
354
335
338
243
236
366
357
284
296
287
304
335
296
374
411
1
162 APPENDIX A.
This curve of birth-rate averages, convex upwards, discloses a con-
tinuous and accelerating decrease with time, i.e., the Malthusian gradient
shews an accelerating increase. This fact is significant, and is worthy
of special notice. If the tendency to reduction of the birth-rate were one
which bore a constant ratio to the rate itself we should have —
(367) d^/dt = - K/3; or dj8/j8 = — Kdt;
j3 denoting the birth-rate, and k a constant. In this case we should
have by integrating —
(368) log. ^= - «:« 4- c ; or j8 = Ce-"'
in which log. C = c, or C = e», a curve which of course is concave up-
wards, not convex. Again, a linear diminution of the birth-rate, viz.,
one of the form —
(369) j8= )8o(l - kt)
is an accelerating reduction of the relative increase of the population by
births, consequently the convexity upwards of the curve implies a still
more rapid reduction than a linear one.
11. The Malthusian law. — ^The question naturally arises whether
the birth-rate phenomena may properly be considered as conformable to
the law enunciated by Malthus, which may be stated thus : — If, as time
goes on, food-production increases in an arithmetical, while population
increases in a geometrical, ratio, the latter must inevitably overtake and
surpass the former.
Let us suppose that from any given moment [i.e., for t =0) the
(possible) increase of food-production is continually in the ratio \ -\- qt,
and that the population increase is continually e'^- Both expressions
are unity for < =0. Let q = Mr. The factor M, we may suppose to be
considerably greater than unity. Thus for small values of t, the value
of 1 -\- qt is greater than e". To determine the value for t, when the
two expressions become equal, we put —
(370) \ +qt = 1 + Mrt =e.''K
By expanding, subtracting unity from each side of the equation, trans-
posing and dividing by \r, we get for Y, the number of years when the
population wiU overtake the food supply.
(371). . Y= 2_(^:il) _ ^ (J ^1^^ ^ i^^^, ^ _i_ ^3^3^ ^^^
NATALITY.
163
The ratio r being very small, this equation may be solved for t by successive
approximations, for which purpose the equation is preferably written in
the form —
(371a)..,
Y^m^^,,,
-rt[l-
3
; t(l + lrt+.
4 5
•)]\'
The values of r range up to about 0.03. The solution of this leads to a
remarkable result, viz., that if the food-supply can be increased in the
ratios 1, 1 -f Mr, 1+2 Mr + . . 1 + tMr, as the years pass, then after
a relatively small number of years there will be a shortage, though in the
interim there will be an excess. The interval of time necessary, with
population increasing continually at any given rate, and the food-pro-
duction increasing by uniform amounts per annum, may be called the
'■ Malthusian equivalent interval."
12. Malthusian equivalent interval. — ^These intervals have been
computed, here, for the cases where the annual increase of food supply
is either 2, 4, 8, or 16 times that at which the population is continually
increasing.
TABLE XIi.— Malthusian Equivalent Intervals corresponding to various
Rates of Increase.
Number of Times
Food Supply Ex-
oeeds Needs of
Population.*
Number of Years (t) Beiore Population Overtakes
Food-Supply, the former inoreasing as e^, the
latter as (1 + Mrt).
M.
H.
r = 0.01
r = 0.015.
1
r = 0.02 1 r = 0.03.
2
1.2664
125.6
83.8
1
62.8 41.9
4
2.3370
233.7
155.8
116.8 : 77.9
8
3.3160
331.6
221.0
166.7 110.6
16
4.2290
422.9
1
281.9
211.4 •. 141.0
* Initially.
The above table shews that even if the possibUity of increasing the
food supply was initially larger, the interval of time elapsing before the
increase of population would overtake that of food supply is small com-
pared with historical or geological periods. Thus it wiU be seen that
the conclusions of II., § 34, pp. 30-32 hereinbefore, are supported from a
somewhat different point of view ; in other words, the general truth of
Malthus' proposition is certain. In short, the ordinary rates of population
increase, small as they may appear to some investigators, are sufficient
^If a table of values of e^ be available, we may rewrite (371) in the form
Ma; = e^ — ■ 1, consequently M = (e* — ■ 1) / a; and this can be solved by trial. The
following values will serve for most oases required.
X = .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 2. 3. 4. 6.
}£ = 1.052 1.107 1.166 1.255 1.297 1,370 1.448 1.532 1.622 1,718 3.195 6,362 13.400 29.485
164 APPENDIX A.
to bring about, in a relatively short time, trouble from over population,
at least in countries where the population density is already appreciable.
In this connection it may be of interest to mention that the rates for
Australia deduced from the populations determined as at the censuses
of 1901, 1911, and from the deaths for that intercensal period, shew that
for the number of males and females at birth to be each constantly
100,000 per annum the constant populations would have to be as follows —
Fob Malhs. Fob Females.
Population (To) .. 5,520,030
Number bom m . . 100,000
Expectation of life (e°) 55,200
Death-rate -^ = — 3- = 0.01812
^0 e°
5,883,742
100,000
58,837
0.01700
Thus if the death-rates and birth-rates were equal, a constant
population of 5,701,886, with a crude masculinity of 0.938184, would
give 100,000 births, ia which the numbers of the sexes would be initially
equal. The masculinity of the actual population based on the aggregates
for the years 1901 to 1910 inclusive was actually 1.115600. The ascer-
tained increases of population due to excess of births over deaths and of
immigrants over emigrants were, however, as follows : —
Population at 31st December, 1900 —
Males 1,976,992 ; Females 1,788,347 ; Total 3,765,339
Population at 31st December, 1910 —
Males 2,296,308; Females 2,128,775; Total 4,425,083
Decennial Increase of Population —
Males, 319,316 ; Females, 340,428 ; Total, 659,744.
Rate of continuous increase—
r^ = 0.0149726 ; r, = 0.017426 ; r^ = 0.0161435.
As already shewn this rate of increase could not, of course, possibly
be maintained over many centuries.
13. The Malthusian coeflacient and Malthusian gradient.^Let us
suppose that in any community unrestricted fecundity would give a
birth-rate B, the actual birth-rate, however, being j3 = iiiB. The value
of in is necessarily less than unity, and is a coef&cient of reduction which
diminishes in the ratio that the birth-rate diminishes. This coefficient
may be called the Malthusian Coefficient, since it measures the degree of
the restriction of fecundity which characterises the community in ques-
tion .1 Thus—
(372) m = /3/B.
1 See the Essay on the Prmoiple of Population ; or a view of its past and
present effects on human happiness ; etc., by T. R. Malthus, A.M. ; Professor of
History and PoUtioal Eoonomy, East India College. In partioular Chan IX
pp. 506-636, 40 Edit., 1807. ^ ^'
NATALITY.
165
BIRTH-RATES OF VARIOUS COUNTRIES, 1860 to 1913.
Hungary
.040 lor Hungary
.035 for Austria.
Prussia.
Austria.
Switzerland.
.020 for Prussia & Switzerland
Netherlands.
Belgium.
Italy.
.020 for Netberlands and
Belgium.
Scotland.
Sweden.
.020 tor Scotland, France,
Sweden and Italy,
itence.
Denmark.
.025 for Denmark.
.030 for England and Wales.
Norway (broken line).
3ingland and Wales.
Ireland.
.020 for Ireland and Norway.
.035 tor General trend West-
ern World.
Australia (heavy line A dots).
Aust. Marriages (light do.
.030 for General trend West-
ern World.
General Trend for Western
World. + + +
.0080 tor Marriage Ilates.
Australia.
.0075.
.0070
.0065
.0060
1860
1870
1880
1890
1900
1010
Pig. 53.
The heavy curve with heavy dots is the general trend of the Australian birth-
rates, the dots denoting the individual annual rates. The light curve with light
dots is the general trend of the Australian marriage-rates in Australia, the dots
denoting the individual annual rates. The light curve with vertical crosses -|- -|- +
is the unweighted average of the various birth-rate curves, the crosses denoting the
mean of the annual values.
166 APPEOTJIX A.
If j8 = /(<), we shall have also Bm = / (t), that is the graphs of the two
are of the same form and differ only in scale. The question of the deter-
mination of this coefficient wiJl be later considered.
The general change in the values either of the birth-rate or of the
Malthusian coefficient, if regarded as characteristic of any country, is
best found by taking the means irrespective of the magnitude of the jpopula-
tions. A mean so found does not, of course, apply to the aggregate of the
populations ; a weighted mean would do so however.
The Malthusian gradient (n) may be defined either as the rate of
fall of the curve representing the birth-rate, or as the rate of fall of the
curve representing the Malthusian coefficient. Until B is determined,
see above, the former definition is the more practical. Thus —
^•^^^' "^--w -ST' °^-mt-^Edr
of which the former is, at present, to be preferred.
14. Reaction of the marriage-rate upon the birth-rate.— If the
marriage -rates were constant in all communities, or were constant at
different periods in the one community, the birth-rates would be properly
comparable as measures of fecundity. The effect of the marriage -rates
depends, however, upon the degree of fecundity characteristic of different
ages of life ; hence exact comparisons of the fecundity are possible only
after a correction is apphed depending upon its variation with age and
the age distribution of the marriages. This question will be considered
hereinafter. It will suffice to observe that there is some sUght indica-
tion of the correlation of the facts exhibited by the curves representing
the birth and marriage rates, Fig. 53, though the effect is easily masked
by the economic factors which influence human affairs : see p. 165.
15. Annual periodic fluctuation of births. — In order to see whether
there was any distinct evidence of a seasonal fluctuation among births,
the numbers of births have been compiled, according to the actual date
of birth, for the three years 1911-1913, in equalised half-months, and,
assuming the rate of increase of population to be uniform, these numbers
were corrected for its general increase during the period. The results
are as follow : —
NATALITY.
167
TABLE XLI
—Seasonal Fluctuations in the Coriected Frequency of Births.
AustiaUa, 1911-1913.
Equalised
Montbs.
January.
February.
March.
April.
May.
June.
Nuptial . .
14,444
14,350
15,045 14,893
15,766
15,058
15,098
15,218
15,132
15,202
15,347
15,446
Ex-nuptlal . .
893
806
831] 835
896
870
872
857
912
887
905
1,003
Totals . .
!Ratio of total
15,337
15,156
15,876 15,728
16,662
15,928
15,970
16,075
16,044
16,089
16,252
16,448
to Aggregate
lor the Year
.9604
.9491
.9942! .9849
1
1.0434
.9974
1.0001
1.0066
1.0047
1.0075
1.0177
1.0300
Equalised
Months.
July.
August.
September.
October.
November.
December.
Nuptial
15,542
15,513
15,367^ 15,216
15,514
15,715
15,183
14,323| 14,787
14,398
14,352
14,778
Ex-nuptial
975
949
986 961
946
903
955
902
872
849
846
867
Totals . .
Baldo of total
16,517
16,462
16,353 16,177
16,460
16,618
16,138
15,225
15,659
15,247
15,198
15,640
to aggregate
for the jrear
1.0343
1.0309
1.0241 1.0130
i
1.0308
1.0406
1.0106
.9534
.9806
.9548
.9517
.9794
The semi-monthly means for nuptial, ex-nuptial and total births
were 15,070,899, and 15,969 respectively. An examination of the results
shews that a fluctua-
tion certainly exists,
and although the
number of ex-nuptial
births is both rela-
tively and absolutely
very small, the agree-
ment of the fluctua-
tion of nuptial with
that of ex-nuptial
births is fairly defi-
nite. The results for
the totals are shewn
by curve A, Fig. 54,
the curved line de-
noting the general
trend of the fluctua-
tion.
In order to further
examine the ques-
tion, the quarterly
results of the births,
as registered, during
the eight years 1907-
1914 were compiled,
and this is done also
for population. The
interval between
~
-
e
/
1
u
3
jf
t
^
/
>
' /'
\
1
ri
i
f
\
/
0.
n
4
kJ
I
?'
\
1
^..^
t ,
\
i''
' i
t\
7
' '
,-»
1
~"
i
>
"
/
1
/
V
/
>
1
/
-a
—
—
y
\
/
\
^
\
n
7
\
11]
y
'
10 -
«. ^
^
^
^''
1
-
\
\-
y
\
0.07 ■
,L
\
r
-
s
/
_
5«
^
r
~
Ja Fe Mr Ap My Jn
Fig.
Jy Ag
54.
Se Go No De Ja
birth and registration has, however, shortened since the introduction
of a maternity bonus : see pp. 151 and 152.
168
APPENDIX A.
The following procedure was adopted. The births registered were
taken out in the several quarters ; these quarters were then equalised,
the numbers being corrected to shew what would have been given by a
constant population, since it was found that the increase of this last was
sensibly at the rate 1 + 0.0247265 t. In this way the values shewn in
Table XLII. hereinafter were obtained. These quarterly results may be
subdivided into monthly values, as explained on the next section, so as
to give the monthly values. These results are shewn by the curve B in
Fig. 54.
TABLE
XUI.— Birtbs Registered. AustraUa, 1907-1914.
Births aa Registered.
Births as Corrected for Equal Quarters
and » Ck)nstant Population.
236.462
243,191
254,141
242,860
241,457
.98891
244,914
1.00307
251,467
1.02987
238.830
.97816
The values for the individual months may be deduced as explained
in the next section, and are as follows : —
123456 78 9 10 11 12
.9807 .9916 .9944 .9936 .9996 1.0160 I.Q333 1.0366 1.0197 .9924 .9922 .9699
and these monthly results are shewn by the small rectangles in curve B,
Fig. 54.
For the greater part of the year, at least, the results are substantially
identical for the two sexes, as a compilation made for the four years, 1907-
1910, shews. The results were as follows : —
TABLE XLm. — Seasonal Fluctuations* of Births, according to Sex.
Australia, 1907-1910.
Males,
Females
or Persons.
Jan.
Feb.
Mar.
April
May.
June.
M
F
P
.9874
.9903
.9889
.9169
.9229
.9198
.9949
.9950
.9949
1.0162
1.0079 1
1.0116 '
1.0064
1.0069
1.0067
.9978
.9859
.9920
July.
Aug.
Sept.
Oct.
Nov.
Deo.
M
F
P
1.0321
1.0170
1.0249
1.0410
1.0583
1.0504
1.0299
1.0437
1.0367 1
1.0378
1.0465
1.0420
.9924
.9760
.9844
.9482
.9479
.9480
* The registration was on the avert^e 38.2 days after birth for the years
1907-1910.
Reverting to curves A and B, Fig. 54, the curve drawn by lines may
be taken as a probable representation of the fluctuation ; since there is
no reason to suppose that the large oscillations are other than accidental.
NATALITY. 169
As the theory of determining the Fourier curves to fit the group
results presents certain special features, it is given hereunder.
16. The subdivision of results for equalised quarters into values
corresponding to equalised months. — When quarterly results are available,
they may (after equalising and also being freed from the annual pro-
gression so as to give, as residuals, only the fluctuation elements) be readily
resolved into monthly values, which have a high degree of probability.
The most convenient form in which to give such results is the height of
the monthly group. Let the mean of the heights of four quarterly groups
be denoted by R, with suffixes corresponding to the quarter (viz., 1 to 4),
and that of the monthly group by r, with corresponding suffices (viz.,
1 to 12). Then the solution can proceed on one of two possible assump-
tions, viz. (a) that the amplitudes of the component fluctuations are
identical, and the epochs are different, or (6) that the epochs are identical
and the amplitudes are diflEerent.^
That is, we may assume either (a) that —
(375) y=a-\-b sin. (a; + j8 ) + 6 am. 2 (a; + y) ;
or (6), that —
(376) y =a +b sin. (x + ^ ) + c sin. 2 {x + j8).
The data are, of course, inadequate in themselves to determine
which assumption should be adopted, and the results are to that extent,
uncertain. But this uncertainty, in general, is of small moment.
In case (a) we have —
(377). J -= bcos.^ = ^{Bi + Bi); m = ~bsin.^ =^{Ri+Ri),
3 3 3 3
(378). .2)= 6co«2y = -(E2+-B4) ; g=-hsin2y=-^y/ —^B.yB^-^Ei,B^)
It will be seen that q is not independent of Z, m and p, since we must
have —
(379) g2 == (^2 _|_ ^2 _ 4 ^2)
From this last, the value ^\/ —1 (RiRt -f RiRs) is deduced. Ob-
serving that ^ ^/3— 1 = — 0.1339746 ; ^ (1 — \/3) = - 0.3660254 ;
^ y'3 = 0.8660254 ; we may put the values of ri to r^ in the following
very convenient forms, viz. :—
(380) n = — 0.1340 Z+ ^ m - ^ P -f 0.8660 g.
(381) rg = — 0.3660 I + 0.3660 m - P
^ See Studies in Statistioal Representation (Statistiool Applioations of the
Ppurier Series), by Gr. H. Knibbs, Joum. Roy. Soo. New South Wales, Vol. xlv.,
pp. 76-110, 1911. In partioular see pp. 88-89.
170 APPENDIX A.
(382) rs =— 4" ^ + 01340m - ^ P— 0.8660 g.
(383) U =- J ^— 01340 m + -i P - 0.8660^.
(384) rs = ^ 0.3660 Z — 0.3660 m + P
(385) re = 1.8660 Z — -^ m — — P + 0.8660 q.
(386) ry =+ 0.1340 Z— ^ m — ^ P + OMGOq.
(387) fa =+ 0.3660 Z — 0.3660 w — P
(388) rs =+ 4 Z- 0.1340m —\ P— 0.8660 g.
(389) rio = + -5^ ^ + 0.1340 m -\- ~ P — 0.8660 g.
(390) rii = + 0.3660 Z + 0.3660 m + P
(391) n% =— 1.8660 Z + ^ "* — 4 ^ +0.8660g.
In case (fc) we have —
(392).... Z= -- 8 6 cos = -^{Rz^R^); m== bsin^=-^ (Ri+ Ri).
(393). . . . P = — I c co« 2j8 = I (2?2+i24) ; 3 = | V (^J (-^3 + -B4)*
+ (JJi + P4)^]-(iJi+ Rs)^\
Again, q is not independent of Z, m, and p, since we have —
(394) *^^=i^;=-^^=^(^^+-^)-^^-
which leads to the value of q above written.
If c = 6, the last expression for q in (394), reduces to that first given,
viz., in (379). It is obvious from this last value for q, that the ratio
cjb is at our disposal, and provided it be so chosen that the whole expres-
sion within the braces is not negative, there wiU be a real value for q. A
unique solution will be that which makes the q term zero in the above
series of equations for monthly values. This is given by making the
expression within the braces in (393) zero. Hence for this we have
(395) j= {Ri+Rs)/VHR3 + Ri)^+(Ri + R*)^'i
If, therefore, the relation between j8 and y, and between b and c are both
unknown, we may, with advantage, write g = in the series of equations
380) to (391). In short, if we assume that c = b then y is determinate.
NATALITY. 171
If this relation be not assumed, but that y = ^ is assumed, we may,
vnthin certain limits, still make the ratio of c to 6 whatsoever we choose,
and, if we have no ground for believing that a particular ratio is to be
preferred, the simplest solution of the whole problem is, making the
epochal angles fi and y identical, to so take the ratio of c to 6 that the
q term will be eliminated from the series of equations for monthly values,
viz., formulae (380) to (391), etc. ; that is, we may determine this ratio
by (SOS)-*^. It may be reiterated that the subdivision of the quarterly
into monthly values by the preceding formulae assumes that the fluctua-
tion involves only terms sin. x and sin. 2x.
17. Equalisation of periods o£ irregular length. — ^In order to apply
the formulse of the preceding section, it has already been indicated that
the crude data must be freed from any annual progression depending on a
progression in population numbers and among the births themselves. It
is preferable to operate, therefore, on rates, i.e., to divide the number of
births (or marriages or deaths, etc.) each month, quarter, or year, as the
case may be, by the mean population of the month, quarter, or year itself.
Even then a correction is necessary, since for precise results it is still
necessary to equalise the period, in fact, if the seasonal fluctuation (or
armual period of oscillation) to be determined be small in amphtude, the
equaUsation is an essential. Both months and quarters differ appreciably
in length.^
For population-numbers and for birth-numbers, the equalising
corrections will necessarily be made in a somewhat different manner. A
table of corrections for the ends of the months or quarters is first formed.
Numbers such as population-numbers and rate-numbers may be called
continuant, B,TaA. those such as numbers of birth, marriages and deaths,
etc . , accretional. For the purpose of corrections it may also be assumed that
the daily values at the terminals of the unequal periods is the mean of the
values for the adjoining periods.^
Then, except for the first and final period, there are two corrections.
For a single leap-year there is no correction at the end of August, and none
at the end of October. The equalised February is always in January,
and excepting as above mentioned the terminal of the equalised month
is always in the month follomng.*
' Suoh a solution has the further advantage of making the deviations from the
averages for the respective quarters a minimum.
' The shortest month is no less than 8 per cent, short of the average, and
shortest quarter 1.37 per cent.
' It is more rigorous, of course, to determine the function, the integral of which
gives the result dealt with, but this process is tedious and ordinarily quite un-
necessary.
* There would have been some advantage if January had had 30 days, instead
of 31, and February 30 days in ordinary and 31 in leap years, instead of 28 and 29
days.
172 APPENDIX A.
Let 8 T and S T be the small periods to be added respectively to
the beginning and the end of an unequal period to make it coincide with
an equalised period, the length of this last being T^. Let also the periods
preceding and following that to be corrected be denoted by T and T' ;
and let the period to be corrected be denoted by T^. Then, the correct-
ing periods Sy, etc., being small, we have very approximately, for
continuant numbers, P, P„ and P', etc., denoting that corresponding
to To,
(397)....Po=P„+ ^ |{P-P„)8r+(P'-P„)8'T;.
and for accretionai numbers, N, N„, N', etc., N^ denoting that corres-
ponding to the period T^,
= Nm+^J-{N+N^)ST+(N„ + N'}S'Tl
The approximate identity of these expressions can readily be estab-
lished.^ In regard to the sign of the corrections it may be observed that
for continuant numbers the value is to be increased when the shift of either
terminal of the unequaUsed period towards the terminal of the equalised
period is in the direction of higher values. For accretionai numbers, the
number is increased for an additive shift, diminished for a negative shift.
18. Determination of a purely physiological annual fluctuation of
birth-rate. — ^The annual birth-rate fluctuation, as obtained in section
15, by means of the formulae of sections 16 and 17, cannot be regarded
as furnishing the variations of the reproductive activity solely due to
physiological causes, which variations may be presumed to repeat them-
selves every year. The distribution of the frequency of marriage, and
therefore of birth, throughout the year is afiected by the fetes observed,
and particularly by the " movable feasts" (Easter, etc.). The number
of years to be included to secure a true mean-determination must embrace
the whole cycle of movement. The extent of this cycle has been referred
to in a paper on the Statistical Application of the Fourier Series, by the
writer.^ But even when this mean result is obtaiaed, what may be
called the physiological fluctuation is not to hand, since the effect of the
" movable feast" is distributed, not eliminated. By a systematic analysis,
^ The question of corrections of this kind has been dealt with at length by me in a
paper read 5th July, 1911, at the Roy. Soc, N.S.W., see its Joum. xlv., pp. 79-85,
wluch treats of the correction of an increasing population, and that for unequal
months, quarters, half and whole years.
" Vide Journal Royal Soo. N.S.W., Vol. xlv., pp. 76-110.
NATALITY.
173
however, of the results for different years in which the place of the mov-
able feast is as different as possible, the effect of this distribution can be
ascertained and corrections applied to ehminate the effect. The diffi-
culty of a perfectly satisfactory solution will be apparent from Fig. 55
hereunder.
' 19. Periodicities due to Easter. — As ecclesiastically defined, Easter
Day is the first Sunday after the 14th day of the paschal " Calendar
Moon," a fictitious ecclesiastical moon, which is from one to three days
later than the real moon. The average position of Easter for the century
1800 to 1899 is April 8.55 days, and for the ceiitury 1900 to 1999 is April
8.89 days, or say for the whole period of 200 years April 8.72 days. In
Fig. 55 the Easters in each decade are shewn on a single Une for the years
1800 to 1999 inclusive. An inspection of the figure shews that the points
lie approximately on a series of 10 slanting lines, four days apart, these
lines progressing at the rate of one half day per decade, and further that
they are inversely symmetrical. For lines a, b, c, and e and a', b', c',
and e' the symmetry is perfect ; for lines d and d' however the symmetry
is not absolutely perfect. It is evident that no means derived from two
decades nor from periods of 19 years, nor from centuries are exactly
comparable.
POSITION OF EASTER FOR 200 YEARS.
March April
2zaitazizisaxii < z i i s 6 ? s a lO n iz u i4 is le i7 is lazozi zzbzazs
-^Yr-^r--- p^-^-t^-^ — ^r--
'\ ' V S, ^, \ 1 S \
. V . , 1 , (1 . . 1 1 1 f 1
Y S, ^^ ^ , \, i\ \ , ^ i''
■^t ^ -s zi i: :e> it^n^ s* ^^-
V ^ . '. %. I' ■ ' i^ A V-
'' , ■ , ^ S,^ 1 S, !,N \>
t±^ =ldS± = ^t^^;STS^=^S"::
U4--1^--A-.-^^_4^^^^^^ + ^ ^- ^
'' \ \ ■ ' v ' . ! ^ \ ^
\ ' '\i ';; ■^, \, 1 :.,, y, :;,
\, t 1 V ' ^ . < \
\4 \: 1 X , , f V,, T
^ i N, ^ { \, '^v N
' ^ < i I ^ t',' , 1 ' , r '■ 1
-l; ,j_^i_^^ -^.-JJ 11 :i^. Jl .
H-f-T-L-K - - -rMi- -\ -^-f - f 1 f - ^ H -hhW^^-^^^mHv^^J
1800' 9
1013
zoa
30-39
40-49
50-59
eo-G3
70-73
80-89
90-93
20-29
50-33
40-49
50-59
eo-es
70-79
80-89
30-99
Fig. 55.
Since the tropical year = 365.2422 days and the synodic lunar
month = 29.530588 days, the Metonic cycle, *19 tropical years is 6939.6018
days, and 235 complete lunations equal 6939.6882 days, differing only
.0864 day from the nineteen years.
174
APPENDIX A.
The following table exhibits the peculiarities for successive decades.
TABLE XLIV.— MEAN POSITION OF EASTER FOB 200 YEARS.*
-
1800
1900.
Easters
Mean of
Mean of
Easters
Mean of
Mean of
DeoEkde.
Mean.
in
March
April
Decade
Mean.
in
March
April
March.
Easters.
Easters.
March.
Easters.
Easters.
April.
April.
0-9
9.46
1
29
9.56
0-9
10.66
2
30.5
13.12
10-19
8.16
3
25.67
13.57
10-19
8.36
3
27
12.86
20-29
8.86
2
28
11.50
20-29
8.06
2
29
10.25
30-39
7.86
3
29
11.43
30-39
9.66
2
27.6
12.62
40-49
9.06
2
25
12.50
40-49
8.96
2
26
12 12
60-59
8.56
3
27
13.57
50-59
7.56
2
27
10.38
60-69
7.96
3
28.67
11.86
60-69
9.66
2
27.5
12.88
70-79
9.36
2
29.5
11.88
70-79
8.66
3
28.33
13.14
80-89
9.46
2
25.5
12.62
80-89
8.66
2
28
11.26
90-99
6.76
2
27
9.25
90-99
8.76
2
30.5
10.88
Means
8.55
2.3
27.48
11.70
Means
8.89
2.2
28.09
11.94
* The complete Eaater Cycle, restoring both the day of the week and of the
month, is known as the " Dionysiaii" or " Great Paschal" period. Its length
is 4.7.19 = 532 years.
To obtain a normal periodic fluctuation it would be preferable,
were it practicable, to combine the results, each for a series of years such
as would give Easter an identical distribution. In the period such a
series is, however, impracticably long. Hence in the case of marriage,
birth-rate, migration, etc., it is necessary to consider the actual effect on
the periodic fluctuation studied. In respect of marriages the effect
of Easter is to reduce the number of marriages in the Lent period (6
weeks) preceding, and to augment them in the preceding and following
periods.
It may be noted that for the fluctuations of annual period in the
marriage frequency, the great length of the Lent period, viz., 6 weeks,
has the effect of throwing the increase of frequency as far back as Febru-
ary. The migration frequency is often thrown back into March. Thus,
as is evident from the preceding table and the diagram, decennial means
will clearly be nearly but not exactly comparable. The data for a thor-
ough study of periodic fluctuation would in these cases have to be weekly
groups.
Xn.— NUPTIALITY.
1. General. — -The phenomena of reproductioivhave a double aspect,
viz., one a sociological and the other a physiological. Thus, from the
standpoint of a- theory of population, both are important. The women
of reproductive age in any community furnish the potential element of
reproduction ; but the resolution into fact depends also upon social
facts as well as upon physiological ; for example, the relative proportion of
married and single, i.e., the nuptial-ratio, even more profoundly affect the
result than physiological variations of fecundity. In Chapter XVIII. of
the Census Report (Conjugal Condition), the numbers of married and un-
married females have been given as at 3rd April, 1911, in Australia.
These will be considered mainly in regard to the child-bearing age, in
dealing later with fecundity.
2. The Nuptial-Ratio. — ^The " nuptial-ratio," j, may be defined as
the ratio of the married, J, to the unmarried, U, which latter may be
taken generally as including the never married, the widowed, and the
divorced. This ratio, J/ U may apply to either sex and to any age,
or age-group, or to the total for aU ages, etc. The nuptial-ratio in any
community may be regarded as a measure of the social instinct, and also
a measure of the reproductive instinct, modified by social traditions as
well as facilitated or hindered by economic conditions. This ratio, for
the case of females, is, of course, specially important in relation to
fecundity.
The significance of marriage in respect of reproductive activity
depends upon the relative frequency of nuptial and ex-nuptial births,
as well as upon the relative proportions of the married and unmarried,
that is, it depends not merely upon the nuptial-ratio, but also upon
nuptial and ex-nuptial fecundity, particularly during the reproductive
period of life. The values of
(399) j = J/U
for various countries are given in the following table for women during
the reproductive period, and for women of all ages, viz., from age to
the end of life.
TABLE XLV. — ^Ratios of Married Women in various Age-groups to Unmarried Women
in the same Groups. Reproductive Ages. Female Nuptial Ratios.
Ages of
Aust. Census, 1911t
C'wlth
Aust.
1908.
England
and Wales.
1901. 1911.
Scotland.
1901. 1911.
Ireland.
1901. 1911.
Bel-
gium.
1910.
Germany.
Women.
Metro
porn.
other.
Tc(tal.
1900. 1910.
10 to 141ncl.
15 „ 19 „
20 „ 24 „
25 „ 29 „
30 „ 34 „
35 „ 39 „
40 „ 44 „
45 49 „
50 „ 54 „
55 „ 59 „
60 „ 64 „
.0000
.0337
.03510
1.0945
1.8201
2.2491
2.5045
2.4617
2.0628
1.5747
1.0622
.0000
.0435
.4892
1.6325
2.8810
3.5996
3.9037
3.6935
3.1420
2.3651
1.5761
.0000
.0395
.4242
1.3613
2.3318
2.8938
3.1586
3.0324
2.6634
1.9470
1.3070
.0001
.0382
.4214
1.2997
2.4698
2.9805
3.1159
3.1068
2.6025
1.8482
1.5815
•
.0000 .0000
.0157 .0121
.3731 .3184
[ 3.0124 ;|;0?«?
} 2.3915 IIS
} 1.3217 {i:«tit
.0000 .0000
.0767 .0145
0.3049 .2758
[ 1-3759 -; Jill
1 2.2854 ||3«|
I 1 aioQ 1 2-0750
r ^■'"■^ 1 1.6795
I 1 nnn? ^ 1-3061
r lOOO^ 1 0.9089
.0000 .0000
.0075 .0063
.1538 .1538
\ .8397 .8137]
|- 1.6777 1.7040 1
}■ 1.4343 1.5443 1
\ .8686 1.0490 j
-0000
.0271
.04482
1.6385
2.8324
3.4697
3.3632
2.8921
2.2601
1.5909
1.0929
.0000 .0000
.0161 .0139
.3977 .3959
1.8172 1.9359
3.63813.8471
4.2516 4.4905
3.80124.0635
3.00863.2488
2.16352.3415
1.48641.5995
.095901.0353
„ 106 „
.5231
.5198
.5218
.5159 .5528
.4293 .45X6
.3643 .3765
.5781
.5200 .5466
• Ages 60 to 61 only. f 3rd April, 1911.
176 APPENDIX A.
The results in the table shew that there are considerable divergences
between populations as regards their nuptial constitution, consequently
even if the individual fecundity were constant, the birth-rates would
differ. The results of the Australian Census of 1911 shew also that there
are striking differences between metropolitan and extra-metropolitan
communities, the marriage-rate being very much higher for the latter ;
and they shew also that the nuptiality is very different as regards the
sexes. See Vol. I., Chap., XVIII., Conjugal Condition, § 6, of the
Census Report.
3. The Crude Marriage-Rate.— The lack of homogeneity in popula-
tions, illustrated in the last section, renders the crude marriage -rate, viz.,
the ratio of the marriages, J, to the population, P, of uncertain signific-
ance. The heterogeneity arises largely from divergences of social life
and tradition, in respect of the relative frequency of marriage, and the
frequency according to age. Inasmuch, however, as ordinarily the
constitution of any population does not materially change, the marriage-
rates for any particular country and for limited periods are comparable
among one another, and their variations may generally be attributed to
variations in the economic conditions of the population in question.
Wars have, of course, a marked effect, see the points marked with
asterisks, on Table XL VI., and also Fig. 56, giving the curve of the mean
of the marriage -rates of a number of important countries. We shall
denote the crude marriage-rate by n ; thus —
(400) n = J/P.
In some countries the marriage-rate is the ratio, not of the " marriages,"
but of the " persons married," to the population. In such cases the rates
will be double those shewn in Table XL VI. hereunder, the which gives
the marriage -rates for the countries for which in Table XXXIX. the
crude birth-rates were given. This also gave the values of the marriage-
rate. In Table XL VI., the mean in the final column is merely the un-
weighted mean, and is therefore not the rate for the aggregate of the
populations. The trend, thus determined, treats each population as
equally important in regard to the revelation of the secular tendency,
if any, of the marriage-frequency. For the constitution of a norm a
weighted-mean would of course be needed.
Fig. 56 illustrates the movement in the marriage -rate, and shews
that movement in its relation to that of the western world generally
(excluding America). Although the general trends shewn by broken
lines of curves A and B, are by no means similar, there are often very
similar fluctuations about this general trend, which appear readily enough
if the general trend be regarded as a basic line about which the minor
fluctuations may be regarded as moving.
NtJPTIALITY.
177
1860
.004S
.0040
.0036
.0030
.009
.008
.007
S .006
I
.0075
.0070
.0035
.0030
.0025
.00015
.00010
.00005
70
Birth, Marriage and Divorce Rates.
80 90 1900 10
00000
■--.
"■••.
'••.
'••.
* ••»-,..
..r- •
'■
I
/^
J
^
z-^
^
f-
v.^
?\
/
r^'/
/
V^
r"
■ V
^
h
1"*
V
1
V
?<i
^s:?2
r^y
^l
"^
''^
"\-H
.....
Ap
• •*.,
Ja
pb
Mr
.. /
^
My
Ju
Jy
Ag,
sp:
Oo
Nv
Cc
W
""^
=^
-«— -
t
>v\->
r,
/«
i
'vs
A
^
\=/
y^
«^
1)
Years.
A a. Decennial mean
of Australian
birth-rates.
A. Crude mairiage-
rates, Australia
.0080
.0075
B. Curve of marriage-
.0070 rates ; mean o<
various c'ntries
Bb. Decennial aver-
ages of birth-
rates ; mean of
various c'ntries
1.5000
1.0000
G. Begistration of
marriages 1008
.6000 —1914, Aus-
tralia.
D. Belative frequency
of divorce.
1860
70
80
90
1900
Fig. 56.
10
20
Yean.
Curve Aa shews the successive decennial means of the birth-rates of Australia,
the central year being changed one year at a time.
Curve A shews the marriage-rates of Aastralia by the zig-zag line ; the fine dots
shew the successive decennial means ; the broken line, closely following the decen-
nial means, indicates the general trend.
Curve B shews the mean of the marriage-rates of a series of countries ; the fine
dots shew the successive decennial means of these ; the broken line indicates the
general trend of the marriage-rates.
Curve Bb shews the successive decennial averages of the means of the crude
birth-rates of a niimber of countries.
Curve C shews the mean annual fluctuation of the registration of marriages in
Australia for the period 1908-1914.
Curve D shews the relative frequency of divorce per unit of population for
Australia, the portion a b being prior to acts facilitating divorce ; b c being the con-
dition immediately following upon the passing of the facilitating Acts ; o d, and d e
being the subsequent trends of the relative divorce-frequency.
As regards birth-i'ates and marriage -rates, it will be observed that
here there is some indication of a correlation between the phenomena.
This correlation will not, of course, be well-marked, since the aggregate
178
APPENDIX A.
of " first births" is not large compared with " all births," But the
trend of the AustraUan birth-rate shewn by Curve Aa is strikingly
similar to Curve A shewing the marriage -rate, and Curve Bb gives some
indication of its connection with Curve B.
TABLE XLVI.
Mairiage-iates for Various Countries — 1860-1913 — ^per 10,000 of the Population.
Teai
<!
^1
GQ
1
1
fi
>>
1
1
1
to
a
1
1
.3
1
<
W
^1
So
1=
1860
84
86
70
79
84
78
82
73
85
801
1861
86
82
68
82
80
73
75
81
71
80
778
1862
88
81
67
81
85
71
74
79
71
88
785
1863
84
84
72
80
87
73
75
83
73
85
796
1864
86
86
72
48
79
87
80
70
57*
84
75
83
*756
1865
83
88
74
55
79
91
91
71
89
85
76
78
800
1866
76
88
74
54
80
78
57*
67
84
84
79
65*
82*
*745
1867
75
83
70
54
79
93
68
61
77
84
78
97
104
781
1868
76
81
67
50
79
89
72
67
65
73
77
73
92
137
772
1869
73
80
67
50
83
90
80
72
57
74
77
74
104
110
772
1870
71
81
72
63
61*
74»
74
70
60
74
80
70
98
98
•740
1871
69
84
72
54
73*
80«
75
73
67
65
73
80
74
95
104
•759
1872
70
87
•76
50
98
103
75
79
70
70
75
83
78
93
108
810
1873
74
88
78
48
89
102
79
77
73
73
81
86
78
94
113
822
1874
72
85
76
46
83
97
76
83
77
73
82
84
76
91
107
805
1876
73
84
74
46
82
91
84
90
79
71
85
84
73
86
109
807
1876
71
83
75
50
79
86
82
82
77
71
86
83
72
83
102
788
1877
73
79
72
47
75
80
78
79
76
69
81
81
69
76
94
753
1878
74
76
67
48
75
78
72
74
73
65
74
78
67
76
96
728
1879
72
72
64
44
76
77
76
70
68
63
74
77
68
78
104
722
1880
72
75
66
39
75
77
70
69
67
63
76
75
71
76
92
709
1881
76
76
70
43
75
77
81
69
64
62
78
73
71
80
100
730
1882
81
78
71
43
75
79
78
69
67
64
77
72
70
83
103
740
1883
84
78
71
43
75
80
81
69
66
66
77
71
68
79
105
741
1884
83
76
68
46
76
81
83
70
69
66
78
72
68
80
103
746
1886
82
73
66
43
75
82
80
70
67
67
76
70
68
77
101
731
1886
79
71
63
42
74
82
79
70
66
64
71
70
67
79
97
715
1887
76
72
64
43
73
80
80
71
63
63
70
70
71
79
90
710
1888
80
72
64
42
72
80
79
71
61
59
71
69
71
80
94
710
1888
77
75
67
45
71
82
77
71
63
60
71
70
73
76
82
707
1890
76
78
69
45
70
82
73
70
65
60
69
71
73
76
82
706
1891
75
78
70
46
75
82
75
71
66
69
68
71
74
78
86
716
1892
67
77
71
47
76
81
75
72
64
57
68
72
77
78
92
716
1893
62
74
66
47
75
81
74
72
65
67
70
73
76
80
94
711
1894
61
75
67
47
75
80
75
72
64
58
70
72
75
80
93
696
1896
62
75
68
51
74
80
73
73
65
59
71
74
78
81
86
713
1896
66
79
71
51
76
83
71
76
67
60
73
75
81
80
81
727
1897
67
80
72
51
76
84
72
79
67
61
75
74
83
81
82
736
1898
67
81
74
50
74
85
69
78
70
62
76
73
83
79
84
737
1899
70
83
75
50
77
85
74
78
71
63
75
74
83
83
91
755
1900
72
80
73
48
78
86
72
78
69
62
76
76
86
83
89
752
1901
73
80
70
51
78
83
73
76
66
61
72
77
87
82
88
745
1902
73
80
71
52
76
80
73
74
64
60
71
76
81
78
87
731
1903
67
79
72
52
76
80
72
74
60
58
71
75
79
78
82
716
1904
70
77
71
62
76
81
75
74
60
59
72
74
80
78
92
727
1905
73
77
68
63
77
81
77
75
58
59
72
73
79
78
86
724
1906
75
79
72
52
78
83
79
77
59
62
75
75
81
79
88
743
1907
79
80
72
52
80
82
78
77
60
62
77
75
80
76
100
764
1908
78
76
68
52
80
80
84
76
61
61
75
72
78
77
92
740
1909
79
74
64
62
78
78
78
76
60
60
74
71
77
76
87
722
1910
84
75
65
51
78
78
79
73
62
61
73
72
79
76
87
729
1911
88
76
67
54
78
80
75
74
63
59
72
72
80
76
93
738
1912
91
78
69
53
79
80
76
73
62
59
73
76
80
74
86
739
1913
87
78
71
51
75
63
59
72
78
704
1914
88
M-I13
758
791
698
487
771
831
761
740
661
634
744
761
753
815
949
NUPTIALITY.
179
4. Secular Fluctuation of Marriage-rates. — ^Fig. 56, embodying the
results on Table XLVI., reveals the fact that the relative frequency of
marriage has been increasing in Australia since 1897, although it has
tended to diminish recently in the old world. It is apparent from a
comparison of the two curves, A and B, that there is no very marked
correlation between the two progressions. The factors influencing the
relative frequency of marriage probably have a very unequal incidence in
different countries. In order to obtain an accurate measure of reaction of
the larger economic influences on the rates, statistics covering long periods
of time will be required. The characteristics of the longer or secular
fluctuations will fully appear only when much more statistical material
is available than exists at present.
The period of the larger oscillations in the data shewn amounts to
about 22 or 23 years in Australia, and about 30 or 31 years for the aggre-
gate of the populations of the western world. The period of the minor
fluctuations is very variable, and is somewhat ill-defined. In Table
XLVII. are shewn the values of successive decennial means for the
marriage-rates, and also for the birth-rates. These are shewn by dots
on Mg. 56.
TABLE XLVn.- -Decennial Unweighted Means of Marriage and Birth-rates, 1860 to 1909.
Marriages per
100,000
of the Population.
Decade
Year.
1860.
1870.
1880.
1890
1900.
Year.*
A
W
A
W
A
W
A W
A
W
740
781
758
746
734 712
687
734
1
729
781
766
739
715 710
697
732
2
724
786
774
732
701 711
707
737
3
722
783
777
727
692 714
719
739
4
721
778
783
725 680 716
729
739
5
812
779
719
773
788
724 i 673 721
740
735
6
799
773
719
770
793
723 1 669 726
749
733
7
782
771
726
767
792
722 668 729
764
732
8
764
773
737
760
778
720
673 730
782
733
9
754
776
747
752
756
717
678 731
802
732
Decade
Year.
Births
per 10
0,000
of the Popiilation.
3,894
3,396
3,634
3,382
3,435 3,179
2,743
3,009
1
3,832
3,397
3,532
3,365
3,382 3,161
2,702
2,984
2
3,793
3,397
3,526
3,343
3,313 3,142
2,683
2.960
3
3,739
3,400
3,532
3,326
3,239 3,122
2,669
2,935
4
3,688
3,396
3,533
3,311
3,155 3,102
2,663
2,915
6
4,141
3,478
3,659
3,397
3,522
3,284
3,082 3,092
2,657
2,890
6
4,102
3,436
3,625
3,381
3,519
3,272
3,005 3,082
2,651
2,862
7
4,059
3,423
3,598
3,401
3,512
3,240
2,932 3,063
2,652
2,827
8
3,997
3,420
3,572
3,393
3,503
3,218
2,863 3,052
2,671
2,776
9
3,955
3,407
3,546
3,382
3,483
3,206
2,788 3,026
2,700
2,732
A denotes the values for the Commonwealth of Australia.
W denotes the values derived from the unweighted means" for the series of
countries shewn on Tables XXXIX and XLVI.
* The moment of time to which the values apply is the beginning of the years
0, 1, 2, etc 9.
180
APPENDIX A.
5. Fluctuation of annual period in the frequency of mairiage. —
Social custom in regard to marriage expresses itself in a fluctuation of
annual period, but the changes in the date of Easter make the results for
any one year not comparable in general to those of any other. The
movement of Easter has been already considered, see Part XI., Natality.
The following results are for the period 1908-1914, and are corrected for
inequality in the length of the month, and for an increasing population.
The table gives the crude and the adjusted data.
TABLE XLVm.
Number of Marriages Registered in the Different Months. Australia, 1908-14.
Period.
Jan.
Feb.
Mar.
April.
May.
June.
July.
Aug.
Sept.
Oct.
Nov.
Dec.
1908-14 . .
Equalised . .
Constant
Population
21,462
21,060
21,325
21,106
22,691
22,924
22,732
22,420
22,599
28,358
28,663
28,817
19,714
19,205
19,271
22,059
23,232
23,268
20,752
20,357
20,434
20,733
20,369
20,299
22,824
23,154
23,022
22,138
21,760
21,579
21,140
21,455
21,343
25,534
25,106
24,790
Batio to
Average . .
.9490
1.0201
1.0057
1.2824
.8576
1.0350 .9093
.9033
1.0245
.9603
.9498
1.1032
These results are shewn, the rectangles and the probable fluctuation,
by curve C, on Pig. 56, and represent the fluctuation of the registration of
marriage. It is not certain that the returns made to the Registrars of
Marriages by those who celebrate them have not also seasonal peculiarities,
and consequently the fluctuation shewn is compounded of the two, and
in reference to the time scale is in advance of the true position. The
components of the curve can be found by applying formulae (90) to (101)
of § 5, part III., Determination of Constants, etc.i
6. G«neral. — Conjugal Constitution of the Population. — The
" general conjugal constitution" of a population is deflned by the number
of persons therein who have never been married ; who are living in the
state of marriage ; or of widowhood, etc. ; or who are living in the state
of " divorced" persons. The actual unadjusted numbers of males and
females in age-groups on the 3rd April, 1911, as indicated by the Census
are shewn on the table of §4, Chapter XVin.,Vol. I., of the Census Report.
These are represented on Pig. 57, which shews both the group-values and
the curves, which give sensibly the same totals. The results as furnished
by the Census are somewhat vitiated by misstatements as to age ; on the
whole, however, they give a fair representation of the change in the
1 See also formulae (376) to (395), § 16, Part XI., Natality.
NUPTIALITY.
181
Fig. 58.
Mg. 57 — The rectangles
shew the total numbers
as at the Australian
Census of 1911 in 5-year
groups, and the ciirves
give approximately the
equivalent areas, the
heavy curves denoting
the results for males, and
the light those for females.
Curves A and B shew
thenumbers of the "Never
married" ; C and D the
numbers of the "married"
E and F the numbers of
the " widowed" ; the
former being for males ;
and G and H (which can-
not be distinguished) shew
the numbers of the
divorced."
Fig.58 — ^The figures,
which illustrate Table
XLIX., shew the asym-
metry of the distribution
for the " never married,
U ; the " married," M ;
the " widowed," W ; and
the " divorced." The
scale of W is ten times
that of U and M, and that
of D is 100 times that of
U and M.
10 29 30 40 60 60 70 80 90
Fig. 57.
conjugal constitution with age. The general significance can be better
grasped from the results shewn in the following table : —
182
APPENDIX A.
TABLE XLIX. — Proportional Conjugal Constitution of the Australian Population,
3rd April, 1911, per 10,000,000 Total Population (Adjusted Numbers.)
Proportion per 10,000,000 of Total
Proportion per 1,000,000 of
Population.
same Sex and Age-groups.
Age-groupB.
Never
Married.
Wid-
Di-
Total.
Never
Married.
Wid-
Di-
Married.
dowed.
vorced
Married.
owed.
vorced
Under 14 M
1,506,806
1,506,806
1,000,000
F
1,467,395
2
1,467,397
999,998
2
14 to 20 M
710,197
5,304
34
4
718,539
992,538
7,413
48
6
F
662,798
35,358
184
18
698,358
949,080
50,630
264
2S
21 to 39 M
875,496
699,580
14,646
1,731
1,591,453
550,123
439,586
9,203
1,088
r
602,222
862,948
24,658
2,265
1,492,093
403,609
578,347
16,526
1,518
40 to 59 M
231,079
746,217
55,057
2,941
1,035,294
223,201
720,778
53,180
2,841
F
116,157
621,059
107,535
2,229
846,980
137,142
733,263
126,963
2,632
60 to 79 M
58,438
194,935
61.309
595
315,277
185,354
618,297
194,461
1,888
F
18,608
124,159
134,718
285
277.770
66,991
446,985
484,998
1,026
80&above M
4,507
10,770
12,301
45
27,623
163,161
389,893
445,317
1,629
F
1,129
3,850
20,424
7
25,410
44.431
151,515
803,778
276
All Ages M
3,386,523
1,656,806
143,347
5,315
5,191,992
652,259
319,108
27,609
1,024
F
2,868,309
1,647,374 287,519
4,804
4,808,008
596,569
342,632
59,800! 999
1
The table is based upon 4,455,005 persons, of whicli 2,313,035 were
males, and 2, 141 ,970 were females ; it shews the distribution of 10,000,000
persons on that basis. The ratios in the second part of the table shew the
proportional distribution in each age group. This distribution is illus-
trated in the small diagrams of Fig. 58, in which U denotes the males
and females belonging to the class " never married " ; M denotes the
" married" males and females ; W denotes the " widowed," 'of each sex ;
and D the divorced of each sex. These small diagrams represent by the
rectangular areas on the left of the median line the males, and on the
right thereof, the females. The scale of U and M is identical ; that of
W is 10 times, and that of D, 100 times as great.
The age at which the married are equal numerically to the unmarried
is about 29.49 for males when the proportion of the total at that age is
0.49557, and 25.27 years for females when the proportion at that age is
0.49699. The difference is 4.22 years, and the mean proportion 0.49629
is close to either. This is due to the fact that the number of widowed and
divorced is very small at the ages in question.
7. Relative conjugal numbers at each age. — ^The progress of the
conjugal constitution with age is completely defined by giving for each
sex, the proportion living at each age, and the proportional division of
each such number according to conjugal condition. In the following
table, which represents the smoothed results for the population of Aus-
tralia at the Census of 3rd April, 1911, the relative distribution of males
and females is shewn in columns II. and III. These numbers multiplied
by 0.2313035 in the case of males, and 0.2141970 in the case of females
(see the preceding section) give the absolute numbers, smoothed. The
distribution of 100,000 of these at each age is given for each conjugal
condition, viz., in IV. and V., the unmarried ; in VI. and VII., the
widowed ; and so on. Thus at each age a complete comparison is
NUPTIALITY.
183
possible of the conjugal state. Assuming the constancy of the conjugal
constitution of the population the results given in columns IV. to XII.
are the probability of the number of males or females which will be found
characterised as never married, married, widowed or divorced, in a total
of 100,000 males or females of each year of age throughout life. Columns
II. and III. shew, for the population of 10,000,000, a probable number of
males or females living at each year of age throughout the lite -period
on the assumption of an unchanging constitution according to sex and
age. As a matter of fact the Australian population, however, has not
reached a " steady" state as regards the constitution of its population.
TABLE L. — ^Relative Conjugal Numbers at each Age. Australia, 3rd April, 1911.
Proportion per 100,000 of any Age in each Conjugal Condition.
Proportion per
10,000,000 of
same Sex.
Age
Last
Never
Married.
Married.
Widowed.
Divorced.
Birth-
day.
Fe-
Fe-
Fe-
Fe-
Fe-
Males.
males.
Males.
males.
Males.
males.
Males.
males.
Males.
males.
I,
n.
m.
IV.
V.
VI.
vn.
VIII.
IX.
X.
XII.
253,554
263,314
100,000
100,000
1
236,741
247,352
100,000
100,000
2
227,662
238,776
100,000
100,000
3
221,173
232,426
100,000
100,000
4
216,158
226,689
100,000
100,000
6
211,030
221,422
100,000
100,000
6
205,544
216,147
100,000
100,000
7
199,236
210,605
100,000
100,000
8
193,611
205,675
100,000
100,000
9
189,232
201,852
100,000
100,000
10
186,115
199,135
100,000
100,000
11
184,835
197,118
100,000
100,000
12
184,813
106,086
100,000
100,000
13
185,860
196,417
100,000
99,998
"2
14
188,588
198,425
99,993
99,958
"7
42
15
192,846
202,463
99,982
99,783
18
215
2
...
16
196,742
206,660
99,945
99,207
55
789
4
17
200,105
209,910
99,842
97,445
156
2,547
"2
8
18
202,552
212,020
99,507
94,363
491
5,621
2
15
1
19
203,339
212,575
98,803
90,089
1,191
9,878
4
27
"2
6
20
202,932
211,646
96,862
84,638
3,111
15,290
23
59
4
13
21
201,908
209,144
93,784
77,311
?>}^S
22,547
54
121
6
?!
22
200,256
205,554
88,724
70,131
11,172
29,634
93
204
11
31
23
197,226
200,885
82,292
61,261
17,537
38,393
153
302
18
44
24
192,582
195,288
76,334
54,327
23,403
45,192
236
418
27
63
25
186,746
189,284
70,235
48,343
29,402
51,018
326
555
37
84
26
180,702
183,033
64,175
44,043
35,349
55,149
426
701
50
107
27
174,619
177,047
58,423
40,529
40,975
58,487
535
857
67
127
28
168,700
171,165
53,325
37,220
45,937
61,599
654
1,036
84
145
29
163,041
165,339
49,526
34,316
49,586
64,280
782
1,242
106
162
30
157,732
159,615
45,773
31,703
53,169
66,641
918
1,477
140
179
31
152,938
154,153
42,050
29,253
56,732
68,827
1,060
1,726
158
194
32
148,316
149,297
38,623
27,299
59,998
70,490
1,210
2,001
169
210
33
144,192
144,913
35,755
25,742
62,703
71,732
1,365
2,303
177
223
34
140,534
141,029
33,532
24,593
64,757
72,623
1,523
2,648
188
236
35
137,417
137,532
32,018
23,352
66,100
73,362
1,687
3,038
195
248
36
134,594
134,166
30,608
22,188
67,326
74,092
M®9
3,462
206
258
37
132,387
131,164
29,495
21,236
68,252
74,546
2,041
^■??i
212
267
38
130,491
128,344
28,536
20,209
69,012
75,059
^■5^i
4,456
219
276
39
128,870
125,725
27,727
19,392
69,612
75,368
2,436
4,957
225
283
40
127,499
123,036
27,035
18,423
70,089
75,785
2,645
^?S^
231
288
41
126,085
120,006
26,296
17,585
70,601
76,023
2,867
f'^°2
236
292
42
124,753
116,766
25,596
16,697
71,060
76,272
?'S®?
^•m
246
294
43
123,297
113,820
24,815
16,111
71,584
76,108
3,345
7,486
256
295
44
121,810
111,075
24,004
15,481
72,121
75,913
3,606
8,311
269
295
184
APPENDIX A.
Relative Conjugal Numbers at each Age. Australia, 3rd April, 1911. — Continued.
Proportion per \
10,000,000 of
same Sex. j
Proportion per
100,000 of any Age in each Conjugal Condition.
Age
Last
Birth-
Never , Marri<.,1
Married. | Manned.
Widowed.
Divorced.
day.
Males.
Fe-
males.
Males.
Fe-
males. Males,
Fe-
males.
Fe-
Males. males.
Males.
Fe-
males.
I.
45
46
47
48
49
120,227
118,632
116,738
114,201
110,863
in.
108,316
105,580
102,742
99,455
95,281
IV.
23,206
22,540
22,062
21,696
21,673
V.
14,831
14,203
13,632
13,064
12,592
VL
72,620
72,959
73,124
73,169
72.924
vn.
75,875
76,693
76,231
74,888
74,455
VllL
3,887
4,195
4,503
4,831
6,191
IX.
9,000
9,911
10,846
11,760
12,668
X.
288
308
311
314
312
xn.
294
293
291
288
285
60
51
52
53
54
106,112
99,890
93,341
86,985
80, '(68
90,165
84,049
78,320
73,129
68,376
21,317
20,762
20,065
19,693
19,471
12,216
11,746
11,362
11,034
10,676
72,747
72,765
72,868
72,763
72,387
73,969
73,183
72,254
71,200
70,127
6,627
6,172
6,767
7,336
7,837
13,645
14,798
16,118
17,511
18,981
309
311
310
308
305
280
274
286
256
238
55
5«
57
58
59
74,798
68,840
62,865
57,474
53.237
63,726
59,053
54,585
50,631
47.302
19,317
19,106
18,994
18,896
18.694
10,316
9,966
9,653
9,316
9,031
72,094
71,915
71,433
70,806
70.294
69,009
67,531
65,772
63,932
62.064
8,300
8,704
9,291
10,012
10,728
20,484
22,342
24,427
26,611
28.770
289
276
282
286
284
192
181
148
141
136
80
61
62
63
64
49,602
46,433
43,873
44,724
39,870
44,622
42,424
40,678
39,258
37,904
18,757
18,920
18,940
18,879
18,781
8,764
8,434
8,166
7,741
7,556
69,406
68,371
67,491
66,677
66.863
60,052
58,108
56,959
53,939
51,669
11,558
12,439
13,313
14,206
15.138
31,063
33,332
36,763
38,202
40.761
279
270
266
238
228
131
126
122
118
114
65
66
67
68
69
38,149
36,378
34,574
32,771
30,912
36,588
35,276
33,782
32,078
30,164
18,382
17,709
17,457
17,620
17,790
7,256
6,870
6,690
6,409
6.068
65,254
64,647
63,624
62,309
61.063
49,263
47,147
44,855
42,859
40.966
16,160
17,542
18,832
20,000
20.979
43,371
45,868
48,363
60,634
52,872
204
202
187
171
188
110
106
102
98
94
70
71
72
73
74
29,096
27,341
25,460
23,562
21,669
28,194
26,359
24,608
22,890
21,121
18,276
18,817
19,358
19,817
20,152
6,861
5,668
5,422
5,185
6,131
59,406
57,938
66,563
55,468
53,990
38,856
36,880
34,906
33,008
31,012
22,140
23,087
23,943
24,687
26,738
66,194
57,367
59,692
61,730
63,784
178
168
186
128
120
89
85
81
77
73
, 75
78
77
78
79
19,861
18,123
16,459
14,639
12,568
19,281
17,400
15,453
13,441
11,545
20,026
19,323
18,126
16,539
16,480
4,868
4,789
4,649
4,558
4,466
52,024
50,334
48,595
47,549
46,784
29,141
27,327
25,442
23,500
21,890
27,841
30,248
33,202
36,824
37,668
65,932
67,849
69,848
71,885
73,591
109
96
78
88
68
69
65
80
81
82
83
84
10,817
9,023
7,263
5,824
4,630
9,762
8,189
6,830
5,640
4,650
14,788
14,960
15,833
16,704
17,647
4,371
4,291
4,236
4,160
4,080
43,965
42,166
41,072
39,347
37,348
20,214
18,566
16,882
15,183
13,495
41,167
42,789
42,976
43,876
44,912
76,366
77,109
78,842
80,667
82,425
80
95
119
74
93
85
86
87
88
89
3,662
2,832
2,166
1,634
1,258
3,842
3,142
2,516
1,975
1,636
18,654
18,473
17,964
17,196
15,808
4,012
3,960
3,891
3,830
3,770
35,419
33,588
31,936
31,746
31,615
12,063
10,288
8,809
7,764
7,264
45,809
47,786
49,900
50.794
52,234
118
153
200
284
343
90
91
92
93
94
968
722
532
363
233
1,186
878
616
416
280
14,732
16,669
16,260
17,867
20,370
3,720
3,681
3,640
3,601
3,563
30,367
28,743
26,016
23,810
22,222
7,204
7,621
8,581
9,882
11,637
54,464
56,688
57,724
58,333
57,408
447
95
S6
97
98
99
169
125
99
78
61
210
159
117
84
61
20,513
20,690
21,739
22,222
21,429
3,634
3,601
8,470
3,446
3,427
20,513
24,138
26,087
27,778
28,571
13,044
14,746
16,630
18,276
20,050
68,974
56,172
62,174
50,000
60,000
■•
■•
100
43
42
20,000
3,411
30,000
21,811
60,000
Total
10,000,000
10,000,000
NUPTIALITY.
186
8. The carves of the conjugal ratios. — ^The smoothed results for
each sex, representing the ratios which the " never married," the
" married," the " widowed" and the " divorced" bear to each other
(given in Table L ) are graphed in Fig. 59, and are represented respectively
by the curves U^ and U/ , M^ and M/ , W^ and W/ , and D^ and D/ .
Conjugal Ratios, Australia, 1911.
1.0
0.9
0.8
0.7
0.6
0.5
0.4
3
[0.3
0.2
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Widows.
10 20 30 40 50 60
Years of Age.
70
80
90 100
Widowers.
Married Males.
Married Females.
TTmnaTried Males.
Unmarried Females.
0. Zero tor
"unmarried,"
" married," and
" widowed "
curves.
0. Zero for
" divorced '
curves.
Fig. 59.
These curves shew merely the proportion ol the unmarried, married, widowed, and divorced
at each age, the number at each age being unity tor males, and also unity tor females. They
thus shew the distribution for each age according to age, but not between one age and another.
The results for males are shewn by small crosses in the figures ; those for
females by small dots. The curves for the " never married" are somewhat
of the type e"*^> where p is large. The critical features of these curves
can be best shewn in a tabular form, and are as in the following table : —
186
APPENDIX A.
TABLE LI.— Critical ieatuies in the freauencies of conjugal conditions.
Aostialian Census, 3rd April, 1911.
•
Proportion
Character of Critical Feature.
Exact
of Total
Age.
Age -group.
Maximum proportion married, males
49.5
0.73100
,, „ „ females
43.0
0.76160
Minimum proportion married, males
95.0
0.00217
,, „ „ females
90.0
0.00063
Equal frequency married and unmarried males . .
29.49
0.49556
„ „ „ „ females
25.27
0.49699
Maximum proportion widowed, m.ales
90.5
0.89100
„ „ „ females . .
93.7
0.58400
Equal frequency unmarried and widowed, males
67.6
0.18600
„ „ „ females
49.5
0.12600
Maximum proportion of divorced, males
52.0
0.003115
females
44.0
0.00295
In general these results are for the smoothed curves represented in
Fig. 59, as may be seen by a reference thereto. The ratios among one
another of the various ratios given in Table L follow no simple law, and
an examination of them was found to lead to no important results.
9. The norms of the conjugal ratios. — ^It is eminently desirable that
a series of curves based upon the aggregate of all populations to be com-
pared, should be tabulated and constructed on some such model as that
indicated here for the population of Australia. Such a norm, representing
the relative frequency of the never-married, the married, the widowed
and divorced for the entire aggregate would constitute the best possible
bases for comparisons of the position of individual nations and peoples.
The international issue of graph paper on which such curves were already
drawn, preferably in faint colour, would enable the statistician to see
instantly the position of his own country in regard to the larger average
in respect of the particular character compared.
10. Divorce and its secular increase. — ^The frequency of divorce is
of sociological interest. The effect of the Divorce Act (55 Vict., No. 37)
of New South Wales, and of Victoria (53 Vict., No. 1056), which came into
force on 6th August, 1892, and 13th May, 1890, respectively, have had a
conspicuous influence in increasing its frequency. In the former State
the frequency was more than quadrupled for about three years ; in the
latter it was tripled, as the result of the operation of these Acts. Table
NUPTIALITY.
187
LII. shews the frequency of divorce per 10,000,000, for the several
States of the Australian Commonwealth for which they were available
up to 1886, and for the whole Commonwealth from 1887 onward.
The populations up to 1886, used to compute the divorce-rate,
correspond to the number of States for which the divorce results were
available, and the number of divorces include the judicial separations.
The results for the successive years are as follows : —
TABLE LII.— Relative Frec[ueueies, per 10,000,000 population, of Divorces and
Judicial Separations. Australia, 1874 to 1913.
Year
Rates* in Decades.
Proportionf of Judicial
Separations.
of
Decade.
1870.
1880.
1890.
1900.
1910.
Period.
Pro-
portion.
237
377
981
1,066
1874-1879
.020
1
179
594
1,052
1,154
1880-1884
.052
2
113
684
1,024
1,464
1885-1889
.062
3
274
1,293
909
1,347
1890-1894
.043
4
140
176
1,261
1,014
1895-1899
.038
5
220
269
1,194
862
1900-1904
.042
6
350
229
1,039
860
1905-1909
.043
7
210
205
1,113
854
1910-1913
.023
8
140
297
1,024
997
1874-1913
.0381
9
120
361
1,043
1,163
•■
* Number per 10,000,000 of population,
judicial separations and divorces together.
f Ratio of judicial separations to
The total number of divorces and judicial separations were 10,194
and 404 respectively, the total thus being 10,598. The relative fre-
quencies, tabulated above, are shewn by the bottom curve in Pig. 56,
viz., curve D. The proportions which judicial separations bear to the
totals appear also in the table. Apparently divorce was increasing at
first approximately at the rate 0.00000165 per unit of the population per
annum, so that the number of divorces (V.) from 1781 to 1890 would be
represented roughly by
(401) V = 0.00000165 P (t - 1870),
in which formula t denotes the year for which the number is required, and
P the population at the middle of the year.
The values according to this formula are denoted by the dotted line
a b on Fig. 56. The relative frequency then rises in 3 years from, say
0.0000330 to the value 0.0001293 ; that is at the rate 0.0000321 per
188
APPENDIX A.
person per annum — the line b c on the figure. The relief afforded
through the change in the divorce acts, having apparently been secured
in the short time mentioned, the relative frequency of divorce fell fairly
regularly until about 1907, viz., at the rate of 0.00000333 per person per
annum. Hence for this period the relative frequency is about
(402).
V = - 0.00000333 P {t - 1893).
This is the line c d on the graph. The relative frequency of divorce then
rapidly increases to about 0.0000100 per person per annum. This is
denoted by line d e on the graph.
11. The abnonnality of the divorce curve. — Owing to the change in
the divorce law being, as shewn, instantly followed by a large increase
in the number of cases, the curve of frequency cannot be regarded as
normal for the larger ages. For the purpose of estimating the rate of
increase, previous to the legal change, the results for a few years before
the change can be used. Similarly the results after the change can be
carried backward to some common year in the changing period. This
gives the following results : —
TABLE Lin.— Shewing Influence of Divorce Acts on Number of Divorces.
Australia.
Average Increase per
Annum (Number).
Number as
per Year
at Change
(1892).
Factor
State.
Before
Legal Change
After
Legal Change
Before
Change.
After
Change.
of
Increase
N.S. Wales ..
Victoria
Commonwealth
(1884-1891) 5.6
(1881-1889) 1.9
(1881-1888) 7.4
(1893-1895) 0.0
(1891-1893) 0.0
(1893-1907) 5.6
69.7*
32.5
116.6
306.7*
91.7
436.6
4.4
2.8
3.7
* Divorces and judicial separations together.
In view of the fact that, as shewn, the change consequent upon the
operation of the Divorce Acts is very marked in the frequency of divorce
between 1890 and 1893, say 21 to 18 years before the (Census cf 1911, and
that there is a remarkable decrease in the proportion of " divorced" for
NUPTIALITY. 189
ages about 55, see the points marked a and a' in Pig. 59 (which would
correspond to ages of about 35 in the year 1891), it seems more than
probable that the left-hand branch of the divorce curves belongs to the
later, and the right-hand branch belongs to the earUer divorce
regime. To obtain the true tendency to divorce according to age of the
parties, these irregular frequencies would, of course, have to be eliminated.
Hence it is desirable to include in the statistics of divorce the age of
petitioners and respondents. See later.
12. Desirable form of divorce statistics. — ^From what has preceded,
it is evident that for divorce statistics to be of high value from the stand-
point of sociology, they should fulfil the following requirements, viz.,
they should include the numbers both of petitions for judicial separation
and for divorce, and should shew for each : — (1) The date and the ground
of the petition ; (2) The action resulting therefrom (granting, refusal,
or other action), together with the date of such action ; (3) The date of
birth both of petitioner and respondent ; (4) and the date of their
marriage. Statistics so kept would furnish results shewing frequency-
according-to-age and age-differences and according to duration-of-
marriage. The sociological value of such statistics is self-evident, for it
would throw light upon the influence of age per se, of difference of age,
and of duration of marriage, and thus would expose the conditions which
are of danger from the standpoint of social stability.
13. Frequency of marriages according to pairs of ages. — ^The fre-
quency of marriage according to pairs of ages can be well determined
only for a considerable number of instances. For example, if assigned to
groups, according to age last birthday, there are, between the ages 12
and 95 for brides, and 15 and 99 for bridegrooms, no less than 7140
groups. As for the last eight years the average number of marriages per
annum was only 37,740, this gives a little over 5 per group on the average,
a number insufficient to indicate the characteristics of the frequency.
For this reason eight years marriages were taken, viz., 301,918, or the
marriages of 603,836 persons, who were married during the years 1907 to
1914 inclusive. Of these marriages the ages of 57 brides were not stated,
though the ages of the bridegrooms were given ; the ages of 19 bride-
grooms were not furnished, though those of the brides were given ; and
in 54 cases neither the age of bride or bridegroom was given. That is,
there were 130 cases (or about 1 in 2322, or the 0.00043058th part)
defective. These are disregarded.
For single year groups the numbers of marriages are shewn in Table
LIV.
190
APPENDIX A.
TABLE LIV.— NUMBER OF MARRIAGES* ARRANGED ACCORDING TO
* The figures denote the number
Ages
of
Bride-
grooms
AGES OF BKIDES.
12
13
14
15 16
17
18
19
20
21
22 23
24
25
26 27
28
29
30
31
32 33
34
35
36
37
38
■• 1
2 8
12 44
28 113
38 208
37 214
65 362
68 268
40 233
32 195
381 15'
25' 158
13
59
233
426
602
1158
870
742
603
460
405
311
247
182
133
94
78
54
50
26
37
27
22
18
13
6
1
3
16
17
8
51
309
778
1033
2076
1703
1494
1170
971
875
607
567
339
287
186
154
114
121
81
76
40
47
28
18
14
22
6
10
14
5
6
10
4
6
30
195
740
1261
2527
2342
2152
1458
1197
985
775
606
439
311
259
193
157
126
93
90
74
57
18
1
3
16
112
395
1075
2301
2384
2550
1885 2267
1824
1627
1238
1036
766
609
402
344
261
215
174
129
101
75
58
71
26
34
21
17
12
15
14
14
4
10
78
327
891
3764
3960
4114
3672
3170
2630
2171
1858
1364
1022
727
604
462
334
304
241
188
181
125
90
70
52
39
41
35
38
30
16
11
6
7
8
2
8
58
194
489
1845
3008
3468
3302
5
30
90
320
1185
1869
3128
3249
2981 3006
2555 2683
2050 2255
1783 1873
1353 1431
986
737
710
513
366
323
272
171
178
113
110
53
70
42
36
23
29
21
15
20
20
21
1088
769
668
553
422
313
273
259
205
128
112
87
64
72
49
37
32
21
37
15
15
5
7
9
12
47
160
738
1269
2032
2989
2821
2540
2141
1957
1484
1145
786
727
512
551
364
300
236
259
164
138
87
88
59
48
45
37
24
32
21
3;
6
24|
124 76
465^ 289
744
1247
1776
42
190;
526, 3071
835! 513
1311 769,
2664 16161 1089
1354
2382
2200
2073
1844
1736
1668
1457
1317
1172
1183
820
841
767
790
549
563
455
432
477
375
323
368
242
263
250
216
177
204
157
139
85
122
84
84
B4
68
57
62
72
45
43
52
26
29
30
27
27
26
1723
1532
1242 1121
982
752
621
536
449
337
327
215
190
161
109
112
27
2
11
28
118
226
309
545
832
948
1098
1320
944
754
673
541
449
387
324
254
354
197
172
116
114
91
65
72'
50
52
49
28
22
9
12
12
12
6
4
29
73
134
231 179
395 228
17!
58 34
28
450
584
730
879
1037
830
703
636
517
407
354
333
250
240
243
180
108
106
81
70
79
56
36
44
29
31
20
12
19
13
29
360
424
490
590
709
742
599
555
461
384
377
362
255
243
208
208
119
131
110
74
30
51
82
132
298
360
409
423
555
472
355
264
270
239
167
193
160
121
142
107
84
31
24
59
84
94
162
190
253
280
314
365
339
497
342
344
288
258
227
216
168
152
117
135
105
96
61
63
46
47
39
34
13
25
17
19
7
15
17
63
61
78
124
154
203
260
229
230
280
317
275
246
228
194
216
169 174
70 75
100 64
132
115
110
101
80
72
62
123
166
168
204
182
225
193
298
235
223
178
178
128
49| 60
49 48
33
34
86
114
132
153
157
173
180
197
233
190
153
187
154
173
106
132!
96
72
93
88
65
44
48
52
28
32
22
19
5
6! 4
22! 11
35 261
48 30'
55 76
82 66
88 66
35
100
100
126
130
157
164
188
153
172
147
118
101
124
86
76
95
87
64
55
58
40
38
24
21
24
36
76
76
94
90
95
103
126
165
160
142
124
73
86
88
37
21
25
26
35
53
60
61
46
67
64
95
94
115
111
189
139
126
76
111
85
80
73
66
69
81
58
54
36
40
26
20
15
15
19
24
31
38
46
48 23
42
50
66
87
75
103
120
151
124
86
97
81
84
78
54
47
60
55
57
34
34
27
20
1 473 489 2412 6907
13246
18140
2023132673
27950 26402 23903,20707
I I
17781
14440
12372
10010
8405 5848
5558
4341
3854
3521
2924
2438
200fi
1688
NUFTIALITY.
191
THE AGES OF THE CONTRACTEIG PARTIES, AUSTRALIA. 1907-1914.
of abuples : not of persons.
AGES OF BBIDES.
Ages
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
69
60
61
62
63
64
65
66
i
2
1
1
1
2
2
67,68
69
70
71
72
73
i
74
75
76
::
77
78
78
79
79
80
80
81
81
82
82
83
84
95
1
1
2;
1
of
Bride-
grooms
"l
1
4
6
8
15
8
8
20
21
26
25
28
27
30
31
43
61
58
61
87
90
56
75
59
66
60
39
32
24
44
33
19
1
3
1
3
1
5
7
9
8
7
6
10
12
19
27
30
24
27
38
37
33
52
42
73
56
78
58
50
39
42
30
27
23
37
24
i
i
2
1
3
2
3
4
6
15
10
6
14
10
20
21
15
18
27
35
30
28
42
46
53
61
40
49
41
36
30
31
40
25
30
"3
1
1
3
7
2
2
3
6
11
10
11
7
16
23
12
20
20
40
33
31
34
29
56
58
55
49
42
41
50
29
27
18
21
'3
1
2
1
3
8
4
4
3
2
6
9
12
7
6
8
20
21
15
23
15
19
32
33
51
42
38
37
35
18
34
18
26
i
2
2
3
1
3
6
3
7
3
6
8
7
10
13
14
19
18
22
26
12
27
29
34
3e
40
27
29
36
19
21
i
2
1
2
4
1
5
2
5
5
3
1
9
6
7
12
24
8
13
13
20
14
13
13
31
27
43
23
36
29
29
23
26
i
1
1
2
2
2
3
4
2
1
2
6
3
7
4
4
9
10
6
10
12
15
20
12
10
31
35
40
32
22
15
16
i
'2
1
2
2
3
1
3
1
i
3
7
2
3
2
11
10
6
6
8
13
19
15
10
24
30
32
24
25
17
19
i
1
'i
2
1
5
5
2
2
2
3
6
7
4
2
4
10
7
15
18
12
16
20
13
17
'i
i
i
2
1
3
1
1
'i
4
4
5
3
9
3
3
5
13
12
8
18
15
16
24
19
19
3
2
3
1
2
6
5
5
5
4
7
10
14
8
15
16
1
1
1
i
i
1
2
'3
2
3
1
3
'3
7
5
8
5
4
9
9
9
12
16
1
'3
"2
i
5
2
6
2
4
4
7
5
2
3
9
7
1
i
i
1
i
2
2
5
3
"7
7
8
9
7
10
i
i
i
i
1
1
2
2
'2
2
2
3
2
7
4
6
3
1
6
4
1
'2
1
i
1
2
1
i
'2
3
4
2
4
6
3
2
8
1
i
'i
1
1
2
2
1
2
1
1
'2
3
3
7
2
3
4
1
4
1
1
i
2
1
3
1
3
1
i
i
i
1
2
'4
1
i
3
2
i
"i
1
1
i
1
1
'2
1
'i
1
'2
3
2
1
i
1
i
2
'2
1
4
'i
2
1
'i
.1
i
i
i
'i
1
'i
i
83
1
1
2
2
2
2
4
2
2
4
2
2
4
1
2
2
1
2
2
2
1
2
2
1
1
1
i
1
1
4
51
239
1205
3353
6438
17374
19977
23655
24918
24650
23494
21012
19384
16113
13392
10349
9745
7712
6796
6066
5345
4411
4530
3737
3252
2336
2437
2058
1745
1847
1675
1350
1363
1146
1113
748
795
647
623
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
4?
48
49
50
51
52
53
54
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
'i
1
1
"%
"\
2
1
2
2
74
75
76
77
84
95
Total.
Ages.
23
15
13
9
5
6
5
7
6
7
4
3
2
6
■3
2
3
1
"1
1
1
22
16
13
15
9
6
4
7
8
4
3
6
5
5
1
"1
"1
"1
17
22
16
12
11
11
7
6
11
8
7
3
2
4
3
1
2
2
1
'i
1
4
20
15
19
22
6
14
13
4
9
6
12
8
6
2
1
2
4
2
2
3
'i
20
29
19
17
12
9
9
4
3
5
8
7
2
2
5
3
1
'i
26
IS
IS.
IS
IS
11
12
5
IC
6
7
C
4
«
2
i
2
]
'4
:
']
21
29
21
22
9
16
9
10
9
6
6
2
6
7
4
6
2
3
1
3
2
1
..
i
i
19
15
g
17
11
11
8
6
7
6
8
6
8
6
4
'4
1
3
'i
'i
i
'i
15
19
12
14
15
17
11
12
10
7
3
6
3
6
6
5
2
S
16
12
9
6
7
10
11
7
7
3
7
4
9
4
1
3
5
'5
1
1
1
2
i
14
13
9
20
11
7
9
15
7
5
XI
4
5
5
4
4
6
2
2
1
11
8
8
15
9
11
4
5
6
9
7
5
3
5
2
6
1
1
1
2
'3
1
'i
10
6
12
10
8
12
3
9
6
6
- 7
4
4
4
4
4
2
'e
1
1
i
i
1
12
10
11
6
6
9
3
9
10
2
6
9
5
3
2
2
1
2
'2
2
1
1
1
'i
12
12
8
6
9
14
6
3
6
4
5
4
4
7
5
6
2
1
2
3
1
1
i
1
1
3
12
14
10
9
7
6
8
5
2
4
6
11
3
'2
'2
3
4
1
1
1
7
8
10
11
14
8
3
6
1
8
5
3
4
3
7
3
1
1
2
1
2
2
1
1
3
1
5
1
6
6
10
10
5
3
1
6
7
6
1
2
2
1
2
2
1
2
4
1
4
8
3
5
8
6
4
6
5
6
9
5
6
5
5
3
1
3
%.
1
2
1
3
1
2
4
1
6
5
2
2
1
'2
3
i
1
1
'2
i
i
i
4
1
2
2
7
2
3
8
7
4
5
6
6
1
3
3
1
4
2
2
i
1
1
4
'2
4
3
2
1
6
5
7
4
7
5
2
6
4
'3
1
1
1
1
1
i
2
i
2
2
2
'5
3
4
4
6
2
6
6
5
1
'4
1
1
'i
5
2
2
2
1
3
5
i
6
8
7
2
3
4
3
2
4
7
3
4
i
1
1
'2
1
3
2
i
3
6
3
3
2
4
1
2
3
1
3
i
1
'i
"i
'3
2
1
2
3
1
2
3
4
4
5
6
4
1
1
'2
2
2
i
'2
3
1
2
2
3
4
2
5
4
4
1
2
3
5
2
i
'i
1
i
1
'2
2
'2
2
1
3
4
4
2
2
2
3
4
3
i
i
i
2
i
2
2
3
1
2
1
1
4
5
3
1
1
'i
2
2
'i
1
2
2
1
1
I
2
1
1
'i
1
i
i
1
2
1
'2
i
i
"i
i
1
1
3
3
3
i
2
1
i
i
'i
1
'i
2
i
1
'2
'i
'i
'i
2
3
1
i
1
1
'i
1
1
1
"%
1
i
i
i
'i
1
i
'i
i
2
2
64
545
489
400
414
286
347
236
227
229
203
219
170
163
154
122
134
70
70
77
64
53
33
33
22
23
28
10
9
8
7
5
3
3
1
1
" 1
1
73
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
96
99
N.S.
1287
1065
948
946
712
663
649
493
481
308
361
238
229
179
189
165
159
98
136
58
93
"
68
95
58
54
51
43
36
21
21
14
11
10
4
6
6
4
2
1
3
1
1111
301918
Total
192 APPENDIX A,
This table exhibits the various irregularities in the data. The num-
bers are not quite trustworthy about the ages 21, for reasons which will
appear later, as it is certain that in some cases misstatements are made by
persons marrying under that age. This table is suitable for the analysis
of the frequency at the lower groups of ages only. For the analysis of the
frequency at the more advanced age groups, a second table of five-year
groups has been prepared. (Table hereinafter).
The frequencies exhibited by this large group of marriages can,
without sensible error, be referred to the beginning of the year 1911
(i.e., to 1911.0), as the moment which they can be regarded as true, and
from which any secular change may be reckoned, or they may be regarded
as contemporaneous with the Census of 3rd April, 1911.
14. Numbers corresponding to given differences of age. — ^The mode
of tabulation in Table LW., though satisfactory in respect of shewing the
grouping according to age-groups for single years, is by no means perfectly
satisfactory for the purpose of very accurately determining the frequency
of conjugal-groups according to various differences of age. It is obvious
that when all bridegrooms, whose age was say x last birthday, and brides
whose age was say y last birthday (x and y being integers), are grouped,
the group contains brides who are one-half year older than the difference
X — y, as well as brides one-half year younger than this difference. This
can be readily seen from the nature of the table itself. To obtain some
rough idea of the defect of such a mode of grouping, we may first divide
the numbers (having regard to second differences) into four parts, so as
to get the probable numbers attributable to each half of the age-period
analysed. These quarter (or half-year) groups, however, will evidently
not agree with what would have been given by an original compilation
into half-year groups, for the reason indicated above ; this will appear
more clearly hereinafter.
To properly determine the law of nuptial frequency according to
specified differences of age the only perfectly satisfactory compilation
would be one in which, for small age-groups of bridegrooms (say) the
tabulation was according to a series of increasing age-differences (of the
age of the bride), positive and negative, and (for complete analysis) a
similar tabulation for small ranges of the age of the brides, with a series
of increasing differences, positive and negative, of the age of the bride-
groom. These two tabulations wovld not give identical results, but if the
age-groups were small, they would be approximately identical. The
data of the table are nevertheless of value, and give a result which is of
high precision in regard to the characteristic features of the surface
representing the relative frequency of marriages for given pairs of ages.
The results given in Table LIV. are for 301,918 marriages occurring
in Australia during eight years, and are drawn from populations (mean
annual), which aggregated to nearly 36 miUions. The marriage rates
were thus as shewn in Table LV., p. 193.
NUPTIALITY. 193
TABLE LV.— Marriage Rates, Australia, Total Period, 1907-1914.
Males
Kates, Males
18,614,557
0.0162195
Females | 17,206,457
I 0.0175468
Persons 35,821,014
0.0168570
These rates may consequently be regarded as representing the pro-
bability of a marriage occurring in a population of males, females, or
persons, constituted as the average for the eight years, 1907 to 1914,
both Inclusive, in Australia. The probability of a marriage occurring
among the never-married, the widowed, and the divorced, cannot be so
well ascertained.
By excluding the unspecified, the probabiUty of marriage for any
pair of ages can be ascertained roughly by dividing the numbers in
Table LIV. by 301,864 ; the quotient is the chance of the marriage
occurring in the group of the pair of ages in question, provided that the
proportions to the whole population of the males and females in eacJi
group is unchanged. Denoting this probability by pxy , the marriage-
rate by r^, and the population-by P, the number of marriages, N.j.,/, to
be expected of bridegrooms whose age last birthday was x, with brides
whose age last birthday was y, is : —
(403). . N^y = Pr^'p^y ; Na;y =P'r',nP=oy ; Nxy =P"r'inPxy ;
P, P'and P" denoting the population of persons, males, and females,
respectively ; and r„, r'^ and r^ similarly denoting the marriage
rates based upon persons, males, or females, respectively. The numbers
of the table would roughly give the chance according to " alleged age,"
not according to " actual age" unless the alleged is also the actual age.
We shall proceed to examine this question.
15. Errors in the ages at marriage. — ^Before analysing the data
giving the protogamic surface, it is desirable to determine the error of
statement at ages earUer than 21. Here it may be mentioned^ that the
curves of apparent frequency of birth at different ages from say 17 to 22
shew that the numbers are doubtless erroneous. The same fact is sug-
gested by the pecuhar irregularities in the numbers graphed in Pig. 60,
which shews the numbers of brides and bridegrooms at all ages ; see
curve A in the figure shewing the result for brides and curve B shewing
that for bridegrooms. The explanation is unquestionably that the
group " 21 years last birthday" contains a number of persons whose
real age was 18, 19 or 20, or possibly even younger than 18.
From an investigation of birth-frequency during the seven years, 1908
to 1914, both inclusive, it was found that the numbers given at
ages 18 to 21 needed to be multiplied by the factors 1.05701,
1.07918, 1.17022, and 0.82704 respectively. (This applies to females
only. There is doubtless also an error for males). Correcting these
factors so as to obtain the same totals, the figures in line (4) below are
obtained ; these are the probable correcting factors to be appHed to the
1 The matter is dealt with fully hereinafter.
194
APPENDIX A.
numbers furnished directly. That is let M' be the true number of
marriages for brides of any given age, and let M be the alleged number :
then m being the factor of correction, we shall have : —
(404) M' = m M,
hence, if the error occur solely through misstatements by persons of 18,
19, 20, and 21 years of age we should have, for each age of bridegroom,
to form corrections of the type : —
(405) . . (M'ls+M'ig+M'zo+M'zi) ={misMis+mi,QMig+r)HoM2o+m2iMzi)
This would be the appropriate scheme of correction^ if corrections for
only one sex were needed. The result would then be as follows in Table
LVI. hereunder : —
TABLE LVI.— Coirectiou of Nambers of Brides of Alleged Ages, 18 to 31.
Australia, 1908-1914.
(1) Age of Bride
(2) Number of Brides . .
(3) Ratio to Total for
Ages 18-21
(4) Factor of Correction
(5) Product of (3) & (4)
18.
13,246
0.1572
1.0572
19.
18,140
0.2152
1.0794
0.16619 0.23229
20.
20,231
0.2400
1.1704
0.28090
21.
32,673
0.3876
0.8272
18-21.
Total, 84,290
Total, 1.0000
Mean, 1.03355
0.32062 Total, 1.00000
These figures imply that there are 5.72 per cent, more brides of 18, 7.94
per cent, more brides of 19, 17.04 per cent, brides of 20, and 17.28 per
cent, less of brides of 21 than admit that they are the ages in question.
Misstatement of Ages.
Ages
15
Curves C and D.
30
35
i
. X0,000
S ^v
■a / \
7 ^ X I^
S ' ^~v
i" t ^K^^
a -.rnn / /' \ \
'^ t\ ^
7,500 h I t \ ^
_, ...^ L.-.\.. .
°. T^ V S
i- I ^ \ '
§ Jl Z ^ -t
t L J£ ^
° i t \ ^
t - ^ \\
° snnn % ^ V
^•^ ^ y-^ ^
3 1 5 vi^ ■
° - Zjl N. L
I 5 ^Ip^^^lt
« ^ i \ ^^
- ^^ SI -;
« t t -K ^^
1 ^ X ^
J5 r ^ ^
^ 7 i s % J
S i5.S0U If , \
^J fc \ ''-I
S 1 7 ^^
c^^ ^
S t
■^1-3 2"^^ ""— —
W 7
oh.^.^_ : : ::::: :::
-t-^5:=^ = 3s==- = =
Ages 15
50
go 25 SO 35 40 45
Fig. 60. ^^^^^ A *°* ^- Fig. 60a.
The curves A and B denote the number out of a total of 100,000 marriages of
brides and bridegrooms respectively ; married at given ages. The dots and circles
represent the original data, the curves themselves being the smoothed result.
Curves C. and D. — The areas of the rectangles shew the nimibers of brides and
bridegrooms, respectively, married at the given alleged ages. The true nmubers
are the areasjo the curves, which furnish the smoothed results.
1 If in any example the result needed a small correction to balance, it should be
made proportional to these last m Jf -quantities.
NUPTIALITY.
195
An attempt has also been made to ascertain, by smoothing, the probable
misstatement on the part of bridegrooms as well as on that of brides.
For the sake of comparison the factors for converting the crude data into
the smoothed results are given for both bridegrooms and brides, and for
males and females from the smoothing of the results of the 1911 Census.
The actual smoothing and its effect is shewn on Fig. 60a, see curves C and
D, the former being the curve for brides, the latter that for bride-
grooms. The areas to the curves give the smoothed results, the areas of
the rectangles themselves shew the crude data. In this way the results
(1) and (2) are obtained.
TABLE LVO
. — Coirection-Factors for Males and Females of Alleged Ages,
18 to 21. Australia, 1911.
Factor of
1
1
Correction
for —
How Obtained.
18.
i
19.
20. :
21.
,1) Males ..
Smoothing of Cm-ve shewing
1
Number of Bridegrooms
1.211
1.137
1.262
0.831
(2) Females
Smoothing of Curve shewing
Number of Brides
0.962
1.054
1.228
0.844
(3) Females
Smoothing of Fecundity
1
Curves
1.0572
1.0794
1,1704}
0.8272
(4) Females
Mean of (2) and (3)
1.010
1.067
1.199 !
0.836
(5) Males ..
Smoothing of Census of
1
Population, 1911
0.9843;
1.0273
0.9955|
1.0283
(6) Females
Smoothing of Census of
1
Population
0.9924;
1.0217
0.9902
1.0504
The indications from the smoothing of the number of brides, with
those from the smoothing of the fecundity curves (see later) are in sub-
stantial agreement, so far as the ages of 19, 20, and 21 are concerned ;
see Unes (2) and (3) in the table above. It will be observed, however,
that they are not in agreement with the Census deduction. An agree-
ment was not, however, to be expected- in the latter case, for the mis-
statements occur in regard to the age at marriage, an occasion on which
there is not infrequently a motive for the misstatement. *
16. Adjustment numbers for ages 18 to 21 inclusive. — ^The actual
adjustment of a table of numbers according to pairs of ages, however,
involves the deduction of a number of brides and bridegrooms; which
shall be equal for each group. It is evident that, inasmuch as the factors
for the two are disparate, different results are obtained if we first correct
by the factors for one sex and then by those of the other, or correct in-
dependently and take means, etc. For this reason the following method,
though not ideally satisfactory, was adopted.
Denoting the correction-factor for bridegrooms (males) of age x by
rrix, and that for brides (females) of age yhjfy, the composite factor (/j.)
1 Chiefly, but not whoUy, owing to the attempt, by persoijs vmder 21 years of
age, to avoid the legal requirements,
196
APPENDIX A.
for the group of brides and bridegrooms of the respective ages, may be
taken as : —
(406).
■ fJ-xy = \/{'>nxfy\
that is, it is regarded as the geometric mean of the two. If we decide
to make the totals of the groups 18 to 21 unchanged, we shall have to
apply a small correction to these factors. Let gxy denote a group of
marriages for the ages in question. If the sum of the products jugr be
equal to the sum of the original groups, no correction will be required.
If it be not equal, then the correction can be distributed in the ratio of
the groups themselves. That is, | denoting the correction, the new
values (g') of the groups will become : —
(407).
■g' =g + ^ =g \^+{G - s i^g) / q\
G denoting the sum of the groups, that is to say, O = Eg. This method
of correcting leaves the entire aggregate unaffected, though it adjusts
its component groups. The results are shewn in the table hereunder.
The ^ correction necessary was very small, amounting to only 18 in
17,862. See Table LVIII.
TABLE LVm. — Coirection of Numbeis of Mairiages for Ages 18, 19, 30, 21.
Australia, 1907 to 1914.
Crude Results.
Factors of Corebotion.
1
CORRECTED RESULTS.
18 19 20 21
Totals.
18
19
20
21
18 19 20 21
Totals
18
19
309 195 112 78
778 740 ; 395 327
694
2,240
Males
Females
Means
Males
Females
Means
1.211
1.010
1.1059
1.137
1.010
1.0716
1.211
1.067
1.1367
1.137
1.067
1.1015
1.211
1.199
1.2049
1.137
1.199
1.1676
1.211
0.836
1.0062
1.137
0.836
0.9750
343 223
837 819
136 79
463 320
781
2,439
20
21
[ !
1,0331,26111,075 891 4,260
2,076 2,527| 2,301 3,764' 10,668
Males
Females
Means
Males
Females
Means
1.262
1.010
1.1290
0.831
1.010
0.9161
1.262
1.067
1.1604
0.831
1.067
0.9416
1.262
1.199
1.8301
0.831
1.199
0.9982
1.262
0.836
1.0271
0.831
0.836
0.8335
1,171 1,469
1,910 2,389
1,328 919
2,3063,150
4,887
9,755
rtia' 4,196 4,723 3,883 5,060 17,862
i ( j
4,2614,900
4,233 4,468
17,862
The effect at the dividing ages of this regrouping is to change the
2,022
S™"P^ 6:897^
912
3,302
23,130
1,395
56,029
8,031
become
into
2,222 1 998
6,939 17,703
3,502
23,172
1,481
; hence the five-year groups
The totals for brides require
55,701
that the original figures in Table LIV. should be corrected by + 65, -|-
177, + 350, and — 592, and the totals for bridegrooms corrected by
+ 87, + 199, + 627 and — 913.
NUPTIALITY.
197
TABLE LIX. — Shewing the Number per 100,000 Bridegrooms, and per 100,000
Brides Married at Griven Ages. Australia, 1907-1914.t
Crude Results.
Adjusted Results .
Age.
Crude Results.
Adjusted Results.
Age.
Bride-
grooms.
Brides.
Bride-
grooms.
Brides.
Bride-
grooms.
Brides.
Bride-
grooms.
Brides.
(1.)
12
13
14
(ii.)
(Ui.)
1
1
24
(iv.)
• 0.0
0.1
0.2
(V.)
0.5
1.5
24
(i.)
55
56
57
58
59
55-59 . .
(u.)
181
162
133
137
95
(iU.)
59
63
55
53
32
(iv.)
184
167
151
136
122
(V.)
72
66
60
12-14 ..
26
0.3
26
48
15
16
17
18
19
1
17
79
428*
1,176*
162
799
2,288
4,409*
6,067*
0.8
M.9
79
428
1,176
162
799
2,288
4,409
6,600
70S
262
760
30 1
60
61
62
63
64
60-64
65
66
67
68
69
65-69 ..
70
71
72
73
74
70-74 ..
75
76
77
78
79
75-79 ..
80
81
82
83
84
80-84
85
86
87
88
89
85-89
90
91
92
93
94
90-94 .
Unspeci-
iied .
115
78
75
76
67
45
19
31
27
23
109
97
86
76
68
42
36
31
27
13-19 . .
1,701
13,725
1,700.7
14,253
2,340*
5,452*
6,615
7,834
8,253
6,817*
10,626*
9,257
8,745
7,917
2,542
4,997
6,868
7,834
8,253
8,020
8,920
9,200
8,745
7,917
21
411
178
436
160
23
24
73
56
54
51
41
32
19
18
17
14
60.0
53.0
47.0
42.0
37.5
21.8
19.5
17 1
20-24 ..
30,940
43,362
30,494
42,802
15.0
8,165
7,782
6,960
6,420
5,337
6,858
5,873
4,783
4,098
3,315
8,190
7,782
7,120
6,290
5,337
6,819
5,843
4,897
4,078
3,297
26
275
100
239.5
86.0
28
29
44
23
23
26
21
19
7
7
5
4
34.0
31.0
28.0
25.0
21.5
10.5
8.5
67
25-29 ..
34,664
24,927
34,719
24,934
5.1
3.6
4,436
3,428
3,228
2,554
2,251
2,784
1,937
1,841
1,438
1,277
4,383
3,603
3,003
2,603
2,278
2,670
2,155
1,760
1,470
1,260
31
137
35
139.5
34.6
32
33
34
17
11
11
7
7
3
1
2
2
1
17.2
13.6
10.9
8.7
7.0
2.8
2.1
1.6
30-34 ..
15,897
9,277
15,870
9,315
1.2
0.9
2,009
1,770
1,461
1,501
1,238
1,166
968
808
785
664
1,995
1,748
1,533
1,346
1,183
1,143
1,003
873
753
643
35
36
53
9
57.4
8.6
37
38
39
9
3
3
3
2
1
1
5.6
4.5
3.6
2.9
2.3
0.7
0.6
0.5
35-39 ..
7,979
4,391
7,805
4,415
0.4
0.3
1,077
774
807
682
580
560
373
426
353
314
1,040
912
800
713
649
547
465
397
343
303
40
41
20
2
18.9
2.5
43
44
2
1
i
1.8
1.4
1.0
0.6
0.3
0.2
0.1
40-44 ..
3,920
2,226
4,114
2,055
0.0
0.0
45
612
522
447
452
380
313
236
220
215
163
589
527
468
413
363
271
241
213
187
163
0.0
46
47
4
5.15
0.3
48
49
0.25
0.15
0.1
0.05
0.0
45-49
2,413
1,147
2,360
1,075
SO
369
248
263
214
. 206
159
102
116
79
76
319
282
251
225
203
138
118
102
90
80
51
52
0.55
53
54
24
37
Nil
50-54 ..
1,300
532
1,280
528
NU
• These have been partially corrected for misstatement o£ age.
for description of Table.
t See Section 17, hereinafter.
198
APPENDIX A.
17. Probability of marriage of bride or bridegroom of a given age,
to a bridegroom or bride of any (mispecified) age. — ^The correction of the
data, as indicated in the preceding section, admits of the construction of a
table shewing in say 100,000 marriages the number occurring for bride-
grooms of any given ages, and for brides of any given ages, the age of the
other partner to the union being unspecified. In columns (ii.) and (iii.)
of Table LIX., hereinbefore, the data are given the corrections referred
to having been applied : columns (iv.) and (v.) are the smoothed results.
The original data are shewn by dots on Fig. 60, the smoothed results by
the curve, the ordinates to which represent throughout the probability
of a marriage occun-ing within one half-year either side of any given age :
that is, they are the values of the integrals : —
K \ xdx
and
V+i
K' \ydy;
Jy-i
see section 19 hereinafter.
18. Tabulation in 5-year groups. — So small a number as 300,000
does not give sufficient data for the determination of the averages for
single years, at the higher ages. Before 25 is reached over one-fourth
of the marriages have been consummated, and before 30, over two-thirds
(exactly 0.277921, and 0.691744 respectively). This leaves for groups of
over 30 years of age only about 93,069 among 6500 groups or an average of
about 14 per group. It is thus necessary to form 5-year groups. These
are shewn in Table LX. hereunder. The corrections, referred to in last
section, change these numbers as follows : —
Oriqenal Data.
Adjusted Data.
3,302 1,395
23,130 1 56,029
1
4,852
92,354 ,
3,502
23,172
1,481
55,701
5,138
92.068
41,193 13,1151
Totals.
41,135
130,909
Totals.
The numbers given in the table itself are the uncorrected data. It
will be seen that they are still small for the higher ages. To determine
the critical features of the surface representing the frequency of marriage
both Tables LIV. and LX. are required. Were these two tables smoothed
they would give the probabilities of a marriage occurring within the year
groups of specified ages or specified quinquennia. None of the groups
is perfectly regular, but the greater regularity of the larger groups
exists only for a limited range of years. The matter will be dealt with
more fully hereinafter, viz., in § 23.
NUPTIALITY,
199
TABLE LX. — Number of Maiiiages Airanged According to Age at Marriage in
Five Year Groups. Australia, 1907-14.
Brides' Aees.
Bride-
Total,*
Katlo
1
1 III
[111
Bride
grooms'
Ages.
10
15
20
25
30
35
40
45
50
55
60
65
70 75 80
10 to 84.
grooms
to
to
to
to
to
to
to
to
to
to
to
to
to to to
to
14.
i9.t
24.t
29.
34.
39.
44.
49.
54.
59.
64.
69.
74.
79.184.
1
Total.
15-19t ..
9
1 3,302
1,395
124
17
3
2
4,852
1,608
20-24t ..
44
23,130
lS6,02g
11,302
1,437
325
6C
22
4
1
. .1
92,354
30,602
25-29 . .
18
10,637
1 50,597
34.896
6,739
1,368
282
7(i
20
1
1
i; ••
104,639
34,673
30-34 . .
1
2,795
15,513
117,366
9,130
2,476
525
146
26
4
1
47,983
15,900
35-39 . .
Si
917
5,134
1 7,298
5,672
3,621
i.oas
1 313
65
15
2
2
24,080
7,979
40-44 . .
1
237
1,576
2,564
1 2,811
2,47 a
1,502
510
112
26
a
1
11,821
3,917
45-49 . .
2
115
598
1,077
1,313
1 1,653
1,27S
859
263
74
36
8
7,277
2,411
50-54 . .
41
183
384
538
1 76S
1 7b4
675
4011
117
37
'20
2
i
3,926
1,301
55-59 . .
11
73
129
197
3ia
36(1
1 445
2fiii
218
65
26
4
2
2,132
706
60-64 . .
6
28
71
79
152
162
1 207
1 208
144
lOR
60
16
2
1,242
412
65-69 . .
1
15
24
43
66
80
1 133
122
113
105
»7
19
7
826
274
70-74 . .
6
16
17
SO
an
47
1 65
41
50
59
28
6
415
138
75-79 . .
1
2
3
8
6
n
13
17
i»1
14
21
25
fl
164
54
80-84 ..
2 2
2
2
K
1 10
7
4
1 9
4
1 8
4
62
21
85-90 . .
1
1
1 4
1
1
1
1
1
12
4
Total*
78
41,193
131,151
75,257
28,003 13,257
6,114
3,462
1,605
790
435
300
103
30
301,785
100,000
Batio ol
Biides
26
13,650
43,459
24,937
9,279 4,393
2,026
1,147
532
262
144
99
34
10
2
100,000
.3313617J
to Total
The heavy faced type
the mark of exclamation ( I ) denotes the maximum on the
* Brides over 85 and bridegrooms over 95, and unspecified cases are omitted,
denotes the maximum on the vertical lines ; "
horizontal lines.
t The values corrected for misstatement of ages, 18, 19, 20, and 21 give the following results :— For 3,302
and 1,395, 3,502 and 1,481 ; and for 23,130 and 56,029, 23,172 and 55,701. In the totals 41,193 and
131,151 become 41,435 and 130,909 : and 4,852, and 92,354 become 5,138 and 92,068. The ratios 13,650
and 43,459 become 13,730 and 43,378 ; and 1,608 and 30,602 become 1,703 and 30,508.
J Factor of reduction to 100,000.
19. Frequency of marriage according to age representable by a
system o£ curved lines. — ^Frequency according to pairs of ages (bride and
bridegroom) can best be represented by a surface, the vertical height of
which, above a reference plane, is the frequency for any pair of ages
denoted by x, y co-ordinates. The numbers marrying in any given
period, whose ages range between x-^ ^k and x -\- \k (for bridegrooms),
and between y — \k and y -\- \k (for brides), as ordinarily furnished by
the data, are denoted by Z, the height of the parallelepiped. This
frequency may, of course, be expressed as for the exact age, or it may be
for the age-groups. When k is not infinitesimally small, the difference
between the two is sensible and important. We shall assume for the
present that the frequency varies only with age (not with time). The
exact (instantaneous) age-frequency denotes the frequency which would
exist if the persons were all of the exact age (x) in question, instead of
being of various ages between x ~ ^ k and x -\- ^ k. The age-group
frequency denotes the frequency with the ages distributed between the
limits referred to. For most practical purposes the latter is the more
important. Suppose the exact frequency, z, for the population P, to be : —
(408).
^=F{x.
y)
then we shall have for any group-value : —
(409) Z = PJj F (x.y) dx dy
200 APPENDIX A.
The group-values usually furnished are for single-year groups, hence
the limits of the integral are « ± i> 2/ rb ^- It may sometimes be more
convenient to use a series of functions of the form : —
(410) y=^Fy{x); or F^(y)
in which case the fixed value of 7 or of X will be the middle of the range
2/ ± i, or .T J: |. Then we shall have : —
(411) Z =PJ Fr(x)dx; or = Pj Fx(y)dy
These last expressions, with fixed values either of Y or of X, are thus
appropriate for representing the vertical or horizontal columns of figures
in Tables LIV. and LX. by means of equations. For the vertical
columns the abscissa is x, the age of the bridegrooms ; for the horizontal
columns the abscissa is y, the age of the brides ; and the constants of
the equations relate only to a particular range of y in the first case, and of
X in the second, as many equations being required as there are ranges
taken. We consider the matter more fully in a later section. This
scheme of representation is practically more convenient than a more
generalised system, it shews for each age of bridegroom (or of bride) the
frequency of marriage with a bride (or a bridegroom) of a given age. (See
part v., § 10, formulae 211 to 216.)
20. The error of adopting a middle value of a range.— ^In dealing with
group-ranges, in the manner referred to in the preceding section, the
results are not strictly attributable to the middle age of the range, nor is
the error of such an attribution by any means always wholly negligible.
The function represents the value of a range of values of the argument,
i.e., for example, all bridegrooms whose age last birthday was x, x being
an integer, or the group of bridegrooms whose age last birthday was say,
between 20 and 24, etc. Suppose, for example, that the progression
of a series of numbers, representing numbers at successive ages is approxi-
mately : —
(412) y = a -\- inx ; so that xy = ax -\- mx^ ;
then the true value of the product of the numbers into the ages is given
by the integral : —
X + 1
(413) \xy dx = a (x -\- i) + m, {x^ -\- x + ^)
Consequently where we require the weighted mean-age, it is necessary to
compare this value with that arising from the supposition that all may be
regarded as of age x -\- \. If we make this last assuniption,then we should
have for the product of the numbers into the age, supposed common to all
(414) a (a; -f i) -f m (a;2 + X -f 1).
The former expression is algebraically greater than this latter one by the
difference of m/3 and m/4, that is m/12, which is sensibly equivalent to a
NUPTIALITY. 201
shift (e) of the central position of the amount m/Yly. Thus, instead of
the central value of the range of ages we should take the " weighted
mean" xa, which is given by : —
(41f5) x^ = x^\ +e =.f +1 +
m
122/
In applying this we may take m as indicated by the mean of the
differences of the groups adjoining on either side. Thus if the groups for
the ages 20 (and less than 21), 21, and 22 were respectively 76,132, and
224, then, instead of taking 21.5 as the mean age-value, i.e., the middle
age of the range 21 (which include everyone whose age last birthday
was 21), we could take m as the mean of 132-76 and 224-132, that is,
m —\ (56 -f 92) ; or, as is obvious, \ (224-76), i.e., 74. Consec[uently by
the rule above, viz. (415),' we have «„= 21.5+74 / (12 x 132) = 21 .54671.
A curve which would give the group-results indicated is 60 + 20^
+ 18p, the origin of abscissae being x = 20, so that ^ = 1 for a; = 21,
and so on. The integral of the curve is 60 ^ + 10 f^ -f- 6^ If we put
^ = a; - 20 we obtain the curve y = 6860 — 700 a; + \%x^ with the
origin at a; = 0, hence the integral between the limits a; = 21 and a; = 22
is 3430 a;2 — 233| x^ -f 4^ a;*, which gives the result 2844J as the
sum of the xy products. Dividing this by 132, the number in the group,
the average age is found to be 21.54671 as before. Let three successive
groups for equal ranges of the variable be denoted by .4, M, and B ; and
let x^ be the middle point on the range of abscissae of the middle group,
M ; "then the mean value required (i.e., in the case under review, the
average age of the persons in the group) is : —
(416) Xfl = x^ + ITT A; ~
24 M.
in which h is the range of the variable comm.on to the three groups. If the
curve of instantaneous values be of the second degree, this last formula
is rigorously accurate. By means of it, the average values can, as a rule,
be written in by inspection, and it can be ascertained where the correction
e = jV^^ ^ (jg — A) / M is. sufficiently large to be taken into account.
21. General theory of protogamic and gamic surfaces. — ^The ages of
husbands being adopted as abscissae, and those of wives as ordinates,
the infinitesimal number dM in an infinitesimal group of married couples,
consisting of husbands, whose ages lie between x and x -{- dx, and their
wives, whose ages lie between y and y + dy, will be : —
(417) dM = Z dx dy = kF (x, y) dx dy.
Thus Z = k F {x, y) is representable by a co-ordinate vertical to the
xy plane. Since Z denotes an actual number of persons in a double
age-group, between say the earliest age of marriage and the end of life,
viz., {xi to Xz) and (yi to ys), it is necessary, if we desire to institute
comparisons between different populations, that Z should be expressed
as a rate, z say : that is, z = either Z/P ; or Z/M ; that is to say, the
202 APPENDIX A.
vertical height wUl represent the relative frequency of married couples
whose ages are, in the order of husband and wife, x and y, in either the
whole popuation P, or the married portion of it M. Thus we shaU have
(418) P,OT M =kff F (x,y)dx dy.
If the value of the double integral be taken for the limits denoting
the range of ages of the married, say about 11 to 105, we shall have either
M/P, or unity, as the result ; according as we denote by the frequency
in reference to the total population or to the total married.
Thus the marital or gamic condition of a community is completely
specified by the gamic surface F {x, y, z), the unique mode of which is the
summit of the conoidal solid represented by (418) above. Its first
principal meridian is the Une joining the modes of the curves x = a, con-
stant, or 2/ = a constant, passing therefore through the unique mode.
The curves, z = any constant less than its maximum value, are necessarily
closed curves, and may be called isogamic contours. The orthogonal
trajectory passing through the unique mode is the second principal
meridian of the surface. The values of x, y, and z for the unique mode of
the surface may be called the gamic mode of the " population," or of the
" married population," according as the constant k, in (418) above,
gives M/P, or unity for the value of the double integral between the
widest age limits.
The gamic characteristics of a population are more briefly, and of
course less completely, defined by the two principal meridians which we
may call its gamic meridians, and the position (and magnitude) of the
gamic mode. Reducing these tn their simplest numerical expression we
have, for the briefest possible statement of the gamic characteristics
of any community the values of x„, y^, and z„ ; and of the skewness of
the profiles of the first and second principal meridians. The sign of the
skewness may be determined by always making the right hand branch of
the curve that for increasing age for the first principal meridian, and
increasing age of the husband for the second principal meridian.
A surface representing the frequency of marriage at particular
pairs of ages we shall call a protogamic surface, and one representing
the number of persons of particular pairs of ages living together in the
state of marriage we shall call simply a gamic surface.
Curves of equal frequency on these two surfaces, we shaU call
isoprotogamic and isogamic contours, respectively, or more briefly,
isoprotogams and isogams, and curves cutting such contours orthogonally
will be called protogamic and gamic meridians.^
Let s denote a distance measured along a slope, so that ds is an
element thereof. Then when —
(419) dz/ds = sin ^
1 The word " isogamy " has aheady been appropriated in a different sense in
biology, viz., to denote the union of two equal and similar " gametes" in repro-
duction. This, however, will obviously lead to no confusion. The isogamy of a
people might be regarded as of two kinds, initial or nuptial isogamy (isoprotogamy),
and characteristic or marital isogamy (or simply isogamy).
NTJPTIALITY. 203
= a maximum or a minimum, the element ds is an element of a meridian ;
such meridians are the principal meridians above referred to ; i.e., the
principal meridians are the lines of greatest and least slope.
22. Orthogonal Trajectories. — ^The general theory of orthogonal
trajectories may be stated as foUows : — ^Let the co-ordinates of a system
of curves (isogams or equal marriage frequency in the case considered) be
denoted by x and y, and those of the trajectory, cutting the system
orthogonally, by ^ and tj ; then, although for any point of intersection
of the two X = ^ and y = yj, dy/dx is not the same as dr)/d^, Since the
tangents to the two curves are at right angles, we have the geometric
relation dy/dx = - d^/dyj or
<**> '+S-'S=''
For any system of curves we have then
(421)..../ {X, y, a) = Op
where a is a constant ; then, employing S/Sa; and 8/Sy to denote partial
differentiation with respect to x and y, we have also
U22) _^ , _V ^ _
'*^^^ 8x + Sy -dx ^"'
an equation by means of which a may be eliminated, so that a relation
may be obtained between x, y and dy/dx. Let this relation be denoted
by:-
(423) i,(x,y, %)=0
This last expression is the differential equation of the system of curves
we require.
For orthogonal trajectories we have i = .i\ rj = y and dy/dx =
— 1/ {dr]/d^), hence the differential equation of the system of orthogonal
trajectories is : —
(424) ^(^,^, - J-) =0
In the system we are considering, the curves (isogams) do not con-
form to any simple specification, hence the present imperfect data do not
indicate any unique system of curves of a simple character. If they
did, it would be preferable to deduce the principal meridians of the surface
by means of the general equation thereto. An examination of the
surface, however, shews that there is no practical advantage in attempting
to express it analytically.
23. Critical characters on the protogamic surface. — ^A review of the
figures in Tables LIV. and LX. reveals the fact that, in general, if we
regard the numbers of marriages corresponding to any given age for
brides (the columns), there is a clearly-defined maximum value ; but
that if we regard those corresponding to any given ages for bridegrooms
(the rows), there are in many cases two or even three maximum values.
204
APPENDIX A.
In this latter case, too, the maximum is often less clearly defined. The
positions of these maximum points and the numbers (frequency) cor-
responding thereto, are important, as they disclose the characteristics
of the surface. There are two ways of estimating the position and fre-
quency at the maximum (or any other point). One is to ascertain the
position and frequency for the maximum of the frequency integral taken
over the range x — ^ to z -\- ^, or over the range y — ^ to 2/ + J ; the
other is to determine those elements for the maximum instantaneous
frequency ; that is to ascertain the point when the frequency for an
indefinitely small range is a maximum (expressed, however, per unit of
age-difference, say one year). The latter only will be ascertained.
By applying formulae (292) to (294), see Part VII., § 11, p. 92, the
position and value of these maximum points (viz., those on the surface
for ages of brides constant that of bridegrooms being variable, or for ages
of husbands constant and that of brides variable), may be obtained.
In this way the results given in the two following tables are deduced, viz.,
Tables LXI., and LXII., and in connection therewith it is to be remarked
(a) that for results of high precision, the quinquermial grouping can be
used only for the small groups at higher ages ; and (6) that the grouping
in fives, not only tends to obliterate characteristics readily discernible
in year -groupings, but gives a frequency of the order of about 25 times
the magnitude of those groupings. Thus for very young ages and for the
older age-pairs, the large grouping gives the best indication.'^
1 The values are obtained in the following way : — The position of the maximvim
of one group (say of bridegrooms) corresponding to the range of another group (say
of brides) is found from the succession of the group-totals of the first, for any one
range of the second, and is attributed to a mean age of the second, computed from
the progression of numbers in the series of group totals of the second. By way of
illustration consider the group of 59, for the age-group 65-69 of brides, and 70 to 74
of bridegrooms ; viz., the following figures : —
Instances
in Group.
Adjoining
Group Totals.
The surrounding group-totals are as shewn.
rf the arrac nf U^Aa^ V^ tol,.>« o„ „* <-V.„ 4^ JI„ „I
3.4.1.1.2
2.1.1.2.2
4.2.0.3.2
7.3.2.5.3
3.0.0.2.4
60
105.97.19
41.50.59.28.6
14.21.25
4
the years, i.e., as 65J, 66J, etc., and of the
bridegrooms as 70J, 714, etc., the actual
weighted-mean ages (deduced from the iudivl-
vidual numbers) are as shewn hereunder.
Slightly different results are obtained if the ages
are deduced from the vertical and horizontal
columns, viz., 97, 59, 21 ; 50, 59, 28 ; and from
the diagonal totals, viz., 105, 59, 25 ; and
19, 59, 14. These different results are for bride
and bridegroom respectively : —
Middle Values
of Groups.
Actual Weighted
Group-means.
Computed from
Vertical Groups,
etc.
Computed from
Diagonal Groups.
Years
67.5
72.5
67.35
72.64
67.48
72.45
67.46
72.40
This series of results shews that the error of assuming that the entire gi-oup is repre-
sentable by the middle ages is not ordinarily considerable.
XUPTIALITY.
205
TABLE LXI. — Critical Positions on the Piotogamic Surface for Teai-gioups.
Marriages in Australia, 1907-1914. (Greatest frequency for various combina-
tions of Age at Marriage).
Mean Age of
Age of Bride-
!
' Proportion of
Brides in
groom for
i Difference of
Maximum
AU Brides
Maximum
Maximum
Age.
Frequency.
of same
Group.
Frequency.
Age -Group.
13.5
21.2
1
7.7
1
0.250
14.7
22.4
7.7
17
0.233
15.7
21.6
5.9
69
0.141
16.6
21.6
5.0
372
.1504
17.6
21.7
4.1
1203
.1742
18.5
21.7
3.2
2164
1986t
.1621
.U92t
19.5
21.9
1 2.4
2600
i .1434
21.8
2.5
2500t
.1364
20.5
23.4
2.9
2573
.1272
.1256
21.5
23.3
1.8
4156
.1266
.1295
22.5
23.7
1.2
3511
.1256
23.5
24.3
1.2
3269
.1239
24.5
24.6
0.1
3040
.1272
25.5
25.7
0.2
2744
.1325
26.5
26.6
: 0.1
2247
.1276
27.5
27.7
0.2
1753
.1214
28.5
28.5
0.0
1328
.1073
29.5
29.5
0.0
1045
.1046
30.5
30.7
0.2
768
.0913
31.5
31.6
0.1
565
.0966
32.5
32.5
0.0
510
.0916
33.5
33.5
0.0
320
.0737
34.5
34.6
0.1
305
.0791
35.6
35.5
0.0
236
.0670
36.5
36.5
0.0
190
.0650
37.5
1 37.9
0.4
167
.0685
38.5
38.6
0.1
194
.0801
39.5
39.5
0.0
153
.0765
40.5
40.3
—0.2
121
.0717
41.5
41.2
—0.3
74
.0657
42.5
43.1
+ 0.6
94
.0730
43.5
45.2
I +1.7
80
.07512
44.5
45.3
+ 0.8
63
.0664
In determining any critical point, however, the ages deduced as shewn above are
not what is required. "^Vhat is definitely sought is the position and value of the
maximum frequency, referred to a mean-age of bridegrooms (a;), (or of brides {y) ) ;
that is the value of y (or of x, respectively) at which the ma xi mum value occurs.
The data from which these are deduced are the series of parallelepipeds the heights
of which may be taken as the group-totals. Thus, the horizontal series of group-
numbers 50, 59 and 28, treated as ordinate-values bounded by a curve, gives 66.13
years as the' age of brides, corresponding to a maximum frequency of 62.18. If the
41 group be included, the maximum wUl be changed to age 67.50 years, and the
frequency to 60.29. The mean age of the bridegrooms should be ascertained on the
vertical line 67.50 for brides, but without incurring sensible error it may be taken
as 72.50 — 5 (97 — 21)+ (24 x 59) =-- 72.23, see this part, section 20, formulae (412)
to (415) ; the factor 5, however, appearing because the unit is 5-years. Bespecting
(with sufficient approximation) X»i = J and i or in years five times these amounts,
or 2i and 12^. This gives 39 Jf and 60^^ as the frequencies at t^e maximum and
miniTmiTTir
206
APPENDIX A.
Mean Age of
Bridegrooms
in Maximum
Group.
Age of Bride
for
Maximum
Frequency.
DiSerence of
Age.
Maximum
Frequency.
Proportion of
all Bridegrooms
of same
Age-Group.
15.5
16.6
1.0
1
0.250
16.5
17.5
1.0
14
.274
17.5
17.7
0.2
60
.250
18.5
18.4
—0.1
318
.264
18.5t
0.0
352t
.272t
19.5
18.9
—0.6
820
897t
.2416
.2654t
20.5
19.5
— 1.0
1279
1496t
.1986
.2117t
21.5
19.7
—1.8
2558
.1472
19.7t
-1.8t
2410t
.1465t
21.5
21.4t
— o.it
32501
.1968t
22.5
21.6
—0.9
4110
.2057
21.8t
—0.7
3424
.1714
23.5
21.7
— 1.8
4250
.1839
22.lt
-1.4t
3508t
24.5
21.8
—2.7
3766
.1511
22.8t
— 1.7t
3333t
25.5
21.5§
— 4.0§
3276 §
.1329§
23. 3t
— 2.2t
3026t
.1225t
21.9
—3.6
3342
26.5
21.6§
— 4.9§
2710§
.1158
23.4t
— 3.0t
2694t
.H47t
21.9
—4.6
2774
27.5
22.6
—4.9
2271
.1080
20.6
—6.9
2230
.1061
21.8
—5.7
2293
28.5
24.3
—4.2
1977
.1199
21.9
—6.6
1973
29.5
24.7
—4.8
1492
.0932
22.0
—7.5
1458
30.5
26.0
—4.5
1195
.0892
21.9
—8.6
1080
31.5
26.2
—6.3
849
.0820
32.5
26.1
—6.4
809
.0830
22.7
—9.8
719
33.5
26.3
—7.2
565
.0733
23.4
— 10.1
557
.0722
34.5
24.5
— 10.0
560
.0823
34.2?
— 0.3?
309?
.0455?
35.5
25.5
— 10.0
486
.0800
36.5
26.5
— 10.0
371
.0694
37.5
27.4
—10.1
332
.0753
37.2
— 0.3
171
.0388
38.5
28.5
— 10.0
364
.0804
38.3
— 0.2
195
.0430
39.5
29.5
— 10.0
246
.0658
39.2
— 0.3
157
.0420
40.5
30.3
— 10.2
217
.0667
41.5
31.5
— 10.0
144
.0617
41.2
— 0.3
76
.0325
42.5
32.5
— 10.0
137
.0562
42.3
— 0.2
90
.0361
43.5
32 9
— 10.6
108
.0625
42.9
— 0.6
94
.0457
44.5
32.7
— 11.8
99
.0567
43.6
— 1.0
57
.0326
J The restilts include corrections for misstatements of age. § These maxima
disappear altogether when corrections are applied for misstatements of age.
NUPTIALITY.
207
TABLE LXn. — Critical Positions on the Fiotogamic Surface, for 5- Year Groups.
Marriages in Australia, 1907-1914.
Maximum age-gronp of
brides
Mean age of brides in
maximum group . .
10-14
•14.3
?
15-19
•18.3
20-24
•21.6
22.5
25-29 30-34
•26.6 ' »32.2
27.3 32.1
35-39
?
37.2
40-44
?
42.2
Age of bridegroom for
maximum freauency
Difference of age
Maximum frequency . .
Proporiiion of all brides
of same age- group
22.9
8.6
46.7
0.600
23.1 t23.8
4.8 2.2
1 1.3
24685 I t72500
124727 ' J72170
0.599 I 0.553
t0.601 I t0.551
27.5
0.9
0.3
36722
0.488
32.1
-0.1
-0.0
9397.6
0.336
37.5
0.3
3716.5
0.280
43.4
1.2
1541.1
0.251
Maximum age- group of
bridegrooms
Mean age bridegrooms
in maximum group
15-19
•18.4
1
20-24 i 25-29
22.3 [ 27.3
1
1
30-34 35-39
32.2 , 37.1
40-44
42.2
45-49
47.3
Age of bride for maxi-
mum frequency
Difference of age
Maximum frequency . .
Proportion of all bride-
grooms of same age-
group
17.8
0.6
3800
t4000
0.783
0.779
1
22.1 23.6
0.2 3.7
59496 51865
{59166
0.644 i 0.496
0.643 1
25.9 27.9
6.3 I 9.2
18290 7465.5
0.381 0.310
32.1
10.1
2837.0
0.240
37.4
9.9
1683.8
0.231
Maximum age-group of
brides
Mean age of brides in
maximum group . .
45-49
?
47.3
50-54
?
52.2
55-59
?
57.3
60-64
1
62.3
65-69
•67.3
67.3
70-74
•71.5
72.1
75-79
•76.50
76.8
80-84
•81.8
82.2
Age of bridegroom for
maximum frequency
Difference of age
Maximum frequency . .
Proportion of all brides
of some age-group
48.3
1.05
887.5
0.254
52.7 ■
0.5
417.1
0.260
57.9
0.6
225.8
0.286
64.9
2.6
111.0
0.255
67.5
0.2
0.2
100.1
0.334
73.7
2.2
1.6
28.5
0.277
78.0
1.5
1.2
8.3
0.280
77.5
-4.3
-4.7
4.3
0.610
Maximum age-group of
bridegrooms
Mean age bridegrooms
in maximum group
50-54
52.1
55-58
57.3
60-64
62.3
65-58
67.3
70-74
72.2
75-79
77.3
80-84
82.4
8!>-89
87.4
Age of bride for maxi-
mum frequency
Difference of age
Maximum frequency . .
Proportion of all bride-
grooms of same age-
group
39.6
12.5
785.5
0.200
46.8
10.5
457.9
0.215
48.8
13.5
213.9
0.172
45.2
22.1
139.0
0.168
62.8
66.1
9.4
6.1
66.6
62.2
0.160
0.150
57.5
72.5
?
32.3
25.9
0.197
0.158
47.4
62.5
72.5
?
10.2
9.6
8.7
0.165
0.155
0.140
47.5
7
4.3
0.360
* Calculated from yearly group results. t It is impossible from the data to determine these valuet
with precision. { With partial corrections for misatatements of age.
Fig. 61 shews the graphs of the maximum values. It is evident
from these graphs that the greatest frequency of marriage is not well-
defined according .to alleged ages. The surface shews ridges on the lines
Aa, Ab, Acde, Afg and Ah. The highest point is for the group bridegrooms
about 23.4, and brides 21.6 years of age, the frequency attaining to about
4,200, or about one seventy-second part (0.013911) of all the marriages.
208 APPENDIX A.
The maximum group is 4114, or 0.13626 of the marriages. These figures
are, however, somewhat uncertain, for reasons which will be pointed out
in the next section.
24. Apparent peculiarities of the protogamic frequency. — Fig. 61
shews, by dots, the positions of maxima on the (vertical) columns,
that is according to the ages of brides ; and, by dots with circles,
the positions of the maxima on the (horizontal) rows, that is accord-
ing to the ages of bridegrooms. If the ages have been correctly given
there is no unique mode on the horizontal lines ; and this is a matter which
demands special consideration. In Part X., § 6, Fig. 42, p. 115, it is
shewn that the number of under-statements by women amounting to
10 years, is quite abnormal ; it does not follow the progressive diminution
which characterises understatements amounting from 1 to 11 years.
In the figure the line bAde would be the characteristic summit if the
greatest frequency of marriage was in the case of parties of the same age.
The Une f g would be the characteristic if a large number of men married
wives 10 years younger than themselves ; while for the line Af to hold
good, very large numbers of men of ages 22 to 31 must marry women of
21 years of age, irrespective of the disparity of age. To give the line of
maxima Ah, a considerable number of men must marry women whose
difference of age is one-half their age above 22. Such characters in a
protogamic surface, are, a 'priori, extremely improbable. They would
also characterise the apparent protogamic surface, if a considerable number
of women, really of ages 22 to 32, all gave their ages as 22, when marrying
men of from 22 to 32 years of age, and if a considerable number of women
of 32 and upwards understated their ages by 10 years. This explanation
probably does not differ very materially from the fact. Hence Tables
LIV and LX must be regarded as of inferior value. It is, of course, much
to be regretted that social organisation does not admit of the social-
psychological fact of conjugal frequency at equal and disparate ages
being accurately ascertained.
25. The contours of the protogamic surface. — ^The tedium of a
rigorous analysis of a surface, when the measure of uncertainty is so large
as is the case with the protogamic surface for Australia, is not warranted.
A rough smoothing of the 5-year groups was, therefore, effected, and
attributing the smoothed values to the centre points of the groups, and a
series of contours for the proportions of 5, 10, 20, 40, etc., in a million
of total marriages of all ages, were inserted by graphic methods. These
gave fairly smooth contours. Regular curves being drawn, so as to
ignore the minute undulations of the contours the results shewn on Fig.
61 are obtained. These represent with considerable precision the actual
data from which they were derived, and will enable such data to be
reproduced. They disclose the frequency distribution, for all combina-
tions of ages,
NXrPTIALITY.
209
Curves of Equal Marriage Frequency. — The Frotogamic Surface.
Ages of Brides.
10 SO 30 40 50 60 70 80 90
100
100
Fig. 61.
Note. — The pairs of ages which give equal frequency of marriage are found
by following the course of any isoprotogam. The frequency indicated is per
million marriages of all ages. The co-ordinates of any two points, whatsoever, on
any isoprotogam are equivalent age-pairs, that is pairs of ages which are
characterised by the same frejjuency of marriage.
The protogamio surface, indicated by the family of curves or isogamic contours,
is not the surface of frequency for indefinitely small ranges of age, but the surface for
5-year ranges of age ; see hereunder. These contours or " isogams" are numbered
5, 10, 20, etc., denoting the doubling of the frequency. The point denoted by an
asterisk near A, is the summit of this surface, i.e., its ordinates are the centre of the
5-year ranges of age for which the frequency of marriage is greatest. From the sum-
mit it falls most rapidly in the directions A, B and A, C, and least rapidly in the
directions A, B and A E, the directions being shewn by broken lines.
The values on the protogamio surface can be thus interpreted : — Assuming that
the frequency of majriage for given pairs of ages, is as in AustraUa during the eight
years, 1907-1914, in every 1,000,000 marriages of brides and bridegrooms of all
ages, the number to be expected in any 5-year group over the range of 2J years
earlier to 2J years later than the ordinates of the point taken, in the case of both
bride and bridegroom, will be that shewn by the corresponding isogam, along which
there will be equal frequency of marriage. Thus, for example, following the varia-
tion with age contour corresponding to 10,240 marriages out of a total of 1,000,000,
the frequency indicated will be very approximately that for the 5-year ranges, the
middle values of which are brides 20 with bridegrooms 37 ; brides 24 J with bride-
grooms 40 ; brides 30 with bridegrooms 42 ; brides 35 with bridegrooms either
41 J or 29 ; brides 37 with bridegrooms either 40 or 33 ; and so on. The contours
thus shew the centre values of a 5-year range of age, at which there is equal frequency
of marriage within the range. That is, if the co-ordinates of any point on a contour
be X and y, the frequency of marriage is for the ages bridegrooms x — 2J to a; + 2^,
with brides y — 2^ to y + 2^. Hence if M be the total number of marriages, the
actual nimiber will be the number on the contour divided by 1,000,000 and multi-
plied by M,
210
APPENDIX A.
Characteristics of the Frotogamic Surface.
Age of Brldesi Age of Bridegrooms.
10 20 30 40 10 20 30 40 SO
S
I
h s
■<S ea
0.8
Aa
'■^'l
^?
A
[^
p
■%
Tk
v^
e ^
y
u
-%
iv^
^v^
V
^
^.A
*>f'
.••
o» c
A nnn
A
A
^
A
3 000
\
1
j^
\a
\
T)
2,000
/
\
J>
\
\
1,000
■
\
1
i
\,
1
\,
}
v.
-^
y
S,
s.*.
10 20 30 40 10
Age of Brides.
Pig. 62.
20 30 40
Age of Bridegrooms.
50
Curves ABC and D shew the various vertical features of the protogamio
surface. Of these : —
Curve A shews the projection of the profile on the y or age-of-brides axis, the
dots indicating the values according to the data, and the continuous line shewing
the probable true position of the surface profile.
The outer Curve B shews the projection of the profile on the x or age-pf-husbands
axis, the dots and circles indicating the positions according to the data. The inner
curve indicates the position of a series of second and fairly well-defined maxima.
All the points shewn are maxima of some kind.
Curve C shews by dots, and a zig-.zag line joining them, the proportion which
the frequency at the various maxima bears to the totals for the same age-groups of
brides. The general trend of this frequency as a function of age is shewn by a broken
line.
Ciu-ve D shews by dots with circles and by a zig-zag Une, the proportion which
the frequencies at the various mSixima bear to the total for the same age-groups of
bridegrooms. The broken line shews their general trend.
Each contour is twice the height of the contour immediately outside
it ; thus the surface rises with great rapidity, and is very steep on the top,
and also the left hand side in the figure. The proportion per million
marriages for a 5-year group, ranging between a; ± 2^ and y ± 2^ is
defined by the numbers written along the contours. The projection on
the y—axis of the ridge running from the top left-hand corner to the
NUPTIALITY.
211
bottom right-hand corner is shewn by curve A, Fig*. 62 ; and its pro-
jection on the x—axis is shewn by Fig. B. The proportion which the
frequency at the maximum bears to the total for the same age-group of
brides is shewn by curve C, and for the same age-group of bridegrooms by
curve D. In these two last curves the zig-zag lines shew the successive
principal maxima, and the dotted Unes the general trend. It is probable
that in a large population, when the ages at marriage are correctly given,
the results would yield regular curves of the types drawn. The contours
do not indicate curves of great regularity, but that is doubtless due
(at least in part) to the inexact statement of age and the paucity of the
numbers for higher ages.
27. Relative marriage frequency in various age-groups. — ^For socio-
logic purposes, a table shewing the relative marriage frequency in various
age-groups is of obvious- importance. Given an Australian population,
constituted as to numbers of married and unmarried in age-groups as
was its population during 1907 to 1914, 1,000,000 marriages are found to
be distributed as follows : —
TABLE LXin.— Relative FieoLuency of Marriage in Various Age-Groups. Australia, 1907-1914.
Age-
group
of
AOE-OKOOT OP BEIDES.
All
Bride-
grooms
10-14
15-19
20-24
25-29
30-34
35-39
40-44
45-49
50-54
55-59
60-64
65-69
70-74
75-79
80-84
85-89
Ages.*
15-19
20-24
25-29
30-34
35-39
40-45
45-49
50^54
55-59
60-64
65-69
70-74
75-79
80-84
85-89
30
146
60
10
7
5
4
3
2
1
1
1
1
11,605
76,788
36,249
9,262
3,039
785
381
136
43
20
7
5
3
1
4,920
184,576
167,668
51,407
17,013
6,222
1,982
607
182
93
43
23
7
6
1
411
37,452
115,639
67,647
24,184
8,496
3,669
1,273
414
209
83
40
13
9
1
56
4,762
22,331
30,265
18,795
9,315
4,351
1,783
686
331
143
66
20
10
2
12
1,077
4,537
8,206
11,999
8,196
5,477
2,499
978
457
219
99
28
14
3
7
199
935
1,740
3,440
4,978
4,239
2,545
1,293
547
265
146
38
22
5
3
73
259
484
1,037
1,690
2,827
2,237
1,425
686
366
186
48
28
10
2
13
66
it
371
872
1,346
1,027
689
431
215
64
33
8
1
3
6
13
50
86
246
388
697
624
431
215
85
29
5
"l
3
7
13
30
80
166
215
351
315
166
92
23
3
"2
3
5
10
27
53
99
199
182
113
73
13
2
"1
3
7
17
50
63
73
47
8
1
"3
6
9
13
21
27
4
1
"2
3
5
7
11
2
"1
1
1
17,048
305,080
346,765
159,019
79,797
39,183
24,058
13,046
7,086
4,169
2,571
1,377
558
202
42
All
Ages*
271
137,324
433,760
249,345
92,906
43,799
20,398
11,358
5,438
2,778
1,465
781
270
84
30
3
1,000,000
• Theae totals are about ten times those in the final oolumns of Table LX., p. 199. Though in substantial agreement
they are not absolutely identical because these results have been slightly smoothed.
The above table is founded upon the results given by a slight smooth-
ing of the actual numbers, and gives the roughly adjusted relative-
frequency of marriage' according to age-groups, based upon the marriages
of the 8-year period, 1907 to 1914 inclusive, the 1911 Census being re-
garded as giving a sufficient indication of the relative numbers of married
and unmarried for the computation of any derivative relations. The
middle point of time would be Jan. 0, 1911, while the Census is April 3rd,
1911. The total marriages were 301,922, or about 37,740 annually;
half of them had occurred by about April 28, 1911, that is 25 days after
the Census, hence a correction is not required,
212
APPE^TDIX A.
28. The numbers of the unmarried and their masculinity.— The
smoothed results of the Census give the following numbers of unmarried
at each age, viz., those shewn in Table LXIV. From these the ratios of
the males to the females {M/F) have been computed ; they are shewn
opposite the letters " Mas." in the Table. From the numbers given the
mascuUnities can be computed of the various age-groups, which are
required hereinafter for the computation of the probabihty of marriage
according to pairs of ages.
TABLE LXIV. — Number of Unmamed Males and Females and the Masculinity
(,M/F) at each Year oJ Age. Australia, 3rd April, 1911.
Year of
AaBS.
Decen-
nium
in Age.
10
20
30
40
50
60
70
80
90
M
F
Mas.
58,648
56,401
1.03984
43,049
42,654
1.00926
45,466
38,370
1.18493
16,700
10,839
1.54073
7,973
4,987
1.598T
5,232
2,340
2.2359
2,152
830
2.593
1,230
360
3.417
370
92
4.0
33
8
4.1
1 M
F
Mas.
54,759
52,982
1.03354
42,753
42,222
1.01258
43,799
34,634
1.26462
14,875
9,659
1.54001
7,669
4,623
1.6588
4,797
2,127
2.2553
2,032
760
2.674
1,190
320
3.719
312
80
3.9
26
6
4.3
2 M
F
Mas.
52,659
51,145
1 02960
42,748
42,001
1.01779
41,097
30,878
1.33094
13,250
• 8,730
1.51775
7,386
4,226
1,7477
4,332
1,938
2,2353
1,922
716
2,688
1,140
280
4,071
266
69
3.85
20
4
5.0
3 M
F
Mas.
51,158
49,785
1.02758
42,990
42,072
1.02182
37,541
26,360
1.42418
11,925
7,835
1.52201
7,077
3,940
1.7962
3,942
1,780
2.2146
1,822
690
2.641
1,080
240
4.50
225
58
4.05
15
3
5.0
4 M
F
Mas.
49,998
48,556
1.02970
43,618
42,484
1.02669
34,003
22,725
1.49628
10,900
7,278
1.49766
6,763
3,707
1.8244
3,642
1,549
2.3512
1,732
650
2.665
1,010
205
4.927
189
48
3.94
11
2
5.5
5 M
F
Mas.
48,812
47,428
1.02918
44,598
42,273
1.03062
30,338
19,600
1.54785
10,177
6,791
1.49860
6,453
3,441
1.8753
3,342
1,363
2.4519
1,622
600
2.703'
920
180
5.11
158
38
4.16
8
2
4.0
6 M
F
Mas.
47,543
46,298
1.02689
45,482
43,915
1.03568
26,823
17,267
1.55342
9,529
6,319
1.50799
6,185
3,212
1.9256
3,042
1,248
2.4375
1,490
550
2.709
810
155
5.23
121
28
4.32
6
1
6.0
7 M
F
Mas.
46,084
45,111
1.02157
46,212
43,813
1.05475
23,597
15,370
1.53526
9,032
5,910
1.52826
5,957
3,000
1.9856
2,762
1,145
2.4122
1,396
500
2.792
690
135
5.11
90
20
4.5
5
1
5.0
8 M
F
Mas.
44,783
44,055
1.01652
46,620
42,854
1.08788
20,808
13,646
1.52484
8,613
5,630
1.52984
5,731
2,783
2.0593
2,512
1,032
2.4341
1,328
450
2.951
560
120
4.67
65
14
4.6
4
1
4.0
9 M
F
Mas.
43,770
43,236
1.01235
46,470
41,020
1.13286
18,677
12,153
1.53682
8,265
5,303
1.55855
5,522
2,570
2.1486
2,302
910
2.5297
1,272
405
3.141
450
105
4.28
46
10
4.6
3
1
3.0
100 and over — Males, 2 ; Females, 1. Totals under 13, 662,764, 611,873 =1.08317.
Note. — ^The masculinity is for the year-groups, and may be assumed to be the
masculinity at age a; + J, where x is the tabular age, viz., the " age last birthday."
NUPTIALITY.
213
The change of masculinity with age follows no simple law, as will be
seen from curve A on Fig. 63. The irregularities after 80 are due to the
relatively small numbers on which the curve is based, and must be re-
garded as accidental. The masculinity diminishes in the earlier years,
because of the greater mortality among males. Its constancy between
the ages 25 and 37 is remarkable, as also is the sudden increase commencing
at 66 years of age, and continuing to 76.
CA S mOQ
I ^^^
a. «.°a
■a w ■«
r.ili
E-l H
Number of Males and Females Marrying and Living
together in the State of Marriage, and the
Masculinity of the Unmarried.
9 180
8 160
7 140
6 120
5 100
4 80
3 60
2 40
1 20
AlteB 10
!L
—
—
/'
\
/
\\
\V"^
B
M-^
>s-
1
iv
\;
i-
i
\
\
i
s \
\
;
1\
\
S
,/
1
D
-\
\.
\
1*1
1
' M
\
,
/-^
r
y
\
^
\
'
A
k
/
\
i A
v
p^
/
4I
L V
A^"
-\
\
/ "
\
\.
y
t^*^
X
V
"^
-"Hill
r- \
.>
-^
k-
Ml
s
\
s
b._^
b^
_J2_
V
-♦^^S
^^
^s
^
^
^
i
3
2 1.1
1 1.0
30 40 50
Mg. 63.
70 80 90 100
Curve A denotes the variation with age of masculinity ('M/F) of the unmarried.
The small lozenge-shaped dots are the values according to the data ; the continuous
line shews the general trend. The scale for the masculinity up to nearly 20 years
has also been plotted on ten times the scale. See Table LXIV., p. 212.
Curve B denotes the number of married females of marriages living with their
husbands in a total of 1,000,000 couples. See Table LXVIII., p. 224.
Curve C denotes the number of married males of various ages living with their
wives, in a total of 1,000,000 couples. See Table LXVIII., p. 224.
Curve D shews the adjusted number of females of various ages, per 100,000
marriages, occupying between 1907 and 1914. See Table LIX., p. 197.
Curve E shews the adjusted number of males of various ages, per 100,000
marriages, occurring between 1907 and 1914. See Table LIX., p. 197.
214 APPENDIX A.
29. The theory of the probability of marriages in age-groups. — ^The
data do not exist for a definite and rigorous determination of the pro-
bability of marriage in age-groups ; nevertheless a fairly accurate esti-
mate is possible by means of a somewhat empirical theory, which will
now be indicated. The deduced results are shewn in Tables LXVI. and
LXVn., see pp. 219 to 222.i
For convenience the adjusted numbers from the Census are given in
Tables LXIV. and LXV. hereinafter ; the corresponding numbers of
marriages occurring in each age-group are also given. The values of q
given in the tables enable the number of marriages likely to occur in each
age-group to be computed when the numbers of unmarried males and
females in the group are known. Thus, q being the tabular number, the
number of marriages, N, may be computed by means of formula (431)
or formula (434) hereinafter. (See next section.)
Suppose that in any age-group there are M unmarried males and F
unmarried females ; and that in a unit of time N pairs of these marry.
The probabiUty wiife F females in the group, of a particular marriage occur-
ring among the M males is obviously N/M ; and with M males in the group,
the probability of a particular marriage occurring among the F females is
similarly N/F. Such a statement of probability, however, lacks general-
ity. To obtain a more general one, an expression is needed which, given a
definitive tendency towards the conjugal state in males and in females,
though not necessarily of the same strength (or potential) in each sex, and
not necessarily independent of the relative numbers of the sexes, nor even
independent of the lapse of time, will give the number of marriages
occurring in a group, constituted in any manner whatever in regard to the
numbers of either sex. We shall call the tendency to marry the conjugal
potential under a given condition. In the case of males let the conjugal
potential be denoted by y, and in the case of females by y'; y and y' vary
with age, doubtless also with time, and (we may assume) with the relative
frequency of M and F. Without d.sserting it to be exactly the law of
variation, we may suppose that the conjugal potential varies somewhat as
some constant, multipUed into some power of the ratio of the numbers of
the unmarried of each sex. Put p for the constant in the case of males,
p' for the constant in the case of females, then the conjugal potentials
are of the type p. f {-^), which function can, for all practical purposes,
probably take the form
(425) y=2,(^)';andy'=p'(-|.)'
formulae in which r and s are indices to be ascertained by experiment.
^ These results are on the basis of 10 million males, and the same number of
females. Hence if they are multiplied by one ten-miUionth of 1,508,623, and
1,277,259 respectively, they will give the absolute numbers, since these were the
number of unmarried males and of unmarried females respectively, on 3rd April
1911.
NUPTIALITY. 215
Thus y = p and y' = p' when the numbers of unmarried of either
sex are equal ; ordinarily they do not differ sensibly therefrom. Again,
if the number of females be large, the y potential is doubtless smaller ;
and if the number of males be large the y' potential is smaller. This
appears to be confirmed by experience. The expressions (425) can be
made to fit the facts by appropriately determining r and s.
From (425) we have at once for the ratio of the conjugal potentials
<-' f'-H'Y-
where w = r + s, from which it is evident that it is not necessary to ascer-
tain r and s individually, but only their sum, w. And if the conjugal
potential vary with age, it could be ascertained only by comparing a series
of results for the one age-group when the numbers of males and females
were very divergent ; all other circumstances promoting marriage remain-
ing constant. For this reason, with the limitations of existing data, We
must assume (which doubtless, as already indicated, is not exactly true),
that, when the numbers of the unmarried of each sex are equal, the
conjugal potential and probability of marriage vary in the same way .
That is
(427) y / y' a: p/p' ;
or the probabiUty of marriage is the effective measure of the conjugal
potential ; or in other words (subject to what has been said above) we
may suppose that, with equal numbers of unmarried males and females,
the frequency of marriage is a normal measure of the conjugal potential.
If we make still another assumption, viz., that indicated hereunder
(in the passages in italics), a crude type of solution becomes possible,
and the problem may then be envisaged as foUows : —
If there be M males in any age-group and F females in any other age-
group, it is obvious that there can hei MF marriages of particular pairs
among these groups : and if a group out of these of N males and N females
be taken, it is similarly seK-evident that they can form N N marriages of
particular pairs. GonsequerUly assuming that the marriage of particular
pairs is equally probable, avd that the relative magnitude of M and F does
not influence the probability, p, then the chance of N marriages occurring
is
(428) pxy =- N^Ny/ (M^Fy)
X and y denoting the age-groups referred to. The value of p cannot
possibly become unity unless M = F = N. This probability does not,
however, enable us to compute the likelihood of N marriages occurring
with particular values for M nd F, since obviously N is not y/p.-\/{M.F),
although that is a solution of eqi ation (428) ?■ Subject to the assumptions
1 For example, given M constant, N would depend upon ^JF, which is certainly
not correct if M be large and F small. In this ease iV would evidently vary as F,
not B& ■^F.
216 APPENDIX A.
made, the function representing the chance of N marriages occurring must
clearly vary approximately as -^{MF), when they are sensibly equal,
and must vary sensibly as F (or M) when M (or F) is relatively very large.
In order to obtain an expression that will readily fulfil the necessary
conditions, we may observe that if we put
(429) N^y=q^y.4> (Mx) . ^ (Fy)
and for ready computation assume that the functions <fi and tfi may, with
sufficient precision, take the form M^ and J?"' ; then | and tj must fulfil
the following conditions, viz. : —
(a) In order to give k N, when the numbers of males and females
are kM and kF (at the same time), we must have ^+tj =1,
so that jfcf . jfc'' = k.
(b) As a consequence of this condition, viz., (a), it follows that
when M = F, ^ = r], and each must be J.
(c) In order that, when M is relatively very great (or small) as
compared with F, N shall vary as F (or M), f (or ij) must
become in such a case sensibly zero, and tj (or f ) must
become sensibly unity.^
{d) The fundamental assumptions require also that the expressions
shall be symmetrical in regard to M and F, i.e., one can be
had from the other by mere interchange.
In practical examples we may have the ratio of Jf to .F varying from
about 0.1 to about 10.0 through a wide range of important ages, so that a
formula, to be of the widest appHcation, should at least embrace this
range. Up to 40 years of age M ranges from about Fto^F. A function
that will fulfil the required condition as above indicated is
F M
(430) N Qc M'>^+P . F^+'
Consequently we may write instead of (429) : —
F M 1 1
(431). . N^y = q^y . M^+^ . F^+P = q^y . M,l,^+<l> =q^y . J-jni+z*
and to find q from the results furnished in Tables LXIV. and LXV. we
have,
N-Xy Nxy
(432) qxy= JH = XI ; or
(432a) \ogqxy = log Nxy - fT—'^^^ ^ ~ YTTj, ^°S ^'
X and y denoting the central values of the age-groups, i.e., a; ± | Jfc, «/ J; \k
where k is the range of the group. The apphcation of this formula can be
greatly facilitated in the following way : — ^Let Sxy = Mx + Fy, that is, let
' 4 is the quantity denoted by <p^, and ij that denoted /Uj on page 132 hereinbefore.
NUPTIALITY.
217
Sx„ denote the total number of single persons in the groups of males of age
X and females of age y, and let the masculinity (or the femininity) of S be
denoted by M/F (or F/M) ; then assuming that the probability is
identical for A males and B females, with that for B males and A females
(which, however, though by no means certain, is not determinable from
existing data), we may compute the value of the ratio
(433)
F M I 1 ; _i_ /
iS
which depends merely upon the masculinity, /x (or the femininity <f) ), and
is independent of the absolute value of S, or of M and F. Consequently
with a table of values of R arranged according to the argument fi (or ^ ),
we have, by simply dividing M by F, (or F by M) and entering the table,
(434).
, Jyxy — ^ ^xy • R/i . <lxy — Say ^ lixy • Qxy
Q itself could be tabulated but for the fact that the masculinity in age-
groups may differ appreciably with the lapse of time. We require, there-
fore, two tables, viz., one for R depending upon the masculinity (or
femininity), and one for q depending on the frequency of marriage for
the age-groups in question. After preparing a table of the values of R,
Table LXV., those of q can readily be calculated. In using the following
table of the values of R, it is, of course, a matter of indifference whether
it be entered with the argument " masculinity" or " femininity."
TABLE LXV.
F M
Values ol R = (M^+^ . F^ + ^) / \S, for computing the effect of
unequal numbers oi unmarried males and females on the
frequency of marriage.
M/F and F/M are interchangeable.
VALrBS OP —
M
F
M
F
M
F
M
F
M
F
M
F
M
F
lOOOC
lOOOC
—
10001
—
—
lUOOl
—
— lOOQC
—
—
lOOOf
—
10000
F
M
B
F
M
fi
F
M
&
F
M
K
F
M
K
F
M
B,
F
M
B
1.0
1.0
10000
34
2.0
.5000
8,399
203
3.0
.3333
6,580
IS^
4.0
.2500
5,278
107
5
.2000
4,359
668
15
.0667
1,481
96
60
.0167
351
1.1
.9091
9,96e
9C
2.1
.4762
8,196
200
3.1
.3226
6,428
147
4.1
.2439
5,171
102
6
.1667
3,691
503
16
.0625
1,385
84
70
.0143
299
1.2
.8333
9,876
ISO
9,746
2.2
.4545
7,996
195
7,801
3.2
.3125
6,281
142
6,139
4.2
.2381
5,069
100
4,969
7
.1429
3,18t
388
2,800
17
.0588
1,301
76
1,225
80
.0125
261
IS
.7692
2.3
.4348
3.3
.3030
4.3
.2326
8
.1250
18
.0556
90
.0111
231
15£
191
13(1
9t
30£
6t
1.4
.7143
9,587
178
2.4
.4167
7.61C
186
3.4
.2941
6,003
132<
4.4
.2273
4,873
93
9
.1111
2,491
249
19
.0526
1,159
61
100
.0100
207
1.5
.6667
9,409
193
2.5
.4000
7,424
18C
3.5
.2857
5,871
127
4.5
.2222
4,78(
9C
10
.1000
2,242
207
20
.0500
1,098
227
200
.0050
102
1.6
.6260
9,216
20C
2.6
.3846
7,244
174
3.6
.2778
5,744
12S
4.6
.2174
4,691
87
11
.0909
2,035
173
25
.0400
871
151
400
.0025
51
1.7
.5882
9,016
204
2.7
.3704
7,070
169
3.7
.2703
5,621
118
4.7
.2128
4,603
84
12
.0833
1,862
146
3U
.0333
186
700
.0014
29
1.8
.5556
8,812
207
2.8
.3571
6,901
163
3.8
.2632
5,503
116
4.8
.2083
4,510
81
13
.0769
1,716
126
4U
.0250
534
111
1000
.0010
20
1.9
.5263
8,605
206
2.9
.3448
6,738
158
3.9
.2564
5,388
110
4.9
.2041
4,43i
79
14
.0714
1,59{;
109
50
.0200
423
2000
.0005
10
In the columns " 10,000 E," the " differences " are also shewn.
218
APPENDIX A.
From the values in the above table, a working table may readily be
constructed so as to avoid tedious calculations of the function R.
When, however, the value of ilf is large, and that of F is small (or
vice versa), the value of N depends mainly on F (or on M). In this case
1 1
it is preferable to use a table of the values of fj.^+i^ (or of ^^+*) with
the argument jj, (or (^) ; see formulae (431) or (432) just given. A table
such as LXVa. will then be required.
The formula to be used wiU be
(435) Nxy= F . R'fi .qx„=M . iJ> . q^y :
1
in which R'^ is the tabular value ^^ + '' and R'^ is the tabular value ^^ + *,
the q quantities being a.s before.
Values of iJ' = 2/x
i+M
TABLE LXVa.
{or computing the effect of unequal numbers
of unmarried males and females on the frequency of marriage.
M F
M F
M F
M F
— or —
R'
— or —
R'
— or —
R'
— or —
R'
F M
F M
F M
F M
10
1.2328
60
1.0694
200
1.0267
700
1.0094
20
1.1533
70
1.0616
300
1.0192
800
1.0084
30
1.1159
80
1.0556
400
1.0150
900
1.0076
40
1.0958
90
1.0507
500
1.0124
1,000
1.0069
50
1.0797
100
1.0467
600
1.0107
2,000
1.0038
The table shews very clearly that as the unmarried females (or
males) become relatively fewer the number of marriages varies more
nearly in the proportion of the number of females (or males).
30 . Masculinity of the unmarried in various age-groups. — ^The results
embodied in Table LXIV., make it possible to compute the mascuUnity
of the unmarried for any combined age-groups, since this affects the
number that may be expected to marry. The masculinities are shewn in
two tables, viz., Table LXVT. and Table LXVII., the former giving the
results for 2-year age-groups for ages 15 to 44 for bridegrooms, and ages 13
to 44 for brides ; and the latter the results for 5-year age-groups for ages
15 to the end of life for bridegrooms, and 10 to the end of life for brides.
From the values of M/F, = fi, (or F/M, =ff>,) the values of F / {M+F)
and oi M / (M+F) may be readily computed if required. Thus^
(436).
F
1
_^
M + F l+[i l+<f>
= <f>2
M
M + F !+<!, l+n
= Ma
1 For other definitions of masculinity and femininity see Part X., § 3, (333) to
(335), and Table XXI., pp. 132, 133 hereinbefore.
NUPTIALITY.
219
TABLE LXVI. — Shewing the MascuUnity of the Unmarried in 2-year Age-groups
(M/F), and the Frobability-hmction 1,000,000 ^qxy for calculating the Number of
Marriages in the Two-year Age-groups indicated.
Aqb of Brides and Nttmbee of Unmabeied Females.
Age
Group.
13-14
84,574
15-16
87,188
17-18
86,667
19-20
79,390
21-22
65,512
23-24
49,085
25-26
36,867
27-28
29,016
15-16
90,080
1.065
3
1.033
15
1.039
30
1.135
15
1.375
13
1,835
4
2.443
3
3.104
2
17-18
92,832
1.098
6
1.064
273
1.071
905
1.170
515
1.417
253
1.891
96
2.518
54
3.199
23
s
<
19-20
91,936
1.087
16
1.054
691
1.061
3,959
1.158
5,081
1.403
3,134
1.873
1,257
2.494
691
3.168
286
s
21-22
84,896
1.004
38
0.974
1,100
0.979
8,420
1.069
14,490
1.296
21,328
1.729
10,494
2.303
5,304
2.926
2,742
i
23-24
71,544
0.846
16
0.821
795
0.826
6,391
0.901
14,652
1.092
26,492
1.458
24,774
1.941
13,917
2.466
7,057
(3
ft
O
25-26
57,161
0.676
15
0.656
696
0.660
4,999
0.720
11,579
0.873
23,148
1.164
26,102
1.551
25,173
1.970
14,407
§
27-28
44,405
0.525
13
0.509
473
0.512
3,867
0.559
9,172
0.678
18,819
0.905
2,1970
1.205
22,699
1.530
20,553
GO
o
29-30
35,377
0.418
8
0.406
312
0.408
2,552
0.446
6,632
0.540
13,383
0.721
15,776
0.960
17,670
1.219
16,710
31-32
28,125
0.333
8
0.323
238
0.325
1,719
0.354
4,436
0.429
9,529
0.573
10,650
0.763'
12,658
0.969
12,193
1
33-34
22,825
0.270
8
0.262
169
0.263
1,398
0.287
3,408
0.348
6,950
0.465
8,707
0.619
9,068
0.787
9,683
35-36
19,706
0.233
7
0.226
211
0.227
1,055
0.248
2,497
0.301
5,450
0.402
6,074
0.534
7,830
0.680
7,804
1
<
37-38
17,645
0.209
6
0.202
102
0.204
733
0.222
1,823
0.269
3,831
0.359
5,156
0.479
5,392
0.608
6,775
39-40
16,238
0.192
5
0.186
153
0.187
453
0.205
1,317
0.248
2,542
0.331
3,153
0.440
4,035
0.560
4,477
41^2
15,055
0.178
3
0.173
45
0.174
275
0.190
678
0.230
1,538
0.307
2,041
0.408
2,390
0.519
2,977
43^4
13,840
0.164
1
0.159
35
0.160
154
0.174
465
0.211
1,082
0.282
1,551
0.375
1,726
0.477
2,427
See Note to the continuation of the Table on the next page.
220
APPENDIX A
TABLE LXVI. — Shewing the Masculinity of the Unmairied in 2-year Age-groups
{M/F), and the Probability-function 1,000,000 ^qxy for calculating the Number
of marriages in the Two-year Age-groups indicated. ( Continued).
Age of Brides
AND Number of
Unmarried Females.
Age
Group.
29-30
22,992
31-32
18,389
33-34
15,113
35-36
13,110
37-38
11,540
39-40
10,390
41-42
8,849
43-^4
7,647
15-16
90,080
3.918
4.899
5.960
6.871
7.806
8.670
10.18
11.78
17-18
92,832
4.038
20
5.048
13
6.142
8
7.081
4
8.044
8.935
10.49
12.140
19-20
91,936
3.999
225
4.999
88
6.083
83
7.013
52
7.967
34
8.849
19
10.39
11
12.022
7
21-22|
84,896
3.692
1,405
4.617
866
5.618
450
6.476
392
7.357
161
8.170
143
9.594
102
11.102
93
<
23-24
71,544
3.112
4.241
3.891
2,010
4.734
1,422
5.457
905
6.199
694
6.886
375
8.086
167
9.356
105
O
25-26
57,161
2.486
7.574
3.108
4,090
3.782
2,318
4.360
1,600
4.953
881
5.502
571
6.460
372
7.475
218
1
1
27-28
44,405
1.931
11,620
2.415
6,269
2.938
4,043
3.387
2,421
3.848
1,877
4.274
1,000
5.018
441
6.807
339
29-30
35,377
1.539
15,147
1.924
8,170
2.341
5,493
2.699
3,430
3.066
2,152
3.405
1,287
3.998
714
4.626
471
1
1
31-32
28,125
1.223
12,319
1.529
10,652
1.861
6,073
2.145
4,133
2.437
2,354
2.707
1,409
3.178
789
3.678
517
33-34
22,825
0.993
9.604
1.241
8,016
1.510
7,563
1.741
5,146
1.978
2,948
2.197
1,760
2.579
1,100
2.985
939
8
35-36
19,706
0.857
8,384
1.072
6,901
1.304
6,836
1.503
6,247
1.708
3,874
1.897
2,746
2.227
1,349
2.577
1,124
a
37-38
17,645
0.767
6,209
0.960
5,546
1.168
5,870
1.346
5,558
1.529
5,695
1.698
3,571
1.994
1,775
2.307
1,354
39-40
16,238
0.706
5,565
0.883
4,343
1.074
4,793
1.239
5,104
1.407
4,966
1.663
4,968
1.836
2,837
2.123
1,725
41-42
15,055
0.655
3,242
0.819
3,783
0.996
3,430
1.148
4,119
1.305
3,323
1.450
3,364
1.701
3,314
1.969
2,123
43-44
13,840
0.602
1,819
0.753
2,792
0.916
3,170
1.056
3,049
1.199
3,186
1.332
3,194
1.564
2,833
1.810
2,871
Note. — ^The upper figures denote the masculinity of the group, that is, the ratio
of the number of all the unmarried males of the 2-year groups of ages (15 and 16 to
43 and 44), to the munber of all the unmarried females of the 2-year groups (13 and 14
to 43 and 44). The lower figures are the values of the probability-function, ^qxy,
for the double 2-year groups, by means of which the number of marriages occurring
annually in the indicated age-groups may be calculated by formulse (431) and (432),
(434), and (436).
NUPTIALITY.
221
TABLE LXVn. — Shewing the Masculinity (M/F) oJ the Unmarried in 5-year Age-
groups (M/F), and the Probability-function, 1,000,000 ^q^y, for calculating
the number of Marriages in the 5-year Age-groups indicated.
Age or Brides and Numbek of Unmarried Females.
Age
Groups.
10-14
211,433
15-19
214,875
20-24
152,967
25-29
78,036
30-34
44,341
35-39
29,953
40-44
21,483
45-49
15,006
50-54
9,734
15-19
229,382
1.085
5
1.067
1,964
1.500
1,027
2.939
150
5.173
,37
7.658
11
10.677
9
15.29
7
23.57
7
20-24
201,906
0.955
26
0.940
13,850
1.320
40,184
2.587
13,821
4.553
3,068
6.741
1,054
9.398
280
13.45
152
20.74
44
09
25-29
120,243
0.569
15
0.560
8,932
0.786
47,073
1.541
46,906
2.712
14,445
4.014
4,309
5.597
1,239
8.013
512
12.35
212
1
30-34
67,650
0.320
4
0.315
3,897
0.442
22,202
0.867
29,880
1.526
21,665
2.259
8,006
3.149
2,305
4.508
921
6.950
260
^
35-39
45,616
0.216
5
0.212
1,906
0.298
10,601
0.585
16,323
1.029
15,687
1.523
12,726
2.123
4,724
3.040
1,970
4.686
633
40-44
36,868
0.174
4
0.176
618
0.241
4,033
0.472
6,800
0.832
8,721
1.231
9,356
1.716
7,127
2.457
3,259
3.787
1,084
o
45-49
29,858
0,141
4
0.139
376
0.195
1,907
0.383
3,439
0.673
4,664
0.997
6,875
1.390
6,452
1.990
5,656
3.067
2,551
1
50-54
21,945
0.104
4
0.102
188
0.144
812
0.281
1,647
0.495
2,415
0.733
3,746
1.022
4,399
1.462
4,794
2.255
4,041
Q
%
55-59
13,960
0.066
5
0.065
98
0.091
401
0.179
858
0.315
1,399
0.466
2,062
0.650
2.932
0.930
3,700
1.434
3,415
o
60-64
9,660
0.046
6
0.045
68
0.063
306
0.123
642
0.218
980 .
0.322
1,715
0.450
1,658
0.644
2,243
0.992
2,668
65-69
7,108
0.034
6
0.033
31
0.046
198
0.091
379
0.160
584
0.237
876
0.331
1,141
0.474
1,514
0.730
1,992
O
a
70-74
5,650
0.027
6
0.026
28
0.037
137
0.072
221
0.127
349
0.189
507
0.263
734
0.376
944
0.580
1,172
<
75-79
3,430
0.016
7
0.016
27
0.022
67
0.044
127
0.077
181
0.115
247
0.160
321
0.229
399
0.352
536
80-84
1,362
0.006
9
0.006
45
0.009
123
0.017
238
0.031
247
0.046
335
0.063
519
0.091
635
0.140
703
85-89
480
0.002
0.002
26
o:oo3
51
0.006
75
0.011
124
0.016
195
0.022
382
0.032
697
0.049
565
90-94
105
.0005
.0005
.0007
.0013
.0024
.0035
115
.0049
173
.0070
229
.0108
339
See Note to the continuafcion of the Table on the next page.
222
APPENDIX A.
TABLE LXVn.— Shewing the Masculinity (M/F) of the Unmarried in 5-year
Age-groups, and the Probability-function, 1,000,000 ^qxy, for calcnlating the
number of Marriages in the S-year Age-groups indicated. {Continued )
Age of Bribes and NtrMBEB of Unmabbibd Females.
Age
Group.
55-59
5,698
60-64
3,645
65-69
2,505
70-74
1,405
75-79
695
80-84
347
85-89
110
90-94
23
15-19
229382
40.26
6
62.93
91.57
163.3
330.1
661.0
2085
9973
20-24
201906
35.43
18
55.39
10
80.60
5
143.7
290.5
581.9
1835
8778
25-29
120243
21.10
32
32.99
31
48.00
23
85.58
173.0
346.5
.1093
5228
GQ
30-34
67,650
11.87
72
18.56
59
27.01
43
48.15
97.34
194.9
615.0
2941
\
35-39
45,616
8.006
260
12.51
113
18.21
64
32.47
8
65.63
131.5
414.7
1983
40-44
36,868
6.470
442
10.11
249
14.72
126
26.24
16
53.05
106.2
335.2
1603
it
O
45-49
29,858
5.240
1,236
8.192
651
11.92
328
21.25
77
42.96
16
86.05
271.4
1298
n
'A
50-54
21,945
3.851
1,934
6.021
1,321
8.760
636
15.62
150
31.58
145
63,24
34
199.5
954.1
a
•a
<
00
S
o
i
65-59
13,960
2.450
3,030
3.830
1,680
5.573
1,147
9.936
338
20.09
290
40.23
164
126.9
607.0
60-64
9,660
1.695
2,835
2.650
2,769
3.856
2,257
6.875
1,039
13.90
405
27.84
319
87.82
107
420.0
w
65-69
7,108
1.248
2,572
1.950
2,585
2.838
2,081
5.059
1,287
10.23
582
20.48
467
64.62
212
309.0
O
O
70-74
5,650
0.992
1.425
1.550
1.437
2.256
1.315
4.021
1.476
8.129
896
16.28
610
51.36
314
245.6
75-79
3,430
0.602
764
0.941
979
1.369
957
2.441
970
4.935
1,094
9.885
958
31.18
406
149.1
465
80-84
1,362
0.239
610
0.374
475
0.544
295
0.969
207
1.960
171
3.925
163
12.38
281
59.22
85-89
480
0.084
341
0.132
205
0.192
99
0.342
59
0.691
45
1.383
31
4.364
20.87
90-94
105
0.184
221
0.288
161
.0419
.104
.0747
.1511
.3026
.9545
4.565
Note. — The upper figures denote the masculinity of the group, that is, the ratio
of the number of all the unmarried males of the 6-year groups of ages (15 to 19) to
( 90 to 94), to the number of all the unmarried females of the 5-year groups (13 to 14)
to (90 to 94). The lower figures are the values of the probabiUty-functiou,
1,000,000 ^qxy, for the double 5-year groups, by means of which the number of
marriages occurring annually in the indicated age-groups may be calculated by
formulae (431) and (432), (434), and (435).
NUPTIALITY. 223
3 1 . The probability of marriage according to pairs of ages. — Assuming
that the " conjugal potential" does not change in any community, the
number of marriages likely to occur among groups of the unmarried of
given ages can be computed by means of formula (434), at least if the
masculinity is at all similar to that shewn in- Tables LXVI. and LXVII.
These tables give also the values of q^y as well as the masculinity.
If the conjugal potentials are the same for A males and B females as
for B males and A females, and the law of variation is, as by hypothesis,
(437). . .'. . .(y +y') °c M'"' . Ff"' = M<f>^- = Ffif'
then the quahfication as to the masculinity being approximately identical
disappears.'^ It is not unimportant, however, to remember that the
fundamental assumption would have to be very erroneous (and that
would seem to be impossible) in order to seriously prejudice the precision
of the result obtained by the application of the formula (434). The error
in any real appHcation of the formula can be a differential one only,
and if the constitution as regards numbers of the population be approxi-
mately therefore that from which it was derived, any defect in the theory
of variation with relative numbers of the sexes, formula (430), has no
sensible effect.
32. The relative numbers of married persons in age-groups. — ^The
Census of 1911 disclosed the fact that the number of married persons
living together on the night of the 3rd April, 1911, was 623,720. The
number of wives absent from their husbands was 112,129, and husbands
absent from their wives 110,053. There were 616,738^ (out of a total of
about 734,000 married couples) whose ages were fully specified, and who
-were living together.
This may not be a perfect sample of the entire population, for although
the date of the Census, viz., 3rd April, is well chosen, the number of
spouses of each age apart at a given moment is probably not sensibly
proportional to the total number. As the totals, however, are only
about one-fifth greater than the number for wiiich the information is
complete, the 616,738 may be taken as fairly representing the popula-
tion. The results are shewn upon Table LXVIII.
1 fi^ and 01 are the same as /i and above ; f,..^ and <f>^ are defined in Table XXI.
p. 132 hereinbefore.
' This number is made up as follows : —
Husbands and wives com-
pletely specified as to age,
and living together . . 616,738 Living to- Living to-
gether but gether but Wives Total
Wife's Age Husband's Absent. Husbands.
Both ages unspecified . . 506 not Age not
stated. stated.
617,244 + 4,108 + 2,368 -|- 112,129 = 735,849
Living together but wife's
age not stated . . . . 4,108 ||
Living together but hus-
age not stated . . . . 2,368 1.19313
Husbands absent . . . . 110,053
X
Total wives .. = 733,773 = 1.18976 x 616,738
224
APPENDIX A.
TABLE LXVm. — Number of Married Persons per 1,000,000 Married Couples, Living Together on
the Night of the Census, 3rd April, 1911. In 5-year Age-groups.
Wives' Ages.
Total,
Hus-
I
1 1
10
bands' X
Ages, t
0! 15
20
25
30
35
40
45
50
56
60
65
70
75
80
8590 95
to
o to
to
to
to
to
to
to
to
to
to
to
to
to
to
to to to
99
1
4 19
24
29
34
39
44
49
54
59
64
69
74
69
84
89 94 99
15-19 .
. 577
347
39
8
3
974
20-24
8 6,771
24,015
7,168
1,090
217
"63
"28
" 6
38,366
25-29
2 3,574
40,354
54,338
11,871
2,015
383
112
44
"11
2
1
112,707
30-34 .
. 1,090
17,907
54,009
54,757
12,145
2,264
516
123
29
11
5
2
142,858
35-39 .
376
5,845
24,489
61,157
47,891
10,786
1,966
379
89
16
11
3
' '2
■]
143,009
40-44 .
130
2,048
9,082
25,695
47,680
44,462
9,936
1,934
452
92
36
10
3
141,660
45-49 .
44
760
3,287
9,610
28,654
43,595
40,083
8,644
1,450
340
96
16
10
131,489
50-54 .
24
258
1,090
3,124
7,694
19,245
35,589
29,716
5,800
1,138
311
50
13
"3
104,056
55-59 .
11
94
334
921
2,380
5,567
13,677
22,851
16,769
3,478
666
154
41
11
66,954
60-64 .
5
45
135
357
798
1,899
4,506
9,790
13,578
10,622
2,330
478
81
18
i'i'.'.
44,645
65-69 .
23
62
156
413
830
1,840
4,081
6,684
9,571
7,639
1,629
292
42
8.. ..
33,270
70-74 .
8
26
58
180
319
718
1,505
2,616
4,405
6,040
4,533
1,004
118
16 6 2
21,552
75-79 .
'. "2
5
23
29
57
131
268
517
820
1,600
2,996
3,322
2,238
399
37 8 ..
12,452
80-84 .
2
2
3
16
24
42
79
152
227
472
751
1,166
1,111
655
84 26 ..
4,801
85-89 .
2
2
6
10
16
28
53
34
148
198
267
183
91 6 ..
1,094
90-94 .
2
2
6
1
3
B
34
37
31
36
18 15 . .
183
95-99 .
2
2
5
5
5
6
3.. ..
28
100-104 .
••
2
1
3
T.otals
1
15-104 1
11,606
91,713
154,087
158,750
145,157
129,598
109,339
79,771
48,584
31,841
21,070
11,593
5,098
1,471
258 52 2 1,000,000
33. Conjugal age-relationships. — For certain estimations it is
important to know, for given ages of husbands, the average difference of
the age of the wives ; and also for given ages of wives the average differ-
ences of the ages of the husbands. These relationships as at marriage,
i.e., initially, may be ascertained from marriage records. They may be
called the protogamic age-relationships. The instantaneous relationships
at any moment, however, are disclosed only by a Census, and may be
called the gamic age-relMionships.
The age-groups, wj^h the age of the husband as argument, and those
with age of wife as argument, lead, it will be found, to different results,
which have no obvious direct mutual relation. Hence this, in common
with other analogous groupings of a non-homogeneous character, must be
independently made, for a reason which we shall now more definitively
indicate. In cases of the kind under consideration two formulae are
needed ; in one the argument is the age of the husband (or bridegroom),
in the other the age of the wife (or bride).
34. Non-homogeneous groupings of data. —If , associated with any
group-range, viz., x^j. to x^ + i say, of any class of elements (ages of hus-
bands in the case under review), there is a class of related elements (ages
of wives), viz., «/fc_a to 2/4+6 say, where a and b, in general, have large
values ; and if, reciprocally, a group-range, 2/4 to yi^^i say, is associated
NUPTIALITY. 225
with the group x^_^ to x^.^^ say, A and B also having large values, the
result obtained from the former will have no simple relation with that
based on the latter. For a result based on the argument x, has not the
same constitution as one based on the argument y. If the distribution
about the mode in such cases be not symmetrical in each, in fact if it be
not similar in all respects, no direct functional relationship subsists between
results for groupings arranged according to the values of x, and those for
groupings arranged according to the values of y. Groupings subject to
this limitation may be called non-homogeneous groupings, and require
special consideration.
3d. Average differences in age of husbands and wives, according to
Census. — In Chapter XIX., Vol. I., § 2, of the Report on the
Australian Census of 1911, results are given for a series of age-groups of
husbands and of wives. The results are also given in greater detail in
Vol. III., Table I., pp. 1106-7. The difference for the central-age of the
group, which is sensibly, though not exactly, the mean-age, of those
included therein, is as shewn on Fig. 64,^ the curve marked A, representing
the excess of the age of husband over the average age of their wives, as
determined from groupings according to the age of the husbands, and
the curve marked B, representing the excess of the age of the -wiie over
the average age of their husbands, as determined from groupings accord-
ing to the age of the wives.
The differences are given in Table LXIX. hereunder. The tangent
line to curve A is coincident with the curve for the ages 40 to 60 inclusive
(beginning point of year) ; hence for this interval the relation is —
(438) D„ = -I- 0.098 a;^, for ages 40 to 60,
D„ denoting the average excess in years of the age of the husband over
the average age of the wives, and X)^ being the age of the husband.
The tangent is coincident with curve B for the ages 30 to 67 inclusive,
and the age of the wife is greater than the average age of the husbands by
the amount Df^, where
(439) !>/,== - 6.275 + 0.058 x„, for ages 30 to 67,
in which x^ denotes the age of the wife. It is obvious from the table that
the assumption ordinarily made is invalid. The characteristics of a table
of values of the differences will be evident from the table itself.
See pa^ 227
226
APPENDIX A.
TABLE LXIX. — ^Differences of the average Age of Wives for Husbands of various
Ages, and of the Average Ages of Husbands for Wives of various
Australia, 1911.
Age
of
Calculated Result, Curve A.
Calculated Result, Curve B.
Hus-
band
Position
Ordin-
Smooth-
Crude
Position
Ordin-
Smooth-
Crude
A;
of
ate to
ed value
value
of
ate to
ed value
value
Wife
Tangent
Curve.
of Dw
from
Tangent
Curve.
of Dh.
from
B.
Data.
Data.
14i
+ 1.42
—5.43
—8.4
15J
1.52
—6.52
— 5.00
—5.0
5..38
—.5.02
—lOAO
10.4
m
1.62
5.27
3.65
0.9
5..32
3.07
8.39
9..1
I'i
1.72
4.52
2.80
2.8
.5.26
2.25
7.51
7.5
m
1.81
3.78
1.97
1.2
5.20
1.76
6.96
7.2
m
1.91
3.48
1.57
1.1
.5.14
1.44
6.58
6.6
20J
2.01
3.12
1.11
0.6
5.09
1.16
6.25
6.2
23
2.25
2.38
—0.13
—0.4
4.84
.80
5.70
5.7
27i
2.70
1.33
+ 1.37
+ 1.2
4.68
—.17
4.85
4.7
30 t
2.95
.90
2.05
4.54
.0
4.54
32J
3.19
.56
2.63
2.5
4.39
.0
4.39
'4.4
37i
3.67
.10
3.57
3.6
4.10
.0
4.10
4.1
40 *
3.92
.0
3.92
••
3.96
.0
3.96
m
4.16
.0
4.16
4.2
3.81
.0
3.81
3.8
474
4.66
.0
4.66
4.7
3.52
.0
3.52
3.4
52i
5.15
.0
5.15
5.2
3.23
.0
3.23
3.1
57i
5.64
.0
5.64
5.8
2.94
.0
2.94
3.0
60 *
.5.88
.0
5.88
2.80
.0
2.80
62^
6.13
+ .08
6.21
6.5
2.65
.0
2.65
2.9
67 t
6.55
.19
6.74
2.50
.0
2.50
67J
6.61
.20
6.81
7.3
2.36
+ .08
2.28
2.3
72i
7.11
0.66
7.73
8.1
2.07
.70
—1.37
—1.3
77i
7.60
1.58
9.18
9.2
1.V8
1.96
+0.18
+ 0.4
82^
8.09
3.14
11.23
11.3
1.49
3.76
2.27
2.2
87J
8.58
5.70
14.28
14.4
1.20
6.70
5.50
4.2
92J
9.07
9.10
18.17
18.6
0.91
12.01
11.10
11.1
97i
9.56
14.90
24.46
22.3
0.62
+ 25.62
+ 25.00
+ 25.0
102i
10.05
29.95
40.00
40.0
—0.33
*f The asterisks and daggers denote the ages between which curves A and B,
respectively, are straight lines.
In the figure the curves A and B are very approximately the smoothed
values. The tangents are shewn by dotted lines ; the data by the dots ;
it is instantly evident that the difference is not constant, but is a definite
function of age. A and B are the curves of the gamic age-relationship.
36. Average differences of age at marriage. — A similar table to the
preceding can be constructed for the ages at marriage . In order to eUmin-
ate the uncertainties due to paucity of data the results for the eight years
1907 to 1914 were combined. The combinations shewed the same tend-
ency as was revealed by the Census, vIts., for the numbers to be unduly large
for the ages ehdii^ with the digits and 6. The niimbers for the purpose
of the following table have, however, not been smoothed ; the smoothing
in the table itself making that .uivaeeegsary.
NUPTIALITY.
227
Differences between Ages of Husbands of any Age and the Average Ages of their
Wives, and between the Ages of Wives and the Average Ages of their Husbands.
Curve C + 20
Curve C + 10
Zero of Curve C
Curve A + 10
Zero of Curve A
o t.^
Curve A - 10
Curve A -20
Ages
/
/
/'
li
^
/
_-■
--
■<'-
^
ii-
-^
i
■y^
^'-'
^
</
''••■"'3
■^
^
y
"^
/
El •
J--
.^
— ^
ij-
.-T-r
—
/
/••■"
/■
^
/ X,'
1
—
,_—
^=
^
r:^
—
—
/^
^■,-
—
—
TT
^
^
—
-■-x>;
'^■r'
— '
= =
^^
—
—
+ 10 Curve U
ZeroofCurveD
TM
T'h
-10 Curves
s|°
Zero of Curves
T'l.
-10 Curve B.
10
20
30
40
50
60 70
80
9D
100 Age6
Fig. 64.
Curve A. — ^Excess of the husband's age over the average age of their wives,
at the 1911 Census. See Table LXIX., p. 226. Oa is the zero for the curve.
Curve B. — ^Excess of the vpife's age over the average age of their husbands, at
the 1911 Census. See Table LXIX., p. 226. Ob is the zero for the curve.
Curve C. — ^Excess of the bridegroom's age over the average age of their brides,
1907-1914. See Table LXX., p. 228.
Curve D. — ^Excess of the bride's age over the average age of their bridegroomst
See Table LXX., p. 228.
The results are shewn by curves C and D in Fig 64. The tangent to
curve C, which is analogous to curve A, is identical with the results for
ages 42^ to 67| years ; thus : —
(440).
.D\
1.745 + 0.266 x^ ; for ages 42^ to 67|.
For curve D, the difference of ages is analogous to curve B. The tangent
is parallel to the age -axis at the distance
(441).
.D' = - 1.76 ; for ages 32^ to 60.
The table shews the differences outside these hmits.
Towards, the ends of the curves the results for all four ciirves are of
course somewhat uncertain. C and D are the curves of the protoganiic
Skge-relationship.
228
APPENDIX A.
TABLE LXX. — ^Difference of the Average Age of Brides for Bridegrooms of various
Ages, and of the Average Age of Bridegrooms for Brides of various Ages.
Age
of
Calculated Result, Curve C.
Calculated Result, C'lu^e D.
Bride-
groom
Position
Ordin-
Smooth-
Crude
! Position
Ordin-
Smooth-
Crude
C;
of
ate to
ed value
value
1 of
ate to
ed value
value
Bride
Tangent
Curve.
of X)'„.
from
Tangent
Curve.
of D\.
from
D.
Du,.
Data D),
13i
__
-1.76
-11.04
. 12'.80
12.80
144
1.76
8.45
10.21
1 10.21
15J
+ 2.38
— 5.35
—2.97
—5.50
1.76
7.10
8.86
•i 9.18
m
2.64
5.08
2.44
2.36
1.76
6.10
7.86
7.86
17J
2.91
4.85
1.94
1.08
1.76
.5.24
7.00
6.95
18i
3.18
4.56
1.38
0.81
1.76
4.50
6.26
6.25
19J
3.44
4.35
0.91
0.37
1.76
3.92
5.68
5.66
20i
3.71
4.08
— fj.37
—0.18
1.76
3.42
5.18
.5.26
23
4.37
3.49
-0.88
-0.49
).7li
2.24
4.00
3.94
27i
o.ol
2..52
3.05
2.72
1.7(j
.70
2.46
2.46
32it
6.90
1.48
5.42
5.35
1.76
.00
1.76
1.76
37i
8.23
.56
7.67
7.67
1.76
.00
1.76
1.72
42^*
9.56
.00
9.56
9.45
1.76
.00
1.76
1.91
47i
10.89
.00
10.89
10.95
1.76
.00
1.76
' 1.66
.52i*
12.22
.00
12.22
12.30
1.76
.00
1.76
1.75
57i
13.55
.00
13.55
13.42
1.76
. .00
1.76
1.31
60 t
1.76
.00
1.76
1
62i
14.88
.00
14.88
15.03
1.76
.06
1.82
..30
67i
16.21
.00
16.21
16.16
;, 1.76
.28
2.04
2.08
72J
17.54
.90
18.44
19.52
: 1.76
.73
2.49
1.31
77i
18.87
2.30
21.17
19.93
1.76
1.54
3.30
5.83
82^
20.20
4.50
24.70
37.05
1.76
3.00
4.76
7.14
871
21.53
8.09
29.62
29.62
1.76
.5.. 30
7.06
97i
22.86
1.76
10.00
*f The asterisks and daggers denote the ^es between which the curves C and D ,
respectively, are straight lines.
37. The gamic surface. — ^The data furnished in Table LXVni. may
be used to construct the gamic surface, on the same principle as was
followed in the construction of the protogamic surface, dealt with in
§ 25 hereinbefore. The results are shewn on Fig. 65, from which it will be
seen that the isogams are more elliptical in form than isoprotogams, and
are more regular; see Fig. 61. The principal meridians AB, AC and
AD, AE are in much the same positions as on the protogamic surface, but
the point of maximum frequency A, and the line of greatest slope are
for higher ages than on that surface. The interpretation of the curves is,
mutatis mutandis, the same as that for the isoprotogams ; in the case of
Fig. 65, however, everything applies to persons " living in the state of
marriage," instead of to " persons at the moment of marrying. "'
NUPTIALITY.
229
Curves oJ Equal Conjugal Frequency.— The Gamic Surface, 1911.
10
Ages of Wives.
30 40 50 60 70 80 90 100
Fig. 65.
Note. — The pairs of ages for which an equal frequency of married couples
existed at the Census of 1911 are found by following the course of any isogam. The
remarks in the footnote to Fig. 01, p. 209, apply, mutatis mutmidis, to the contours
of the Gamic Surface.
38. Smoothing of surfaces. — ^Let it be supposed that the nature of
statistical data is such that the most suitable representation is by means
of the heights of series of parallelepipeds, as for example, in the case just
considered, of the numbers of marriages of bridegrooms between given
age limits and of brides between the same or other given age limits. For
simplicity we may assume that the combination is according to age last
birthday, and thus is in single year groups. Since the general equation
of a surface of a second degree will involve nine constants,we can deduce
the constants of a surface representing its integral between the limits
a; = 0, 1, 2, and 3, and y = 0, 1, 2, and 3, the deduced expression will
give totals corresponding to those of the nine contiguous groups. By means
of the corresponding surface equations, deduced from these, for lines
parallel to the a;-axis, or parallel to the 2/-axis, we can find the hei^t to
230 APPENDIX A.
this surface, along the four edges of the central parallelepiped. If this
operation be then repeated, making each of the four adjoining parallele-
pipeds the central ones in a group, we shall obtain a second series of values
for the distances along the four edges to the surface ; if these do not
differ very greatly then the means of each pair of values may be taken,
in general, as the smoothed result. In this way the greater part of the
entire surface can be dealt with, and the series of verticals to the surface
thus found will have reduced the original irregularities, and may be
regarded as a first smoothing of the surface, conforming, however, a.s
nearly as pqpsible to the general series of group-heights. The results so
obtained, however, are " instantaneous values," that is, they are the
heights corresponding to the ranges .»; to r + dx, and «/ to y + di/.
U the numbers be very irregular the process above indicated is
extremely tedious, and of little value. It may then be preferable to
regard the group results as vertical ordinates with the central values of the
group-ranges as the horizontal co-ordinates. The procedure then in-
volves the independent smoothing of a double system of curves, and the
taking throughout of the means of the pairs of verticals so found. The
whole procedure is then repeated, with the means thus obtained, until the
smoothing is satisfactory. The criterion of good smoothing is that the
" accumulated deviations" in either of the two directions (at right angles
to one another) do not attain to appreciable values, and that they
alternate in size. It should be noted that smoothing in this way does
jiot give " instantaneous values," that is where k is the extent of the range,
the heights now denote values true for the ranges | fc on either side of
the values x and y, these being the ordinat€s of the centre of the ranges .
There is another possible scheme of solution, viz., to ascertain the
constants of an equation, which will give at once the group values for
groups of the same double-range, the arguments being the ordinates of
the centres of the groups. The method is analogous to that treated for
a surface in Part V., § 10, formula (211) to (216), pp. 72-73, and the
solution by a process analogous to that indicated in the section
immediately following, will give the group-height for any value of x
and y, the range being a; ± i ^, y ± ^k.
39. Solution for the constants of a surface representing nine
contiguous groups. — ^The most general expression for a surface, every
section of which parallel to the .r-axis and parallel to the y-axis is a
curve of the second degree is
(442). .z=A+Bx+Cy+Dxy +Ex^+FxY + Gy^ +Hx^y + Ixy*
NUPTIALITY.
231
Let the values of the groups be denoted by the letters I, m.
according to the following scheme : —
y = 3
.0 = 2
* = 3
The integral of the above, divided by xy, the area of the base, is ;
(443). . . . ^fJF (X, y, k) dxdy =4+ ^Bx +Y^y +X^^2/
+ jEx^+jFy^ + j Gx^y + 1 Hxy^ + j Ix^y^
from which we deduce, by putting x {or y) successively 1, 2, 3, and making
y (or x) equal 1, 2, or 3, the following values of the constants J to Z in
terms of Z, m, t. The results are : —
(444). ... A = - (q+ 0-2^3)+ {n+l - 2m)+ (I - p) - 3 (s - p)
+ (< - ?) + 3 (r - 0) + (p - o) - -2- (m - Z)
(445)....5 = 3(g+o - 2p)- 3(»i + I - 2m) + 8(s -p)-3(t - q)
- 5 (r — o) - 2 (jj - o) + (m - Z)
(446). . . . C = 2 (g+o-2^)+2;)-2(p-o)+9(s-p)-9(r-o)-3(«-9)
(447). . . .D = 4 (2j-o)-6(^+o— 2p)-24 (s-p)+15(r-o)+9(t-g)
(448) . . . . .0 = - (w + i - 2m)
{q^o-2p)-3[s -p)
+ I (« - 9) + I ('• - o)
■232 APPENDIX A.
(449). . ■ . -f = I (r - O) _ |- {« _ p) + A (< _ 9)
(450).... G!= - ^(t-q)- ^ (r-o)+9(s-p)+^{q+o-2p)
(451). ...H=U(8 -p) - ^-^(r -o)- ~(l -q)
(452). . . . I = I (t - g) + -^ {r - 0) --^ (s - p)
It will be .seen that the arithmetical labour of deducing the constants
of a surface which will exactly reproduce any square system of 9 contiguous
group-values, is very great, and ordinarilj- prohibitively so. In general,
therefore, less rigorous methods have to be adopted, and are ordinarily
quite satisfactory, particularly in view of the fact that in practical
calculations values according to a given double-range are required.
40. Naptiality and conjngality norms. — It would appear desirable to
establish decennially, what may perhaps be called a nuptktUty or pro-
togamic norm, and also a conjugality or gamic norm, on the basis of an aggre-
gation of the marriages of a large number of populations for the former ;
and of the Census results for the latter. The norms should preferablj'
shew single-year re.sults up to 24 years for brides, and 29 years for bride-
grooms ; and up to 34 years for wives, and 39 years for husbands,
respectively.
The protogamic norm will reflect the trend in regard to the early
institution of marriage, and the gamic norm the modification of this by
change in longevity, the frequency of divorce, etc. These norms could
include the curves of the totals according to the- age of the males (bride-
grooms and husbands), and according to the age of the females (brides
and wives), and could include also the frequency of the group-pairs.
The norms of the conjugal state, '■ never married,'" " divorced,'' and
" widowed," might, with advantage — as well as those of the " married" —
also give the frequencies according to group-pairs.
41 . The marriage-ratios of the unmarried. — ^It has already been shewn
that the probability of marriage depends, among other things, upon the
relative numbers among the unmarried of the sexes. So long, however,
as a population does not greatly change its constitution according to sex
and age, the crude probability of marriage according to sex and age may be
regarded as varying approximately as the aimual rate. This probability
maybe called the peithogamic coefficient^ for the sex and age in question.
It will be further discussed in Part XIII. in connection with fecundity.
1 From vcWu to prevail upon, (flfiffii the Goddess of Persuasion) and yafUKoi,
of or for marriage.
Xra.— FERTILITY AND FECUNDITY AND REPRODUCTIVE
EFFICIENCY.
1 . General. — ^The phenomena which directly concern the measure of
the reproductive power of the human race will be dealt with in this part.
These phenomena are in general complex, the variation of the repro-
ductive power being in part of physiological origin, and in part of the
result of the reaction of social traditions upon human conduct. This
will appear in any attempt to determine the laws of what has been called
bigenous^ (better, digenous) natality, or natality as affected by the ages
of both parents, as distinguished from those affecting merely monogenous
natality, or natality as related to the producing sex. In deducing the
most probable value for certain of the phenomena it will be necessary to
minimise the effect of misstatement of age. This can probably be done
more effectually than would at first sight appear probable. The final
results, however, must be subject to some small degree of uncertainty.
The question of the reproductive efficiency of a population has in
part been dealt with in Parts XI. and XII., dealing with Natality and
Nuptiality ; this, however, is derivative and depends in its turn upon the
age -distribution and conjugal condition of the producing sex.
Many questions concerning the measurement of fertility and fecundity
can be settled with sufficient precision without recourse to a differentia-
tion depending on the age of the father, the better in Australia, perhaps,
inasmuch as the decay of virility with age is not well marked, and in this
aspect the digenous fertility stands in marked contrast with that of
Hungary.
2. Definitions. — ^It is desirable , initially, to define the sense in which
several terms will be used hereinafter.
Monogenous fertility and monogenous fecundity will denote the
fertility and fecundity of the female considered without regard to the age
of the associated male.
Digenous fertility and digenous fecundity will denote the fertility and
fecundity of the female, as modified by the age of the associated male,
and therefore is considered in relation to the ages of both males and females .
Consequently computations of monogenous fertility or fecundity will be
based upon the age of the female. It foUows from this, that two popula-
tions will be (i.) exactly, or (ii.) approximately, comparable, only when
the conjugal age-relationships are (i.) sensibly identical, or (ii.) are
similar.
1 By Joseph Korasi, see Phil. Trans. Lond. B., 1895, p. 781.
-34 APPENDIX A.
Isogeny will deuote either equal fertility or equal fecundity, the former
to be called initial isogeny or isoprotogeny ; the latter general isogeny, or
characteristic isogeny, or simply isogeny.
A curve, passing through a series of pairs of ages plotted as co-
ordinates, in such a manner that it will pass through all ages which give
either equal initial or equal general fertility or fecundity, will be called
an isogen as appUed to either. The curves may therefore, Ln the cases
considered, be called isoprotogens , and isogens.
The terms '' fertiUty" and " fecundity"'^ though ordinarily sensibly
identical in meaning, have sometimes been assigned different meanings
by statisticians, one being employed to signify the qualitative, and the
other the quantitative, aspect of reproductivity.
Owing to their phonic resemblance the words " sterility" and
" fertility" are the more appropriate to employ in order to denote the
difference between producing or non-producing; while "fecundity,"
which biologically is used without quaUficative to imply producing in
great numbers (a meaning which requires the qualification "great"
when fertility is used), is obviously the more appropriate word to denote
" multiple fertihty."!
1 In Latin, although " fertiUtas" and " feovmditas" have no marked difference
of meaning, the latter word seems to be the preferable one for denoting frequency of
bearing offspring. The root of fecundus is " feo" (obsolete), or PE = Greek (pu ;
e.f. Sanskrit bhu ; Zend bu ; see 0i)u Liddell and Scott's Greek-English Lexicon,
8 Edit., p. 1703.
The root of "fertUis" is "fero"=Greek root 0e/) : e.f., Sanskrit "bhar"; Zend
"bar"; A.S., "bear-n"; the radical meaning being to bear or carry. See LiddeU
and Scott op. cit., p. 1662.
In regard to "sterilitas, " o.f., Sanskrit "stari" (vacca sterilis).
In other languages the following correspondence might be suggested : —
Enghsh. French. Italian. German. Danish. Swedish.
Fertility ; FertiUty ; Fertility ; Fruehtbarkeit or Frugtborhed Frukteamhet;
Gebarfahigkeit
Fecundity. F^eondite. FeconditiL Ergiebigkeit or Avledygtighed Afvelsamhet.
Fruehtbarkeit
Inasmuch ' Fruehtbarkeit," " Frugtbarhed" and " fruktsamhet" ought, if possible,
to be appropriated to the one meaning, the first suggestion as regards the German
is to be preferred. That is, it is better to adopt " Fruehtbarkeit" for fertility and
•' Ergiebigkeit " for fecimdity.
KOrOsi suggests "Ergiebigkeit der Ehen." " iluttersohaftsfrequenz " and
" Maternitatsfrequenz " refer only to cases of maternity.
J. Matthews Duncan, in his " Fecundity, fertility, sterility and aUied topics,"'
1866, 2nd Edit., 1871, has used " fecundity" to imply the qu&lity of producing
•' without any superadded notion of quantity," and " fertility or productiveness"
" the amount of births as distinguished from the capability to bear." For the reasons
indicated in the text, it is better to adopt the terms " sterile " and " fertile" as
contrasted, that is, as meaning " non-productive" and " productive" without
reference to quantity, and the term ' fecund" as conveying the idea of quantity.
The matter seems of sufficient importance to abandon Duncan's usage.
FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 2:ir
Physiological or potential fecundity is, at. present, not ascertainable :
what is discoverable is only actual fecundity. Both rise to a maximum
and fall away, the latter .very early in life, while it is improbable that this
is true of the former. The difference is theoretically (and of course
practically) important. , The following definitions make the matter
clear : —
(i.) Physiological fecundity at a given age is the probability that a
female of that age, subject to a definite degree of physiological risk,
uniform for all ages, will reproduce.
(ii.) Actual fecundity at a given age is the probability that a female
of that age, subject to average actual risk (as modified by social traditions,
etc., and also by reproduction itself, and not necessarily uniform for all
ages), will reproduce.
Inasmuch as physiological fecundity is probably not identical in
populations of different races or nations, or even in populations of differ-
ent localities and times, and is, moreover, dependent upon general health
and mode and standard of living, the obtaining of its measure is in a
high degree important, though at present impracticable.
Actual fecundity is, naturally enough, different for married and un-
married females. While it does not, even with married females, measure
without correction the urgency of the reproductive impulse, or in un-
married females measure the force which this impulse opposes to restric-
tions created by social environment, it throws, as we shall later see,
important light on this question.
3. The measurement of reproductive efficiency. — ^The determination
of an unequivocal method of measuring the reproductive efficiency of a
population is not without difficulty for the following reasons, viz., that —
(a) The life of women varies in duration ;
(6) The reproductive period is only a limited portion of it ;
(c) FertiUty and fecundity are neither uniform for all ages, nor for
all women ;
(d) It appears to be qualified by the age of the associated males ;
(e) Marriage and child-bearing initiate at different ages ;
(/) Reproductive efficiency must take account of the duration of
life of the children ; and that
(g) The exercise of the reproductive function is subject to ad-
ventitious influences.
By way of enforcing the penultimate point, it may be noticed that gener-
ally a high birth-rate is associated with a high rate of infantile mortality,
and the rate measured by taking account only of survivors at the end of
one year or other prescribed period may give quite a different indication
to that derived from births only. The following outline of various
schemes of measurement, some of which have already been dealt with,
will indicate the nature and limitations of each.
236
APPENDIX A.
Rate JtEASCRF.n by-
Numerator.
Denominator.
Deduced
Result
known as-
Keniarks.
Total births, B \ Total popula-
ation. P
; Crude btcth-
I rate, B/P
Is dependent on age, sex, and conjugal
constitution of total population, and there-
fore not strictly comparable as between
different populations ; it measures merely
one element determining increase.
Total births, B
Total female
population, F
Birth-rate re-
ferred to total
number of
women, B/ F
la dependent on female population onlj'
and is affected of course by the age and
conjugal condition of that population.
Tot-al liirths, B \ Female popula- j Birth-rate re- Indicates reproductive efficiency of all
tion of repro- ferred to "women within the reproductive period,
ductive age women of re- Owing, however, to the Umits of this period
i (viz., from productive being iU-deflned at the initial and terminal
about 10 to age only ages, to the largeness of the number of
60), F', say BJ F' women at those ages, and to the fact that
itis dependent on the age-constitution with-
! in the group chosen to represent the repro-
ductive age, the rate is not as definite as is
I ' desirable. The denominator, however. Is
a good crude measure of the potential of
I I reproductiveefficiencyof the population.
Births in each I
age-group, B^
The women in
same groups,
F,
Birth-rate re-
ferred to
women of
each age-
group In
question,
Is uncertain for comparison because the
ratio of married to umarried women may
vary, and the relative frequency of mater-
nity in each is not identical.
Nuptial births iu
each age-
group, B\
JIarried women
in same age-
group, M^
Nuptial mater-
nity rate for
each age-
group, B'^/M^
Shews only the average frequency of
maternity (average probability of mater-
nity) for married women in each age-group.
E>:-nuptial births
in each age-
group of un-
married
women, B",.
Unmarried
women in
age-group.
Ex-nuptial
maternity
rate for each
age -group.
Shews only average frequency of mater-
nity (average probability of maternity) for
unmarried women in each age-group.
Appropriately
weighted sum
of birth-rates
of the married
and un-
married
Unity
llodifled
*' Nuptial
Index of
Natality"
This attributes tlie reproductive facts of
an existing population to a supposititious
" standard" population, in which the re-
lative number of married and unmarried
females is the general average (norm) for
the groups of populations to be compared.
The comparison so attained may be re-
garded a suitable comparative measure of
reproductive efficiency (natality).
4. Natality tables. — The preceding methods of measuring productive
efficiency are all more or less defective. A more satisfactory scheme is
to construct a monogenous age-group " natality table" for married, and
one for unmarried, females . Such tables shew for each age the probability
of the occurrence of a birth and the average number of children per con-
finement : see hereinafter. This, without doubt, is a more definite
method, and stands in much the same relation to statistics of births, as a
mortahty table does in relation to statistics of deaths. It is, however,
not perfectly satisfactory, because, as already indicated, it would appear
that the age of the father as well as that of the mother affects the probabil-
ity of maternity. This will be dealt with hereinafter. Tables of digenous
natality, i.e., double-entry tables, shewing the natality for every com-
bination of age, are more complete and exact, and would be perfectly so,
if the fertility at any age were unaffected by the number of previous con-
finements. This, however, is probably not the case. These matters will
be dealt with in the various sections and tables hereinafter in this part.
FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 237
5. Norm of population for estimating reproductive efficiency and the
genetic index. — ^In order to eliminate the effect of variations in the con-
stitution of populations, it is desirable to establish on as wide a basis as
possible the norm of its female conjugal constitution, preferably for every
5 years of the reproductive period. This norm would shew for a total of
1,000, 10,000 or 100,000, etc., women of all reproductive ages, the number
aged 10-14, 15-19, 55-59 ; that is from the 10th to the
59th year of age inclusive.* For each age -group there would be (at
least) two classes, viz., the "unmarried" which might include widows and
divorcees not remarried, and the " married." If, then, to these numbers
in the age-groups of the " married" we attribute the nuptial birth-rates*
and compute the births, and to the " unmarried" we similarly attribute
the ex-nuptial birth-ratesf, which are actually experienced by any popula-
tion considered, we shall have comparable measures ; and the aggregate
(divided if desired by 1,000; 10,000 or 100,000, etc.) will be the " Index
of Natality" based on the women of reproductive age. In short, the
birth-rates actually experienced in the various age-groups of females of
reproductive ages, for a series of populations to be compared as regards
reproductive efficiency, are attributed to a common standard population
(the norm). The sums in the various cases are the comparable measures of
reproductive efficiency. Symbolically this may be described as follows : —
Let^i andp'i, p^ and p'^, etc., denote the ratio of the married and of
the unmarried respectively in age-groups 1, 2, etc., to the total number of
women married and unmarried of reproductive ages in the norm or
standard population ; that is, to the total of all the reproductive groups
of that population. Then the sum pi+p2+- ■ ■ ■p'i+p'2-\-- . . • =1-
Hence the index of natality, v, which measures reproductive efficiency,
is simply —
• (453) v=I:1:{pP)+2TAp'P')
where |8 denotes the nuptial, and j3 ' the ex-nuptial, birth-rate based upon
the numbers of the married and unmarried respectively, and not upon the
total population of each group. In practice these results may of course
for convenience be actually multiplied by 1,000, or any higher number.
This index of reproductive efficiency we shall call the genetic index.
It is formed in a manner identical with that adopted to determine the
index of mortality.
6. The NataUty Index. — ^Following a procedure similar to that dealt
with in last section, let gji and g''i, 9^2 3'Udg''2, etc., denote the ratio in the
standard population of the married and unmarriedj respectively to the
* By dividing the nuptial births in each age-group by the mean number of
married women in that group, b„,/M.
f By dividing the ex -nuptial births in each age-group by the mean number of
unmarried women in that group ; 6„ /U. When desirable to distinguish them
" never married " may be used instead of " unmarried," the latter would include
" widowed " and " divorced."
I See preceding , note.
238
APPENDIX A.
total of the standard population. Then these quantities will be smaller
than^i, p'l, etc., in the ratio of the sum of all females of reproductive
age in the standard population to the total standard population, male
and female. Hence if we attribute to each age-group-ratio the birth-rate
experienced in the population to be compared, we get a total also smaller
in the same ratio. This then would give the nataUty -index v ' That is —
(4M)....v' =S(qP)+S{q'^') =
P'
where P' denotes the females of reproductive age in the norm, and P
denotes the total population, male and female, in the norm.
7. Age of beginning and o! end of fertility. — ^The determination of the
age at which fertility begins and ends is of importance, and also the range
of the reproductive period, which, of course, may not extend in individual
cases from the initial age to the terminal age for a large population.
What will be discussed here is the latter. The limits may best be deter-
mined from the usual statistical data by considering the nature of the
frequency as the limits are approached. Keeping in view the fact that
the numbers from which the experience is drawn do not vary appreciably,
the absolute numbers may preferably be used for judging the age-terminals
We get, therefore, for the old-age limit the following results for the period
from 1st January, 1907, to 31st December, 1914, for Australia, the popula-
tion being nearly o milUons.
TABLE LXXT
— ProbabiUty o£ Birth
in
Old-age, Australia, 1907 to 1914.
Age of Mothers
' i 1
1 , <
1
Line Nuptial and
48 49 50 51 ' 52
53 54 55 ; 56 , 57
58 1 59
60
Totals.
No. Ex-nuptial.
1
1
j
1 No. of births ill
i 1
' ; 1 1 ' '!
8 years
322 113 39 13 6
5| 3, 2| li 11 0'
506
2 Decrease at the
1
■
rate of e"
319
117.S| 43.2. 15.9
5.8i
2.1
0.8, 0.3
0.1 0.04 0.014
.0053
.0020
504.613
3 1 Decrease at
1
1
j varying rate . .
322
113.4
42.0 16.4
6.8
3.0
1.4 0.7
0.4| 0.2 0.1
0.069
0.053
606.522
4
JElatio of decrease
2.84 2.70 2.66 2.42 2.28
2.14 2.00 1.86 1.72 1.58 1.44 1.30
5
" Equivalent
1
' ,
nnmhei" of
,
1 1
married women
16938
16105|15113;13898|12759 11716
10819 ,9940
8989 8071; 7269 6608
6033
6
Probability per
I i
1
100,000*
2,377
877' 323 117i 59
53' 35i 26
14 15
7
Harried women
! i
of same age
1
; ' '
per annutnt . .
2,377 877 320! 117 71;
1
49 351 25
171 10 4 ? 2' ? 1
• Crude result. t Smoothed result, see formula (464).
The above results indicate that towards the end of the child-bearing
period the numbers decrease (above 48 years of age) roughly at about
the rate e*, where x is the number of years ; see line 2. This at least
holds from 48 to 52, when it would appear that the decrease is much more
slow. A closer correspondence can be had by forming the numbers
according to a formula varying the rate of decrease such as —
(455) »^^i = {2.84-0.14 (a;- 48) i n,
where n^ denotes the number of mothers of age x, last birthdaj"-.
FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 239
The figures in line 1 in Table LXXI. are 8 years' experience of
nuptial and ex-nuptial births with women of from 48 to 60 years of age
in Australia. During this time there were 476 of the former to 26 of the
latter, the number of married and unmarried females of the ages men-
tioned being respectively 136,781 and 21,615, giving one case of maternity
in 287.3 and 831.3 women, respectively. The frequency of maternity
with unmarried women between the age-Hmits in question is thus 0.346
that of married women (or that of married women is 2.89 times that of
unmarried women). If, therefore, we add to the number of married
women 0.346 times the number of unmarried that will be the total
" equivalent number" of married women to whom the cases of maternity
can be ascribed. These, divided into one-eighth^ of the numbers on line
1, give the crude probabilities of maternity for married women of the
ages in question. The values, as calculated from the data, are given in
line 6 ; the smoothed values obtained from these are given in Une 7.
Although a probability is given for age 58, the actual fact is that in over
7,000 possible cases (see line 5) no birth occurred ; 57 is the greatest age
at which a birth actually occurred. The values are shewn as curve A
and on a larger scale, as curve B on Pig. 66. It will be noted that the
continuation of the curve for ages 49 to 51 (see a b) on the figure, suggests
that 53 is the age at which the value approaches z^ro, point c, and the curve
for ages, 51 to 60, b d in figure, seems to be quite a different curve. No
simple exponential relation, however, will bring these two curves under a
single formula. 2 See page 244 for Pig. 66.
Por .the lower limit we have the following data, viz. : —
TABLE LXXn.— Probability of a Birth in Early Age, Australia,
1907-1914.
Line
Age.
11
12
13
14
15
16
17
18
1
Nuptial births, 8 years
•
4
30
170
1,138
4,062
11,761
2
Ex-nuptial births, 8 years
5
21
126
537
1,500
2,980
4,504
3
4
Total births, 8 years . .
Ratio of ex-nuptial to
nuptial births . .
5
00
25
5.2
156
4.2
707
3.16
2,638
1.32
6,942
0.73
14,265
0.38
5
Married women . .
1
18
93
349
1,145
2,651
6
7
8
f Never married " women
Probability of nuptial
maternity per annum per
1,000
Probability of ex-nuptial
maternity per annum per
1,000,000 unmarried
women
42,222
42,001
1.6
42,071
? 500
6.6
42,484
? 208
37.1
43,273
228
155.1
43,915
408
427
43,813
443
850
42,854
576
1,313
1 Approximately, see § 8, p. 240.
' Results deduced from the initial value 2377 by means of the formula —
"x+l ={2-75-0.15 (a;-48)} ,.^.
would bo in substantial agreement with (455), and are as follows : —
2377 864 332 136 59 27 14 7 4 3 2 1
They are less probable, however, than Jbpse. given on line. 7 in the table.
240 APPENDIX A.
The results on line 8 do not need smoothing. Those on line 7 for the ages
13 and 14 are, of course, very uncertain, the normal values would probably
be much smaller than 200. It is evident from the above, that the cases
of ex-nuptial maternity throw most light upon the question of the com-
mencing age of fertility. These are shewn on line 2, and will be given
very nearly by the equation. ^
(456) nx_i = ; 1.50 + 0.50 (18— x); n^.
The results are shewn as curve D, and on a larger scale as curve E, on
Fig. 66, on page 244.
The general result of the investigation as to the terminal condition.s
is that the null-points can be taken as say 11 and 60, the values being
very small from ages 53 onward, and from 1 1 to 12.* The initial null -point
is consistent with the curve of frequency of the first menstrual appearance,
which would give a null-point of about 9 years' and a maximum just after
16 years of age are attained. The curve as shewn in Fig. 66, curve C,
gives, according to AVhitehead, the group-numbers of single year age-
groups for a tots^l of 4,000 cases under observation. These group-num-
bers are shewn by small circles, see p. 244.
8. The maternity-frequency, nuptial and ex-nuptial, according to
age, and the female and male nuptial-ratios. — Let g, m, and u, denote
respectively the number per annum (i.) of brides, (ii.) cases of nuptial
maternity ; and (iii.) cases of ex-nuptial maternity, and also let M and U
denote the number of married and " never married " women among
whom the latter occur. These numbers are given for each age from
12 inclusive onward, in Table LXXIIL, see columns (ii.J, (iii.), (iv.), (\'i.),
and (^^i.), or g, m, u, M and TJ.
The numbers are for 8 years, and the mean population from which
they are drawn is about 8.0406 times that of the moment of the Censu.s.
viz., 3rd April, 1911. Hence the epoch can be regarded as the date of the
Census, and the numbers have been divided by 8.0406 to obtain the annual
equivalent.
1 If we take 4500 as the number of ex-nuptial births for the age 18, we shall obtain
4.0, 27.7, 145.6, 545.8, 1500.3, 3000.0, and 4500, instead of the numbers ehevm
on line 3 in Table LXXII.
2 At Budapest, J. Kdr&si records two mothers at 54, one at 56, and one at 57
in 4 years ; vide, Phil. Trans. 1895, B., p. 794. In Edinbvu-gh and Glasgow Matthews
Duncan records for the ages 51, 52 and 57, and for an aggregate of 16,301 married
mothers, 2, 4 and 1 respectively, p. 9 of his "Fecimdity, Sterility, &c." 1871 Edit.
C. Ansell in 1874, vide his " Statistics of Families," regards an alleged case at 59 as
needing verification. Tauffer, of Budapest, in 2083 cases, records one at 54. lu
handbooks of Forensic Medicine, Casper -Liman mentions one case at 54 ; one is men-
tioned by Hofmann at 55 ; see Phil. Trans, loc. cit. C. J. and J. N. Lewis' " Natality
and Fecundity," published 1906, out of 84,971 cases of births in Scotland in 1855,
give for the ages 15, 16, 17 and 50 and upwards to 58 ; the following results, viz. : —
Ages 15. 16. 17 ; 50. 51. 52. 53. 54. 65. 56. 57. 58.
Numbers 3. 23. 132; 16. .5. 7. 1. 3. 2. 1. 1. 2.
' See " Sterility and Abortion," Whitehead, p. 46, or M. Duncan, op. cit., p. 32.
FERTILITY, FECUNDITY. AND REPRODUCTIVE EFFICIENCY. 241
The ratio (e) of ex-nuptial to nuptial cases of maternity is found by-
dividing the values in column (iv.) by those in column (iii.) in Table
LXXIII. That is to say—
(457) e = u / m.
The ratio of " brides" to " unmarried " females, or to females
" never married " given in column (viii.) of the table, may be called the
"female nuptial ratio " (g) according to age, and is given by —
(458) 9 =g/ U
the total number of brides being the same as the number of marriages J
in (400), p. 176. Suffixes will denote the age to which the ratio refers.
The values a are the probabilities of marriage according to age of the
unmarried. This probability corresponds to a mean of the marriage-
rates of 0.008403, and to a marriage rate over all the eight years of
0.00842863.^ For any particular year the distribution according to age
will therefore approximately be in the ratio of the crude marriage rate for
the year in question to that above ; expressed ordinarily, say as —
^*^^^ ^' ^u-qM^
n being calculated as indicated by (400), p. 176.
The greatest number of never married appears to be for the year be-
tween the ages 16.32 to 17.32, the number being about 43,950. Similarly
the greatest number of brides appears to be for the ages 21.90 to 22.90,
the number being about 27,955.
The curve shewing the number of brides of each age is curve F,
Fig. 67, and that shewing the number of the females "never married" is
curve G of the same figure ; G' and G" shew the terminal values on a
larger scale. The circles with crosses denote the positions of the data
when corrected for the error of statement of age at marriage ; see pp.
193-6 hereinbefore. The crude results are shewn by circles on E', G, G ' and
G". It will be seen from these terminal values that there is considerable
regularity in the curve even for advanced ages (see p. 244).
The " male nuptial ratio," according to age, is, similarly to (458)
and (459)—
(^««) t,=./F;or(461) ^'=y- ^^^Z
The values are given in Table LXXIII., the crude results being shewn
in column (xiv.). The curve shewing the number of bridegrooms of each
age is curve W, Fig. 70, and that shewing the unmarried males is curve V
of the same figure. V ' and V ' ' shew the terminal values on a larger scale .
The smoothed values of the probability g', and u' are given in columns
(xviii.) and (xix.) of Table LXXIII.
Expressed per thousand, as is usual, 8.42863.
242
APPENDIX A.
TABLE LXXni.-Shewing the Numters of Brides and Bridegrooms and the Cases of Nnptaal and Ec-nuptial ^^^^^ I^
1907-191^ Australia, and the Numbers of Married and Never Married Males and Females, at t^eCwwus of 3rd AjwU,
1911. Shewing SthrProbabUities of Marriage for Never Married Males and Females, and the Probability of Nuptial
and Ex-nuptial Maternity, and Ratios Dependent upon these.
>.
es
■—
^
o
IS
e
"C*^
s§
CJ
T-l
^
u
<
1
■a
Unspecified, 111. Total including the Unspecified, 301,922. For notes see next page.
FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 243
9. Nuptial and ex-nuptial maternity and their frequency-relations. —
The crude rate, according to age, of nuptial and of ex-nuptial maternity
is found by dividing the number of cases of maternity of each kind by
the number of married or of " unmarried " or " never married " women.
That is to say ^j, u, and § denoting the probability of maternity,
according to age, respectively of the married, the never married, or of
both combined, we shall have : —
(462) V =m / M;
(463) u =u / U;
(464) ^ = (m -\- u) / (M + U).
The relation, according to age, between the ex-nuptial and nuptial
rates, is —
(465)....e = «/i. = ^/^ = -^ . _
These crude rates and their ratio to each other are given in Table LXXIII.
for the whole reproductive period in columns (ix.), (x.) and (xi.). The
smoothed values are given in columns (xv.), (xvi.) and (xvii.).
The graphs of the numbers of cases of nuptial and of ex-nuptial
maternity are shewn respectively by curves H and I, on Fig. 68, the dots
in the former case, and the crosses in the latter, denoting the crude
results. The ratio of the ex-nuptial cases to the nuptial cases are shewn
by curve J, and on a larger scale by curve J, ' Fig. 68. The nuptial and
ex-nuptial maternity-rates are shewn on the same figure by curves K and
L, the dots in the former, and the small circles in the latter indicating the
crude results (see p. 244).
It should be noted that m and M in (462), etc., are not necessarily
homogeneous, since each will contain, though in unequal proportions,
primiparous and multiparous women, and these will have been subject
to risk for unequal periods. Moreover the multiparse may have given
birth to very different numbers of children. If, therefore, the probability
of maternity is affected by previous issue, the value of p must be regarded
as merely a cr%de probability. An exact probabihty would have to be
defined in categories according to the age, the number of previous issue,
and the length of exposure to risk. This wiU appear more clearly in the
theory of fertility and sterihty. For this reason the values given of jj and
u in Table LXXIII, are for the " average risk" of the " average married
woman" or the " average never married woman" during twelve months,
and takes no account of variation of the " risk" according to the age of the
husband. In section 11 hereinafter it will be seen that the maxima
vary.
Notes to Table LXXIII. on preceding page.
• If the corrections referred to in Part XII., § 15 and 16, pp. 193-6, be applied, these
numbers become 14,004; 19,580; 23,678; 26,927; see formula (407). This will change the ratios
in column (viii.) from .03844 to .04064; .05500 to .05937 ; .06557 to .07674 ; and. 11733 to .09706.
t The maximum is for the cenfroZaffe 18.73, that is for the group of ages 18.23 to 19.23, and
the amount is 0.4849.
t The maximum is for the central age 22.50 ; that is for the group of ages 22.00 to 23.00, and
the amount is 0.01885.
I The ex-nnptial births are attributed to the " never married," but may, pertiaps, be equally
well attributed to the " unmarried," that is the " never married " together Mith the " widowed"
and " divorced."
244
APPENDIX A.
Terminal Frequencies of Fertility; Frequency of Nuptial and Ex-nuptial Maternity;
Probability of Marriage of both Sexes at each Age; etc.
Fig. 70.
Fig. 67.
,10 so 30 40
60 70 80
llfjl
COODD
Li h ^ ^ %^
10 13 14 16 Carves A,B>C.
48 SO 63 64 Caires D,E
Fig. 69.
Fig. 66.
Fig. 68.
Fig. 66. — Ourvea A and B shew the terminal age of fertility. Curves D and E
shew the initial age of fertility. Curve C shews the frequency of the ap-
pearance of menstruation according to age.
Fig. 67 — Curve F shews the niunbers of brides at various ages. Curves G, G' and
G"'shew the numbers of the " never married " at various ages.
Fig. 68. — Curve H shews the number of cases of nuptial maternity, and Curve I
those of ex -nuptial maternity at each age. Chirves J and J' shew the pro.
portion of ex-nuptial to nuptial cases of maternity at each age. Curve K
shews the nuptial and L the ex-nuptial rates of maternity at each age, the
ex -nuptial rate being determined by attributing the births to the " never
married."
Fig. 69. — Curve M shews the ratio of the ex-nuptial to the nuptial rates of
maternity at each age. Curve N is the ratio of the brides at each age to
the " never married females " of the same ages. Curves O and O' are
similarly the ratio of the bridegrooms at each age to the " never married
males " of the same ages, curve O' being displaced one division (10 years) to
the right so as not to be confused with curve N.
Pig. 70. — Curve W shews the number of bridegrooms of each age, and V, V and
V" the number of " never married males " at each age.
In all the above cases the age is the "age last birthday."
Fertility, fecundity, and reproductive efficiency. 245
10. Maximum probabilities of marriage and maternity, etc. — ^The
position and amount of the maxima determined from the smoothed
results in columns (xv.) to (xix.) of Table LXXIII. are as follow : —
Table LJilXIV. — Maximum Probabilities, Marriage and Maternity.
Maximum probability of — Year -group from — Ainoi.mt.
Age. Age.
Nuptial maternity 18.45 19.45 .0486
Ex-nuptial maternity . . . . 22.00 23.00 0.01835
Ratio of ex-nuptial on nuptial Probably no maximum value point of
maternity inflexion at —
25 to 26 0.0510
Marriage of women . . . . 24.52 to 25.52 0.12632
Marriage of men .. .. 27.5 to 28.5 0.11320
The maxima are for the two heterogeneous groups " nuptial" and
" ex-nuptial" aggregated according to age merely. In the next section
it will be shewn that the maxima are dependent upon age at marriage.
The largest number of marriages of brides would appear to be for the
ages 21.9 to 22.9, and to be about 28,000 in 8 years ; and the largest
number of marriages of bridegrooms, for the ages 24.8 to 25.8, the number
being about 25,000 in 8 years, the total mean population aggregated for
the years in question being 35,821,000 persons. The largest number of
cases of nuptial maternity occurred for ages 26.12 to 27.12, the number
being about 55,500 in 8 years, and the ratio at the crude maximum con-
sequently 0.3182. The largest number of cases of ex-nuptial maternity
occurred for the ages 19.5 to 20.5, the number being about 5,400 in 8 years,
and the ratio at the crude maximum of cases, therefore, 0.01691.
The question of a more accurately defined maximum wiU be con-
sidered hereinafter.
11. Probability of a first-birth occurring within a series of years after
marriage. — ^To determine the variation of initial fertility with age, the
initial probability of maternity may be deduced by ascertaining primarily
the number of women at different ages who were married during a given
period. Then, tracing these through the first portion of their married
life, the respective periods which elapsed after marriage before they gave
birth to their first living child may be ascertained.
Tor this purpose the six-year period, 1909-14, was brought under
observation, the experience being all cases in the Commonwealth of
' Australia within a series of years, viz., 6 after marriage. Owing to mis-
statements regarding age, however, the number of brides registered at
each age during the several years under observation required correction.
It was found that, if the actual numbers of brides registered at ages 18,
19 20 and 21 years were accepted, without adjustment, anomalous
results would be obtained. Evidently serious errors existed owing to
brides of 18, 19, and 20 years overstating their age as 21, and therefore
the numbers of brides upon which the rates of fertility should be founded
246
APPENDIX A.
needed correction. A special type of smoothing of the number of brides
of 18, 19, 20 and 21 years to remedy the misstatement of age had there-
fore to be adopted.^
A similar misstatement of age had evidently occurred in the case of
mothers (registered as being 19, 20, and 21 years of age), who gave birth
to a first-bom child during the period 1909-14, and the numbers conse-
quently had also to be smoothed, so as to eliminate the effect of mis-
statements in the age of mothers.^
Tables were compiled shewing the mean number of brides of each
age in any year and in the year immediately preceding ; and for the same
ages the number of first confinements in successive years of duration of
marriage. Assuming then that the migration elements balanced each
other, the table gave a series of results shewing for the years 1909 to
1914 inclusive the aggregate number of brides of each age at marriage to
which the aggregate number of first confinements could be referred,
hence the ratio of the latter to the former gave the probability required.*
' The justification for this smoothing is really that the probability of a mis-
statement of age is very great, and the probability of some physiological or other
cause, for the anomaly, is relatively negligible.
^ The following are the unadjusted and adjusted figures : — •
Age.
Nuptial First Births, according to Suc-
cessive Years of Duration after Marriage.
Number of Brides to whom the Births may-
lie ascribed, according to Successive
Years of Duration after Marriage.
Total
0-1
1-2
2-3
3-4
4-5 5-6
Total 0-1
1-2
2-3 1 3-4
4-5
5-6
18 7,568
5,899
1,291
262
81
29
6
10,159 10,159
8,331
6,513! 4,735
3,039
1,484
7,568
5,899
1,291
262
81
29
6
10,736 10,736
8,802
6,880
5,003
3,213
1,571
19
11,625
9,071
1,943
429
118
48
16
13,838 13,838
11,364
8,899
6,463
4,156
1,998
11,228
8,761
1,877
414
114
46
16
14,902 14,902
12,227
9,557
6,917
4,457
2,177
2U
13,596
10,141
2,618
556
202
56
23
15,496, 15,496
12,737
9,978
7,244
4,657
2,241
14,400
10,741
2,773
589
214
59
24
18,100
18,100
14,860
11,630
8,453
5,475
2,675
21
17,507
12,613
3,699
823
262
81
29
24,850
24,850
20,309
15,838
11,520
7,498
3,702
17,100
12,320
3,613
804
256
79
28,
20,600
20,600
16,848
13,158
9,588
6,264
3,002
The upper number is that furnished by the registration records, the lower is
that which was obtained after adjustment. The only adjustment deemed essential
as a preliminary is for these ages, viz., 18 to 21. For aU other ages the results are
as given by the unadjusted data.
» The
following illustration of the method of compiling will sufiSce :—
Year.
Age
at
Mar-
riage.
Mean
No. of
Brides
for Year
and pre-
ceding
Year.
Number of First Confinements in suc-
cessive Years of Duration of Marriage.
Duration of Marriage, 0-1.
0-1
1-2
2-3
3-4
4-5
5-6
Age.
Brides.
Confine-
ment.
Ratio.
1909
26
27
26
27
26
27
(Frou
1,864
1,563
2,076
1,616
2,268
1,781
1 these th
1,002
835
1,047
853
1,171
967
s totals
443
417
551
444
645
527
on the
160
107
219
149
212
178
right w
71
66
79
73
101
88
ere fon
54
39
46
41
ned.)
15
22
26
27
13,637
11,054
7,279
5,721
0.5338
0.5176
Duration of Marriage, 1-2.
L910
26
27
11,068
9,004
3,095
2,566
0.2795
0.2850
1911
Duration of Marriage, 2-3.
26
27
■ 8,571
6,971
800
619
0.0933
0.0888
I*ERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 247
The probabilities so ascertained are shewn on Table LXXV. up to 6
years. The crude results are shewn by the dots on Pig. 71, on which the
curved lines give the smoothed results, the corresponding numerical
values appearing on the right hand side of the table.
TABLE LXXV.— Probability o£ a Nuptial First Birth occurring within 6 Years of
Marriage, Based on Australian Data, 1909 to 1914.
Crtjde Results. Adjusted Results.
Probability of Giving Birtli to a
First Child
Probability of Giving Birth to a First Child for
Age
lor a Duration of Marriage of —
a Duration of Marriage of —
Age
last
last
Birth-
less
less
less
Birth-
day.
than
1-2
2-3
3-4
4-5
5-6
than
than
0-1
1-2
2-3
3-4
4-5
5-6
day.
lyr.
yrs.
yrs.
yrs.
yrs.
yrs.
6 yrs.
1 yr.
yrs.
yrs.
yrs.
yrs.
yrs.
yrs.
11
.0000
.0000
.0000
.0000
.0000
.0000
.0000
11
12
.1308
.0963
.0217
.0066
.0030
.0020
.0012
12
13
.2568
.1881
.0433
.0131
.0060
.0039
.0024
13
14
.3781
.2755
.0647
.0195
.0091
.0058
.0035
14
15
.3324
.1233
.0470
.0353
.0278'
.4946
.3585
.0860
.0258
.0121
.0076
.0046
15
16
.4352
.1042
.0424
.0177
.0149
.0075
.6219
.6063
.4370
.1073
.0321
.0150
.0093
.0056
16
17
.4979
.1271
.0413
.0128
.0141
.0053
.6985
.6975
.4985
.1263
.0377
.0176
.0108
.0066
17
18
.5495
.1467
.0381
.0162
.0090
.0038
.7633
.7770
.5485
.1455
.0432
.0199
.0123
.0076
18
19
.5879
.1535
.0433
.0165
.0103
.0073
.8188
.8414
.5800
.1664
.0497
.0229
.0138
.0086
19
20
.5934
.1866
.0506
.0253
.0108
.0090
.8757
.8856
.5950
.1854
.0551
.0252
.0153
.0096
20
21
.5981
.2144
.0611
.0267
.0127
.0093
.9223
.9176
.5958
.2051
.0614
.0280
.0168
.0105
21
22
.5919
.2301
.0675
.0299
.0151
.0122
.9467
.9429
.5908
.2247
.0673
.0306
.0182
.0113
22
23
.5800
.2425
.0783
.0314
.0173
.0094
.9589
.9619
.5819
.2423
.0730
.0331
.0195
.OlSl
23
24
.5545
.2466
.0827
.0344
.0231
.0130
.9543
.9730
.5688
.2569
.0785
.0354
.0206
.0128
. 24
25
.5314
.2636
.0815
.0375
.0235
.0158
.9533
.9771
.5533
.2679
.0831
.0378
.0216
.0134
25
26
.5338
.2795
.0933
.0404
.0254
.0081
.9805
.9750
.5357
.2754
.0872
.0402
.0225
.0140
26
27
.5176
.2850
.0888
.0458
.0252
.0141
.9765
.9667
.5168
.2795
.0903
.0423
.0233
.0145
27
28
.5037
.2677
.1013
.0465
.0260
.0126
.9578
.9530
.4967
.2813
.0922
.0439
.0240
.0149
28
29
.4548
.2774
.0836
.0359
.0198
.0107
.8822
.9330
.4766
.2792
.0929
.0446
.0245
.0152
29
30
.4686
.2421
.0898
.0498
.0224
.0107
.8834
.9075
.4545
.2751
.0930
.0448
.0247
.0154
30
31
.4602
.3084
.1003
.0447
.0238
.0178
.9552
.8745
.4310
.2668
.0923
.0446
.0245
.0153
31
32
.4191
.2464
.0873
.0368
.0220
.0132
.8248
.8381
.4073
.2571
.0907
.0440
.0240
.0150
32
33
.4057
.2422
.0825
.0428
.0217
.0194
.8143
.7938
.3789
.2463
.0883
.0426
.0231
.0146
33
34
.3310
.2526
.0928
.0353
.0232
.0204
.7553
.7411
.3487
.2319
.0843
.0407
.0217
.0138
34
35
.3036
.1950
.0771
.0387
.0113
.0155
.6412
.6748
.3123
.2135
.0784
.0382
.0198
.0126
35
36
.3024
.1820
.0724
.0395
.0236
.0061
.6260
.6063
.2768
.1935
.0718
.0354
.0178
.0110
36
37
.2241
.1910
.0741
.0341
.0173
.0000
.5406
.5367
.2423
.1730
.0650
.0315
.0157
.0092
37
38
.1919
.1576
.0634
.0252
.0105
.0144
.4630
.4662
.2088
.1520
.0573
.0276
.0134
.0071
38
39
.1844
.1391
.0406
.0275
.0087
.0000
.4003
.3946
.1755
.1303
.0490
.0237
.0110
.0051
39
40
.1436
.0986
.0520
.0131
.0049
.0000
.3122
.3245
.1426
.1082
.0415
.0198
.0088
.0036
40
41
.1323
.0870
.0336
.0194
.0076
.0000
.2799
.2558
.1111
.0863
.0333
.0158
.0070
.0023
41
42
.0756
.0627
.0211
.0135
.0000
.0073
.1802
.1951
.0855
.0656
.0254
.0119
.0053
.0014
42
43
.0669
.0665
.0131
.0051
.0000
.0083
.1599
.1411
.0634
.0474
.0178
.0080
.0037
.0008
43
44
.0384
.0462 1 .0064
.0030
.0000
.0000
.0940
.0937
.0441
.0321
.0116
.0041
.0014
.0004
44
45
.0258
.0066
.0086
.0000
.0000
.0000
.0410
.0622
.0296
.0220
.0070
.0022
.0012
.0002
45
46
.0400
.0199
.0147
.0035
.0012
.0006
.0001
46
47
' *
.0252
.0131
.0094
.0019
.0005
.0003
.0000
47
48
■"
.0159
.0093
.0056
.0007
.0002
.0001
48
49
■■
.0095
.0062
.0029
.0003
.0001
.0000
49
50
.0026
.0031
.0000
.0000
.0000
.0000
.0057
.0053
.0040
.0012
.0001
.0000
SO
51
.0028
.0023
.0005
.0000
51
52
.0013
.0011
.0002
52
53
.0006
.0005
.0001
53
54
.0002
.0002
.0000
54
55
.0001
.0001
55
The probabilities in the table apply to the total number of women
married at the given ages, not to the survivors after the series of years under
observation. The probabilities are of course cumulative, that is to say
248
APPENDIX A.
the probability, qJ)„, that a first birth will occur before the end of the
n-th year after marriage, is the sum of the probabiUties that it wiU occur
during the fibrst, during the second, etc., up to and including the w-th
year. Or
(466) .
■oPn = oPi + iPz +
-iPn
12. Maximum probabilities of a first birth. — ^From the smoothed
results in the table, it will be seen that, as the interval to the first birth
increases, the age of maximum increases. Thus the greatest probability
of a first birth within the first year from marriage is for age at marriage
21.24, during the year succeeding that of marriage it is at age 28.47,
and so on as shewn in the following table, viz., LXXVI.
TABLE IiXXVI. — Shewing the Age of Mazimuin Fiobability of a Fiist Biith.
AustraUa 1909-1914.
Interval Years. 0-1 1-2 2-3 3-4 4-5 5-6 0-1 0-2 0-3 0-4
0-5
0-6
0-7"
0-8
0-9
0-10
0-11
Vertex at (years)
Corresponding to
Median Age at
Marajage
Or to Median Age
at Birth
Probability
By Formula (467)
I
20.74 27.97 29.62 29.75130.03 30.14 20.74 23.75 24.52 24.91
30.25'30.53 30.64 21.24
I I
33.75 35.03 36.14 21.74
21.24 28.47
21.74 29.97
.5962
.2813
30.12
32.62
.0931
.0448
.0247.0154
.5962
25.06 25.16 25.22 25.26 25.30 25.32 25.33
I I tl I t
24.25 25.02 25.41
I i
25.25 26.52 27.41
.8259.9050
840 .920
9421
.947
25.56 25.66 25.72 25.76 25.80
28.06 28.66 29.22 29.76 30.30
9637 .9772 .9859 .9916 .9953
.960 .968 .973 .977
25.82 25.83
30.82,31.33
.9978
.9998
• The ratios 9050/8259, 9421/9050, etc., are 1.0958, 1.0410, 1.0229, 1.0140, which oontinued.are 1.0089,
1.0058, 1.0038, 1.0025, 1.0020, the factor of the last two figures converging to 52/80. Xhis, however, would give
1.0011 for 0-11. It is more probable, however, that the probability is of the type oP'„ + ^='oP'„ + m)/ (1+ m)
where m may perhaps be taken even as unity, implying that the residual chance is reduced about one half each
year. The matter requires special investigation.
t These correspond to the values of the vertices on Fig. 71.
It is worthy of note that the above results for 0-2 up to 0-8 are
roughly given by the formula —
(467).
•oP'« = 1 -
0.16
-1 •
in which n is the total duration of marriage.
The figure (7 1 ) and table shew cleatly that the maximum is a function
of the duration of marriage as well as of age. To find the maximum
value for any durations to t the line of vertices C D on Fig. 71 must be
followed, or during year-intervals < to i -|- 1, the line C E must be
followed. Thus for age 20 last birthday, the duration is to 0.93, the
probability is about 0.555. The graphic solution may also follow the
method indicated in Fig. 71a, which needs no comment when examined
in connection with Table LXXVT.
FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 249
Probabilities of a flist-bitth during first 6 years of marriage.
Interval from nuuriage.
i-O ■'e-i!a 25J%^j >vi7fe
icr ^'-'■'
^■^ '4
iti^d^A
J/ 1
,J^.,,,..r X
^ 4 '" a
' iivrX"^
^\ f" IS Fig.
, /!!i--^-
^h y ■ -'i r^^-
--i--i ^.
. v-i- - i ^
fi U '^'^'^T^tet,^'4,>.3;i
A^ l--/--.^,,.. J^ i
_____] _/:?^^N,
.\ n.^^^-^-i ,^
.5--- -/-^ -.--
1 \ 1 aijlInoQt 2o;;au5t ,0
--W--\-\-
" lis
VW -
.4-- /. — --1-V
__JL.
__._
1 ,___ u. ,,
\ \^ :— -
■^ ' ^ \^
i :-x::-:'z:::X:-
--^-^- "i
i "" \
1\\ ^^
M j' / VT'^;:7-a,;wArta:
,,7.„.,v, y;^ r^fi;^:;^/
1 /^ J3--'"' ^^
.i-=;:===^:;====igi
]0
45
25 30 3S 40
Ages of mothers at marriage.
Fig. 71.
13. Determination of the co-ordinates of the vertices. — ^The repre-
sentatioTi of group-totals by means of integral functions of the values of
the central abscissa of the group-base (central value of the interval) has
been referred to in Part V., § 10, pp. 72, 73. In curves of the type which
has just been considered, the results about the vertices may be closely
represented by a curve of the second degree, and the curve itself may be
regarded as defining the curve of group-totals for all values of the central
abscissa (the abscissa of the middle ordinates of the group). In such
instances the co-ordinates for the maximum-group may be very accurately
ascertained from the tabular maximum together with the tabular values
on either side of it. Let the maximum tabular value denote the point
M on the curve, and the adjoining tabular values denote the points A, B,
viz., the points on either side. Then, if the difference of the mean of the
ordinates of the points A and B, and the ordinate of M be denoted by h,
and the half difference of the ordinates of B and A be denoted by I, that
is if —
(468) A = 2/m - i (2/6 + ya) ; and i = 4 (y^ -y^) ;
then we shall have —
12. I
(4b9) Vmax ^= Vm H tt" i •^max = *m "T
2h
250 APPENDIX A.
The proper maximum is greater than the tabular maximum by the amount
l^/4}i, and its abscissa lies between that of the tabular maximum and the
next highest tabular quantity distant from the former by the amount
l/2h.
The positions of the vertices have been computed in this way. It
remains to be noted, however, that when the value of the abscissa indi-
cates merely the " age last birthday," it is necessary to add the amount
J to the value given by the formula in order to refer the co-ordinates to the
middle values of the group-abscissae. Thus, in Fig. 71, the curves are
plotted with the argument " age," i.e., last birthday, hence the vertex-
value 20.74, see curve 0-1, and the maximum 0.5962, refer to the group of
brides whose ages ranged between 20.74 years of age and 21.74 years of
age. The middle value of the range is 21 .24, but the average value is not
that. The probability 0.5962 applies to the brides whose ages were
between 20.74 and 21.74. Of 10,000 such, 5,962 would give birth to a
first child within one year of marriage.
14. Average age of a gioup. — ^The error of adopting the middle value
of any range has been considered in Part XII., § 20, pp. 200-201. It is
sometimes preferable to relate the values of the dependent variable, not
to the middle values but to the average values of the independent variable.
In such a case formula (416), p. 201, may be used. Let A, M, and B be
three group totals on equal bases k (equal intervals on the axis of ab-
scissae). The values of the co -efficients of a rational integral function
of the second degree — the graph of which wiU represent, viz., the areas
standing on the equal bases, the group-totals — may be found by the
formulse of Part V., §§ 1 to 9., pp. 64-72. The weighted mean abscissa
of the middle group may be denoted by x'^. If then we make the origin
at 0, so that A is the integral of the equation a -f 6a; + cx^ between the
limits and k, M the integral between k and 2k, and B between 2k and
3fc, then we shall have —
(470) x„,= - ^' '
■•^"'~ a + lbk + ^ck^
which may be put in the very simple form —
(471) x^^€ =x,^+ ^k(B -A)/M.
This is the same formula as (416). In general, therefore, it is sufficient
to find the value of the abscissa to which a group may be referred by using
the value of the group and of these on either side : see the results as to
average interval in §§ 21, 24, etc., hereinafter.
15. Curves of probability for different intervals derived by pro-
jection. — ^Reverting to Fig. 71, it may be noted that the probabihties
of a first birth between 1 and 2 years, 2 and 3 years, etc., after marriage
FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 251
may be derived for each age approximately by projection if the ratio of
the aggregates and the position of the maximum are known. For ex-
ample the faintly-dotted curve is the curve for the interval between 1 and
2 years after marriage derived by projection from that up to 1 year (0-1).
The difference between the two curves is nearly negUgible. The following
are thte relations between the curves : —
Let X, y be the co-ordinates of any point P on a curve, and let x', y'
denote the co-ordinates of what may be called the corresponding point
P ', on a curve derived therefrom by drawing the line P P ' Q to cut the
axis (OX) of abscissse in the point Q, so as to make the angle of inter-
section therewith, XQP, equal to 6, and also the ratio QP '/QP equal to
p. Then, if 6 and p be constant, the derived curve will belong to a
family of curves of the type of the original, but differing therefrom in
" skewness" if 6 be not 90°. The co-ordinates of any point P', viz., of
the " corresponding point" on the derived curve are simply related to
those of the point P on the original curve from which it was derived,
being given by the equations —
(472) y'=py\ x'=x-y(l -p)cot.e.
Hence if the equation of the original curve be f{y) = F(x), that of the
derived curve will be —
(473) /(— )= F(x' -ky'};
in which k — cot. 6 {I — p)/p.
To determine whether the succession of probabilities for 0-1, 0-2, 0-3,
etc., and 0-1, 1-2, 2-3, etc., are rigorously derivable by projection would
involve data embracing larger numbers and free from all uncertainty
as to the effect of migration thereupon.
1 6 . Numbers of fiist-biiths according to age and duration of marriage.
— ^There were in Australia during the years 1907-14 inclusive, 220,021
cases of nuptial first births . The records of these were compiled according
to " age last birthday," and duration of marriage." Multiplying the
numbers as compiled by a factor, that would make the total 1,000,000, the
results are as shewn in Table LXXVII., compiled for single months of
duration of marriage from 1 to 12 months, and for single years of duration
of from 1 to 26. The table thus furnishes the distribution of 1,000,000
nuptial first births according to age and duration of marriage. The
figures for the months are of course only one-twelfth of the figures which
would be comparable to the yearly values. This distribution may be
called the nuptial protogenesic distribution.
252
APPENDIX A.
TABLE LXXVn.— Shewing the Number in 1,000,000 Nuptial First-births of Births occuiring for all
Births occuiiing in Australia during the Years
[STERVAL AITEK MaEEIAGE DUEINQ WHICH A BlETH OCCURS.
AOE OF
•
MOIHEBS.
O-I
1-2
2-3
3-4
4-5
5-6
6-7
7-8
8-9
9-10
10-11
11-12
0-1
1-2
a-3
3-4
4^5
5-6
6-7
mths.
mths.
mths.
mths.
mths.
mths.
mths.
mtlis.
mths.
mths.
mths.
mths.
year.
years.
years.
years.
years.
yrs.
yrs.
12
IS
"
5
"
5
"
"■
"
5
"15
4
14
32
14
23
18
9
9
5
5
9
124
15
45
73
109
68
91
77
68
36
9
27
23
626
18
16
382
423
514
532
568
541
486
295
91
95
73
91
4,091
209
"l8
9
17
959
1,073
1,336
1,663
1,532
1,836
1,859
1,250
532
704
450
304
13,498
1,054
73
5
9
18
1,523
1,773
2,754
3,163
3,613
3,891
4,468
2,950
1,636
2,118
1,359
1,082
30,330
3,417
377
50
"'5
19
1,886
2,227
3,513
4,272
5,127
6,054
6,790
4,609
2,850
3,909
2,972
2,345
46,554
7,794
1,054
155
32
20
1,754
2,236
3,172
4,104
5,086
5,995
7,068
5,590
3,254
6,154
4,777
3,318
52,508
11,921
J'S?^
382
64
27
9
21
1,877
2,309
3,454
4,640
5,704
6,981
8,331
6,613
4,613
9,122
6,845
4,772
65,261
16,125
2,950
682
227
36
23
22
1,532
1,827
2,682
3,445
4281
5,740
7,254
6,159
5,077
11,981
8,935
i'Z08
65,621
22,225
3,995
1,118
395
132
32
23
1,113
1,523
1,941
2,909
3,854
4,740
6,263
5,413
6,177
11,953
9,226
6,331
60,443
24,316
5,672
1,859
677
236
95
24
986
1,104
1,586
2,086
2,886
3,441
4,959
4,231
4,045
11,467
8,726
6,313
51,830
24,261
6,413
2,268
904
377
191
25
768
818
1,273
1,573
1,873
2,830
3,754
3,250
3,895
10,549
7,986
5,645
44,234
21,988
6,954
i'®l2
1,040
550
277
26
691
677
1,027
1,168
1,500
2,154
2,836
2,732
3,272
9,031
7,222
4,877
37,187
20,670
6,613
1^12
1,268
782
459
27
382
432
718
895
1,136
1,463
2,234
2,143
2,373
7,649
5,909
4,177
29,533
18,419
6,009
^W
hill
800
427
28
491
345
691
736
964
1,232
1,595
1,677
2,091
6,372
5,140
3,450
24,784
15,315
5,508
H^i
h^t
695
541
29
305
282
414
600
568
845
1,177
1,245
1,613
4,536
3,659
2,795
18,039
12,281
4,436
2,263
1,218
727
455
30
227
255
282
418
432
677
877
1,034
1,404
4,191
3,336
2,304
15,457
10,221
4,113
2,077
1,182
732
477
31
209
159
241
395
359
450
641
691
945
2,600
2,245
1,782
10,717
7,549
2,936
1,618
868
532
377
32
177
182
227
264
373
405
523
627
786
2,145
2,000
1,354
9,063
6,680
i-^^S
1,300
823
577
441
33
173
168
195
150
273
345
441
436
564
1,654
1,468
1,082
6,949
5,086
2,032
1,154
650
441
395
34
73
105
191
141
218
256
335
327
382
1,264
1,027
773
5,112
3,918
1,712
800
586
373
323
35
105
100
132
127
227
177
268
318
314
1,014
863
559
4,204
3,530
1,314
714
423
282
268
36
95
55
150
132
118
150
223
255
264
750
777
450
3,419
2,654
1,232
577
423
268
259
37
55
91
36
105
64
114
141
155
182
609
527
345
2,424
2,034
1,041
459
282
259
168
38
41
59
59
45
68
86
105
150
136
400
364
332
1,845
1,786
786
477
314
232
159
39
73
32
32
68
82
86
123
109
82
309
259
232
1,487
1,427
641
377
200
114
100
40
41
27
64
39
27
45
64
59
64
168
164
136
918
1,114
523
300
195
109
82
41
14
27
36
27
41
23
59
32
45
109
105
68
586
600
286
173
114
68
55
42
9
14
27
32
45
41
14
27
18
55
50
73
405
386
318
159
91
68
23
43
5
14
18
27
14
18
23
18
14
41
36
36
264
295
132
105
36
64
??
44
14
14
4
14
14
14
14
27
9
14
138
209
82
45
50
18
18
45
18
9
9
5
5
27
18
18
109
91
55
50
14
14
3
46
5
9
5
19
27
18
23
9
5
47
4
4
5
9
22
14
18
14
5
5
4
48
5
5
9
5
4
5
49
5
5
5
15
5
5
4
50
51
4
••
.. *
S
4
9
■•
52
•■
■■
4
4
•Age at
Maximum
20.0
20.1
20.1
21.0
20.2
20.6
20.9
21.2
22.4
22.4
23.0
22.80
23.6
25.1
26.0
27.6
27.4
28.0
Mean Interval
0.0
0.1
0.2
0.3
0.4
0.5
0.5
0.6
0.7
0.8
0.9
1.0
1.5
2.5
3.5
4.5
5.5
6.5
Age at
Marriage
20.0
20.0
19.9
20.7
19.8
20.1
20.4
20.6
21.7
21.6
22.1
21.8
22.1
22.6
22.5
23.1
21.9
21.5
do., smoothed
20.0
20.0
20.0
20.1
20.2
20.3
20.4
20.8
21.6
21.7
21.8
21.9
22.0
22.3
22.7
22.7
22.3
22.0
Frequency at
Max.t(crude)
1,890
2,291
3,437
4,527
5,800
6,980
7,933
6,565
5,065
12,100
9,226
6,700
24,600
6,960
2,620
1,395
803
541
do.,(3mooth'd
1,890
2,290
3,440
4,530
5,800
6,980
7,930
6,560
5,070
12,100
9,230
6,700
72',520
24,600
6,960
2,620
1,400
800
640
Totals
Smoothed . .
16,041
18,452
26,929
33,880
41,156
50,722
63,043
52,472
45,742
111053
86,569
61,789
607848
247676
71,816
29,354
14,908
8,541
5,700
Batio Max.
to Total . .
.1178
.1241
.1277
.1337
.1411
.1377
.1258
,1250
.1108
.1089
.1066
.1084
.0993
.0969
.0893
.0939
.0936
.0947
• Age at beguming of year of maximimi. Add 0.5 year for the median age of the
FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY.
2S3
durations of Mairiage up to 26 Years with Women of Ages 13 to 52 inclusive. Based upon 220,021
1907-1914 inclusive. Unadjusted Numbers.
Interval after Marriaoe durino which a Birth Ooours.
ToTAi Number op
First Births.
AOB OF
7-8
yis.
8-9
yrs.
9-10
yra.
10-11
yrs.
11-12
yrs.
12-13
yrs.
13-14
yrs.
14-15
yrs.
15-16
yrs.
16-17
yrs.
17-18
yrs.
18-19
yrs.
19-20
yrs.
20-21
yrs.
21-22
yrs.
22-23
yrs.
23-24
yrs.
24-25
yrs.
25-26
yrs.
1-26
years.
0-26
years.
Mothers.
14
9
36
82
136
200
232
345
305
391
327
300
323
250
232
195
136
123
77
14
36
36
32
18
5
9
5
4
'i8
59
55
118
145
200
255
864
227
191
191
205
177
164
86
132
59
45
36
23
18
5
14
4
'is
14
65
77
127
127
827
177
191
177
136
150
95
86
123
64
32
41
36
14
19
.5
"9
9
18
18
64
36
82
114
118
805
123
145
105
145
77
50
64
32
27
9
4
14
18
5
27
23
86
45
100
77
132
73
100
77
82
59
32
23
9
9
5
5
4
5
14
18
41
55
32
91
109
100
105
50
77
73
91
27
18
14
4
5
9
14
41
27
68
82
114
45
59
18
45
59
9
9
9
9
5
9
9
41
36
27
41
32
41
32
27
14
I
18
9
4
5
9
23
36
68
41
36
23
23
14
9
18
14
4
9
5
23
18
36
45
9
27
23
9
5
9
14
5
9
9
14
32
64
18
18
18
18
9
4
4
5
14
9
9
27
23
14
23
5
4
"4
5
14
9
9
5
14
9
87
5
4
4
li
14
9
5
4
"5
9
14
14
9
5
••
"4
4
4
5
5
5
14
9
5
5
4
"5
4
"9
5
4
18
236
1,141
3,849
9,035
14,371
20,057
27,906
32,918
34,582
33,600
32,688
30,215
26,703
22,222
19,975
14,860
13,526
10,885
8,836
7,614
6,362
5,034
4,595
3,455
2,710
1,601
1,285
802
537
281
117
74
27
18
9
4
19
124
644
4,327
14,639
34,179
55,589
66,879
85,318
93,627
93,361
86,412
77,834
69,875
59,748
51,487
40,261
35,432
25,577
22,589
17,834
13,948
11,818
9,781
7,458
6,440
4,942
3,628
2,187
1,690
1,066
675
390
136
96
32
33
17
4
4
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
S3
34
35
36
37
38
39
40
41
-42
43
44
45
46
47
48
49
50
51
52
30
7.5
22.5
22.0
391
390
30
8.5
21.5
21.9
264
285
31
9.5
21.5
21.8
227
225
32
10.5
21.5
21.7
205
175
34
11.5
22.5
21.6
132
■ 140
33
12.5
20.5
21.5
109
115
35
13.5
21.5
21.4
114
93
35
14.5
20.5
21.3
41
75
36
15.5
20.5
21.2
68
60
38
16.5
21.5
21.1
45
47
38
17.5
20.5
21.0
64
36
38
18.5
19.5
20.9
27
27
40
19.5
20.5
20.8
14
20
39
20.5
18.5
20.7
14
15
41
21.5
19.5
20.6
14
11
45
22.5
22.5
20.5
14
8
9
20'.4
9
6
?
2'o'.3
5
5
?
20'.2
9
4
•Age at
Maximum
Meanlnt'rv'l
Age at
Marriage
do., sm'tiied
Frequencyat
Maxt(crude)
do.,(sm'tlied
3,872
2,691
1,991
1,486
1,005
920
820
631
620
332
442
332
322
221
264
232
193
137
140
105
101
59
72
56
51
41
37
23
25
9
16
14
9
392,152
1,000,000
1,000,000
Totals
Smoothed
.101
.106
.112
.117
.139
.140
.150
.170
.186
.178
.187
.193
.198
.208
.216
.216
.240
.313
.444
Ratio Max.
to Totals
maximum 12 montlia. t The freijuenoy at tlie maximum is for the age.
254
APPENDIX A.
The detailed results for the successive years shew considerable
regularity in the frequency of fifcrst births even for individual ages, as for
example the births, for ages 23 and 25 during the tenth month and first
year after marriage, were respectively as follows : —
Interval.
Year. 1908. | 1909. j 1910. 1911. | 1912. 1913. j 1914.
1908-1914.
Months
10-11
Number (23)
Number (25)
Corresponcling
Marriages . .
239
195
32,480
232
184
32,704
261 302
237 249
34,127 36,953
328
288
39,815
314
296
42,078
354
308
41,808
290.0»
251.0'
37,138*
Tears
1-2
Number (23)
Number (25)
Conesponding
Marriages
622
559
31,440
685
631
32,510
688 698
604 654
33,163 '35,183
1
860
757
38,037
888
813
40,814
909
820
41,870
764.3*
691.1*
36,145*
* Ayeiage for the period 1908-1914.
The significance of these figures, which are taken at random, is
seen, when the " corresponding marriages" (i.e., the marriages
earlier, by the proper interval, than the year indicated) are taken
into account. The interval in question is about 10 J months in the
one case, and 18 months in the other. Thus for the two upper
numbers the figures adopted for 1908 are those for 1907, plus one-eighth
of the difference between them and those for 1908, and so on ; and for
the lower numbers the mean of the figures for 1906 and 1907 ; and simil-
arly throughout. The ratio of each number to the seventh of the total
shews the degree of correspondence since the whole of these ratios are
relative, and the vertical columns should be identical for exact correspond-
ence . The ratios corresponding to the six lines above are : —
Interval.
Year.
1908.
1909.
1910.
1911.
1912.
1913.
1914.
Number (23)
.82
.80
.90
1.04
1.13
1.08
1.22
Months
10-11
Number (25)
CJorresponding
.78
.73
.94
.99
1.15
1.18
1.23
Marriages
.87
.88
.92
.99
1.07
1.13
1.13
Number (23)
.81
.89
.90
.91
1.13
1.16
1.19
Yeare
1-2
Number (25)
Corresponding
.81
.91
.87
.95
1.09
1.18
1.19
Marriages
.87
.90
.92
.97
1.05
1.13
1.18
Seeing that the original numbers are very limited, the agreement is
remarkably good, and confirms the utility of Table LXXVII., and the
utility of the graphs of the protogenesic surface, to which surface refer-
ence wiU now be made.
FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 255
17. The nuptial protogenesic boundary and agenesic surface.— If the
relative numbers of first-births, after different durations of marriage and
for different ages of women, given on Table LXXVII., are regarded as
vertical (z) ordinates, with the ages of women and duration of marriage
as the other two ordinates {x and y), the surface so defined may be called
the nuptial protogenesic surface or surface of nuptial primiparity. In the
graph of such a surface the area for which the ordinates are zero may be
called the agenesic region, or the surface of absolute steriUty ; and the
boundary between the two may be called the agenesic boundary.
The values of x and y for all points on the boundary between the
agenesic region and the protogenesic surface are the ages and correspond-
ing durations of marriage which define the existence of perfect steriHty.
Thus with a duration of marriage of say 6| years there were no cases of
first-births among women of 19^ years of age in the records extending
from 1908 to 1914 ; see Table LXXVII. or Fig. 72.
The Protogenesic Surface. Australia, 1908-1914.
,£3
I
01 3m e
rtjD. 05 18 Olyr 5 lo
Duration o£ Marriage.
Fig. 72.
15
2Syr5
The contours represent equal frequency of first-births with varying age and
duration of marriage. The area outside the contour is the agenesic region. The
figures on the contours are per million first births, for all women of age x last birth-
day, and for durations of marriage « to * + 1, where t is expressed in months on
the left hand part of the figure and in years on the right hand part.
The characteristic features of the protogenesic surface are shewn
in Figs. 72 and 73. On Fig. 72 this surface is defined, by contours, on
extended lateral scale for to 18 months after marriage, and on a smaller
256
APPENDIX A.
Profiles of the
Frotogenesic Surface.
3 6 9 12ino, 18
2000)
loot
J^D
IDOCO
5000
N
^
«l4Zi
42
18,-32
lateral scale from to 27 years after marriage, and in both cases for the
whole nuptial reproductive period, say 13 to 52 first-births. A vertical
frequency of 1 on the right hand side of Fig. 72 corresponds to the
frequency of 12 on the left hand side. The line
of maxima is shewn by a broken line on the
figure. In Fig. 73 the vertical sections of the
protogenesic surface are shewn for each age from
13 to 27 years (" age last birthday"), and for
the 5-year groups 28-32, 33-37, 38-42, 43-47, and
for the group for all ages from 13 to 52.
The frequencies of first-births, which are
identical on any contour, are indicated by figures.
These are per million total first births for intervals
of a month of duration of marriage on the left-hand
side of the figure, and for intervals of a year's
duration of marriage on the right hand side. The
" age" indicated is always to be taken as the
" age last birthday," or what is the same thing,
and more general, for the age x to the age x-\-\.
It will be seen that these contours constitute
a family of curves for which there is no simple
mathematical specification. The unique maximum
shewn by a small contour hke an " 0" on the left
hand side of the figure and by an asterisk on the
right hand side.
The profiles of the protogenesic surface, shewn
on Fig. 73, from to 18 months, are the curves
shewing the frequency at various ages, for a total
of a million first -births at all ages, and for the first
18 months after marriage. These curves have
characteristic similarities, indicated by the points
letters o, b, c, d, on the figure. The similarities
are important since they shew that there is a
remarkable regularity in the interval between
marriage and first-birth in women of different
ages. The curves drawn are not for instantaneous
group-values, viz., for the groups x to «+ dx, but
are the values for mensual groups, the abscissae
for which are referred to the middle of the month.
100 000
1)000
12 mo.
Duration of Marriage.
Fig. 73.
18. Curve of nuptial protogenesic maxima.
— The curved broken hne on Fig. 72, shewing the
ordinates for greatest frequency of first-birth, can
be replaced by a regular curve, which will give the
actual values of these ordinates with sufficient pre
cision. Adopting as the argument the " age last
FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 257
Age
20
25
30
35
40
Calo. Value
0.0
2.12
6.73
13.23
21.37
Graph Value
0.0
2.2
7.0
13.0
21.5
birthday," that is the initial value of the age where the range is from x to
x+l, and for the corresponding initial value of the duration y, where the
duration meant is from 2/ to «/-fl, we have —
(474) y = 1.45 |i = 1.45 [x - 20)S
^ being the " age last birthday" less 20. This gives the values on the
upper line, while those on the lower are scaled from Fig. 72 : —
(Initial value of
the duration.)
The maximum frequency per million total births, where the age is
" age last birthday," and the duration is from y to y-\-\, cannot be ex-
pressed by any simple mathematical formula. The values, however, are
given at the bottom of Table LXXVII.
19. Ex-nuptial protogenesis. — -The previous issue is not ascertained
in the case of ex-nuptial births, and the point of time to which the interval
corresponding to duration of marriage should be referred cannot be
defined. Hence no comparison can be made with nuptial protogenesis.
20. Average age for quinquennial age-groups of primiparae. — ^The
following table gives the average age of mothers of first-births in quin-
quennial groups : —
TABLE IjXXVIII. — Average Age o£ Mothers, First-births, for Quinanennial Groups.
Age-group
14
15-19
19
20-24
25-29
30-34
35-39
40-44
45-49
50-52
Average Age . .
14.36
18.78
18.77
22.61
27.19
32.06
37.08
41.74
40.31
41.84
Middle Age
14.0
17..50
16.51
22.50
27.50
32.50
37.50
42.50
47.50
46.50
Difference
+ 0.3fi
+ 1.28
+ 2.27
+ 0.11
-0.31
-0.44
-0.42
-0.76
-1.19
-4.66
The differences between the middle and average ages are obviously
too large to be neglected, and therefore it is always necessary to
decide whether the average value or the middle value of the ranges of
the argument (age-group ranges) shall be used. In general the middle
value is the more convenient.
21. Average interval between marriage and a first-birth, a function
of age. — The data furnished in Table LXXVII. shew that the average
interval between marriage and first-births is a definite function of age. *
1 T. A. Coghlan, in his " Child-birth in New South Wales, a study in statistics,"
has given results (see his Table VIII., on p. 26) for the average period from " marriage
to birth of first-child" for " post-nuptial conceptions only." He introduces an ad-
justment for the non-stationary character of the population from which they are
derived, see p. 26. His main result, however, is wholly erroneous, and the true
result is inconsistent with his conclusion, viz., that for married women between the
ages of 17 and 39 the average period between marriage and a first-birth is only 19.4
months, and the range between 18.3 and 21.5 months. The matter will be referred to
more fully later, see pp. 271-2, and particularly the note on the latter page.
258
APPENDIX A.
If age-groups of primiparaB be formed, it is found that the mean ages of
the groups and the average intervals between marriage and first-births
are as shewn in the third colunm of the Table hereunder, viz., LXXIX.,
see also Figs. 74 and 75. The average values of the ages and of the
corresponding intervals are as follows : —
TABLE LXXIX.^AveTage Ages and Average Interval between Marriage and
Fiist-births.
(L) Age of Hairied Women . . Under 20
(U.) Average Age .. .. 18.77
(iii.) Average interval between
Hairiage and Fiist-birth
(crade data) . . 0.623;
(iv.) Average interval by for- |
mula (smoothed data) ; 0.604
(V.) Difference (data-calc.) .. +0.019
20-24
22.61
0.994
0.991
+ 0.003
25-29
27.19
1.483
1.502
—0.019
30-34
35-39
40-44
32.06
37.08
41.74
2.026
2.862
3.501
2.100
2.766
3.420
—0.074
+ 0.096
+0.080
45-49
46.31
4.048
4.209
—0.161
The values on hue (iii.) are fairly well given by the simple formula : —
(475) i = 0.0437 x + 0.01221 ;^i-s
where i is the average interval between marriage and the first-births, and
;^ is 11 years less than the average age, a 5-year group, that is to say, the
age 1 1 is taken as the zero of x ■ This age has not been arbitarily adopted,
but, as is shewn by the line OP on Fig. 74, is indicated as the minimum
age to which reproductive facts should be referred. (See Table LXXII.,
p. 239 and p. 268).
The small crosses in Fig. 75 are the results for individual years of ao^e
last birthday, computed by means of the formula (475) ; see p. 268.
There is a fairly definite indication that the continuation of the
curve should be as shewn by the broken line in Figs. 74 and 75, terminating
therefore at about age 55. There are, however, so few births at ages
greater than 45, that this part of the curve cannot be regarded as yet
weU determined or determinable : see p. 268.
The following Table, LXXX., gives the results in greater detail, and
furnishes also smoothed values of the approximate average interval i
between marriage and first-births for all first-births ^^•ithin a year of
marriage, and for aU ages during the reproductive period. Since formula
(475) refers to the average age, it will not give the quantities in the Table
LXXX :—
!■ The intervals are only approximate. They have been calcvilated by assuming
that the births in each month during the first 12 months may be referred to the middle
of the months, and those during the intervals of from 1-2 years onwards may be re-
ferred to the middle of the year. The change in rapidity of births is so great during
the year after that of marriage that a correction is necessary for rigorous accuracy.
That the difference is appreciable is obvious from the following results :
1st Births
to 9 mths
Approximate average interval, age 22
Average interval more rigorously calculated . .
5.52
5.53
1st Births
to 12 mths
7.S3
7.54
All First-
births.
1 11.70 months
I 10.88 months
The intervals are found more rigorously hereinafter for birtlis occurring not earlier
than nine months after marriage.
FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 259
TABLE LXXX. — ^The Fiotogenesic Indices, according to Age. (Appioximate Average
Intervals between Marriage and First-births)* Australia, 1908-1914.
AVERAOB INIEBVAI.
Births Occurring
within 12 months
Births
Occurring
after
12mths.
Marriage.
All First-births.
Age of
Mother
at Birth.
after
Marriage.
Interval
for Age-
Crude
Smo'th'd
Crude
CrudS
Smoothed
Smoothed
Group.
B^sult.
Result.
Eesult.
Besnlt.
Result.
Result.
Crude
Eesult.
Years.
Months.
Montiis.
Montlis.
Montlis.
Months.
Years.
Months.
10
0.00
0.00
11
0.85
0.071
12
1.72
0.143
13
5.i7
2.88
18.00
8.38?
2.61
0.217
4.03
14
3.39
3.49
3.39
3.53
0.294
15
4.41
4.06
18.00
4.80
4.47
0.373
(7.48)
16
4.49 '
4.60
19.85
5.33
5.44
0.453
17
5.12
5.11
19.15
6.21
6.44
0.537
7.55
18
5.64
5.58
19.54
7.20
7.47
0.623
19
6.07
6.03
19.94
8.32
8.53
0.711
20
6.56
6.44
20.57
9.57
9.62
0.802
21
6.89
6.82
21.19
10.25
10.73
0.896
22
7.53
7.11
21.51
11.70
11.91
0.992
11.93
23
7.77
7.48
22.84
13.08
13.10
1.092
24
8.04
7.77
23.99
14.42
14.33
1.195
25
8.26
8.02
25.29
15.61
15.60
1.300
26
8.36
8.24
26.69
16.94
16.91
1.409
27
8.55
8.43
27.75
18.20
18.26
1.522
17.80
28
8.52
8.58
29,46
19.38
19.65
1.638
29
8.61
8.71
30.95
20.94
21.08
1.757
30
8.75
8.80
33.68
22.81
22.56
1.880
31
8.64
8.86
34.39
23.60
24.08
2.007
25.15
32
8.55
8.89
37.44
25.85
25.65
2.137
33
8.49
8.88
39.41
27.38
27.26
2.272
34
8.46
8.85
42.38
29.96
28.93
2.411
35
8.30
8.78
43.75
31.14
30.64 ,
2.553
36
8.30
8.68
45.85
32.73
32.41
2.700
37
8.44
8.55
47.51
34.82
34.22
2.852
34.34
38
8.45
8.38
52.24
39.70
36.09
3.008
39
7.99
8.19
50.15
37.47
38.02
3.168
40
7.71
7.96
49.74
39.09
40.00
3.333
41
7.62
7.70
55.74
42.82
42.03
3.503
42
7.32
7.41
56.15
44.50
44.13
3.677
42.01
43
7.26
7.08
53.32
41.90
46.28
3.856
44
6.87
6.73
59.69
48.99'
48.50
4.041
45
7.46
6.34
55.77
42.13
50.77
4.231
46
8.25
5.92
?8.92
69.50
53.11t
51.50t
4.426t
4.292t
47
5.90
5.47
64.60
50.55
55.51
50.55
4.626
4.212
48.58 .
48
11.50
4.98
52.00
46.21
57.98
46.22
4.832
3.851
49
7.17
4.47
54.00
33.93
60.52
40.50
5.043
3.375
50
7.50
3.92
66.00
36.75
63.12
30.50
5.260
2.542
51
3.34
18.00
18.00
65.79
18.00
5.483
1.500
28.54
52
9.50
2.73
9.50
68.53
9.50
5.711
0.792
53
71.34
3.00
5.945
0.250
54
74.23
1.20
6.185
0.100
55
77.18
0.00
6.432
0.000
7.491
nonths
29.06 m.
15.95 J
nonths.
24.20
years
27.34 yrs.
25.43 3
^ears.
• Based on a total of 220,021 hirtlis. t These values from ages 46 to 55 are merely
' extensions of the curve. t Tliese values are probably iairly reliable.
260 APPENDIX A.
The yearly groups may with advantage be referred to the " age last
birthday," instead of the middle-age value, which is approximately the
"age last birthday plus J." Let then | denote the " age last birthday,"
less 10 ; the intervals are found to be very accurately given in months
and in years respectively by the following formulae, viz. : —
(476). . . .i' = 0.83641 +0.01062^ +0.000198P, and for months ;i
(476a) ...A" = 0.06971 + 0.000885^ + 0.0000165^, for years : ^
f is of course expressed in years in either case. The values may be readily
computed by taking the interval for age 10 as zero, and the smoothed
results for 20, 30 and 40, and applying formulae (199) to (199c), see
Part v., § 7, p. 69, and remembering that the coefficients b, c, d vary
inversely as the variable, and as the square, and the cube of the variable,
respectively. To develope the table we may calculate the values for
11, 12 and 13 (i.e., for | = 1, 2 and 3), or calling the leading differences for
10 years as Di, D^, and Dg, the leading diflEerences di, d^ and ds can be
found by the formulae^ —
(477) di = O.lDi - 0.045I>2 + 0.02852)3
(477a) ....tfe = .. O.OID2 - 0.009X>3
(4776) ....^3 = •• •• O.OOID3
We have also, for the coefficients of the equations above : —
(478) b = rfi 1^2+ ^ds
(478o) c = .. ^dz— i^ds
(4786) .... d = .. .. ^ds
The agreement between the crude values and the values by formula
(476) for the average interval between marriage and first-birth is remark-
ably close throughout, the curve applying as far as age 45. Beyond this
age the values for the extrapolated curve are given as well as those of
the probable value of the interval.
22. The protogenesic indices. — -The average interval, calculated as
shewn in the preceding section (viz., by formula (484) in the section next
following, § 23, but omitting the correction term e) is not rigorously
1 These formulae are for the " approximate" average iut«rval ; aoe the preceding
note.
» See Text Book. Institute of Actuaries, Part II., Chap. XXIII., Art. 22, p. 443,
Edit. 1902.
FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 261
exact, but is sufficiently approximate to be used as an index of the fre-
quency distribution throughout the interval. We shall call the interval
so calculated the protogenesic index for married women of the age in
question, and for all ages, the general protogenesic index}
Table LXXX. is thus a table of protogenesic indexes rather than a
table of average intervals, though the intervals are approximately correct.
We shall now consider methods of correctly estimating the interval.
23. Exact evaluation of the average interval from a limitefd series of
group-values. — The average interval may be determined with a higher
degree of approximation from the series of group-values for equal ranges
themselves by formulae developed as follows : — Since the group-values
can often^ be reproduced with sufficient accuracy by a rational integral
function we have, in such cases, for the value (a;^) of the interval (the
distance to the centroid vertical) : —
(479) . ,
^xydx ^aa;2+i6a;3-|-eto. ^ bx^+cx^+ ^doi^-{-^ex^-{-.
jydx
=*x-'
ax+ibx^+etc. ^ 12a+Qbx+4cCX^+Mx^+^hx*+
in which last expression we may substitute, by means of formulae (195)
to (197), see Part V., § 5, pp. 67, 68, the values of the groups themselves
for a, b, c, etc. This will give a series of formulae according to the number
of groups taken simultaneously into account. We may take the common
value and the ranges as unity : if it be k the value deduced will then be
multiphed finally by A.
It will be convenient to call the group values A, B, G, etc., hence if
n of these are included, n will be the value of x. That is to say, in formulae
in which D appears, x will be 4. From (479) we thus obtain the following
series of formulae, viz : —
(480) a;^ = l +\- ~f_^B ' *""" ^ ^ ^''
9 A + G
(481) a;™ = li + -g . ^ j^ b + G ' ^°^ ^ ^ ^ '
.482^ X -2+^ -19A-BB + 3G + 19D _
(***^) a:™ -^+45. ji ^ B + G + D '
for a; = 4 ; and
125 -5A-2B+2D+aE .
(^83) '''^ = ^^ + m- A + B+G+D+E :'^''''' = ^
* To fully define the term it should be preceded by the term " nuptial " ; but
for obvious reasons this may be always understood.
* But not invariably : see latter part of Section 24.
262 • APPENDIX A.
If the common range be k, these expressions should of course be
multiplied by that quantity.
From these formulae multiply-infinite series of formulae may be
developed, and such development can be effected by processes similar to
those indicated in Part VI., § 2, and Table VT., pp. 75 to 77.
A practical way of applying the formulae is to calculate by an
approximate method and make the necessary correction, if it be sensible.
Thus :—
_ (^+3^+5g+7Z>+etc.) ^
^ ' '^'" ~ 2{A + B + + D+etc.) + ^
where e is a small quantity. For the value of e, we have, from (480) to
(483) :—
(485) . . €2 = ^ . A I T> ' when there are two quantities only.
I j4 + C
(486) . . es = g-. Aij^ip > when there are three quantities only.
(487). .., = - . ^^s+C + D
there are four quantities only.
when
i±iiSi\ 1 - 49 ^ + 385 - 38Z) + 49^ , ^^
(488).. C6 =288- A + B + C +D + E ' ^^en there
are five quantities only.
i7„ denoting the sum of n successive groups, A, B, etc., these expres-
sions may be put in the arithmetically more convenient form hereunder,
viz. : —
(489).... 62 =0.16(S - A) /Zz; es = 0.125((7 - A) / S^;
ei= { 0.18{D -A)—0.2S{G - B)\ /Si
(490). . . .£5 = I 0.17014 (E~ A)- 0.13194 {D - B)^ / S^
Whenever each group-value in a series is not greater than say 2 to 2J
times an adjoining group-value, the preceding formulae give fairly' good
results, and may be used for a succession of three, four, or five groups in a
way which will now be indicated.
24. Evaluation of group-intervals for an extended number of groups.—
To apply the preceding formulae to a large number of groups it wiU be
convenient to adopt the following notation. Let A, B, 0, etc., be
denoted by Ai, A%, ^3,etc.,andletaIsoa;' = x^ for A]_ to A^ reckoned
from the beginning of A^, x" ^ x^ for say ^^^ito ^™ , etc., reckoned
FERTILITY, FECUNDITY. AND REPRODUCTIVE EFFICIENCY. 263
not from the beginning of A^.^i, but from the beginning of Ai ; and
so on. Let also A', A", etc., denote the totals of the various series of
groups in question ; that is, let A' = Ai -{- Az + etc. ; A" = A^^x
+ ^A+2 + sto- ; 3'nd so on.^ Then the value for the entire series is : —
(4yi)....9a:„- -^' ^ ^"_^ ^»' ^etc. ~ E A'
Consequently, if a; ' =w' -\- e', where w ' is an approximate value of x ' and
e ' is the correction to make it exact, we shall have for the true value of
(492) X - ^^^'""'^ + ^^^'''^
y^^^i o'*'«> — E A' E A'
in which E(A'e') = A'e' + A"e" + etc. Let the factors 1/6, 1/8,
1007/90, 3051/90, 49/288, 38/288, on formula (485) to (488)
be denoted by ai, az, etc., and generioally by a', a", etc. Then,
since when e', e", etc., are multiplied respectively by A', A", etc.,
their denominators disappear, we have, for the total correction e^ say, the
sum of the numerators divided by the sum of all the groups. Thus
a A', a" A", etc., denoting the numerators, we have :^
v*^"*;' o-'^m ~ E A' E A'
that is to say, the approximate value of the average interval, found by
multiplying each group by the middle value of its interval, and dividing
the sum of all the products by the sum of all the groups, merely requires
the correction found by multiplying each group by its correction co-
efficient (a), and dividing by the sum of the whole of the groups. Hence
formulae may be developed to embrace the corrections by multiplying the
individual groups by factors, and these factors are readily found by
summations. Thus we obtain the following, viz. : —
(494).. oa;m =(0.375^1+1.5.42+2.625 43 +3.375^4+ 4-5^6
+ 6.625^8 +eto.) / EA.
the series of coefficients being in threes ; thus the coefficient for the third
term from any term of the series is 3 greater than that of the term from
which it is reckoned. Further, : —
(495). -oXm =(0-31 Ai + 1.73 Az + 2.26 A3 + 3.68 Ai + etc.)
/ EA ; and
(495a). .oCBm = (0.32986.41,+ 1.63194^2 + 2.5^3 + 3.36806^4
+ 4.57014^6 + etc.)/ 27 4;
the series of coefficients being respectively in fours and in fives : thus
the coefficient of the fourth term in the one case, and of the fifth term in
the other, from any term in the series, is 4 greater in the former case and
5 greater in the latter, than the coefficient of the term from which it is
reckoned.
1 It is of course immaterial what nvmiber of groups are combiaed.
264 APPENDIX A.
25. Average interval for curves of the exponential type. — In cases
where A^ is very small (or very large) compared with Ai, the preceding
formulae are not very accurate.^ In general, if the curve giving the groups
be approximately of the tyipe e^™*, and the groups be also very different
in magnitude, it is preferable to proceed as follows : — ^
Let Ai, Az be two adjoining groups ; these can be satisfied by the
equation : —
(496) y = Be'"', or «/ = e» + ''-" ;
in the former of which, therefore, B = e*. Similarly three adjoining
groups, Ai, A2 and A^ may be satisfied by the equation : —
(497) y = A + Be'""
Putting Ai the group for the range to 1 ; A^ the group with the range
1 to 2 ; A^ the group with the range 2 to 3 ; we have from these equations
the following, viz. : — From (496) : —
(498) ^ = i^ = e» ; or 6 = 2.3025851 log^^ ^
and this applies to a whole series of groups if the ratio A^^i / A„ be
constant. Also : —
^*^^> ^ - e& _ 1 - (e6 _ l)e» ^ (e» - 1) e"" -^*"'-
the final equation in (499) being true only if A3/ A^ = A^/ Ai = n, say.
From (497) we have, similarly to (499) : —
(500). .Ai = A+B{e'>-l)/b; A^ = A -\- B (e» - 1) e'>/b ;
A3 = A + B{e'> —1) e^'> /b;
and consequently : —
(501).... (^3 - A^)/{Az -^i)=e»,
or 6=2.3025851 log.iQ{(A3 ~ Az)/{A2. - ^1)}; etc.
1 For example, if there be two groups, on equal bases 0-a;, x-2x, one of which is
three times greater than the other, the straight line (which in such a case would be the
assumed curve, giving areas equal to the groups), would start at the terminal (or 2a!)
of one of the groups. If one is greater than 3 times the other, it will fall within one
of the rectangles. The question has been exhaustively considered by Prof. Karl
Pearson, see Biometrika, Vo. I., pp. 265-303, Vol. II., pp. 1-23.
' As the formulae of this section are of general application x has been used
for the independent, and y for the dependent variable.
FEBTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 265
Writing n for e*, we have also : —
(502)..B=b(Az-Ai)/(n-l)^ = b{A3-2Az+Ai)/{n-l)^,eto.;
and
/^rwov A A B, ,, . Az- Ai Aiu— Ao
(503) A = Ai— -J- (n-1) = Ai r-^ = -^ q-^ ,etc.
n-1 w— 1
Thus the constants b, B and A in (497) are determined.
In applying these formulae to ascertain the average interval, four
bases will require specially to be con^ddered, viz., when the factor b is
positive, and when it is negative, the range being either to 1, or 1 to 2
in both cases. For the ascending and descending branches respectively,
these cases correspond to the curves Be'"' and Be~'"' For the purpose
in view (496) is suitable, and the results, to be tabulated for various
ratios of Ai/ A^ or Az/Ai, will be the groups B jy", B j'^, B^e-^",
and B fe~'"'. The mean interval lies between the centre of the group-
range and the side on which the groups have higher values. For the more
general case, that is when three values are satisfied, we should have to
determine
/xydx _/x(A + Be'"')dx _^Ax^+B{{bx—l)e'"'-{-l}/b^
(504:).. x^= jp^ = y(^_|_5e»«)(^a; = Ax + B (e»* - I) / b
If A, however, be taken as zero, this last will become
(bx — 1) e"^ + 1 xe^" 1
(505) o^m =
6 (e»* - 1) e** — 1 b
which function is the basis of the tabulation hereunder for ratios of Az to
Ai and for ranges of x=l and 2, by applying (498). It may be noted
that the value of (605) = for a; == 0. In the table hereunder, LXXXI.,
the four cases above referred to are as follows : —
Case I.
„ II.
„ III.
,. IV.
^i> -^i*; Origin 0; Bange — k ; Tabular Interval computed from 0.
; „ k—2k; „ „ „ „ *.
^a< Ai; „ ; „ 0— i ; „ „ „ „ 0.
,> ; „ }i—2k; „ „ „ „ *.
These four cases are illustrated by Fig. 78, hereinafter.
The necessary formulae for calculating the required values are simple
it we put .^2 = « -^i, viz. : —
(506). . „<=I+ ^-|; i»^2=2+^j-i; . .^.,x', = p+^-l ;
formulse which are convenient for computing tabular values.
For negative values of 6, in which case Az is less than Ai, it is
arithmetically convenient to use the ratio Ax/ Az= m, so that m=\/n,
and put p = — b, then the preceding formulae become : —
„ , »w . 1 ■ „ „ m 1 „ m , 1
It may be easily verified that p-ia;'j, + p-ia^'j, = 1.
266
APPENDIX A.
By means of the preceding formulae Table LXXXI. has been
computed : it will serve for readily estimating the position of the
centroid vertical for any group by means of the relative magnitudes of the
adjoining groups. The determination of that vertical from the relative
magnitudes of the groups on either side of any group in question gives
results of a fair degree of precision.
To satisfy three groups by means of (497) we have for the value ai A
in terms of Ai to A^, : —
(508) .
A =
AiAs - Al
Ai + A3
2Ao
instead of (503) : hence we can subtract this quantity from the groups and
we then obtain : —
(.509).
A'l = Ai— A; A' 2 = Az
A ; etc. ; etc.
these reduced groups, denoted by accents, conforming to the relation
A's/A'2 = A'2/A\.
The value of the average interval is therefore : —
i(l+3 + ..2p — 1}A + A 1 o*'i+ ^'212^2 + to p terms
(510)..„x'p=
A + ^2 +.
.to p terms.
Results so computed have a high order of precision. If A, and A'l, etc.,
be expressed in ratios to ^1+ etc., as unity, the denominator of course
disappears.
TABLE LXXXI. — Abscissae of the Centroid Verticals of Gronps Bounded by the Carve
Bel": and Be-''^. For the Computation of Average Intervals, etc.
Ratio
Ratio
Ratio
A,/Ai
Case I.
Case in.
A./A,
Case I.
Case ni.
A,/A,
Case I.
Case III.
or
or
or
Ai/A,
Ai/A,
Ai/A,
1.0
.50000
.50000
4.0
.61199
.38801
9
.66988
.33012
1.25
.51857
.48143
4.5
.62085
.37915
10
.67682
.32318
1.5
.53370
.46630
5.0
.62867
.37133
11
.68297
.31703
1.75
.54639
.45361
5.5
.63563
.36437
12
.68848
.31152
2.0
.55731
.44269
6.0
.64189
.35811
13
.69346
.30654
2.25
.56685
.43315
6.5
.64757
.35243
14
.69800
.30200
2.5
.57531
.42469
7.0
.65277
.34723
15
.70216
.29784
2.75
.58290
.41710
7.5
.65754
.34246
20
.71672
.28328
3.0
.58976
.41024
8.0
.66196
.33804
25
.73100
.26900
3.5
.60177
.39823
8.5
.66606
.33394
50
.76479
23521
For case II. add unity to the value lor case I., and lor case IV. add unity to the value lor case in
Applying the various formulae to the results given on the penultimate
line on Table LXXVII. for all first-births, 12 months or more after
marriage, the following results are obtained : —
FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 267
By formula (484), neglecting the correction e, 29.06 months (Index).
„ (494), applied through same range, 28.18 „ (Interval)
(495) „ „ „ ^ 28.00 „
(506) „ • „ „ 27.721 „
By graduating and using monthly values for
the groups up to 48 months 27. 70^ ,, ,,
26. Positions of average intervals for groups of aU first-births. —
The positions of the average intervals (abscissae of the centroid verticals),
computed on the basis of the results shewn on the penultimate line of
Table LXXVII., wiU probably be found approximately true for any popula-
tion. By means of Table LXXXI., they may be readily found.
TABLE LXXXn. — Average Intervals'" in Months for First-biiths, for Various
Ranges of Inteival. Australia, 1908-1914.
Bange
Aver-
Bange
Aver-
Bange
Aver-
Bange
Aver-
Bange
Aver-
Bange
Aver-
ol
age
oi
age
oJ
age
of
age
of
age
of In-
age
Int'rval
Value.
Int'rval
Value.
Int'rval
Value.
Int'rval
Value.
Int'rval
Value.
terval
Value.
mouths.
months.
years.
months.
years.
months.
months.
months.
months.
months.
years.
months.
0- 1
.051
0- 1
7.51
12-13
149.76 .
0-3
1.70
0- 6
3.68
- 1
10.34
1- 2
1.52
1- 2
16.35
13-14
161.69
a- 6
4.65
0- 9
6.41
- 5
17.35
2- 3
2.53
2- 3
28.95
14-15
173.67
6-19
7.40
0-12
7.51
-10
19.60
3- 4
3.62
3- 4
39.22
15-16
185.73
9-12
10.34
years
-15
20.43
4- 5
4.52
4- 5
53.39
16-17
197.74
years
0- 5
13.07
-20
20.72
5- 6
5.52
5- 6
65.53
17-18
209.69
0-1
7.51
0-10
14.63
-25
-26
20.79
6- 7
6.50
6- 7
77.61
18-19
221.68
1- 5
22.36
0-15
16.19
20.80
7- 8
7.49
7- 8
89.63
19-20
233.67
6-10
81.12
0-20
15.38
- 5
22.36
8- 9
8.55
8- 9
101.67
20-21
245.66
10-16
142.96
0-25
15.43
1-10
25.82
9-10
9.53
9-10
113.71
21-22
257.67
15-20
203.05
0-26
15.43
1-15
27.13
10-11
10.62
10-11
125.73
22-23
269.65
20-25
261.40
1-20
27.59
11-12
10.63
11-12
137.77
23-24
24-26
281.58
293.49
25-26
305.43
1-25
1-26
27.71
27.72
• These will be sensibly true for any distribution at all similar to that shewn in Table LXXVII .
and in Table LXXXIII. hereinafter.
The above results have been computed by using graphic graduation^
v^here necessary, by means of the values given in Table LXXXI., and by
formula (416), p. 201. In general the computed values proved to be
sensibly identical. A result intermediate between the extreme values has
always been taken, regard being had to the general probabilities of each
case.
1 These last results are the most accurate ; the value for the month 11-12 is
taken into account in the graduating ; in applying formulte (494) and (495) and
(506) it is not considered.
2 It is impossible in the absence of monthly data to determine the position of
the centroid vertical with great precision. By graphic graduation conforming to
the 11 to 12 months group and to the 1-2, and 2-3 years groups, the result, 16.46
was obtained. By extrapolating the 10-11, 11-12 months group -results, adopting
this extrapolation for the year -group 0-1, and conforming to this fictitious year-
group and the actual year -groups 1-2 and 2-3, the result is 16.25 by formula (510).
Adopting the extrapolated result and the group 1-2 only, gives 15.91 ; while the
exponential curve conforming to the group 1-2 and 2-3 only, gives the result 16.79.
The groups 1-2, 2-3 and 3-4, treated by formulse (508) and (510) give 16.63. After
consideration of all the circumstances I have adopted 16.35 as the result which I
believe to be nearest the correct value. Similarly the results 28.95, 28.93 and 29.11
were obtained for the group 2-3 ; of these the first was adopted.
268
APPENDIX A.
Average Issue and the Frotogeaesic Indices.
Fig. 75.
Durations of Marriage (Interval between marriage and first birth).
iutervaL 0^5 10 15 20 25 30 35_
1 L. y ■■' .--'
1 . -y^ .-"-.i'- .•• .•' Ai
1 _.. ^y\r y ^K /-^y^' t
1 -.,.:.-,.' -,.i-.^-:; ^l::^- '^'v::
1 -.ii./_.S— 4- y ^^- 7j- . .
« -i;:.-.:: -.^^-— .r - ^''-,c2: \-i
a -. _^2^_ _. _,^^^ y / -Ml
1 .J:i_..^:_.,^ L y y 4 ¥
g.^- t\ 1..:/ X iJ'- ^^ ^i '
o3 ._i:'o ^'''' _^^^ ,''■■ ^•'f [. . 5; =
li -1-^.^ -_.: _.j^ _ J : """"^ , ::. ._
f ij.i __,.:_. .,/_V • .J:.._...4
1 Jj-...^:__,.^^.3i____ • ..i^ ^\
i 3.Jo._,^Z_^L_ ,,i^' L_..i
■| 1. .!..:::_..: y-k^
1 ! .L,a._.r: .^^^^^ \___\
2 '-■' - "tx^ --i-
1 __,..a ,.J|?:^ ._!
S 1 ^ .-J^?^" ^
-- . o,^>-^*^' - — \J
< A.ges. 10
15
20 25 30 35 40
Age of Mothers at birth of child.
45
50 55
Fig. 74.
FiG.74. — The lower curve OPQ is the curve of the protogenesio indices (or
approximately computed average intervals between marriages and the first-births)
according to the age of mother at the birth of the first child.
!FiG. 75. — ^The upper series of lines are graphs of the average number of children
bom to aU mothers under 20 years of age, to mothers of from 20 to 24, 25 to 29,
etc., and to mothers of all ages — ^who come xmder observation — according to
duration of marriage. The fine dots give the crude results. The parallel broken
lines indicate that the average raie, of increase is nearly independent of the age
of the mothers, and is dependent on the duration of the marriage.
27. The unprejudiced piotogenesic interval. — The protogenesio
interval gives unequivocally a measure of what may be oalled the modified-
fertility of married women, that is fertihty as modified by physiological
and social conditions, by Malthusianism, etc. It is evident that first
births are likely to give the best available indication of the physiological
element in fertihty ; that is to say, the ratio of cases of nuptial-maternity
at any age to the total number of nuUiparous women, is a better indication
of variations with age of physiological fertihty, than would be the indi-
cation given by later births. But what have been called "prejudiced
cases" should obviously be excluded, viz., cases where maternity, being
expected, leads to marriage. For this reason the interval obtained by
excluding such cases is not only appreciably longer, but also gives a truer
idea of the normal probability of maternity, other things being equal.
Results were published in New South Wales in 1899, purporting to shew
FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 269
that, when prejudiced cases were excluded, the^ " average period from
marriage to the birth of a first child" was, for unprejudiced mothers of
from 17 to 39 years of age, about 19^ months, individual cases ranging
between 18.3 and 21 .9 months. ^ In order to definitely ascertain whether
there was any justification for the statement, the New South Wales
statistics, upon which they were based, were examined and recompiled ;
the data are given in Table LXXXIII. hereunder.
Table LXXXIII. — ^Interval between Marriage and First Births occurring later than
9 Months after Marriage.— New South Wales, 1893-98.
Interval (mths.)
Interval Years.)
^
9
10
11
1
2
3
4
5
6
7
8
9
10
IT
1213
14
15
16
17
18
20
22
t,o
to
to
to
to
to
tn
to
to
to
to
to
to
to
to to
to
to
to
to
to
to
to
10
11
12
2
3
4
5
6
7
8
9
10
11
12
13 14
15
16
17
18
19
21
23
13
14
15
3
1
1
5
16
U
11
7
20
1
54
17
f,9
51
42
132
9
286
18
IS-i
137
lOP
380
37
i:
1
800
19
273
256
159
641
73
17
1
••
1,420
?,n
3!?0
337
209
772
129
36
9
6
1,818
?,1
47(1
425
292
1,026
172
41
,11
1
1
..
2,439
?,?,
Mi
521
365
1,181
210
61
22
4
6
1
2,915
■PA
48S
498
357
1,205
249
79
37
17
7
1
1
2,934
Zi
453
431
265
1,031
245
96
40
13
9
5
1
■•
■•
2,589
9.fi
419
382
246
925
205
85
41
17
10
6
5
2
1
2,344
9fi
34?
294
240
8U1
205
83
41
17
19
4
5
1
1
2,054
9,1
?,43
264
. 185
650
176
74
38
34
15
lU
2
7
2
1,700
9.H
?,3?
185
153
549
142
86
32
27
15
18
4
5
3
1
1
1,453
29
141
145
103
417
122
52
43
13
12
9
6
8
3
2
2
1,078
an
133
131
83
343
124
46
37
32
16
14
6
6
3
5
3
1
983
!t1
68
83
62
248
76
32
20
lU
13
7
4
7
6
5
1
643
W,
5?,
S3
48
209
78
27
12
19
9
9
14
7
8
3
3
4
1
556
»»
48
47
41
142
66
25' 13
8
8
8
6
6
4
3
3
1
2
431
34
33
45
31
117
43
16
13
5
6
1
2
2
4
1
4
4
4
1
1
333
3!>
33
29
25
95
34
16
11
6
2
4
8
4
5
3
4
1
1
281
3A
29
K5
12
90
32
7
12
6
4
1
4
7
3
4
4
I
2
1
1
246
37
1?,
22
9
59
20
8
8
6
2
2
1
2
3
. ,
2
1
1
1
159
38
10
1?,
8
58
13
14
3
7
3
1
3
1
2
1
1
1
138
39
11
8
13
47
15
9
5
1
1
1
2
2
1
1
.. ..
1
118
4Qi
3
5
37
15
6
2
3
1
1
1
2
1
2
..3
1
83
41
2
9.
10
7
3
1
1
1
1
... 1
31
^9.
3
6
3
17
12
3
1
1
2
1
. . . .
. ,
49
43
1
3
2
H
3
2
1
1
1
1
1
. . ' . .
24
44
2
6
1
1
1
1
12
4.'i
1
1
9
1
,.
12
46
3
1
1
34
30
13
15
"5
2
5
2
4,561 4,407
3,075 11229
1
2,515 928 453
256
L58
107
76
70
44
6
6
1'
1
2
27,993
Further, to ascertain whether any material difference existed
between the results for New South Wales for the period 1893-8, and for the
whole of Australia for the period 1908-14, the latter were also computed,
and are shewn in the same table. On Fig. 79 the intervals for successive
ages are shewn by a light zig-zag line, and for the Commonwealth by a
heavy zig-zag line. The two are evidently substantially identical, as the
figures in Table LXXXIV. also shew.
1 See note on page 257, hereinbefore.
270
APPENDIX A.
Table LXXXIV. — Protogenesic Interval or Average Interval elapsing between
Marriage and First-birth, for all First-births occurring not Earlier than 9 months
after Marriage. New South Wales, 1893-8 ; and Australia, 1908-14.
Agel
of '
' INTERVAL.
Age
INTEEVAL.
Age
Interval.
Age
Interval.
of
Mother
of
Mother
of
Mother
Mother
1
last
last
last
last
EBirth-
N.S.W.
lAust.
Birth-
N.S.W.
Aust.
Birth-
N.S.W.
Aust.
Bh1>h-
N.S.W.
Aust.
day.
day.
day.
day.
years.
months.
months.
years.
months, months.
years.
months.
months
years.
months.
months
-13
—
13.83*
23
16.10
16.25
33
28.87
30.70
43
—
49.38
14
—
10.14*
24
16.88
17.20
34
30.10
33.64
44
—
57.35
15
—
12.09*
25
17.30
18.23
35
32.21
35.04
45
—
47.49
16
—
13.65
26
28.28
19.49
36
35.91
36.92
46
—
58.70
17
13.48
12.72
27
19.71
20.54
37
32.34
38.55
47
—
—
18
13.93
13.46
28
20.91
21.92
38
33.00
43.62
48
—
—
19
14.40
14.05
29
22.07
23.39
39
28.47
42.14
49
—
—
20
15.03
14.46
30
24.65
25.42
40
43.88
43.77
50
—
—
21
14.71
14.76
31
25.15
26.28
41
48.57
51
—
22
15.04
15.02
32
30.09
28.94
42
49.83
52
* Depend upon 9, 14, and 68 cases only.
The above table and Pig. 79 indicate that there has been no materia]
change in the interval between marriage and first-birth during the
elapsed 15 years, and also that the average period is not constant but is
a function of the age when tabulated according to "age of mothers," that
is, according to age at maternity. It will be shewn later that when the
TABLE LXXXV.— Approximate Protogenesic Index for
(These results are only approximate, the table being constructed from the data in Table
Ages
of
Number of each Duration of Marriage, the total being 1,000,000,
Mothers
at
Mar-
riage.
0-9
months.
9-12
months.
1-2
yrs.
2-3
yrs.
3-4
yrs.
4-5
yrs.
5-6
yrs.
6-7
yrs.
7-8
yrs.
8-9
yrs.
9-10
yrs.
10-11
yrs.
11-12
yrs.
12-13
yrs.
13-14
yrs.
14-15
yrs.
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
"lO
110
576
3,832
12,040
25,771
37,328
38,259
44,522
37,997
32,933
25,336
20,054
16,057
11,798
9,822
7,049
5,614
4,090
3,564
2,745
2,048
1,768
1,442
943
749
687
450
304
227
151
88
46
5
13
5
4
5
14
50
259
1,458
4,559
9,226
14,249
20,739
27,624
27,510
28,506
24,180
21,130
17,735
14,962
10,990
9,831
6,627
6,499
4,204
3,064
2,436
1,977
1,481
1,096
800
468
282
178
113
50
63
14
9
5
'I
4
4
18
209
1,054
3,417
- 7,794
11,921
16,125
22,225
24,316
24,261
21,988
20,670
18,419
15,315
12,281
10,221
7,549
6,680
5,086
3,918
3,530
2,654
2,054
1,786
1,427
1,114
600
386
295
209
91
27
14
9
5
4
18
73
377
1,054
1,968
2,950
3,995
5,672
6,413
6,954
6,613
6,009
5,508
4,436
4,113
2,936
2,532
2,032
1,712
1,314
1,232
1,041
786
641
523
286
318
132
82
55
18
18
5
9
5
50
155
382
682
1,118
1,859
2,268
2,550
2,482
2,650
2,454
2,263
2,077
1,618
1,300
1,154
800
714
577
459
477
377
300
173
159
105
45
50
23
14
5
9
32
64
227
395
677
904
1,040
1,268
1,373
1,432
1,218
1,182
868
823
650
586
423
423
282
314
200
195
114
91
36
50
14
9
5
4
5
27
36
132
236
377
550
782
800
695
727
732
532
577
441
373
282
268
259
232
114
109
68
68
64
18
14
5
5
4
9
9
23
32
95
191
277
469
427
541
455
477
377
441
395
323
268
259
168
159
100
82
55
23
32
18
5
5
4
14
9
36
82
136
200
232
345
305
391
327
300
323
250
232
195
136
123
77
14
36
36
32
18
5
9
5
4
18
59
55
118
145
200
255
264
227
191
191
205
177
164
86
132
59
45
36
23
18
5
18
14
55
■ 77
127
127
227
177
191
177
136
150
95
86
123
64
32
41
36
14
19
6
•
9
9
»
18
18
64
36
82
114
118
205
123
145
105
145
77
50
64
32
27
27
5
9
4
14
18
5
27
23
86
45
100
77
132
73
ion
77
82
59
32
23
9
9
5
5
4
5
14
18
u
32
91
109
100
105
50
73
91
27
18
14
4
5
9
14
41
27
68
82
114
45
59
18
45
59
9
9
9
I
5
9
9
41
36
27
41
32
41
32
27
14
5
18
Totals
348,437 259,411
247,676
71816
29354
14908
8,541
5,700l 3,872
1
2,691
1,9911,486 1,005 920 631
' 1
332
PBRTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 271
tabulation is according to " age at marriage," there is a great approach to
constancy of the interval, though the distribution according to interval is
very different for different ages.
28. Frotogenesic Index based on age at and duration of marriage. —
The protogenesic indexes as determined in the preceding sections, viz.,
§§ 21, 22, 26 and 27, are based upon the ages at maternity. For certain
purposes, however, they might with advantage be based upon the ages
at marriage, and for exact results the evaluation of the index would of
course require a compilation according to those ages, and cannot be
quite satisfactorily deduced from the results given in Table LXXVII.
A very fair approximation, however, can be obtained by reconstructing
that table (see pp. 252-3), and the simplest fgrm which this reconstruction
can take is to treat the results in columns 1-2, 2-3, etc., years as re-
spectively applicable to " ages at marriage, 1 year, 2 years, etc., earUer
than that in the age-column. Such a compilation will be sufficiently
accurate to disclose the general characteristics of the protogenesic indices
for ages at marriage. This has been done in Table LXXXV. hereunder,
which is self-explanatory when compared with Table LXXVII.
Australia, 1908 to 1914 based on Age at Marriage.
LXXVII. by moving the successive columns upwards, 1, 2, 3, etc., places respectively).
iaoludlng those Born within 9 Months of Marriage.
Protogenesic
Index, or
Proto-
genesic
Quad-
■ ratio
Index.
.(Crude).
Ages
of
15-16
16-17
yrs.
17-18
yrs.
18-19
yrs.
19-20
yrs.
20-21
yrs.
21-22
yrs.
22-23
yrs.
23-24
yrs.
24-25
yrs.
25-26
yrs.
9 mtha. ■
to
26 yrs.
Approximate
Average
Interval.
Moth'r
at
Mar-
riage.
Crude.
Smooth'd
9
4
5
23
36
68
41
36
23
23
14
9
18
14
4
9
5
23
18
36
45
9
27
23
9
5
9
14
5
9
9
14
32
64
18
18
18
18
9
4
4
5
14
9
9
27
23
14
23
5
4
4
5
14
9
9
5
14
9
4
'6
4
5
18
14
9
5
4
'6
5
9
14
14
9
5
4
4
4
5
5
5
14
'6
9
5
5
4
'6
5
4
9
5
12
41
106
601
2,167
7,190
16,424
27,406
39,196
54,645
64,922
65,125
61,525
56,667
50,539
42,252
35,220
27,283
22,621
17,248
14,060
10,728
8,882
7,045
5,526
4,454
3,405
2,427
1,565
863
623
410
181
117
38
18
10
10
8
4
134.0
80.9
66.0
57.4
33.3
29.5
24.8
23.5
22.9
22.3
21.1
21.0
20.8
20.7
21.2
21.2
21.1
21.4
21.3
21.1
21.7
21.2
21.6
21.9
21.4
21.5
21.2
19.8
21.8
19.5
20.7
19.9
20.8
19.5
20.0
14.3
14.3
10.5
14.3
10.5
134.0
88.0
67.0
58.0
34.8
28.5
25.3
23.6
22.4
21.6
21.2
21.0
20.9
20.8
20.9
21.0
21.1
21.2
21.3
21.4
21.4
21.5
21.5
21.6
21.6
21.5
21.4
21.2
21.0
20.7
20.5
20.4
20.0
19.3
18.3
If.O
15.4
13.5
11.3
8.8
6.0
159.0
101.9
82.5
81.8
47.8
44.3
35.8
34.0
32.6
31.6
29.4
29.2
28.0
27'8
28.4
28.5
27.7
28.1
27.9
26.6
27.8
26.7
26.5
26.7
25.9
25.8
25.2
22.9
25.2
22.0
23.9
22.4
24.1
24.6
22.9
14.7
14.7
10.5
14.7
10.5
12
13
14
15
16
17
18
19^-
20
21
22
2i
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
62
332
221
232
137
105
59
56 41
23
9
14
651,563
Totals
272 APPENDIX A.
Much more accurate results would be secured by that reconstitution
of the data, which would be possible if monthly or quarterly graduations
for at least the first 3 years after marriage were used. Such gradua-
tions would have to be both for the horizontal and vertical values, and
when effected, the sub-divided numbers would admit of a new table
being compiled, giving with considerable exactitude the required numbers
of births occurring after various durations of marriage, borne by women
of various ages at marriage (instead of ages at maternity). The general
characteristics of the values determined from such a table will, however,
not differ materially from those in the table pp. 270, 271.
In the final columns of Table LXXXV. are given the crude and
smoothed protogenesic indexes or approximate of protogenesic intervals
according to age, with the argument " age at marriage." These are quite
different in form from those deduced with the argument " ages at
maternity." The values exhibit considerable regularity and require
relatively little smoothing. As might be expected a priori, the interval
decreases rapidly as the age at marriage increases, until the age 20 is
reached, when it is 21 months. It remains sensibly constant tiU age
46, and then rapidly diminishes. It is evident that it must necessarily
have a small value at the end of the child-bearing period.
The protogenesic index, or the protogenesic interval, determined
according to " age at marriage," is perhaps to be preferred to one or the
other based on the "age of mothers" {i.e., age at maternity). The average
" period elapsing between marriage and the birth of the first child of
post-nuptial conception" is evidently not the same for all women marrying
at ages below 40 years, as had been stated,'^ but is a function of age, and
is very nearly constant for a long period, viz., from about 22 to 45 years
of age. The maximum frequency is about age 23.4 or 23.5, but cannot
be very accurately ascertained without a special compilation.
29. Protogenesic quadratic indices and quadratic intervals. — The
fact that the protogenesic indexes or the protogenesic intervals are sensibly
identical through a wide range of ages, notwithstanding the "scatter"
of the distributions varies enormously, necessitates the adoption of a
second and different index, or of a second and different type of " interval."
This wiU of course be of the nature of a higher moment since the higher
the power the greater the influence of the distribution on the product.
It will in most cases be sufficient to employ the second power of the
"duration of marriage," and to use the quadratic index, viz., that
» T. A. Coghlan, " Childbirth in New South Wales," 1899, p. 26, says : " .
but where a marriage proves fertile, as the following table shews, the period elapsing
from marriage to the birth of the first child of post-nuptial conception averages the
same for all women marrying at ages below 40 years. This average period is 19.4
months, ranging between 18.3 and 21.5 months." In the table referred to the
results are grouped under " age of mother," not under " age at marriage," but
the text might suggest that what is implied is " age of mother at marriage" (age of
brides). The table shews that from age at marriage 21 to 45 the average interval
is sensibly constant, and only slightly larger than that deduced by Coghlan if in
his Table VIII. " age at marriage" be substituted for " age of mothers."
FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 273
analogous to the radius of gyration in mechanics. That is, we shall
require the value of G where its square is given by : —
(511) (?2 = /^^ /(^) ^^-
J f {X) dx
When /(a;) is a rational integral function {a -\- bx + etc.), this gives —
(512). ...oGl = T*'+ ^^-^ '"" ,,
^ 12a + 66x + icx^ + 3dx^ + J|- ex* + ....
a formula which is appropriate when the graphed areas extend from the
origin. The values of b, c, etc., can be ascertained from the group-totals,
see, for example, by formulae (195) to (197d), etc., pp. 67, 68. When the
gra,ph-totals are not continuous to the origin, the solution is a matter of
integrating between the same limits in both numerator and denominator.
If the limits be x — ^k to x-\-^k, that is, if the middle of the group-range
be taken as the value x in the formula, then it is easy to shew that
(513). . . . Gl = — ^ -, ^ ^°
a + bx + c{x^ + ^F) + d{x^ + ^Bx) + etc.
Gm being the radius of gyration of the figure standing on the range
referred bo, viz., x ± ^k. This formula can be readily recast into arith-
metically convenient forms.
When the function is a simple exponential one (5e*^), we have : —
n / 2 \ 2
(514). . . .,Gl = ^-— ^ (^1 _ _J + — ; or generally
(5l5)....,G;=^^[p-f^)+l^
in which n = e*. These are also suitable only for the figure starting from
the origin. When the limits of the integral are p and q, we shall have
(516).. ...«.• = '"' (^-|)-'K^ -I) +f.='g--^'+ ^-
w* — «.''
in which last expression s = p — 1/6 and < = g — 1/6. When the values
of the squares of the several " radii of gyration" have been obtained, the
radius of gyration of the whole series of groups is given by : —
(517).. Gl=={AiGi+A2G2 + eto.)/{Ai+A2+eto.)=i:{ AG) /a A
Ai denoting the number in group 1, .^2 in group 2, and so on.
The protogenesic quadratic index is computed in a manner analogous
to that for computing the simple protogenesic index : that is by multiply-
ing the square of the middle value of the successive yearly ranges of
* This may be seen by adding l/b' to the first term, thus making the terms in
brackets perfect squares when multipUed by q and p respectively ; and then
multiplying both numerator and denominator by e- «.
274 APPENDIX A.
duration by the number in the group : that is in formula (491) a;'^,
a/'*, etc., is written instead of x' , cb", etc., a;', etc., here denoting the
durations of marriage.
30. Correction of the protogenesic interval for a population whose
characters are not constant. — When a population is increasing, all other
facts remaining the same, the first-births, after a given duration of marriage
{%), are drawn from a smaller population than are those for any lesser
duration and presumably also from a smaller number of marriages. For
comparative purposes, therefore, they need to be " corrected" so as to
agree with what would be shewn by a constant population. Thus, were
the ratio of first-births to marriages constant, it might very properly be
assumed that the number of first-births to be expected would vary
roughly as the ratio of the total marriages (marriages at all ages) for the
period i years earlier, to the total number for the period being compared.
Thus, if J-i be the total number in the former case, and J the total number
in the latter, the correction to be applied would be^ : —
(518) 1 -t- Ci = J /J.j
a quantity ordinarily greater than unity, i.e.., Cj is ordinarily a positive
factor since populations generally are increasing.
We may, however, envisage the problem more rigorously as follows :
Let M, with suffixes shewing the age, denote the number of mothers
of first-born children, and / the number of women marrying, from which
they were derived. Then in the case of a " constant population," in
which also the relative frequencies of nuptial first-births were constant,
the former number would bear a constant ratio to the latter, for any age
in question ; that is to say, for any age and at any time we should have
M/ J = fi, 3. constant. Actually this ratio, however, is not quite con-
stant, hence, rigorously, the number of nuptial primiparse must be
taken as : —
(519) M^ = ^fifJ^ = J^ .f{x,t)
In short we cannot take the marriages as the basis of the correction, but
we should take what may be called their Malthusian equivalent ; that is
the number of marriages so reduced (or increased) as to be of equal
productive efficiency : thus, ju, J must replace J, and fj, is not a constant .
The character of ^ may not be simple ; it is probably a function also of the
interval elapsing before birth, i.e.,
(520) /x = f {X, i, t).
The form and constants of this function can be ascertained only by
computing jx for differing ages with different intervals and at different
times. Thus, instead of (518) we should write : —
(521) 1 + Ci = fi'J/{iM'.i. J.i)
1 This was pointed out by Sir (then Mr.) T. A. Coghlan, Childbirth in New
South Wales, 1899, p. 26. He used this correction, which, however, would not be
completely satisfactory if the " Malthusian coefficient" were increasing.
FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 275
in which ;u,' denotes : — (a) the value of /x for a given age and interval,
(when J and care to be ascertained for a given age and interval), or (6) : —
its value for the total for all ages and for a given interval, (when J and c are
required for the total of all marriages). For Australia the ratio M/ J is
known only since 1893. During the period 1893-1914 it ranged between
.790 in 1903, and .901 in 1912, for first-births and women of all ages (see
hereinafter). As this average 0.0156 per annum for the 9 years interval
between the years mentioned, it is of the same order as the yearly increase
of population, and in the case cited would increase the correction. It may
fall or rise 0.03 in one year. This term may be negleeted, however,
because its effect is relatively negligible when the correction is large, so
that it has very little influence on the result computed by ignoring it.
This is shewn by the results in the following table : —
TABLE LXXXVI. — -Correction to the Computed Average Inteival between Maiiiage
and First-biith when Population is Increasing.
Factors to be multiplied into the
When the increase per unit per annum is,
computed average interval be tween
marriage and first-birth when the
correction for increase is ignored
0.010
0.015
0.020
0.025 0.030
See (511) to (514).
Multiply
the compi
ited interi
tbX by the factor : —
(a) When the first-births after 12
months are taken into account
1.0195
1.0294
1.0395
1.0500
1.0604
(6) When the first-births after 9
months are taken into accoimt
1.0132
1.0199
1.0267
1.0338
1.0408
(c) When all first-births are taken
-
into account . .
1.0083
1.0125
1.0168
1.0213
1.0257
It is to be remembered that the epoch to which the results refer is
(sensibly) the middle of the year of observation, and that the intervals
are 0, 1, 2, etc., years.
Since the relative numbers for different intervals will probably differ
from those of Australia but slightly for most countries, we obtain the
following very simple rules : — (i) If the ratio of first-births to marriages
increase continually at the rates indicated in Table LXXXVI., or
(ii.) if that ratio be constant, and the number of marriages increase con-
tinually at the rates in the table, or (iii.) if the sum of the ratios in ques-
tion be as indicated in the table, then —
The correction to the interval for all
first-births occurring more than twelve
months after marriage is
For all first-births occurring more than
nine months after marriage the correc-
tion is . .
For all first-births occurring
marriage, the correction is
after
in which r denotes the rate of increase.
Twice the rate of increase.
1 + 2r
The rate of increase plus
one-third.
1 +llr
The rate of increase less
one-sixth.
1 + %r
276
APPENDIX A.
31. Proportion of births occurrii^ up to any point of time after
marriage. — The rate of occurrence of first-births, for different intervals
after marriage, is well shewn by giving the proportion of the whole which
have occurred up to any given time. The following table furnishes the
proportions in question : —
TABLE LXXXVU. — Shewing Fiopoition of Nuptial First-births occurring up to any
point of time after Marriage.
Up
TO EHD OF MONTH.
AGE OF
MOTHEBS.
1
2
3
4
5
6
7
8
9
10
11
12
15
.0699
.1832
.3525
.4581
.5994
.71891 .8245
.8804
.8944
.9363
.9720
.9798
20
.0262
.0597
.1071
.1685
.2445
.3341' .4398
.5234
.6721
.6641
.7356
.7851
25
.0099
.0204
.0367
.0569
.0810
.1176 .1659
.207«
.2577
.3932
.4958
.5683
30
.0070
.0140
.0229
.0362
.0493
.0685 .0945
.124C
.1631
.2761
.3677
.4346
35
.0080
.0156
.0284
.0395
.0562
.0723; .0958
.1216
.1485
.2338
.3083
.3577
40
.0120
.0201
.0324
.0466
.0606
.0748! .0977
.1161
.1340
.1884
.2376
.2781
45
.0128
.0281
.0510
.0536
.0714
.0714 .0867
.102C
.1122
.1658
.1888
.2194
13-52 . .
.0160
.0345
.0614
.0953
.1365
.1872 .2502
.3027
.3484
.4595
.5461
.6078
13-52
i j
1
1
Proportion ol
first year's
1
'
births dur-
ing month
.0264 .0304, .0443
.0557
.0677
.0834 .10371 .0863
.0753
.1827: .1424
.1017
Proportion of
'
t
first year's
'
births up
to end of
i
months . .
.0264 .0.)68 .1011
.1568
.2245
.3079 .4116 .4979
.57321 .7559 .8983
1.0000
V
P TO E
ND OF YEAK.
AGE OF
MOTHBE.S.
1 i 1
2 3 4
5
6
10
15
20
26
15
1.0000 '..
(7.
20
.9634 : .9928 .9985
.999.
> ' .9999
1.0000
(I!)
25
.8608 , .9402 .9729
.986.
S .9933
.9996
1.0000
(i8)
30
.7278. .8458 , .9043
.936'
' 1 .9565
.9928
.9998
1.0000
(22)
35
.6455 .7631 , .8223
.86K
i .8869
.9562
.9932
.9997
1.0000
40
.5704 .7052 ' .7842
.8314
I 1 .8585 ! .9202
.9676
.9916
(=5)
1.0000
45
.4974 .6276 .7270
.790f
; .8316 .9107
.9541
.9745
1.0000
13-52 . .
.8555 ' .9273 .9567
.97ie
.9801 ' .9943
.9988
.9998
1.0000
13-52
Proportion of
first year's
births dur-
ing mont'^i
1
Proportion of
i
first year's
1
1
births up
to end of
1
month
1.4075 1.5254 1 1.5739 '
1.5984
1 1.6125 1 1.6359
1.6431
1.6448 1.6451
1
This table is interpreted as follows : — Taking the upper line, 13-52,
0.0160 of all nuptial first-births occur within one month of marriage,
0.3484 occur before the end of the ninth month after marriage, and
0.6078 before the end of the twelfth month. Again, of the nuptial first-
births occurring, with women of all ages, during the year of marriage,
0.5732 are bom before the end of nine months, and all births exceed
those bom during the first twelve months by only 0.6451. This is shewn
on the last line of the table.
32. Range of the gestation period. — In order to accurately estimate
the cases of first-births properly attributable to pre-nuptial insemination,
the range of the normal gestation-period must be taken into account as
FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 277
well as the frequency of premature live births. Contrary to popular
opinion this gestation-period has a considerable range .^ The following
data represent the best available results : —
TABLE LXXXVIII.— Relative Freqiuency of Births after Different Periods, between
the last Menstruation and Parturition.
Authorities.
Duration
Days.
Reid.*
Hannes.f
Hannes.J
Hannes.
Various!
Reid, with
500 Cases.
561 Cases.
314 Cases.
875 Cases.
51 Cases.
Hannes.
241-250
56
36
16
28
1
41
251-160
59
37
13
29
20
44
251-170
150
141
111
130
210
140
^71-280
317
325
366
340
510
329
Maximum
(days)
(277.77)
(277.73)
(277.02)
(277.42)
(274.64)
(277.58)
281-290
269
271
258
267
160
268
291-200
97
121
118
120
100
109
301-310
24
50
76
59
?
41
311-320
18
14
22
17
?
18
321-330
10
5
19
10
?
10
Total
Average!
Duration
1,000
1,000
1,000
1,000
1,000
1,000
277.2
279.2
281.9
280.3
276.5
278.8
Note. — ^The oases for 241 to 251, 316-330, have been obtained by extrapolating Eeid's curve.
* See Hart, Edinburgh Medical Journal, 1914, New Ser. XII., p. 401 ; also
Journ. Edin. Obstetr. Soc, XXXVIII., pp. 107-134; 1912-3. Biometric analysis
of some insemination-labour and menstrual-labour curves in certain mammalia.
The distribution of Eeid's results according to the normal curve of probability for
a table of frequency is unquestionably unsatisfactory, as an examination of the
original data will shew. The distribution does not conform to the normal curve.
The average is given as 278.3 ; it should be 278.84 ; there is an arithmetical mistake
in the original calculation.
t Zeit. f. Geburt und Gynak. LXXI., 1912, p. 524. Die korperliche En-
twioklung der Frucht in ihrer Beziehung zur bereohneten Schwangerschaftsdauer.
Walther Hannes. Children 3000 to 4000 grammes weight.
X Same authority, children above 5000 grammes weight.
§ Interval reckoned from coitus, certain. These i .elude 51 cases reported by
Desormeaux, Girdwood, Montgomery, Rigby, Lockwood, Lee, Dewers,' Beatty
Skey, Mcllvain, Ashwell, Clay and Reid.
The average durations indicated are not exactly identical with
the maximum frequency, since the frequency curves are very sHghtly
asymmetric.
If Hannes' cases are combined with Reid's, a total of nearly 1400
is obtained. If the result be " smoothed," so as to agree with the final
column of Table LXXXVIII., the result shewn in Table LXXXIX.
on next page is obtained.^
1 Other values are as follows : — Hippocrates, repl dxTa/i-^vov, generally within
280 days ; Hansen, 128 cases, 272.5 days after coitus ; see Handbuoh der
Physiologie by Hermann, VI., 2., p. 73, 1881 ; M. Zbllner, after menstr., &st-births
279.1, second births 282.0 ; see Zur Kenntniss und Berechnung der Schwanger-
schaftsdauer, Jenenser Dissertation, 1885, p. 6. Hasler, 195 cases, 281.0 ; after
coitus 665 cases, 272 days ; Glusing, after menstr., 279.6 ; Wiirzburger Dissertation,
1888, p. 15 ; Voituriez, 274-8 after menstr. Thgse de Paris (Lille), 1885, p. 62 ;
Winckel, 274.8, Lehrbuoh d. Geburtshiilfe, p. 78, 1889 ; Ahlfeld, 270.4 after coitus,
Monatsohr. f. Geburtskr u. Frauenkr., XXXIV., p. 304. 1869.
278
APPENDIX A.
TABLE LXXXE.— Shewing the Frequency per diem per 100,000 Births occurring-
between the 240th and 332nd day after the Termination of the Menstrual Period.
i
Batioot
Ratio of
Eatio of
Batioof
Eatio ol
Day
No.
Aggre-
gate.
Day
No.
Aggre-
gate.
Day
No.
Aggre-
gate.
Day No.
Aggre-
gate.
Day
No.
Aggre-
gate.
240
297
.00297
260
675
.09012
280
3,429
.56930
300 657
.93538
320
Ill
.99424
241
303
.00600
261
733
.09745
281
3,318
.60248
301 597
.94135
321
101
.99525
242
310
.00910
262
807
.10552
282
3,196
.63444
302 546
.94681
322
91
.99616
243
318
.01228
263
911
.11463
283
3,014
.66458
303
496
.95177
323
81
.99697
244
327
.01555
264
1,052
.12515
284
2,847
.69305
304
455
.95632
324
71
.99768
245 1 338
.01893
265
1,305
.13820
285
2,676
.71981
305
420
.96052
325
61
.99829
246 349
.02242
266
1,548
.15368
286
2,504
.74485
306
389
.96441
326
51
.99880
247 : 361
.02603
267
1,784
.17152
287
2,332
.76817
307
361
.96802
327
40
.99920
248 1 374
.02977
268
2,015
.19167
288
2,160
.78977
308
334
.97136
328
30
.99950
249 ! 388
.03365
269
2,246
.21413
289
1,988
.80965
309
304
.97440
329
20
.99970
250
404
.03769
270
2,470
.23883
290
1,816
.82781
310
277
.97717
330
15
.99985
251
420
.04189
271
2,689
.26572
291
1,644
.84425
311
252
.97969
331
10
.99995
252
437
.04626
272
2,913
.29485
292
1,477
.85902
312
227
.98196
332
5
1.00000
253
455
.05081
273
3,132
.32617
293
1,320
.87222
313
207
.98403
333
254
474
.05555
274
3,420
.36037
294
1,189
.88411
314
188
.98591
240
255
496
.06051
275
3,455
.39492
295
1,077
.89488
315
171
.98762
to
looiooo
256
521
.06572
276
3,501
.42993
296
976
.90464
316
156
.98918
333
257
551
.07123
277
3,511
.40564
297
885
.91349
317
143
.99061
258 ; 587
.07710
278
3,506
.50010
298
804
.92153
318
HI
.99192
259 : 627
.08337
279
3,491
.53501
299
728
.92881
319
.99313
Maximum frequency occurs on tlie 277.67th day. Average (240 to 332 days) = 279.28 days.*
• If tlie average date be found in the usual way (t.«., from the weighted mean), it will prove
to be 278.78. But the births occurring on the nth day range between n and n + I, hence the
average is about n + i, consequently the 278.78th day Is from 278.78 to 279.78 ; hence the
average interval U> 279.28 about.
It would appear from these results that the most frequent interval
between the termination of menstruation and parturition, and the average
interval, may be regarded for practical purposes as identical, and may be
taken as 278 days on the average for births of children of ordinary weight,
and that only 2 or 3 days need to be added in the case of the birth of
heavier children. For first-births the interval is about 3 days shorter.
From insemination to parturition the interval is slightly shorter, perhaps 5
or 6 days on the average. In view of social custom, however, the interval for
first-births may be taken as say about 14 days longer than the 278, or
about 292 days in all. Making allowance for live births occurring after
210 days from insemination, and for the fact that 40 per cent, of births
occur between the 261st and 278th day from the last menstruation,
(see Table LXXXIX.), we may take 274 days, or 9 months, as the period
to be rejected as uncertain as regards post-nuptial conception.
The frequency-curve for the interval between the termination of
menstruation and parturition is curve E on Fig. 76, see later, page 284.
33. Piopoition of births attributable to pre-nuptial insemination. —
It is evident, from the preceding table, that there is a certain period during
which it is not possible to ascertain what proportion of births should be
regarded as attributable to pre-nuptial insemination.^ The numbers
^ T. A. Coghlan in 1899 based his computations on the assumption of a 9-
months iaterval, see Childbirth in New South Wales. He points out that ia the years
1893-8, the nuptial first-births registered were 41,384, of which 13,366, or 32.3 per
cent., were " due to pre-nuptial conception." It may be observed that pre-nuptial
insemination may have characterised some cases where birth occurred in the tenth
or even eleventh month after marriage, and a small nvimber of births may be attribut-
able to cases of post-nuptial insemination from 200 to 240 days after marriage, and a
considerable number from 240 to 270 days. However, the jjercentage he deduced
for New South Wales in 1893-8 seems, on the whole, to be confirmed by the present
investigation for Australia, 1908-14.
FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 279
per million nuptial first-births for women of all ages bom during various
intervals after marriage are shewn on the penultimate line of Table
LXXVII.
By plotting the groups of first-births occurring monthly from 1 to
12 months, and drawing a continuous curve giving the same totals, re-
sults are obtained analogous to those shewn on Pigs. 76 and 77. On the
former figure the part of curve A, marked f, g, g', h, denotes the boundary
of the groups, which may be attributed to pre-nuptial insemination.
The curve i, i', j shews the boundary of the groups which may be at-
tributed to post-nuptial insemination. On Fig. 77 the curve k, 1, 1 ', m,
denotes the pre-nuptial insemination quota, and the curve n, n', o, p,
the post-nuptial quota ; see page 284.
By fixing the position of that part of the curve shewn by the dotted
lines in the figures referred to, it would appear that about 0.634 of the
births occurring during the 9th month after marriage are to be attributed
to pre-nuptial insemination. Thus, about 0.952 of the first-births occur-
ing within 9 months of marriage are due to pre-nuptial insemination.
This is equal to 0.546 of all first-births occurring during the year of
marriage, and 0.332 of all first-births, in every case for women of all ages.
These ratios, it will be seen from Fig. 73, are a fairly definite function of
the age of the mothers ; and this function could be ascertained by treating
the group-results given in Table LXXVII. in the manner above described.^
34. Issue according to age and duration of marriage. — The recording
of the number of children borne by married women of various ages, and
after various durations of marriage, furnish data of value in any attempt
to ascertain the law of increase " according to age and duration of mar-
riage." But it is to be kept in view that the immediate results from
such data apply only to those who thus, through maternity, come under
observation, and does not aipply to married women generally. That is to
say, if averages be formed these averages are not averages for all married
women of the given ages and durations of marriage. During the seven
years, 1908-1914, 805,015 mothers came under observation in Australia,
their total issue being 2,675,291, or an average of 3.3233 each. The results
are shewn in Table XC. hereunder, the averages being found as follows : —
Let jm"a, denote the mothers of age-group x — k/2 to a; -|- k/2, and of
duration of marriage i — 1 to i, and let the total issue of these be iCx',
then the average, ^Ca, is given by : —
(522) ca, = i,Gx / im^'x
and these are the averages which have been tabulated.^
1 The attributing of the whole of the births occurring during the 9 months
after marriage to pre-nuptial insemination, gives a, result somewhat too great.
Nevertheless it is clear that for practical purposes it is a satisfactory rule for eliminat-
ing the so-oalled " prejudiced" from the " unprejudiced" oases, to assume that, on
the average, births occurring less than 9 months after marriage are " prejudiced."
' The original data will be found in the Population and Vital Statistics of
AustraUa for the years 1908-1914, Bulletins 14, 20, 25, 29, 30, 31 and 32.
280
APPENDIX A.
TABLE XC. — Shewing the Average Number of Children Bom to those who Bear
during Varying Intervals after Marriage, based upon the Experience of Australia
during the Years 1908-1914.
Dura-
Age-groups. (Age at Birth of Last Child.)
ation
of
-19.
20-24.'
25-20.
30-34.
35-39.
40-44.
4.5- 1 AU Ages.
1
:
Totals, All Ages.'
Mar-
riage.
Jlothers.
Issue.
Years.
VVEBAGE NUMBER OF CHILDREN.
n-1 1.006 1.010
1.016
1.030
1.051
1.029
1.142
1.013
134,171
135,996
1-2 j 1.250; 1.157
1.085
1.087
1.089
1.113
1.151
1.125
61,213
68,906
2-3 1 1.9251 1.882
1.747
1.700
1.627
1.454
1.545
1.802
64,229
115,759
3-4 2.145' 2.171
2.087
2.039
1.997
1.923
1.786
2.107
70,317
148,160
4-5 2.4661 2.622
2.520
2.441
2.401
2.207
2.041
2.525
59,407
150,009
5-6 2.701 3.020
2.919
2.825
2.803
2.870
2.153
2.906
53,275
154,836
6-7 •2.750 3.401
3.339
- 3.194
3.216
3.038
3.000
3.290
47,250
155,476
7-8
3.000 3.776
3.731
3.576
3.544
3.447
2.846
3.655
41.713
152,461
8-9
. . 1 4.105
4.126
3.954
3.883
3.820
3.142
4.018
37,115
149,129
9-10
. . ' 4.292
4.514
4.330
4.271
4.149
3.940
4.374
32,170
140,725
10-11 .. '4.347
4.910
4.705
4.600
4.619
4.318
4.726
29,607
139,942
11-12 1 4.950
5.256
5.122
4.965
4.954
4.931
5.091
25,887
131,795
12-13
4.571
5.541
5.513
5.329
5.319
5.037
5.443
23.372
127,226
13-14 i
5.790
5.868
5.725
5.608
5.761
5.718
20,339
117,691
14-15 } . .
6.131
6.269
6.091
6.056
5.721
6.156
17,572
108,160
15-16 ■
•6.24
7.434
6.453
6.324
6.493
6.494
15.217
98,827
16-17 i . .
5.59
6.967
6.859
6.688
6.844
6.844
13,271
90,836
17-18 j
;;
5.16
7.239
7.401
6.985
7.282
7.193
11,617
83,539
18-19
5.00
•7.371
7.679
7.431
7.291
7.575
10,073
76,308
19-20
••
7.480
8.018
7.865
7.775
7.926
8,520
67,530
20-21
7.111
8.418
8.282
8.168
8.329
7,424
61,839
21-22
6.192
8.824
8.750
8.449
8.751
5,988
52,403
22-23
5.60
9.154
9.230
8.962
9.191
4,726
43,437
23-24
9.609
9.503
9.171
9.483
3,561
33,770
24-25
16.00
9.265
9.973
9.700
9.884
2,664
26,330
25-26
•9.053
10.450
10.500
•9.932
1,809
17,967
26-27
9.105
10.730
10.773
10.16
1,146
11,637
27-28
7.000
10.860
11.150
10.54
643
6,781
28-29
11.260
11.480
10.71
383
4,102
29-30
11.210
11,840
10.75
192
2,064
30-31
12.00»
•12.220
12.51
77
963
31-32
13.00
11.770
9.51
45
428
32-33
10.00
12.460
12.94
17
220
33-34
14.80
7.80
5
39
AllTJura-
tions
1.202 1.760
2.643
3.837
5.341
6.997
8.565
3.3233
Totals all
dur'tions
i
Mothers 29,371 185,694
239,066
181,191
118,310
46,705
4,678
805,015
805,015
Issue . . ' 35,292 326,868
i
631,954
695,220
626,641
326,095
40,181
2,675,291
2,675,291
Owing to the limited data, the values are not reliable for the age-group 45, nor for the values
shewn by the asterisks and those for greater durations of marriage.
The table shews that, for all ages, the average total issue of married
women, with various durations of marriage, who each year appear in the
Australian maternity records, increases approximately at the rate of one
child in 2.745 years, or 0.3643 'of a child per annum. The results are
graphed in Fig. 75, p. 268. The parallel dotted lines in the figure shew
that the rate of increase of the total issiie according to the duration of marriage
is identical for all ages, at least for the greater part of the range of duration.
That the graphs approximate so closely to straight hnes, and, moreover,
to parallel straight hnes, is remarkable.^ These hnes may be defined by
equations : —
(523) c"x =aa:+ bi = 0.6667 -|- 0.3643i, approximately ;
1 This characteristic can no doubt be deduced, but no explanation of an
elementary nature can be offered.
FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 281
in which only ax is dependent on the age of the mothers, being about
|, and b is constant for all ages. The more exact values of a are given in
Table XCI. hereinafter.
The results ^hewn in Fig. 75, p. 268, and detailed in the table referred
to, can be referred in a general way also to the age-groups, that is to say, if
yx denote the average issue for mothers of a given age-group for all dura-
tions of marriage, then the number is as shewn in Table XCI. The average
ages for these age-groups, as shewn in the table, are found on the sup-
position that the distribution of the cases of nuptial maternity occurring
during the period 1907-1914, in Australia, apply. This distribution is
given in Table LXXIII., p. 242, and the average ages of each age-group
have been calculated strictly^ : these are as given hereunder.
TABLE XCI. — Shewing the Total Issue foi Mothers in various Age-groups, for All
Durations of Marriage ; the Constants of Formulae for Computing this Number,
and the Differences between the Observed and Computed Numbers. Australia,
1908-1914.
Age-group
Average age . .
-19
18.92
20-24
22.87
25-29
27.46
30-34
32.35
35-39
37.29
40-44
41.91
45-
46.29
13-52
Average number of
children, all dura-
tions of marriage
Smoothed resultt
1.202
1.242
1.760
1.751
2.643
2.636
3.837
3.895
5.341
5.413
6.997
6.994
8.565
8.764
3.3233
The above
crude and
smoothed (Crude)
results are
equivalent
to dura- (Smooth-
tions for ed)
all ages of:
1.37
1.48
2.90
2.88
5.33
5.31
8.60
8.76
12.73
12.93
17.28
17.27
21.58
22.13
Crude
Smoothed
Values of Ax for age-
group
Value of 6
.6515
.3643
.7909
.3643
.7778
.3643
.6921
.3643
.6646
.3643
.5977
.3643
.4939
.3643
.7029
.3643
Calculated
Values of
. Ax+bu
when u =
and the
value of ft
g 1
1 2
■a 3
^4
^ 5
1.016— .010
1.380— .130
1. 744+. 181
2.109 + .036
2.473— .007
e
1.155— .145
1.520— .363
1.884— .002
2.258— .087
2.622— .000
1.052— .036
1.416— .331
1.781— .034
2.155—068
2.519 + .001
1.056— ,026
1.421— .334
1.795— .095
2.159—120
2.524— .083
e
1.029 + .021
1.393- .304
1.758— .131
2.122— .125
2.486—085
0.962 + .067
1.326- .213
1.691— .237
2.055— .132
2.419^.212
0.858 + .284
1.223— .072
1.587— .042
1.951— .165
2.315— .274
1.067— .054
1.432— .307
1.796 + .006
2.160— .053
2.524+. 001
t The smoothed result conforms to a rational integral equation of the fourth degree.
i e is the quantity which, added to the tabular value (calculated), makes it identical with the data.
The smoothed results for the average number of children, according
to age, for all durations of marriage, are given by : —
(524) yx = l +bx+cx^ + dx'' + ex* ;
in which x =* — 13, and the values of which for 2 J years' intervals are
as follow : —
TABLE XCII.— Shewing the Effect of "Age of Mothers " upon the Total Issue for All
Durations of Marriage. Australia, 1908-1914.
Ages at bkth of last
child, in years . .
Children* ..
13
1.000
15.5
1.019
18
1.160
20.5
1.413
23
1.770
25.5
2.221
28
2.760
30.5
3.378
33
4.070
35.5
4.829
38
5.650
40.5
6.528
43
7.460
45.5
8.441
48
9.470
Difference for 2i yrs
0.019
0.141
0.S53
0.357
0.461
0.539
0.818
0.692
0.759
0.8S1
0.878
0.93S
0.981
1.029
• That these are given by a curve of the fourth degree, can be readily seen by taking the values
for 13, 18, 23, etc.
^ That is, the numbers are referred to the exact average for the year of age, not
merely to the age for the middle point.
282 APPENDIX A
la the above table the differences for 2^ years shew that for all
durations of marriage, differences of age have much less infliience than
differences in duration. To obtain this relationship exactly, it is necessary
to compile for each age, and for given durations of marriage the total
issue. For all age-groups the general result is 0.3643 a child per year,
that is 0.9107 for 2| years. Prom the above table, however, it would
appear that this value is not attained for " all durations of marriage"
until, almost exactly, age 40.
Such results as are referred to, are dependent upon the combination
of two things, viz. : — (a) The age-effect proper, and (6) the fact that for
the higher ages the average of the durations of marriage are greater, and
thus, throughout the range of observation, the conditions are not homo-
geneous.
35. Initial and terminal non-linear character of the average issue
according to duration of marriage. — An inspection of Fig. 75, p. 268, and
the results given in the preceding table, shew that there is a more or less
systematic departure from hnearity at the terminals of the graphs repre-
senting " issue according to duration of marriage." The table reveals the
fact that the character of the differences, according to age, and for various
durations of marriage, between the values according to formula (523),
and the individual results are as follow : —
(i.) For the first year of duration of marriage, the computed
total issue for ages under 35 is too great, and for ages over
35 is too small,
(ii.) For the second year of duration of marriage, the computed
total issue is invariably too great, the maximum difference
being at about age 24.
(iii.) For the third year of the duration of marriage, the computed
issue is less than the actual for the younger ages, but soon
becomes greater, the maximum difference occurring at
about the age 43 or 44.
(iv.) The same remarks apply to the fourth year of the duration of
marriage with the exception that the age is later than 45.
(v.) In the fifth year of the duration of marriage, the differences
are small until the age of 40 is reached, when the computed
result becomes markedly greater than the actual.
The relatively large differences for the various age-groups character-
ising the second year of the duration of marriage are due to the fact that
the length of the period, which must necessarily intervene between a
first and second birth, does not admit of so wide a " scatter" of the cases of
maternity as to make the result uniform ; thus the average for the second
year is in defect. This consequence is one which will (and does) tend to
vanish for longer durations of marriage, owing to the fact that any want of
coincidence of the intervals between birth and birth must more markedly
characterise the points of time in proportion to their remoteness from the
first year of duration of marriage. Owing to the fact that the period of
FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 283
gestation alone is three-fourths of a year, and the period of lactation a
considerable part of a year, and to the fact that so great a proportion of
births appear in this year, it follows that the second year of duration
must necessarily disclose a falling off in the apparent average. As time
goes on, however, this apparent defect will tend to disappear, as will be
clearly seen by a reference to Fig. 75, p. 268.
The character of the curves at their terminals for the longer durations
may be fairly well ascertained by combining the terminal values. This
has been effected as follows : — In the series shewn on Fig. 75 the two
differences between the three last averages of the issue of curve for under
20, are taken, and similarly the four differences between the five last
averages of the issue, etc., the number of values (averages of issue) being
respectively 3, 5, 8, 9, 10, 12, and 12. The means of the differences,
the numbers of which are respectively 7, 7, 6, 6, 5, 5, 5, 4, 2, 2, are taken,
the results being as follows : 1-0.230, +0.285 + 0.582, +0.106, + 0.153,
—0.105, — 0.162, + 0.060, —0.246, +0.845, —0.489. The accumulated
results compared with the successive multiples of 0.3643 furnish the
co-ordinates of the average terminal shape. This gives : —
.364,
.230
.729
.515
1.093
1.097
1.457
1.203
1.822
1.356
2.186
1.251
2.S50
1.089
2.914
1.149
3.279
.903
3.643
1.748
4.007
1.259
Diff.
Smth'd
.134
.025
.214
.100
— .004
.225
.254
.400
.466
.625
.935
.900
1.461
1.225
1.765
1.600
2.376
2.025
1.895
2.500
2.748
3.025
The differences shew the amounts by which the successive points fall
short of the line defined by the formula (523) . As is shewn by the smooth-
ed values, the defect from the linear condition, once it initiates, increases,
on the average, very approximately as the square of the duration from the
initiating point onward. This average defect ij is expressed by the
equation : —
(525). 7] == 0.025 P
I denoting the duration reckoned from the initiating point. This
point may approximately be found as foUows : —
Average age at, birth
Initiation of droopf
Difference . .
18.9
6.0
12.9
22.9
10.0
12.9
27.5
15.0
12.5
32.4
18.0
14.4
37.3
24.0
13.3
41.9
29.0
12.9
46.3
33.?
13.3
Aver. 13.1
• ».«., Age oi mother at birth of children. f Years of duration of marriage.
In these results the first line gives the average age of women at the time
of maternity, and the second line gives the points where the droop from
the linear relationship commences : the positions of these points being
estimated from the graphs. Fig. 75, p. 268. The differences give a sensibly
constant age, which is seen to average 13.16, hence the droop implies
that the fecundity of those who are characterised by early marriage and
late motherhood is less than the average for those who may be regarded
as falling into the normal place.
284
APPENDIX A.
Fig. 76.
Fig. 79.
140.000
2 16
Fig. 77
._los. Curve ' C, , 18 monlbs after marriage
years Cntvea C,C
Fig. 78.
Fig. 76. — Curve A denotes the frequency, according to duration of monthly-
groups, of first-births, viz., the number of cases in a total of 1,000,000 first-births
for all durations of marriage (see Table LXXVII., pp. 252-3). ■ T?he curve f, g, g', h,
denotes the relative numbers attributable to prenuptial insemination, and the curve
i, i', and j, etc., the relative numbers attributable to post-nuptial insemination.
Curve B denotes the frequency, according to duration, of yearly groups, with a
less extended lateral scale, the point g" thereon corresponding to g on Curve A.
Curves B', B" and B'" are plotted on a larger vertical scale, y' and y" being
the same point as y, and z ' and z ' ' the same point as z.
Curve E is the curve of relative frequency of birth, according to the interval
after the last menstruation, see Table LXXXIX., p. 278.
Fig. 77. — Curve C shews the relative maximum frequencies according to age
(i.e., for any age). The points 1, 1' and m, and n, n', o and p have the same signific-
ance as points g, g' and h, and i, i' and j in Fig. 76, curve A, and the point k corresponds
to f.
Curves C ', C", are an extension of curve C, the lateral scale being altered. The
point p' is the same as p, q' as q, etc.
Curve D denotes the ratio, according to age, of first-births, to married women.
It appears to be compounded of two curves, viz., u, u' and v, v', w, s. Curve D ' is
plotted on a larger scale, the point s ' being identical with s.
Fig. 78 illustrates the formulae for determining the exponential curves so as to
make the shaded areas equal to the areas of the rectangles Aj and Aj, in order to
determine the positions of the centroid verticals, etc. See formidse (496) to (510),
pp. 264-5.
Fig. 79 is the graph of the approximate average intervals to between marriage
and the " unprejudiced" first-births for New South Wales, 1893-1898, and for the
Commonwealth, 1908-1914 ; the light zig-zag line marked W denoting the result for
the former, and the heavy zig-zag line marked T denoting that for the latter. The
figures denote months, and the lateral divisions denote two years' duration.
FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 285
36. The polygenesic, fecundity, and gamogenesic distributions. —
As we have seen, there are two ways in which records of issue, according
to age and duration of marriage, come to hand, viz. : —
(i.) When, at the registration of births, the age, duration of
marriage, and "previous issue" are also registered ; and
(ii.) When, at a Census, the age, duration of marriage, and total
issue are ascertained.
There are certain differences between these. In (i.) the total age-range
covered is that of the child-bearing period only ; in (ii.) the age-range
is from the earliest age of maternity to the end of life. In (i.) the cases
come under observation ditrmgr a period of time ; in (ii.) they come under
observation at a given moment. Hence, to deduce (ii.) from (i.) it is essential
that the necessary records of births, migration, and deaths should extend
over a long period of time, and even then, the deduction of (ii.) from (i.) is
by no means simple. Both records are, however, of value statistically
and both yield appropriate measures of fecundity, though on the other
hand both require corrections if they are to represent what would have
been furnished by a " constant population."
If, on a plane, the ages of mothers [x) be plotted as abscissae, and their
duration of marriage {y) be plotted as ordinates, and if then verticals to
this x^z-surface be drawn denoting the number of cases of maternity,
corresponding to each age and duration, the surface so defined may be
called the genesic distribution at maternity, or simply (i.) the polygenesic
distribution. Similarly if the verticals denote the number of children
recorded at any moment as having been borne by women of any age and
duration of marriage, the distribution may be called the general genesic dis-
tribution, or (ii.) the fecundity distribution.''- The fecundity-distribu-
tion-contours, or lines denoting equal issue for various ages and dura-
tions of marriage, can be drawn by means of formula (523), together
with the values of the constants given in Table XCI., the values of
the durations (according to age) where the linear condition ends, see
§ 35, and formula (525). If 11 be assumed to be, the earliest age of
what may be called " extraordinary marriage," and 14 be assumed
to be the earliest age of " ordinary marriage," and if also the generally
approximate result, be adopted, viz., 0.6667 + 0.3643 i, the plan of the
polygenesic surface will have for a limiting boundary the line y ^ x — 11;
its surface will, for the major part, be (approximately) a plane,
steepest at right angles to the axis of abscissae (age), and making an
angle 6 with the xy plane, the tangent of which angle is 0.3643. For any
age X, .the line on the surface denoting increasing durations of marriage,
1 The assigning of the word " polygenesic" to the one, and " fecundity" to
the other distribution, is, of course, somewhat arbitrary : the terms might, of course
have been interchanged.
286 APPENDIX A
rises uniformly till it attains the value y = x — li. For greater durations
than this the surface will droop. Between the axis and the contour-
line representing say the third or fourth child, the surface is somewhat
irregular.
If the distribution is based on the ages at marriage and the duration
of marriage, it may appropriately be called the gamogenesic distribution.
The abscissae then are the ages of mothers when married (i.e., " ages at
marriage"), and the ordinates, as before, are the duration of marriage.
37. Diminution of average issue by recent maternity. — ^Returning to
the results shewn in Tables XC. and XCI., for the second and subsequent
years of duration of marriage, it may be noted that they are important
in any attempt to ascertain what may be called the unmodified fertility -
ratio. When the fertiUty-ratio is found by merely dividing the total
number of cases of nuptial maternity at any age by the number of married
women at the same age, the quotient is " modified" by the fact that they
are not ail at equal risk. If the fertiUty-ratio is to shew what is due to
change of age alone, or rather, to change of age, unmodified by the effect
of a recent birth, but unaffected as to all other factors, a certain proportion
of the married women should be subtracted from the total. We shall
first consider the question of estimating the diminution of average issue
by recent cases of maternity.
Formula (523), shewing the general rate of increase in the average
issue, (since it is derived only from aU cases of maternity coming under
observation for each duration), gives what may be called " the unmodified
rate of increase" for what also may be called " the fertile section only"
of the whole body of married women ; see § 34, hereinbefore. Con-
sequently the differences of average issue for successive durations of
marriage, although an indication of, do not give a very exact measure of
the proportions of women who are virtually removed from risk. These
proportions are doubtless better defined by the differences between the
observed average and the average issue computed upon the assumption
of constant average rate of increase per year of duration. Hence the
ratio of the diminution in the cases of maternity for any given age-group
and for any given duration of marriage may at least approximately be
foundjas follows : —
Let c" be the average number of children (or average issue) on the
supposition of a uniform increase, and c the actual number, each with
suffixes to denote the duration of marriage and age . Then the diminution-
ratio, that is the amount by which any previous births wUl have dimin-
ished the actual record of cases, will presumably be c/c" But this
diminution-ratio appUes only to the cases in which maternity has occurred.
FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 287
Consequently if the values of this fraction be formed, for successive years
of duration, commencing not from marriage, but from the number for
th^e first year of duration of marriage, it will furnish a rough estimate of
the correction necessary, if it be desired to ascertain, from the number of
cases actually occurring, the number of cases that would have occurred
had the whole of the women in any age-group been at full average risk.
If to the values of c ", f or duration to 1 , given in the top line of Table
XC, successive multiples of 0.3643 be added, and the sums, so formed, be
subtracted from the values on the second, third, etc., lines of that table,
we shall obtain the figures shewn on Table XCIII. on next page. These
figures afford a fairly good indication of a systematic effect, according to
duration, that is, of an effect which varies with age. This variation is not
the same for each duration, and appears to change somewhat irregularly
with age. The mean of the changes gives a fairly regular curve (see the
upper part of Table XCIII. ).^ The individual graphs for the various
durations, however, appeared to shew that the adoption of this general
average for each series, was of doubtful validity, and for this reason a
different linear change according to age was adopted for each duration.
In any attempt to estimate the diminution of the numbers at risk
by means of the falling off in the average issue, according to duration, it
is probably desirable to take the adjusted results in the upper part of
Table XCIII. This will give — .186/.364, + .177/.729, etc., for age
18.92, — .217/.364, +.177/.364, etc., for age 22.87; and so on. The
results are shewn in Table XCIV. If we call the tabular value c"',
the ratio p of the altered risk to the average risk is given by : —
(526) p = I + c'" / 0.3643 = 2.745 (0.3643 + d").
The value of 1 — p will be required ; it is consequently : —
(527) 1 -p = - 2.745 c'".
Since c"' is negative, if for any duration of marriage fewer women than
the average have given birth to children (owing to a recent birth, etc),
then this last expression is positive. Table XCIII. shews the deviations,
according to age and durations of marriage up to four years ; from the
general rate of increase.
'■ The curve can be very closely represented by the curve a+ bx + cX" , where
n is greater than 1. Smoothed, the values would be about + .000, —.031, —.072,
-.124, -.183, -.265, -.422.
288
APPENDIX A.
TABLE XCIII.— Shewing the Average Effect of a recent Maternity upon the Average
Issue (Number of Children) Corresponding to Various Durations of Marriage,
and of a Consequent Correction.
Mothers
Excess ( + ) or Defect ( — ) in the Average Number of Children, on an
Average (Linear) Increase according to Duration of Marriage.
group.
Years.
-19
20-24
25-29
30-34
35-39
40-44
45-
Duratlon of Marriage.
Crude Besults.
•1-2. 2-3. 3-4
-.120
.217
.295
.307
.326
.280
-.355
+ .190
.143
.022
—.059
.153
.304
—.326
+ .046
.068
—.022
.084
.147
.199
—.449
4-5. Mean.
—.007
+ .027
.210
—.054
+ .047
.062
—.046
.124
.107
.183
.279
.265
—.558
.422
Aver. Difference for an age-difference of 10 yrs. — .077
Mothers
Age-
group.
19-
20-24
25-29
30-34
35-39
40-44
-45
Adjusted Results.
tl-2.
-.186
.217
.252
.292
.328
.363
-.396
2-3.
3-4.
+ .177
+ .078
.106
.031
.023
—.024
—.070
.086
.154
.141
.237
.197
—.815
—.249
+ .049
— .014
.086
.168
.241
.313
—.382
—.120
-.158
Aver-
age
Age.
18.92
22.87
27.46
32.65
37.29
41.91
46.29
Average Increase in the Average Number of Children.
Crude Results.t
Adjusted Eesults.§
.244
.675
.220
.321
.245
.690
.240
.453
.147
.725
.289
.451
.147
.680
.275
.446
.069
.662
.340
.433
.088
.650
.310
.429
.057
.613
.339
.402
.053
.600
.34.5
.402
.030
.538
.370
.404
.032
.530
.380
.365
.084
.341
.469
.284
.019
.440
.415
.328
.009
.394
.241
.255
.011
.330
.450
.271
Age.
17.5
22.5
27.5
32.5
37.5
42.5
47.5
Values of 1 — P
= — 2.745 c'"
Moth-
ers'
Age.
jt=l
12
13
14
15
16
17
18
19
20
21
22
23
24
25
30
35
40
45
50
+ .364
.385
.407
.428
.449
.470
.491
.512
,534
.555
.576
..597
.618
.639
.745
.850
.956
1.062
+ 1.167
*=2.
J;=3.
4=4.
— .777
.728
,679
.629
..580
,531
.482
.433
.384
.335
.286
.236
— .187
+ .059
.304
.550
.800
+ 1.041
.376
.343
.310
.277
.244
.211
.178
^148
.113
.080
.047
.014
.151
.316
.480
.645
.810
— .304
.261
.218
.174
.131
.088
.044
— .001
+ .043
.086
.129
.346
.563
.780
.997
+ 1.214
* These results are found by adding multiples of 0.3643 to the figures in the first row of Table XC,
and then subtracting them from the figures for the corresponding duration in the successive columns.
t These results are the linear smoothings of the crude results. The linear adjustments are made
by using the " average" ages, and can be regarded only as fairly satisfactory. The total number of
cases of maternity analysed is, however, large ; viz., 805,015.
t These rows are the differences of the columns in Table XC.
§ The adjustments follow no general law : tlie first is on a curve jle' **, the second is A' — Bx',
the third. A" + B'x, and the fourth A"'—E'x — Ca;', the intervals x^—x^, etc., between the age
groups being taken as always ol equal value, i.e., the adjusted values are for 17.5, 22.5, etc.
The above table appears to shew that the period of time over which
the influence of a case of maternity extends on the average, follows no
simple law, and is by no means negUgible for some years, especially as
regards the later portion of the child-bearing period. The whole method
is not quite satisfactory, but is the best available, until the record of the
procreative history of a large number of married women is to hand, giving
the intervals between marriage and the births of successive children
preferably compiled for intervals of single months from at least one to
sixty, and for somewhat larger intervals (quarters, half-years, or years),
to the end of the child-bearing period. Such statistics would reveal
accurately the characteristic of the frequency of maternity according to
duration of marriage, and would allow of the ratio p referred to in formulae
(528, 529) hereinafter being more exactly ascertained .*
^ As far as I am aware such a statistic has aot been compiled, although it is of
considerable ituportanoe.
FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 289
38. Crude fertility according to age corrected for preceding cases of
maternity. — The ratio [m/M), between the number of nuptial mothers
(m) of a given age -group during a given period of time, to the total number
[M) of married women of the same age-group, is not the true monogen-
ous-fertility -ratio, inasmuch as the M married women are not homogene-
ous as regards the maternity-risk (p ) to which they are subject. Obvious-
ly m/M is too low a value for women whose fertility remains in abeyance,
and is too high a value for women who have just borne children. The
survivors after the lapse of k years of the married women of age x last
birthday are 2/j,+ft / L^.^ Consequently if p^. is the average risk for
the jfcth year after a birth (calling the year of birth 0), the corrected fertihty
ratio (p") is given by the eq^uation : —
(528)
1^
'^ *^ ilf^- |m^_i .y^ (l-pi) (i-r^_i)-fm^_2.^ (l-p2)(l-2r,_2+etc. }
(1— fcr^..^) denoting the rate at which the mothers of age x — k have
increased in k years. This may perhaps be ordinarily taken as the same
at all ages, and as the rate of the population increases. The above
formula may be put in the following form, viz. : —
(529) .
V --W
"■■'-i^'iT;<'-"'"-"'+-+™i?-&"^'""'-'"'+-}
and the ratios of the m/M quantities in the denominator do not need to
be very exactly computed. It will always be abundantly accurate for
the purpose in view to assume that : —
(530) i/^/Vft = 1 - P fe + ^x-k)
a formula which is satisfactory through a fairly large range for ^.^ Since
the quantity between the braces in (529) is positive and small,
its effect is to increase the value of p" The correction is important
in any attempt to ascertain the age of greatest fertility, con-
sequently the values given in Table LXXIII., p. 242, are those with which
we are mainly concerned ; see columns ix. and xv. therein. The values
of the factors (k) of m/M in the denominator of (529) can be readily
tabulated for say r =0.01 and 0.03.
^ Iix denotiixg the mean population living in the year of age x : as in the ordin-
ary actuarial notation.
2 For example from Australian Life Tables for 1901-1910, Report of Census,
Vol. III., pp. 1217-8, we have for ages 40 and 30, from the L values 0.93986, and from
the a values 0.93815, i.e., for so large a value of h as 10, the error is less than 0.002.
290
APPENDIX A.
The value of the L, p and r terms are as follows for Australia : —
TABLE XCIV. — Shewing the Factors Beauired to Correct the " Grade Feitility-
ratio," for Preceding Cases of Maternity. AustraUa, 1908-1914.
Values of (1— *r) L^/L^.^
Values of « when r = .01.
Age of
Mother.
r = .01«; i = lto4.
r = .01 and .03 ; * = 1 to 4.
1.
2.
3.
4.
1.
2.
3.
4.
15
20
25
30
35
40
45
.9879
.9868
.9858
.9849
.9840
.9830
.9821
.9761
.9739
.9720
.9701
.9683
.9663
.9845
.9843
.9813
.9584
.9556
.9529
.9500
.9472
.9527
.9590
.9450
.9414
.9378
.9340
.9304
+ .423
.414
+ .527
.516
+ .630
.617
+ .733
.718
+ .837
.820
+ .940
.921
+ 1.043
1.022
—.662
.635
—.422
.405
—.182
.175
+ .057
.055
+ .294
.282
+ .531
.509
+ .787
.736
—.331
.311
—.171
.160
—.013
.012
+ .144
.135
+ .301
.282
+ .456
.428
+ .611
.573
—.290
.288
—.084
.077
+ .122
.112
+ .326
.299
+ .528
.484
+ .728
.687
+ .927
.850
* To find the values for any other value, r ' say, of r, multiply the tabular values by (r '— r) / r.
t To find the values for any other value of T, multiply by 0—rk) / (1 — .Olt). Thus, for r=.02
the multipliers of the successive columns are 0.9899, 0.9796, 0.9891, 0.9583 ; and if r= .03 the successive
multipliers are 0.9797, 0.9592, 0.9381, 0.9167.
The above values are very approximately given by : —
(531) (l—kr) L^ /L^-^. = 1 — 0.000188A; (47.7 + x) ;i
and those for the correcting factors e by : —
(532) . . . . ei = 0.02070 {x + 5.43) ; (532a)
(5326). ...es= 0.03140 (a;— 25.54) ; (532c)
Formula (525) may thus be written : —
(533)
€2 = 0.04763 (a;— 28.91);
€4 = 0.04057 (a;— 22.15).
1
-^ (/carnal + +«rtw^t)
k being the tabular value given in Table XCIV. (in which r = .01 and
r = .03), and the probabihty of maternity ascertained by this last formula,
will be free from the effect of recent cases of maternity : that is the crude
probability must be multiplied by the fraction foUowing m/M.
39. Age of greatest fertility.— When the probabilities according
to age of maternity have been corrected so as to represent what would be
given if aU women were at equal risk, then the age of greatest probabihty
may be regarded as the age of greatest fertihty. Applying formula (533)
to the data in Table LXXIII., p. 242, we have the following results
about the maximum : —
1 More exactly the valvies of the constant to be added to x are 47.60 46 81
47.63, and 48.63, and of the coefficients to be multiplied into k are 6 OOOlQ^iq'
0.0003866, 0.0005700, and 0.0007433. "-uuuiMrfrf,
FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 291
TABLE XCV. — Shewing Corrections to the Fertility-ratio for Ages 13 to 23, when
Allowance is made for Preceding Cases of Maternity.
Age of
Values ol k when 4 = 1 to 4.
Factor
Kin
= ?"
Fertility-ratio.
Mothers.
1.
2.
3.
4.
Crude.
Smoothed.
Crude.
Corrected.
13
+ .374
—.727
—.371
—.342
1.039
1.001
.5?
.52?
14
.394
.681
.341
.304
1.011
1.013
.2055
.2076
15
.414
.635
.311
.266
1.013
1.024
.2269
.2299
16
.434
..589
.281
.288
1.012
1.036
.4063
.4112
17
.466
.543
.251
.190
1.048
1.048
.4316
.4521
18
.475
.497
.221
.152
1.066
1.059
.4776
.5093
19
.496
.451
.191
.114
1.077
1.071
.5022
.5409
20
.516
.406
.160
.077
1.092
1.083
.4540
.4956
21
.536
.369
.130
.039
1.074
1.094
.4375
.4700
22
.656
.313
.100
—.001
1.106
1.106
.4167
.4596
23
+ .677
—.267
—.070
+ .037
1.123
1.117
.3813
.4283
Although the values of k are of the same order of magnitude, yet
within the range shewn, the values of the successive ^m-terms rapidly
diminish, so that although there is no theoretical justification for stopping
at & = 4, the inclusion of later terms would but slightly afiect the result
(at least in the second place decimals).
The factors K shew that about the age of maximum fertihty the
correcting factors to give the fertility, unprejudiced by previous cases
of maternity, increase linearly with age, and are represented very ap-
proximately by thrB formula : —
(534) K = 1 + 0.01163 {x - 12.91).
The values for these factors, so computed, are the smoothed values in the
preceding table.
A smoothing, independent of that already given in Table LXXXIII.,
gave, as the maximum for the uncorrected fertiUty-ratio, 0.483 ; and a
similar smoothing of the corrected values gave 0.517, the maxima and
corresponding ages being : —
Uncorrected, age, 18.8,i 0.483 ; corrected, age, 19.0, 0.517.2
In the method outlined, of correcting the crude fertiUty-ratio (proba-
bility of maternity), equal " weight" is attributed to the values of k.
An examination of Fig. 75 shews, however, that the " weight" to be
attributed should probably decrease with increase in the value of k (that
is with the number of years elapsed since a previous birth). Moreover,
the change in the numbers of married women and cases of maternity is
so rapid at the ages of maximum fertility that the age divisions should be
less than one year, and the ages need to be very exactly given, which
unfortunately they are not. For these reasons great exactitude in regard
to the correction is at present impracticable.
40. Fecundity-correction for infantile mortality .^The frequencies
of child-bearing as between two populations are, like their birth-rates,
rigorously comparable as accurate measures of fecundity, only when their
infantile mortahty-rates are identical, and the crude frequencies require,
1 The result in Table LXXXIII. was 18.23 years.
' The factor, according to (534) above, gives, on multiplying into, 0.483, 0.5168.
292
APPENDIX A.
therefore, a correction, to reduce the risk of maternity to an equality ;^
see Part XI., §§ 4-6, pp. 145-152. It has been shewn that the infantile
mortaUty correction to birth-rate is, on the whole, about ^g = ^
(l-fO.OSS/i) ; see p. 145. If, therefore, there were two equal populations
of say married females (M), of equal fecundity (/), but with different
rates of infantile mortaHty, we should have for the cases of maternity (m)
occurring therein, respectively : —
(535) mi = fM (1 + kiiJLi), and m^ = fM (1 + hfiz) ;
whence it follows that
(536).
•/ =
nti
mz
M(l+ Vi) -^ (1 + V2)
Thus the correction is always very small, and, in general, is practically
negUgible.
41. Secular trend 0! reproductivity. — ^The crvde reprodiictivity may
be measured by the ratio of the number of confinements to the number of
persons at uniform risk ; thus the
Nuptial and Ex-nuptial Maternity-
Ratios, etc.
.008
Curve A is the ratio of nuptial con-
finements to all married women.
Ciirve B is the ratio of ex-nuptial
confinements to "unmarried" women of
12 years of age and upwards.
Curve C is the ratio of the ex-
nuptial to the nuptial confinement rates,
the range being between .038 and .059.
Curve D shews the variation in the
average number at a birth.
Curve E shews the variations in the
survival factor for the first year of hfe.
crvde nuptial reproductivity is the
ratio of nuptial confinements to
the total number of married women,
and similarly, Uie crude ex-nuptial
reprodiictivity is the ratio of ex-
nuptial confinements to the total
" unmarried," which here wiU
include the " divorced " and
" widowed." The ratios are
" crude," since no corrections have
been applied for age-differences
in the female population, and
it is obvious from columns ix.,
X., XV., and xvi. of Table LXXIII.,
p. 242, that fertility greatly varies
with age. For this reason, whenever
the age-distribution is not identical,
the results are not strictly comparable :
they do not rigorously measure
the degrees of reproductivity, or of
malthusianism, operating. Con-
sequently, for strict comparisons, a
properly determined index of initial
reproductivity would have to be
computed, see §§3 to 6, pp.
235-239.
Neglecting this, however, for
the present, and restricting the consideration to the crude initial nuptial
1 It may be noted that after deducting the period of gestation and the puerperal
period, there remains about one-sixth of a year during which mothers of the first-
sixth of any year of record may give birth to a second child even in the same year
and the chance of this occurring is increased by the death of the child born '
FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 293
and ex-nuptial reproductivities, the results are set out in Table XCVI.
hereunder ; see columns (ii.), (iii.), and (v.) thereof. The results are
shewn also by curves A and B of Fig. 80, the former curve denoting the
nuptial, and the latter the ex-nuptial .frequency of maternity. The figure
shews that while the nuptial and ei-nuptial rates by no means run identi-
cally, they yet exhibit, on the whole, similarity of trend, the ex-nuptial
rate being roughly 0.05 of the nuptial. The exact fluctuations of the
ratio of the ex-nuptial to the nuptial rate are indicated in column (v.) of
Table XCVI., and are shewn as curve C in Pig. 80. The dotted lines on
curves A and B shew the general trend of the phenomena.
TABLE XCVI.— -Shewing the Secular Changes of Nuptial and Ex-nuptial
Reproductivity. Australia, 1881 to 1914.
Ratio of
Nuptial
Confine-
ments to
Married
Women.*
Batio of
Infantile
Year.
Ex-nuptial
Confine-
ments to
Number of
Unmarried
Eatio of
Births to
Total Con-
finements.
Eatio of
Ex-nuptial
to
Nuptial
Bates.
MortaUty
(Ratio of
Deaths of
Children
during first
Survival
GoefflcientB
for end of
First Year
Women.t
12 Months)!
(i-) ,
(ii.)
(iii.)
(iv.)
(V.)
(vi.)
(vii.)
1881 . .
.2285
.00950
1.00865
.0416
.1165
.8835
1882 . .
.2206
.00891
1.00779
.0404
.1357
.8643
1883 . .
.2245
.00870
1.00847
.0388
.1222
.8778
1884 . .
.2305
.00893
1.00875
.0380
.1260
.8740
mean 1-4
.2269
.00901
1.00842
.0397
.1251
.8749
1885 . .
.2301
.00918
1.00873
■ .0399
1292
.8708
1886 . .
.2274
.00946
1.00866
.0381
.1271
.8729
1887 . .
.2285
.00957
1.00852
.0419
.1164
.8836
1888 . .
.2271
.00983
1.0102i
.0433
.1164
.8836
1889 . .
.2206
.01008
1.00989
.0457
.1319
.8681
Mean 6-9
.2267
.00962
1.00920
.0418
.1242
.8758
1890 . .
.2216
.01021
1.01005
.0461
.1082
.8918
1891 . .
.2181
.01026
1.01030
.0470
.1155
.8845
1892 . .
.2133
.01060
1.00865
.0497
.1058
.8942
1893 . .
.2072
.01034
1.01008
.0499
.1149
.8851
1894 . .
.1947
■.00961
1.00931
.0494
.1031
.8969
Mean 0-4
.2110
.01020
1.00968
.0484
.1115
.8886
1895 . .
.1916
.00947
1.01008
.0494
.1012 •
.8988
1896 . .
.1788
.00935
1.00900
.0558
.1126
.8874
1897 . .
.1770
.00914
1.01066
.0517
.1048
.8952
1898 . .
.1700
.00879
1.00997
.0586
.1272
.8728
1899 . .
.1697
.00894
1.01086
.0527
.1167
.8833
Mean 6-9
.1774
.00914
1.01011
.0636
.1125
.8875
1900 . .
.1691
.00905
1.01078
.0535
.1002
.8998
1901 . .
.1668
.00865
1.01095
.0519
.1037
.8963
1902 . .
.1625
.00826
1.01060
.0508
.1071
.8929
1903 . .
.1513
.00807
1.00997
.0533
.1105
.8895
1904 . .
.1554
.00859
1.01079
.0553
.0825
.9175
Mean 0-4
.1610
.00852
1.01062
.0630
.1008
.8992
1905 . .
.1524
.00861
1.01076
.0565
.0819
.9181
1906 . .
.1527
.00868
1.01112 ,
.0568
.0836
.9164
1907 . .
.1527
.00864
1.00962
.0566
.0814
.9186
1908 . .
.1506
.00857
1.00969
.0569
.0780
.9220
1909 . .
.1506
.00837
1.01024
.0556
.0718
.9282
Mean 6-9
.1518
.00867
1.01029
.0565
.0793
.9207
1910 . .
.1511
.00801
1.01040
.0530
.0751
.9249
1911 . .
.1541
.00818
1.01033
.0531
.0680
.9320
1912 . .
.1632
.00821
1.01037
.050S
.0708
.9292
1913 . .
.1609
.00805
1.01025
.0500
.0720
.9280
1914 . .
.1598
.00766
1.01038
.0479
.0713
.9287
Mean 0-4
.1678
.00802
1.01036
.0509
.0714
.9286
* That is, to all married women, irrespective of age.
t That is, to " never-married," " widowed," and " divorced," of 12 years of age and upwards,
taken together.
t The infantile mortality as given is not the ratio of deaths registered as under one year of age,
in any year, to the births registered in the same year, but are those given in a paper " On the im-
provement m infantile mortality, etc.," read before the Australasian Medical Congress in September,
1911 (see p. 672 Journ.), and are related to the number of births of the "equivalent year."
42. Crude and corrected reproductivity. — It has been shewn in Part
XI., § 6, see Table XXXV., that the crude birth-rate gives only the
initial reproductivity, and that, owing to the measure of infantile
294 APPENDIX A.
mortality, the residua], after the first 12 months have elapsed, is more
sigmfic8.nt than the birth-rate as regards the increase of the population.
The necessary correction is secured by multiplying by a " survival factor."
The principle may be extended for various purposes. Thus survival
factors (cr) maybe calculated for the commencing school-age, the ages of
puberty or nubility, the commencing age of miMtary service, the age of
highest average economic efficiency, and so on. In actuarial notation
these factors are denoted by Ix/lo^ *nd for brevity's sake may be denoted
by ax- To compare two populations for survivals, S, up to any age x,
we have, therefore, B denoting the births : —
(537) Sx= Bk:/lo = Bax= B — Dx
in which Dx denotes the aggregate of the deaths (of the native-born) up
to age X. When x = 1, the values of a are unity, less the rate of infantile
mortality taken for the "equivalent year." For rates, these quantities
must be divided by the mean population of the period covered by the
births.* The more rigorous treatment of this question has already been
dealt with in Part XI., §§ 7 to 9, pp. 152-180 ; see also Tables XXXVI.
and XXXVII. The infantile mortality varies, however, considerably
from year to year, see column (vi.) in Table XCVI., which gives the rates
calculated approximately for the "equivalent year."^ If y denote the
infantile mortahty (see p. 151, hereinbefore), a being the survival factor,
then we have : —
(538) a = 1 — y; ory = l — a;
as on (352), p. 151. This, of course, differs according to sex, with time,
as is shewn in Table XCVI., and according to locality. The highest
value of the survival-factor for Australia was 0.9320 in 1911. For the
period 1901-10 for the Commonwealth of AustraUa it was 0.90490 for
males, and 0.92047 for females,^ corresponding to infantile mortahties of
0.09510 and 0.07953. We thus arrive at the conception of a survival-
value for a birth-rate, that is, the birth-rate reduced to its value at age x,
and this survival-value may be averaged for the whole of life, i.e., integ-
rated for all ages. Such an integral will constitute the best general
measure of the reproductivity. It is equal to the average period lived
multiplied by the birth-rate. Or if o) denote the greatest possible age,
then : —
(539) 2*0 =-p\axdx
o
and Eq is the reproductivity of the population taken as a whole. If o-q
be unity, and the unit of x be, one year, then the value of (538) will be the
' Vide a paper (by the author): " The improvement in infantile mortality ; its
annual fluctuations and frequency according to age, in Australia." Journ. Aus-
tralasian Medical Congress, Sydney, Sept. 1911, pp. 670-679.
• See Life Tables, Census Report, Vol. III., pp. 1215 and 1217.
FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 295
birth-rate multiplied into the number of years expressing the length of
life lived on the average ; consequently the product of the birth-rate
into the " expectation of life at age 0," may be taken as the most service-
able expression of the reproductivity.i
The value given by (538) may be regarded as the crude reprodwctivity.
The birth-rate j8 is ordinarily computed as for the total population, but
may also be based upon the total female population, upon the female
population of child-bearing ages, or upon the married of child-bearing
ages plus a reduced number of the unmarried, equating them to the
nuptial condition. Let the ratio of the fertihty of women at full risk
(or otherwise if desired), at any age x, to the fertility at the age at which
it is a maximum be denoted by fx : then the actual number of married
women of all ages may be reduced to an equivalent number of women at
the age of maximum fertihty by multiplying by this quantity. With
these can be included also the unmarried, with whom in Austraha the
fertility is about one -twentieth of that of the married. The corrected
reproductivity may be given in the form of a birth-rate, viz., j8e : —
(540) I3e = B /S {fxMx + .A Ux)
in which 2 denotes " sum, "/and/' are the ratios for the fertilities of the
married and unmarried respectively, referred to the greatest fertility of
the married, and M and U are respectively the numbers of the married
and the unmarried, who together give birth to B children. This measures
the ratio of the actual births to a fictitious number of mothers of highest
fertihty, and hence birth-rates so computed shew the variations of the
extent to which potential fertility is actualised. These, of course, may
be further reduced to their survival values.
The mode of comparing reproductive efficiency by means of an index,
viz., the genetic index or first natality index, has already been indicated ;
see § 5, p. 237, hereinbefore.
43. Progressive changes in the survival coefficients. — The survival-
factors are by no means constant, as is shewn in column vii. of Table
XCVI. As tabulated, they are merely unity, less the ratio of the deaths
under 12 months to the births in the same year. This, as shewn before.
1 Actuarially, the quantity :-
ex
= T^x / Ix = \ Ix dx -i- Ix
when a; = 0, may, when multiplied by the birth-rate, be adopted as the measure of
the reproductivity of a popjilation. Since this is obtained from the mortalities at
suooeasive ages, it ia not quite homogeneous, as it is aSeoted by the vitality of
migrants, and, moreover, the mortality of the older part of the population is affected
by their earlier history, and may not therefore represent future experience. If
J„ = I. then e„ = To = So / P-
296
APPENDIX A.
is not quite correct, see pp. 155-160, but the correction is of no moment
for the present purpose. It is worthy of note that the infantile mortality
is roughly about 0.5522 of the rate of confinements of married women, as
is shewn by comparing the means. The means (see Table XCVI.) 0.2269,
0.2267, etc., multiplied by the above fraction gives the follomng results : —
Period
1881-4
1885-9
1890-4
1895-9
1900-4
1905-9
1910-4
Infantile mortality
As^ computed from the
nuptial confinement rate
Survival factor divided by
ratio of nuptial confine-
ments
.1251
.1253
.5513
.1242
.1252
.5479
.1115
.1165
.5284
.1125
.0980
.6342
.1008
.0889
.6261
.0793
.0838
.5224
.0714_
.0871
.4525
The ratio is therefore not uniformly constant.
The infantile mortality is decreasing, but nevertheless shews a fairly
definite fluctuation, see curve E, Fig. 80, which shews it on a large vertical
scale ; its Hmiting value is, of course, unity. '
XIV.— COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY.
1. General. — In dealing with the more complex elements of fertility
and fecundity, it will generally be necessary to distinguish between the
nuptial and ex-nuptial cases, and since their frequency is very different,
some simple method of correlating and comparing the two wijl have to
be devised.^
Often it is necessary to distribute unspecified cases, since, in double-
entry tabulations, the cases are often partially specified, and the neglect
of partially-specified and wholly-unspecified cases will often lead to
material error.
There is another general matter of importance, viz., the corrections
required in statistics of duration, if they are required to represent the
results which, other things being equal, would have been furnished by a
constant population. This will receive attention in § 3, pp. 298-9.
2. Correspondence and correlation. — It is often possible to see the
essential identity of two curves by mere change of scale, or by systematic
deformations (anamorphosis) of one in order to bring it into agreement
with another. This fact is of value in the graphs of various vital
phenomena.
For example, any attempt to make the widest possible comparisons
of population phenomena requires the construction of world-norms for
the human race. But such an attempt involves the consideration of
physiological and general correspondence of human developments. In
connection with marriage, fertility, fecundity, etc., and their signific-
ance, for instance, this demands the consideration of the following,
viz. : —
(a) The average ages of puberty, nubility, etc.
(b) The frequency-distribution about those ages ;
(c) The fertility and fecundity at different ages ;
(d) The characteristics of the decay of fecundity at the end
of the fertile period.
* The determination of a type-formulae to be adopted for any two curves,
the ascertaining of their constants, and of the " skewness" of each curve will serve
to exhibit their degree of correlation. This can also be expressed by a correlation
coefficient ; see " Statistical Methods," by C. B. Davenport, 1904, and the mono-
graphs of Prof. Karl Pearson, W. F. Sheppard, G. U. Yule, De Vries, W. Pahn
Elderton, Gini, Savorgnan, and others.
298
APPENDIX A.
r ^
/p^
^
o^yir. k^
v^^
^:x
Suppose, for example, curve A, Fig. 81, represents the average fertility
according to age of women of one part of the world and B that of
another part. Let x, x' , x", etc., denote the abscissa, of the initial
point, that of the mode, and that
of the terminal point of the curve
A, or of curve B, the particular
curve being indicated by the sufiix a or
b. Then the simplest correspondences
are those where xjxi, = x Jx i,= xl'^/o^'j,,
etc., or where xj, — x^ = xfj, — a-'^^ etc.,
i.e., where the abscissae of the correspond-
ing critical points of the curves are in a
constant ratio, and the ordinates are also
in a constant ratio, or where the
abscissae of the critical points differ
by a constant. Correspondence of this
character may be called planar, because the curve B can be derived from
the curve A by parallel linear projection on to a plane inclined to that
on which A lies. If the two curves in question be represented by
y^ = Fa (x) ; Vb = Fb {^) then planar correspondence may be defined
as follows :■ —
The points on curve B are in planar correspondence with those on A
when —
Say Age.
Fig. 81.
(541).
■Vb = kFaimXa +q)
k, m, and q being constants : when k ot m or both are functions of x^,
then the correspondence is nort-planar. If these functions of x^ are not
simple, the correspondence becomes less significant.
This method of envisaging the problem has advantages over the
system of determining a mere numerical " coefficient of correlation,""^
because it is often possible to construct one curve from the data of the
other. Moreover, it is not without value to examine how far the graphs
of phenomena, which might have been imagined a priori to be identical,
or convertible by oblique projection with change of scale, differ. Later
nuptial and ex-nuptial fertility, according to age, will be compared.
3. Conections necessary in statistics involving the element of dura-
tion. — ^The type of corrections necessary to be applied to the data of
statistics involving the element of duration, depends upon the purpose in
view. Two types are of special importance, that which aims at presenting
the results, in the form in which they would have been given by (a) a
constant population, and (6) by a population increasing according to
some definite law, which for general comparative purposes is preferably
• See Galton's graphic method, F. Galton, 1888, Proe. Roy. Soc. Lond., XLV.,
136-145. Davenport, Statistical methods, p. 44, 2nd Edit., Lond., 1904. See
also Pearson's, Yule's, and other papers on the subject.
COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY.
299
the norm of increase, i.e., the characteristic of the increase of the whole
of the populations to be compared. The latter involves the smaller
corrections, and has the advantage that for many purposes the corrections
will be negligible. Let it be supposed that the population is an increasing
one : the data will then be characterised as follows : —
(i.) The data for longer durations, drawn therefore from a
smaller population, will be smaller (all other things
being equal) than would characterise a constant
population of the size from which the more recent data
are drawn. Hence the necessary correction is a factor
1+e, where e is positive.
(ii.) If the numbers of individuals have been taken into
account for earlier dates, they can be deduced from the
survivors, provided (a) that a correct mortality table is
available, and (6) that migration has introduced no
(material) modification.
(iii.) If the data are related to events occurring with a varying
rate (as in cases of birth, marriage, death, etc.), the
rate at which they occur must be determined according
to the duration in question.
The type-formula for correction is as follows : — Let N denote the
number given at any point of time, that is, let N denote the survivors
after the duration i, from N' persons ; then if, in origination, N may be
presumed to vary with the population, we shall have, on making allow-
ance for the fact that these are only survivors, and that what is required
is a result which shall either coincide (i.) with the final magnitude of the
population, viz., at the date from which i is reckoned, or (ii.) with a
definite rate of population growth (the rate of normal increase) : —
(542).. J\r' = Nei'iL^.i/Lx = Nei^^ll + i(g'a:.i+g«)],* approximately.
"■ See formula (530), p. 289. The notation is the ordinary actuarial notation.
It is fairly obvious that Lx-i/Lx must equal 1 + ^ {^x-i +qx ) i approximately.
It will be found that, through a large range, this latter and arithmetically more
convenient form is sufficiently accin:ate for correction purposes to the data of
statistics of duration. For example, if 12 be taken as the lowest age (it is the age
of least mortality for Australian females), and successive intervals of 10 years
from this be also taken, the following results are obtained, viz. : —
x-i and X
12-22
12-32
12-42
12-52
Exact formula
1.03114
1.0933
1.1861
1.3133
Approx. formula . .
1.03110
1.08^3
1.1500
1.2888
Even the final difference is ordinarily of no moment, since, as a rule, the numbers
to which it would have to be appUed are very small.
300
APPENDIX A.
In this p will denote in case (i.) the absolute rate of increase, and in case
(ii .) the excess over the normal rate of increase. Ceitain events, however,
for example births, marriages, and deaths, migration, etc., occur with a
rapidity which fluctuates on either the positive or negative side of the
general rate of increase of the population, in which case it inay be necessary
to introduce, into equation (542), a factor depending on the fact in
question.
4. Distribution of partially and wholly unspecified quantities in
tables of double-entry- — If a series of quantities. A, B, C, etc., and A',
B', C, etc., fuUy specified so as to permit of proper double-entry, and
others, a, a', etc., and a, a ', etc., specified so as to permit only of single
entry, and again a third set at not specified, so as to permit of entry under
either of two series of headings, be tabulated or arranged as hereunder,
and totalled, the result will be as shewn syrubolically in the following
table : —
TABLE XCVII.—
Scheme ot a Donble-entry Tabulation of Defectively Specified Data.
Arguments
y
y'
2/"
y'"
etc.
etc.
Specified as
regards x only.
Totals.
X
A
B
C
D
etc.
etc.
a (6)
S + o (S + 6)
x'
A'
B'
c
D'
etc.
etc.
a' (6')
S' + a'(S' + 6')
3i'
A"
B"
C"
D-
etc.
etc.
a" (6")
S" + a' (S" + 6")
etc.
etc.
etc.
etc.
etc.
etc.
etc.
etc.
etc.
Specified as
regards y only
"W
a' (;3')
a"(^')
a"'(r')
etc.
etc.
a-(O)
[a + a'+ ..] + <-
(;8 + ;8'+ ..)+
Totals
T +«
T' + a'
T" + a'
T"'+a"'
etc.
etc.
[a+a'+..] + w
(6 + 6'+.. ) +
SS + Sa + Sa + w
ST + Sa+2o + u
In this type-table, the horizontal and vertical totals of the fully-specified
quantities are respectively S, S', etc., and T, T', etc., but the aggregates
of the rows are S + a, etc., and of the columns are T + a, etc. (i.e., for^
the fully specified quantities together with those specified as regards one
particular only). The totals T + a are specified as regards the " argu-
ments" in the horizontal headings, and the totals S + a are specified as
regards the " arguments" in the vertical headings. Thus the grand total
is i7S ( = Z'T) -\-Za-\- Sa + a> , and this is the sum of either of the series of
totals, viz., that of the final column or that of the final row.
In order to distribute the quantity wholly unspecified, it is necessary
to add a portion of oj to the" (vertical) columns, and a portion thereof to
the (horizontal) rows, so that the corrected values of A, B, A',
B' etc., shall equal the grand total, and so that the adjustment
COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY. 301
shall be the most probable. Such adjustment can be effected as follows :
It is assumed that the division of the quantity w into two p'arts, viz.,
CO ' and w", proportional to the aggregates of the a and a quantities
respectively, is the most probable apportionment of the doubly-unspeci-
fied quantity among the two, and further, that if these divisions, u> ' and
w", be again subdivided proportionally to the individual values of
a, a', etc., and a, a', etc., the result will be the most probable sub-
division. Let —
(543) CO = w' +co" ; a.nd Q = la + Ea; then
(544) w ' = Za .oi/Q\ and m" = Ea . oj / Q ;
consequently the amounts of the corrections to the a and a quantities
are ascertained by multiplying each of them by the ratio oy/Q, or what is
the same thing, the required result is attained by multiplying by this
factor increased by unity. Calling the adjusted numbers b and j8
respectively, their values are : —
(545) b = a(\ + u) / Q); /3 = a (1 + oj / i3>.
Similarly, if these 6 and ^ quantities are distributed proportionally to
the A, B, C, etc., quantities, and the A, A', A", etc., quantities
respectively, the required corrections are : —
(546)..A+a = A(l + |-+|-); B + b = B (1 + -|-+-|-' );etc.
(547). .A' + a' = A' (1+ |i+| ) ; B'.+ b'= B' (1+1'+|) ; etc.
and so on. The additive quantities, A6 / S, Aj8 / T, etc., are most
readily computed separately, and are then added to the fully -specified
quantities. By the process indicated, both series of singly -specified
quantities, and the unspecified quantities are suitably distributed, the
adjusted table consisting of the values A + a, B + b, etc. ; and A' + a',
B'+b',etc.
The process indicated is also valid when the distribution should be
made on other bases.
Let a = aj + a2 ; b = bj -f b2 ; etc., a' = a'l -j- a'2 ; etc., etc.,
the subdivisions being the values of A6/S, Aj8 /T, etc. Then, if the
fundamental supposition that the corrections are proportional to A, B,
etc., A', B', etc., be not satisfactory, any function of these quantities
302
APPENDIX A.
may be substituted, in which ease S and Twill be 2<j)(A), and i7^(A),
the former denoting the sum of the values of 0A, ^B, etc., and the latter
the sum of i/tA, ^A', etc. The process is identical in all respects with
the preceding one, when the substitutions of i^A for A, etc., have been
made.
In general, this method of distribution not only gives results of a
very high degree of probability,^ but has also the advantage of being
arithmetically very convenient.
5. Unspecified cases follow a regular law. — In general, the number of
unspecified oases in any compilation exhibit great regularity. It will be
sufficient to take two examples, which may be obtained from Tables
CXIII. and CXIV. hereinafter.
According to the former Table, out of 733,773 wives, 21,151 made no
statement as to the duration of marriage, but stated the number of
children borne by them ; 12,073 stated the duration of marriage, but
omitted to state how many children were borne by them, and 3747 gave
no information as regards either particular. See Census Report, Vol. III.,
pp. 1140-1. In the latter table, out of the same number, 5432 stated the
number of children borne by them, but did not state their ages ; 15,477
stated their ages, but did not state the number of children borne by them ;
and 343 gave no information as regards either particular : see Census
Report, Vol. III., pp. 1136-7.
The regularity of distributions of the partially-specified cases is
shewn by forming the ratios of the unspecified to the completely specified
in the same category. The results are as follow : —
TABLE XCVm.— Exhibiting the Regularity of the Ratios of those who Fail to
Specify Particulars completely to those who do not so Fail.
Eatio to total
who fully
Specify, of
PAKTioniAM Specified— ISSUE to the Numbek of—
those who
omit to State
1
2
3
4
5
6
7
8
9
10
Duration of
Marriage
Age
.0253
.0066
.0220
.0068
.0232
.0069
.0273
.0077
.0303
.0081
.0347
.0084
.0369
.0086
.0386
.0079
.0406
.0087
.0421
.0085
.0433
.0089
Paktichlars Specified— Issdb to the Number of—
11
12
13
14
15
16
17
18
19
20
21, etc.
Duration of
Marriage
Age
.0452
.0094
.0539
.0081
.0554
.0094
.0464
.0066
.0465
.0088
.0370
.0105
.0652
.0074
.0490
.0236
.0488
.0000
.0625
.0000
.1081
.0208
* The ground of assurance as to this js indicated in the next section.
COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY.
303
Table XCVin. — ^Exhibiting the Regularity of the Ratios of those who Fail to
Specify Particulars completely to those who do not so Fail — continued.
Hatio to
total oJ those
who lully
Duration of Mahruqb.
Specify, o£
those who
Omit to State
0-4
5-9
10-14
1.5-19
20-24
25-29
30-34
35-39
40-44
45-
No. of ChUd-
ren borne
.0216
.0204
.0174
.0104
.0142
,0137
.0130
.0145
.0124
.0151
A(JES OP Wives.
14
15
18
17
18
19
20
21-24
25-29
30-34
35-39
40-44
No. of Child-
ren borne
.0000
.0000
.0148
.0105
.0107
.0161
.0234
.0215
.0209
.0198
.0201
.0212
AsEs OF Wives.
45-49
50-54
55-59
60-64
65-69
70-74
75-79
80-84
85-89
90-94
95-99
100-04
No. of Child-
ren borne
.0207
.0236
.0238
.0263
.0290
.0349
.0378
.0842
.0824
.0625
.0000
2.0000
Prom the above results it is evident that the number who fail to
specify " duration of marriage" is a fairly definite function of the " number
of children borne," and also that the number who fail to specify the
" number of children borne" is also a fairly definite function of the
" duration of marriage," (see the upper part of the Table). Also, the
lower part of the table shews that those who omit to state the number of
children borne is a fairly definite function both of the " duration of
marriage, "'and — as might consequently be expected — of the " age of the
wives." These facts justify, pro tanto, the distribution of the unspecified
cases, and there is little reason to doubt the result, after distribution, has a
much higher degree of probability than that which rejects all partiaUy-
specified cases.
Other tabulations disclose, in an equally striking way, the regularity
of the numbers of the unspecified, and confirm the desirability of ad-
justing tabulations generally, in the manner indicated, before using the
results. The use of the fully specified tabulation as proportionally
correct is obviously not satisfactory.
6. Number of children at a confinement — a function of age. — By
dividing for each age the number of cases of confinement into the number
of children born, during a sufficiently long period, the average number of
children at a confinement is found to vary with age : that is, B denoting
children born, and M the number of their mothers : —
(548).
.1 +ex= B^/M,=f{x)
e- denoting the excess over unity. Seven years' experience give the
following results, the figures, however, being confined to oases of twins,
that is, the third child in cases of triplets is not taken into account : —
304
APPENDIX A.
TABLE XCIX. — Shewing Excess due to the Occuirence of Multiple Births according
to Age in the Average Number of Childien Born per Confinement, Australia,
1907-14.
1
Nuptial Confinements.
Ex-nuptial Confinements.
Ratirt. 1
Ratio-
Con-
fine-
Excess*
over
Con-
fine-
Excess*
over
Age.
ments.
1 Child.
Crude.
Smooth-
ed.
ments
1 Child.
Crude.
Smooth-
ed.
12
..
.0000
5
.0000
13
4
.0006
21
.0007
14
30
.0013 1
126
.0014
15
170
.0019 1
537
1
.0019
.0021
16
1,138
2
.0018
.0026
1,500
2
.0013
.0028
17
3,962
12
.0030
.0032
2,980
11,
.0037
.0035
18
9,761
35
.0036
.0038
4,504
15
.0033
.0042
19
18,071
94
.0052
.0045
5,3 17t
22
.0041
.0049
20
25,159
148
.0059
.0051
5,272
30
.0057
.0056
21
35,326
203
.0057
.0058
5,008
32
.0064
.0063
22
43,353
254
.0059
.0064
4,231
36t
.0085
.0070
23
50,322
333
.0066
.0070
3,848
30
.0078
.0077
24
53,175
394
.0074
.0077
3,182
26
.0081
.0084
25
54,259
453
.0083
.0083
2,548
19
.0075
.0091
26
55,006t
447
.0081
.0090
2,161
20
.0093
.0098
27
53,735
494
.0092
.0096
1,785
27
.0151
.0105
28
53,244
509
.0096
.0102
1,699
20
.0118
.0112
29
49,200
539
.0110
0.109
1,410
7
.0050
.0119
30
47,980
555
.0116
.0115
1,356
17
.0125
.0126
31
40,199
484
.0120
.0122
851
10
.0118
.0134
32
41,528
565t
.0136
.0128
956
13
.0136
.0146
33
37,426
508
.0136
,0134
812
15
.0185
.0162
34
34,362
486
.0141
.0141
779
13
.0167
.0186
35
31,349
445
.0142
.0147
688
17
.0247
.0200
36.
29,399
496
.0169
.0154
636
12
,0189
.0190
37
. 26,213
419
.0160
.0160
544
7
.0129
.0160
38
24,664
380
.0154
.0163
555
9
,0162
.0135
39
20,790
326
.0157
.0158
436
5
,0115
.0115
40
17,023
232
.0136
.0145
383
6
.0016
.0102
41
12,252
173
.0141
.0129
201
3
.0149
.0083
42
11,012
126
.0114
.0114
205
.0000
.0068
43
7,457
85
.0114
.0101
155
1
.0065
.0056
44
4,746
37
.0078
.0088
85
1
.0118
.0045
45
2,755
21
.0076
.0075
58
.0000
.0036
46
1,389
10
.0072
.0063
36
1
.028
.0028
47
684
4
.0058
.0052
17
.0022
48
310
1
.0032
.0042
12
.0016
49
106
.0000
.0032
7
.0011
50
34
.0000
.0023
5
.0007
51
12
.0000
.0016
It
.0004
52
6
1
.1666
.0009
.0002
53
4
.0005
1
.0001
54
3
••
.0002
1 ■■
•■
Total
897,618
9,271
-/ '- —
.001032
54,913
428
,000778
• Triplets are included in the reanlt. t Maximum for confinements,
excess for multiple birtliB,
% Maximum
COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY.
305
The results shewn indicate that the increase with age, x, is as follows :
(549) j8"„ = 1 + 0.00064 (x — 12) ; up to age 37.
(560) j8", = 1 + 0.00070 [x^ 12) ; up to age 30 (?)
The excess of over is the quantity e in (548). Probably the age 37 could
be adopted in both cases. For later ages, Table XCIX. must be con-
sulted, and the values are uncer-
tain. As the numbers are small
this fact is, however, of little
moment. Later, the case will be
more fuUy analysed, for example, in
regard to the duration of marriage,
etc. Curves A and B, Fig. 82, shew
the nuptial and ex-nuptial results
respectively. The nuptial maximum
is 0.0163 for age 38.4, and the ex-
nuptial maximum is 0.0201 for age
35.5. The numbers after age 44 in
the nuptial, and after age 36 in ex-
nuptial cases are so small as to
make the results for later ages
doubtful, and the somewhat wide
dispersion of the ex-nuptial results
then probably is large for the later
.025 r
.020
.010
.015
.010
.000
1
'(
\
'■
...
/'■
A
'
A
s
"
■
y
^-^
\
/*"
y
P
^
^
^
^\
y
^'
^•
y
/
\
^
10
20
30
40
50
Fig. 82.
Curve A shews by a continuous
line the smoothed curve of the
excess over 1 at a. birth, according to
the age of the married mother ; and
Curve B similarly shews the results
for unmarried mothers. The dots
and circles shew the crude results.
7. Relative frequency of multiple births. — For the period 1881 to
1915, the relative frequencies of twins, triplets, and quadruplets were as
follow : —
TABLE C— multiple Births, Australia, During 35 Years, 1881-1915, and for Other
Places.
Population
Aggregate.*
Female
Population
Aggregate.*
Con-
finements.
Cases of
Twins.
Cases of
Triplets.
Cases of
Quad-
ruplets.
Cases of
Quin-
tuplets.
ITuiul>ers
Batios ..
113,900,167
34,208,424
53,955,512
16,204,832
3,329,594
1,000,000
102.02
12,064
665,919
32,636
9,802
118.25
6,527
276
82.9
.00846
1
53.2
5
1.50
.00015
.0188
1
Year.
Authority.
Total Births.
Con-
finements
Cases of
Twins.
Cases of
Triplets.
Cases of
Quad-
ruplets.
Cases of
Quin-
tuplets.
1871-80
1872-80
Neefet
Prinzlngt
Enlbbs
50,000,000
63,000^00
German Em-
pire
1,000,000
1,000,000
1,000,000§
12,080
11,677
12,856
156
143
124
1.8
1.3
1.33
0.2511
• Sum of the mean annual populations of the Australian States for which the necessary birth
statistics were taken o*t.
t ZuT Statistik der Mehrgeburten. Jahr. f . Nat. u. Stat., 1877, Bd. XXVni., p. 174.
} Medizinischen Statistik. H, Prinzing, p. 65.
§ Confinements 12,013,134 ; Twins 154,444 ; Triplets, 1489 ; Quadruplets, 16 ; in the
German Empire.
t Based on 15,965,391 children born, excluding still-births about 15,758,822.
306
APPENDIX A.
Quintuplets have been reported by Volkmann,^ Dasseldorf ; by
A. Bemheim,* Philadelphia; by Horlacher,' Wiirttemberg; by Nyhoff,*
Groningen ; in 30 cases collected by the last-named, the majority were
bom at between 4 and 5 months.
Sextuplets are reported by Vassali,* and Vortisch, Alburi,' and
sextuplets at Hameln in Westphalia in 1600' ; no cases, however, so far
as I am aware, have been reported in Australia.
The observed frequency of multiple births is as follows : —
TABLE CI. — Relative Frequency of Twins in Various Countries.*
Coiintry.
Period.
Frequency.
Country.
Period.
Frequency.
Australia
Switzerland . .
1881-1900
.0126
Spain
1863-70
.0087
Germany
1901-1902
.0127
Roumania
1871-80
.0088
Baden
1891-1900
.0128
France . .
1899-1902
.0109
Prussia
J,
.0129
Belgium . .
1890, 5,
.0111
Netherlands . .
.0129
1900
Hungary
.0131
Italy
1891-1900
.0117
Wiirttemberg
19
.0132
Kuasia
1887-91
.0121
Norway
1876-1880
.0133
Bavaria . .
1891-1900
.0123
Sweden
1871-80
.0146
Saxony . .
.0123
Finland
1891-1900
.0147
Austria . .
1896-1900
.0126
* The results other than for Australia are given in H. Prinzing's " Handbuoh der
medizinischen Statistik, p. 64.
The frequencies, however, have wide ranges of values. Thus, in
Italy, they ranged in the period 1892-1899 through .0080 for BasiUcata, to
.0148 for Venice. For rough approximations the order of frequency
with which twins, triplets, etc., occur, is as follows : —
8. Uniovular and diovular multiple births. — Observations as to the
frequency of what may be called uniovular and diovular production
of twins shew (i.) that the sexes are the same where the twins are produced
by the division of a single ovum ; (ii.) that this occurs in about one-iifth
or one-fourth of the cases, these being recognised by the fact that they
have common chorion ; and (iii.) that where the twins are produced from
two ova, the sexes may be identical or otherwise, these being recognised
by the fact that the chorion is divided.
Zentral bl. f. Gyn., 1879, p. 17.
Deutsche med. Wocheuschrift, 1899, p. 274.
Horlacher, Wfirtt., Korr. Bl. 1840.
Zeitsohr. f. Geb. u. Gyu., 1903, Bd. Iii., p. 173. •
Anatom. Anzeiger, Bd. x., No. 10.
Munch, med. Wochenschr., 1903, No. 38, pp. 1639-40 a photograph is given.
Date of birth, 9th January, 1600.
Deutsche med. Wochenschr., No. 19, 1899, p. 312.
COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY.
307
Statistics for an examination of this question are not available in
Australia, but are available for the German Empire. The data for 1906
to 1911 inclusive are as follows : —
TABLE on.— Frequency oJ Multiple Births (German Empire, 1906-11).
Confinements.
2
Males.
Pairs.
2 Fe-
males.
3
males.
2m.,lf.
lm.,2f.
3 Fe-
males.
Males
Born.
Females
Born.
Total
Quin-
tuplets.*
12,013,134
49,426
58,382
46,637
343
390
395
361
28
36
Children Born.
Total Cases of Twins.
Total Cases of Triplets.
Total Cases of
Quadruplets.
12,170,604
10,000,000t
154,444
"128,563
1,489
1,239.5
18.
13.3
3.
2.54
• This is based upon 15,965,391 children horn ; or about 15,758,800 conflnements in 1872 to
1880, during which time 4 quintuplets were born, t This would give the proportion 3.05. As is
evident for the number of children, the twins must be multiplied by 2, the triplets by 3, etc.
The proportion (^) of uniovular cases can be deduced at once from the
preceding figures. Let /x denote the masculinity, defined as the ratio
of the difference of the pairs of males and pairs of females to their sum ;
see (335), p. 132. Obviously, the uniovular cases are in the ratio (l+/x)
pairs of males to (1 — ju,) pairs of females. Th^ diovular cases are in
the same ratio as regards the same pairs, and the mixed pairs are equal
to both combined, that is they are : —
TABLE cm. — Theoretical Distribution oJ Diovular and Uniovular Cases Among
Cases of Twins.
Total T
2 males : Male and female + Female and male : 2 females
1 + M : 1 + 1 : 1 - At
l+yu.: + : I — iJ.
Of the total there are ^ uniovular and (1-
quently —
-^) diovular cases : conse-
(551).
.^ =
JI/ + F — P
Jf + .F + P
and
IJ- =
M
M -\- F
M denoting the number of pairs of males, F the pairs of females, and P
the cases of one of each sex. The above results thus give f = 0.24397
and n = 0.029023.
Direct observations according to Weinberg^ and Ahlfeld^ gave
respectively for the relative frequency of uniovulate cases 0.21 and .0172,
but it would appear from the preceding result that a sufficiently extended
number of cases could be expected to give a higher ratio.
* Beitrage zur Physiologie und Pathologie der Mehrlingsgeburten beim Menschen.
Arohiv f. ges. Physiol., 1901, Bd. Ixxxviii, p. 346 ; Neue Beitrage zur Lehre von den
Zv^illingen. Zeit. f. Geb. u. Gyn., 1903, Bd. xlviii., H. 1.
» Zeit. f. Geb. u. Gyn., 1902, Bd. xlvii., p. 230.
308 APPENDIX A.
A similar investigation may be applied to the more limited results
for triplets. Neglecting the masculinity tendency, it is obvious that for
the triovular and diovular cases the proportions of cases in each
category will be respectively : —
TABLE CIV. — ^Theoretical Distribution oi Diovular and Triovular Cases Among
Triplets.
Total T
3 males : 2 males and 1 female : 1 male and 2 females : 3 females
(M) (P) (Q) (F)
T(\- r)
Ti'
.125 .375 .375 .125
.25 .25 .25 .25*
* It is assumed that when the births m.f.m and f.m.f occur, the chance of the
two males or two females being uniovular is zero. If this condition were not
physiologically impossible, it is easy to see (by exhaiostive enumeration) that the
probabilities of the four cases would be 0.2 : 0.3 : 0.3 : 0.2.
An examination of the individual figures for each year shews that
the differences are too great to give any ground for deducing masculinity
to be other than zero. Hence we may take means adopting : —
352 : 392.5 : 392.5 : 352 instead of 343 : 390 : 395 : 361.
and this gives for the series of triovular and diovular births respectively :
20.25 : 60.75 : 60 : 75 : 20.25 and 331.75 : 331.75 : 331.75 : 331.75,
or 162 triovular and 1327 diovular births in all ; or ratio of diovular
cases of no less than 0.8912 of the total, the triovular being 0.1088.
Thus it follows that triovulation is a mtich rarer occurrence than the pro-
duction of uniovular twins, that is, the ratio of triovulation in triplets to
diovulation is 8.20. From the above we obtain by symmetrically in-
cluding all the data : —
(552) i'= i^Z{M + F)-{P + Q)] /{M + P + Q + F).
Thus, according to the recent experience of the German Empire, we have
for 10,000,000 cases of confinement, 31,365.5 cases of uniovulation
production of twins among the twins, and 1104.6 cases occurring among
the triplets. We may assume at least the same ratio for the cases of
quadruplets and quintuplets, which will give, say, 14.1 for both combined.^
Hence the ratio t, of occurrence for all cases of uniovular production of
twins (i.e., appearing as twins or as portion of triplets, &c.) : —
(553) t, = 0.0032484.
or, say, 13 cases in 4000, or 1 case in 308.
1 In quadruplets there are 16 possible orders in which births may occur, and
in these 24 possible cases of uniovulation. Since, however, the number of males and
females are unequal — 28 and 36 — the possible cases have not occiu'red, and hence
we may regard the 16 quadruplets and 3 quintuplets as roughly expressing the
probable number of cases. Sohroeder (Lehrbuoh der Geburtshulfe, 10° aufl.) gives
for twins 1 : 89, triplets 1 : 7910, quadruplets 1 : 371126.
COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY.
309
9. Small frequency of triovulation. — ^The preceding analysis appears
to shew that the triovular cases are only 162 in 12,013,134 confinements.
The probabihty of triovulation, ^ ', therefore, would appear to be : —
(554).
.r = 0.00001348.
or, say, I case in 74,000 confinements, though triplets occur at the rate
of 1 case in 8068 confinements in the German Empire. This subject
might well form the result of more definitive study when the data are
adequate.
10. Nuptial and ex-nuptial probability of twins according to age. —
The probability, in any nuptial or in any ex-nuptial confinement, of the
occurrence of twins has been ana-
Frequency of Twins according to Age.
.010
.006
.000
.010
.005
.000
Age
^
C^
•^
^
^
i
A^*.
°r*
k
K
^
>f*^
V
<•
/
y
.\
~^
10
so 30
Fig. 83.
40
60yrs.
lysed from an aggregate of the Aus-
tralian data- from 1908 to 1914, both
inclusive. It must, of course, be
in substantial agreement with the
result found for e in § 8. Table CV.,
columns (ii.) and (vi.), give the
number respectively of nuptial
and ex-nuptial confinements (totals
897,618 and 54,913) occurring in
AustraUa ia 8 years, and the num-
bers of twins corresponding to each,
viz., 9187 and 422. These are
shewn by curves A and B, Fig. 83,
the dots denoting the individual
results for nuptial cases, and the
firm Unes the smoothed results ; the
values for the latter being given in
column (v.) of the table. The ex-
nuptial cases are denoted by circles,
and where the numbers were small, the quinquennial aggregates only
were graphed. The rate of increase per year of age up to age 37 is for
nuptial and ex-nuptial cases respectively.
Curve A represents the ratio of the
number of cases of at least two births to
the number of nuptial confinements.
Curve B represents the same ratio
for ex -nuptial confinements.
Curve C represents the number of
cases of three or more at a birth to the
number of cases of two or more.
(555).
= 0.000632 {x — 12) and e', = 0.000668 (a; — 12)
X being the age of the mother. Beyond the age in question the results
can be taken from the table. The ratios for all ages are— nuptial,
0.010234, and ex-nuptial, 0.00768. The general result is (i.) that with
increase of age (and possibly duration of marriage) the frequency of twins
increases linearly, till the end of the ordinary child-bearing period is
approached, and (ii.) this increase is slightly greater for ex-nuptial cases,
viz., about 5.7 per cent, greater. The ex-nuptial relative frequency of
310
APPENDIX A.
twins for all ages combined is exactly 0.75 the nuptial relative frequency.
Since in the ex -nuptial cases the confinements are probably on the whole
not repeated, the result would appear to be due to age. This matter will
be further considered later.
TABLE CV. — Shewing Probability according to Age of the Occunence of Nuptial
and Ex-nuptial Twins, and of Triplets, based on 8 Years' Australian experience,
1907-1914.
Age.
Nuptial
Con-
fine-
Cases
of
Nuptial
Frequency of
Nuptial Twins.
Ex-
Nuptial
Con-
fine-
Cases
of Ex-
Nuptial
Frequency of
Ex-nuptial
Twins.
All
Twins.
All
Iriplets
Batio
of
Triplets
to
Twins.
ments.
Twins.
Crude
Smo'th-
ments.
Twins.
Crude.
Smo'th-
ed.
ed.
(i.)
(ii.)
(iii.)
a
(iv.)
(v.)
(vi.)
(viii.)
(vu.)
(ix.)
(X.)
(xi.)
(xii.)
12
.0000
5
.0000
.0030
13
4
.0006
21
.0007
.0035
14
30
34
.0013
126
152
"0
.0660
.0013
.0039
ie
170
"o
.0019
537
"1
.ooio
.0020
.0044
16
1,138
2
.oois
.0025
1,500
1
.0007
.0027
.0049
17
S,962
12
.0030
.0032
2,980
9
.0030
.0033
.0054
18
9,761
36
.0037
.0038
4,604
16
.0036
.0040
.0058
19
18,071
94
.0052
.0044
6,317
23
.0043
.0047
.0063
33.102
144
14,838
50
.0337
i94
"1
00.52
20
25,159
147
.0068
.0051
5,272
27
.0051
.0053
1
.0068
21
35,326
202
.0057
.0067
6,008
33
.0066
.0060
3
.0072
22
43,353
254
.0059
.0063
4,231
34
.0080
.0067
1
.0077
23
50,322
329
.0065
.0069
3,848
32
.0083
.0073
6
.0082
24
53,176
392
.0074
.0076
3,182
24
.0075
.0080
3
.0086
207,335
1,324
21,541
150
.0696
1,474
14
.0095
25
64,269
452
.0083
.0082
2,548
21
.0082
.0087
1
.0091
26
65,006
434
.0079
.0088
2,161
19
.0087
.0094
3
.0096
27
53,735
487
.0091
.0095
1,785
26
.0100
8
.0101
28
53,244
506
.0095
.0101
1,699
23
.0107
5
.0105
29
49,200
538
.0109
.0107
1,410
8
.0114
2
.0110
265,444
2,417
9,603
96
.oioo
2,6i3
19
.0076
30
47,980
548
.oii4
.oii4
1,356
14
.0i20
7
.0115
31
40,199
485
.0121
.0120
851
13
X127
4
.0119
32
41,528
659
.0135
.0126
956
11
.0134
8
.0124
S3
37,426
505
.0136
.0133
812
15
.0140
3
.0129
34
34,362
488
.0142
.0139
779
15
.0147
4
.0133
201,495
2,585
4,754
68
.0i43
2,663
26
.0098
35
31,349
436
.0140
.0145
688
17
.0i54
10
.0138
36
29,399
488
.0166
.0162
636
9
.0160
12
.0143
37
26,213
414
.0158
.0168
544
7
.0167
6
.0148
38
24,664
377
.0153
.0161
665
8
.0168
4
.0152
39
20,790
324
.0156
.0156
436
6
.0163
3
.0167
132,415
2,039
2,859
46
.oiei
2,085
34
.0163
40
17,023
226
.0133
.0143
383
6
.0150
6
.0162
41
12,262
171
.0140
.0127
201
3
.0134
2
.0166
42
11,012
123
.0112
.0112
208
.0118
2
.0171
43
7,457
85
.0114
.0099
156
1
.0104
.0176
44
4,746
36
.0076
.0086
85
1
.0090
1
?
52,490
641
1,029
11
.0167
662
U
.0169
45
2,756
21
.0076
.0074
58
.0077
7
46
1,389
10
.0072
.0062
36
1
.0064
?
47
684
4
.0060
.0051
17
.0052
?
48
310
1
.0032
.0041
12
.0042
?
49
106
.0031
7
.0032
?
5,244
36
130
1
.0077
■37
.0000
SO
34
.0022
6
.0023
?
61
12
,
.0016
1
, ,
.0016
?
52
6
1
.17'
.0009
.0009
7
53
4
.0005
1
.0005
7
54
3
59
"l
.0002
■7
.0002
"1
7
.0000
Not"
Stated
Totals
897,618
9,187
.01023
54,913
422
.00768
9,609
105
.01093
11. Probability of triplets according to age. — ^The results of the
8 years, 1907-14, gave the following results for nuptial and ex-nuptial
twins and triplets, viz. : —
COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY.
311
Twins.
Triplets.
Nuptial.
Total.
Ex-nuptial.
Nuptial.
Total.
Ex-nuptial.
Numbers
Ratio
9,187
1.0000
(9,609)
422
.0459
98
1.000
(105)
7
0.071
The numbers are too small, however, to establish that the frequency of
the occurrence of triplets ex-nuptially is between 50 and 60 per cent,
greater than nuptially.
If the frequency be related to the number of twins, it is roughly
given by the smoothed results in column (xii.) of Table CV. We shall
call the probability Pg /P2 say, t^. Thus we shall have : —
(556) . . . .T3= 0.0030 + 0.00047 {x — 12); or = 0.00047 {x — 5.6)
the second form, however, being without meaning till the age of child-
bearing. The firm line, curve C, on Fig. 83, denotes the increase ; the
crosses represent the group results used in deducing this.
12. Probability o£ twins according to duration of marriage.— Given
a birth, the probabihty of a second child being born is found by dividing
the number of twins, including triplets, by the number of confinements
tabulated according to duration of marriage. Thus, column (v.) in Table
CVT. is found by dividing the figures in column (iii.) by those in column
(ii.). The crude results are shewn by the dots in Fig. 84, and the smoothed
results by the firm line, curve A. For the form of the initial part of the
curve see § 14, and also Fig. 85 hereinafter.
13. Probability 0! triplets according to duration of marriage.— The
probability of a Jhird child being born may, as before, be referred to
the number of cases where a second
child has been born. This probability
is found by dividing the number of
triplets by the number of twins, in-
cluding the triplets, etc. But the
numbers to be dealt with are so small
and irregular that the expedient was
adopted of forming groups of eleven.
As no correction was apphed for the
systematic error of the grouping, the
curve represents the ratio of 11 -year
groups of duration of marriage, the
argument being the central years of
the group. The results are shewn on
Fig. 84, curve B, and the data are
shewn in Table CVI., and seem to
indicate the change with duration of
marriage is sensibly a Unear one
through for the major part (presum-
ably) of the child-bearing period.
.030
.010
.000
.010
.000
■y^
■'•B
\,
r^
•\
V.
/
><^
A-^
^
V'
\'
•\
10 so
Duration of Marriage.
30
40
[years.
Fig. 84.
Curve A denotes the frequency
of the birth of two or more children
to the number of confinements.
Curve B denotes the ratio of
11-year means of the number of
triplets to the number of cases of
two or more children.
312
APPENDIX A
TABLE CVI. — Probability of Twins* and Tripletsf according to Duration of
Marriage. Australia, 1908-1914.
Dura-
Ratio of Twins to
Ratio of Triplets to
tion
Con-
Twins
Confinements.
Twins (Groups of U).
of Mar-
finements
including
Triplets.
Triplets.
riage.
Crude.
Smoothed
Crude.
Smoothed
(i.)
(ii.)
(iii.)
(iv.)
(V.)
(vi.)
(vii.)
(viii.)
0-1
134,171
1,129
9
.0084
0084
.0073
1-2
61,213
460
3
.0075
.0075
.0073
2-3
64,229
465
4
.0072
.0072
, ,
.0073
3-4
70,317
564
3
.0080
.0080
.0073
4^5
59,407
551
2
.0093
.0090
, ,
.0073
5-6
53,275
504
4
.0095
.0098
.0074
.0073
6-7
47,250
468
1
.0099
.0106
.0072
.0075
7-8
41,713
492
3
.0118
.0113
.0078
.0080
8-9
37,115
466
7
.0125
.0120
.0077
.0087
9-10
32,170
417
3.
.0130
.0126
.0088
.0095
10-11
29,607
404
5
.0136
.0132
.0112
.0102
11-12
25,887
328
2
.0127
.0138
.0115
.0109
12-13
23,372
352
5
.0151
.0143
• .0125
.0117
13-14
20,339
273
2
.0134
.0148
.0130
.0124
14-15
17,-572
281
6
.0160
.0152
.0128
.0131
15-16
15,217
228
9
.0150
.0154
.0138
.0138
16-17
13,271
196
2
.0148
.0152
.0139
.0146
17-18
11,617
159
1
.0137
.0149
.0153
.0155
18-19
10,073
139
.0138
.0145
.0152
.0158
19-20
8,520
117
2
.0137
.0139
.0164
.0158
20-21
7,424
89
2
.0120
.0132
.0149
.0149
21-22
5,988
76
.0127
.0124
.0087
.0121
22-23
4,726
46
1
.0097
.0114
.0083
.0095
23-24
3,561
35
.0098
.0103
.0105
.0068
24r-25
2,664
34
.0128
.0092
.0043
25-26
1,809
22
.0122
.0080
.0028
26-27
1,146
8
.0070
.0067
.0016
27-28
643
2
.0031
.0054
.0010
28-29
383
4
1
.0104
.0041
.0006
29-30
192
.0028
.0003
30-31
77
.0016
.0002
31-32
45
.0010
.0002
32-33
16
.0006
•
.0001
33-34
5
.0004
.0001
34-35
.0003
.0001
35-36
1
.0002
.0000
TotaU
805,015
8,308
77
.010320
.00927
•■
* That is, of two or more occurring at a birth,
two are bom.
t That is, of third child in any case where
14. Remarkable initial fluctuation in the irequency of twins, accord-
ing to interval after marriage. — -There i» no known ground for supposing
that the ratio of the number of twins to the number of confinements in
which they occur, can in any way depend on the interval after marriage,
at leaist, if that interval be small. The results in Tables CVII. and CVIII.
COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY.
313
hereunder for the years 1908 to 1915 and 1908 to 1914 respectively, shew,
however, that apparently the dependence exists. The average for the
first three months after marriage equals that of the third three months,
and both are very much above the average. The second and fourth
periods of three months are about equal. These results are shewn by
curve C on Fig. 85.
TABLE CVn. — Shewing Variation in the Fiec[uency of Twins during the First
24 Months after Marriage. Australia 1908-1915.
Twins Born during
Interval after
Marriage of Months
Confinements during Intervals
Ratio of Twine during Intervals
Year.
after Marriage of Months
after Marriage of Months
0-3
3-6
6-9
9-12
12-24
0-3
3-6
6-9
9-12
12-24
0-3
3-6
6-9
9-12
12-24
1908
16
24
34
56
\ 60
1,533! 3,152
,4,006
7,007
6,298
.0104
.0076
.0085
.0080
.00951
.0068]
1909
21
2«
• 44
62
48
1,799 3,556
4,139
7,307
6,973
.0116
.0073
.0106
.0085
1910
19
29
58
as
48
1,888 3,659
4,474
7,500
6,919
.0101
.0079
.0129
.0079
.0069
1911
15
31
49
64
56
1,987, 4,075
5,220
7,877
7,400
.0076
.0076
.0094
.0081
.0075
1912
27
82
60
61
60
2,119 4,458
5,827
8,899
8,518
.0127
.0072
.0103
.0069
.0071
1913
17
34
61
66
65
2,107, 4,502
5,916
9,301
9,142
.0081
.0076
.0103
.0071
.0071
1914
14
32
j>8
60
63
2,080 i 4,268
5,897
9,185
9,247
.0067
.0075
.0098
.0065
.0069
1915
28
46
51
76
82
2,023, 4,149
5,828
8,795 8,953
.0099
.0111
.0088
.0086
.0091
Totals
157
254
415
504
482
15,536 31,819
1
41,307
65,871' 63,450
.01010
.00798
.01005
.00765
.00760
Thus the proportion of twins for all pre-nuptial conceptionB is high. It is
to be noted, however, that the proportion of ex-nuptial twins over all is
low (see Table CV.), and it is not unUkely that the initial high rate, and,
in general, the higher rate for the cases due to pre-nuptial insemination
is due to the transfer, owing to the peithogamic influence, of what might
have been ex-nuptial to the nuptial cases. To obtain the fluctuation
more exactly, the results were taken out monthly, from 1908 to 1914,
according to interval after marriage.
TABLE CVm. — Shewing Variations in the Freaueney of Twins foi each Interval of
One Month after Marriage (First Births only), and of Triplets. Australia
1908-14.
Interval* . .
Twins
Confinements
0-1
39
3,529
1-2
40
4,059
2-3
50
5,925
3-4
55
7,455
4-5
70
9,055
5-6
83
11,160
6-7
85
13,870
7-8
109
11,545
8-9
170
10,064
9-10
195
24,434
10-11
146
19,047
11-12
87
13,595
0-12
1,129
133,738
Katio
.0110
.0098
.0084
.0073
.0077
.0074
.0061
.0094
.0169
.0080
.0076
.0064
.00844
Intervalf
Twins
Confinements^
1-2
400
54,497
2-3
141
15,801
3-4
58
6,458
4r-7
59
6,413
7-11
17
2,209
11-26
7
905
1-26
682
86,283
0-26
1,811
220,021
Interval'
Triplets
Twins
0-1
8
1,12s
1-26
6
682
Batio ..
.0073
.0089
.0091
.•0092
.0077
.0078
.00790
.00823
Ratio
.0071
.0088
• Months. t Years. First births.
314
APPENDIX A.
tl
s
V,
s)\
■\3-
—-
—
/
/
\
0-3
c ,
J
J
\
A
LJ
n
t""
-—
-
_
.020
d
I
" .010
.000
Q 150
I 100
|Zi 50
S 10 15 20 25 30 35 40
Duration of marriage, [months
Fig. 85.
Curve A denotes the actual
number of twins in Australia during
7 years' experience.
Curve B denotes the ratio of
cases of births of 2 or more children
to oases of confinement.
Curve C denotes, similarly to
curve B, the group ratios for three
months, however, instead of one.
The ratio for 1-4 i^i .0078, and for
5-26 is .0087. The numbers for the
lesser subdivisions are doubtless too
small to rely on the results. The
results shewn are for first births
only ; but for the smaller durations
the distinction is without meaning.
Fig. 85 shews the results, curve A
denoting the actual number of twin
births, and curve B the frequency with
which twins occur.
15. Frequency of twins according
to order of confinement. — ^From the
frequency of the occurrence of twins
according to previous issue, an estima-
tion according to order of confine-
ment can be made by taking account
of the probability of twins or
triplets, &c. From the frequency
according to previous issue, it may be
deduced that the probability of twins is approximately as follows : —
Previous Confinements
Probability (about)
.0082
1
.0096
2
.0107
.0117
4
.0124
5
.0180
6
.0134
7
.0136
.0138 .0139
10
.0140
We have also from the general result that the frequency of single births,
twins, and triplets in Australia was, for 1908-14,
799831 : 8247 : 77 1
1 : 0.010311 : 0.000096 |
or roughly, say,
10,000 : 100 : 1
The probability of twins occurring twice, 2P2, ^, therefore, approxi-
mately identical with that of the occurrence of triplets, ^3, that is : —
(557).
!!?'2 = fl = Pa' approximately.
The number entered under will be correct. That is, the cases " accord-
ing to previous issue," and " according to previous confinements" are
identical. But in every case where there were twins or triplets, etc., at
the first birth, the cases would be tabulated under " previous issue,"
2 or 3, etc., respectively, instead of under 2 ; and similarly mutatis
mutandis for all later columns in the " according-to-previous-issue"
COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY.
315
tabulation. We therefore must add the appropriate numbers, and deduct
equal numbers from later columns. The precision of the result will, of
course, never be of a high order.
The data are given in the upper part of Table CIX., and the approxi-
mate restatement according to the order of confinement forms the lower
part of the table.
TABLE CIX. — Frequency of Multiple Births according to Previous Issue. Australia
1908-14.
Previous Issue (upper table), or Order of Confinement (lower table.)
Numbers.
1
2
3
4
. 5
6
7
8
9
Cases ot at least
2 children . .
Cases ot at least
3 children ..
Mothers of at
least 1 child
Ilatio ot twins
to mothers
1,811
12
220,80?
.0082C
1,357
10
167,091
.008121
1,325
7
125,779
.01053
1,094
7
92,116
.01188
834
8
65,343
.01276
591
5
46,156
.01280
477
9
31,733
.01503
306
2
21,918
.01396
22"
14,72'
.01541
! 127
i 1
r 9,671
L .01313
Ac-
cord-
ingto
order
of •
Con-
fine-
ment
" Twins
Mothers
Corres-
ponding
L Batio
1,811
220,807
.0082C
1,386
169,851
.00816
1,337
126,377
.01058
1,096
92,083
.01190
I
831
65,099
.01277
590
45,683
.01292
467
31,253
.01494
302
21,467
.01407
21S
14,28'
.0152
i 122
I 9,254
5 .01318
lUtio Triplets
Smoothed . .
.000056
.000062
.000074
.000088
.000106
.000130
.000158
.000193
.00023
> .000286
Numbers.
10
11
12
13
14
15
16
17
18
19
20
21
22
Cases ot at least
2 children . .
Gases of at least
3 children . .
HotheiB ot at
least 1 chUd
Batio of twins
to mothers
79
6,694
.01387
39
3,181
.01226
21
1,665
.01261
9
814
6
388
(
14'
! 1
) 1
L 59
.0
25
6
3
1
1
1
1
0.01388
Ac--
cord-
ing to
order .
of
Con-
fine-
ment
r Twins
Mothers
Corres-
ponding
'- Batio
74
5,378
.01378
37
2,964
.01248
19
1,530
.01242
8
740
.01081
5
340
.01471
12
.0157
1 1
r 52
> .01923
21
5
2
1
1
1
1
Since the correction system affects the number of twins and the
mothers in the same way, it obviously cannot produce any appreciable
difference in the ratios, though it may alter the numbers. This is seen
in the results given in the table above. If the number of triplets be
smoothed, the result shewn in the final line is obtained. But the numbers
are too small to lead to any reliance upon their value, though they con-
firm in a general way the dictum that multiple fecundity increases with
the issue, thus also with age and duration of marriage.
316
APPENDIX A.
TABLE ex.— Shewing Secular Variation in the Frequency of Twins and
Triplets.
Australia, 1881-1915.
No. of
Satio of Twinst
Katio of Triplets
Conflne-
Cases of
Cases of
Cases of
to Confinements.
to Twins, etc.t
Year.
raenta
2 or more
3 or more
4 or more
dotal).*
CluldTen.
Children.
Cbildien.
Crude.
Smootbed.
Crude.
Smoothed.
"oT
(ii.)
(iii.)
(iv.)
(V.)
(vi.)
(vii.)
(vlu.)
(ix.)
1831
63,818
645
7
.00864
.0080
2
64,069
496
3
.00774
.0082
3
68.135
675
,2
.00843
.0084
i
72,832
629
8
.00863
.0086
5
76,026
661
3
.00869
.0087
.0063
.0063
6
79,009
682
2
.00863
.0088
.0066
.0066
7
83,085
704
4
.00847
.0090
.0064
.0066
8
86,393
875
6
1
.01012
.0096
.0066
.0062
9
87,195
859
3
.00985
.0099
.0068
.0070
1890
91,030
910
5
.00999
.0102
.0072
.0076
1
91,734
941
4
.01026
.0103
.0083
.0081
2
91,980
784
12
.01023
.0102
.0082
.0082
3
90,379
899
11
1
.00994
.0100
.0080
.0081
4
86,384
797
7
.00922
.0096
.0081
.0081
5
91,225
907
12
1
.00994
.0094
.0085
.0085
6
86,526
775
4
.00896
.0094
.0089
.0088
7
90,614
960
6
1
.01069
.0099
.0085
.0089
8
88,993
883
4
.00992
.0104
.0086
.0088
9
90,244
971
9
.01076
.0107
.0088
.0087
1900
92,057
985
7
.01069
.0108
.0084
.0086
1
92,826
1,005
11
.01082
.0107
.0089
.0088
2
92,852
972
12
.01046
.0104
.0088
.0092
3
89,060
877
10
1
.00984
.0102
.0098
.0095
4
93,973
1,005
9
.01069
.0104
.0093
.0097
5
95,060
1,012
11
.01064
.0107
.0099
.0099
6
97,867
1,083
5
.01106
.0107
.0100
.0100
7
100,161
961
13
.00949
.0102
.0099
.0099
8
110,491
1,066
6
.00963
.0098
.0100
.0097
9
112,921
1,142
14
.01011
.0100
.0096
.0096
1910
116,609
1,189
13
.01028
.0102
.0092
.0093
1
120,967
1,236
14
.01021
.0102
.0093
.0089
2
131,726
1,360
16
.01024
.0101
3
134,343
1,369
8
.01019
.0101
4
136,676
1,406
11
.01029 .0102
6
133,444
1,417
10
.01061 .0104
Totls
3,221,694
32,917
281
6§
.010217
.00863
* That is, nuptial and ex-nuptial. t Including triplets and auadiuplets. t That is, the
ratio of 9-year groups of triplets including quadruplets to O-year groups of twins, including triplets.
% Batio of quadruplets to triplete = 0.018.
16. Secular fluctuations in multiple-biiths. — ^The ratio of multiple
births to confinements would appear a priori to be independent of time,
but it win be seen from Pig. 86
Secular Fluctuation in Relative
Frequency of Births and Twins
and Triplets.
« 3 .005
..000
<1.010
is
4 t .005
.000
c
V
'^'
J ~
'
Irac
b.
'■^V'
^■^
-^1
.036
.030 C
.025 C
.020 C
.015 C
1880
Fig.
1900
86.
10
Curve A denotes the smoothed
secular fluctuation of the ratio of births
of two or more to the number of con-
finements.
Curve B denotes the ratio of 11 -year
groups of births of three or more to the
number of births of two or more.
Curve C denotes the crude birth
rate and number of births per unit of
the general population.
that there are indications of a
definite secular fluctuation, see
also Table CX. above. The
number of confinements which
constitutes the basis of the experi-
ence is more than doubled in the
35 years under review (see column
ii.), and the number of twins
(which includes triplets and quad-
ruplets) is large. The aggregate
experience includes 3,221,594 con-
finements, in which there were a
total of 32,917 births of two or
more children, a total of 281 births
of 3 or more children, and 5
quadruplets. These give the ratios
shewn in the table. In Pig. 85,
curve A is the smoothed secular
fluctuation-curve of the twins ;
curve B that of the triplets (which
COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY.
317
were grouped in nines) ; while curve C shews the fluctuations of the
crude birth-rates for the same years. The individual values are shewn
by dots. It will be observed that on the whole the frequency of twins and
triplets rises as the frequency of births diminishes.
17. Comparison of nuptial and ex-nuptial fertility. — ^In columns (x.)
and (xvi.) of Table LXXIII., p. 242 hereinbefore, the crude and smoothed
ratios for ex-nuptial fertility, attributed wholly to the " never married,"
were given. The crude results are repeated in column (ii.) of Table XCI.
hereunder. If attributed to the " unmarried," which includes the
widowed and divorced, the results in column (iii.) are obtained, and the
corresponding smoothed results are shewn in column (iv.). Reference to
the table shews that the maximum fertility is nuptially attained at about
the year of age 18.3 to 19.3, and is about 0.484. The maximum fertiUty
is ex-nuptially attained, however, only at about age 21.5 to 22.5, and is
about 0.0182 ; that is to say, the maximum is about 3.2 years later, and
the proportion at the maximum is only 0.0376, or say 3/80ths. For all
ages from 12 to 57 we have for nuptial-fertility ratio 0.1704, and for the
ex-nuptial ratio 0.00993. Hence
Nuptial and Ex-nuptial FertiUty-ratios. the proportion of the averages
is 0.05828. It is obvious that
the initial parts of the curves
representing the nuptial and ex-
nuptial fertility-ratios are not
Ukely to be identical, because
the nuptial denominator for
early ages will be small, and the
ex-nuptial denominator will be
large. Curves A and C, Fig. 87,
denote respectively the nuptial
and ex-nuptial curves. By the
process indicated in § 2, p.
298, the results in columns (vi.)
and (vii.) of Table CXI. are
obtained ; these are shewn in
Fig. 87 by curve C ; hence the
curves are not in planar correspondence. If, however, the curve A be
corrected for the effect of previous births, the two curves come into
closer correspondence^ ; that is, ex-nuptial fertility has, in general, nearly
the same characteristics as nuptial fertility, excepting that the greater
measure of restraint operates to make the maximum occur later, and to
enormously reduce the ratio.
Age 10
Curve
ratio.
Curve
curve A.
Curve
ratio.
Fig.
A denotes the nuptial fertility
B is the oblique projection of
C is the ex-nuptial fertility
1 It is obvious that the ex-nuptial curve does not need the same correction,
since oft-repeated ex -nuptial maternity is not likely to occur.
318
APPENDIX A.
TABLE CXI. — Comparison of Nuptial and Ez-nuptial Fertility-iatios according to
Age. Australia 1907 to 1914.
Ratio of Ex-nuptial Births to —
Batio of
Nuptial
Births to
Ex-nuptial Bate
Computed by
Age of
Mother.
the "Never
the " Unmarried."
the
Oblique Projection."
Married."
Married.
Crude.
Crude.
Smoothed.
Smoothed.
Rate.
Age.
(i-)
(ii.)
(iii.)
(iv.)
(V.)
(vi.)
(vU.)
12
.0000
.0000
.0000
13
.0001
.0001
.0001
U
.0004
.0004
.0004
.207
.0077
V5.4
15
.0016
.0016
.0016
.227
.0085
16.5
16
.0043
.0043
.0043
.301
.0113
18.0
17
.0085
.0085
.0085
.458
.0171
20.1
18
.0131
.0131
.0131
.483
.0181
21.3
19
.0162
.0162
.0158
.479
.0179
22.3
20
.0172
.0172
.0174
.464
.0174
23.2
21
.0181
.0181
.0181
.443
.0166
24.0
22
.0173
.0171
.0181
.416
.0156
24.9
23
.0183
.0182
.0177
.381
.0142
25.7
24
.0176
.0174
.0172
.352
.0132
26.4
25
.0163
.0161
.0163
.333
.0124
27.3
26
.0157
.0154
.0154
.319
.0119
28.1
27
.0147
.0143
.0149
.307
.0115
29.0
28
.0157
.01.52
.0145
.293
.0110
30.0
29
.0145
.0139
.0141
.274
.0102
30.9
30
.0157
.0150
.0136
.256
.0096
31.7
31
.0111
.0104
.0131
.241
.0090
32.6
32
.0138
.0128
.0127
.225
.0084
33.5
33
.0131
.0119
.0123
.210
.0079
34.4
34
.0135
.0121
.0119
.197
.0079
35.3
35
.0129
.0113
.0114
.185
.0069
36.3
36
.0127
.0109
.0108
.174
.0065
37.2
37
.0116
.0097
.0101
.164
.0061
38.1
38
.0125
.0101
.0093
.149
.0056
39.0
39
.0103
.0082
.0083
.130
.0049
39.9
40
.0097
.0074
.0070
.108
.0040
40.7
41
.0055
.0041
.0054
.087
.0033
41.6
42
.0060
.0043
.0042
.067
.0025
42.4
43
.0049
.0033
.0030
.0.50
.0019
43.3
44
.0029
.0019
.0020
.033
.0012
44.2
45
.0021
.0013
.0013
.020
.0007
4.5.1
46
.0014
.0008
.0008
.010
.0004
46.0
47
.0007
.0004
.0004
.005
.0002
47.0
48
.0005
.0003
.0003
.003
.0001
48
49
.0003
.0002
.0002
.001
.0000
49
50
.0003
.0001
.0001
.001
.0000
50
51
.0001
.0000
.0000
.000
.0000
51
• The oblique projection brings the maximum points into arbitrary agreement, tlie values for the
ages indicated also being determined thereby. The rates lor these ages are tound from those of the
nuptial curve by using tlie projection-ratio.
COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY.
319
The difference between the nuptial and ex-nuptial probabilities of con-
finement are more comprehensively indicated by a decennial table. In
Table CXII. hereunder these are given as the number of cases respectively
occurring per 10,000 married and per 100,000 "never married" women.
The rates, based upon the numbers of the " unmarried," are somewhat
smaller.
TABLE CXn. — Shewing the Probabilities of Nuptial and Ex-naptial Confinement
and their Ratio, for Five-Year Age-gioups. Australia 1907-1914.
"
Batio oi
Probab-
No. of
No. of
Probab-
No. of
No. of
Probab-
Probab-
ility of ex-nuptial
to nuptial matRpn-
Married
Cases o(
ility of
" Never
Un-
No. of
ility of
ility of
ity
Age
Women
at
Nuptial
Confine-
Matern-
ity*
Married"
Women
married
Women
Cases of
Bx-nup-
Matern-
ityt
Matern-
ity
Groups.
Census
ment in
during
ai
at
tial Con-
during
during
Based
Based
1911.
8 Years.
1 Year.
Census
1911.
Census
1911.
finement
1 Year.
1 Year.
upon the
Never
Married.
upon the
Un-
Married.
11-14
19
34
2,226
168,778
168,778
152
11
11
.0005
.0005
16-19
8,637
33,245
4,791
214,875
214,905
14,889
862
862
.0180
.0180
20-24
65,506
208,667
3,962
152,967
153,514
21,695
1,765
1,759
.0445
.0444
25-29
109,832
267,886
3,036
78,036
79,918
9,696
1,546
1,510
.0509
.0497
30-34
112,532
204,093
2,257
44,341
47,903
4,822
1,353
1,253
.0600
.0555
35-39
104,825
134,481
1,597
29,953
35,888
2,909
1,208
1,009
.0757
.0632
40-44
94,917
53,143
697
21,483
30,325
1,040
602
427
.0865
.0613
45-49
82,263
5,280
80
15,006
27,172
131
108
60
.136
.075
50-54
60,939
60
1.2
9,784
23,463
7
9
3
.73
.025
55-60
38,905
4
.12
5,698
20,063
?
?
* Probability per annum per 10,000 married women of same age-group.
t Probability per annum per 100,000 "never married" women of same age-group.
18. Theory of fertility, sterility and fecundity. — The fertility-ratio
ov probability of maternity in a unit of time may be defined as the proportion
of cases, which, subjected to a given degree of risk for a unit of time, result
in maternity ; and similarly, the sterility ratio or probability of maternity
is the arithmetical complement of the probability ; or calling these
respectively p and q, p-\-q = 1. If instead of " a unit of time," we write
" varioics given periods of time," we arrive at the conception of a varying
degree of fertiUtyor sterility, which for brevity, we may call the fertility, q,
or the sterihty, 0. That is to say, instead of making a sharp qualitative
cleavage between the fertile and the infertile or sterile, both are to be
regarded as varying quantitatively. Any compilation shewing the
frequency of cases of maternity according to duration of marriage reveals
the propriety of this mode of envisaging the question. But we have
seen that fertility decreases after a certain age, hence age must also be
taken into account. Further, the " degree of risk" varies with the age
of the husband. Hence, if x denote the age of the wife, y that of the
husband, i the duration of the risk, we have : —
(558) q =f{x,y,i) ; and s = 1 — q
Fertility and sterility in the sense indicated are determined by the
question of a single case of maternity. If instead of this we substitute
" result in n cases of maternity," or " result in the bearing of n' children,"
we arrive at the quantitative conception of fecundity. It is not unlikely
that the " degree of risk" varies with the number of previous births.
If so, we must write (x, y, i, n) in this last equation.
320
APPENDIX A.
If the total number of married women of age x be denoted by xM,
the duration of their marriage be denoted by a suffix i, the number of
nulliparae, primiparae, and multiparse up to « by the suffixes 1,2, . . . n,
then we can have compilations of the types
(559).
^M =^Mo + ,Mi+....^Jfi
(5601.
,M = ^M'o + ;.M'i + ^M'„
that is, compilation according to age and duration of marriage, or according
to age and " issue." It is at once evident that an exhaustive compilation
according Ui x, y, i and n is out of the question, since the individual
numbers in each " parcel" would be too small. Hence, serviceable tables
must ignore some of the factors.
In some countries fertihty probably varies but slightly with the age
of the husband, and in all the distribution according to the age probably
does not materially vary. Hence, by ignoring the issue, tables of
" fertility and sterility " and of " fecundity" may take the following
forms, the partial tables serving all general practical purposes : —
Tables of Fertility and Sterility (effect of " Previous Issue" being Ignored).
Arguments of complete tables.
Argument of partial tables.
(i.) Age of wife, with (ii.) age of
husband,
(iii.) Duration of marriage.
(i.) Age of wife only (i.e., with hus-
bands of all ages),
(ii.) Duration of marriage.
The tables themselves should shew, for each combination of age and
duration of marriage, the proportion of married women who have borne
one child.
Tables of Fecundity (effect of " Previous Issue" being Ignored).
(i.), (ii.), and (iii.) as above.
(i.) and (ii.) as above.
The tables themselves should shew, for esich combination of age and
duration of marriage, the proportion of married women who have borne
n children, where n is successively 0, 1, 2, 3, 4, etc., etc.
Such tables wiU need to be for small age-groups (say for single years),
and for durations of marriage, which change by smaU amounts (say one
year), inasmuch as the age and duration change together, and the effect
of age is considerable,
COMPLEX ELEMENTS OP FERTILITY AND FECUNDITY.
321
19. Past fecundity ot an existing population. — The past fecundity
of any population as at a particular moment is given by a census, both
according to " duration of existing marriage" and according to " age."
The usual tabulation according to existing marriage ignores the fact that
the record is incomplete, and that for deduction purposes a previous
marriage may to some extent modify the fecundity. The results in
Tables CXIII. and CXIV. hereunder are deduced from the Census tabula-
tions by applying the method outlined in § 4, p. 300, to the crude results.
The aggregates for the same " issue" are not, of course, in agreement
since in the one case the numbers according to the issue from existing
marriages are recorded, and in the other, the numbers according to age
include all previous issue.
Numbers who boie 1, 2..n Children; also Proportion found to be Sterile.
Age of Wife.
thoua-
ands.
(
1
"30-
s
S
10^
3
iTTo
V-
-e
d ^
ft/
•|a
30 70
//
V
•e
a
Ik
>
s
\
fi «
1 20 60
o
A
v^
1«
5000 ^
\
\
o
s
S 10 50
;;;f
p
£-
^
Ss
\
\zo
-4
4000 'S
SC
^
=^>
\
^_
S
\.—
X
■<
~ 40
-S
>-
=^
^'^
P^
4-
•0
3000 g
^ W
7
u
1 30
1 *
0-4
9
1
w
2000 ^
/
s
^
\
^ S 20
.-'
(
#
\\
\
|2i
«
/
\ <
r
\
^
^y
\
fc 10
/
-^
^
»?s
3^
^
^
'
,„___^
—
^
^
,A
-^
^
is-.
I
■
2
1
> ,
i •(
(
5«
g
x
^
^*
is
^
= ^
!-^
)
'
1
1 <
s
9 •
N
r f
uml
I
9
jerof
10 1
Child
88.
I L
lien
1 1
3 1
« I
S 1
e 1
T I
e
Curves a to i shew numbers who bore to n children during durations of
marriage to 4, 5 to 9, 10 to 14, etc., see Table CXIII.
Curves a' to j' shew the numbers who bore to n children according to age
and without regard to duration of marriage ; curve a' denoting all under 20 ;
curve b' all aged 20 to 24 last birthday ; curve c' all aged 25 to 29, etc. ; see
Table CXIV.
Curva-s 15 tp 20 shew numbers of wives who bore to n. children for ages
15 to 20 last birthday ; see Table CXIV.
These curves are valid only for integral values of the abscissa (number of
children).
Curve A shews the proportion of wives according to age, but of all durations
of marriage, who proved sterile.
322
APPENDIX A.
Table CXm.-
-Shevnng Issne of 1,000,000 Wives according to Duration of Existing
Duration Existing
NUMBEB OP WiTEB WHO HAD OIVBII BlETH TO ChILDBBK TO THE NUMBEE 01" —
Marriage.
0.
1.
2.
3.
4.
5.
6.
7.
8.
9.
Under 5 years . .
5-9 years
10-14 „
15-19 „
20-24 „
25-29 „
30-34 „
35-39 „
40-44 „
45 and over
73,765
23,504
16,031
9,586
7,374
5,082
2,947
1,904
1,055
970
82,436
28,564
15,059
8,821
8,465
3,806
2,038
1,212
800
585
37,904
50,165
22,961
13,150
9,714
5,450
2,869
1,438
778
606
6,874
47,053
27,141
15,427
12,803
7,701
3,586
1,921
948
821
469
24,421
28,897
16,200
13,916
9,413
4,684
2,600
1,206
1,094
23
7,800
22,421
14,542
13,278
10,078
5,581
3,080
1,582
1,513
1,776
13,774
13,072
12,088
10,095
5,977
3,478
2,024
1,883
344
6,325
10,191
10,253
9,162
8,338
3,884
2,330
2,374
'87
2,285
6,814
8,338
8,043
8,223
4,354
2,611
2,859
"7
718
3,767
6,602
6,858
5,734
4,101
2,618
3,067
Totals for existing
marriage
142,218
149,584
144,833
128,055
100,900
79,898
64,146
51,179
41,492
33,260
Total per million
(or all ages . .
123,995
146,153
145,107
124,239
103,088
82,140
67,029
53,803
44,026
35,392
• This does not include children by previous marriage , or ex-nuptial children ; it shews the relative
frequency of issues of a given number according to " duration of marriage."
t The actual total number of wives was 733,773, of which 3747 gave no information either as to
durationof marriage or as to number of children ; 12.073 gave no information as to number of children,
but stated their age ' and 21,151 gave no information as to age, but stated the number of children.
The 3747 were distributed proportionately to the partially specified totals, the two parts being 1362
Table CXIV
.—Shewing Issue of 1,000,000 Wives according to Age, at
NffMBEE OF Wives to whom had been Bobit Childeen to
THE HVKBER OF —
Age of
wives.
1.
2.
3.
4.
6.
6.
7.
8.
9.
10.
13
1
14
18
7
, ,
. ,
. ,
, ,
, ^
16
92
34
, ,
, ,
, .
16
249
207
14
17
879
701
61
6
..
18
1,445
1,723
298
19
1
1
19
2,088
3,002
747
123
8
20
2,987
4,751
1,765
320
22
3
21-24
19,474
29,892
19,136
7,882
2,215
439
108
18
4
25-29
25,137
38,640
38,232
24,981
14,161
6,362
2,357
749
215
S3
7
30-34
18,429
26,026
30,571
27,374
21,084
14,291
8,831
4,645
2,082
838
284
35-39
14,383
15,169
20,990
21,917
19,799
16,043
12,429
8,728
5,785
3,568
1,945
40-44
12,037
10,458
14,208
16,019
18,525
14,877
12,835
10,073
7,847
5,986
4,208
46-49
9,516
7,165
9,619
11,466
12,822
11,945
11,484
9,762
8,827
8,749
6,228
50-54
6,888
4,240
6,378
8,848
7,85(S
8,276
8,145
7,926
7,351
8,484
5,843
55-59
4,171
2,377
2,755
3,338
3,991
4,408
4,599
4,811
4,882
4,427
4,083
60-64
2,938
1,408
1,601
1,803
2,340
2,583
2,965
3,113
3,290
8,106
2,953
65-69
1,913
881
928
1,107
1,307
1,808
1,941
2,101
2,285
2,118
2,175
70-74
1,057
603
503
851
723
891
1,005
1,169
1,168
1,294
1,239
75-79
474
214
296
269
314
424
512
484
562
584
587
80-84
164
61
86
110
113
149
182
181
171
167
155
85-89
48
12
12
22
27
49
39
36
49
28
38
90-94
10
12
9
7
6
8
9
9
7
4
6
95-99
1
1
1
100-104
1
Totals
123,995
146,153
145,107
124,239
103,088
82,140
87,029
63,803
44,026
35,392
28,248
Totals
million
for
existing
mar-
riaiges
142,218 149,5841
144,833
123,066
100,900
79,896
64,146
51,179
41,492
33,280
26,328
The actual total number of wives was 733,773, of which 343 gave no information as to age, or as to
number of children ; 15,477 gave no information as to number of children, but stated their age ; 6432
gave no information as to age, but stated the number of children. The 343 were divided into two
groups, viz., 254 and 89, these being distributed proportionately among the partially specified totals.
The total additions thiu become for the several ages and age-groups : 0, 0, 0, 6, 11, 28, 70, 167, 1228
COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY.
323
Marriage* at Census of 3rd April, 1911, Australia (Based upon 733,773 Wives.)t
Ntimbbr of Wives who had given Bikth to Childkeit to the NnMBER of—
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21
and
over.
Totals.
"l
175
1,741
4,530
5,172
4,845
3,722
2,675
3,467
"63
726
2,364
3,455
3,313
2,789
2,077
2,676
"io
277
1,295
2,145
2,223
2,211
1,560
2,819
"6
99
606
1,199
1,301
1,237
978
1,433
'29
250
592
668
672
514
788
"9
115
282
317
319
287
421
"l
39
157
145
171
141
202
'io
50
67
87
59
88
"6
18
29
29
23
44
"3
6
23
4
23
"l
6
4
9
1
1
"6
6
14
10
22
200,471
183,722
153,846
114,452
109,821
88,571
58,677
39,213
23,981
27,246
26,328
17,463
12,040
6,859
3,513
1,750
856
341
149
59
22
58
1,000,000
28,246
18,826
13,035
7,488
3,834
1,927
941
379
182
68
36
66
1,000,000
and 2385. The luoreased numbers thus become : — For Age-groupa as indicated in table, 3358, 2903,
2074, 1371, 1210, 943, 591, 439, 230, 316 ; in all 13,435. For numbers of children as indicated in
table, 2796, 2569, 2816, 2614, 2373, 2142, 1828, 1521, 1298, 1076, 876, 604 493, 288, 125, 62, 24, 17,
6, 2, 1, and 5 ; in all 23,536. These aggregates o£ unspecified and partially specified were then dis-
tributed proportionately to the original numbers, see Vol. III., p. 1140-1, Census Eeport.
Census of 3rd April, 1911, Australia.
(Based upon 733,773 Wives.)
NUMBEIl OP
Wives to whom had been Born Children to the Number or-
21
Totals.
11.
12.
is.
14.
15.
16.
17.
18.
19.
20.
and
over.
••
*■
••
1
25
126
470
1,446
3,485
5,966
. 9,848
78,943
5
3
148,792
97
31
7
3
3
1
153,375
907
380
160
67
- 16
3
1
142,270
2,437
1,443
766
328
178
78
20
13
1
3
129,939
3,458
2,241
1,287
685
327
156
73-
26
8
8
5
112,052
3,699
2,611
1,482
754
356
196
67
33
8
8
13
83,756
3,021
2,166
1,313
669
335
171
74
36
21
7
8
51,656
2,206
1,592
904
1,759
1,000
548
298
144
70
33
6
7
12
34,172
1,305
762
410
236
113
51
21
11
4
8
22,855
656
463
238
114
58
13
13
8
11
12,681
357
347
194
105
53
14
4
7
6
1
1
5,809
116
76
47
22
10
4
6
3
1,823
24
16
6
4
1
3
1
413
4
1
1
1
92
1
4
••
1
18,826
13,035
7,488
3,834
1,927
941
379
182
68
36
66
1,000,000
17,463
12,040
6,859
3,513
1,750
856
341
149
59
22
58
1,000,000
2252 2203, 2071, 1993, 1677, 1431, 888, 648, 476, 316, 157, 81, 23, 4, 0, 2 ; in aU. 15,731 : and for the
numters oi children as indicated in the table, 591, 717, 730, 693, 602 490, 415,^09, 277, 218, 181, 128,
76 51 18 12,7,2,3,0,0,1; in all 5521. These aggregates for the unspecified, together with the
partiaily-Bpecifled, were then distributed proportionately to the original numbers ; see Vol. Vnj.
Census Report, pp. 1366-7,
324 APPENDIX A.
The results given in Table CXIII. are shewn by curves {a) to (*) in
Fig. 88 ; and those in Table CXIV. are shewn by the curves (a' ) to (/ )
in the same figure, the single year results of the latter table being marked
15, 16, ... . 20. Interpolated curves would give the results for any
other 5-year age or duration ranges.^
The curves of frequency of cases, according to number of issue, for
the 5-year, or for the single-year age-groups, are of the same type, and are
essentially dimorphic : strictly they give values only for integral values of
the variable.^ Thus they could no doubt be fairly well represented by
curves of the type : —
(561) y = Aer^" + Bx^+o"
in which x has the values 0, 1, 2, 3, . . . . etc.
20. Fecundity during a given year. — A different type of compilation
is necessary to reveal what may be called the " existing fecundity." The
existing nuptial fecundity is shewn by the number of married women in
each age-group, the number who failed to bear a child during the year,
and the number who bore the wth child where w = 1, 2, 3, .... etc.
This is deduced from two sources, viz., (i.) from the Census record for
the numbers of married women ; and (ii.) from the records of one year or
for a series of years (1908-1914). The grand total of those who bore a
child during the whole period of 7 years, if divided by 7.0666, gave a
result substantially identical with that for the year 1911, which may be
regarded as satisfactory.* This is seen from the close agreement of the
numbers in the two upper portions of Table CXV. It is evident, therefore,
that the vital statistics results for the Census year represent fairly satis-
factorily the general case, and a 3 or 5-year result with the Census year as
middle year would ordinarily be quite satisfactory.
' It is clearly desirable that Census results should be compiled for single years,
as soon as public appreciation of the value of a correct statement of age leads to
accuracy.
2 Statistical results furnish a number of examples of this character : for example
the numbers of families living in houses with 1, 2, 3, ... n rooms, etc.
' If the rate of change of the proportion married be supposed linear, the married
female population at the Census is to the aggregate of married females a^s 1 : 7.1272.
The ratio of the number of brides is 1 : 6.9473. Theratioof females is 1 : 7.1077,
and of population 1 : 7.1160. It is obvious, therefore, that the ratio 7.0666 is
very nearly correct.
COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY.
325
TABLE CXV. — Shewing foi various Age-groups and for all Durations of Marriage
the Number who, during the year, bore the nth Child, where n = to 10 ; and
the Total of those who bore a Child later than the 10th. Australia, 1911 and
1908-1914.
Age of
Xotal
Married
Women
*
No. who
Bore a Child
during
the Year.t
No. who
Bore no
Child during
the Year.
Number for which the Chh.p Born was the —
Order
Mothers
1st
2nd
3rd
4th
5th
Later
than
5th.
6th
7th
8th
9th
10th
Later
than
10th.
Speci-
fied.
-19
20-24
26-29
80-84
85-40
40-44
45-
8,716
65,959
110,591
113,810
105,550
95,578
82,988
4,146
25,957
88,817
25,682
16,839
6,768
718
114,570
40,002
76,774
87,628
88,711
88,810
82,220
8,456
18,089
9,271
3,632
1,279
303
20
619
7,717
8,672
4,327
1,539
816
24
53
3,642
7,109
4,522
1,997
405
29
4
1,085
4,727
4,328
2,277
531
36
246
2,419
3,501
2,243
722
40
62
1,554
5,342
7,476
4,479
561
50
1,093
2,527
2,143
740
48
8
336
1,565
1,848
777
64
4
86
745
1,388
771
70
29
317
970
706
72
8
181
591
607
86
2
57
541
878
221
14
166
65
30
28
7
3
Totals
582,632
113,917
468,715
31,000
28,214
17,757
12,988
9,171
19,474
6,601
4,598
8,059
2,094
1,423
1,699
313
Numbers Gorrebfondinq to the Above Babes upon the Totals for the Period 1908-1914.
-19
•8,716
§4,156
114,560
3,410
676
66
4
20-24
65,959
26,277
89,682
13,248
8,043
3,578
1,102
246
60
48
10
2
U
25-29
110,591
33,831
76,760
9,317
8,703
7,065
4,748
2,468
1,530
1,041
348
101
29
8
3
30-34
113,310
25,639
87,671
3,592
4,817
4,624
4,281
8,504
5,321
2,523
1,529
757
326
123
63
35-40
105,550
16,742
88,808
1,259
1,490
1,963
2,274
2,293
7,463
4378
2,130
1,865
1,416
968
580
504
40-44
95,573
6,609
88,964
288
312
418
547
666
717
746
749
677
579
910
45-
82,933
663
82,270
22
19
21
38
36
532
48
60
65
76
74
, 209
Totals
582,682
113,917
X
468,715
31,136
23,560
17,735
12,989
9,213
19,284
6,507
4,558
8,090
2,076
1,864
■
1,689
~
FBOFORTIONS 10 lOIALB OF SAME AOE ; 1911 BEBULTS.
-19
100,000 47,668
52,432
39,651
7,102
618
46
^
^0
161
20-24
100,000 39,353
60,647
19,768
11,700
5,522
1,645
373
94
'^76
• 12
6
1*6
E«o
6
2B1
25-29
100,000 80,579
69,421
8,384
7,842
6,428
4,274
2,187
1,405
15988
S304
78
26
"7
2
59
30-34
100,000 22,665
77,885
8,205
8,819
3,991
3,820
3,090
4,714
2,230
1,381
657
280
116
50
26
35-40
100,000| 15,958
84,047
1,212
1,458
1,892
2,157
2,125
7,083
2,030
1,751
1,310
919
560
513
26
40-44
100,000
7,076
92,924
317
381
424
556
755
4,686
774
813
80V
789
635
918
7
45-
100,000
860
99,140
24
29
35
44
48
676
58
77
84
87
104
266
4
Froportiokb to Totals of same Aqe ; Based upon the Totals for tee Period 1908-1914.
-19
100,000
47,682
52,818
39,128
7,766
757
46
20-24
100,000
89,838
60,162
20,085
12,194
5,425
1,670
373
91
V3
16
3
u
U
26-29
100,000
30,591
69,409
8,425
7,870
6,388
4,298
2,232
1,888
941
315
91
26
7
3
30-84
100,000
22,627
77,873
3,170
3,810
4,081
3,778
3,092
4,696
2,226
1,349
668
288
109
66
85-39
100,000
15,862
84,189
1,193
1,412
1,860
2,164
2,172
7.071
2,018
1,767
1,342
917
550
477
40-44
100,000
6,916
93,085
302
326
487
672
697
4,581
750
781
784
708
606
952
4&-
100,000
799
99,160
27
23
25
40
43
641
58
72
78
92
89
252
• Adjusted numbers, see Census Report, Vol. II., p. 19, and also Vol. III., pp. 1136-7. The numbers given
are the Census numbers adjusted and multiplied by a factor to make them agree with the mean female population
of the year. t In cases where a woman bore twice in the same year, she has been counted twice. The results
in this column are obtained from the vital statistics of the year 1911. t The actual figures throughout have
been multiplied by a factor (viz., 0.141509 = l-f- 7.0666), so as to make this total, 113,917, to agree with the total
above : hence, if the distribution for 1911 were identical with that of the seven-year period 1908-1914, the figures
in the several columns would be identical. They are approximately so. § The whole of the numbers in the
column are those for 1908-1914, multiphed by 0.141509. II These numbers are obtained by subtracting the
totals of those who bore children from the total number of married women.
3^6
APPENDIX A.
TABLE CXVI. — Shewing the Number of Married Women at each Age, the Number
oi Cases of Maternity, and the Number for all Durations of Marriage, who had
not given Birth to a Child. Australia 1907-1914.
Wives at
1911 who
■en Birth
hUdren.t
Bange
Proportion of
^s^*:
Proportion of
Age
last
No. of
Married
Women
in Years
of
Dura-
Married
Women who
had not given
Age
last
No. of
Mauled
Women
No. of Wives a
Census 1911 wli
had given Birtl
to no Children
Married
Women who
had not given
Birth-
at
No. Cas
Maten
1907
tions of
Birth to a
Birth-
at
Birth to a
day.
Census
1911.t
No. of
Census
hadgiv
tono
Mar-
riage,
(up to)
Child.
day.
Census
1911.
ChUd.
Crude.
Smooth
ed.
Crude.
Smooth-
ed.
(i.)
(ii.)
iu.
(iv.)
(V.)
(vi.)
(vii.)
(i.)
(ii.)
(iv.)
(vl.)
(vii.)
13
1
0.5
1
1*
1.0000
1.0000
14
18
19
3.7
4.2
13
14
2
.7222
.7388
.8140
■•
••
15
93
21.2
67
"s
.7204
.6530
'56
9,468
769
.08i7
16
349
141.9
183
4
.5244
.5330
56
8,557
678
.0815
17
1,145
494.7
498
5
.4349
.4450
57
7,675
581
.0814
18
2,551
1,219
1,061
6
.4159
.3820
58
6,912
.531
.0813
19
4,499
2,261
1,531
7
.3403
.3403
59
6,293
501
.0814
8,637
4,137.8
3,340
.3867
38,905
3.060
.0786
20
6,933
3,150
2,192
"8
.3162
.3075
'60
5,746
479
.0815
21
10,100
4,423
2,772
9
.2744
.2815
61
5,277
458
, ,
.0816
22
13,047
5,428
3,422
10
.2622
.2580
62
4,871
435
.0820
23
16,521
6,306
3,973
11
.2405
.2365
63
4,505
412
, ,
.0823
24
18,905
6,669
4,123
12
.2181
.2165
64
4,161
382
.0827
66,606
25,976
16,482
.2516
24,660
2,166
.0882
25
20,683
6,811
4,123
■i3*
.1993
.1990
'65
3,829
353
.0837
26
21,620
6,903
3,958
14
.1831
.1825
66
3,502
319
.0842
27
22,180
6,751
3,678
15
.1658
.1670
67
3,194
283
.0848
28
22,584
6,691
3,448
16
.1527
.1524
68
2,880
247
.0854
29
22,765
6,192
3,238
17
.1422
.1424
69
2,621
211
.0861
109,832
33,348
18,446
.1679
16,026
1.413
.0882
30
22,784
6,042
3,034
■i8*
.1332
.1339
■70
2,365
190
.0868
31
22,726
5,065
2,849
19
.1264
.1266
71
2,099
168
.0876
32
22,542
5,240
2,684
20
.1191
.1203
72
1,867
146
.0885
33
22,421
4,722
2,540
21
.1133
.1147
73
1,652
129
.0896
34
22,059
4,338
2,416
22
.1095
.1101
74
1,444
115
.0908
41S,632
25,407
13,523
.1202
,
9,427
748
.0793
35
21,700
3,958
2,299
'23
.1059
.1062
'75
1,224
96
.0921
36
21,350
3,721
2,195
24
.1028
.1029
-76
1,004
82
.0934
37
21,000
3,315
2,101
25
.1000
.1000
77
818
70
38
20,560
3,118
2,017
26
.0981
.0979
78
650
59
39
20,215
2,629
1,942
27
.0961
.0959
79
510
48
104,835
16,741
10,564
,1007
4.206
355
.0844
40
19,851
2,148
1,880
'28
.0947
.0942
'so
397
38
41
19,457
1,548
1,823
29
.0936
.0927
81
317
30
42
19,026
1,386
1,766
30
.0928
.0913
82
241
23
43
18,543
939
1,710
31
.0922
.0900
83
184
17
Pi
44
18,040
595
1,653
32
.0916
.0888
84
140
13
es
94,917
6,616
8.832
.0930
1.279
121
.0946
-ts
45
17,554
346
1,577
■33
.0898
.0877
■ '85
105
10
M
46
17,064
174.2
1,494
34
.0876
.0868
86
80
8
V
47
16,554
85.6
1,403
35
.0847
.0860
87
56
6
u
48
15,975
38.7
1,306
36
.0817
.0852
88
35
5
a
49
15,216
13.2
1,203
37
.0791
.0845
89
24
4
p
82,363
657.7
6,983
.0848
300
33
i.i6o
50
14,303
4.2
1,116
'38
.0780
.0837
'90
20
3
51
13,162
1.5
1,049
39
.0797
.0832
91
16
2
52
12,088
0.9
981
40
.0812
.0827
92
12
1
53
11,100
0.6
914
41
.0823
.0823
93
9
54
10,286
0.1
847
42
.0823
.0819
94
7
60,939
7.8
4,907
.0806
64
6
.0937
95-100
21
• Actually extends to about 1 year greater than shewn. t Graduated.
21. Number of married women without children, all durations oi
marriage. — ^The relative numbers of married women of each age, and for
all durations of marriage, who are without children, are readily determin-
able by means of a Census. That for 1911 gave the results shewn in
Table CXVI. above. The smoothed results in column (vii.) of the
table are shewn by curve A on Eig. 88. The ratio very rapidly falls to
the value of about one-fourth, which is attained during age 22 ; one-eighth
COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY.
327
is reached during age 31 ; one-tenth during age 37 ; and the minimum
during age 58, which age is, of course, somewhat uncertain. After the
age of that minimum the results are very uncertain. Apparently the
curve will require several terms of the type Ae""* to empirically represent
it, thus the ratio being denoted by a, and the age ^ being reckoned from
say 12 or 13, the ratio will be of the form : —
(562).
.0- = J. + 5e-»f G+e-of +....+ ZP
22. Sterility-ratios according to age and duration of marriage.— The
effect of the age of the husband being ignored, the number of cases of
sterihty, (or more strictly of childlessness,)^ according to duration of
marriage, for women of different ages in Australia was found from the
Census of 1911 to be as shewn in the following table : —
' Physiological sterility is the condition, not merely of childlessness, but of
childlessness due either to failure to conceive, or to retain the fertilised ovum the
full time. The data of ordinary statistics cannot conclusively establish the frequency
of physiological sterility, since what is given are merely measures of childlessness.
A number of instances are given in the " Handbuch der Medizinischen Statistik,"
by Friedrich Prinzing, Dr. Med., 1906, Cap. III. ; " Die sterilen und kinderloaen
Ehen," pp. 30-40.
The following estimations of sterility may be mentioned
*
No. of
Elapsed Period
No. of
Marriages
after
Sterile
Ratio.
Authority.
under
Observation.
Marriage.
Cases.
Dresden Returns
27,911
5 years
672
0.02407
Dresden Returns
27,911
10 years & more
134
0.00480
AusterUtz, Prag, 1891-
1900
3,920
Not stated
295
0.0753
Hofmeier
2,220
Not stated
?
0.147
Lier and Ascher
2,500
Not stated
?
0.090
Huizinga (Groningen) . .
1,180
Not stated
?
0.115
Verrijn Stuart, Nether-
lands
9,443
16 to 21 years
?
0.131
Do., poorer classes
1
Not stated
?
t0.141t0.110
Do., middle classes
1
Not stated
?
• 0.162 t0.109
Do., well-to-do classes
?
Not stated
?
t0.160t0.126
"I" Town. { Country.
*Other results are: — Spencer, Wells & Sims (Great Britain), 0.125; Duncan
(Glasgow and Edinburgh), 0.163; Ansell, 1919 cases. Married Women, 0.079 ; A
Swedish County, 0.100 ; Massachusetts, 1885, 0.176 ; Women over 50, 0.119.
The whole of the above statements are, of course, defective, inasmuch as sterility
is a function both of duration of marriage as well as of age, etc.
328
APPENDIX A.
Table CXVn. — Sterility according to Age and Duration of Existing Marriage.
Australia, 8rd April, 1911 (Censns).
DTJaATION OF BXISTING MAKMAGB.
AQE
OF
TTiTDEE 5 Tears.
5 TO 10 Tears.
10 TO 15 TeAES.
15 TC
20 Tears.
20 TO 25 Tears.
AT
Time
OP
Cbssds
II
•si
§3
11
Ss
fl
ll
fl
jJ
H£
fl
%3
Hfi
fl
Sa
s
03
oS
g
"'
oy
00
3S
s
m
Under
•14
1
1
1.000
•14
13
18
.722
•15
67
92
.728
•1«
179
338
.530
•17
490
1,044
.469
•18
1,042
2,512
.415
•19
1,496
4,270
.350
•20
2,114
6,69S
.316
7
261
.027
21-24
13,378
43,424
.308
474
11,926
.040
25-29
14,724
45,67S
.322
3,004
47,785
.063
346
10,587
.»33
1
21
.048
30-34
7,398
19,735
.375
3,998
38,675
.103
2,07fi
40,121
.052
262
8,594
.030
2
21
.095
35-39
4,099
8,11S
.505
2,885
16,992
.170
2,693
32,715
.082
1,348
31,792
.042
279
9,324
.030
40-44
2,597
3,575
.726
2,120
6,731
.315
2,096
14,568
.144
1,723
24,408
.071
1,250
32,477
.038
45-49
1,753
1,865
.938
1,712
2,74S
.623
1,517
5,280
.287
1,199
9,253
.130
1,365
22,758
.060
50-54
893
894
.999
1,10S
1,187
.933
1,08S
1,865
.584
803
3,230
.244
803
7,780
.103
55-59
431
431
1.000
6'C
531
.998
701
779
.900
S6«
1,088
.520
530
2,265
.234
60-64
247
247
1.000
255
255
1.000
332
332
1.000
420
447
.940
411
791
.520
65-69
140
140
1.000
117
117
1.000
173
17^
1.000
198
199
.995
305
337
.905
70-74
64 64
l.OOO
74
74
1.000
91
91
1.000
100
100
1.000
127
128
.992
75-79
20 20
1.000
28
28
1.000
37
37
1.000
38
38
1.000
63
53
1.000
80-84
3
a
1.000
7
7
1.000
11
11
1.000
4
4
1.000
17
17
1.000
85-89
1
1
1.000
2
2
1.000
3
3
1.000
2
2
1.000
4
4
1.000
DUEATION OF EXISTING MAKEIAGB.
Age
25 TO 30 T
''EAES.
30 1
35 1
fEABS.
35 1
40 1
''EARS.
40 TC
45 T
EARS.
Otes 45 Tears.
WiTES
AT
TIME
OP
Cmavt
oS
ll
&4
1^
Is
i|
^1
1^
ll
i|
m
il
^1
1i
to
oS
li
35-39
1
8
.125
I
40-44
221
8,075
.027
45-4S
1,005
28,66«
.035
165
6,55e
.025
50-54
955
17,237
.055
663
20,004
i)33
121
5,081
.024
55-59
456 4.762
.096
511
9,574
.053
432
12,606
.034
91
3,087
.029
60-64
346
1,590
.218
284
2,946
.096
388
6,437
.060
250
7,789
.032
59
2,209
.027
65-69
292
522
.559
213
978
.218
' 189
2,120
.08S
22«
4,083
.055
214
6,630
.032
70-74
188
206
.913
125
295
.424
• 106
569
.186
101
1,150
.088
221
5,660
.039
75-75
72
72
1.000
68
73
.932
59
121
.488
4«
305
.151
116
2,994
.039
80-84
15
15
1.000
23
23
1.000
25
26
.962
18
57
.316
46
945
.049
85-8S
2
2
1.000
5
5
1.000
e
9
1.000
5
8
.625
14
200
.070
90-94
1
1
1.000
1
1
1.000
e
41
.146
95-99
1
1
1.000
••
1 ■■
• The results are from Cenaug Eeport HI., p. 1136. The general resnlts are obUined from an
nnpablished series of compilations according to age-groups, and duration-of-marriage groups. In
neitiier case were the " unspecified" dislaributed ; such distribution, however, can affect Vbe results
only very slightly.
An examination of the results given in the table shews that initially
the sterility-ratio decreases ; it attains a minimum, and then increases ;
see particularly the duration of marriage to 4 years (i.e., under 5 years).
The initial fall may be regarded as the normal decrease of cluldlessn^s
with increase of the duration of the risk. From the minimum onward,
however, the cur re shews the true measure of steriJity for a given duration
of marriage, and for any age terminating the given duration of marriage.
COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY.
329
The curves on Fig. 89 are the sterility-ratios according to age, each
curve denoting a separate range of duration of marriage. By projection^
Fig. 90, shewing the curves of equal sterility, is derived. From these, the
correlative durations of marriage and ages, corresponding to any degree
of sterihty, can be at once seen. The dots give the positions as determined
from the data,^ the curves throughout are smoothed.
Fig. 89.
t6 ' Sr~" BT
Age of Wives
Fig. 90.
In Fig. 89 the ordinates to the evirvea denote the degrees of sterility : the
abscisses denote the age corresponding to the duration of marriage shewn on any
curve in question.
In Fig. 90, the intersections of the curves with the lines of equal sterility on Fig.
89, are projected, to the ordinates -line corresponding to the mean of the range of
durations, viz., 2.5, 7.5, 12.5, etc. years. Smoothed curves have then been drawn
shewing the probable position of the curves of equal sterility.
Curve A in Fig. 90 denotes the sterility -ratio according to age at marriage
where the duration of marriage is 20 years.
On Fig. 90 they represent the projected results, and the lines drawn
among them, the smoothed general results deduced therefrom. Thus the
1 It has been assvimed that the group-results for the ranges 0-5, 6-10, 10-15, etc.,
are sensibly correct for the durations 2.5, 7.5, 12.5, etc., as is evident from Fig. 90.
This is not quite exact ; the error is not large, however, and the inherent limitations
of the determination of the ratio render the measure of uncertainty of but little
moment.
2 The three broken lines crossing from Fig. 89 to Fig. 90, indicate the scheme of
projection. Thus, the point b, viz., the intersection of the curve assumed to repre-
sent a sterility of 0.3 for a duration of marriage of 12.5 years, is found in the graph
(plan), Fig. 90, as the point b', viz., on the line parallel to the axis of age at the distance
(ordinate) therefrom 12.5, and similarly for point a and c and a' and c'.
330
APPENDIX A.
new curves so obtained represent completely the steriUty-ratios according
to age taken in conjunction with past duration of maniage.'^
It is obvious that tables shewing average sterility can be constructed
(i.) according to ajge at marriage and time since elapsed ; and
(ii.) according to age attained after the given interval between it and
marriage.
As, however, the one differs from the other merely by the whole
amount of the duration, it is immaterial in which form they are set out.
In the following table (CXVIII.) the former method is adopted ; Figs. 89
and 90, however, give the age attained after a given duration of marriage.*
TABLE CXVm.-
-Shewing for varions Ages and Durations of Marriage the Degree
of Sterility experienced. Aoslralia, 1911
.
COBBESPONDING DUBATIONS OF MabEIAGB (iN
Yeabs).
Sterility-Ratio.
5
10
15
20
25
30
35
40
45
When Tim Age at Mabbiaoe
is:—*
.025
13.8
15.5
16.6
17.1
17.1
16.7t
16.0t
15.lt
.050
19.3
21.3
22.9
23.7
24.3
24.3
24.4
27.6
.075
, ,
23.1
24.9
26.3
27.2
27.8
28.0
28.1
31.9
.100
25.8
27.6
28.8
29.6
30.1
30.3
30.9
34.1
.150
29.3
30.7
31.8
32.4
32.6
32.5
33.4
37.9
.200
31.6
32.7
33.8
34.1
34.3
34.5
35.4
.250
33.4
34.3
35.1
35.4
35.7
35.9
36.8
.300
34.9
35.5
36.2
36.6
37.0
37.2
37.9
.350
34.0
35.8
36.5
37.1
37.5
37.9
38.0
38.6
.400
35.1
36.7
37.2
37.8
38.3
38.7
38.8
39.6
.450
36.1
37.5
38.1
38.7
39.2
39.7
40.0
40.8
.500
37.1
38.7
38.8
39.5
39.8
40.4
40.9
41.7
.600
39.0
39.8
40.1
40.7
41.0
41.4
41.7
42.6
.700
40.8
41.4
41.6
42.1
42.2
42.5
42.8
43.7
.800
42.5
43.0
43.1
43.3
43.5
43.7
43.7
44.6
.900
44.5
44.5
44.6
44.7
44.0
44.0
45.0
46.0
.950
45.9
46.1
46.1
46.2
46.3
46.3
46.3
47.0
.975
47.6
47.7
47.7
47.8
47.7
47.6t
47.6t
48.0t
1.000
51.6t
51.5t
51.4t
51.3t
51.2t
51.lt
50.9t
50.7t
• The table is thus Interpieted : — Heading horizontally, it the age at maniage was say 16.6 years,
and the duration of marriage was 20;year8, 0.025 would be the proportion without children. Similarly
it tt\e age at marriage was 17.1 years, and the duration of marriage was either 25 years or 30 years,
or reading yertically, for the duration of marriage of 15 years, if the age at marriage were 15.5, tlien
0.025 woiud be sterile ; if the age were 21.3, tlien 0.050 would be sterile ; and so on.
t The apparent anomaly in these results may possibly be explained by tlie more fertile not living
sufficiently long to be included in the category of those whose duration of marriage attained the numlter
of years indicated.
The steriUty-ratios given in the table for durations of marriage 0-5,
do not accord very closely with those deduced by the method of Part
1 Strictly these curves represent the mean of 5-year groups, both as regarda
duration of marriage and age. The corrections to make them instantaneous
results, however, are small.
' Data have not been compiled which would enable these resvdts to be worked
out with very great precision. For this it would of course be necessary to compile
according to single years both as regards age and duration of marriage ; and give
results according to " age at marriage" and " duration of marriage" instead of
existing age.
GOMPLEX ELEMENTS OF FERTILITY AND FECUNDITY. 331
XIII., §§ 11-13, pp. 245 to 250. The probabiUty of a birth, and that of
childlessness should together equal unity : For 0-6 years the agreement,
however, is closer; see Fig. 71, p. 249, or the values given in Table LXXV.,
p. 247. As, however, the results for the shorter durations are necessarily
somewhat uncertain, these differences are not remarkable. It may be
pointed out the results indicated in Table LXXI., p. 238, shew that for
the age 51 the probabihty of a birth is 1.17 per thousand, hence the final
value should probably be 0.999, rather than 1.000. But tables of this
kind are, of course, probably never reUable to this order of precision.
23. Curves of sterility according to duration o£ marriage. — ^The
steriUty-ratios determined from the age of the married woman only, are
based upon the assurryption that fertility is independent of the age of the
husband : this is shewn hereafter not to be the case. Or we may regard
the results as true for the average condition {i.e., the condition including
husbands of all ages ) . Continuing this assumption and taking the curve for
a duration of marriage of 20 years, it is found that the proportion sterile
who are married at the ages 11, 12, . . . 51 respectively are as shewn in
Fig. 90, Curve A. The ordinate at age 11 is not necessarily zero, but
owing to the fact that marriages at that age usually arise from special
circumstances, the value of the sterility-ratio is practically zero.^ The
curve has a point of inflexion, for marriages at ahoatehge40,{i.e.,d^yjdx^=0
for X = 40), and the sterility-ratio changes most rapidly at about age 28
(i.e., d\/ dx^ = for a; = 28). The curves of steriUty can be obtained
by plotting the ages in the vertical columns in Table CXVIII., as abscissae,
and the value of the observed steriUty as an ordinate. For every given
duration of marriage there will be a different curve.
24. Fecundity according to age and duration of marriage : various
distributions and ratios. — ^As already pointed out, fecundity is a function
of the age of the husband and of the wife, as well, of course, as of the dura-
tion of marriage. It has been shewn herein also, for various durations of
marriage, that on the average (i.g., the results being for husbands of all
ages combined), and for those only who come under observation in cases of
birth, the number of children borne, according to duration of marriage
(i), is about I + TT * ; see formula (523) of Part XIII., §§ 34, 35, and
Table XC, pp. 279-283. The surface of representation of this is, for
the most part, sensibly a plane. It defines the polygenesic^ distribution,
see p. 285 ; and thus may be called the polygenesic surface. In the case
of this distribution differences of age have much less influence, if any,
than differences in duration of marriage. It is important to bear in mind,
however, that this distribution, as above stated, applies only to a limited
^ That is the marriages are what have been (somewhat ill-advisedly) called
prejudiced" — and do not represent the average liability of becoming fertile.
» The word " polygenesis" has been used to indicate the origination of a race
arising from several independent ancestors or germs. The above use will, however,
lead to no confusion, and is consistent with the general mode of word construction.
The word polyphoroua (from ro\v^6pos = bearing many) is used hereinafter for a
different function.
332
APPENDIX A.
number of married women, viz., those whose total fecundity ha/ppe7is to
come under review through repeated child-bearing. In Part XIII., § 36,
p. 285, the total number of children borne by married women of given
limits of age and duration of marriage has been called the " general
genesic," or " fecundity" distribution. For many purposes, however, it
is desirable to know the number of mothers (a) instead of the number of
children (say, z'= kz, k = Q, \, 2 . . . n) being the number borne by
each woman) . It is also preferable to relate the number of married women
to the exact number, k, of children borne by each. Let, therefore,
^m, im, zin . . . ^m denote the number of married women who bore
0, 1, 2 . . . n children respectively, the range of whose ages are between
Xq and xi, xi and xz, etc., and the range of whose durations-of -marriage
are t^ and <i, fi and t^, etc., the ages and duration limits, however, being
quite independent. Then the various quantities of importance may be
embraced by the following distributions, which will hereinafter be de
fined, viz.: — (i.) The age-genesic distribution, (ii.) the'durational genesic
distribution, (ni.) the age-fecundity distribution, (iv.) the durational
fecundity distribution, (v.) the age-polyphorous distribution, (vi.) the
durational polyphorous distribution, (vii.) the duration-and-age-fecundity
distribution, (viii.) the age-and-duration fecundity distribution, (ix.) the
duration-and-age polyphorous distribution, and (x.) the age-and-duration
polyphorous distribution. The ages may be those at the moment of
enumeration or at the moment of marriage : for given purposes ei^er may
be required.
These distributions are most clearly defined by means of a symbolic
table. Table CXIX., shewing the two types of possible compilation of the
results exhibiting the degree of fecundity characteristic of a community.
With the aid of this table the various types of distribution — essentially
ratios — ^are readily symboUcally represented.
TABLE CXIX. — Scheme of Compilation according (i.) to Age, and (ii.) to Duration of
Marriage, exhibiting the Characteristics of the Fecundity of a Population.
Either
(i.) (ii.)
•A-ge- Duration-
Total Number of children borne by each married
woman where the age is given, or during the
existing marriage where the duration is given.
Totals
of
(hori-
Group, Group.
1
2
k
■■
n
zontal)
rows.
Xo to X,
Xi to Kj
!Kjl-l to Xp
a:,-! to x.
«„ to t^
«i to «,
tp-i to tp
ts-x to is
ki^i
hi^p
k'^t
^1
Mp
M,
Totals of (vert.) columns
,M
^M
,M
k^
••
„ilf
M
COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY.
333
To distinguish clearly between (i.) and (ii.), the m and M quantities
are not accented for the former, and are accented (m' and M') for the
latter. The surface, the z co-ordinate to which is the height above the
xk, or the xt plane, as the case may be, is : —
(563).
.z=fi{z,k); or a = /a {t, k).
in which, of course. A; is a variable.
The following table, viz., CXX., gives symbolically the definition of
each distribution.
TABLE CXX.— Types oi Distribution.
DOTTBLB EnTBY DATA.
Age Distribution.
Group Formulae.
Durational
Distribution.
(i.) Age genesio =
(iii.) Age fecundity =
(v.) Age polyphor-
ous . . =
% = *%/ ^
\i = k'^plk^
k^'plk^' = -'df
^m'jM'^ = -'dp
= (ii.) Durational
genesio
= (iv.) Durational
fecundity
= (vi.) Durational
polyphorous
The equations of the continuous surfaces
for the above are z = f (x, k)
Equations of the continuous surfaces
for the above are z = / («, ik).
TbipiiB Entry Data.
A table required for each range of duration.
A table required for each age -group.
Age Distributions.
Group Formulse.
Durational
Distributions.
(vii.) Duration and age,
fecundity =
(ix.) Duration and age
polyphorous =
-d-ap^k^'p/^'p
k-^'"p/k^"=-a4f
k-'"'p/p^'"=-a.dp
= (viii.) Age and
durational
fecundity
= (x.) Age and
durational
polyphorous
The equation of the o
for each range of dur
jntinuous surface
a,tion is z = / (x,k)
The equation of the
for each range of c
continuous surface
uration is z=f{t,k)
25. The age-genesic distribution. — ^This distribution furnishes at
once the means for determining how a given total of married women may
be " partitioned" according to (i.) age, and (ii.) the number of children
borne by them. These ratios, multipUed by 1,000,000, are given in Table
CXIV., pp. 322-3, for various age-Umits. The ignored elements are th^
durations of marriage and the ages of the husbands.
26. The durational genesic distribution. — ^This distribution similarly
furnishes the basis for ascertaining how a given total of married women
may be subdivided according to (i.) duration of existing marriage, and (ii.)
334
APPENDIX A.
number of children borne by them. The ratios multiplied by 1,000,000
are given in Table CXIII., pp. 322-3. The ignored elements are the ages
both of the wives and their husbands.
27. The age-fecundity distribution. — ^This distribution represents
the relative numbers, according to age, of married women who bore a
given number of children : thus.it enables the relative frequency according
to age of those who bore any given number of children to be compared,
as between one community and another, a fact which wUl be immediately
obvious from the table hereunder. The ignored elements are the dura-
tions of marriage and the ages of the husbands.
TABLE CXXI. — Shewing, for Wives of all Durations of Marriage combined, and for each Total Number of Children borne by
them {i.e., 1, 2, 3 ... . to n), the Proportion Contributed by each Age-group indicated.
Australia, Census 3rd April, 1911. Age-Fecundity Distribution.
Ages
of
No.
Batio
Katio of the Number in a given Age-group to tlie Total of all Ages, of those who bore h Children, where k is
successively 0, 1, 2 . . . n.
of
Wives
to
Total
SO
at
Census
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20 over
20
13
1
00000
00000
0000
14
18
00002
00002
0001
0000
16
92
00013
00013
0006
0002
16
345
00047
00047
0017
0014
0000
17
1,061
00145
00145
0048
0047
0004
0000
18
2,557
00348
00348
0101
0115
0020
0000
IB
4,376
00596
00597
0146
0201
0052
0010
0001
20
7,224
00985
00985
0209
0318
0122
0026
0002
0000
21-24
57,896
07890
07894
1378
1991
1313
0630
0215
0054
0016
0001
0000
25-29
109138
14874
14879
1802
2596
2495
2008
1377
0782
0357
0142
0050
0013
0002
0003
0000
30-34
112523
15335
15337
1376
1720
2085
2200
2048
1754
1316
0881
0478
0240
0107
0052
0023
0008
0007
0000
0002
35-39
104619
14258
14228
1146
1049
1456
1766
1922
1959
1871
1636
1372
1025
0699
0492
0305
0219
0175
0009
0003
40-44
95,392
82,237
61,447
37,900
25,065
13000
11207
08374
05165
03416
12994
11205
08376
05166
03417
1026
0896
0669
0442
0314
0742
0510
0304
0171
0100
0981
0658
0378
0193
0113
1285
0922
0549
0274
0151
1597
1212
0770
0381
0229
1785
1449
1003
0533
0318
1894
1888
1816
1467
0886
0565
1793
1888
1653
1090
0726
1724
1911
1526
1864
1874
1441
1045
1319
1876
1952
1534
1150
1117
1745
2019
1043
1728
2005
1778
0893
1787
1973
1744
1448
0094
0177
0189
0170
0152
0080
0171
0205
0191
0147
0408
1878
1918
1918
1838
0841
1308
2057
1776
1776
0000
1190
1429
2857
0952
0625
3125
1875
2500
1875
0556
45^49
1697
1199
0683
0440
0833
50-54
1840
1241
0857
1944
55-59
1645
1343
1667
60-64
1284
1667
65-69
16,640
02268
02285
0215
. 0063
0065
0094
0131
0175
0280
0370
0506
0585
0761
0840
0972
1033
1069
0123
0118
1470
1215
1906
0000
1111
70-74
9,297
01267
01268
0125
0035
0037
0051
0070
0108
0140
0215
0265
0355
0422
0467
0499
0605
0582
0057
0064
0327
0654
1190
0000
2222
75-79
4,254
00580
00581
0058
0016
0021
0023
0031
0052
0076
0090
0121
0154
0197
0814
0253
0243
0260
0024
0016
0122
0373
0476
0000
80-105
1,691
00230
00233
0025
0006
0007
0011
0014
0028
0031
0043
0052
0055
0062
0071
0073
0054
0062
0005
0003
0121
0000
13-105
733773
1.00
1.00
104761
109720
106195
90218
73962
58482
47045
37540
30537
24399
19317
12805
8841
5023
2575
1280
625
245
107
42
16
36
Totals
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00 1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
Note. — The flguies marked with an asterisk are the maxima in the vertical columns and those underUned are the maxima in the horizontal Unes.
t The figures though very approximate to those in the column to the left are obtained from a wholly different distribution of unspecified and
partially specified cases.
The figures in the body of the table are, of course, decimals. They are not deduced from those given in Table CXIV., pp. 322-3, but from the results
of a more detailed distribution of the unspecified quantities for various age and duration-of-marriage groups.Isee Table CXXIII., p. 338-9 later.
COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY.
335
Fig. 91 shews the characteristics of the age-fecundity surface, the
age being that at the time of the Census. If compiled according to
the " age at marriage" the form of con-
Age-Feoundity Surface. tours is, of course, materially changed.
1^
[D '-'
<D-ta
> &
m c3
<ar0
Number of Children borne.
Fig. 91.
2S. The durational fecundity dis-
tribution. — This distribution is exactly
analogous to that preceding, the argu-
ments in the table being Umits of dura-
tion of marriage and the number of
children borne. The values could be
obtained roughly from the data in Table
CXIII., by dividing the numbers of wives
who bore a given number of children by
the corresponding total (i.e., of the vertical columns, see pp. 322-3).
More accurately it could be found from the data- given in Table CXXIII.
hereinafter (pp. 338-9), the results for all ages being added together for
the required numbers.
The ignored elements are the ages of the wives and of the husbands.
If instead of being made out for all ages, durational fecundity surfaces
are determined for various age-groups, their characteristics wiU not
markedly differ, as might be inferred from Table CXVIII., p. 330
hereinbefore.
29. The age-polyphorous distribution. — ^The data which give the
age-fecundity distribution by dividing the tabular numbers by the totals
according to the number of children borne, give also the age-polyphorous
distribution if divided by the totals of the respective age-groups, see
Table CXXII. hereunder, in which the
required ratios are given. The distribu-
tion thus shews the relative frequency
with which married women in any given
age-group bear 0, 1, 2 . . . etc., children.
Age-FolyphoTous Surface.
The ignored elements are the dura-
tion of marriage and the age of the
husbands.
> ^
o >;
1 j
\<^^
hJ&v
^^^■^
" N ^ S T
Y ^ g s ^
n V Ss^
c^-
^ ^
-§%^
j,iss|
n SSS
^^5^
^Si
, ^) ^
:^:s_-
:^^5
\
\^^
J^f ^k-
^^
\^
V
5 n
:^5V '>
-■a-
/I-»5
.
/
SI
■ ,
III
1
16
Fig. 92 shews the characterisics of
the age-polyphorous surface, the age
being that at the time of the Census. If
compiled according to the " age at marriage"
is, of, course, materially changed.
Number of Children borne.
Fig. 92.
the form of the contours
336
APPENDIX A.
TABLE CXZn. — Shening. for all Durations of Marriage combined, the Relative Numbers of Married Women of given Age-gronps who
bore 0, 2, 3 . . . to n Children. Australia, Census of 3rd April, 1911. Age-polyphorous Distribution.
Ages
No.
of
Wives
Batio of the Number who bore the kth Child to the total Married Women of the Age-groups indicated, where k =
Wives.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Over
20
Total.
13
1
1.000
1.0
14
18
7200
2800
..
1.0
15
92
7280
2720
••
1.0
16
345
5300
4410
0290
1.C
17
1061
4690
•4850
0420
0040
1.0
18
2557
4149
•4939
0849
0055
0004
0004
..
1.0
19
4376
3497
•5034
1252
0203
0014
..
1.0
20
7,224
3034
•4828
1789
0324
0022
0003
1.0
21-24
57,896
2495
•3772
2408
0980
0275
0055
0013
0002
00003
1.0
25-29
109138
1730
•2610
2428
1660
0933
0419
0153
0049
0014
00003
00005
00004
00001
1.0
30-34
112523
1281
1678
•1967
1764
1346
0912
0550
0294
0130
0052
0018
00055
00018
00003
00002
00001
00001
1.0
35-39
104619
95,392
1147
1126
1100
0853
1478
1093
•1522
1215
1359
1095
1095
0842
0934
0587
0743
0400
0574
0240
0440
0129
0309
0060
0177
0026
0104
0010
0055
0004
0024
0001
0013
0005
0001
0001
1.0
40-44
•1238
1.0
45-49
82,237
61,447
1142
1140
0680
0543
0849
0652
1010
0806
•1091
0927
1030
•0956
0971
0829
0896
0700
0825
0567
0730
0438
0589
0292
0407
0188
0290
0106
0164
0056
0083
0028
0039
0013
0021
0006
0008
0002
0004
0001
0001
0001
0000
0001
1.0
50-54
0918
1.0
55-59
37,900
1222
0495
0540
0653
0743
0823
0847
•0878
0878
0799
0734
0539
0384
0236
0118
0058
0031
0012
0005
0003
0001
0001
1.0
60-64
25,065
1312
0437
0479
0542
0677
0742
0828
0845
•0884
0834
0805
0588
0475
0257
0149
0078
0037
0020
0007
0001
0001
0002
1.0
65-69
16,640
9,297
1353
1408
0415
0417
0417
0419
0508
0498
0581
0559
0617
0682
0788
0713
0837
0869
•0928
0858
•0930
0883
0878
0647
0644
0516
0474
0312
0327
0165
0161
0094
0044
0022
0008
0008
0005
0005
0000
0000
0002
0008
1.0
70-74
0869
0079
0043
0009
1.0
75-79
4,254
1425
0416
0524
0487
0531
0712
0825
0790
0872
0881
•0893
0555
0527
0287
0157
0073
0024
0007
0009
0005
1.0
80-105
1,691
1532
0373
0473
0597
0609
0875
0905
•0958
0934
0798
0710
0538
0385
0160
0095
0035
0012
0012
1.0
13-105
Nos.
733773
104761
109720
106195
•
90218
73962
58482
47045
37540
30537
24399
19317
12805
8841
5023
2575
1280
625
245
107
42
16
36
1.0
Batio
100000
14277
14953
14472
12295
10080
07970
06411
05116
04162
03325
02633
01745
01205
00685
00351
00174
00085
00033
00015
00006
00002 00005
1.0
Besnlt as by
cxnr.t
14222
14958
14483
12306
10090
07990
06415
05118
04149
03326
02632
01746
01204
00686
00351
00175
00086
00034
00015
00006
00002 00006
1.0
jq'Qte — ^Tbe figures marked with an asterisk are the maxima in the horizontal lines, and those underlined are the maxima in the vertical columns
excei>ting in the case of column 0, where .1126 is the minimum. '
t The figures though very approximate to the line above are given by a wholly different distribution of unspecified and partially specified cases.
The figures in the Ixidy of the table are, of course, decunals. They are not deduced from those given in Table OXTV., pp. 322-3, but from the results of a
more detailed distribution of the unspecified quantities for various age and duration-of-marriage groups, see Table CXXm., pp. 338-9 later.
30. The duiational polyphorous distribution. — ^The data from which
the durational fecundity is derived furnish also the numbers required for
the computation of the durational polyphorous distribution, viz., that
which shews for given durations of marriage, or between given limits of
duration of marriage, the relative frequency with which given numbers
of children are borne. The ignored elements are the ages of the wives
and of their husbands. Thisftable hasjnot been computed, but the
necessary data are given ia Table CXXIII. hereinafter.
COMPLEX ELEMENTS OP FERTILITY AND FECUNDITY. 337
31. Fecundity distributions according to age, duration of marriage
and number of children borne. — The fecundity distribution tables, so far,
are of the type z =f(x,y), but if age, duration of marriage and number of
children borne, be simultaneously taken into account, then the distribu-
tion-frequency is of the type z = f (w, x, y), and cannot be represented by
a single three-dimensional graph, for example, height contours upon a
plane. It is necessary in fact to have a graph for each value of w
adopted in the tabulations.
The exigencies of tabulation, of course, also require that a separate
table of the values of z shall be given for each value of one co-ordinate
(say w), for the values given by double entry of the other two (say x and y).
In Table CXXIII., hereunder, the results are tabulated for single
years of age from 13 to 20, (last birthday), for the ages 21 to 24, and then
for every five year age-group onward. The table gives, for existing
marriage, the number of wives, of various ages and durations of marriage,
who failed to give birth to children, or who gave birth to 1, 2, 3, etc.
In the tables as originally compiled, there was a considerable number
of unspeciiied cases, viz., the following : —
Class (i.), the larger class, in which the ages were specified.
Class (ii.), a relatively small class, in which the ages were not specified.
In each of these were three sub-classes as follow, viz. : —
(a) in which the duration of marriage was not specified;
(6) in which the number of children was not specified;
(c) in which neither the duration of marriage nor the number
of children was specified.
It was consequently necessary to efEect a distribution in order to get
anything like the most probable results.^
The method of distribution was that outlined in § 4, Table XC VII.,
and formulae (543) to (547). That is to say, sub-class (c) was first dis-
tributed proportionately among sub-classes (o) and (6), and sub-classes
(a) and (6) of Class (i.) were distributed proportionately among the fully
specified cases. In Class (ii.) the corrected sub-classes (a) and (6) were
then proportionately distributed among the fully specified corrected
groups of Class (i.). The details of the distribution shewed that the
result was very satisfactory judged by the regularity of the ratios (see
§ 5 hereinbefore).
1 The method of adopting the fully specified cases as characteristic of the
whole, involves merely multiplying each by the ratio of the totals. An examina-
tion of actual results shewed that recourse to this procedure was unsatisfactory.
It rejects part of the evidence available. To distribute the partially specified
oases is, therefore, much to be preferred.
338
APPENDIX A.
TABLE CXXm. — Shewing, for Varions Durations of existing Marriage, the Number of Wives in Various Age-groups who bore k Children,
where k = Q, 1, 2, etc. Australian Census, 3rd April, 1911.
Nnmlier of Wives to whom had been bom Children to the Number of :-
-
\A
Age 13
Age 14.
Age 15.
Age
16.
Age 17.
Age 18.
a g
1
Total.
1
Total
1
2
Total
1
2
3
Total
1
2
3
4
5
Total.
0-5
1
13
5
18
67
25
92
183
152
10
345
498
514
45
4
1,061
1,061
1,263
216
14
1
1
2.556
5-10
••
1
1
Totals
1
13
5
18
67
25
92
183
152
10
345
498
514
45!
4
1,061
1,061
1,263 217I
14
1
1
2,557
Age 19.
Age 20.
Age 21-24.
1
2
3
4
Total.
1
2
3
4
5
Total.
1
2
3
4
5 1 6
7
8
Total.
0-5
1,530
2,203
548
84
3
4,368
2,185
3,445
1,178
140
5
6,953
13,947
20,116
9,708
1,499
104
4
0, 45,378
5-lC
5
3
8
7
43
114
94
11
2
271
493
1,725
4,232
4,175
1,482 3141
73 lOj ..! 12,506
10-15
1
51
1 4| 1 2| 12
Totals
1,530
2,203
548
89
6
4,3761
2,192
3,488
1,292
234
16l 2
7,224l
14,440
21,841
13,94o! 5,675' 1,591^319' 771 n' 2' 57,898
•25-29
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
30
Total.
0-5
15,392
20,412
10,234
1,632
144
5
47,819
5-l(
3,127
7,205
14,778
14,40C
7,689
2,358
500
78
17
1
50,153
10-15
36(
866
1,48C
2,082
2,35C
2,212
1,171
453
133
31
4
4
11,145
15-2C
1
1
5
1
4
2
4
1
1
1
21
Totals
18,880
28,482
26,493
18,119
10,183
4,576
1,675
533
154
33
5
4
1
109.138
•30-34
i
0-5
7,788
8,17^
4,099
675
55
1
20,792
5-11
4,185
6,678
11,08S
10,308
5,759
1,988
466
97
17
2
40,587
10-lE
2,16(
3,456
6,051
7,827
8,093
7,022
4,372
2,087
727
204
377
49
13
5
2
42,077
15-21
27i
566
906
1,037
1,241
1,244
1,348
1,122
710
155
52
15
a
2
1
1
9,051
20-2E
2
C
1
2
1
2
3
5
2
2
2
• •
-.
22
Totals
14,414
18,87e
22,140
19,849
15,149
10,257
6,189
3,306
1,459
585
206
67
20
4
2
1
1
112,525
•35-35
0-5
4,39E
2,76-
1,307
20i
2'.
5
8,701
5-11
3,061
3,06(
4,50S
4,06f
2,22J
78e
203
6C
24
]
1
17,994
10-lE
2,835
3,21(
5,45<
6,288
6,087
4,94E
3,240
1,511
585
207
57
24
3
2
34,443
15-21
1,425
2,03.
3,458
4,424
4,782
4,596
4,297
3,54«
2,59t
l,39i
66«
281
106
39
8
1
33,649
20-2E
295
43E
739
944
1,092
1,128
1,062
1,019
985
89i
62e
324
161
68
37
11
2
1
9,824
25-3(
) ]
C
1
1
2
1
1
1
..
8
Totals
12,002
ii,5oe
15,462
15,927
14,213
11,458
8,803
6,138
4,190
2,500
1,350
630
270
110
45
12
2
1
104,619
•40-44
0-5
2,811
787
244
2!
4
..
3,869
5-lC
2,30(
1,52;
1,53(
l,06i
571
232
53
7
;
]
7,288
10-16
2,23(
1,97(
2,607
2,626
2,435
1,821
l,04e
48e
177
71
25
5
15,506
15-2(
1,836
1,99!
3,06S
3,654
3,859
3,359
2,931
2,227
1,442
81!
387
177
65
27
10
4
25,865
20-25
1,32J
1,570
2,587
3,634
4,156
4,165
3,984
3,531
3,03<
2,541
1,814
957
55S
264
110
56
17
3
3
34,308
20-35
23E
287
385
585
782
864
890
835
81!
775
721
55C
363
232
109
6C
33
7
A
r
1
?,
8,539
30-35
2
4
1
1
4
1
1
1
15
Totals
10,739
8,140
10,422
11,590
11,809
10,441
8,908
7,087
5,476
4,207
2,948
1,689
988
524
230
120
50
10
9
1
2
95,390
•45-49
0-5
1,901
112
11
1
1
2,026
3,029
5,738
9,914
24,368
30,339
6,816
7
5-10
1,882
624
309
139
46
21
5
1
2
10-15
1,646
1,105
1,001
854
553
305
168
63
2!
!
i
2
15-20
1,281
1,055
1,479
1,567
1,516
1,195
826
406
277
132
59, le
12
3
1
17
7
»
1
(1
1
20-25
1,444
1,478
2,312
2,973
3,364
3,130
2,870
2,297
1,666
1,16!
749' 38S
195
88
31
122
68
22
A
V
S
<^
25-30
1,061
1,053
1,631
2,441
3,050
3,276
3,515
3,250
3,018
2,652
2,100 1,450
91J
50S
263
88
32
21
7
3
1
1
30-35
175
170
242
339
437
• 545
601
738
772
70<
690, 542
422
268
165
(
35-40
1
1
2
1
1
1
..
Totals
9,390
5,598
6,985
8,314
8,968, 8,472
7,985
6,817
5,764
4,663
3,601
2,402
1,543
868
460
227
, 107
46
14
6
6
3
82,237
•50-54
0-5
969
1
, ,
970
1,308
2,099
3,520
8,363
18,424
21,319
5,435
9
5-10
1,221
68
13
4
2
'
10-15
1,224
451
208
126
45
25
11
4
4
1
15-20
877
600
619
565
406
219
119
51
3(
20
71 2
5
* •
••
20-25
862
748
1,055
1,289
1,291
1,098
782
552
352
16t
83, 44
23
11
4
2
1
25-30
1,019
769
1,237
1,742
2,144
2,329
2,244
2,145
1,761
1,264
847! 454
26!
110
49
22
13
e
1
30-35
704
590
749
1,076
1,580
1,893
2,156
2,275
2,38C
2,414
2,049, 1,439
977
646
265
11?
64
26
14
1
1
^
35-40
40-45
129
111
128
151
2
229 304
331
477
1
540
624
634 560
509
2
339
1
189
1
98
49
1
le
17
6
1
3
1
Totals
7,005
3,338
4,009, 4,955
5,637, 5,868
5,643
5,505
5,067
4,489
3,620 2,499
1,784
1,007
508
240
j 128
47
22
e
3
7
61,447
• Ages at date of Census.
COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY.
339
TABLE CXXm. — Shewing, for Various Durations oi existing Marriage, the Number of Wives in various Age-groups who bore h Children,
where ^==0, 1, 2, etc. Australian Census, of 3rd April, 1911. — Oont.
s^i
Number of Wives to whom had been born Children to the Number oJ :-
-
tl
1
2
3
4
5
6
7-
8
9
10
11
12
13
14
15
18
17
18
19
20
o
20
Total.
•55-59
0-5
466
466
5-10
578
1
579
10-15
770
60
21
3
1
1
856
15-20
630
285
174
75
37
8
4
1
1
1,215
20-25
576
388
387
390
292
184
112
73
32
26
5
5
2
1
2,472
25-30
493
398
488
625
721
762
661
433
264
143
93
48
16
11
9
1
1
5,167
30-35
544
369
510
751
889
1138
1,176
1,286
1,174
927
707
402
215
118
53
23
10
2
10,294
35-40
463
318
398
558
784
898
1,069
1,311
1,583
1,600
1,515
1,161
873
509
261
111
64
26
9
10
4
4
13,529
40-45
113
57
69,
71
92
128
188
222
275
331
462
425
348
254
126
83
44
19
10
2
2
3,321
45-
•• 1
1
1
Totals
4,632
1,876
2,047|
2,473
2,816
3,118
3,211
3,326
3,328
3,028
2,783
2,041
1,454
893
449
218
119
47
19
12
4
6
37,900
•60-64
0-15
905
905
15-20
477
22
499
20-25
459
190
i30
74
22
5
4
1
885
25-30
380
231
265
270
248
168
97
45
29
9
5
4
1,751
30-35
312
2C3
267
320
428
445
417
330
211
142
88
35
11
5
3
1
3.216
35-40
422
235
302
397
551
690
829
802
888
687
524
304
220
56
38
17
12
8
4
1
1
2
6,988
40-45
270
170
207
253
381
468
620
796
887
1,019
1,088
832
642
401
207
105
45
21
7
1
1
3
8,424
45-
64
44
29
45
67
84
108
145
201
234
315
297
319
183
125
72
3b
18
8
2
1
1
2,397
Totals
3,289
1,095
1,200
1,359
1,697
1,860
2,075
2,119
2,216
2,091
2,018
1,472
1,192
645
373
195
92
45
19
4
3
6
25,065
•65-69
0-15
456
_^
456
15-20
213
1
214
•20-25
340
30
4
1
1
376
25-30
328
115
73
50
19
4
4
1
594
30-35
234
139
180
157
139
102
55
39
26
13
6
2
3
1,095
35-40
204
154
161
233
278
339
335
241
178
103
56
15
16
3
2
4
2,322
40-45
245
117
159
214
278
272
485
554
591
509
399
246
146
69
43
22
11
3
4,363
45-
231
134
116
191
253
308
433
557
749
803
1,009
813
694
447
230
131
63
33
13
8
4
7,220
Total8
2,251
690
693
846
967
1,026
1,312
1,392
1,544
1,428
1,470
1,076
859
519
275
157
74
36
13
8
4
16,640
•70-74
O-20
351
351
20-25
13£
]
140
25-3C
205
18
1
1
225
30-35
145
75
74
24
19
8
3
2
1
1
347
35-40
119
84
84
116
105
72
29
32
14
3
1
1
660
40-45
110
71
9S
126
138
191
187
141
100
73
31
IS
6
6
1
1
1,293
45-
240
139
133
196
263
367
444
633
693
788
784
579
435
298
150
72
39
8
7
5
8
8,281
TotalB
1,309
388
390
463
520
633
663
808
808
865
816
598
441
304
150
73
40
8
7
5
8
9,297
•75-79
0-SC
27S
,
278
30-35
7£
i
C
C
1
84
35-40
70
32
26
5
(
2
(
]
C
2
144
40-45
50
38
68
62
SS
35
37
2C
i
6
C
1
2
C
1
361
45-
129
103
134
140
181
266
314
315
363
367
380
235
224
120
67
31
9
3
4
2
3,387
Totals
606
177
223
207
226
303
351
336
371
375
380
236
224
122
67
31
10
3
4
2
4,254
80-105
0-3S
117
117
35-4(
3(
]
39
40-4G
27
17
11
1]
6
2
]
]
1
76
45-
77
45
69
90
98
1
146
152
161
157
iss
120
91
65
27
16
6
2
2
1,459
Totals
25S
62
80
101
103
145
153
162
15E
13E
120
91
65
27
16
6
2
2
1,691
= 1
104,76]
L 109,72(
) 106,19E
9C,21S
i 73,962
58,48!
! 47,04E
> 37,54(
) 30,535
24,399
19,31'
12,80.
) 8,84]
5,02C
2,57!
1,28C
) 62.
) 24£
10
1 i.
: i(
5 3
3 733,773
^^1
• Ages at date ot Census,
340 APPENDIX A.
From the data furnished, distributions (vii.) to (x.) can readily be
computed.
32. The duration and age-fecundity distributions. — ^For a series of
duration-of-marriage-groups these distributions are obtained by com-
puting, for successive age-groups and for each number of children borne,
the relative frequency of the mothers within the indicated age-limits
who bore a given number of children tathe total mothers of all ages (which
are included) bearing the same number of children. These results may be
obtained by a re-arrangement of the data in Table CXXIII., pp. 338-9.
The distribution is (yii.) of Table CXX., p. 333.
The ignored element is only the age of husbands.
33. The duration and age-polyphorous distributions. — ^These, for a
series of duration-of -marriage groups, are obtained by computing for a
series of age-groups the relative frequency of the mothers within the age-
group who bore a given number of children to the total of all mothers in
the same age-group {i.e.-, who bore to « children). The results may be
obtained by the same re-arrangement as is required for the distribution
referred to in § 32, the present distribution being (ix.) in Table CXX.,
p. 333. The ignored element is, again, the age of the husbands.
34. The age and durational fecundity distributions. — By dividing
in each age-group the number of mothers who bore any given number of
children, and whose duration of marriage was between given Umits, by
the total number of mothers who bore the same number of children (i.e.,
for all durations of marriage in the age-group in question), the ratios in
Table CXXIV. hereinafter are obtained. Each series of ratios is the age
and durational fecundity distribution for the fundamental age-group.
This case is (viii.) in Table CXX., p. 333. The only ignored element is the
age of the husbands.
35. The age and durational polyphorous distributions.— As in the case
of the distributions immediately preceding Table CXXIII., pp. 338-9
furnishes the required data. The series of divisors in each age-group are
the totals for the indicated Umits of duration of marriage. Thus for
married women of a given age and a given duration of marriage, the relative
frequency of giving birth to 0, 1, 2 ... w children are obtained, and
these are shewn in Table CXXV. below. This case is (x.) in Table CXX.,
p. 333, and the only ignored element is again the age of the husbands.
COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY.
341
TABLE CXXIV. — Shewing, for a Series of Limits of Duration of Existing Marriage, and according to the Age groups given
in the Table, the Ratios of Married Mothers who bore k (where /c = 0, 1, 2 .... 20, and " over 20") Cliildren, to the
Total Number who, for all Durations of Marriage, Bore that Number. Census 3rd April, 1911. Australia.
Duration and Age Fecundity Distribution.
Dura-
tion
Proportion oJ the Number ot Women who, within the Indicated Limit o( Duration of Marriage, Bore k Children to the Total
Number o£ Married Women who Bore le Children, where k =
of
Mar-
riage.
1
2
3
4
6
6
7
8
9
10
11
12
13
14
16
16
17
18
19
20
over
20
0-5
6-10
1.000
1.000 9988' 9533
.. 1 0012 0467
5714! 1.000
4286|
Nos.
•13-19
3,363
1.00
4,162
1.00
820] 107
1.00 1.00
7i 1
1.00, 1.00
0-5
6-10
10-15
9699
0301
9308
0692
7147
2853
2774
7225
0001
0678
9291
0031
0125
9844
0031
9480
0520
9091
0909
1.00
N03.
*ao-24
16,632
1.00
25,329
1.00
15,232
1.00
5,909
1.00
1,607
1.00
321
1.00
77
1.00
11
1.00
2
1.00
0-5
5-10
10-15
15-20
S153
1656
0191
7166
2530
0304
3863
5578
0559
0901
7947
1149
0003
0141
7651
2308
0000
0011
5153
4834
0002
2985
6991
0024
1463
8500
0037
1104
8637
0259
0303
9394
0303
.9000
.1000
1.00
1.00
Nob.
•25-29
18,880
1.00
28,482
1.00
26,493
1.00
18,119
1.00
10,183
1.00
4,576
1.00
1,675
1.00
533
1.00
154
1.00
33
1.00
5
1.00
4
1.00
1
1.00
0-5
5-10
10-15
15-20
20-25
6404
2903
1503
0189
0001
4330
3538
1833
0299
0000
1862
6006
2733
0409
0000
0341
6193
3943
0522
0001
0036
3802
5342
0820
0000
0001
1938
6846
1213
0002
0753
7064
2178
0005
0293
6313
3394
0000
0117
4983
4866
0034
0034
3488
6444
0034
2379
7524
0097
0194
0776
0030
2500
7500
0000
5000
6000
1.00
1.00
1.00
Nos.
•30-34
14,414
1.00
18,876
i.oo
22,140
1.00
19,849
1.00
15,149
1.00
10,257
1.00
6,189
1.00
3,306
1.00
1,459
1.00
585
1.00
206
1.00
67
1.00
20
1.00
4
1.00
2
1.00
1
1.00
1
1.00
0-5
6-10
10-15
15-20
20-25
25-30
3662
2550
2360
1185
0243
2405
2659
2790
1768
0378
0845
2913
3528
2236
0478
0127
2554
3948
2778
0593
0016
1568
4283
3365
0768
0004
0686
4314
4012
0984
0231
3682
4881
1206
0098
2462
5777
1660
0003
0057
1396
6196
2351
0000
0004
0808
5572
3592
0004
0007
0423
4933
4637
0000
0381
4460
5143
0016
0111
3926
5963
0000
0182
3546
6182
0090
1777
8223
0834
9166
1.00
1.00
s
Noa.
•35-39
12,002
1.00
11,506
1.00
15,462
1.00
15,927
1.00
14,213
1.00
11,458
1.00
8803,
1.00
6,138
1.00
4,190
1.00
2,500
1.00
1,350
1.00
630
1.00
270
1.00
110
1.00
45
1.00
12
1.00
2
1.00
1
1.00
0-5
5-10
10-15
15-20
20-25
25-30
30-35
2617
2142
2082
1710
• 1232
0217
0966
1871
2425
2456
1929
0353
0234
1468
2502
2945
2482
0369
0020
0921
2266
3163
3135
0506
0003
0484
2062
3268
3519
0662
0002
0222
1744
3217
3989
0828
0000
0060
1174
3291
4472
0999
0004
0010
0686
3143
4982
1178
0001
0005
0323
2633
5541
1496
0002
0002
0169
1947
6040
1842
0000
0076
1313
6152
2446
0014
0030
1048
5666
3256
0000
0658
5658
3674
0010
0515
5038
4428
0019
0435
4783
4739
0043
0333
4667
5000
•
3400
6600
3000
7000
3333
6667
0000
1.00
1.00
Nos.
•40-44
10,739
1.00
8,140
1.00
10,422
1.00
11,590
1.00
11,809
1.00
10,441
1.00
8,908
1.00
7,087
1.00
5,476
1.00
4,207
1.00
2,948
1.00
1,689
1.00
988
1.00
524
1.00
230
1.00
120
1.00
50
1.00
10
1.00
9
1.00
1
1.00
2
1.00
0-5
5-10
10-15
15-20
20-25
25-30
30-35
35-40
2025
2004
1753
1364
1538
1130
0186
0200
1115
1974
1886
2640
1881
0304
0001
0016
0442
1434
2117
3310
2335
0346
0000
0001
0167
1027
1885
3576
2936
0408
0000
0001
0052
0617
1690
3751
3401
0487
0001
0025
0360
1410
3695
3867
0433
0000
0006
0210
1034
3594
4398
0758
0000
0001
0092
0684
3370
4767
1083
0003
0003
0050
0482
2890
5236
1339
0000
0019
0283
2494
5688
1514
0002
0008
0164
2080
5832
1916
0000
0008
0067
1595
6074
2256
0000
0078
1264
5917
3735
0006
0034
1014
5853
3088
0011
0022
0674
5717
3587
OOCO
0749
5374
3877
0000
0654
6355
2991
0652
4733
4565
0714
4286
5000
0000
4000
6000
2000
6000
2000
6667
3333
N03.
•45-49
9,390
■ 1.00
5,598
1.00
6,985
1.00
8,314
1.00
8,968
1.00
8,472
1.00
7,985
1.00
6,817
1.00
5,764
1.00
4,663
1.00
3,601
1.00
2,402
1.00
1,543
1.00
868
1.00
460
1.00
227
1.00
107
1.00
46
1.00
14
1.00
5
1.00
6
1.00
3
1.00
0-5
5-10
10-15
15-20
20-25
25-30
30-35
35-40
40-45
1383
1743
1747
1252
1231
1455
1005
0184
0000
0003
0204
1351
1797
2241
2304
1768
0332
0000
0032
0519
1544
2632
3086
1868
0319
0000
0008
0254
1140
2601
3516
2172
0305
0004
0004
0079
0713
2266
3763
2773
0402
0000
0043
0373
1871
3969
3226
0518
0000
0019
0211
1386
3977
3820
0587
0000
0007
0093
1003
3896
4133
0866
0002
0008
0059
0695
3475
4697
1066
0000
0002
0044
0370
2810
5378
1390
0000
0019
0339
2340
5660
1752
0000
0008
0176
1817
5758
2241
0000
0038
0129
1503
5476
3853
0011
0110
1092
5433
3366
0010
0079
0964
5217
3720
0030
0083
0917
4875
4125
0000
0078
1016
5000
3828
0078
1276
5320
3404i
0000
0455
6363
3183
0000
1667
8333
0000
6667
3333
0000
4286
4286
143U
Nos.
•50-54
7,005
1.00
3,338
1.00
4,CC9
1.00
4,955
1.00
6,697
1.00
5,868
1.00
5,643
1.00
5,505
l.OU
5,067
1.00
4,48Q
i.m
3,620
1.00
3,499
l.OfI
1,784
1.00
1.007
1.00
508
1.00
240
1.00
128
l.OC
47
1.00
22 6
1.00 1.00
3
1.00
7
l.OU
342
Al-PENDIX A.
TABLE CXXI7. — Shewing, for a Series of Limits of Duration of Existing Marriage, and according to the Age groups
given in the Table, the Ratios ot Married Mothers who bore fc (where i = 0, 1, 2 20, and " over 20") Children
to the Total Number who, for all Durations of Marriage, bore that Number. Census, 3rd April, 1911. Australia.
Duration and Age Fecundity Distribution — contimied.
Dura-
tion
Proportion ot the Number of Women who, within tlie Indicated Limit of Duration ot Marriage,
Bore ft Children to the Total
Number ot Married Women who Bore ft Children, where ft =
o£
Mar-
riage.
1
2
3
4
5
6
7
8
9
10
11
12
13
14 15
1
16
17
18
19
20
over
20
0-5
1006
1-
5-10
1248
0005
0000
10-15
1662
0320
0103
0012
0004
0000
0003
15-20
1360
1519
0850
0303
0131
0026
00] 2
0003
0000
0003
20-25
1241
2068
1891
1577
1037
0590
0349
0219
0096
0086
0018
0024
0014
0011
25-3C
1064
2122
2384
2527
2560
2444
2059
].sn2
0793
0472
0334
0235
0110
0123
0200
0046
0084
30-35
1174
1967
2491
3037
3157
3650
3662
3867
3528
3062
2540
19711 1479
1322
1181
105t
0841
0426
35-4C
lOOC
1695
1944
2256
2784
2280
S329
3942
4757
5284
5444
5688
6(104
5700
5813
5092
5378
5532
4V3V
0833
1.00
6667
40-45
0244
03C4
0337
0288
0327
0410
0586
0667
0826
10,13
1660
2082
2393
2844
2806
3807
3697
4042
5263
0167
0000
3333
t45
■■
0004
OCOO
0000
0000
Nos.
4,632
1,876
2,047
2,473
2,816
3,U8
3,211
3,326
3,328
3,028
2,783
2,041
1,454
893
449
218
119
47
19
12
4
6
*55-59
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
l.UO
1.00
l.CO
1.00
0-15
2752
15-1!
145(
0200
20-25
139(
1735
1083
0544
01 3C
0027
001 5
00fi6
25-3(
1156
21 IC
2208
1987
1461
0903
0467
0212
0131
0043
0(125
0027
30-33
094!
1854
2225
2355
2522
23J2
20] f
1.5.S7
09C2
0679
0426
0238
0092
0078
0080
0051
35-4C
128S
2146
2517
2921
3247
371C
3996
3785
4007
SWfi
2597
2065
1846
0868
1019
0872
1304
1333
2105
2500
3333
3333
40-46
082(
lEsa
172C
1862
2245
2516
29R8
3656
4003
4873
5391
5652
,f,386
6217
5550
.5.S85
4892
4667
3684
2500
3333
5000
t45
C195
0402
0242
0331
0395
0452
0520
0684
0907
1119
1561
2018
2676
2837
3351
3692
3804
4000
421 J
500C
8334
1667
Nos.
3,28f
1,095
1,200
1,359
1,697
1,860
2,C75
2,119
2,216
2,091
2,018
1,472
1,192
645
373
195
92
45
19
4
3
6
•50-64
l.OC
1.00
1.00
1.00
1.00
1.00
1.00
l.UO
l.CO
1.00
1.00
1.00
1.00
l.CC
1.00
l.UO
1.00
1.00
1.00
1.00
1.0c
1.00
&-l£
2026
15-2S
G94I
0014
2C-2I
151(
0436
005f
0C12
OOOC
OOIC
25-31
-145'-
1667
105;
0596
019(
003;
0036
0007
30-3E
104:
2014
259S
1856
143f
0994
0426
0286
0168
0091
0041
0019
0035
35-41
0901
2232
232i
2754
2875
3304
255E
1731
1163
0721
0381
0139
0186
0058
0073
0255
40 -4£
108S
1696
2294
253C
2875
2651
3697
3986
3828
.'(565
2714
2286
170(1
1323
1564
1401
1486
0833
1.0*;
l.CO
0000
1.00
1-45
1026
1942
1674
2258
2616
3002
3300
4002
4851
5623
6864
7556
8079
8613
8363
8344
8514
9167
Nos.
2,251
690
693
846
967
l,826l 1,312
1,392
1,544
1,428
1,470
1,C76
8.59
519
275
157
74
36
13
8
4
•6.5-6!
l.OC
1.00
1.00
l.CC
l.CO
l.CO
l.OC
i.oo
1.00
1.00
l.OC
l.OC
1.00
1.00
l.CO
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0-2C
) 268]
20-2E
1065
002f
25-31
1561
0464
0026
0022
30-3E
1101
193;
1897
051S
0365
0047
0045
0025
0012
0011
35 -4(
0911
2165
2154
2505
201!
1137
0437
0396
0173
0035
0C12
0017
40-4E
0S4(
183C
251J
2722
2558
3017
2821
1745
1238
0844
0386
0301
0136
0197
00061
137(1
0250
t45
183S
3582
3410
4233
5058
5799
6697
7834
8577
9110
9608
9682
9864
9803
1.00
8630
9750
1.00
1.00
1.00
0000
1.00
Nos.
1,30{
388
390
463
520
633
663
808
808
865
816
598
441
304
1.50
7S
40
8
7
5
8
•70-74
1.0C
1.00
- 1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
l.OC
1.00
l.CO
l.CO
1.00
1.00
1.00
1.00
1.00
1.00
l.CO
1.00
C-3t
458?
30-3E
130'
022«
OOOC
OOOC
0044
35-4C
1156
180£
1166
0241
0265
0066
noor
0030
0000
00.53
40-46
0826
2147
2825
2996
168]
1156
1054
0.595
0216
O160
0006
0043
000(1
0164
0000
flflon
1000
t45
2129
5819
6009
6763
8010
8779
8J46
9375
9784
9787
1.00
9957
1.00
9936
1.00
1.00
9000
1.00
1.00
1.00
Nos.
606
177
223
207
226
303
351
336
371
875
380
236
224
122
67
31
10
»
4
2
•75-79
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
l.OC
1.00
1.00
1.00
l.OC
1.00
1.00
0-36
4517
35-4(
1467
0016
40-45
104S
0270
1375
1089
0486
0132
0065
006?
0063
t45
2973
0714
8625
8911
9514
9968
9935
9338
9937
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
Nos. *
259
63
80
101
103
148
153
162
158
135
120
91
65
27
16
6
2
2
80-106
1.00
l.OC
1.00
l.CO
1.00
l.CO
1.00
1.00
l.OC
l.OC
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
Grand
Total
104761
109720
106195
90,218,
73,962
68,482 47045
37540
30535
24399
19317
12805
8,841
5,023
2,575
1,280
625
245
107
42I 16
36
Ages.
t 45 and over.
COMPLEX ELEMENTS OP FERTILITY AND FECUNDITY.
343
TABLE CXXV. — Shewing, for Various Durations of existing Marriage, the Proportion of Women of Various Groups
of Ages, who bore 1, 2, 3 ... «. Children, the Total for each Age-group between the Limits of Duration of
Marriage being Unity. Australia, Census of 3rd April, 1911. Duration and Age-polyphorous Distribution.
Duia
tion
of
Mar-
Proportion of tlie Total ot Women within the Indicated Limit oJ Duration o£ existing Marriage who bore
Children to the Number of *, in which i =
Total
No.
for the
riage
and
Age.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
" 19
20
OTer $
20 H
Dura-
tion.
0-5
5-10
3975
.4931
L097(
111
3 012
1555(
LOOO!
i333;
)000]
i ..
..
■*
■••
••
1.00 8,441
1.00 9
•13-lS
396J
492:
)097(
)012f
iOOOS
iOOQ]
..
ITOO 8,450
0-5
5-10
10-15
308C
039)
450S
1384
,208(
1340]
)03i;
334]
083C
0025
116£
4168
0247
0833
065-
3333
0006
0833
066s
■■
••
••
••
••
1.00 52,331
1.00 12.777
1.00 12
♦20-24
2554
389C
233£
0906
0247
0049
0012
0001
oooc
..
l.OC
65,120
0-5
6-10
10-15
15-20
3218
0624
0323
0476
426S
1437
0776
OOOfl
214C
2947
1328
0476
0341
2871
1868
2380
0030
1533
2109
0000
0001
0470
1985
0476
oioo
1051
1906
o6i5
0406
0952
0663
0119
1906
0660
0027
0476
0004
0476
0004
OOOO
0476
1.00
1.00
1.00
1.00
47,819
50,153
11,145
21
* 25-29
1730
2610
2428
1660
0933
0419
0153
0049
0014
0003
00005
00004
00001
1.00
109,138
0-5
5-10
10-15
16-20
20-25
3745
1031
0515
0302
0909
3931
1645
0822
0620
0000
1972
2732
1438
1000
0454
0325
2540
1860
1146
0909
0025
1419
1924
1370
0454
0001
0490
1670
1374
0909
oii5
1039
1489
1365
0024
0496
1239
OOOO
0004
0172
0784
2273
0660
0049
0416
0909
ooio
0171
0909
0663
0057
0909
0661
0016
0661
0002
0662
0661
0661
••
1.00
1.00
1.00
1.00
1.00
20,792
40,587
42,077
9,051
22
•30-35
1281
1678
1967
1764
1346
0912
0650
0294
0130
0052
0018
00055
00018 00003
00002
00001
00001
1.00
112,526
0-5
5-10
10-15
15-20
20-25
25-30
5051
1701
0822
0423
0297
1250
3180
1700
0932
0605
0443
0000
1502
2502
1583
1027
0751
1260
0233
2262
1826
1315
0961
0000
0028
1238
1767
1420
1112
0000
0006
0437
1435
1366
1148
0000
oii3
094]
1307
1081
1250
0033
0349
1054
1037
2500
0013
0170
0772
1003
OOOO
0661
0060
0414
0914
1250
0660
0016
0198
0636
OOOO
0007
0084
0329
1250
0662
0032
0164
OOOO
0660
0010
0069
1250
0662
0038
0661
0011
0004
0062
1.00
1,00
1.00
1.00
1.00
1.00
8,701
17,994
34,443
33,649
9,824
8
*35-40
1147
1100
1478
1522
1359
1095
0842
0587
0400
0240
0129
0060
0026
0010
0004
0001
OOOO
OOOO
1.00
104,619
0-5
5-10
10-15
15-20
20-25
25-30
30-35
7265
3156
1443
0710
0386
0273
0000
2034
2090
1274
0773
0458
0336
0000
0632
2099
1682
1187
0754
0451
0000
0059
1466
1692
1413
1058
0685
0000
0010
0783
1570
1492
1212
0916
1334
03i8
1174
1299
1214
1012
0000
0073
0675
1133
1161
1042
3667
loio
0313
0861
1030
0978
0667
0004
0114
0557
0884
0959
0667
0661
0046
0317
0740
0908
OOOO
o6i4
0150
0529
0844
2667
0003
0068
0279
0644
OOOO
0025
0163
0426
0667
ooio
0077
0271
0667
0004
0032
0128
0667
0661
0016
0070
0005
0039
0661
0009
0661
0007
0660
0661
0662
1.00
1.00
1.00
1.00
1.00
1.00
1.00
3,869
7,288
15,506
25,865
34,308
8,539
15
.40-44
1126
0853
1093
1215
1238
1095
0934
0743
0574
0440
0309
0177
0104
0055
0024
0013
0005
0001
0001
OOOO
3625
3556
1.00
95,390
0-5
6-10
10-15
15-20
20-25
25-30
30,-35
35-40
9383
6213
2869
1292
0593
0350
0257
0000
0563
2060
1926
1064
3606
0347
0249
1429
0054
1021
1745
1492
0943
0538
0355
OOOO
0005
0459
1487
1582
1220
08O5
0497
OOOO
0005
0152
0964
1529
1380
1005
0641
1428
0069
0632
1205
1285
1080
08OO
OOOO
0016
3293
0833
1178
1157
0888
OOOO
0663
0110
0470
0943
1071
1083
2867
0667
0O5O
0279
0684
3996
1133
OOOO
ooie
0133
0477
0874
1036
1429
0005
0060
0307
0692
1012
OOOO
0003
0016
0157
0482
0795
OOOO
o6i2
0080
0301
0619
1429
0003
0036
0167
0393
1428
0661
0013
0087
0242
o6i7
0050
0029
0007
0028
0010
3003
3009
3007
0661
0002
0002
0660
0001
0001
3661
3001 (
3000 (
3661
3000
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
2,026
3,029
5,738
9,914
24,368
30,389
6,816
7
*45-49
1142
068O
0849
1010
1091
1030
0971
0829
0700
3567
0438
0292
0188
0106
0056
0028
0013
3006
0002
0001
3001
3000
1.00
82,237
0-5
5-10
10-15
15-20
20-25
25-30
30-35
35-40
40-45
9890
9335
5831
2491
1031
0553
0330
0237
OOIO
3520
2149
1705
0894
3417
3277
3204
0099
0991
1759
1261
0672
0351
3236
0031
0600
1605
1452
0946
0505
0278
2222
o6i5
0214
1153
1544
1164
0741
0421
oiig
0622
1313
1264
3888
0559
3052
3338
3935
1218
1012
3610
o6i9
0145
066O
1164
1067
0878
nil
3020
3085
3421
3956
1117
0994
0665
0O57
0198
0686
1133
1148
0020
0099
0460
0960
1166
0006
0053
0246
0675
1030
06i4
0028
0145
0458
0937
2223
o6i3
0060
0256
0624
1111
0665
0027
0124
0348
1111
0662
0012
0056
0182
•• ■
0661
0007
0030
0090
1111
0003
0012
0029
00005
0007
0013
00664
0009
0661
0002
0661
0005
1111
1.00
1.60
1.00
1.00
1.00
1.00
1.00
1.00
970
1,308
2,009
S,520
8,363
18,424
21,319
5,435
9
•60-59
1140 (
3543
3652
08O6
0927
1
0956
3918
0896
0825
073O
0589
0407
0290
0164
0083
0039
0021
0008
0004
0001
OOOO
0001
1.00
61,447
344
APPENDIX A.
TABLE CXXV. — Shewing for Various Durations of Existing Marriage the Proportion of Women of Various
Groups of Ages, who Bore 1, 2, 3 ... n Children, the Total for each Age-group between the Limits of
Duration of Marriage being Unity. Australia, Census of 3rd April, 1911. Duration and Age-polyphorous
Distribution — continued.
Duia-
tionol
Mar-
Proportion of tlie Total of Women witliin tlie Indicated Limit of Duration of Existing Marriage who Bore
Children to the Number of Ir, in which ]c =
Total
No.
riage,
and
Age.
1
2
3
4
5
6
. 7
8
9
10
11
12
13
14
15
16
17
18
19
20
over
20
for the
Dura-
tion.
0-5
5-10
10-15
15-20
20-25
25-30
30-35
35-40
40-45
45 and
over
1000
9983
8995
5185
2327
0954
0528
0342
0340
o6i7
0701
2346
1570
0770
0358
0236
0172
0245
1432
1565
0944
0495
0294
0208
0035
0617
1578
1210
0730
0412
0214
o6i2
0305
1182
1395
0864
0579
0277
0000
0066
0744
1475
1105
0664
0385
o6i2
0033
0453
1279
1143
0791
0566
0008
0295
0838
1250
0979
0668
0000
0129
0512
1140
1157
0828
0008
0105
0277
0900
1183
0997
0000
0020
0180
0687
1120
1391
1.000
0020
0093
0391
0858
1280
0008
0031
0209
0646
1048
0004
0021
0115
0376
0765
o6i7
0051
0193
0380
0002
0022
0082
0250
0002
0010
0048
0132
0002
0020
0057
0007
0030
0007
0006
0003
0000
0003
0006
1.00
1.00
1.00
1.00
1.00
1.00
1,00
1.00
1.00
1.00
466
579
856
1,215
2,472
5,167
10,294
13,529
3,321
1
•54^59
1222
0495
0540
0653
0743
0823
0847
0878
0878
0799
0734
0539
0384
0236 0118
0058
0031
0012
0005
0003
0001
0001
1.00
37,9b0
0-15
15-20
20-25
25-30
30-35
35-40
40-^5
45 and
over
1.00
9559
5186
2170
0970
0604
0320
0267
0441
2147
1319
0631
0336
0202
0184
1469
1513
0830
0432
0246
0121
0836
1542
0995
0568
0300
0188
0249
1416
1331
0789
0452
0280
0057
0959
1384
0988
0556
0350
0045
0554
1297
1186
0736
0451
ooii
0257
1026
1148
0945
0605
0i66
0656
1271
1053
0839
0052
0442
0983
1209
0976
0029
0267
0750
1292
1314
0023
0109
0435
0988
1239
0034
0315
0762
1331
ooio
0080
0476
0764
0009
0054
0246
0521
0003
0024
0125
0300
o6i7
0053
0146
0009
0025
0075
0006
0008
0033
0001
0001
0008
0001
0001
0004
0003
0004
0004
1.00
1.00
1.00
1.00
1.00
1.00
1.00
100
905
499
885
1,751
3,216
6,898
8,424
2,397
•60-64
1312
0437
0479
0542
0677
0742
0828
0845
0884
0834
0805
0588
0475
0257
0149
0078
0037
0020
0007
0001
0001
0002
1.00
25,065
0-15
15-20
20-25
25-30
30-35
35-40
40-45
45 and
over
1.00
9953
9043
5522
2137
0379
0562
0320
0047
0798
1936
1269
0663
0268
0185
oioe
1229
1644
0693
0364
0161
0027
0842
1434
1003
0490
0265
OOOG
0320
1269
1197
0637
0350
0026
0067
0932
1460
0623
0427
0067
0502
1443
1112
0800
o6i7
0356
1038
1270
0772
0238
0767
1354
1037
oiio
0443
1167
1112
0055
0241
0915
1397
o6i8
0065
0564
1126
0027
0069
0335
0961
o6i3
0152
0619
0009
0099
0319
o6i7
0050
0181
0025
0087
0007
0046
0018
0011
0000
0006
1;00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
456
214
376
594
1,095
2,322
4,363
7,220
•65-69
1353
0415
0417
0508
0581
0617
0788
0837
0928
0858
0883
0647
0516
0312
0165
0094
0044
0022
0008
0005
0000
0002
1.00
16,640
0-20
20-25
25-30
30-35
35-40
40-45
45 and
over
1.00
9929
9112
4179
1803
0852
0382
0071
0800
2161
1273
0549
0221
0044
2133
1273
0758
0212
0044
0692
1758
0974
0312
0547
1591
0129
0419
0087
1091
1427
0587
0086
0433
1446
0707
0058
0485
1090
1008
0029
0212
0773
1103
0028
0045
0565
1255
0015
0240
1248
06i5
0139
0922
0046
0692
0047
0474
0000
0239
0008
0115
0007
0062
0013
0011
0008
0000
0013
1.00
1.00
1.00
1.00
1.00
1.00
1.00
351
140
225
347
660
1,293
6,281
•70-74
1408
0417
0419
0498
0559
0682
0713
0869
0869
0930
0878
0644
0474
0327
0161
0079
0043
0009
0008
0005
0000
0008
1.00
9,297
0-30
30 35
35-40
40^5
45 and
over
1.00
9405
4861
1385
0381
0476
2222
1053
0304
0000
1806
1745
0396
0000
0347
1717
0413
oiig
0417
1052
0534
0i39
0975
0785
0000
1025
0927
0069
0554
0930
0000
0222
1072
0i39
0166
1084
0000
1122
0028
0694
0000
0661
0055
0354
0000
0198
0000
0092
0028
0027
0009
0012
0006
••
1.00
1.00
1.00
1.00
1.00
278
84
144
361
3,387
•75-79
1425
0416
0524
0487
0531
0712
0825
0790
0872
0881
0893
0555
0527
0287
0157
0073
0024
0007
0009
0005
..
1.00
4,254
0-35
35-40
40-45
45 and
over
1.00
9744
3553
0528
0256
2237
0308
1447
0473
1447
0617
0658
0672
0263
1001
0i32
1042
0i32
1103
oisi
1076
0925
0822
0624
0446
0185
0110
0041
0014
0012
••
•■
1.00
1.00
1.00
1.00
117
39
76
1,459
80-105
1532
D373l0473
0597 0609
0875
0905
0958
0934
0798
0710
0538
0384
0160
0095
0036
0012
0012
1.00
1,691
• Totals for ages indicated. Ages at the time of the Census.
COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY.
345
36. Fecundity-distributions according to age at marriage. — By sub-
division, according to duration of marriage, of the numbers in Table
CXXIII., pp. 338-9, and subsequent rearrangement, tables can be prepared
giving very approximately the distributions corresponding to the ages at
marriage ^- As this involves the relative numbers marrying at successive
ages, it is essential to know the frequency of marriage at given agesi
This is furnished by Table LIV., p. 190 2. The results are as follow : —
TABLE CXXVI. — Shewing the Relative Number of Marriages according to Ages of
Brides. Australia, 1907-1914, ' and the Average Age for each Year Group.
Alleged age (last
birthday)
12
13
14
15
16
17
18
19
20
21
22
23
24
25
Mean age
Ko. of marriages
per 1,000,000
12.66
3
13.66
13
14.67
242
15.67
1,620
16.61
7,992
15.57
22,885
18.54
43,889
19.52
64,027
20.52
81,033
21.49
90,337
22.49
92,609
23.49
87,491
24.49
79,199
25.49
68,610
Alleged age (last
birthday.
26
27
28
29
30
31
32
33
34
35
36
37
38
39
Mean age
No. of Marriages
per 1,000,000
26.49
58,749
27.48
48,897
28.48
40,286
29.48 30.48
33,259 26,627
31.49
21,480
32.49
16,927
33.49
14,553
34.49
11,548
35.49
10,451
36.49
9,415
37.49
8,444
38.49
t
7,540
39.49
6,702
Alleged age (last
(birthday)
40
41
42
43
44
45
46
47
48
49
50
51
52
53 to
95
Mean age
No. of marriages
per 1,000,000 -
40 49
5,931
41.49
5,225
42.49
4,584
43.49
4,003
44.49
t
3,481
45.49
3,014
46.49
2,598
47.49
2,230
48.49
1,906
49.49
1,623
50.49
1,375
51.49
1,160
52.49
t
975
7,064
' * Smoothed for misstatement of age.
reciprocals of 1.105, 1.110, 1.115, etc.
t Smoothed to a curve by a multiplier changing regularly, viz., the
The preceding table shews that, from the age 18 onwards, the average
age is, sensibly, the age last birthday plus one half-year, and no serious
error wiU result if it be so taken even for the ages earlier than 18. Hence
a correction can be readily made for the effect of mortality, and asynthetic
table prepared in the following way : —
Let a, h, etc., denote the marriages at ages (last birthday) Xi, xz, etc.,
reduced for a half-years' mortality ; a', b', etc., these reduced for one
and a half years' mortality ; a", b", etc., the same reduced for two and
1 Original compilation according to age at marriage is, of course, the best
method of obtaining the proper numbers.
^ This gives 8 years' experience in Australia of the frequency of marriage at
different ages, the total oases being 301,918.
. » These numbers are deduced from those shewn on pp. 190-191 by distri-
buting the 111 unspecified cases.
346
APPENDIX A.
a half years' mortality, the mortahty being both of husbands and wives,*
and so on. Then, ignoring migration, the numbers according to age, as,
at a census, and for a given duration of marriage, will be as shewn in the
following table, viz. : —
TABLE CXXVn. — Scheme of Compilation of Numbers according to Duration of
Marriage.
Duiations
Aqe at Census.
of Marriage.
»!
«2
■■is
«*
Xn
Xe
X-,
etc.
0-1 ..
1-2 ..
2-3 ..
3-4 ..
4-5 ..
a
b
a'
c
b'
a"
d
d
b"
a"'
e
d'
c"
6"'
a"
f
e'
d"
c'"
6'v
9
r
e"
d'"
etc.
etc.
etc.
etc.
etc.
The total numbers of married women for durations of to 5 years,
5 to^lO years, etc., are consequently : —
(564). .0^6 = (a) + (a' + b) + (a" + b' + c) + {a"' + b" + c' + d)
+ (aiy + 6" + c" + rf' + e) + (6'' +c"' + ...+/) + etc.
(565). .bMio = (a^) + (a'' + 6') + (a™ + 6" + C) + etc.
(566). .10-^15 = {a-^) + (a"' + 6'^) + (a*" + 6^' + c^) + etc.; etc.; etc.
It is obvious that a synthetic table can be prepared by means of which
the partition can be effected of a group of married women between given
limits of age and duration of marriage : in this way the mean age of any
element may also be readily ascertained. Obviously the successive
quantities vertically are, with sufficient precision for the purpose in view,
respectively — in actuarial notation : —
m^i^—^l'x) ; ■rr^x-x (1— k'«-i)-P'*-i •P'x;m^-^ (^-yx-^)-P'x-,'P'x-i'P'x
m denoting the number of marriages, according to the age of the woman,
^ For rigorous results the fact must be taken into account that the death of
hiisbands also removes the women from the category " married." Hence the
correction for mortality includes the probable number of deaths of wives, and of
husbands, diminished, however, by the joint deaths, which are counted, of course,
once only.
COMPLEX ELEMENTS OF FERTILITY AliTD FECUNDITY.
347
<lit the probability of a woman^ of age x either dying or becoming a
widow within one year, and ^j, the probabiUty of Hving in wifehood one
year.*
Adopting the roughly approximate method we obtain from the data
in Table LIV., pp. 190-l,»the figures shewn in Table CXXVIII. hereunder.
TABLE CXXVin.— Shewing Ezample;!of Computation of Distribution of Numbers
according to Age at and Duration of Marriage.
a&
AOBS AT CENBUB.
^1
12
13
14
12-14
15
16
17
18
19
15-19
20
21
22
23
24
20-24
0-1
1-2
2-3
3-4
4-6
1
4
1
73
4
1
78
5
1
.489 2,409
73 488
4 73
6,898
2,404
488
73
4
13,227
6,878
2,398
486
73
18,084
13,190
6,858
2,391
485
41,107
23,033
9,821
2,955
563
20,198
18,028
13,149
6,837
2,384
32,616
20,132
17,969
13,106
6,815
. 27,898
; 32,502
20,062
1 17,906
il3,060
26,351
27,795
32,382
19,988
17,840
23,854
26,249
27,687
32,256
19,910
130,917
124,706
111,249
90,093
60,009
0-5
1
6
78
84
567
2,975
9,867
23,062
41,008
77,479
60,596
90,638
111,428
124,356
129,956
616,974
5-6
6-7
7-8
8-9
9-10
1
4
1
73
4
1
78
5
1
483
72
4
1
2,376
481
72
4
1
6,791
2,368
479
72
4
13,012
6,766
2,359
477
72
17,771
12,962
6,740
2,350
475
40,433
22,649
9,654
2,904
562
5-10
1
5
78
84
560
2,934
9,714
22,686
40,298
76,192
In the above results it is obvious that the age at marriage is at once
approximately, though not exactly, obtained for each sub-group by
subtracting the " duration" from the " age." The general result may be
represented as follows : —
Let s, with appropriate suifixes, denote an element of 8, the total
between given limits of age and duration of a series of groups of s : then
(567).
x.t^of,t' = *1 + «2 + • • . . etc.;
+
s'l + s'z + • • • • etc.
+
etc. -|- etc. -f . . . . etc.;
1 If husbands and wives were of the same age the probability of mortality
which takes both into account would be approximately 1 — ^q — ^q + Iq q for
one half-year, the suffixes m and / denoting male and female respectively. Cor-
responding changes must also be made in the p factors.
* For greater rigour account must be taken of the exact interval ; the half-
year and year is not exactly correct, because the distribution is not uniform. This
refinement, however, is not called for, because migration and other irregularities
prejudice the data to a much greater extent.
' Similarly those given in Table CXXVI. could be'used, andjwould perhaps be
more reliable as they are smoothed results. The table includes only allowances for
deaths of wives : the deaths of husbands have been omitted from consideration.
34S
APPENDIX A.
X and x' denoting the age limits, and t and t' the duration limits. Con-
sequently if Q be any given total of a series of groups, and g be the value
of any component group, its approximate value is given by^
(568).
.g = G.s/S
the suffixes being the same for g and s.
For greater precision the values of s must be taken in Table CXXVIII.
as modified not only by death but also by migration. In this way tables
compiled according to the ages as at the Census can be reconstructed to
furnish results according to the ages at marriage. The recasting of the
ages may be effected as follows : —
Let Xc and Xg denote respectively the ages at which fertility com-
mences (say 11 or 12), and ends (say 58 or 59) ; <„ and t^ the limits of any
duration of marriage adopted in compilation, Xi and x^ being also any age
limits adopted, as at the moment to which the compilation refers (the
Census) ; then the whole range of ages, x ' , at marriage is given by : —
(569),
.a;'i to x'-i ^ (a;i — <j) to {x^ — <„)
because on the inferior side an age will be included less than the lower age
limit by the whole amount of the longer term of the duration, and on the
superior side an age which is less only by the shorter term of the duration.
1 The group syntheses (a) in Table
CXXVIII. further extended ; (6) those
obtained by taking no account of
deaths, and (o) those given by the
Census are respectively as a, b, and c
hereunder, a and 6 being reduced so as
to give the same total as at the Census.
Owing to Census defects the
Census results (c) cannot be regarded
as absolutely correct; and owing to
migration effects the synthesis results
(a) or (6) will, of course, materially
differ from the Census. It is evident,
however, that the general correspond-
ence between the Census and the
synthetic results is sufficiently well
established over a wide range of dura-
tions and ages, and that the corres-
pondence furnishes a sufficient reason
for relying upon the subdivision of
the group-totals into their elements,
especially for the earlier ages, and
lesser durations of marriage. For age
20, viz., the age at which the misstate-
ments are known to be large, the
results are by (a) 7158 ; 66 : (6)
7157; 67: (c) Census, 6953 ; 271.
Corrections for mortality are pro-
bably an unnecessary refinement.
Dura-
Ages. .
Totals.
tion.
11-14 15-19| 20-24 25-29 | 30-34
Synthesis (a)
0-5
5-10
10-15
15-20
20-25
8
7,523
8
50,199
7,398
8
49,51S
49,166
7,239
8
20,894
48,291
47,925
7,052
8
128,139
104,863
55,172
7,060
8
8
7,531
57,605
105,927
124,171
295,242
Synthesis {bi
0-5
5-10
10-15
15-20
20-25
9
7,342
9
49,224
7,347
9
48,760
49,224
7,347
9
20,637
48,750
49,224
7,347
9
125,967
105,330
56,580
7,356
9
9
7,356
56,580
105,330
125,967
295,242
Synthesis (c)
0-5
5-10
10-15
15-20
20-26
19
8,422
9
52,331
12,777
12
47,819
50,158
11,145
21
20,792
40.587
42,077
9,051
22
129,383
103,531
53,234
9,072
22
GTtl.
19
8,431
65,120
109,143
112,629
295,242
COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY.
349
Hence it follows that if the age-groups and durations both change by
a constant, the range of ages at marriage will be always as above (569).
Since ages outside the limits Xc and Xg have no significance as regards
fertility they may be ignored and consequently the earUest age of marriage
may be taken say, as 11, and the latest say, as 58.
If x'2 = a;'i + 1 and <j + <„ + 1, the range of ages extends over two
years. The subdivision by applying the synthetic results can con-
sequently give only approximate results and cannot sensibly attain to
the accuracy of " direct compilation according to the age at marriage."
37. Complete tables of fecundity. — Still disregarding the age of
husbands, complete tables of fecundity are based, as in the case of sterihty,
on the age at — and duration of — ^marriage. They give the proportions
of those married at each age who bear 0, 1, 2, 3, etc., children, after
the lapse of given durations of marriage. They are most serviceable if
developed in the following way, viz. : —
Arguments for
each Table.
Age at marriage ; and
duration of marriage
(for the child-bear-
ing period only).
Body of Table.
Proportion — ^for each increase of 1 year of
age, and for each increase of 1 year in
the duration of marriage — of the grand
total of married mothers who bear
children (Sterility table) ; who bear 1
child; who bear 2 children; etc., etc.
From such a table as the above the derivative tables, previously
indicated, can be readily prepared. Tables compiled on the basis of age
at marriage could be distinguished as
gamogenesic, etc., see p. 285. Fig. 93, and
the notes thereto, exhibit in perspective
the nature of the gamogenesic surfaces
representing the proportion of wives who,
having married at a given age, have borne
during a duration of marriage of t years
k children.
38. Digenesic surfaces and diisogenic
contours. — ^If the husband's age be not
ignored fecundity relations become greatly
increased in complexity. For example, instead of a maternity rate or a
birth-rate according to the age of wife, we have a series for each age of the
350 APPENDIX A.
husbands ; the compilation-table becomes one of double entry,, and the
various fertiUty and fecundity-relations become correspondingly multiplied.
If the ages of husbands and wives constitute respectively the abscissae
and ordinates of verticals, the heights of which represent the particular
birth-rates, maternity-rates or else that characterise the combinations of
ages in question, the surface defined by the totality of the verticals may be
called a digenesic surface. That is to say, a vertical 2 of a digenesio sur-
face is represented by.: —
(570) z=^.F(x,y)= p^, or p^
where pxy is the birth-rate (or !p:^y is the maternity rate) for the group of
wives of the ages y to y -\- dy, the ages of whose husbands he between
X and x-\-dx. To avoid circumlocution let the case be restricted to the
consideration of birth-rate only. The curves z = constant, or lines of
equal birth-rate on this surface are diisogenic contours (they have been
called by Korosi and Galton'^ isogens). Any series of ages x, y, x', y' ,
x",y" , etc., for which ^^y ^ constant may be called the diisogenous ages.
The system of orthogonal trajectories which define the lines of the most
rapid increase or decrease of birth-rate for any points through which they
pass, may be called the meridians of these points.
The diisogeny of communities has not yet been generally investi-
gated. Korosi has examined the question for the population of Budapest.
For Australia the results are given hereinafter, and differ materially
from the results for Budapest.
39. Diisogenic graphs and their significance. — Owing partly to
paucity in the number of instances when they are distributed into small
age-groups, coupled with the fact that even " physiological fecundity" is
probably by no means uniform in the human race, and the further fact
that the intentional restriction of fecundity is operative in widely different
degrees, the crude data, distributed say in year-groups, do not give very
definite indications of the exact position of the contours, though they
reveal unmistakeably that the birth-rate is not only profoundly affected
1 See, " An estimate of the degrees of legitimate natality as derived from a
table of natality compiled by the author from his observations made at Budapest."
By Joseph KorSsi, Phil. Trans., Vol. 186, Pt. H., pp. 781-875, 1896.
" Isogens," by Francis Galton, Proc. Roy. Soc, Lond., Vol. 55, p. 18.
The question had engaged the attention of a large number of persons, for
example, A. N, Kiaer, 1876. Stieda, R. Boeckh, BertiUon, Keefe, and ol3»ers.
COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY. 351
by the age of the mother but is also by no means unaffected by the age of
the father. It is also evident that, for a considerable range of single-
year age-groups, the relation that j8 is constant
when X -\- my is constant, is approximately true,
a point to which we shall recur ; see (576) here-
inafter. Thus the problem to be solved is that
of determining, from the somewhat irregular
surface indicated by the crude data, the more
regular surface which ideally defines the general
characteristic of birth-rate as related to
"' ' T the ages of the husband and wife, viz., x, y.
Fig. 94. The magnitude of the accidental differences
between single-year age-groups is so large that
meticulous precision is out of the question. Hence, using Umited ranges
of age we may proceed as follows in order to " smooth" the surface.
Let K, L, M, etc.. Pig. 94 denote crude values of the birth-rates for
the age-groups a; to a; -f l,ytoy-\- 1, etc. Then approximately
{511).. k = i{K+L+P + Q); 1=\{L+M+Q+B)- etc., etc.
If this does not give a sufficiently smooth surface we can reconstitute
a smoothed value of Q, Q' say, from k, I, p, and q, thus :^
(572). . Q'=^(h+l+p+q) = ^ {4:Q+2{L+P+R+ V) + {K+M+ U+W)}
In this last the weight assigned to the values L, P, etc., and K, M,
etc., vary reciprocally as the square of the distance to the centre of the
group-square from the centre of Q. If the results are extremely irregular
it may even be preferable to adopt : —
(573) C = i (K+L+M+P+ Q+R+ U+ V+W)
instead of the preceding formula. The smoothed values being to hand,
the contours may readily be drawn. When deemed necessary small
corrections can first be applied to the heights for any systematic error
introduced by the process of smoothing. •
Since the group heights are too small for the central value when the
surface is convex upwards, too great when it is concave upwards, a
limitation which is accentuated when the mean of a number of heights is
formed, as in (571) to (573). This error is analogous to that dealt with in
Part IX., §§8 and 9, formulae (311) to (323), and Table XV. The cor-
rections may be ascertained as soon ^s the su^J^pe is approximately
determined.
352 APPENDIX A
Tiz/c to z«, are the vertical heights of the centres of the squares
in Mg. 94, for the average height Zq for the whole area of 4 squares
embraced between the lines joining the points K, M, W, U, would be
rigorously
(574) Zo = ^{l6Zj+4(zi+«,+«r+«y) + (z*+2m+Z«,+Z„)}
provided the sections of the surface are curves of the third or a lesser
degree ; or, if the four component surfaces K, L, Q, P, etc., were " ruled
surfaces," the height Zq would be, also quite rigorously : —
(575).... Zo = ^{4Zj+2(z,+Z,+Z,+Zj,) + (3ft+2m+««>+Zu)}
If the external factors, therefore, are made unity in (574) and (575)
the internal will be, respectively, ^, -g-, -^, and -|-, -§-, and ^. It
is evident from these results that the elimination of systematic error
involves in aU cases the assignment of a high " weight" to the central
value. But it is equally certain that if the central values be considered
liable to deviations from the general trend of the surface, which, compared
with the systematic errors introduced are small, we may practically reach
a better result by emplojdng (571) or even (573).^
Another and more satisfactory method of obtaining values of ^^v
is to smooth the series of the values of the type K, P, U, etc. ; i.e., with
y constant ; and independently those at right angles thereto, viz.,
K, L, M, etc., i.e., with x constant. The means of the two results for
each point are then adopted as a jQrst smoothing, and the process repeated
as often as is found necessary. This leads to more rigorous results, but
can be readily employed only when the original results do not deviate
largely from the general trend of the surface.
40. Diisogens, their trajectories and tangents. — ^The general nature
of surfaces such as are here under consideration has been indicated in
Pt. XII., §§ 21 and 22, pp. 201-203, and the fundamental formulsB of
orthogonal trajectories have been given. The system of contours upon
such surfaces (diisogens) probably do not conform to any simple geo-
metrical specification ; the present imperfect data certainly do not point
to their representation by any system of curves of a simple character,
though the settlement of this question must remain for more extended
investigation and more accurate data. At any point {x, y) whatsoever,
dy I dx furnishes the relation by means of which the birth-rate equivalence
1 The question of the adjijstment of such values, has been systematically
treated by E. Blaechke, Ph. D., see his " Methoden der Ausgleiohung von
Wahrsoheinlichkeiten," Wien, 1893. See also Phil. Trans., Vol. 186, 11., pp. 870-5,
1895. See also Part XII. herein, § 39, pp. 230-2.
COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY. 353
of pairs of ages may be expressed in the form K^ = a; + OA y =
a constant. For we shall have, for the direction of the tangent to a
diisogen, dy/dx = tan 6 — 1/6? say. Hence it follows that
(576). .X — Ay cot Q = X ^ Ay -5— = G ; that is x-\-OAy = G
If k be the recripocal of K then kG will be the constant value of
the birth-rate for the diisogen in question. Ordinarily dx/dy is negative.
Parallelism of the tangents of diisogens to the a;-axis would imply
that the increase of the age of the husband had no influence whatever on
the birth-rate, while the parallelism of the tangents to the y-axis would
denote that the age of the wife had no influence. If, therefore, the age
of the wife has, in general, the preponderating influence, the diisogens
must make a smaller angle with the a;-axis than with the y-axis. If
the diisogens are inclined 46° to each axis, then the birth-rate is constant
when x -\- yis constant.^
41. Digenesic age-eauivalence in two populations. — As already
shewn, the diisogens or their orthogonal trajectories determine the cor-
relative changes in the ages of husbands and wives which give equivalence
of birth-rate, i.e., diisogeny. The diisogenic factor G in formula (576) for
any pair of ages {i.e., of husband and wife) is the coefficient which must
be multiplied into the age of the wife so that the product, plus the age of
the husband, will be continually proportional to the birth-rate. It holds,
of course, only for a moderate range of age-differences about the point
for which it is ascertained. Thus the expressions : —
(577) j; — • «/ -5- = constant ; x ~ — y = constant,
apply only to a limited region. For two populations the differential
coefficients are not identical. Hence, for a given difference of age in the
wife, the equivalent difference of age in the husband is not the same.
The factor to make one equal the other may be called the masculine
factor of age-equivalence, E. Similarly the factor to make the difference
in the wives' age equal, for a given difference in the age of husband, may be
called the feminine factor of equivalence, E'. Suffixes can be used to denote
the ages (of. husband and wife) to which these factors exactly apply.
1 Roughly speaking this representa the general character of the relation
indicated (on Table 3, facing p. 852, Phil. Trans., Vol. 186, Pt. II.), by KorOsi.
Thus, for quite a large range of ages, the birth-rate would appear, according to that
authority, to depend merely upon the sum of the ages of husband and wife, and not
upon their individual ages. This condition may be called equilateral diisogeny, and
is probably not a general condition.
354
APPENDIX A.
Let 8 y denote any small difference in the age of wives at the point x, y,
common to the populations A and B, the tangents to the diisogens
making the angles 6^ and ^j, respectively, with the 9;-axis. Then since
8a;, = hy cot da and Sa;;, = 8y cot 0j, we have
(578). .
..E
8a-6 Sycotdi Gft tan 9„ dFa(x)/dx
Sxa Syootda Ga tan ^j dFt(x)/dx
Similarly —
(579)..
hy„ hx tan 6^ (?„ 1
hya Sx tan d^ G^ E
that is, the masculine and feminine factors of age-equivalence are recip-
rocals.
42. Birthrate-sauivalences for given age-differences. — The factors
of age-equivalence merely disclose the equivalent differences of age for
two populations for a given age-difference in either sex, but not the birth
rate equivalence. This latter depends
not only upon the direction of the
tangents to the graphs of the
diisogens in plan {i.e., upon the tan
gents to their horizontal projections),
but also to the angle of slope i/r of
the orthogonal trajectories. The
tangent to any point Q, on a
trajectory will be required. The
angle it makes with the z-axis will be
^ so that 1 + ^ = 90°. The follow-
ing procedure will always be abund-
antly accurate for determining the
age-equivalence and digenesic effi-
ciency for any point Q the co-ordin-
ates of which in plan are x,, y^. Let P' P P", Q' Q Q", and R' E R"
in Fig. 95 be three diisogens (the values of which are known), crossed
by the orthogonal trajectory P, Q, R, which in general is, of course, a
curve of double curvature (tortuous curve). Let this trajectory be
projected orthogonally on to the horizontal plane X Y passing
through P : this proj ection is the broken line P q r , the proj ections of short
stretches of the diisogens being similarly the broken lines q' q q" and
r'rr"; P'PP"is itself in the plane of projection.
Let the curved hne Pq be denoted by Xi, and the curved line Pr, of
which Pq forms part, by X^, measured along the curve ; and let also the
difference of birth-rates for P and Q {i.e., Qq) be denoted by 8i, and the
Fig. 95.
COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY. 355
difference for P and R be denoted by 83 ; then we may assume that the
curved triangle, P Q R r q P in relation to lengths along the curved axis
P q r, is, with sufficient precision, given hy S = bx + cy^, and therefore
that the tangent at the point Q is dh/dy = 6 + Sc^. Thus we shall
have : —
When 82 = 281, this becomes
81 y-l — 2X2^1 + 2X-^
and when, in addition, X2 = 2Xi, the expression becomes, of course,
(582) tan i^ = Si /^x =82/^2
The direction of this line of slope tangentially passing through Q,
and making the angle ^ with the horizontal plane, is shewn by the pro-
jection S q T, which is tangential to p q r, passing tangentially through
the point q. It, of course, makes the angle d' with the X axis. Con-
sequently the angles d' and (/«, or their complements Q and ^, give all the
necessary relations required.
Since the line, Qq, in the figure =81 = /J, — jSj,, viz., the difference
of birth-rates indicated by the diisogens at P and Q, the horizontal
equivalent thereof, Sq = s, say, measured in the direction of the tangent
to the orthogonal trajectory at Q is : — •
(583) s = 81 cot i/* = (i3j — j8p) tan I,
since ^ + 1^ = 90°. Thus, in plan, the rate of change of the birth-rate
at any point a;, 2/ on a diisogen can be ascertained from the position of the
diisogens on either side, and the position of the orthogonal trajectory
through the point. Thus the age-equivalence of this difference of birth-
rate is to be found by dividing by the sine and cosine of the angle which
the orthogonal trajectory makes with the co-ordinate axes, 9 and 9' ;
their sum, 9 -{- 9' = 90°. Consequently the masculine birth-rate-
equivalence, H say, for wives of the one age, is : —
(584) H = (jS, - jSp) tan ^ oosec 9
since 1 /sin 9 = cosec 9, and the feminine birthrate-equivalence H', for
husbands of the one age, is
(585). . . ■ . , fl' = (;8, - jSp) tan ? sec 6
356
APPENDIX A.
We thus have, from these two equations, for two populations, A and
B for any common small difference of birth-rate, the ratio : —
(586) Hb^taii^t, cosec 6^ _ __, i^ b
H„
tan ^„ cosec 6^ ' H\
H'j, tan ^6 sec Bj,
tan ^a sec $„
These relations, however, can be determined very readily from
appropriate graphs of the populations.
43. Diisogeny in Australia. — Diisogeny is doubtless best exhibited
by the maternity rates, not the birth-rates, the ratios to be ascertained
being the proportions which the number of cases of maternity bear to the
number of women at risk in any age-group with husbands of any age-
group.
In order to ascertain the nuptial maternity rates of Australia accord-
ing to pairs of ages, the nuptial cases of maternity have been taken out
for the seven years 1908 to 1914 inclusive, that is, for the Census year
1911, and for the three years before and after that year. In order to
relate these cases of maternity in age-groups to the numbers of married
couples in the same age-groups at the Census, they have been divided,
not by 7, but by a number which gave the true average, viz., 7.13143.^
The results thus obtained are shewn by the uppermost of the figures in
Table CXXIX. hereunder. Thus the results used are equivalent to a
total 5,232,988 married women, among whom maternity was experienced
814,617 times. This gives an annual maternity rate of 0.15567. But of
this number of married women, 7.6368 per cent, were 60 years of age and
over, and 12.7667 per cent, were 55 years of age and over, so that about
87 per cent, were of child-bearing age. Hence the birth-rate for married
1 This figure was ascertained in the following way ; — ^The number of females in
the years 1908 to 1914 inclusive were multiplied by a linear changing ratio (deter-
mined from the intercensal period 1901-1911) in order to obtain the numbers of
married women during the years in question, the results being as hereunder : — ■
1908
2.0187C6
X
.33355
= 677,377
1309
2.038512
X
.33818
= 696,148
1910
2.103318
X
.34081
= 716.832
1911
. 1 2.156781
X
.34344
= 740,725
1912
. 1 2.224484
X
.34607
= 769,827
1913
2.301011
X
.34870
= 802,363
1914
. 1 2.361643
X
.35133
= 829,716
Tot.ll N
0. of married women
in 7 years
5,232,988
No. at Census date
Total birthg in 7 years
Births in Census year
738,773
Total population
Census population
All females, 7 years
Census females
Total married
females, 7 years
Census year (wliole)
_ 31,697,28.=
~ 4,455,005
_ 15.224,455
2,141,970
= 7.11498
= 7.10769
_ 5.232,988
734,226
= 7.12722
= 7.13162
= 7.12758
This was found to agree with other deductions as to the number of years, viz., 7-|- e
where e was a small fraction (as shewn above) varying between 0.10769 to 0.13162.
The actual divisibn used was 7.13143, the reciprocal of which is 0.140224. This,
multiplied into the births during the 7 years, gave the uppermost figure shewn in
the table,
Complex elements of pertiliTy and fecundity.
35?
women of 13 to 54 years of age inclusive was 0.17845, or for women of
13 to 59 years of age inclusive, 0.16854. Korosi's results were 46,926
children from 71,800 married couples, in 4 years, that is 0.16339 per
annum. 1
The numbers of husbands and of wives recorded in the Australian
Census of 3rd April, 1911, were not equal. It was deemed probable that
the number of wives recorded would be the best basis for determining the
distribution according to the age of the married women at the Census :
in this way the numbers exposed to risk are ascertained in each age-
group. The adjusted distribution^ gives the numbers which constitute
the denominators of the ratios.
In general there is a considerable number of cases for each pair of
age-groups adopted ; the table discloses the number. It is evident,
however, that in extreme instances the numbers are small, and the
maternity rates consequently ill-determined.* They may be regarded,
however, as well ascertained where the number of mothers has been
shewn in heavy figures.
The age-distribution as at the Census probably differs but little from
the average distribution over the 7 years, which yielded the births : hence
the ratios ascertained may be accepted as very closely representing the
true amounts. The results are shewn in Table CXXIX. hereunder.
f^ 1 The average crude birth-rate for Australia for 1908-14 was .02745, and for
Hungary for 1908-12, 0.3632. Apparently the Budapest matemity-rate is not
larger than that of Australia.
2 The following is a conspectus of the data; — ^
Unspecified as
Husbands whose
Husbands whose
Total
regards
Wives were
Wives were
Wife's Age.
with them.
Absent.
Unspecifledas regards
4,108
620,846
11,084
husband's age . .
2,368
506
2,874
1,045
3,919
Wives whose husbands
were with them . .
619,106
4,614
623,720
112,129
735,849
Wives whose husbands
were absent
108,892
1,161
110,053
Total Wives . .
5,775
733,773
, ,
The adjustment was effected as follows : — ^The 506 doubly unspecified cases
were divided into 185 and 321, that is in the proportion of each to their sum, and
those were distributed proportionally among the wives and husbands unspecified.
Next the 1161 wives, unspecified as regards age, were distributed proportionally
among the 108,892 whose ages were given, thus making up the total 110,053. A like
proceeding was followed in the case of the 1045 husbands, unspecified as regards
age, so as to make the total 112,129. The individual totals were then reduced by
multiplying throughout by 0.981485, so as to form the same aggregate 110,063 as
in the case of the wives. One half of each was then distributed proportionally to
the individual original numbers, thus making the grand total 733,773. See Table
I., pp. 1106-7, Vol. III., Census Report.
' In general, tables prepared in this manner have the advantage that it may at
once be seen whether any change of a ratio, necessary to make it conform to a general
law, is probable or otherwise. A result like that shewn for ages of husbands 17,
and of wives 16, viz., 1.55, is of course not impossible, but it would not be true for a
very large number of cases.
358
APPENDIX A
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•*rteq
00 in to
CO cam
000
OOC
=!£
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R*S
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^-"3
= 3
' 3
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^^
(N
aeus
Oi^t-
o-*o
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M
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- ^-
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rH
rtlOOO
lOiACC
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in
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i-t
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; I ;
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eo
. . .
; ; ;
rH
•saSv
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s
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S
SS
in
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vn
ss
i
ss
sss
O"*
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s
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2St S5§ §"§1
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1i
sa
■"5
39
53
a*
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°!1 >>
aS
0.3
SS
11
§5
"i
P
360
APPENDIX A.
Diisogenic Surface. Australia.
Ages of Wives = y.
S 30 EL9B CSOet'lllSIUIIlUqEO
^^ .§ ^' % J' £^ M- t I ^
« t s'gr^t^'?^!!^!!
i 'S^^N^^^-^^^ ^^^ ^S ■"■ ^ ^ ""■ -- '
"" H ^ ^ 'Vsn""^ ^S^^^ J^'^^SS 5 ^^S^ i;^^ S^ ,'
~? [ ^^ \\ V V \ ^ ^^ o \ ^ ^^^ ^^^ ^ ^ ^ t* '^"^ "^ "^ '
" t -RiJEJ \rfcS3S5VvSsl|||||l§vsNv^4
^t ?^z:S7■-■4Iat^^S^^'SSSs5|5S5 5^'^^X
Xti ■jzm^iii'i - \2ux\^^^^^
\\J/ .'///// 1 .\ V h-.'--l- -1 -Yv -vi •■- i- - '
'zl-MtiiJt^ntutui i--i-ll-li l-
y^^M-tutT-' 2ZQE52' JIhLZJQL Attttt
'^MtttuJt'Jt^ltlttutut^
J. \s±htTt%im
"" ^ WittkujL^xninmjjt'ihtt'iM
^ MtTittfi1fit%tt^
^t lKlttlt^JU:ttuMttJUtH t H
i %it'Zt$^ttumtttt%t4-tt-----l--- t
imih'Uitttitrrnitht 1
? ^titi.itktitjjtt'tit
. ^ xttitttttmtiittz^it
'^'''iqtiiitittLuut^
" G jtiittiUt il-itt it
•a ' \ 1 'i //M/// ' / /
1 « ^j,^ mttl-tltlittMitttt -.-
s J. luLtiu-tutt^i-iittuit I
1 ' tU%1-tutttiUMttJmii
t ' tizrittltjitr'^ :tlifi
" z -\z7tl-ttJ-li^ll21-ttUt 14
a " jszt^ttTittiiiiittmtiti
^ a \%z.7ttTttiittt<i^ iXVXX-
^t ^%u7ttlltLL,Z,_.^J4it1t':,t.
S %%^^7ltiZ,iTll11ttl-t%itt
1 7 ^Zvthlltttttttlltlt t
5 ^ t^V'^tftt'i^^^Aitttit ip
^ , %^-^^t^-iil1.l2tittntt,tt
z ^^^)^^titttL/Lrr-i:iitiitt,_^_..
h^^t^-,t-,~<~ij^j4l-ttt it' t ^1
^^5^4 yiiiltttt'tiifT: ...
" / / \^ T" ''hi ^ J 1 1 1 I
^ S^j^^777^7777777/M7i
" ^ v^ ' ^t t t, ''LLttitti-^
" 7 ^^^-t^21JJJJ.^-,J4UPti-i
^ \^^^'^ ^/-.ttftirritrz ttii
"-- ^ ^ ^tlj^ 'tttz^ij^ ttl
^' ^ ^ ^^^^ ^/^^44^-i7utt^ '~i
i A ^-■^-'"^/^^Z^ Ttlltu^t^ t
5 £=.,3^^,^/:: ^ I^ u _njrii ^
- T'^ y .^^^il4 4-71-ttu 'i^'
1 - ^^--i-'^^/V 11 ttUt'l t-4 LJL
Z ' 2 / / t-i-lTtn ^t Z"
7 '^ I'-' l/ / jrtttttti tl rr
^Z kv ^.££^^7^^5^_^^_ifcj Sh I^iT
2' A% -s ^7 ittTim-tjit-rt^
« 2i ^5 2 ^ ^ ■a^'^'^3^ 23r7 r-.t^
Fig. 96.
The maternity rates shewn, denote the ratio of the average niimber of cases of
matemity occurring to the niunber of wives at risk, whose ages are shewn at the
top line of the figure, their husbands' ages being shewn at the left hand side of the
figure. The dotted lines roughly represent the major part of the surface between
the principal meridians AB and AC.
The results tabulated above, and slightly smoothed, are shewn on
Fig. 96, in which the heavy Unes, viz., the birth-rates -05, -10, -15, ... .
•45, are first ascertained from the tabular results, and then smoothed.
The thin Unes, shewing differences of -Ol in the matemity rate, are then
drawn in (having regard to second differences). The diisogens 045, 0-35,
COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY.
361
0'30, and 0'25 exhibit some peculiarity. This is probably not due to
physiological differences in fertility ; the dotted lines are beUeved to
better represent the character of the physiological law (modified, of course,
by the incidence of social traditions). The broken Une A B is one of the
principal meridians of the surface, and denotes the ages where small
differences in the ages of the husbands have no effect on the fertility.
The broken Hne A C is the other principal meridian, and denotes the ages
where (very) small differences in the age of the wives have no effect. As
the higher ages — during the fertile period of woman's life — are reached,
the age of the husband has apparently very little influence at least from
35 to 65 years of age. The graph. Fig. 96, however, requires no inter-
pretation.
Diisogenic Surface. Budapest.
of Wives = J/.
M. Diisogeny generally. — ^Korosi's results ^ for Budapest (as re-
reduced by me) are shewn on Fig. 97. The results for individual ages
shew great irregularity, but were computed as indicated hereinbefore,
§§38 and 39. The irregularities doubtless would disappear with larger
numbers. It will be seen in Fig. 97 that at age 36 for husbands and 30
for wives, the direction of the diisogen is inclined 45° to the axes x and y
for a considerable length thereof and that, for an extended range of ages
of husband and wife, the relation roughly holds that the birth-rate is
1 Phil. Trans., Vol. 186, Pt. IL, pp. 781-875.
362
APPENDIX A.
constant when the sum of the ages of husband and wife are constant, and
further that it decreases with increase in this sum.^ The statement
acquires greater generahty, however, if put in another way, viz. : —
For ages greater than that of the maximum fertility of women and
for those combinations of ages of husband and wife which are most com-
mon, the fertiUty-ratio may be regarded as represented — very roughly
of course — ^by straight lines : that is to say, x and y being respectively
the ages of husband and wife at the time of the birth, the fertility -ratio is
constant when kx -\- y \s, constant. These constant values are typically
represented by the lines ab, cd, ef , etc., on Fig. 97, and by a'b', c'd', e'f,
etc., on Fig. 96. The pairs of ages, x and y, which give identical fertihty-
ratios, may be called corresponding age-pairs. They do not, of course,
actually lie on straight lines, as is evident from either Fig. 96 or Fig. 97.
Moreover the fertility-ratio (and thus the value of k) diminishes with increase
of the sum of the corresponding age pairs (the aEge of maximum value having
been passed). Obviously, also, k differs for various populations. ^ A
rough general comparison of the Budapest (Korosi's) results for Budapest,
and those for Australia is best indicated by shewing the position of the
lines of " corresponding age-pairs" according to the value of the fertility-
ratio (birth-rate ) .
TABLE CXXX. — Comparison of Approximate Lines of Eaual Fertility according
to Fairs of Ages ; Australia and Budapest.
^•"'h'S
AxrsTB.Ai.iA (Mateinity lates).
Bddapest (Birthrate).
Inii
■5 a jai
Intersection Point of Lines
Intersection
Point of Lines
•
1
of Equal Fertility.
of Eqnal Fertility.
ti to
a r°5 «
X =
-50; w=-t-55i
When
a; =
y =
X =
± 0;
» = -1- 55i
When
x =
V =
II
a-2«S£
Angle
= fl
Tangent = k
Angle = e
Tangent = *
g«Wj^
Obsd.
Gale.
Obsd.
Calc.
Obsd.
Calc.
Obsd.
Calc.
. /
. /
Tears
. /
• /
Years.
000 ..
4.0
9.0
001 ..
4.33
4.28
.0792
.0781
51.54
10.23
9.68
.1833
.1757
65.5
St
0.05 . .
7.2
6.21
.1233
.1113
49.34
14.56
13.53
.2667
.2472
55.5
Qr
0.10 ..
8.40
8.42
.1525
.1530
47.88
18.26
18.46
.3333
.3398
55.5
o.p
0.15 . .
10.48
11.3
.1908
.1953
45.96
23.52
23.39
.4425
.4379
55.6
m n
0.20 ..
13.27
13.24
.2392
.2382
43.54
29.64
28.32
.5750
.5437
65.5
Icl
0.25 . .
15.49
15.45
.2833
.2820
41.34
36.62
33.25
.7228
.6598
56.5
a
0.30 ..
17.47
18.6
.3208
.3269
39.46
40.48
38.18
.8632
.7898
55.5
0.35 . .
19.50
20.27
.3606
.3729
37.47
45.0
43.11
1.0000
.9385
56.5
ef
0.40 . .
22.24
22.48
.4120
.4204
34.90
48.30
48.4
1.1301
1.1132
56.6
cd
0.45 . .
25.66
25.9
.4862
.4695
31.19
52.42
52.57
1.3127
1.3246
■56.5
ab
0.50 ..
?
?
?
?
?
1
The two systems of lines are: —
For Australia^ ^j ( a; -f 50) -f 2/ = 55.5 ; and
For Budapest ; k^ix -\- 0) -\- y = 55.5.
* Ot X -\- Ky = constant, see (576). Galton and KdrSsi's suggestion that
le — 1 is an extremely improbable one, and is not borne out by the data, ae KOrosi'g
own results shew.
* As already mentioned, the number of cases of maternity reviewed by KiirOsi
was 46,926, occurring in the years 1889, 1890, 1891 and 1892, and these were attri-
butable to 71,800 families according to the Census of 1st January, 1891, giving a rate
of 16.339 per 100 families per annum over all. See p. 790, op. oit. The number
is of course, insufficient to determine the surface with great accuracy.
COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY. 363
It would appear that the directions of these hnes vary about equally
for equal changes of rate since they are given by the formulae.
(587) ..0 = 4° + 47V, for Austraha ; 6 = 9° + 97f °r for Budapest.
These values are approximately correct for the regions within lines
AB and AC on Figs. 96 and 97, but not outside those regions. '
45. Multiple Diisogeny. — ^The equal frequency of twins, or of
triplets, etc., according to pairs of ages may be called multiple diisogeny,
the series of ages giving equal frequency being in this case also known as
" corresponding pairs." The twin digenous surface, triplet digenous
surface, etc., are the surfaces defined by the terminals of the a co-ordinates
corresponding to the frequency of twins, triplets, etc., the x and y co-
ordinates representing as before the ages of the husband and wife re-
spectively.
In order that the results may be unequivocal, the ratios to be used
should be those of the number of births " of at least n -\- 1 children," to
the number of births " of at least n children." That is, the ratio of twins
should be to the cases of maternity ; the ratio of triplets should be to
the cases where there were at least twins ; of quadruplets to at least
triplets, and so on. Suppose in a population P there were : —
A ' cases of maternity in which only one child was born ;
B' „ „ „ two children were born ;
,, ,, „ three, ,, „
and let A' -\- B' -{■ G' -\- etc. = M, the total cases of maternity. Then
the maternity ratio for the population is : —
(588) mi = {A' + B' + C + etc.) / P = M/P.
The twin ratio, so taken as to include all mothers who had at least
two children at a birth, is : —
(589) m2 = (5' + C" + etc.) / Jf = B/M.
The triplet ratio, or that based on all mothers who had at least three
children at a birth, is : —
(590) ms = (C '+etc.) /(£' + C" + etc.) = C/B,
and so on.
364
AJPPENDIX A.
In this system we have : —
(591) mi = M/P ; mi mg = B/P ; mi m2 mj = C/P ; etc. ;
that is, the population multiplied by the product of the ratios mi . m2 . . .
m„ gives the number of women bearing at least n children. The ratio
m„ is thus the relative frequency with which a woman — ^who in any child-
birth has given birth to n children — will have given birth to the (w + 1)
child on the same occasion.
For the 9 years, 1907 to 1915, in Australia there were in all 1,042,588
cases of maternity ; 10,630 cases of twins and triplets, and 100 cases of
triplets : that is 1,031,858 single births, 10,530 cases of twins, and 100
cases of triplets. The ratios and their degree of fluctuation are shewn in
the following table : —
Table CXXXI. — Shewing Freciuency of Occurrence of Twins and Triplets (Nuptial
and ex-Nuptial Cases combined). Australia, 1907-1915.
1 Cases
Twins
Ratio of
Ratio of
Ratio of
Ratio of
Year
of
including
Twins to
Triplets to
Twins,
Triplets,
1900.*
. Matern-
Triplets,
Triplets.
Cases of
Cases of
5 Year
5 Year
ity.
etc.
Matern-
ity nia
^wins.
wig
Average.
Average.
7
109,305
1.042
13
.00953
.01247
8
! 110,491
1,065
6
.00963
.00663
—
- —
9
1 112,921
1,142
14
.01011
.01225
.00996
.01057
10
115,609
1,189
13
.01028
.01093
.01011
.01053
11
120,957
1,236
14
.01022
.01132
.01021
.01034
12
131,726
1,350
16
.01025
.01185
.01024
.00946
13
134,343
1,369
8
.01019
.00584
.01031
.00870
14
136,576
1,406
11
.01029
.00782
—
—
15
133,444
1,417
10
.01062
.00706
—
Totals
1,105,372
11,216
105
.01015
♦
.00936
—
—
• 1908-1911 gave 0.010311 and 0.00931. See p. 314 herein.
The 5-year averages shew the regularity of the ratios, and justify the
combination of the results of a series of years for the purpose of examining
the characteristics of multiple diisogeny.
46. Twin and triplet treauency according to ages. — ^The data for
determining the ratios mg and m^ according to formulae (589) and (590)
are given immediately by the records of births, and — unlike the maternity
ratios mi, formula (588) — are independent of the Census results. The
COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY.
365
numbers of unspecified cases are negligibly small. There is some un-
certainty in the numbers for ages 18 to 21 inclusive, owing to misstate-
ments as to age. The following table, based upon 1,035,439 mothers,
10,533 twins, and 104 triplets, gives the available results for 9 years.
TABLE CXXXn-
-Shewing the Freciuency of Twins and Triplets in Cases of Maternity,* according to the Age of the
Mother and of the Father. Australia, 1907-1915.
Aget
Ages of Mothers.
Groups.
Mothers
Under
19
. 19
20
21-24
25-29
30-34
35-39
40-44
45-54
All Ages
of
Mothers.
25-39
Triplets -^
Twins
25-39
Triplets -i-
Twins
All Ages.
Fathers
Under
19
0i3
527
.0057
0;8
380
.0211
0;11
907
.0121
19-20
0;11
2,413
.0046
0;9
1,923
.0046
0;11
1,584
.0069
0;4
2,154
.0002
0;2
335
.0059
0;37
8,409
.00440
0;2
335
.0059
21-24
0; 33
7,985
.0041
0; 55
9,439
.0058
1; 83
12,159
.0068
5; 366
54,749
.0067
2; 137
15,413
.0089
1; 21
1,630
.0129
275
.0218
9; 701
101,650
.00690
3; 164
17,318
.0095
.0183
.0128
25-29
0;13
4,664
.0028
0;26
6,446
.0040
0;55
10,271
.0054
2; 603
94,452
.00636
7 ; 1,075
117,756
.00913
6; 269
21,278
.0126
2; 34
2,723
.0124
0;5
287
.0174
17 ; 2,080
257,877
.00807
15 : 1378
151,757
.00908
.01088
.0082
30-34
0;3
1,204
.0025
0;7
1,773
.0039
0;18
3,355
.0054
3; 251
39,785
.00631
7:928
103,618
.00895
5 ; 1,097
84,976
.01291
6; 219
13,976
.0157
0;19
1,263
.0150
21 ; 2,542
249,950
.01017
18 ; 2,244
202,570
.01108
.00525
.0083
35-39
0;0
407
0;3
594
.0051
0;5
1,039
.0048
0;73
12,564
.00581
3; 446
46,867
.00951
7; 905
72,990
.01240
10 ; 838
54,557
.01536
3; 67
5,707
.0117
0;2
156
.0128
23 ; 2,339
194,881
.01200
20 ; 2,189
174,414
.01255
.00548
.0098
20-44
0; 4
643
.0062
0;26
4,093
.0064
1;143
15,664
.00913
4; 429
35,051
.01224
5; 752
48,109
.01563
5; 282
22,723
.01241
0;3
590
.0051
15 ; 1,639
126,873
.01292
10 ; 1,324
98,824
.01339
.00831
.0092
45-49
0;3
218
.0138
0;9
1,379
.0065
2; 58
5,516
.0105
0;152
11,936
.01274
8; 357
23,485
.01520
2; 229
19,418
.0118
0; 18
2,703
.0067
12 ; 826
64,665
.01278
10 ; 567
40.937
.01385
.0176
* .0145
50-54
0;2
509
.0039
1; 14
1,593
.0087
1;56
3,639
.0154
3; 94
6,596
.0143
0;72
7,775
.0093
0;14
1,687
.0083
5; 252
21^799
.01156
5; 164
11,828
.01386
.0304
.0198
55-59
;;
0;4
619
.0064
0;9
946
.0095
1;22
1,837
.0120
1;29
1,828
.0158
0;5
677
.0074
2; 69
5,907
.01168
1; 35
3.402
.01028
.0285
.0289
60-64
0;2
216
.0092
0;6
342
.0175
0; 5
488
.0102
0;8
540
.0148
0;1
116
.0086
0;22
1,702
.01292
0;13
1,046
.01242
.0000
.0000
65-89
0;9
292
.0308
0;6
536
,0112
0;15
828
.01811
0;-15
828
.01811
.0000
.0000
All ages
o£
Fathers
0;63
17,200
.00366
0;108
20,555
.00525
1;179
23,269
.00612
10 ; 1,334
20J,685
.00636
23 ; 2,803
307,597
.00913
24 ; 2,953
233,080
.01267
35 ; 2,333
152,582
.01529
11 ; 711
59,541
.01194
0;43
5,929
.00725
104; 10533
1,035.439
.01017
AU
Triplete-H
= .00988
.0099
25-39
0;16
6,275
.0025
0;36
8,813
.0041
0;78
14,665
.00532
5; 927
146,801
.00631
17 ; 2,443
268,241
.00913
18 ; 2,271
179,224
.01267
18 ; 1,091
71,256
.01531
3; 91
7,257
.01254
0:.2
166
.0128
Triplets
-i- Twins
25-39
.0000
.0000
.0000
.0054
.0063
.0079
.0164
.0323
.0000
Triplets
-i-Twins
All ages
.0000
.0000
.0056
.0075
.0082
.0081
.0150
.0155
.0000
.0099
^
•■
* The table shews for various age-groups of mothers and fathers the numbers of oases of maternity, and of twins and triplets occurring
during a period of 9 years. The first number is the number of triplets ; the second — divided from the first by a semi-colon — ^is the
numter of twins ; the numbers beneath, viz., on the second lines are the " cases of maternity" ; the numbers on the third lines are
the ratio of the ocoiirrenqes of twins to the eases of maternity. These ratios ^le calculatesj by tl)e fonnula (589). t The ages a^f
"ages last birthday."
360
APPENDIX A.
An examination of the individual columns in the table for any given
age-group of wives discloses the fact that there are no systematic differ-
ences for various ages of the husband. This is confirmed by the com-
bination of the results for considerable groups.^ The age of the husband,
though it has an unmistakable influence on the maternity ratio, has no
influence whatever on the twin-ratio. It is equally clear that the age of the
wife is correlated with the frequency of twins.
The graph of the results indicates that the initial part of the curve
(i.e., wives' ages up to 20 inclusive) does not conform to the general
curve (owing perhaps to misstatements of age). The curve has a maxi-
mum at about 37J years of age (i.e., age 37 last birthday), and is nearly
a straight Une almost up to the maximum .value.
The following table gives the probabilities for the exact ages, not
" age last birthday."
TABLE CXXXnL — Shewing, accoiding to Age of the Mother, the Relative Frequency
with which at least a Second Child is Boin.* Australia, 1907-1915. Twin-ratios.
Age of
Twin
Age of
Twin
Age of
Twin 1
Age of
Twin
Mother
Ratio.
Mother.
Ratio.
Mother.
Ratio.
Mother.
Ratio.
11
.00100
22
.00605
; 33
.01299 1
44
.01070
12
.00137
, 23t
.00659
34
.01370 ,
45
.00997
13
.00180
24
.00714
35
.01440
46
.00937
14
.00217
25
.00770
36
.01499
47
.00880
15
.00260
26
.00827
i 37
.01526
48
.00823
16
.00305
27
.00885
38
.01526
49
.00772
17
.00352
28
.00944
! 39t
.01502
50
.00725
18
.00400
29
.01007
40
.01470
51
.00680
19
.00449t
30
.01075
i 41
.01380
52
.00636
20
.00500
! 31
.01146
i 42
.01260
53
.00593
21
.00552
32
.01221
1 43
.01155
54
.00551
22
.00605
33
.01299
1 44
1
.01070
55
.00510
* The table shews the ratio of cases of birth of tvro or more children to cases of maternity, the
age being exact (i.e., not age last birthday).
t The ratios have been ascext^ned with great precision for all the ages from 23 to 39. Later they
are less accurate, bnt the number of cases is relatively small.
t The ratios are somewhat uncertain owing to misstatements of age.
1 For example the following results were obtained : —
Agbs of Wives.
AOES OP HTTSBAiros.
Under
19.
19.
20.
21-24.
25-29.
30-34.
35-39.
40-44.
45-54.
Under 30
.0039
.0054
.0062
.0064
.0091
.0127
.0133
.0174
25 to 39
.0025
.0041
.0053
.0063
.0091
.0127
.0153
.0124
.0128
All ages ol husbands . .
.0037
.0053
.0061
.0064
.0091
.0127
.0153
.0119
.0073
These clearly establish the fact that the age of the husband has no influence whatever.
CX)MPLBX ELEMENTS OF FERTILITY AND FECUNDITY.
367
47. Apparent increase of frequency of twins with age of husbands. —
If ages greater than 40 be left out of consideration, and the material for
ages (wives and husbands respectively) of 25 to 39 years of age alone be
embraced, the values of the twin-ratios according to the ages of husbands
and wives are given approximately by the following expressions, viz. : —
(592) mz = 0.0034 + 0.000228 x; m'a = —0.0076 + 0.00060 y.
These give the following results for husbands and wives respectively : —
Ages of Husbands
Data
Formula
Ages of Wives
Data
Formula . .
l»-20
.0059
.0080
Under 19
.0025
.0038
21-24
.0095
.0086
19
.0041
.0041
25-29
.0091
.0097
20
.0053
.0047
30-34
.0111
.0108
21-24
.0063
.0062
35-39
.0125
.0120
25-29
.0091
.0089
40-44
.0134
.0130
30-34
.0127
.0119
45-49
.0139
.0142
85-39.
.0163
.0149
50-54
.0139
.0154
40-44
.0125
.0179*
55-59
.0103
.0165
45-54
.0128
.0224*
60-64
.0124
.0176
65-89
.0181
.0188
* The straight line does not hold good for these ages.
This increase with the age of the husband is not, however, due to
any influence the husband may be supposed to have upon multiple-
births, but wholly to the fact of association in pairs according to age.
The smaller coefficient 0.000228 (as compared with 0.000600) arises from
the greater " spread" of the ages of the husbands. ^
Although the attribution of the increased frequency with age to
the husband is physiologically meaningless, nevertheless for rough
estimates the method is valid, and so long as it is remembered that the
effect is not due to increasing age of the husbands, there is no objection
to this' method of estimation.
48. Triplet diisogeny. — The numbers of triplets shewn on Table
CXXXII. are quite insufficient to determine with any exactitude the
digenous relations of triplets. The age-.groups are too small. But if 30
be made a dividing age we get the following result : —
Wives.
Wives.
Husbands.
Ages under 30.
Ages 30 & over.
Ages under 30.
Ages 30 & over
Ages under 30
17 ; 2497
342650.
9 ; 335
25236.
.0068
.000050
.0269
.00036
Ages 30 and over
17 ; 2011
241775.
61 ; 5603
424284.
.0084
.000070
.0109
.000144
Numbers.
Ratios.
I If the ages were identical of husband and wife throughout, the maximum
effect would be on a line making an angle of 45 degrees with either axis : conse-
quently the ratio of multiple births if attributed to either sex would yield the same
result.
368
APPENDIX A.
As it has no influence on the occurrence of twins, it may be assumed
as extremely unlikely that the age of the husband has any influence on
the occurrence of triplets. This is confirmed by the above partitioning
of the results, which shews opposite apparent influence. The results
given in the final column of Table CXXXII. may therefore be taken as
exhibiting the influence of the age of the wife. This influence can be
expressed
(593).
.ms = —0.0044 + 0.00047 y.
y being the exact age of the wife.
The results as ascertained from the data and as given by the formula
are :-
Age
20
21-24
25-29
30-34
35-39
40-44
45-49
Data . .
Formula
.0056
.0050
.0075
.0064
.0082
.0085
.0081
.0109
.0150
.0132
.0155
.0155
.0000
?
Having regard to the number of available cases it is certain that the fre-
quency of triplets increases with the age of the wife. The rate of increase
0.00047 agrees well with that of the rate of increase 0.00060 in the case of
twins. On plotting the results according to the age of the husband it
was found that the points on the graph constituted a curve, not a straight
fine.
49. Frequency of twins according to age and according to order of
confinement. — The relation between the frequency of the birth of twins
according to age and according to order of confinement can be roughly
seen from the results given hereunder. According to the order of con-
finement the frequency is very closely given by the equation : —
(594) m'z = 0.0082 + 0.00114 n — 0.0000185 n^-^,
the calculated and observed results being respectively : —
Previous confinements
Formula
Data . .
Corresponding age
1
2
3
4
5
6
7
8
9
.0083
.0094
.0105
.0115
.0123
.0129
.0135
.01?8
.0140
.0141
.0082
.0096
.0107
XI] 17
.0124
.0130
.0134
.0136
.0138
.0139
26.78
28.26
29.93
31.32
82.24
33.01
33.58
33.86
34.14
34.28
10
.0139
.0140
34.42
And if the age corresponding to these values be inserted from Table
CXXXIII., the values on the final line are obtained.
50. — ^Unexplored elements of fecundity. — ^To distinguish between the
effect of previous births and age upon the frequency of maternity, of twins,
etc., more comprehensive data are required than at present exist fop
COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY. 369
Australia. The effect is one which, so far as the maternity-ratio is
concerned, reflects social tradition in a larger measure than the physio-
logical law ; the latter is modified but not obUterated. In the case of
twins, triplets, etc., the physiological laws doubtless alone operate.
The records necessary to ascertain the characteristics of digenous
masculinity at birth exist for only two years, and disclose the fact that the
variations according to age are too large to admit of satisfactory analysis,
unless say 10 years' material is available.
Assertions from time to time have been made to the effect that the
characteristics of first-born children are often sharply differentiated from
those of later children. If in the record of cases of mortality the ages of
father and mother were also given, and the order of the birth of the
deceased, the data for the consideration of this question as regards length
of life according to age and to place in order of birth would be available.
This question, however, belongs more properly to the subject of
mortality, and will not be further considered in this monograph.
XV.— MORTAUTY.
1. General. — Human mortality may be considered statistically
under two aspects, viz.,
(i.) A general one ; that is, the aspect which has regard to the
aggregate mortality from all causes of death ; and
(ii.) A particular one ; that is, the aspect which takes account o"f
mortality from particular causes or by particular modes of
death.
Both will be referred to.
Deaths from particular causes or by particular modes are, in general,
functions both of age and time, i.e., an individual rate of mortality /x',
viz., the ratio of the number of deaths D ' from a particular cause to that
part of the population P' subject to the risk of such death, is : —
(595) ,x' =D' / P' =f{x,t)
X denoting age, and t time. It also varies with sex. In estimating the
general rate of mortality it is convenient, although in many respects
unsatisfactory, virtually to regard all persons in the population as equally
subject to the risk of death from each cause ^ : hence the general rate of
mortality may be regarded — subject to some liraitations — as made up of
the sum of the supposititious rates 8 ', 8", etc., from each cause ; that
is to say, the general rate of mortality is the ratio of total deaths during a
unit of time to the total mean population, or algebraically : —
D' D" D
(596) S = 8' + 8" + etc. = — + _ + etc. = — .
D = D' -\- V + etc., denotes the total deaths, and P the mean popula-
tion during the unit of time in question. This rate is known as the crude
death-rate, and is obviously inconsistent with (595). If the age-distri-
butions of all populations were substantially identical, this method of
evaluating the rate of mortality could be regarded, for many purposes, as
fairly satisfactory.^ Inasmuch, however, as each particular rate, )u, ', etc.,
is a special function of age, the general rate 8 is obviously also dependent
upon the age and sex distribution. The distinction may thus be drawn
between class mortality and general mortality, " class" denoting any
section of the population, defined in any appropriate way.
1 For example : — Females only are liable to death from say misadventures in
parturition, and that only between certain age-limits ; and children are alone liable
to death from diseases associated with dentition, etc., etc.
' Throughout statistics conceptual precision has often to be sacrificed in
order to express results simply, but the simplicity thus attained is usually more or
less misleading.
MORTALITY. 371
Comparisons, the purpose of ■which is to measure, in a crude way,
the virulence of the death-forces as between one population and another,
may therefore be made on the basis of a common age and sex distribution, that
is, the rates of mortality for each age or age-group, actually experienced
by each population, may be applied to a " standard," or preferably to a
" normal," population, the resulting totals giving the comparison
required. That is to say, if the fact of variation with time {t) be
ignored (though this variation will probably not be even approximately
identical for any two populations), the quantity to be ascertained will
be that indicated in (699) hereunder.
Let the proportion of the normal population between the ages x and
x-\- dx be : —
■ri,j.dx = 1
and let the instantaneous rate of mortality (the so-called "force of
mortality") at age x be : —
(598) ,/x^ = J. {^), and ^,^^ = J^ {x)
for populations 1 and 2 respectively ; then adopting the same function
(597) for both populations and applying (598) to each, we have : —
(599) /^'o =j'^(f^.-Vx)dx = j"{f^(x) .f^(x)} dx;
with antecedent suffixes denoting whether the result applies to population
1 or 2.
These quantities may be called the mortality-coefficients of the re-
spective populations, and generally they will differ somewhat from the
" crude death-rates." If the age-distribution of a population happens
to be sensibly identical with the "standard" or "normal" distribution, the
mortality-coefficient would of course be sensibly the same as the crude
death-rate" ; and it might otherwise also agree with it, but only accident-
ally. For arithmetical convenience it is usual to compute an index-of-
mortality, by attributing to " standard" or " normal" groups the death-
rates actually experienbed in the corresponding groups of the population
under review.
The preceding rectification of the crude death-rate for the purpose of
comparisons , is but one of the possible methods . Its significance depends
virtually upon a common distribution of causes of death, these differing
only in frequency of operation. If two countries had the same age-
distribution, but one was characterised by violence of the diseases which
caused mortality in the earlier, and the other by those which caused
piortality in^the l(iter years of life, the results would differ even for the
372 APPENDIX A.
same differences of rate in each disease, inasmuch as with the same rates
the diseases characteristic of the earlier years of life levy a larger toll
than those -characteristic of later years. In short, the influence or
" weight" of a cause of death varies, according to the relation of its
incidence with age. Crude death-rates and the indexes of mortality are
therefore both of restricted appUcation, and need to be interpreted with
full regard to their inherent limitations.
The frequency according to age of the occurrence of disease is ver>'
diverse, consequently in the aggregate of mortality from all causes the
pecuUar incidence of each is to a great extent masked ; and as regards the
secular trend of mortality the intervention of epidemics may produce great
irregularities.
Many diseases have a well-defined annual period, while others have
not ; these periods, however, are not identical in phase. The aggregate
of the deaths from all causes, therefore, gives a less definite indication of
an annual period. Inasmuch as diversity of phase and of ampHtude do
not wholly obliterate the periodicity, the general death-rate, viz., 8 =
D / P, i.e., the deaths divided by the number of the population, is as
follows : —
(600) . . 8 =D/P =D I (t) / 1 + ao -f 2„^i a„ sinn (^ + aj } / j PJ^{t) \
in which 6 v:> ei fraction of a unit of time (say of a year), w = 1, 2, 3, etc.,
and both D^ and Pq are means over a unit of time, as at a particular
epoch. Thus the graph of a death-rate, extending over several units of
time (years), ir; made up of a non-periodic curve — representing the general
trend — ^upon which is superimposed a periodic curve repeating itself
during each unit upon a scale varying with the death-rate itself. ^
2. Secular changes in crude death-rates. — ^The general lowering of
the general crude death-rate in the western world has been remarkable,
and is best exhibited by deducing the general trend of the rates for each
country. The death-rates for Austraha are shewn in Table CXXXIV.,
from 1881 to 1915, for males, females, and persons ; see columns (ii.) to
(iv.). In order to partially eliminate the irregularities of results for single
years, quinquennial means were formed, see columns (viii.) to (x.), and
the smoothing of these for "persons" gives the values in column (xiv.),
the maximum value 0.01570 being that for the year 1884 and the minimum
0.01066 that for the year 1911. This fall to about two-thirds of its earlier
value in 27 years is remarkable, and is accounted for not only by a stiU
greater decrease in infantile mortaUty, but also in general mortaUty up
to 60 or 65 years of age. It is worthy of note that the year 1895 was
characterised by a halt in the decrease exhibited by the general trend of
the death-rate.
1 So long, of course, as the character of the periodicity is maintained.
MORTALITY.
373
The rates of infantile mortality are given in columns (v.) to (vii.), the
quinquennial means in columns (xi.) to (xiii.), and the smoothed result or
general trend in column (xv.). Here again the fall has not been continu-
ous, see values for 1894-5. The character of the lowering of the rates
doea not therefore fall under any law susceptible of simple mathematical
expression.
TABLE GXXXIV.— Shewing Secular Changes of the Death-rates, and of the Infantile
MortaUty-rates in Australia, from 1881 to 1915.
Year.
Death Bates
X 100,000.
Infantile Mortality
Kates X 10,000.
(Quinquennial Mean
Death Rates
X 100,000.
Quinquennial Mean
Kates of Infantile
Mortality x 10,000.
Cieneral Trend
of Death Kates
(Smoothed)
X 100,000.
1 Trend of
! Morflity
lothed) 1
.0,000. 1
General
Infntile
(Smc
X 1
Males.
re-
males.
Per-
sons.
Males
Fe-
males.
Per-
sons.
Males.
Fe-
males
Per-
sons.
Males.
Fe-
males.
Per-
sons.
(i.)
(ii.)
(iii.)
(iv.)
(V.)
(vi.)
(vii.)
(viii.)
(ix.)
(X.)
(xi.)
(xii.)
(xiii.)
(xiv.)
,(xv.)
1881
1,589
1,328
1,469
1,232
1,095
1,165
1,636
1,348
1,504
1,372
1,203
1,293
1,528
1,293
1882
1,746
1,419
1,596
1,446
1,265
1,357
1,675
1,380
1,540
1,363
1,195
1,284
1,552
1,284
1883
1,654
1,381
1,529
1,302
1,138
1,222
1,708
1,404
1,569
1,353
1,186
1,274
1,569
1,272
1884
1,804
1,460
1,646
1,348
1,168
1,260
1,722
1,417
1,582
1,342
1,176
1,263
1,570
1,261
1885
1,747
1,434
1,604
1,360
1,221
1,292
1,689
1,397
1,555
1,330
1,166
1,251
1,562.
1,251
1886
1,659
1,392
1,537
1,348
1,189
1,271
1,676
1,381
1,541
1,316
1,155
1,238
1,546
1,238
1887
1,583
1,317
1,461
1,235
1,091
1,164
1,651
1,386
1,520
1,300
1,140
1,222
1,526
1,221
1888
1,589
1,300
1,456
1,251
1,072
1,164
1,611
1,336
1,485
1,281
1,122
1,203
1,500
1,203
1889
1,672
1,385
1,540
1,400
1,234
1,319
1,603
1,323
1,474
1,260
1,101
1,182
1,474
1,185
1890
1,554
1,287
1,431
1,152
1,009
1,082
1,570
1,289
1,440
1,237
1,078
1,159
1,444
1,163
1891
1,618
1,328
1,484
1,232
1,074
1,155
1,553
1,274
1,424
1,212
1,053
1,135
1,410
1,138
1892
1,419
1,144
1,291
1,142
970
1,058
1,496
1,223
1,369
1,188
1,024
1,108
1,368
1,112
1893
1,502
1,227
1,374
1,240
1,072
1,149
1,459
1,186
1,332
1,167
998
1,086
1,326
1,086
1894
1,386
1,128
1,266
1,107
952
1,031
1,419
1,147
1,292
1,158
993
1,076
1,292
1,074
1895
1,372
1,102
1,245
1,099
921
1,012
1,403
1,131
1,276
1,161
997
1,079
1,280
1,078
1896
1,414
1,135
1,283
1,202
1,045
1,126
1,411
1,139
1,285
1,177
1,012
1,096
1,282
1,096
1897
1,342
1,065
1,212
1,126
967
1,048
1,416
1,145
1,289
1,196
1,031
1,115
1,284
1,114
1898
1,540
1,267
1,412
1,364
1,175
1,272
1,404
1,130
1,275
1,204
1,038
1,125
1,283
1,125
1899
1,411
1,156
1,291'
1,246
1,085
1,167
1,395
1,157
1,263
1,198
1,034
1,117
1,273
1,117
1900
1,314
1,026
1,178!
1,086
915
1,002
1,403
1,123
1,270
1,181
1,019
1,097
1,255
1,097
1901
1,366
-1,064
1,222,
1,122
947
1,037
1,362
1,086
1,231
1,145
993
1,062
1,231
1,062
1902
1,383
1,102
1,249
1,142
997
1,071
1,322
1,052
1,194
1,103
946
1,019
1,201
1,020
1903
1,837
1,080
1,215
1,183
1,025
1,105
1,302
1,037
1,176
1,053
892
970
1,172
971
1904
1,212
988
1,105
891
756
825
1,269
1,019
1,150
1005
852
930
1,144
920
1905
1,214
950
1,088
906
724
818
1,235
994
1,120
952
800
878
1,118
872
1906
1,201
973
l,092i
901
760
833
1,212
974
1,098
887
734
813
1,098
827
1907
1,211
977
1,099'
884
734
811
1,200
957
1,084
867
711
791
1,085
792
1908
1,224
981
1,107,
855
697
778
1,188
952
1,075
849
702
777
1,076
770
1909
1,151
906
1,033
787
642
716
1,184
946
1,070
820
671
748
1,070
751
1910
1,154
924
1,043
817
675
748
1,192
947
1,074
804
650
729
1,087
733
1911
1,182
940
1,086
759
■ 607
685
1,186
941
1,069
790
641
718
1,066
718
1912
1,251
984
1,123
801
630
717
1,189
946
1,072
791
640
717
1,070
708
1913
1,193
953
1,078
788
653
722
1,200
944
.1,079
776
626
703
1,079
702
1914
1,167
927
1,051
791
635
715
..
• .
..
1915
1,208
916
1,066
743
605
675
■•
••
••
••
The results in the above Table are shewn in Curves A and B of Fig. 98,
the dots shewing the quinquennial means and the continuous line the
general trend. The correlation between the two curves is fairly well
indicated, because, although the ratio of the annual number of cases of
deaths of children under 1 year of age, to the annual number of deaths of
all ages is somewhat variable, there is some degree of general correspond-
ence when a mean is taken over a number of years. See Fig. 98, p. 377.
374
APPENDIX A.
The following example sufficiently illustrates the variable character
of the ratio of infantile to total deaths, shewn in lines (a) and (6) here-
under : —
TABLE CXXXIV.A.— *Ratio x 10,000, of Infantile to Total Deaths,
accoiding to Sex. Australia.
Year .. . .j 1902.| 1903.i 1904
Males (a) ..'2,155:2,206 1,890
Females (6) . . 2,4771 2,469J 2,039
Females (c) . . 2,410| 2,295; 2,202
Males (d)
(d) H- (c) = (e)
2,138
.887
2,078 2,024
.905 .919
1905.
1906.
1907.
1908.
1909.
1910.
1911.
1912.
1913.
1914.
1,930 l,97l{ 1,925
1,834
1,804
1,880
1,720
1,798
1,832
1,871
2,030
2,099 2,041
1,915
1,907
1,966
1,787
1,874
1,973
1,955
2,123
2,056 2,003
1,960
1,930
1,905
1,886
1,873
1,864
1,858
1,976
1,923! 1,893
1,863
1,839
1,819
1,807
1,798
1,792
1,788
.931
.935
.945
.951
.953
.955
.958
.960
.961
.962
1915.
1,672
1,798
1,855
1,786
.963
♦ The figures on lines (a) and (b) are the ratios of the annual nunjhers of male and of female
infantile deaths to the annual number of total male and of total female deaths respectively. The
figures on lines (c) and (d) are the smoothed ratios for females and males respectively. The figures
on line (e) are the ratios of male to the female ratios as determined from the smoothed ratios (c)
and (d).
Although the ratio oscillates between somewhat wide limits, the
female ratio is invariably higher than the male-ratio : the general death-
rate of females, however, is lower than that for males.
These results indicate that the proportion of infantile deaths to total
deaths for both sexes is rapidly decreasing ; the decrease for females being
more rapid than for males. This is best seen by forming quinquennial
means from which the general trend can be readily ascertained. The
magnitude and general trend of the ratios of infantile to total mortality
in the case of females and also in the case of males, are shewn respectively
by curves M and N in Fig. 98, p. 377.
3. Secular changes in mortality according to age. — The death-rate
for any age-group is the ratio of the number of deaths per unit of time
(per annum) therein to the average number of persons in the group
during that unit, i.e., to the number at risk. ^^ This ratio is markedly
different for the two sexes. The following table, viz., CXXXV., based
upon the censal results and intercensal experience since 1881 ^ shews that
for nearly all ages a remarkable diminution in the death rates has taken
place. That this must be so is obvious from the results given in
Table CXXXIV.
In a later Table, viz., CXXXVI., the average, also according to age,
of the ratios between the death-rates of the sexes is given. These average
ratios are the ratios of the sum of the four ratios given in each age-group
for females to those given for males, and may be referred to the epoch
1900.0 for all comparisons as to any possible change with time. ^
1 Actuarially, the ratio of the number of deaths experienced by persons be-
tween given limits of age to the total number of units of time (years of life) lived within
those age -limits by the population considered.
2 The results for 1911 are reaUy based upon the deaths occurring during the
nineyears 1907to 1915inclusive. The actual populations for these years are assumed
to be distributed according to age as at the Census of the middle year, viz., 1911
which must be substantially correct.
MORTALITY.
375
It is obvious from the table that estimations of the frequency of death
based upon tables compiled on the experience of past years are erroneous,
if applied at the present time. * We shall investigate hereinafter the law
of change.
TABLE CXXXV. — Shewing the Mean Death-rates in Age-groups deduced for Various
Epochs, and niusirating their Secular Changes. Australia, 1881 to 1915.
Average ratio ol
MALES. RATE X 100,000.
Females. Bate x 100,000.
Female to Male
Age
Death Rate.
or
Age-
1881-
.1891-
1901-
1907-
1881-
1891-
1901-
1907-
Data.
group
1891, say
1901, say
1911, say
1915, say
1891, say
1901, say
1911. say
1906.0
1915, say
Age.
Sm'thed
1886.0
1896.0
1906.0
1911.0
1886.0
1896.0
1911.0
result.
0-0*
25,439
23,473
19,341
16,360
21,340
19,333
15,562
12,867
0.0
t.l866
14,366
12,738
10,112
8,540
12,414
10,786
8,349
6,862
0.5
.8395
1
3,576
2,685
1,804
1,559
3,427
2,519
1,684
1,389
1.5
.9371
2
1,379
982
677
642
1,336
963
631
575
2.5
.9524
3
891
628
441
409
834
617
412
382
3.5
.9477
4
692
497
350
301
648
488
325
300
4.5
.9571
0-4t
4,549
3,777
2,801
2,455
4,035
3,276
2,365
2,023
2.5
.8614
S-9
384
310
222
222
355
293
201
202
7.5
.9236
10-14
253
219
192
173
235
192
171
163
12.5
.8973
15-19
528
366
300
256
406
315
272
221
17.5
.8372
20-24
793
541
410
364
597
447
370
341
22.5
.8326
25-29
870
651
473
431
781
586
468
432
27.5
.9349
30-34
890
737
552
508
813
703
539
475
32.5
.9416
35-39
1,007
902
714
666
976
847
674
586
37.5
.9374
40-44
1,236
1,029
918
841
1,090
836
746
641
42.5
.8233
45-49t
1,591
1,311
1,222
1,120
1,262
1,000
890
794
47.5
.7525
50-54
2,085
1,737
1,522
1,511
1,568
1,273
1,044
1,050
52.5
.7199
55-59
2,803
2,454
2,091
2,153
2,037
1,793
1,497
1,473
57.5
.7157
60-64
3,717
3,624
3,095
3,174
2,694
2,677
2,293
2,177
62.5
.7231
65-69
5,528
5,207
4,708
4,678
4,423
3,753
3,619
3,471
67.5
.7587
70-74
7,488
7,104
7,584
6,972
6,218
5,704
6,074
5,523
72.5
.8069
75-79
11,778
11,686
11,845
10,900
10,076
9,967
9,378
9,162
77.5
.8350
80-84
15,275
16,210
16,450
16,815
14,490
13,984
13,306
14,575
82.5
.8704
85-89
27,169
26,041
27,372
26,783
24,227
21,960
22,836
21,701
87.6
.8427
90-94
24,661
26,917
30,677
30,896
28,455
26,497
29,433
28,960
92.5
1.0026
95-99
45,050
37,500
36,974
39,111
32,207
45,941
41,188
38,319
97.5
.9938
100-4
24,188
39,844
33,724
113,043
18,621
47,312
39,224
107,229
102.5
1.0075
• Nominally at the instant of birth, but not really so. Foi: the first week after birth the
curve is quite distinct from the general death-rate curve after that period. The values given are
deduced from the results for the five age-groups, to 4 inclusive, by formula (197), p.68 herein.
If computed on the basis of A* = A + Ox+ Bo* , see C. H. Wicliens' Journ. Austr. Assoc. Adv.
Sel. XIV., 1913, p. 535. The values for will be .27640, .26330, .22790, .19460 and .22740,
.21470, .17840 and .15090. But true values of ix, are really much greater than these.
t Between these limits (inclusive) the ratio is 0.8593.
t The ratio of death-rates using .27640 -h etc., to .22740 + etc., is 0.8017.
4. The changes in the ratio of female to male mortality according to
time and age. — ^The ratio of female to male mortaUty, according to time,
may be deduceci from the rates given in Table CXXXIV., and those
according to age from the rates given in Table CXXXV.
To avoid the irregularities of individual years the former ratio is
obtained by dividing the results in column (ix.) by those in column (viii.),
Table CXXXIV. The quotients are given in Table CXXXVI., and are
shewn by the dots on curve C, Fig. 98. This is the ratio for general
mortality. For infantile mortality the results in column (xii.) of Table
CXXXIV. are divided by those of column (xi.), and these are shewn by
' Thus the actuarial tables used by insurance societies err on the side of con-
servatism ; they are based upon death-rates which are nove excessively high.
376
APPENDIX A.
small crosses on curve D, Kg. 98. The firm lines denote the general trend
of these results. They give some indication of correlation with the general
and infantile death-rates, see Curves A and B, and the difference between
the two curves is less marked ; see Mg. 98, p. 377.
TABLE CXXXVI.— Shewing Ratios of Female to Male Death-rates, and Female to
Male Rates o£ Infantile Mortality. Based upon Quinquennial Means. Aus-
traUa, 1881-1913.
Year
Ratios of Female to Male Death- |
Ratios of Female to Male Rates of
of
Rates (Curve C)
1
Infantile Mortality (Curve D).
De- i
cade. '
1
1880.
1890.
1900.
1910. ;
1880.
1890.
1900.
1910.
.821
.800
.795
1 .872
.863
.809'
1
.824
.820
.797
.793
.877 ! .869
.867
.811
2
.824
.818
.796
.796
.877 .862
.858
.809
3
.822
.81:!
.796
.788
.876 .855
.847
.807
4
.823
.808
.803
.876 .858
.848
5
.827
.806
.805
.877 .859
.840
6
.824
.807
.804
.878 .860
.828
7
.840
.809
.797
.877
.862
.820
8 1
.829
.805
.801
. . !
.876
.862
.8.29
9
.825
.829
.799
.874 .863
.818
*2.0
.7925
.8089
*2.5 '
.8160
.7986
.8766
.8630
.8565
*3.0
.8232
;
*7.5 ;
.8290
.8112
.8013
1
.8762
.8611
.8270
* These are means of five quinquennial means, except in two instances where they are means
of four quinquennial means.
That the ratio of female to male mortahty varies with time, having
changed from 0.824 in 1881 to 0.788 in 1913, shews that life-tables for
males and females, based on experience dating many years back, can no
longer represent the facts with sufficient exactitude.
The curve, shewing the ratio of female to male mortaUty according to
age, may be deduced from Table CXXXV., and in view of the overlap or
the partial overlap of the 1907-15 results on those of 1901-11, the epoch
to which the ratio may be referred is 1900. ^ These ratios are based upon
the sums of the four ratios for each sex, given in the table. The result is
shewn in Kg. 98, curve K. There are two maxima and two minima in
the curve, at the ages indicated in the table ; see p. 377.
The dotted curve L, from which the curve K departs during the
reproductive period of life, is symmetrical about an axis passing through
the age 47. It is not unlikely that this departure from the curve L is
due to the vicissitudes of reproduction ; see the reference hereinafter to
the gestate force ofmartality.
^ Though not strictly exact, this assumption is sensibly correct.
MORTALITY.
377
IVIortality Cuives and theii Relations. Australia.
Curves A, B, C, D.
Year 1880 3 4 6 8 1890 S 4 6 8 1900 2 4 6 8 1910 S 4 6
Ages 10
Cuives H, I, and J.
90
Curves H, I, J.
Curve A shews the trend of the quinquennial means of the annual death-rates for " persons" from
1880 to 1913 for the Commonwealth of Australia : the dots shew the quinquennial means themselves:
see Table CXXXIV., p. 373.
Curve B similarly shews the trend of the quinquennial means of the infantile mortality rates : the
dots shewing, as before, the quinquennial means ; see Table CXXXIV., p. 373.
Curve C. — The dots shew the ratios of the quinquennial means of the death-rates for females to
the quinquennial means of the death-rates for males, and the continuous line shews the general trend
of these results ; see Columns viii. and ix. of Table CXXXIV., p. 373.
Curve D. — The minute crosses shew the ratios of the quinquennial means of the rates of female
infantile mortality to the quinquennial means of the rates of male infantile mortality^ and the continu-
ous line shews the general trend of these results ; see columns xi. and xii. of Table CXXXIV., p. 373.
Curves E. — The firm lines are the graphs for males and the broken line the graphs for females,
of the results given in the vertical columns of the lower part of Table CXXXVIII., p. 379.
Curves E. shew the changes in the ratios of decrease of mortality for ages to i, the firm line
indicating the results for males and the broken line those for females ; see pp. 378-380.
Curve a shews the mean of the results for ages to i, so reduced that the mean agrees with curve
J ; see pp. 379-380.
Curves H and I are drawn through the terminals of ordinates representing the means of the factors
of decrease and increase. They shew the effect of age ; see pp. 379-380.
Curve J may be regarded as the corresponding line for " persons." The scale needs modification.
The line denoting unity may be taken at 0.9547: thus 0.9 ' and 1 '.0 are the correct places for 0.9 and
1.0 in relation to the curve ; see p. 380.
Curve K. — The ratios of female to male mortality according to age, are shewn by curve K, the
data being indicated by the dots, and the smoothed result by the continuous ciurve. The smoothed
results are given in Table CXXXVII. ; see p. 377.
Curve L. — This curve is symmetrical about age 47, and is coincident with curve K from age 62
years onwards ; for its significance see p. 376.
Curve M. — The broken lines joining the points shew the ratio of female infantile to total female
deaths for successive years. The dots shew the quinquennial means of these, and the firm line shews
their general trend ; see p. 374.
Curve N. — Similarly the fine zigzag lines are the lines joining the points defining the ratios of
male infantile deaths to total male deaths for successive years. The dots shew the quinquennial
means of these, and the firm line their general trend. The ratios of the ordinates to curve K to the
ordinates to curve M, are given in line (c) in Table CXXXIV. A ; see p. 374.
378
APPENDIX A.
TABLE CXXXVn.— Shewing for the Period of 1881 to 1915 the Average Ratio of
Female to Male Moitality, accoiding to Age. Australia.
DATA.
SMOOTHED RESULT.
Age-
Ratio
X Age-
Batio j^gg. Ratio
Atesaqe Ratios of Fbmales to Male Sbath-bates.
1000 ;group.
!
1000^ ^""P- j lOOOil Age.
Ratio
xlOOO
Age.
Ratio
X 1000 Age.
Ratio
xlOOO
0.0
817
15-20
837
65-70 1 759
0.0
817
^.«A
35.0
944
(»)
7 10
70.0
773
%.
0-1
840
20-25
833
70-75
807
5.0
894
127
40.0
882
Of >
75.0
810
f 10*
1-2
937
25-30
935
75-80
835
7.0
903t
• 01
45.0
787
eis
80.0
850
ISO*
2-3
952
30-35
942
80-85
870
10.0
883*
«8a«
47.0
753
•'"t
85.0
892
• f s*
3-4
948
35-40
937
85-90
843
15.0
836*
!>••
50.0
730
ttt
90.0
935
• ss«
4-5
957
40-45
823
90-95
1.003
17.5
834t
tit
55.0
713
Sflfi
95.0
980
• «0*
5-year
means
45-50
753 95-lOC
994
20.0
839
.w.o
710
100.0
10!',6
10S8*
0-5
861
50-55
720 1100-105
1.007
25.0
894
797
60.0
716
1 1t
10!^
1044
104««
5-lC
924
55-60
716 1
30.0
944
7S8
620
724*
7t4*
10-15
897
60-65
723
32.5
950t
7tt
65.0
741*
7< 1*
15-20
837
65-70
759
35.0
944
7 10
70.0
773*
SIO*
* Curve of ratios identical witli curve L in Fig. 98, shewn by broken lines.
t MaYimiim values, t Minimnm values.
(a) Columns (a) are the values to curve L shewn by broken lines in Fig. 98. This curve is
symmetrically situated about an axis, passing through the axis of abscissae at age 47.0, For the
significance of curve L reference should be made to the text.
5. Secular changes in mortality vary with age. — ^For any age or
group of ages, let /xq denote the mortality at a particular date, adopted
as time origin ; and let p denote its rate of change — ^the sign being
negative if it be decreasing — so that
(601).
■n't = /-tpe^' = jtioe*^*^'-
The last form is necessary only if p be Twt constant. It will be found
probably in aU cases that p is a function of time, and it is also a function of
age. The results for small age-groups are of course irregular, so that
it is only in extended age-groups that the laws of the secular changes
according to age and time are rendered obvious and unequivocal. This
can be seen by an analysis of the results given in Table CXXXV.,^ and
it is important to know whether for any given age p is sensibly constant
for any sensible period.
The analysis is effected by forming a series of sums of age-group-
results from Table CXXXV., and calculating the coefficients which,
multiplied into the results of any period will give those of a later period.
1 For example the sum of the rates to 49 gave the following indication :
Year (o)
1886.0
p 1896.0
P 1906.0
Males m
Males .(c)
FemalesW
FemaleB(e)
.12101 X
.12118 X
.10550 X
.10636 X
.8314 = .09843 x
.8050 = .09755 x
.8052 = .08495 x
.7909 = .08412 x
.7928 = .07804
.8050 = .07853
.7882 = .06696
.7909 = .06659
P 1911.0
V(.8219) = .07036
V(.8050) = .07046
V(.7679) = .05868
V(.7909) = .05922
Data
Computed
Data
Computed
The constant ratios .8050 and ,7909 therefore reproduce the results fairly well, for males and females
respectively, though with a decennium as unit for the ratio-value, we find the value of the ratio is
i„/J = 0,8052 — 0,000127t — 0,0001573{'
for females, t being expressed in years reckoned from 1886,0, The results are computed by takine the
square root of the quantities .8314 and .7928 : allowing each the weight 2 and 9016 = V( ^19)
the weight 1. This gives 0,89723, the square of which is .8050. The factors to divide into 12101
.09843, etc., are respectively 1, .8050, .64802 and .58142 ; the division gives .12101 12227 12043 and
.12101 the mean of all being .12118 from which by inverting the procedure the above values for malpa
are deduced ; similar resiilts give ,10636, etc., for females. i"»ics
The values, foimd as shewn, suggest that, for the purpose of obtaining values for successive dates
multiplication by a factor and its powers, or say an annual guingwnnial or decennial coefficient oi
I'ortofion, has advantages over the employment of differences.
MOBTALITY.
379
The quantities in columns ii. to iv., and vii. to ix., of this table, for males
and females respectively, are deduced for the corresponding series of age-
groups shewn ; the ratios are assumed to be true for the centres of the
ranges of ages, an assumption which is sufficiently exact for the purpose
in view.^
TABLE CXXXVin.— Shewing the Changing Ratios for different age-groups as
between different dates. Australia, 1881 to 1915.
MALES.
FEMALES.
AGE GROUPS.
1886
1896
1906
Ratio
1886
1896
1906
Ratio
to
to
to
Means
to
to
to
to
Means
to
1896
1906.
1911.
Total.
1896.
1906.
1911.
Total.
(i.)
("•),
(iii.)
(iv.)
.8459
(V.)
(vi.)
(vu.)
(vlii.)
(ix.)
(X.)
(xi.)
00 . .
Moe'
.90772
.9165
1.0528
.95182
.89722
.8268
.9050
1.0474
..
.9416 «
.83102
.8445
.9019
1.0361
.93212
.87982
.8219
.8891
1.0291
1 ..
.8665"
.8197*
.8642
.8473
.9733
.85732
.8175 2
.8248
.8349
.9654
2 ..
.8439 i"
.81822
.9480
.8544
.9815
.84902
.8094 2
.9112
.8456
.9777
3 ..
.8395 2
.83802
.9274
.8565
.9838
.86012
.81712
.9272
.8563
.9901
4 ..
.8469 2
.83922
.8600
.8464
.9723
.86782
.81612
.9231
.8582
.9923
Means
• .8832
.8523
.8817
.8705
1.0000
.8864
.8395
.8725
.8649
1.0000
0-4
.9112*
.86122
.8765
.8843
.9251
.90112
.84962
.8554
.8714
.9138
[5-14] . .
[14-24]
.91132
.88472
.9541
[.9092
[.8600
.90672
.87582
.9543
[.9038
[.8908]
.8286'
.88482
.8732
.87162
.31792
.8754
5-24
.85642
.88472
.9030
.8770
.9175
.88582
.90182
.9043
.8955
.9392
25-49
.90982
.91532
.9193
.9139
.9561
.8983 2
.91382
.8827
.9014
.9453
50-64
.95302
.92652
1.0194
.9557
.9998
.95482
.91742
.9723
.9433
.9893
65-79
.98382
1.0029'
.9343
.9815
1.0268
.9683.2
.99092
.9520
.9741
1.0216
80-104 . .
1.03662
1.00542
2.5304
1.1229
1.1747
1.14812
.96802
1.4448
1.1354
1.1908
Means \ i
.9418 „
.9327
1.0305
.95588
=1.0000
.9592
.9236
1.0019
.95351
= 1.0000
.94362
.93432
.96352
.92472
The quantities shewn in the table for the 10-year intervals are the square roots of the quantities
originally given. In the totals these are counted twice. In the means 1 denotes the arithmetical
mean, 2 the mean of the squares.
The irregularities of the results are doubtless due in part to actual
irregularities in the death-rates themsdves, and ia part to errors in the
data. They shew unmistakeably that the death-rate up to age say 60
decreases with time, and that, at any rate above age 80, the rate for males
increases with time. The results exhibiting this are illustrated by curves
E, F, G, H and I, Fig. 98, E shewing the six results given in Table
CXXXVIII., for males by firm lines, and the six results for females by
broken lines. The thick line divides the values under unity, viz., those
1 Let a series of quantities, a and A, be respectively the numerators and
denominators vrhich give the ratio for any range of the variable. Then it is assumed
that I
W ( ^2(i)/(-is4)= o„/ A,„ = a,/A,
where a„ and A„ are the values for the middle range, the suffix notation being
— k, — 1, 0, 1 .... k. Obviously in general such an assumption is
invalid ; the true range is that which gives a value of a' / A' equal to a„, / A„.
Later the assumption will also be made that the mean of a series of ratios may also
be ascribed to mid-point of the entire range. The error of such an assumption is
best illustrated by setting out the two results thus : —
(^)..p,„ = (p, +P, +..+ P„)/« = (^^ + ^4-....+ -Jl-)/n
(7)
■P m =
tti -f- fflg -F . ,
^1 -I- ^2 -I-..-I- An
Although in general />',„ is not equal to p,n, if the successive ratios are in arithmetical
progression, they are in agreement, and /j„, = a,,,/ Am above. If these successions
of ratios are sensibly linear in their changes, the error will be negligible.
380 APPENDIX A.
which represent a decrease, from those which represent an increase (on the
upper side). It would appear from this figure that the change is some-
where between 70 and 80, and that the rate of decrease of mortaKty
unmistakably diminishes as age increases.
Curve G shews the mean of the results multiplied by a factor so as to
make the average agree with curve J. Curves F shew the changes in
the ratios of decrease for ages to 4, the firm hne denoting the results for
males and the broken Hne those for females. Curves H and I are drawn
through the ratios, to the total, of the means of the factors of decrease
(or increase) : they illustrate the general correspondence in the male and
female cases of the effect of age, the curve J being the probable general
indication, i.e., for persons. The line denoting unity may be taken as
at 0.9547 : thus the broken Kne at 1.0474 will be reaUy unity in relation
to the curve. ^
It is obvious that advances in hygiene, therapeutics, and social
condition will be marked by diminished mortality. Whether that wiU
extend over all ages or wiU characterise all but the older ages, depends
upon whether the term of life is virtually sharply fixed or not. We shall
consider the matter further in a later section.
6. Fluent life-tables. — For many purposes (much of insurance
business for example) the ordinary tables of rates of mortality (fi^ or m^),
of probabilities of living or dying within a year (p or g ), or of expecta-
tions of fife (gj.), of the population survivors (Ix), at age x, etc., are satis-
factory because they represent not only a considerable body of past
experience, but also are 'on the safe side' for the major part of the uses to
which they are applied (determination of insurance premiums, etc.).
For the accurate prediction of Ufe, however, existing tables are not at all
satisfactory, because, representing past experience, they take no account
of the fact that the rates of mortality for the major part of fife are rapidly
diminishing, that is the probability of lite is increasing for every age, say
up to 60 for both sexes in many and probably in all, civilised countries.
Hence for estimations of the true probabihty of Ufe, for the evaluation of
payments for annuities, etc., existing lite-tables are seriously defective.
To avoid this difficulty it is necessary to constrwct fluent life-tables,
extrapolated for as many years as may seem safe. Such tables are, to the
extent they are extrapolated, prediction tables. In these, past experience
is brought under review in two ways : that is (o) as to the values of the
various functions as they existed at a given moment, and (6) their trend,
or variation with time. As the variation with time is not linear probably
an annual coefficient of variation would best attain the object in view, and
could be readily appUed.^
1 Any resulting "error of scale" may almost be ignored.
2 Thus if this were 0.993 for example, the values of the factors for successive
years would be — to three decimals — 0.993, 0.986, 0.979, 0.972 . . .0.9454, the last
being for the 8th year forward. A linear diminution of .007 would have given .9444.
MORTALITY.
381
It is only by means of fluent life-tables that accurate predictions of
survivors for any given age can be ascertained. In Fig. 99, shewing the
change of death-rates with time, the dots denote the values according to
the data : the system of curved hnes shews what may be regarded as the
general trend of the mortaUty-rate for. the various age-groups. The
results for individual age-groups are irregular, but they unmistakably
point to a diminution of the type e""**, where t denotes the period elapsed,
m however having a different value for each age and sex. This index
factor (m)'has no simple relation to age or to the magnitude of the mortal-
ity-rate itseK, but is probably related to the two combined ; that is, it
is a function of fx, and x. We shall first deal with the method of evaluat-
ing it, and it wiU simphfy the matter if m be not treated as a function of
time as in the final form of (601).
Change of Rates of Mortality aecoiding to Age and Time.
Males
Females
\ 'J^A^
let -
X -
■0*2 -.^
. I
flJ
«4
■040-^4----^
- 5-79 12-^r
^^-^
^.A~-
^f-
•038 . -»lrj
~'-- — ,i!ta|
'■^ ^ds I
^'t.
— —
N 70-74 ^^ \
•034 - ^
^7
^.7,1 1
-T
.-65;:i6r ^.^
V
JH
v —
^^^
02 i-
-:U.H
.J
tt
7^
§5^;^;=
::== ==^^i
J^-^
i.=
•028 -Y^^~"V
!= = = ===. 00^\-
\
-02C \. V
P "^
N \
(^ ff
^
\ "
rt i..
\
'^l
1 -022 t %.
V \l
\
1 A ^
)> Aa
\
\
S ~%.
N
- -ozo \^
\ ^,
\
\
\> ^ '
\
s
L. ^
N ^ -
\
N
.MR \ \.
S \ _
\~A
N
\-. ^
\ S
N.
.riM S»
^ ^^
s
■014^ Ng
^ ^
§1 n
s
^^ vV
1:^
\
''"^S^v "^
-, ^N 5^
: s.
\
Vv ^.
^.. -^ ^5a
S&^\
s
^-, ^^
^^^ ^ .—
\
i^ ^
^.
v^- ^.
i^~^ 4iMs
V.^;
^
^\
^. l^ ^.^ X'-w^
.\ v;
•^
^^
■""'^C^^^-s
■^•^ ^
^
y^
.004 4^ ^:i
^^ "~~- ■-- -fii
1^
^
¥^- ""S
^^•>
i;
= S
-^ ==- ^
H~~"
Zero •ooo _
r« 10 m 1! )o 1
20 31 4 ) 1 8( 90
l< 10 1
)
n
r
3
1) 40
Zero
Note. — The
scale for the older
ages, viz., .00 to
.16, is shewn
between the graph
for the males and
that for the
females, the zero
corresponding to
.028 in the graphs
for the lower ages.
The curves shew
the general trend
of the improve -
ment.
Zeio
30 4U
Year (to which rate of mortality applies).
Fig. 99.
The dots shew the rates of mortality according to the data ; the curved lines
denote the general trend. The scale of the upper part of the graphs is shewn in the
middle, the divisions representing ten times as great a quantity as in the lower part.
The extrapolation of the curves to the year 1940 give an indication of the con-
tinuation of the improvement.
382 APPENDIX A.
7. Determination of the general trend of the secular changes in
mortality. — ^The results given in Table CXXXV., shewing a decrease with
time — except for very great ages — ^in the rates of mortality, are best studied
in Fig. 99. As this figure, however, gives only the rates of mortaUty as
ordinates, and the epochs to which they refer as abscissae ; and does not
shew the ratio of the improvement, it is necessary to evaluate this ratio.
To do this the mortality at any epoch m^nst be divided by that at some
epoch of reference. Thus we may assume — see Fig. 99 — ^that over
greater or lesser stretches of time, the curve of variation of the inortaUty
is of the form (601) with m constant ; that is : —
(602) fit = fto^* > hence log m = log fi,Q + ''* '"? ^
The logarithmic homologue of this relation being a straight line, as shewn,
the values of fi^ and r may be found by the " method of least squares."
Or, put Bt = fit / fJ-O' thea, reckoning t from the year for which fig is
taken, the general trend of the change in mortality can be computed by
the following formulae, the derivation of which from (602) is obvious.
(603) r'loge= U"^ + '^ + eto)
In this expression n is one less than the number of dates for which jj, is
known : r' is, of course, the mean value of r. Having found r' log e
the mean initial value of the rate of mortaUty is : —
(604). .log fi'o=-\log fig+log fii+log iiz + --—r' log e («i+<2+-)} /(n+l)
and /x is the mean value to be substituted for the original jtio to compute
later rates; that is, the general trend may then be taken as fi't = fi'^ (c*")'
the value of e*" being determined according to the unit of t {i.e., for a year,
a quinquennium, a decennium, etc.)."^
Within what limits an assumption of the relation expressed by (602)
may be supposed to exist is of course to be ascertained by graphing the
results on a suitable scale.
8. Modification of the general trend by age. — In order to discover
the relation between age and the present secular improvement in mor-
taUty, it will suffice to take the terminal values only into account ;^
provided we restrict ourselves to the most consistent results. The
improvement for 25 years has therefore been computed, and is as follows,
the tabulated results being the values of ^^25 / fJ-o '■ — ■
1 The following instance will suffice to disclose the significance of the method :
Year .. .. 1886 1896 1906 1911 Sum of Squares
(a) Date 01379 .00982 .00677 .00642 of residuals.
(6) Adopting terminals .01379 .01016 .00748 .00642
(c) By (603) and (604) .01386 .00993 .00711 .00602
(b) - (a) 00000 + .00034 + .00071 .00000 .00000062
(c) - (a) .. + .00007 +.00011 +.00034 -.00040 .00000029
The values of e" for a unit of 5 years, by (6), i.e., adopting terminal values
0.8582 : by (c), i.e., by above method 0.8463,
MOBTALITY.
383
TABLE CXXXIX. — Shewing the secular improvement for 25 years in the Bates of
MortaUty. AustraUa, 1886-1911.
Agel
2
3
4
7.5
12.5
17.5
22.5
42.5
47.5
67.5
Males . .
Females
Batio*
.436
.405
1.076
.466
.430
1.084
.459
.458
.1.002
.435
.463
.940
.544
.533
1.021
.684
.651
1.051
.485
.544
.890
.459
.571
.804
.680
.588
1.157
.704
.629
1.119
.846
.785
1.078
• Ratio of male to female ratio of improvement. The smaller the ratio the greater the
diminution of the mortality.
These results shew (i.), that in general the diminution of mortality
is more marked in young life than in old ; and (ii.), that the diminution is
not identical for mules and females.
Changes in rates of mortaUty, whether due to causes outside human
control or otherwise, may be regarded as due to changes in the relation
between the human organism and its environment. Factors known to be
operative in various organisms, and which are possibly operative in the
human case, are : —
(i.) Evolution of the protective reaction between the organism
and its environment,
(ii.) Changes of the food supply in amount and quality,
(iii.) Changes due to the reactions of the organism to economic
conditions, in respect of its nutritional and neural apparatus,
etc.
(iv.) Changes in individual and general hygiene, in therapeutical
and surgical knowledge, and in prevailing traditions which
affect the vitality of the organism ; etc.
For our present purpose it is not material whether the change is what
may be called internal — as (i.) above — or external : either or both may
be regarded as changes in environment, i.e., provided they are regarded as
either actual or virtual changes. In short, the effect upon the death and
morbidity rates, of any given change in human environment, necessarily
varies with the modifiability or " plasticity" of the human organism. The
plasticity, however, is not the only element which iufluences the results.
The rate of a general improvement in environment will probably be
masked to soine extent by evolutionary disturbances, as, for example, by
dentitional and puberal changes and, ia the case of females, by the de-
mands made on the organism by the exercise of the reproductive function.
Hence, a priori, it is not to be expected that the secular variation of
mortaUty according to age wiU reveal any simple progression with age.
Moreover, to maintain the same rate of improvement for the ages of least
mortaUty, as for those of greater mortaUty, is probably, from the nature
of the case, very difi&cult.
Let Bx denote the ratio of change in jUj. in a given unit of time ;
R being supposed to vary only with age (x). Excepting at the age of
minimum mortaUty, a given value of yu is characteristic of two ages, viz.,
x>ne less and the other greater than^this'minimum age. Since the plasticity
384 APPENDIX A.
of the organism} diminishes with age, a given (absolute) change in environ-
ment will tend to have less effect on the later than on thg earlier age,
other things being equal. It follows, therefore, that, in so far as plasticity
alone is concerned, B^ will be greater than Bx+k- If the plasticity degrade
continuously with age we may suppose that it could be expected to
vary probably either as 1 /(a; + 0)*+"* or else as 1 / c"** , the value of a in
the former representing the interval between fertilisation and birth, or
say 0.75 year, since the plasticity is initially a maximum, and is greatest
in utero. Consequently if it were necessary to take plasticity alone into
account the reciprocal of the last quantity should be a factor distinguishing
between the equal values of /x. for different ages. The former expression,
it is found, does not represent the facts ; the latter possibly would do so
but for the other elements influencing the result. For the purpose
of analysing these complex relations between age, the change in the rate of
mortahty, and the magnitude of that rate, we shall make use of the Census
Life Tables for AustraUa for 1881-1890, and 1901-1910, see Census Report,
Vol. III., pp. 1209-1218. For exact ages and 1, the ratios of ^^ ^re
used, and for the purpose in hand it wiU be abundantly accurate to take
Mi = 1 ('"^z-i + "^x) for ^g^s 2 and above 2,^ m being the central death-
rate for each age in question. In order to fix upon values of the mortahty
with which to associate the ages and ratios of change, the geometric means
of the mortahties used in computing the ratios have been adopted,
which is consistent with the first form of formula (601). It will also be
assumed that the tabular values may be referred to the central point of
time of the period from which the data are derived.*
As already defined, Bt denotes the ratio of change for the time,
that is Bt = nt //-lo a-s before, see formula (603). But there will be some
advantage in fixing our attention upon the ratio of improvement rather
than upon the ratio of reduction of mortahty. Thus if there be no im-
provement (diminution) in the death-rate with the lapse of time, the
quantity considered should be 0, and on the other hand the vanishing of
death altogether would be denoted by unity. Let B denote this ratio of
betterment (or of improvement), then : —
(605) oBt = 1 - ^Bt = (fio -/x() //xo
1 The fixation of plastic elements, by means of which the growth and recon-
stitution of the cellules of the organism are ensured, or anabolism , and the production
of heat and energy by the oxidation of dynamic elements, or kataboUsm, constitute
together the metabolism of the organism. The rate of metabolism or of waste and
repair may appropriately be said to measure the plasticity of the organism. The
plastic and dynamic elements, for example, the albumins, fats, hydrocarbons, etc.,
require also the presence of mineral salts and vitamines, in order to properly fulfil
their nutritive and dynamic functions. The modifiability of the organism may of
course be affected by its environment as well as by age : but its potential modifiability
may be regarded as the measure of its plasticity.
2 The error of this assumption is, of course, nearly negligible for most purposes
for almost any ages, and for the present purpose is quite negligible. The central
death-rate is the number of deaths occurring between any age limits divided by the
mean population.
» That is, the table for the period 1881-1891 can be regarded as referable to the
point of time 1886.0, and the table for 1001-1911 to the moment 1906.0.
MOKTALITY.
385
with suffixes to denote the age to which the formula refers. As afeeady
indicated, the magnitude of B will be influenced by various circumstances.
For example, the ratio of improvement will probably be low (and as a
matter of fact is low) for those ages which are characterised by the lowest
rates of mortaUty^ ; that is for the ages when vitality is greatest a favourable
advance in the environment will produce a relatively small effect. To
analyse the effect of the value of the death-rate upon the improvement
we may divide B by the geometric mean of the death-rates measuring'
the change ; that is by : —
(606) ixm = -\/(f*o M*)"'
and call the ratio of the betterment to this quantity, A, or the relative
betterment,^ thus : —
(607).
. A( = £( / /im = (1 — /^t / A*o) / ViH-oH't
Since the Umits of B are and 1, this quantity can attain to considerable
magnitude when t is considerable, and is therefore a sensitive measure of
any improvement in the rate of mortahty.
The following Table gives for males and females the values of fim, B,
and A, the values for fig and iit being those given by the analysis of the
Census results for thirty years, and the interval being referable therefore
to the period between 1886.0 and 1906.0. For values of E, if required,
we have simply 1 — B.
The values of B are shewn in Fig. 100, curves B and B' ; in which
also the mean death-rates \/(/'^o/^t) ^^^ shewn, viz., curve A male, and
curve A' female. These exhibit the following characteristics : —
Curves o£ Relative Improvement for 30 Years in Death-rates.
Initial
Point.
1st Maximum
Age. Amount.
1st Minimum.
Age. Amount.
2nd Maximum.
Age. Amount.
Remarks.
Males
Females
0.175
0.215
2.8 yrs. 0.508
2.7 yra. 0.520
12.8 yrs. 0.209
13.2 yrs. 0.224
23.3 yrs. 0.491
24.5 yrs. 0.400?
Later values,
are irregular
Upon plotting the ratio of the betterment, viz., the values of A for males
and females, we obtain the results as shewn upon Fig. 100 by curves
C and C", representing the ratio of improvement in the case of males, and
curves C and C", representing the ratio of improvement in the case
of females. These exhibit the foUowing characteristics : — •
Ratio of the Relative Improvement to the Death-rate for 20 Years.
Initial
Value.
1st Maximum.
Age. Amount.
Minimum.
Age. Amount
2nd Maximum.
Age. Amount.
Remarks.
Males
Females
0.70
1.07
9.2 yrs. 164.6
9.5 yrs. 176.4
13.8 yrs. 94.8
(13.8 yrs. 108.3)
16.8 yrs. 109.4
(16.8 yfc. 100.5)
Results after-
wards irregular
1 This corresponds with the age at which the reproductive function commences
to unfold, viz., at about age 12. Probably what may be called the age of effloresenoe
of the organism is generally its period of highest vitality.
' This is suggested by the word iSeXn^ucris, i.e., betterment ; /3 is already
appropriated for birth-rate, etc,
386
APPENDIX A.
The values for age cannot be deemed to closely represent the f £icts ;
to obtain these a table of deaths occurring on successive days after birth
would be needed, and not merely extrapolated results based upon succes-
sive years. For all other ages, however, they represent the facts with
considerable accuracy.
It will be convenient to call the ratio A the mortality improvement
ratio.
TABLE CXL. — Shewing the Mean Mortality, the Relative Improvement in Mortality
in 20 Years and the Ratio of this Relative Improvement to the Mean Mortality for
Males and Females. Australia, 1886.0 to 1906.0.
1
MAItBS.
Femaies.
Exact
Maies.
Fbmaies.
Bxact
ilmprovemeBt.
Mean ;
Death ^ , Ratio -
Mean '
Plas- Death '
mprovement.||
ilmprovement.
Mean :
Death ' Ratio
Plas-
Mean
Death
Improvement.
Age. !]
Ratio
Age. ,]
Batio
'
Bate
Ee-
to ;„
tieity ; Kate
Be-
to
Rate !
Re- 1
to
ticity
Bate
Re-
to
1896.0 ative. i
DeathlCurve ' iS96.0l
latire.; Death
1896.0 lative. i
Death
Cnrve
1896.0
lative.
Death
1
Rate
1
' Rate, i! ^
1
i
Bate.
Bate.
25100
1751
.70
279.0 20150! 2151
1.07: 45 1
01217!
244
20.1
27.7;; 00963
310
32.2
1
04640
45l'
9.7
265.0 04270 463
10.8 46
01283
239
18.6
26.4! 00988
312
31.6
2
01750
499:
28.5
251.8 01660: 514
31.0:, 47
01353
239
17.7
25.0' 01018
317
31.1
3
00796
507
63.7
239.2 00752! 519
69.0; 48
01426
244
17.0
23.8 01055
321
30.4
4
00559
500
89.5
227.2 00522; 503
:
96.4 49
01503
249
16.6'
22.61 01095
322
29.4
5
00441
488
110.7
215.9 OOlOft 492
120.3i 50
01583
251
15.9
21.5; 01140
320
28.1
6
00354
469
132.5
305.5 003241 470
145.0i 51
01668;
256
15.3
20.4
01190
319
26.8
7
00299
449
150.1
194.8; 002691 432
160.61 52
01758
260
14.8
19.4
01249
317:
25.4
8
00267
423:
USA
185.1 0023.=)' 395
168.11 53
01855
263
14.2
18.4
01319
316
23.9
9
00243
387
riCl.3
17.-).8 00215; 377
175.3! 54
01964
267
13.6
17.5
01398
310^
22.2
10
00222
331
ir,.'!.l
167.0 00201' 354
176.11 55
02081,
268
12.9
16.6
01488
304'
20.4
11
00208
264
126.9
1.58.7 OOinS 305
1.58.0 56
02209'
266
12.0
15.8! 01586
295
18.8
12
00206
219
106.3
150.7 0019.5i 244
125.1 57
02349
259
11.0
15.0l! 01694
284
16.8
13
00216
210
S7.2
143.2 00204 225
110.7 58
02503
246
10.6
14.21 01814
267
14.'7
14
00241
223
95.01
136.0 00221' 238
107.7; 59
02667;
229
8.6
13.511 01948
244
12.5
15
00284
282
on.3
129.2 00243 257
105.8 60
02842
212
7.5
12.9
02081
217
10.4
16
00335
359
100.9
122.8 00273 282
103.3 : 61
03030
192
6.3
12.2
02249
185
8.2
17
00385
421
109.3
116.6 00300 306
9S.;i 62
03234
172
5.3
11.6, 02418
155
6.4
18
00429
450
104.9
110.8 00343 333
97.1 63
03465
156
4.5
11 0: 02605
127
4.9
19
00466
466
100.0
105.3 00375 357
95.2 : 64
03746
149
4.0
10.5 02831
115
4.1
20
00490
476
97.2
100.0 00404
373
92.3 65
04098
155
8.8
9.9 03137
138
4.4
21
00531
483
01.9
95.0 00431
380
88.2 66
04520
166
3.7
9.0 03514
179
5.1
-■>2
00556
487
87.6
90.3 00457
379
82.9^ 67
04971
167
3.4
9.0 03898
201
5.2
23
00577
491
85.1
85.7 00479
371l
77.5 68
05423
151
2.5
8.5 04253
194
4.6
24
00598
490
81.9
81.5 00504
372
73.8|| 69
05862
118
2.0
8.1, 04596
165
3.6
25
00616
485
78.7
77.4 00535
383
71.6: 70
06285
069
1.1
7.7 04945
116
2.3
26
00630
473
74.9
73.5 00565
396
70.1 71
06721
013
2
7.3! 05314
053
1.0
27
00642
458
71.3
69.8;! 00593
403
67.9 72
07240
—030
—.1
6.9; 0574C
—005
—0.1
28
00651
440
67.6
66.3 00619
398
64.3 73
07883
—051
—.7
6.6 06260
—042
— .7
29
00660
426
63.5
63.0 00638
392
61.4 j 74
08647
—052
—.6
6.3 06893
—049
— .7
30
00668
. "1
61.5
59.9 00652
382
58.6 , 75
09484
—044
: — .5
6.0i, 0763
—033
— .4
31
00680
392
57.6
56.9 00664
361
54.41 76
1035
—032
—.3
5.7 0843
—006
— .1
32
00696
374
52.6
54.0 00676
338
50.0 77
1126
—021
—.2
1 5.4 092e
-1-019
-1-0.2
33
00714
360
50.4
51.3 00692
324
46.8 78
1222
—012
— .1
1 5.11' lOlC
042
0.4
34
00736
352
47.8
48.8ji 00714
315
44.1 ' 79
1321
—008
; — .1
4.91' llOS
[ j,
: 064
, 0.6
35
00763
342
44.8
46.31 00737
314|
42.6 80
1422
—Oil
— .1
■ 4.6 121c
) 085
.7
36
00795
330
41.5
44.1- 00762
317,
41.6,1 81
1530
—013
— _■
4.4' 1321
' 104
.8
37
00826
318
38.5
41.8 00790
321
40.6 82
1645
—018
— !i
4.2 1 144(
) 121
.8
38
00862
303
35.2
39.7; 00816
3241
39.7 1 83
1775
—02s
— -^
4.0! 156!
) 13J
.9
39
00902
289
32.0
37.7| 00838
323;
38.5. 84
1915
—037
';
3.8| 170!
> 141
.8
40
00943
281
29.8
35.9;' 00859
316
36.8: 85
3.6;' ..
41
00987
274
27.8
34.l! 00876
3071
35.0
86
3.4 ..
1
1 .'.
42
01037
266
25^
32.4
00896
30ll
33.6
87
3.2'' ..
1 :;
43
01094
258
23.6
30.7
00919
30i:
32.7
88
^
3.1
44
01154
250
21.7
29.2
, 00941
307
32.6
89
1
i
2.9, . .
jl
MORTALITY.
387
Moitality Curves and their Variation with Time.
Ages
,mfi
>.
.
5
J)
25
1
-■
—
(70
fC
/
\
\
'
160 ^
1
/
/•
>
I
/
150 'j
1^
,/
/
i'-
!
[^
C
/ /
1
E
HO :
1^
1
[t
l'
j
1
•
■
130
/
c
■■.
1 L
I j
■
120
li'J
1
\
pLI
|^•■
1
\
'
1
■■
li ■
,'
\
C
1
'A''
/'
.^
\
100
\\
'1
\^
/'-
\
■•.
\\
,}
-~
\
ll'
90
c
c
^N,
N
//>\
■^
1
w
!
s
/
1
'
s
/
\
[/
1
...
s.
L
^
/
■'*'
-■
60 A
R j ^
V
1 i
-
(
f 1
"OJ '
^J -ij
\
B
1
17 ''
\\
\-
<0
m '
'
V,
\
/
\fd
v-
R
A
/
«
J ,
-\
>.
/
/
V
/
i~~
A
V
^
^-'
'.,
\
/
_^
-ST-
A
\
/
t^^
A
^
('.
7(
n
fi
r
A
V
R
IT
n'-
...
\
^
N
D
WO
.0 ^
s\
•■
E
n' i
(rof(
r
c
c
c
c
1)
H
K
^
■
10 2
:i
II
4
[)
i
J
b
u
7
)'"-
8
J -
9
10
n
Curves A and A' represent
the geometric means, according
to age. of the rates of mortality
lor 1886.0 and 1906.0, for
males and femalesrespectively.
Curves B and B' are the
ratios of the diminution in the
^ rate of mortality in 20 years to
a the geometric mean of the
S rates, in the case of males and
8 females respectively.
° Curves C and C are the
o ratios of the improvement
m (last mentioned) divided by the
~ geometric mean rates of mor-
« tality.
Curves C" and C" are the
curves C and C respectively,
plotted to an extended horizon-
tal scale but with the same
vertical scale.
Curves B and E' — the
plasticity curve — shews, in a
roughly approximate way, the
general trend of the mortality-
improvement ratio : see § 10.
Fig. 100.
Ages.
9. Significance of the variations in the mortality improvement ratio.-
The following relation between the changes in mortality and in the
mortaUty-improvement-ratio is important.
The variations of the curve of the mortality improvement ratio are
reciprocal to those of the mortality itself ; that is, x and 17 being the
ordinates to the mortaUty-improvement-ratio curve, and x and y the
ordinates to the mortahty curve, we have, practically for aU ages,' : —
(608).
.tj'/t = Ky /y' ; or s = K /r
17, 17' and y, y' being successive ordinates, and s and r their respective
ratios.^ K, however, is not a constant ; nor is it any simply expressed
function of x, though generally it is a Httle less or a little greater than
unity.
* Certainly for all ages for which themortality ratio caa be very accurately
evaluated.
' That is ri and y are values for x, and r/' and y' values for a; -|- 1.
388
APPENDIX A.
This reciprocal relationship reveals the fact that as the mortaUty
at the beginning of life decreases with the successive years, the relative-
improvement-ratio increases in very similar proportion. This reciprocal
movement of the mortaUty-ratio, as compared with the mortality-im-
provement-ratio with increasing age, probably continues throughout life,
and certainly continues till at least age 70. The values of the coefficient
K in (608) above, are given in Table CXLI., Km denoting those which
apply to males and Kf those which apply to females. The ratios s=r]'/rj
and 1/r = y/y' are also shewn, viz., by the smaller figures between the
values of rj and y respectively. This coefficient K may be called the
beUiotic coefficient.^
TABLE CXLI. — Shewing the ratios between the mean mortalities and the mortality-
impiovement-ratios for successive ages. Australia, 1886-0 to 1906-0.
Ratio
Ratio
Ratio
of
Values of K I
of
Values of K
of
Values of K.
Exact
Ratio
Mor-
Exact
Ratio
Mor-
Exact
Ratio
Mor-
Ages
from
ol
Mean
tality
Im-
Ages
from
of
Mean
taUty
Im-
Ages
from
of
Mean
taUty
Im-
to
Mor-
prove-
Fe-
to
Mor-
prove-
Fe-
to
Mor-
prove-
Fe-
(x).
talities
ment
Males.
males.
(I).
talities
ment
Males.
males.
(«).
talities
ment
Males.
males.
a/r)
ratios,
(s)
K,„
^/
(lA)
ratios,
(s)
^,„
^r
(1/r)
ratios.
(»)
^^
^^
0-1
5.409
13.945
2.578
2.154
28-29
.986
.940
.953
.983
56-57
.941
.918
.976
.952
1-2
2.651
2.934
1.107
1.110
29-30
.988
.968
.980
.973
57-58
.938
.962
1.026
.930
3-3
2.198
2,232
1.016
1.011
30-31
.982
.936
.953
.945
58-69
.939
.809
.862
.914
3-4
1.424
1.401
.984
.970
31-32
.977
.913
.934
.936
59-60
.938
.869
.927
.888
4-5
1.268
1.237
.976
.978
32-33
.975
.958
.983
.959
60-61
.938
.850
.906
.854
5-6
1.246
1.197
.961
.956
33-34
.970
.949
.978
.972
61-62
.937
.837
.893
.837
6-7
1.184
1.133
.957
.919
34-35
.965
.936
.970
.997
62-63
.933
.849
.910
.821
7-8
1.120
1.055
.942
.914
35-36
.960
.927
.965
1.009
63-64
.925
.882
.954
.904
8-9
1.099
1.005
.914
.955
36-37
.963
.927
.964
1.012
64-65
.914
.950
1.039
1.202
9-10
1.095
1.024
.935
.939 1 37-38
.958
.913
.953
1.009
65-66
.907
.970
1.069
1.295
10-11
1.067
.780
.731
.862'! 38-39
.956
.911
.953
.997
66-67
.909
.916
1.008
1.126
11-12
1.010
.838
.830
.800
39-40
.956
.930
.973
.978
67-68
.917
.929
.795
.966
12-13
.954
.914
.958
.926
40-41
.955
.932
.976
.970
68-69
.925
.824
.891
.853
13-14
.896
.977
1.090
1.054
41-42
.952
.924
.971
.981
69-70
.933
,545
.584
.670
14-15
.849
1.045
1.231
1.079
42-43
.948
.920
.971
.997
70-71
.935
.931
.427
.459
15-16
.848
1.077
1.270
1.090
43-44
.948
.918
.968
1.023
71-72
.928
.926
16-17
.870
1.022
1.175
1.085
44r-45
.948
.927
.978
1.009
72-73
.918
.917
17-18
.897
.959
1.069
1.089
45-46
.949
.928
.978
1.006
73-74
.912
.908
18-19
.921
.953
1.035
1.072
46-47
.948
.948
1.000
1.015
74-75
.912
.903
19-20
.951
.972
1.022
1.045
47-48
.949
.963
1.014
1.013
75-76
.916
.905
20-21
.923
.946
1.025
1.019
48-49
.949
.974
1.026
1.003
76-77
.919
.910
21-22
.955
.953
.997
.998
49-50
.949
.958
1.009
.993
77-78
.921
.914
22-23
.964
.972
1.009
.979
50-51
.949
.967
1.019
.997
78-79
.925
.914
23-24
.965
.963
.998
1.003
51-52
.949
.965
1.017
.993
79-80
.929
.916
24-25
.971
.961
.990
1.030
52-53
.948
.959
1.011
.997
80-81
.929
.916
25-26
.978
.952
.973
1.034
53-54
.945
.958
1,014
.982
81-82
.930
.917
26-27
.981
.951
.969
1.017
54-55
.944
.946
1.002
.982
82-83
.927
.918
27-28
.986
.948
.961
.987
55-56
.942
.935
.993
.980
83-84
.927
.920
The ratios in the Table (1/r) are the values of the mortality at any age divided by the mortality
at the age greater by one year ; that is. the tabular values are the quantities M» //*.v+i.
The tabular ratios of the mortality-improvement-ratios are the values obtained by dividing the
mortality-improvement-ratio for any age by that of the age less by one year ; that is the tabular
values are the quantities X . /\
The coefficient K is that quantity which, multiplied into the ratio of the
reciprocally the ratio of the mortality-improvement-ratios.
mean mortalities, gives
' From /SeXTiwTiKos, bettering or amending.
MORTALITY. 389
If the value of the ratio y is required for a single unit of time (1 year),
we have, on the assumption of a geometrically progressive decrease in
mortality, fi^ — fi^ ; consequently : —
(609). .5, = [1 -(/Li//x„)%„ and A,= (1 r- i^.^J/VC/^o ^^^)
(610) I* = l^t/f^o' and,i^=/*of
The form of the expression for A is independent of the unit of time,
though of course its numerical vulue is dependent on that unit.
10. The plasticity curve. — ^If we except the period between exact
ages 14 and 17, the. beltiotic coefficient continually decreases in value
from age 10. If a curve be drawn representing the general result, it is
found (from the 20 years' improvement in the mortahty conditions)
that it is fairly well represented by the equation y = 278.95 (0.95)^
This curve, viz., E and E' on Fig. 100, may be called the plasticity curve,
and its ordinates are given in Table CXL. The amount by which the
beltiotic curve {i.e., the curve of the mortality -improvement-ratio) falls
short of the plasticity curve, does not, however, and least of all initially,
constitute a measure of the great difficulty of attaining to the limit,
which plasticity would admit of, were it not for the great difficulty of
initial adjustment to a new environment, and to the exhaustion of energy
involved by puberal developments. For the analysis of these questions,
however, the available data appear to be inadequate,^ and they will not
be further discussed here.
No simple relation expresses the variation of the constants 278.95
and 0.95 with the unit of time over which the improvement is measured.
11. Bate of mortality at the beginning of life. — The mortality
at the beginning of Ufe is probably considerably affected by local cir-
cumstances ; consequently for the first two weeks and perhaps even the
first month of life it would be difficult to assign any particular law of
change of mortafity with age.^ Statistics for Saxony gave a first minimum
rate at 8 days, and a lesser maximum 15 days, and those for Sweden gave
1 It may be noted that for the relative improvement to be unity we must have
;li, = in (605), that is to say, death must vanish. But no diminution of mortality
in a geometrical ratio can reach zero, for though ii„ i' may be as small as we please,
it cannot become zero with f positive and t finite : moreover, when the death-rate
is large the value of \ cannot be great with any practicable change of death-rate.
^ See " The improvement in infantile mortality : its annual fluctuations and
frequency according to age in Australia." by G. H. Knibbs, Journ. Australas. Med.
Congress., Sept. 1911, pp. 670-679 ; see also " Die Sterblichkeit im ersten Lebens-
monat, Zeit. f. Soz. Mediz., Leipzig Bd. v., p. 175, 15th April, 1910.
390
APPENDIX A.
a somewhat similar indication^, while Austrahan records do not lend
any support to this recrudescence of the rate of mortality. Prussian
statistics shew a minimum rate for 9 days and a rise to 14 days. ^
The statistics in Austraha are imperfect, and some distributing was
necessary owing to the want of precision in stating the exact interval after
birth. The defective statement of _ age does not, however, afEect the
deductions hereinafter. In the following table the results for the fractions
of the first day are merely computed : the rates, calculated without
regard to migration, the effects of which are nearly neghgible, and are not
accurately ascertainable, are determined by deducting the deaths from
the total births in order to ascertain the numbers of survivors.
The rates so found shew that from the end of the first day the law
of mortality is expressed by /x^. = [i-^/x, for 5 or even 6 days. The
generality of this expression can be extended, if it be put in the following
form, viz. : —
(611). ,/Xj. = jiti [1 +/(«)]/«, consequently 1 +/(a;) = a;/Xj.//i,
and / {x) for the first 5 or 6 days is zero. The shorter expression indicates
that after the first 24 hours, and for about the first week of life the
probability of death diminishes as the length of time lived, reckoned from the
moment of birth. The following rates are computed for " persons" {i.e.,
males and females) from the records of about 500,000 births and the
deaths which resulted from them.
TABLE CXLn.— Death-iates per diem at the Beginning of Life. Based upon
499,674 Births, and the Deaths occurring therein. Austialia, 1909, 1910, 1912
and 1913.
Age-
Death-
Death-
Death-
Death-
group
rates
Age
rates
Age
" rates
Age
rates
or Age
per
Days.
per
Days.
per
Days.
per
Days.
Diem.
Diem.
Diem.
Diem.
0*
.015000
4
.001416
40
.0002237
200
.0001233
0- *
.014061
5
.001137
50
.0002117
225
.0001142
i- *
.012355
6
.000975
60
.0002035
250
.0001063
i-*
.010143
7
.000853
70
.0001948
275
.0000986
^:*
.007934
8
.000767
80
.0001875
300
.0000923
j_l*
.006330
9
.000703
90
.0001804
325
.0000865
0-1*
.009404
10
.000653
100
.0001740
350
.0000821
1
.005743
15
.000497
125
.0001594
365
.0000802
2
.002927
20
.0003961
150
.0001464
3
.001899
30
.0002678
175
.0001337
1095
.00001084
4
.001416
40
.0002237
200
.0001233
• Approximate estimates only. There are no available statistics for the accurate estimation of
the frequency of death during each of the first 24 hours of life.
' Op. cit., p. 676. The results are given on graph No. 7 on the page men-
tioned.
» See Handbuch d. Med. Statistik., Fr. Prinzing, 1906, pp. 281-2 ; also G.
Lommatzsch. Zeit. f. saohs. Stat. Bureau, 1897, Bd. xliii., p. 1.
MORTALITY.
3dl
We may take the mean of \x./x for the first 5 days as the value of the
mortahty at the end of the first day ; this gives the rate 0.005729 per
diem. Using this to determine 1 + / («), we find that its values are as
follow : —
TABLE CXLin.— Shewing the Values o£ xfxx /ii-,, that is 1 + / (x) in (611).
1 + / (a;)
1 + / (a;).
1 + / (X).
Exact
Exact
Age,
Exact
Age,
Age,
Days.
Crude.
Smooth-
ed.
Days.
Crude.
Smooth-
ed.
Days.
Crude.
Smooth-
ed.
1-5
1.0000
1.0000
30
1.4023
1.4023
125
3.4779
3.4373
6
1.0211
1.0200
35
1.4442
1.4442
150
3.8331
3.8067
7
1.0422
1.0438
40
1.5619
1.5620
175
4.0841
4.1022
8
1.0710
1.0714
45
1.7038
1.7043
200
4.3044
4.3342
9
1.1044
1.1028
50
1.&476
1.8466
225
4.4851
4.5131
10
1.1398
1.1380
55
1.9894
1.9889
250
4.6387
4.6493
12.5
1.2280
1.2280
60
2.1313
2.1312
275
4.7329
4.7532
15
1.3013
1.3013
70
2.3802
2.3930
300
4.8333
4.8352
17.5
1.3526
1.3526
80
2.6182
2.6330
325
4.9071
4.9057
20
1.3828
1.3828
90
2.8340
2.8500
350
5.0157
4.9749
25
1.3990
1.3990
100
3.0372
3.0450
365
5.1096
30
1.4023
1.4023
125
3.4779
3.4373
From 5 to 10 is a second degree curve, the 1st difif. for a unit being = H- '0200, 2nd di£f. = -l-
.0038. From 40 to 60 is a straight line, the common difference for a unit being + .02846. Th»
curve from 60 to 120 is a second degree curve, the 1st did. being -1-0.2622 and the 2nd diff.
-0.02238. From 125 to 350 is a third degree curve, the first rank of difference being + 0.3694,
- 0.0739, and + 0.0104, the last being the common differences.
The results in the above table shew that although for the first few
(five) days the death-rate diminishes as the duration of hfe, this rapid
rale of diminution is not continued, but the rate falls off more slowly — and
on the whole'continually — tiU the minimum death-rate occurs.
Ages
c,»,E. 3.or
1 1 1 1 M 1 1 1 rnrnu. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 nm
Mortality Curves.
Curve A A shews the values of
1 + Hx) for 90 days, see (611) p. 390.
Curve A' A', shews on a smaller
scale the values of 1 -1- f (x) for 360
days.
Curve B B is the curve of rates of
mortality for 360 days ; the dotted
line shews what would be the curve if
the hyperbolic law held throughout.
Curve C is the curve of rates of
mortality for males, and curve D B
is that for females.
Curves C and D* are the same as
C and D, except that the vertical
scale is increased tenfold.
Cmve B B is the curve of mas-
culinity of the rates of mortality
according to age ; see Table CXLVI.
Curve F F is the curve of the
ratios of the rates of mortality for
males " not married " to those for
married males, according to age.
Curve G G is the curve of the
ratios of the rates of mortality for
females " not married" to those for
married females, according to age.
ays-
Fig. 101.
392 APPENDIX A.
The characteristics of the dimiaution of the initial death-rate may be
summed up as follows :• —
(a) For the first 24 hours of life satisfactory data do not exist to
determine the characteristics of the death-rate (see below).
(6) From the end of the first to the end of the fifth day the rate
varies inversely as the duration of life.
(c) From the end of the fifth day the rate of diminution rapidly
faUs off till about the 20th day, then less rapidly till the
30th day, then the rapidity of the falling off of the rate of
diminution approximates to what it was from the 5th to the
20th days, but after that decreases slowly and fairly
regularly.
{d) No simple function expresses these changes in the variation of
the death-rate, and £hey probably differ somewhat in
different countries.
If the expression (611) is put into the form : —
(612) ^. = ^e«(-i)'
this can be fitted to a considerable range of the curve, provided that minor
fluctuations are ignored. It cannot, however, represent with sufl&cient
accuracy a year's results. To fit any two points on the curve besides the
origin we have : —
(613) log^M: =Mog^^
^ log 2/ ° X— \
in which 2/ = x^j./fji^ = 1 + f (x). When b is foxmd a can be readily
obtained from (612).
For the values of (i for fractions of the first day it may be assumed
that the curve is /Xo^"'' For this to give .995729 at the end of the first
day we must make jno = 0.015573 (per diem), and this would be the
mortality for x=0, viz., at the moment of birth, and is equivalent to a
death-rate of 5.684 per annum. This may be put ia another way, viz!.,
it is equivalent to a rate of unity per 64.21 days {i.e., 365 -=- 5.684), and
implies that such a rate, if operating uniformly for that period on a group
of children for 64.21 days, the group being kept constant, would in that
time account for the death of all bom.
12. Composite chaiacter of aggregate mortality according to age. —
Before dealing further with the variation with age of the rate of mortality,
it is desirable to review the nature of the aggregate rate of mortaUty.
The general rate of mortahty for any age, Dj,/Pj=/ij., viz., the aggre-
gate number of deaths of persons between given infinitesimal limits of age
occurring in a unit of time, divided by the average number of persons of
the same age ^ (the average being taken over the unit period in which the
' In practice D and P are taken between limits x and x', say, in which case /t
is not given but instead the average over the range. The difference is dealt with
later.
MORTALITY.
393
deaths occurred) is made up of the rates from each cause, and if regarded
from the summation point of view — see (596), p. 370 — ^is compounded of
a series of rates, the graphs of which are by no means similar. For
example, in " causes of deaths," Nos. 31 and 32, the real number at risk
are those shewn in hne 2 below, the variation with age is quite unlike the
variation with age of the total mortahty, and is by no means identical in
the two cases, as wiU be at once seen from a Table given hereinafter.
The results are as follows : —
1. Age-group
10-14
15-19
20-24
25-29
30-34
35-39
40-44
45-49
50-54
Ttl.
2. Oases of Maternity
1907-1915*
211
54,527
262,866
317,815
238,746
155,813
60,970
6,075
79
3. Cases of Puerperal
Septic semia
1
96
370
515
459
307
112
12
1?
1872
4. Ratio
.00474
.00176
.00141
.00162
.00192
.00197
.00184
.00197
.01266 ?
6 Cases of other Ac-
cidents ol Preg-
nancy & Labour
4
161
587
816
799
829
397
55
0?
3648
6. Ratio . .
.01896
.00295
.00223
.00257
.00335
.00532
.00651
.00905
.00000 ?
• Actually births. These are, however, only slightly too great. The correction may be neglected
for the present purpose.
The results shew that out of a total of 100 deaths at all ages from
puerperal septicaemia and other accidents of pregnancy and labour, 34
will arise from the former cause, and .66 from the latter ; and also that the
distribution according to age differs considerably for septicaemia ; the
proportion dying at different- ages remaining more nearly constant
than in the case of deaths from other accidents of pregnancy and labour.
The fall to a minimum occurs at about age 23.4, when the ratio is about
0.00219. The minimum in the case of septicaemia is at about age 23.1,
and the ratio is about 0.00139, the proportion of the deaths from other
accidents of pregnancy, etc., being here 0.61 of the two combined.^
Causes of death may be classified, as regards their relative frequency
according to age, as foUow, viz.^ : —
(i.) Normal, viz., those in which the relative frequency is similar
to the relative frequency of death from all causes combined ;
» See formula (292) and (294), p. 92 herein.
' The causes of death given in a Table hereinafter may be classified accord-
ing to the scheme indicated, and are as follows, viz. : —
Glass (1.) Hoima], — 9. Influenza ; 12. Spidemlc niseases ; 16a. General Diseases ; 18. Cerebral
Hsemorrhage, etc. ; 18o. Other Diseases of the Nervous System ; 20. Acute
Bronchitis ; 22. Pneumonia ; 23. Other Diseases of the Respiratory System :
24. Diseases of the Stomach; 25. Diarrhosa and Enteritis; 27. Hernia and
Intestinal Obstruction ; 28o. Diseases of the Digestive System ; 29. Acute
Nephritis, etc. ; 80a. Other Diseases of the Genlto-Urinary System; 32a. Diseases
of the SUn and Cellular Tissue ; 322>. Diseases of the Organs of Locomotion ;
35. Violent Death ; 38. Ill-deflned Diseases.
Class (il.). — Infantile, Sub-classes (o), (ft) and (c). — 5. Measles (ft); 7. Whooping Cough (a);
8. DiphtheriaandCroup(6); 14. Tubercular Meningitis (6) ; 15. Other Forms of
Tuberculosis (c) ; 17. Simple Meningitis (a) ; 33. Congenital Debility and Mal-
formations (a) ; 33a. Other Diseases of Infancy (o).
Class (ill.). — Senile.— 16. Cancer and other Malignant Tumours ; 19. Organic Diseases of the
Heart ; 19a. Other Diseases of the Circulatory System ; 21. Chronic Bronchitis ;
28. Cirrhosis of the Liver ; 34. Senile Debility.
Class (iv.). — Median. — 1. Typhoid Fever : 13. Tuberculosis ; 26. Appendicitis, etc. ; 31. Puer-
peral Septicaemia ; 32. Accldente of Pregnancy and Labour ; 36. Median ;
30. Non-cancerous tumours of the female genital organs.
Organic diseases of the heart and other diseases of the circulatory system are
hardly to be included in the " normal" series, because the death-rate in the first year
of life is not very great.
394 APPENDIX A,
(ii.) Infantile, viz., those which characterise iafancy only ;
(iii.) Senile, viz., those which characterise old age only ;
(iv.) Median, viz., those which characterise middle age only.
The infantile causes of death may be subdivided into three sub-
classes, viz. (a) those in which the mortality is greatest in the first year of
life ; (6) those in which it is later than the first year ; and (c) those in
which the mortaUty is greatest in the first year, but is followed by an
irregular mortahty for all ages.
It is obvious that, apart from variations in the distribution according
to age, and general differences in local salubrity, epidemics will cause
differences in mortahty rates according to age, hence to be representative
of a country, the deduced mortality rates must be taken over a sufficient
period of time. The results in the Table. CXLIV. hereinafter are based
upon 9 years' experience, viz., from 1907-1915 in Austraha, and the
distribution of the population, according to age and sex ia assumed to be
as at the Census of 3rd April, 1911. Before analysing these results it
will be necessary to consider the character of curves of organic increase or
decrease.
13. The curve of organic increase or decrease. — The curve e" (or
e"*) and its variants, may, for obvious reasons, appropriately be called
the curve of organic increase or (orgswiic decrease). In considering its
appHcation to the increase of population by birth or the reduction of
population by death, etc., certain characters of the curve deserve notice,
and will now be considered. If to adapt it to a given instance, the ex-
pression be put in the more general form hereunder, we may note that : —
(614) ^ =^6"^+° = (^e'')e»* = ^'e"* =4'm*
in which m = e" Hence the addition of a constant to the index of e
affects only the vertical scale of the graph of the curve, while n affects its
horizontal scale. If w be constant the final form in the above expression
is satisfactory, but it it change with x, then the appropriate expression is —
(615) y = ^e«^*W = ^e"^-»*W = ^e"'^ = A^{xf
and the form of ^{x) will be determined by the law of change in n'.
Geometrically this is equivalent to changing the a;-scale as x increases.
In order to ascertain the form of <fi{x) the quantities, group or other,
may be set out as shewn hereunder, and the quotients B/A = b, C/B=c,
etc., computed. If b, c, d, etc., are not equal, then the curve Se** will
not satisfy the data. If on computing also the values of b (c— 1) / (b— 1);
c (d— 1)/ (c— 1) , etc., it is found that they are not equal, the curve A-{-
Bi"" will also not satisfy the values. The last step may be shortened by
putting the above ia the form (c — 1)/(1 — 1/b). Similarly, if the
quotients of the differences of the ordinates are not equal the equation
will not apply.
MORTALITY. 395
Scheme of Examination of Data.
Value, of . . Xx x.^ x^ x^ x^ , etc.
or Range between x^^ to x^ , x^ to x^ , x^ to x^ , x^ to a;„ , aSj to x^ , etc.
Group value .. A ,.B , C , D , E , etc.
Ratio B/A, etc. b c d e f
R«ciprocal8 A/B, etc. 1/b 1/c 1/d 1/e 1/f
„, Rat- ,A-n ya-yz yi-ys ,
or Ratio of differencea ; ; etc.
y2-yi 2/3-2/2
If the values are increasing in the order A . . . E, n is positive, if
diminishing n is negative, x being regarded as positive throughout.
There is, of course, no universal guide for deciding what form of
function to adopt, but if b, c, d ascend by a common difference, the
function will be of the form 5e''^<i+''^' ; if by a common multiple, it
will be of the form Be^''' ; if by a common power of x, of the form
£e**'' , and so on. Successive values of m = e" may therefore be
analysed on the same principles as the original data.
14. Exact value of abscissa corresponding to the quotient o! two
groups. — ^It is obvious that if there be two distributions {e.g., the number
of deaths occurring in a population of a given magnitude in a given unit
of time, both set forth according to age) and the average over a range of
the variable be ascertained {e.g., the average death-rate of all persons
between ages xi and xz) the quotient found, by dividing one group by the
other group, with the same range, will, in general, be the exact value for
some given value of the abscissa ; and ordinarily this value must not be
referred to the middle of the range in question, when high precision is
desired. Let the two distributions be denoted by G and H, and let G be
the numerator group and H the denominator group ; and let the five
quantities xi, Xg , X|^, Xm. and xz denote respectively : — (i.) the value of
the abscissa at the beginning of the group ; (ii.) the abscissa of the mean
ordinate of the group Q, and (iii.) that of H ; (iv.) the abscissa where the
ratio becomes exact ; and (v.) the value of the abscissa at the end of the
range. That is, if gym and ^j/m denote the ordinates at Xm for the two
• distributions, then we must have : —
(616) G/H= ^yjy^
The following laws hold as to the position of Xm in relation to Xg and x^
the latter being the abscisssB of the ordinates equal to the respective
means of the two distributions"; —
(i.) The two distributions increase Unearly, ^ then x^ =i(^i+^2)-
(ii.) The relative increase of the two distributions, though not
linear, is identical throughout {i.e., they increase in the
same proportion) ; then Xm is the common abscissa of the
means of the group-ordinates.
1 Provided, however, that the prolongation of the bounding lines does not
meet on the axis of the abscisssB, since, in this case, the required ratio holds for
any value of x ; that is to say, in this case xm is indeterminate. This, however,
does not vitiate the adoption of the middle of the range, or indicate that it should
not be adopted.
396 APPENDIX A.
(iii.) The ordinatesof one distribution are constant throughout; the
values of the other are variable ; a;„ is the abscissa of the
ordinate equal to the mean of the ordinates of the variable
distribution.
(iv.) The relative increase of the ordinates of distribution G is more
rapid than that of the ordinates of distribution H ; then
Xg 7>X^^X/,.
(v.) The relative increase of the ordinates of distribution G is less
rapid than those of distribution H ; then Xg <iXm<X|^.
(vii.) Where the distributions G or H include maxima or minima
(either one or both), no general law apphes as regards the
value of the abscissa Xm, and it may have more than one
value. In general also the position of Xm in such cases is
not accurately determinable from the group-data.
The most general supposition that can be made regarding the curve of
instantaneous values which, between given limits, will satisfy a particular
group-value (not near a maximum or minimum, at a point of inflexion, or
very near the terminals) is that it is approximately represented by
Be'"' or Bw' in which c* or w is the ratio of any group to the adjoining
group. A curve of this type will satisfy three groups G.i, G and Gi, in
ascending or descending order of magnitude, see Part XIII., §25 (508),
p. 266. Let the value of G/Q.\ be %, and of G\/G be m, then the value
of ni = e*, which will give a curve satisfying the three group totals, is ^ : —
(617) tn = Ml (W2 - l)/(wi —1)
The common quantity y^ to subtract so as to get three groups with the
common ratio in between the second and first, and third and second, is : —
(618) ;^ = (? (»2 _ Mi)/(1 -f WiWa _ 2wi)
where G is the central group, and the position of the ordinate corresponding
to the group -height, G / {x' — x) say, is wholly dependent upon itt,^
which should be substituted for either riy or W2.
The abscissa of intercept of the group-rectangle with the curve of
distribution is obviously independent of the scale, or of the zero of the
1 It is important to bear in mind when high accuracy is desired, that the mean
of TCj and n, is not necessarily at all near the value m. For example suppose the
groups are 1000, 1200, and 2040, the value of n^ is 1.2, and of n^, 1.7. The arithmetic
mean is 1.45, and geometric 1.42829. While the value of m is 4.2. For the ratios
(1000 — 937.5) : (1200 — 937.5) : (2040—937.5) are identical, and are 4.2. Similarly
the groups 2040, 1200, and 1000 give for n^ the value 1/1.7= 0.0588235, and for that
of Wa the value 1/1.2 = 0.0833333 give 1/4.2 = 0.0238095, as may be seen by apply-
ing the formula for the value of m. The position of the abscissa of the ordinate
to the curve corresponding to the mean height is the same in either case, as also is
the position of the centroid vertical.
2 The value of a;^ is that of the abscissa of the point (or points), where a line
parallel to the axis of abscissae and distant therefrom the average of the group-
ordinates cuts the bounding curve (the curve of distribution) ; or — in a graph —
the abscissa of the intersection of the group rectangle with the curve of distribution.
MORTALITY. 397
^gure. We suppose the range to be to 1, and the curve tohey = Ae"",
in which e" is m. Hence since log. m = a log. e, we have by integrating
between the limits and 1 : — -
(619) t/™ = e"""' = 1 (e» — 1) = m*"' = I-2S-? (m - 1)
a logm
Consequently by taking logarithms of this last form of the equation, we
have, since log.io 2.3025851 = 0.3622157 :—
(620^ X = logio("t - 1) - log,o(log,o in) - 0.3622157
^ ' *" log,„ttf
To find the value of x^ when there are two curves we have, writing ^ for
Xm for convenience, and in order to distinguish this case from the previous
one : —
Aje^-D/a ACj . therefore (llkZlllA- "^^
*
But 1/a divided by 1/6 = log. xtlb/ log. nto consequently : —
(622) C^y= l^iilL* Jl!a^; , and therefore
^ Vmft/ logiUa mj — 1
(log ma- 1)— log (lUft— l)+log(log,liu)— log(logt Ilia)
(623).. I =
log lHa — log 111 J
which put in suitable form for computation with Briggsian logarithms is
(623a) I =
logio('ii<t— 1)— logio(i"i»— l)+logio logiQ iiift-logio logiQiiitt— 0.3622157
logio iito — logio in 6
The fraction ^ can thus be readily tabulated in a table of double entry,
with the arguments xtla and nij. If in these last formulae (623) or (623a)
we put nifi = 1, we get (620). If (623) is used it is important to note that
the Napierian logarithms are to be used where indicated. Formula (622)
may be regarded as the fundamental equation for the determination of
Xm or f
The preceding formulae are unsuitable when nta = ntj. But by
putting itta = ntf + ^ where h iaa, very small quantity, we obtain, after
expanding both sides of the following equation, viz. : —
^111 +h\i log 111 ill — 1 + ft
(«24) (-IT-) =
log (in + h)' 111 — 1
which is (621) recast, and remembering that the ' powers of h are
negligible : —
(625) r= "^ '
m — 1 log. 111
I ', therefore, is not really indeterminate, when nta = Mt;,, as might have
easily been wrongly inferred from (623). When both jjTa and ntj are 1
it may readily be shewn that ^' = 0.5 by expansions applied to (625),
which is but a special case of the following, viz. ; —
398
APPENDIX A.
If tlto = l/nij, then we have at once from (622) that f = i- And
finally if itta = fentj we have
(626) fc^= {{km,, -l)logiU6}/!(i — l)logA;mj}
by means of which also tables may be constructed.
The following table will enable the value to which any group-ratio
should be referred to be readily found, after the values of vHa and nts for
the two distributions have been ascertained. Where the original dis-
tributions are increasing with x the value f given in the table is read from
the left-hand toward the right ; where diminishing, with increase of x,
from the right-hand toward the left. Thus if ,4Ra= 1/nto and JH j=l/m6,
then : —
(627).
^M = l-^„
consequently the table may be entered for the reciprocals of both^ the
ratios without altering the result provided the point to be determined is
taken either ^^ from one end of the range or |^ from the other.
TABLE CXLIV. — Shewing the values of {, viz., the relative Distance from the
Initial Value of any Range on the Axis of abscissae, to the Ordinate, to which the Ratio
of any Two Groups should be ascribed, the whole Range being regarded as
Unity. Values of f.
ma
or
.05 .10
,15 .20 : .25
.5
1.0
1.25 1.50
1.75; 2.0
3
4
5 7.5
10
15
20
25
1 ma
40 or
ms
'
1 mi
.05
.281! .302
1
.315' .324: .3321 .357
.383 .392i .399
1 "
.406! .4111 .427
.439
.447; .462
.474
.488
,500 ,508
.525
.05
.10
.. .323
.337' .347; .355 .382
.4081 .417i. 424
.430' .436, .452
.464
.473 .488
.500
,514!, 526, ,534
.550
.10
.15
.351; .3611.369; .396
.4231 .432
.440
.446
.451. 467
.479
.489 .504
.516
.530' ,542 .550
.566
.15
.20
.. .371
.3801 .406
.434' .443
.451
.457
.463! .479
.491
.500 .515
.527
.541 .553 .561
.577
.20
.25
;
.388 .415! .443' .452
.460
.466
.472' .488
.500
.509 .524
.536
,560 .561, 569
.586
.25
.50
.443
.471 .480
.488
.495
.dOOi .516
.528
.537 .552
,564
.578! .5891 .597
.613
.50
1.0
.500 .509
.517
.523 .529 .545
.557
.566 .581
,5921 .607
.617 ,624; .639
1.0
1.25
. . .319
.526
,532
.538 .554
.566
.575 .590
,601
,614
.625 ,633 .648
1.25
1.50
■ '
i
.534
.540
.346; .562
.574
.582 .597
.608
,622
,634 ,641 .656
1.50
1.751 .
I ••
.547
.5321 -568
.580
.588 ,602
,614
,628
,639 ,646* ,661
1.75
2.0 1 .
••
.557, .573 .585
.594 ,607
,618
,632
,643, ,650, .665
2.0
3
1 •■
.. 1.590; .602
,611 ,624
,635
.649
,659 .666! .680
3
4 1 .
,..'.. .612; .621 .634
,645
,658
,668i .6751 .689
4
r 5 ! .
1 .6291 .642
,653' ,666
,676 .683 ,696
5
|7.5 i
, , ,658
,669
,681
,691 ,6981,711
7.5
10 .
,677
.689
.698: ,705' ,718
10
15
....
,701
,710 ,716j .729
15
20 j .
.719 ,724; .737
20
25 ! .
, , ,7271 ,742
25
40 i .
' 1 ' '
1
,. .. '.755
40
If X, xm and x' are respectively the beginning of the range, the point at which the ordinates to
the two distributions are in the exact ratio of the corresponding groups, and the end of the range,
c is tlie ratio of tlie distance x to x,„ to the distance x to a;'.
Most of the quantities in the table liave been directly calculated, and are less than .0005 in error.
The greatest error In the interpolated part of the table will be about .002.
Let h and k be any two small quantities such that m 'a = ma + h, and »»'» = mi, — k, then by
expanding and neglecting powers of h and * higher thsip the first we obtain :—
mb
k
»»«»»«
-f,
m^mf,
m^ m,,
^M^ m/y
wliich shews that if we add (or subtract) any small quantity to (or from) ma to get a tabular value we
must subtract (or add) m^ Imi^ times the quantity from (or to) m^ in order to obtain the true value of \
This follows from the fact .that the s quantity alone is approximately unity in all practical cases, and
gives the required quantity by a single interpolation.
If reciprocals be taken of halh m^ and m^ , the value of t is the arittunetical complement of i,ts
value for ma and mi , i.e. : —
«„
+ ^1.1.
= 1
* It hardly needs to be added that the table must not be entered for one reciprocal
only.
MORTALITY.
399
15. Absence of climacterics in mortality. — A general impression
exists that death is more than normally frequent at some age between the
ages of 50 and 60 in the case of males, and an age between 40 and 50 in the
ease of females. The crude figures for individual years are very irregular,
and no precise deduction can be based upon them until they are smoothed.
The sums for 9 years (1907-1915) from Australian records are as follows : —
Ages
Deaths of Males . .
Smoothed Means . .
48
2841
49
2723
2912
50
3667*
2925
51
2323
2922
52
3071
2926
53
2827
2921
54
2987
2960
55
2903*
2902
66
3010
2938
57
2781
2931
Ages
Deaths of ITemales
Smoothed Means . .
37
1402
38
1619
1629
39
1540
1475
40*
1608
1534
41
1206
1489
42
1695
1478
43
1396
1470
44
1449
1527
45*
1673
1493
Ages
Deaths of Males . .
Smoothed Means . .
58
3011
2930
59
2705
2873
60
3636'
2896
61
2232
2893
62
2896
2959
63
2995
3032
64
3068
3182
65
3905*
3260
66
3044
3307
67
3286
Ages
Deaths of Females
Smoothed Means . .
46
1423
1534
47
•1524
1590
48
1601
1619
49
1741
1593
50*
1782
1635
51
1319
1635
52
1733
1684
53
1601
* If the errors be supposed to accrue mainly through ages 1 and 2 years above and below the
true age, the true value can be found approximately by taking the mean ol 5 years, since the results
are usually linear for small ranges of age. The excess over this can then be distributed among the ages
plus and minus 1 and 2, according to the probabilities of the case. The general trend can then be
found. One-third of the excess over one-fltth was added to the age below, one two-thirds to the age
above. The quinquennial means so corrected were formed, and are as given opposite "smoothed
means."
The instances where the age ends in the integers and 5 shew the usual
defect, viz., a tendency to give approximate ages ending in those numbers.
When the necessary distribution of the excess, however, is effected no
indication exists of climacterics (marked in the death-rates of males or
females) in the range shewn. This is evident from the following rates
deduced from the adjusted population results of the Census, after making
allowance for the difference from the mean of the 9 years included, ^ viz. :
Ages (Males)
Death-rates'. .
Smoothed
Ages (Females.
Death-rates'..
Smoothed
49
0124
0124
38
00607
00598
50
0130
0130
39
00598
00613
51
0138
0138
40
00635
00628
62
0148
0148
41
00632
00643
53
0159
0160
42
00646
00658
54
0173
0174
43
00668
00673
65
0183
0180
44
00701
00690
56
0202
0198
45
00702
00711
57
022(
0218
46
00740
00738
58
0241
0240
47
00789
00773
50
0255
0253
48
00830
00818
60
0276
0267
49
00852
00873
61
0295
0292
62
0319
0318
50 51
00924 00991
00928 00983
63
0344
0345
62
01031
01038
64
0377
0373
53
65
0404
0402
0430
0432
It is worthy of note that the actual number of deaths of males
oscillates very little on either side of the average, 2921, between the ages
49 and 62 ; and also that the actual number of deaths of females between
the ages 38 and 45, and between 45 and 52, oscillate but small amounts
on either side of the averages, viz., 1499 and 1598 respectively. The
death-rates of course all increase appreciably.
16. Fluctuations of the ratio of female to male death-rates according
to age. — The average ratio of the female rate of mortality to the male rate
of mortahty for 1886 to 1915, according to age, is given in the final column
of Table OXXXV., p. 375, and this ratio for the death-rate for all ages is
^ The ratio of males and females as at the Census of 191 1, and for the sum of the
mean populations, gave the following results : —
Sum Mean Male Populations 1907-15-H Males at Census 1911 = 21,150,358 -r- 2,313,035 -- 9.143985
„ Female ,, „ -^ Females „ „ = 19,620,889 -h 2,141,970 = 9.160207
The divisors used to obtain the mean numbers were,9,H40 arid 9.1602 for males
^pd females respectively.
400
APPENDIX A.
given in Table CXXXVI., p. 376. It is analysed in Table CXXXVII.
p. 377. The fluctuations with time of the rates according to age are shewn
in the following table : —
TABLE CXLV. — Shewing the Variation in the Ratio of Female to Male Mortality-
rates according to Age ; 1886 to 1911. Australia.
Aqe-qbotip.
Epoch.
Age-
group.
Epoch.
1886.0 ! 1896.0
1906.0
1911.0
1886.0
1896.0
1906.0
1911.0
1
2
3
4
0-4
5-9
10-14
15-19
20-24
25-29
30-34
35-39
.864 i .975
.958 ; .938
.969 ' .981
.936 ; .982
.936 .982
.887 .867
.924 ! .945
.929 1 .877
.769 1 '861
.753 ! .826
.898 1 .900
.913 1 .954
.969 I .939
.826
.933
.932
.934
.929
.844
.905
.891
.907
.902
.989
.976
.944
.803
.818
.896
.933
.996
.824
.913
.884
.862
.935
1.002
.936
.880
40-44
45-49
50-54
55-59
60-64
65-69
70-74
75-79
80-84
85-89
90-94
95-99
100-104
.882
.793
.752
.727
.725
.800
.830
.855
.949
' .892
1.154
.715
.770
.812
.763
.733
.731
.739
.721
.803
.853
.863
.843
.984
1.225
1.187
.813
.728
.686
.716
.741
.769
.801
.792
.809
.834
.963
1.139
1.163
.763
.709
.695
.684
.686
.742
.792
.840
.867
.810
.937
.979
.949
0-4
6-14» . .
.887 .867
.926 .911
.844
.898
.824
.897
15-49»
50-89*
.854
.816
.865
.786
.894
.769
.870
.765
* Average of rates merely.
The results in the table shew that, on the whole, there is a decrease in the
ratio of the death-rates for females, i.e., the environment or its effect has
become more favourable in the case of females than in the case of males in
Austraha. This result is well defined for ages to 4 ; fairly well-defined
for ages 5 to 14, and is not well defined for later ages. Moreover for all
ages the results are rather irregular. The reciprocal of this ratio, viz., the
male divided by the female rate for the years 1907-1915, is given in Table
CXLVI. hereinafter, and is shewn on Fig. 101 by curve E.
17. Rates of mortality as related to conjugal condition. — ^The effect
of conjugal condition upon death-rates is well marked, and is shewn in the
following table, based upon three years' results, viz., 1910-1912. For
convenience of comparison the mortahty results for nine years for all
males and all females is repeated^ in the table, the masculinity ratio
between the death-rates being also given ; see curve E, Fig. 101. The
ratio of the death-rate of the " not married" to that of the married
fluctuates considerably fron^i age-group to age-group, both for males and
females. For males between 20 and 85 years of age the death-rate is
considerably higher for the " not-married." For females the advantage
lies with the " not married" until the child-bearing period has been passed,
after which it lies with the " married." The exact age at which the death-
rates become equal in the case of females is probably about 43 years.
^ Those results were given in Table CXXXV., p. 375, and the average '
inity" of the death-rate was also given for each ags-group.
femin-
MOBTALITY.
401
TABLE CXLVI. — Shewing in Age-gioups the Average Death-rates of all Males and
all Females, 1007-1915, and of Married and Unmarried Males and Married and
Unmarried Females,' 1910-1912. Australia.
1907-1915.
Ratio
MortaUty, Males, 1
MortaUty, Females.
Mortality Bate.
Male
1910-1912. 1
1910-1912.
Age-
Age.
to
Group.
Fe-
Not
Not
Males.
All Fe-
male
Married.
Married.
Ratio.
Married.
Married.
Ratio.
males.
Kate.
(m)
(«)
u/m
m'
u'
u'/m'
..
0.43
.08540
.06862
1.2445
1 ..
1.46
.01559
.01389
1.2226
2 ..
2.48
.00642
.00575
1.1153
3 ..
3.49
.00409
.00382
1.0712
.01034
.00858
4 ..
4.49
.00301
.00300
1.0037
'
0-4 ..
1.98
5-9 ..
7.42
.00222
.00202
1.0949
•
10-14 ..
12.60
.00173
.00153
1.1315
-
15-19 ..
17.58
.00256
.00221
1.1597
.00257
.00251
0.977
.00388
.00215
0.554
, 20-24 . .
22.49
.00364
.00341
1.0699
.00244
.00378
1.549
.00418
.00321
0.768
25-29 ..
27.46
.00431
.00432
.9984
.00329
.00504
1.532
.00454
.00390
0.859
30-34 ..
32.50
.00508
.00475
1.0684
.00405
.00682
1.684
.00482
.00429
0.890
35-39 ..
37.50
.00666
.00586
1.1363
.00564
.00896
1.589
.00616
.00506
0.821
40-44 ..
42.51
.00841
.00641
1.3108
.00752
.01099
1.665
.00643
.00609
0.947
45-49 ..
47.47
.01120
.00794
1.4101
.01039
.01535
x.4m
.00765
.00974
1.273
50-54 ..
52.39
.01511
.01050
1.4394
.01406
.01917
1.363
.01065
.01237
1.162
65-59 ..
57.45
.02153
.01473
1.4615
.02039
.02709
1.329
.01459
.01778
1.219
60-64 ..
62.48
.03174
.02177
1.4578
.02947
.03614
1.226
.02097
.02416
1.152
65-69 ..
67.46
.04678
.03471
1.3479
.04523
.05266
1.164
.03529
.03766
1.067
70-74 ..
72.41
.06972
.05523
1.2624
.06730
.08364
1.243
.05504
.07433
1.351
75-79 ..
77.27
.10900
.09162
1.1898
.10721
.13670
1.275
.09348-
.11829
1.265
80-84 ..
82.15
.16816
.14575
1.1537
.16415
.20613
1.256
.14615
.14664
1.003
85-89 ..
86.96
.2678
.2170
1.2342
.2640
.2199
0.883
.2106
.2385
1.133
90-94 ..
91.88
.309
.2895
1.0669
"
95-99 ..
96.70
.391
.3832
1.0207
-.354
.327
0.924
.328
.505
1.541
lOO&over
101.4?
1.13
1.07
1.0542
All ages'
.001194
.000945
1.2636
The " married" include " widowed" and " divorced."
The graphs of the ratios of the death-rates of the " not-married" to
the death-rates of the married are curve G for males, and curve F for
females, Fig. 101.
18. Exact ages oJ least mortality. — The ages of least mortality appear
to vary shghtly, but cannot be ascertained with a very high degree of
precision, owing to the limitations of the data. They may be taken,
however, to be as follows : —
Males (Year) .
1886.0
1896.0
1906.0
Females
1886.0
1896.0
1906.0
Exact Age
12.0
11.8
11.0
12.0
12.2
10.6
Beath-rate . . .
.00232
.00213
.00178
.00223
.00178 .
.00159 .
•General trend
(.889)'
.00232
.00206
.00183
(.830)
.00223
.00185
.00154
t or
(.896)
.00231
.00207
.00185
(.823)
.00225
.00185
.00152
* Allowing twice the weight to the ratio for the 20- years period to that for the 10-years period.
t Allowing eoLiial weight to the 20-years and 10-years periods. This result is less probable than
the former.
These results, deduced from the values given in the Lite Tables of the
Australian Census, Vol. III., pp. 1209 to 1218, are probably nearly
correct, and indicate a minimum mortahty at " exact age" 11.6 for both
males and females, and not only a less mortahty, but also a greater im-
provement in the case of females : the factors for 10 years being respect-
ively 0.889 and 0.830. These for 25 years would be respectively 0.745
402 APPENDIX A.
and 0.628. The ratios of female to male deaths were — ^from the data —
0.961, 0.836, 0.893 ; from the general trend— 0.961, 0.898, 0.843. Thus
the improvement increases at the rate radicated by multiplying by
0.99362 per annum, or by 0.938 for a 10-year period, that is as 0.959,
0.899, 0.843, at 1886, 1896 and 1906.
19. General theory of the variation of mortality with age. — From
time to time attempts have been made to present a rational theory of
the variation of the death-rate with age.^ On the other hand it has
been held that such attempts are merely efforts to clothe what is really
an empirical " fitting of the curve," with a rational guise. ^ It is certain,
however, that, in a general way, the aggregate of the menaces to lite may
be subsumed under certain elementary conceptions, which we now proceed
to indicate. Actuaries have adopted the term "force of mortality," to
denote the death-rate at a given age (i.e., between the ages x and x -}- dx).
This may be deemed to be composite, and to consist of several forces of
mortahty which, operating over a considerable range of ages, have their
maximum effect, however, at different ages. Thus the deaths D^ of
persons whose ages are (sensibly) a;, in a population P^ of the same age
(within the same limited age hmits) is : —
(628) D = P (fi' + [i" + [i'" + etc.) = Pfi
(with the same suffix— denoting age — ^throughout).
The following conception of the nature of the hfe-and-death struggle
of a hving organism represents the phenomena in a general way.*
^i.) The plasticity of an organism, i.e., its modifiabihty ia reacting
to its environment, is a maximum at its origin,* and con-
tinually diminishes during life ;
' Prof. Karl Pearson, adopting the "Vision of Mirza" conception, suggests an
analysis by means of systems of progressive eUmination, viz., first of deaths due to
" old-age mortality," then those due to other elementSj and finally those due to
infantile mortality. He divided the deaths into five elements, viz. (i.) those from
old-age mortality (mode, at age 72^, mean, age 67J) ; (ii.) those from middle-age
mortality (mode, age 42); (iii.) those from mortality of youth (mode, age 23);
(iv.) mortality of childhood (mode, age 23) ; and (v.) infantile mortality. This last
should start — so it is averred — at — 0.75, i.e., nine months before birth. The
"recorded deaths" are the post-natal, which to the ante-natal are as 246 : 605.
See K. Pearson, " The Chances of Death," etc.. Vol. I., 1897, pp. 1-41.
■" Prof. Harald Westergaard says : — (See his paper on the "Scope and Method
of Statistics," Joum. Americ. Stat. Assoc, Vol. XV., Sept., 1916, p. 254): — "Several
mathematicians have ewed in thinking that it would be possible to find a mathe-
matical law of mortality, a physiological law, as it were. We have several formulas
of this kind, by Lambert, Moser, Gompertz, Makeham. For a certain period of life
Makeham's formula is exceedingly practical, but after all it is only a beatttiful formula
of interpolation."
' It may be added that the similar problem in ictero, though important to
physiologists, can be solved only when a sufficient number of women, having become
competent and interested observers of their own careers, supply the necessary data.
♦ This is probably a measure of the rapidity of metabolism in the organism.
MORTALITY. 403
( ii.) In virtue of its plasticity an organism is both vulnerable and
recuperable.
(iii.) On " birth," i.e., on the introduction into a new environment,
the inimical force, i.e., the difficulty of adjustment to the new
envirormient, is very great, but this difficulty diminishes
continually and with great rapidity. The initial difficulty of
adaptation to the new environment may be called the natal
force of mortality.
(iv.) This falls off so rapidly that it may be regarded as operating
for the very hmited number of days^ that constitute what
may be called the initial or natal adaptative effort.
(v.) The adaptation having been established, a new condition
supervenes during which the mortality is markedly less, and
characterises what may be called the infantile adaptative
effort, covering roughly the first twelve months of life.
(vi.) The inimical force, now greatly diminished as a consequence
of successful initial adaptation, may be called the infantile
force of mortality. This, hke the natal force of mortaUty,
also degrades, but nothing like so rapidly.
(vii.) Since organic Ufe is maintained in virtue of its plastic endow-
ment (adaptabihty to its environment) the inimical forces
(or measure of the difficulty of adaptation) increase con-
tinually from the moment of birth (or more strictly from the
moment of origin in utero). This growing incapacity for
adaptation may be called the senile force of mortality.
(viii.) The pressure put on the organism of social lite (education, etc.),
and by the arising of puberty, and the assertiveness of the
reproductive forces, constitute an inimical force, character-
istic of the period of life commencing in childhood and
vanishing at the end of the disturbing (reproductive) period.
This may be called the genesic force of mortality.
So far the consideration has embraced both sexes, but in the case of
the female another force must be assumed, when a differentiation between
reproductive and non-reproductive females is taken into account, viz.,
the following : —
(ix.) The exhaustion and general dangers of reproduction, initiating
on reaching puberty, and continuing till the end of the
reproductive period, constitute an inimical force which may
be called the gestate force of mortality.
1 From what has preceded, see § 1 1 of this part, it would appear that this ig
fibpvit 5 days,
404 APPENDIX A.
(x.) There exist ako dangers to life whicli are of a purely casual
nature. The aggregate of these may be called the ad-
ventiti(yus force of mortality. This, however, probably need
not be separated from the other forces of mortality. ^
These several forces of mortality can be so evalued as to be additive in
character, as in formula (628), so that the (average) aggregate force of
mortaUty is their sum. This aggregate of inimical forces thus gives the
measure at any age of the force determining the rate of death for persons
of the age in question. We shall later refer again to this element. We
may 'also suppose that there are in addition what may be called special
forces of mortality. The indication given may be regarded as the condition
of things when general hygiene is fairly satisfactory throughout life. Not
only, however, do iadividual instances differ from this, but so also do the
characteristics of particular communities. ^
^ The conception of life as a play between conservative and inimical forces has
been presumed by some to be inadequate. Thus although the rapid diminution of
the " natal" and " infantile" forces of mortality may be supposed to measure the
quick and slower elements of the adaptation attained, the ' ' secular' ' force of mortality
specially characteristic of old age, to measure the decrease of adaptability, and the
" genesic" and" gestate" forces of mortality — analogous to one another — ^to measure
the stress put on the organism by the play of the reproductive function, and its
consequential effects, entering as it were, as a disturbing faictor the effect of which
ultimately vanishes, yet there is another factor, acting throughout life, which, as
Gompertz considered, is apparently independent of the progressive deterioration
with age of the organism, and of course independent also of its adaptativeness. This
chance element, viz., the "adventitious force of mortality," would, of course, include
death by accident or misadventure, is certainly not a constant ; it is a function of
age, and differs strikingly £is between the sexes. The real vicissitudes of life of a
chance nature are, however, not on the average uniform, and probably are not very
dissimilar in relative frequency to the relative frequency of death from such causes as
have already been indicated. To the extent this is so they may, of course, be
regarded as embraced in the other inimical forces.
In considering the whole question, it is to be remembered that we are not really
dealing with individual lives, but with a multiplex-organism, viz., an aggregate of
lives or population ; and we are measuring the progressive reduction of that organism
by the elimination of theoretically infinitesimal elements (removal by death). And
from this point of view it is obviously very doubtful whether the conception of an ad-
ventitious force of mortality is necessary at all, and it is certain that to the extent the
relative frequency conforms to the other types of inimical forces it may be regarded
with advantage not as merely masked by, but included in, them. Prof. Wester-
gaard says, however, op. cit., p. 254 : " If we seek a formula for the combined
effects of all the causes in action, we run the risk of overlooking some, which it would
really be exceedingly important to take into consideration."
2 In Saxony, for example, the mortality apparently falls till the eighth day after
birth after which there appears to be a recrudescence of mortality till the fifteenth
or sixteenth day before the final continuous fall of the infantile mortality. In Sweden
the mortality shews a less marked and irregular recrudescence till the twelfth or
thirteenth day. In Australia the rapid fall continues till the end of the fourth or
fifth day, then continues at somewhat the same rate for twelve months. Thus for
the period of high infantile mortality Australia seems norma 1 for good infant hygiene,
Sweden more normal than Saxony, and Saxony abnormal, i.e., the infant hygiene is
probably not at all good. From this it is obvious that each large population will
probably have to be treated independently in regard even to the form of the curve
representing the earlier stages of the force of infantile mortality. This has already
been shewn, see pp. 389-392.
MORTALITY, 405
20. The Gompertz-Makeham-Lazaras theory of mortality. — ^In
1825, B. Gompertz,^ suggested that death was possibly " the consequence
of two generally coexisting causes," viz. : (i.) Chance, without previous
disposition to death or deterioration ; and (ii.) Deterioration, or lessened
abihty to withstand destructive agencies. Assuming that exhaustion
of the resisting power to disease proceeds in constant ratio for equal
increments of age, that is, that the force of mortality increases in geometrical
progression, he deduced his well-known formula, viz. : —
^'''^ ^^=-w: = ^^^
B and c being constant, independent of the age (x), and determinable
from the data of a mortahty table, Ix being the number living at the age
X, and dlx the change in l^ in the time {i.e., change of age) dx.
In January 1860, Makeham, having examined a number of mortahty
tables for the ages 20 to 80, found it was necessary to modify the Gom-
pertz formula. He shewed that, for the age-period mentioned, the so-
called " force of mortality," /x j;, as given in several mortality tables, could
be closely represented without changing the constants of formula (629)
by adding a constant A, viz., by an expression of the form : —
(629a) 11^ = A -{■ B<f.
in other words, the force of mortahty, assumed by Gompertz to be a
geometrical progression, should, according to Makeham, be represented
by a geometrical progression plus a constant.
^ Gompertz shewed that if the chance of disease were equal at all ages, and
if its effect were independent of age, then it would follow that the number of Uving
and dying, as the age increased in arithmetical progression, would decrease in
geometrical progression. But, if liability to death increased with age, the number
living would diminish faster than in geometrical progression. He observed that,
although the hypothesis was not an " unlikely supposition with respect to a great
part of life, the contrary appears to take place at certain periods " ; see his paper
" On the Nature of the Function expressing the Law of Human Mortality," read
before the Roy. Soc, Lend., 16th June, 1825, and appearing in the Phil. Trans., 1825,
pp. 513-585. He had given an earlier paper (June 29th, 1820) at the same society,
entitled "A Sketch of an Analysis and Notation applicable to the Estimates of the
Value of Life Contingencies." Phil. Trans., Pt. I., 1820, pp. 214-294. This obviously
led to the later ones ; see also a supplement to both read 20th June, 1861, and pub-
lished in the Phil. Trans., 1862, Vol. 152, pp. 511-559. Prof. De Morgan discussed
Gompertz's view ; see " On a Property of Mr. Gompertz's Law of Mortality," Journ.
Inst. Actuaries, Vol. VIII., July 1859, pp. 181-184 ; and also Phil. Mag., Nov. 1839.
To represent number living at age x. Prof. De Morgan used : —
l^ = eS+6«"' = dgi' where d = ek ,g — ef> ,q = eo
Later, viz., 1839, Ludwig Moser published in Berlin his " Die Gesetze der
Lebensdauer.
406 APPENDIX A.
Later he discovered that a further modification, viz., the introduction
of a term Cx, that is, an arithmetical progression, gave the formula a
wider extension. Thus his second modification was the expression : —
(6295) .jn^ = A + Gx + Be'.
The significance of expressions of this type is seen at once from (630)
hereinafter, that is : —
r B
(629c). . loge y = —jBc'dx =K - ^^— ^ ^ -^ or «/ = kg<^
according to Gompertz ; or
(629«i). .log« y= — ](A + B(f)dx= K— Ax— <f; or y=ks'g'"
log c
according to Makeham's first modification of Gompertz's formula ; and
(629e). . \ogey = —UA-\rCx+B(f)dx = K— Ax—^Cx^— (f;
01 y= ks" h"' g'""
according to Makeham's second modification.
In these K is merely an integration constant, and is equal to loge k,
and loge g = —B/loge c ; log^ s = — A ; log^ h= — JC.
More recently WUhelm Lazarus, of Hamburg, ^ and later Vitale
Laudi, ^ of Trieste, in order to embrace results for earlier ages, abandoned
the arithmetical progression represented by the term Cx, and introduced
in its place a second geometrical progression making the form of the
instantaneous rate of mortaMty.
(629/) /x^ ^ A + Gb'' + B(f.
C. H. Wickens has shewn that, for Austraha, infantile mortality
from birth to age 5 is well expressed by a formula of the type of Makeham's
second modification of the Gompertz formula ^ ; see also § 20 hereinafter.
An expression is general, however, if it cover the whole range of Ufe
with the one series of constants for any particular epoch : this none of the
formulss wiU do with the number of terms adopted. Before further
developing the matter we shall consider the nature of a constant popula-
tion, the death-rates of which are also constant.
1 See Uber Mortalitatsverhaltnisse xmd ihre Ursachen, Hambvirg, 1867.
Lazarus' paper was translated by T. B. Sprague, M.A. ; see Journ. Inst. Act., Vol.
XVIII., pp. 54-61 ; 212-223. T. S. Lambert published an article on Longevity, in
1869, New York.
^ In a publication, " Die Eeohnimgsgrundlagen der k.k. priv. Assiourazioni
General! in Triest," a very concise exposition is given of the biologic basis of the
fundamental formula under the title " Die biologische Begriindung der Ausgleichungs-
formel nach Lazarus," See § 2, pp. xxiv.-xxix.
' See " Investigations concerning a Law of Mortality," C. H. Wickens, A.I.A.,
Journ. Aust. Assoc. Adv. Sc. XIV., pp. 526-536.
MORTALITY. 407
21. Theory of an " actuarial population." — Consider an indefinitely
large group of persons, who bom at a given moment, are then subject to
death, the rate of which (governed solely by age) is characteristic of their
environment (and period). If this group be neither increased nor
diminished by emigration, and as age increases be lessened only by death,
the proportion of survivors at each age may be regarded as furnishing the
relative numbers of what may be called an acUtarial population, A popu-
lation so constituted plays an important part ia actuarial investigations
as to the probabihty of death according to age, and l^s also been called a
" constant population."
Let the ratio of such a population after the period x has elapsed, P^
say (the members now being all of age x) to the initial population, viz.,
that at age 0, P^ say, be denoted by y^ ; that is, let yx=Px/Po > then
initially «/,-or (y^) = 1 and I—?/,, will denote the ratio of the aggregate of
deaths up to the age x.
' Let [jL=<l>(x) denote the rate per unit of time*^ at which death occurs
at the " exact age" x ; then the number dying in a unit of time, whose
ages are between x and x + dx, is the number living between those age-
limits, multiphed by the rate of dying, that is, yfj. dx. ^ Thus if /x be re-
garded as positive
(630) — dy= yfidx ; or — = —0 (x) dx
y
By integration we obtain : —
(631) log 2/ = — /^ (a;) da; : or «/ = e"-^''' W<*^
Equations (630) and (631) are the bases of the theory of an " actuarial
population." The number of survivors at each age obviously depends on
the form of ^{x), and is completely determined when that function is
known. Various forms that have been adopted for ^{x), and their
integrals have already been given, formulae (629) to (629/).
The probabihty at birth, of hving to age x is y^, as given by (631)
above. The probabihty of dying before age x (vj,, say), is the arithmetical
complement of the probabihty of hving, viz., l—y^ ', that is : —
(632) v^ =l-2/^ = l_e
'S<t>{x)ix
Similarly the probabihty {p^) of persons of age x hving to age a;+l
and {q^) that of dying before that age, are respectively : —
(633) 'Px={yx+\)/yx ; a.rAqx= {yx — yx+ii/Vx = l—Px-
The average of the death-rates (M) of persons dying between ages Xi and
xz IS : —
1 rX^ 1 r^'-i
(634) M= \ u,dx= ) <l,(x)dx
^ ' Xz — Xi Jxr Xz — Xi JxJ^
^ Which may be a day, month, year, etc., but is usually a year
408 APPENDIX A.
When the range of ages is a unit (or 1 year) we shall denote this quantity
by the letter (m ) . The group-rale of mortality ( M ) for persons dying between
the ages x^ and x^, is the ratio of the total deaths between the ages in
question occurring in a unit of time, to the average population from which
the deaths are drawn ; that is : —
(635)..Jlf=— ^^ ^-
-ty" ^;h^C'-'* '■"■''
When the group-rate of mortahty is taken through a range of age, of one
year only, it is known actuarially as the central death-rate (m) of the year
in question. In this case X2, — X\ is unity, and disappears in the above
expression, and 2/2 = 2/1 + 1- Since through the greater part of life /i
does not change very rapidly, /Xa._,_j is approximately equal to rrix, the
group-range being really from exact age x to exact age a;-|-l,'and M for a
range of h years will — to a very rough approximation — be km.
If, for so small a range of age as one year, it be assumed that the
mean population is the mean of the populations at the beginning of, and end
of the year of age (which is sensibly correct for a considerable range of the
table^) the following relations hold between m, m, and /u, : —
(636). . . .m=cologei) = - log {l-q)=q+W+W+ia*+ ■ ■ ■
(637). . . .m = 2g/(2-g) = q^ (\-^)=q+W+W+iq^+ ■ ■ ■
(638).... ^+i = = g ± e
in which last expression e is usually very small, but is not readily sus-
ceptible of any general expression, and must be specially determined for
the very early and very late stages of hfe.
The instantanexyus rate of mortality /x, at any exact age, is, of course,
not immediately furnished by statistical data, but has to be deduced
therefrom. It is the value of M in equation (634) or oi M ia. equation
(635) when % and x% become identical.
22. The relation between the mortality cuive and the probability
of death. — ^The relation between ju ^ and q^ xasby be established as follows :
For any Umited range of ages, excepting during the first year of life, the
instantaneous mortahty may be put in the form A -)- B m^, the integral
of which is C + A;^ -|- Bm'^/loge m. If three successive values of [i for
equal changes of age are ijlq, [ii and fiz, then we shall have ^ : —
(640) m = ti±=Utj. ; m - 1 = ^-"^^^ + ^ q =„
/^i — Mo Ml —Mo
1 See Census Report, Vol. III., pp. 1215-6, for exsimple.
- 2 For greater precision, three values one place earlier, fi- 1, Md and /jlj^ can also
be taken, and the corresponding values of A', B', lit' and 11' computed, entering the
corresponding values of /i (the suffixes of which are unity less than those given).
Thus for the stretch /i„ to /t j the mean of the two results will — in general be more
accurate than either. Geometrically this is very simple ; Itl is the ratio of the
differences of the rates of mortality : so long as the diSerences are in constant ratio
the one value Itl applies.
MORTALITY. 409
If the successive values of /x are for ages k years apart, then the values
of A and B remain unchanged for values one year apart, but the value of
m for 1 year, nii say, is nii = m/ or ni/ = nio- This quantity is, of
course, always positive, being greater than unity for an ascending curve,
and less than unity for a descending curve. The use of the quantity ii,
enables log ra = log (1 + n) to be conveniently expanded in the series
n — Jn^ + Jtt*— etc., which is convenient when n is small.
Although the above expression for the curve is simple, yet when the
value of the integral is apphed to (635) in order to find the average popula-
tion, it leads to an arithmetically intractable expression. ^ Consequently
a direct general expression for p^, and g'^. is not readily obtained in terms of
IX. It is usual in actuarial computation to compute the Gompertz-
Makeham constants from the values of /^ and to find p and q from the
values of y (that is, from l]^ and lx+\). The relation can, however, be
obtained in quite another way. Put
(641) q =Y<i = y ' Ui^x + i^x+i) ;
then y is a correction factor to what is ordinarily an approximate value.
■It can be computed and entered in a double-entry table for a suitable
range of values of k = fix+i /H'xi ^^'^ * suitable range of values of fi.
Such a table would admit of q being readily and accurately found by
interpolations, and would simphfy the computations of life-tables.
Similarly tables could be constructed ra which the arguments were m^;,
and the ratios k ' = m^+i /w^c. This, however, wiU not be dealt with
further in this article.
In actuarial notation,^ l^, the relative number of persons living at
age X is so expressed that Zg is 100,000, 10,000, or some such large unit ;
1 Put K = e-« ; ct = e-A ; ;3 = e-B/log, m, then
yg-{cH-Ax+Bm''/log,m}^^ = K/o^^/S'"'' dx = KJ e-' dx
Let a = mt and pm^ — ^x^, consequently taking logarithms Itl'^ lege ;3 = t .
Differentiating nt'^ lege tit loge $.dx = dv = v loge m dx (by substitution).
r^ = m'^X = (m'^)* = (i;/loge /3)* ; consequently
/a^ /3""^ dx =/(»/loge pf 'e^-dv/vloge m = (ipg^ p)t log, m -^^" "'"^ '^'" '•
the value of the integral being e" [««-! — (« — 1) vt-2'+ (t — 1) (« — 2) i;«-S _ etc.]
which gives the required values, if it be remembered that when x = 0, 1, etc;
V = loge /3, m loge ft etc., respectively.
Neglecting the C term we have, however, by expansion,
/e-Ux =/[l - Ax + iA^x" - JASx»+ ■ • • • -(^ + Ate - iA^'te")""'^
-I- i (/32 - JA/Sax) 1'*'' + i/S^m''^ - etc.] dx-
which can be integrated term by term.
2 Ordinarily actuarial methods are based upon the algorithms of " finite-
differences." The connection between these and infinitesimal methods has been
dealt with in a paper entitled " On the relation between the theories of compound
interest and life contingencies," by J. M. Allen, F.I.A. ; see Joum. Inst. Act., Vol.
xli., pp. 305-337 ■ see also discussion pp. 337-348, and particularly that by D. C.
Praser.
410 APPENDIX A.
dg; denotes the number of persons dying between ages x and x -{- 1 : and
similarly : —
(642) p^= '-^^ ; and^, = ^W ^ 4
(643) wij. = j-^- ; andp.
d~ , 2 — m,
%
kW 2 + m^
Also, since p and q are arithmetical complements,
2wi» 1 din d I, , ,,
(644) ^^=2-^rV.= '^^=-v^ = -^^'''^'*^
23. Limitations of the Gompertz theory and of its developments. —
The conception put forward by Gompertz, and the modifications of that
conception by Makeham, have, as already stated, been again modified
by Laudi and Lazarus, who, to embrace results for earlier ages, replaced
Makeham's result by putting for the value of the number Uving : —
(645) y^= h= kH^s-b'\ ^
More recently C. H. Wickens has shewn that a similar type of expression
fits ages up to age five. *
These formulae, however, cannot be made to conform to the whole
range of facts, viz., from the earhest to the latest ages, as has already
been shewn. Although Mr. Wickens has shewn that, in a general way, a
curve of the Gompertz-Makeham type represents the facts for the first
few years of life, the formulae given do not conform to the details of the
first twelve months of life : in short, it is not consistent with the ".natal"
^ It is generally assumed that mx=iix+i approxiniately, the approximation
being quite as accurate as the data. Approximations of greater precision are given
in the Text Book of the Institute of Actuaries II., 1887 Edit., p. 25, and by later
writers.
*Laudi and Lazarus gave the value 1.291219 to H and 0.4 to 6, which give factors
that become sensibly unity when a; is 15. Thus the term for age 15 is 1.000000274,
and for age 20 is 1.00000000281. For age 1, however, the tenuis as high as 1.1076433.
' Mr. Wickens' formula gives for the h term (Iv') and for ages up to 5, the follow-
ing values for h, viz., 1.00056 for males, and 1.00037 for females, which are obviously
not in good agreement with the value 1.1076433 mentioned above for age 1.
Mr. Wickens, putting —
4 = - loge « = -^ (a - i7) ; B = - lege c logg g = ^^ °^^ ; and
C = — 2 loge h = 7/ M ; in which
a = — (log* + \ogh) ; p= — {c— l)logg ; and 7 = — 21og ft;
M denoting loge, i.e., 0.4342945, gives the result in the form : — ■
Ma: = 0.00816 - 0.00113a: + 0.21971»-i "^aa; . formates.
/i.'x = 0.00645 - 0.00074a; + 0.17199 »-i*«"!a;; for females.
the fit being excellent. See also "Assurance Magazine" (Joum. Inst. Act.), Vol. X.
pp. 283-5, 1862. (Letter dated 15th August, 1861). '
MOKTALITY. 411
or "infantile" forces of mortality, as is evident from Fig. 101, p. 319.
Formulae of that type can, of course, be made to represent the earlier
features of the curve of mortality, viz., the natal and infantile and earher
forces of mortality, that is, the Lazarus-Laudi scheme of addition can be
extended, the effect of the earlier terms disappearing when we please.
This would give a complete general formula of the type
(646) iM^= a + b(f -\- ^y" + BC +. ..+Sx'+ Tx^+ . .
It will be found hereinafter that a somewhat different conception already-
outlined in § 19 really accords with the facts, and it may be added
that Westergaard's opinion that the formulae are really empirical, is
substantially supported by the analysis.
24. Senile element in the force of mortality. — The senile element in
the force of mortahty may be assumed to operate from birth, with, of
course, increasing potency as age advances. Initially it clearly ought to
be zero, and thus it can be simply expressed by ^ : —
(647) fig =Sx\ or log jUg = log jSf + s log x.
Applying this gives for male and female senile death-rates, re-
spectively :■ —
• f Males u =0.0i» 06100a;=-9"i ; females =0.0i» 2826a;«-"8» ; or
(648) I _ _
( „ log ;li = 13.78533+5.9671 log x ; „ = 14.45117+6.6189 log x,
and they give a common value 0.99844 for the annual rate of mortality
at age 111.40 years. These formulas give the results in columns (ii.)
and (vii.) of Table CXLVIII., p. 413. To find log. 8 we have :—
(649) . . log ;Sf = log /ii — log xi [(log H2 —log /ii)/(log a;2-log x^)],
1 This was decided upon empirically after examining the applicability of other
forms. For example, if senile impairment accumulated at a constant rate, it would be
expressed by /Se"*. If the accumulated effect accelerated with age, a linear accelera-
tion would require Se"^'. Thus the index 2 would be too great if the acceleration
T*ere greater than linear, and too small if it were more rapid than linear. Se<^'"
was examined: this gave ^/is / ifii = e'ix'^—x'J and d/j. / dx = jStrse"'*^ a;«-l.
The value of a- may be readily found by means of a table of values of the x term (in
brackets) for various values of s. Taking S = .0021852, o- =0.00064, and « = 2
gave, for " persons "
Age 0. 7.5 12.5 17.5". . . 52.5 57.5 62.5 67.5 72.5 77.5 82.5
Calc. Ii .0022.0023.0024.0027 .0128.0181.0266.0404.0632 .102 .170
Data M — — — — .0129.0182.0268.0345.0631 .103 .162
The result is obviously too high for early ages, when s was made unity. Formulae
of the type Sx"'"' are inappropriate between and 1, because x'" has a minimvim at
X = .03678794,1.6., 1/e, when its value is 0.6922007. S (e<^a^ - 1) would, however,
probably be satisfactory from age 1 onwards.
412
APPENDIX A.
s being the quantity in the square brackets. It would probably be pre-
ferable to adopt a mortality curve for the older ages, passing among the
points given by the relatively meagre and
uncertain data, than to follow them closely,
since the general indication is probably
the more reliable.
1-0
O-OL
x 4
1
1
1
1
i
05
1
1
1
1
M
1
1
^
t
1-7
M
2
f^oca.
Fig. 102.
The curves shew the senile
element in mortality.
The fit of the formula to the data
is shewn on Fig. 102, on which the lines
represent the logarithmic homologues of the
senile curves, the small circles denoting
the data for males and the small squares
those for females.
25. The force of mortality in earlier
childhood. — The rates of mortaUty from age 1.5 to 7.5 were as follow
from 1907-1915 in AustraUa: —
Ages (years) 0.5 1.5 2.5 3.5 4.5
Males .. .07608 .01550 .00641 .00408 .00301
Coefficient .204 .414 .637 .738
Females .06156 .01380 .00574 .00381 .00300
Coefficient .224 .376 .652 .791
5.5
6.5
.731
.667
7.5
.00220
.00200
No elementary function will satisfactorily represent these results
with precision. 1 The following results, however — empirically found —
reproduce the data almost exactly, and shew the fluctuating character of
the value of e"* in the expression e"**, which would represent the curve: —
TABLE CXLVn.— Bates of Mortality in Childhood. Australia, 1907-15.
Exact
Males.
Females
Exact
Males.
Females
Exact
Males.
Females
Age.
Age.
Age.
0.5
.07608
.06156
4.5
.00304
.00300
8.5
.00208
.00185
.408
.03104
.428
.938
.935
.970
.970
1.0
.02635
5.0
.00285
.00281
9.0
.00202
.00179
.500
.01552J
.524
.940
.935
.970
.970
1.5
.01381
5.5
.00268
.00262
9.5
.00196
.00174
.596
.610
.946
.935
.970
.970
2.0
.00925
.00842
6.0
.00253
.00245
10.0
.00190
.00169
.692
.681
.952
.935
.970
.970
2.5
.00640
.00574
6.5
.00240
.00229
10.5
.00184
.00164
.762
.772
.962
.935
.970
.970
3.0
.00488
.00443
7.0
.00229
.00214
11.0
.00179
.00159
.832
.862
.970
.935
.970
.970
3.5
.00406
.00382
7.5
.00221
.00201
11.5
.00173
.00164
.852
.878
.970
.955
1.000
1.000
4.0
.00346
.879 1
.00335
.895
8.0
.00214
.970
.00191
.965
12.0
.00173
.00154
HOTE. — The small figuies aie the ratios which multiplied Into the values immediately ahove
them give those immediately below.
^We have seen in § 11 that, for the greater part of first year of life, /t =
/'o [1 +f (^)] A> ^^^ *^** toward the end of the first year f {x) ia large — about 4 — -
compared with unity. Also it is evident from curve A', Fig. 101, that it is approxi-
mately a constant at about 320 days to perhaps 400 days, thus /t, = S/i^ /x, and would
appear to have become constant at least for some range of x. Such, however, is
not the case. If it were we should have x/ix =Jc [1 + f (x)] a, constant. We obtain,
however, the following results : —
.03808 .02325 .01602 .01428 .01354 — — .01650
.03078 .02070 .01435 .01333 .01350 — — .01500
which shew that 1+J (x) is not expressible by any simple relation. The results for
males for 24, 3^, and 4 J years can be expressed by fix = Moe""**. and for females this
expression is also fairly approximate.
MORTALITY.
413
26. Genesic and Gestate elements in mortality.— If the infantile
and juvenile, and the senile elements of the mortality be subtracted from
the totals, the residuals will constitute the
genesic element in the case of males, and
the gestate elements in the case of
females. The rate of diminution seemed
to be constantly 0.97 per half-year (see
Table CXLVII., p. 412) from age 8.5 to
11.5 for both sexes. This is equivalent to
0.73752 for 5 years, and the adoption of
this gives the results in columns (iii.) and
(viii.) of Table CXLVIII. This may be
regarded as the measure of degradation
of the power of adjustment to environ-
ment. The residuals smoothed as shewn
on Fig. 103, are given in columns (iv.) and
(ix.). On this figure the heavy curve, M, denotes results for males, and the
light one, P, results for females. The computed mortaHty curves and
those given by the crude data, are shewn in columns (v.) and (vi.) for
males, and columns (x. ) and (xi. ) for females. The agreement in general is
fair up to 62.5 years. Afterwards the results diverge somewhat. It has,
however, to be remembered that these divergencies are not really large,
and do not make large differences as between the computed and actual
numbers of deaths.
Fig. 103.
The curves shew tlie genesic (M)
and gestate (F) elements in mor-
tality.
TABLE CXLVin
— ^Illustrating the component-elements of the Force of Mortality.
AnstraUa, 1911.
'*"-
Male Eates ol Mortality, x 100,000
Female Rates of Mortality, x 100,000.
SenUe
Element.
Juvenile
Element.
(Jenesic
Total.
Senile
Element.
Juvenile
Element.
Gestate
Sm'thd.
Total.
Sm'thd.
(Com- Ob-
(Com- 1 Ob-
puted.) served.
puted.) 1 serve d.
(i.)
(ii.)
(iii.)
(iv.)
(V.)
(vi.)
(vii.)
(viii.) '
(ix.)
(X.)
(xi.)
2.5 ..
640
640
641
574
574
574
7.5
221
221
220
201
201
200
12.5 . .
163
10
173
173
149
4
153
153
17.5 ..
2
120
133
255
255
110
120
230
220
22.5 . .
7
89
252
348
364
3
81
268
352
341
27.5 . .
24
65
343
432
432
10
60
366
436
433
S2.5 . .
64
48
413
525
508
29
44
435
508
475
37.5 . .
151
36
467
654
666
74
33
467
574
586
42.5 . .
318
26
497
841
841
170
24
462
656
641
47.5 . .
617
19
484
1,120
1,122
354
18
426
798
796
52.5 . .
1,121
14
387
1,522
1,522
686
13
356
1,055
1,057
57.5 . .
1,930
11
220
2,161
2,161
1,254
10
215
1,479
1,479
62.5 . .
3,173
8
3
3,184
3,179
2,177
7
3
2,187
2,181
87.5 ..
5,022
6
5,028
4,693
3,623
5
3,628
2,201
72.5 . .
7,693
4
7,697
7,034
5,814
4
5,818
5,580
77.5 . .
11,455
3
11,458
11,136
9,041
3
9,044
9,379
82.5 . .
16,635
2
16,637
17,387
13,674
2
13,676
15,026
87.5 . .
23,632
a
23,634
27,557
20,188
2
20,190
22,492
92.5 . .
33,926
1
33,927
31,673
29,161
1
29,162
30,007
97.5
45,071
1
45,072
40,475
41,314
1
41,315
39,873
02.5 . .
60,744
1
60,745
1.23393
57,531
1
57,532
1.16876
27. Noim of mortality-rates.- — A study of mortahty rates for the
same country at different times, and for various countries, shews that the
real nature of the mortality curve will probably be revealed only by
414 APPENDIX A.
obtaining a norm of mortality rates on a wide basis. Such a norm would
necessitate a compilation for a large series of populations, of the foUowiog
data, viz. : —
(a) Infantile deaths according to hours for the first week of life ; then
according to days for the first month of life : and according
to weeks for the balance of the year.
(b) Deaths in childhood according to months for the second year ;
and according to quarters for the third year and afterwards ;
(c) annually — or better semi-annually — ^tiU 15.
Afterwards the annual number of deaths.
The " number Uving" would preferably be deduced for the first 12
months (making corrections, however, for migration), by subtracting
the deaths from the recorded births. Afterwards, or at any rate after the
second year, the census data would in most cases be preferable to use.
The combination of a large number of results, viz., all deaths in any
age-group, and the sum of the populations in the same age-group from
which such deaths were drawn, would probably disclose the true laws of the
incidence of death. Only in large bodies of figures can it be hoped that
the minor chance influences will counteract one another.
28. Number of deaths from particular causes. — ^The actual numbers
of deaths according to sex and age, which occurred ki Austraha during the
9 years 1907-1915 from various causes, were as shewn in the following
table, viz.. No. CLXIX., their relative frequency from all causes together,
but retaining the age-groups, that is their ratios to the totals for the same
sex, being shewn on the last two fines, see pp. 416-417 : —
29. Relative frequency of deaths from particular diseases according
to age and sex. — ^If for each sex and for each age-group in that sex, the
number of deaths from each cause be divided by the total deaths from all
causes, the quotients are the relative positions of the disease as rewards
their contribution to the totahty of deaths. Thus they measure the
gravity of the incidence of any disease in question. This has been done
and the' results are shewn in Table CL., on pp. 418-419,
MORTALITY. 415
30. Death-rates from particular diseases according to age and sex. —
It has already been pointed out that the incidence of death according to
sex, has diverse characters as regards its relation to age ; see § 12, p. 393
hereinbefore. If the ratio of the number of deaths which occur in one
year from any disease, in any age-group, and for either sex, to the average
number of persons of the same sex in the age-group be found, this ratio
will be the annual death-rate for the particular disease in question. ^
Thus the ratios are exactly analogous to the values with accents in (628)
of § 19, p. 402 ; that is, they are the individual components of the death-
rate for the same sex and age-group. They represent the ratio of the
number of persons of a particular age-group who will (probably) die
of the particular disease in question during the one year. These ratios,
multiphed by 1,000,000, are shewn in Table CLI. and are thus the (partial)
death-rates for each disease and for the two sexes, see pp. 420-421.
The forms of the rate-of-mortahty curves for each disease are shewn
on Fig. 104, the heavy hues denoting the curves for males and the lighter
line those for females. They illustrate the marked differences in the
incidence of death as between the sexes for the same disease, and accord-
ing to age as between different diseases.
31. Rates of mortality during the first twelve months of life.— The
incidence of death during the first twelve months of life is so varied that
the means for the successive years 0, 1, 2, 3, etc., cannot be regarded as
giving a satisfactory indication in regard thereto. Even in the first
month of hfe, the frequency of deaths greatly varies for the successive
weeks therein, so that a month is clearly too large a unit to adopt for
rigorous results. Consequently, a tabulation for the first four weeks is
necessary as well as for each of the succeeding eleven months. The
population on which the ratios were based was 399,823 male births, and
38,027 females, which was reduced by the deaths themselves and increased
by the net immigration of the same sex. ^ ■
1 The sum of the mean populations for each sex and for the 9 years under review
were distributed according to the Census of 1911, the middle year. This gave the
divisors by means of which the rates were computed.
• The immigration is by no means wholly negligible for accurate results : thus
it was estimated to be — for each sex — 267 for the eleventh to the twelfth month,
while the deaths were : males, 933 ; females, 768. Its neglect does not, however,
obviously make a large error, since the deaths are drawn mainly from those born in
the country under consideration,
416
APPENDIX A.
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lao
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l-tiH
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ss
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422
APPENDIX A.
TABLE GUI.— Shewing the Numbers Dying per Month during each of the First Four Weeks o! life per 100,000 of the same Sex
and Age, and during each of the First 12 Months of Life per 1,000,000 of the same Sex and Age. Australia, 1910-1915,
Aqe at Death.
Aqe at Death.
TOTAL
1
2
3
' 1
2 3
4 5
6
7
8
9
10
11
week
weeks
weeks
mth.
mtiis.
mths.
mths, mths.
mths.
mths.
mths.
mths.
mths.
mths.
No.
Maies— Cause.
Under
and
and
and
Under , and
and
and
and 1 and
and
and
and
and
and
and
Under
1
under
under, under'
1 1 under
under
under
under
under
under
under
under
under
under
under
1
week.
2
3
1
Mth. 1 2
3
4
5
6
7
8
9
10
11
12
Year.
weeks
^veeks
mth.
mths.
mths.
mtlis.
mths.
mths.
mths.
mths.
mths.
mths.
mths.
mths.
8
Whooping Cough
2
3
13
21
108
304
254
205
148
117
117
123
78
65
100
73
1,702
28
Pulmonary Tuberculosis . .
5
8
8
8
16
8
11
16
5
3
5
93
29
Acute Miliary ,,
Tubercular MeningitiB
3
5
5
3
8
3
5
32
30
"l
"o
"1
"5
5
31
24
18
21
35
24
35
35
35
38
305
31
Abdominal Tuberculosis
1
3
10
10
32
21
18
16
11
11
11
13
32
188
34
Tuberculosis of Other Organs
5
3
3
3
8
3
3
26
35
Disseminated Tuberculosis
3
11
3
3
3
5
3
29
37
Syphilis
■33
'17
'27
'24
252
2i6
173
74
48
48
24
19
30
8
13
919
61
Meningitis
36
17
12
10
187
117
141
184
167
149
205
190
206
196
143
197
2,084
71
Convulsions
405
187
80
40
1,685
203
160
162
98
88
115
86
75
107
70
76
2,985
89
Acute Bronchitis
24
64
55
62
521
' 542
250
169
148
74
88
77
67
48
81
49
2,143
91
Broncho-Fnenmonia
17
45
40
41
362
466
335
240
198
186
235
174
212
161
191
235
3,011
92
Pneumonia
43
37
29
31
347
276
165
162
172
117
139
130
145
172
124
97
2,058
104
Diarrhoea and Enteritis
64
151
186
172
1,446
1,826
2,324
2,505
2,427
2,331
2,125
1,843
1,663
1,423
1,183
1,157
22,303
109
Hernia, Intestinal Obstruction
37
17
8
8
169
36
37
63
74
101
99
86
62
62
24
22
839
150
Malformations
776
195
91
52
2,632
334
220
179
106
74
69
94
43
43
22
49
3,983
151
Congenital Debility, Icterus
and Sclerema
7,193
990
649
355
21,681
2,046
1,122
724
511
430
357
227
185
124
89
130
28,410
152
Other Diseases peculiar to
early infancy
1,713
184
79
24
4,690
99
52
5,003
153
Lack of Care
29
2
2
80
21
10
^22
114
Other Causes
480
232
192
109
2,432
750
464
424
384
363
241
354
358
337
340
6,854
10,852
2,143
1,461
952
36,600
7,281
5,765
0,171
4,534
4,100
3,998
3,347
3,193
2,821
2,428
2,519
Population of males at the be- S SSS2 ' SSSS^gSSSSS
ginning of each period allow- *1.®. ^'^'^ <oioa>o«ooi*i©i-*t»o
ing for migration (on which g SSSg SgSRSSffKSgg
the results are based). co««o9ot o3oo«cop3«oomcoimoo
Total Deaths (Males) on which gSISS S8§SSiS§S3S?S8
results are based ....<» o>. ", ". ^ « » t- lO S ^1 S o § § S
en
Aqe at Death.
Aqe at Death.
TOTAL
1
2
3
1
2
3 4 5
6
7
8
9
10
11
week
weeks
weeks
mth.
mths.
mths.
mths, 1 mths.
mths.
mths.
mths.
mths.
mths.
mths.
No.
Females— Cause.
Under
and
and
and
Under
and
and
and
and ' and
and
and
and
and
and
and
Under
I
under
under
under
1
under
under
under
under under
under
under
under
under
under
under
1
week.
2
3
1
Mth.
2
3
4
5 6
7
8
9
10
11
12
Year.
weeks
weeks
mth.
mths.
mths.
mths.
mths, , mths.
mths.
mths.
mths.
mths.
mths.
mths.
8
Whooping Cough
1
2
20
24
132
398
310
202
162
85
108
122
152
122
106
106
2,008
28
Pulmonary Tuberculosis . .
1
2
8
3
3
8
14
8
14
8
11
8
85
29
Acute Miliary
3
8
3
8
22
30
Tubercular Meningitis
14
22
22
27
19
22
25
33
42
39
264
31
Abdominal Tuberculosis
"1
"3
5
16
14
25
16
19
3
17
8
22
3
151
34
Tuberculosis of Other Organs
1
3
3
6
35
Disseminated - Tuberculosis
5
I
3
3
11
37
Syphilis
'22
12
'20
'u
i76
135
65
107
57 1 41
58
28
17
11
25
17
739
61
Meningitis
40
27
14
7
222
87
102
128
140
140
135
169
150
172
139
137
1,715
71
Convulsions
286
116
43
29
1,151
146
117
87
82
93
94
75
69
103
70
67
2,157
1,756
2,357
1,657
18,456
89
Acute Bronchitis
15
40
57
35
379
357
250
158
132
80
72
72
69
81
50
47
91
Broncho-Pneiunonia
14
28
33
32
283
372
277
191
153
154
163
122
164
194
153
132
92
Pneumonia
25
22
26
17
228
173
106
137
118
146
149
136
103
133
136
95
104
Diarrhoea and Enteritis . .
37
96
106
95
835
1,432
1,876
2,144
2,197
1,937
1,666
1,544
1,328
1,368
1,159
992
109
Hernia, Intestinal Obstruction
18
13
9
99
32
11
27
57
77
77
53
44
25
19
11
530
150
Malformations
597
122
62
40
2,003
203
128
104
99
92
80
47
36
61
56
59
2,970
151
Congenital Debility, Icterus
and Sclerema
5,579
809
518
289
17,375
1,418
878
710
419
352
265
243
155
142
128
114
22,226
152
Other Diseases peculiar to
early infancy
1,329
147
64
24
3,750
73
38
i
3,863
79
153
Lack of Care
30
74
5
Other Causes
343
152
130
58
1,670
649
343
355
293 223
254
252
283
306
328
306
5,262
8,337
1,586
1,095
675
28,391 5,483 1 4,536 4,394 1 3,967 j 3,485
3,173
2,902
2,620
1 2,773 1 2,436
2,141
Population of fematesat the be-
^SS^'S'SSSKSS""^
_
9
ginning of each period allow-
» S § S 3 S g 3 S S S fe S S S 2
ing for migration (on which § S S e: £ g g" S 2 S" -f '-' o- °>" oo 00"
the results are based). SSoSSS SggSSgSSgg^g
TotaIDeaths(Females)onwhich gSgS SSS'-'-°<=>'aaoiaao'a
results are based .. .. § S § K o » i 3 S S S 1 1 g g S
*"*
eq
rH
r-t
iH
ri
r-l
iH
s
MORTALITY.
423
{(016-
152
Wosf— t— I Wosl— J—
Hz:35i Irf— 150.
% 3 61 9 lit .
^j
004-
t
T
-71
00
1
00
1
1
8S
61
■^
=
3 6 9 li
;00^'4
-t
92
— L
r
-jY
0004-
o'
f ^
/
"v.
, .
■^
1
a?
N
^•^
V
-
—
m
i 3-6- 9 izmths.
.QO''^'*'
1 1
nn\
■iv"!
L
s^
-*;
=».
■0004
Fig. 104.
The 13 figures ruled into rectangles are death-rates for the first 12 months of life,
the rates being shewn by the figures on one of the horizontal lines. The 38 figures ruled
into smaller squares shew the death-rates for all ages of the diseases indicated by the
numbers. For the index to the above curves see next page.
424
APPENDIX A.
Index to Coives in Figure 104.
Death-bates foe all Ages.
+1. Typhoid Fever
6. Measles.
7. Whooping Cough.
8. Diphtheria and Croup.
9. Influenza.
12. Other Epidemic Diseases.
13. Tuberculosis of the I.ungs.
14. Tuberculous Meningitis.
15. Other forms of Tuberculosis.
16. Cancer and other Malignant
Tumours.
16a. Other General Diseases.
17. Simple Meningitis.
18. Cerebral Haemorrhage and
Softening.
18o. Other Diseases of the
Nervous System.
19. Organic Diseases of the
Heart.
190.
the
the
Other Diseases of
Circulatory System.
20. Acute Bronchitis.
21. Chronic Bronchitis.
22. Pneumonia.
23. Other Diseases of
Kespiiatory System.
24. Diseases of the Stomach.
25. Diarrhoea and Enteritis
(all ages).
26. Appendicitis and TypUitis.
27. Hernia, Intestinal Obstruc-
tion.
28. Cirrhosis of Liver.
28a. Other Diseases of the
Digestive System.
29. Acute Nephritis and
Brigfat's Disease.
30. Non-cancerous Tumours of
Female Genital Organs.
30o. Other Diseases of the
Genito-urinary System.
31. Puerperal Septicsemia.
32. Other Accidents of Preg-
nancy and Labour.
32ff. Diseases of the Skin and
Cellular Tissue.
326. Diseases of the Organs of
Locomotion.
33. Congenital Debility and
Malformations.
34. SenUe Debility.
35. Violent Death (Suicide ex-
cepted).
36. Suicide.
38. Unknown or Hl-de fined
Diseases.
fli
Broncho-Pneumonia.
150.
Malformations.
92.
Pneumonia.
151.
Congenital Debility.
104
Diarrhoea and Enteritis.
152.
Other Diseases peculiar to
109.
Hernia and Intestinal Ob-
Early Infancy.
struction.
A.
Other Causes.
Death-bates fob First Yeae op Life.
•8. Whooping Cough.
37. Syphilis.
61. Meningitis.
71. Convulsions.
89. Acute Bronchitis.
* These numbers, on Fig. 104, are identical with those of the "Detailed Nomenclatures
of Diseases" of the International Ojmmission, Session July 1909, at Paris.
t These numbers, on Fig. 104, are identical with those in Table CXLIX. to CLI.,
and where not marked "a" are those of the "Abridged Nomenclature" of diseases of
1909, where " a " or " b" added it denotes that the balance for the class in question is
included.
The form of the mortality curves during the first year are given on the
upper part of Fig. 104 ; see the Index thereto.
32 . Annual fluctuation of death-rates. — The frequency of death from
particular causes, and therefore generally, is afiEected by the season of the
year, and though in the aggregate of deaths from all causes the seasonal
effect is somewhat masked, it is not whoUy obhterated. To ascertain
rigorously the character of the annual periodicity, either generally or
from a particular " cause," of death it is necessary to obtain the rates for
smaU units of time, say equalised months ; thus the rates 8i, 82, ... .
812 must be obtained : these are sensibly iadependent of the fluctuations
in the deaths and population during the month. Inasmuch, however,
as deaths occur very rapidly in the first few days of hfe, any periodicity
in birth-rate involves the death-rate ; that is to say, the constitution of
the population is not quite homogeneous, and a correction is — ^theoretic-
ally — ^necessary. The correction, however, is so small that it may be
neglected. These last observations apply, mutatis mutandis, also to
deaths from certain particular causes. The annual fluctuations of birth-
rate, and the mode of solving have been indicated at length in Part XI.,
§§ 14-19, pp. 166-174. General factors for reducing the values given for
calendar months to the values for equahsed months must be so apphed as
to have regard to the average values at the beginning and end of the
months.
Table CLIII. depends upon a total of 252,443 deaths of males ^, and
185,367 deaths of females occurring in an aggregate population of
> For example there were 3529 deaths from typhoid in the 9 years, of which
473 occurred in the month of January. These, when corrected, for the growth of
population during the year, and altered so as to give the result for the exact twelfth
of the mean length of the year, gave the basis for the calculation of the results in
the table.
MORTALITY.
425
over 21,000,000 males and nearly 20,000,000 females. The numbers
given in the table correspond to a population of 10 millions in each case.
In Table CLIV. the proportions of deaths occurring in months of
equal length, when the population is constantly the same, are given.
Algebraically if b and e be the equalising corrections at the beginning
and end of the month to D, the number of deaths, and P be the sum of the
populations of the corresponding month for the whole period under review,
the results in Tables CLIII. and CLIV. are respectively : —
(650).
.8 = {D+b+e)/P ■ (651) p = 128 / US.
TABLE CLIII. — Shewing Average Number of Deaths due to Various Causes, per 10,000,000 Males,
and per 10,000,000 Females respectively of all Ages during each Equalised Month of the Year.
Based upon 9 Years' Experience (1907-1915) in Australia.
Cause op death.
Sex
Jan.
Feb.
Mar.
April.
May.
June.
July.
Aug.
Sept.
Oct.
Nov.
Dec.
Year.
Typhoid Fever
M
221
231
247
223
167
108
63
54
60
51
92
164
1,671
F
12S
152
156
118
97
80
41
26
24
25
50
88
986
Whooping Cough
M
77
64
43
45
61
63
66
79
85
74
77
77
811
I''
IOC
83
71
63
64
63
92
94
102
102
94
89
1,017
Diphtheria and Croup
M
81
76
99
123
149
153
142
109
102
85
83
89
1,291
i'
95
88
86
147
151
165
145
128
116
90
107
83
1,401
Influenza
M
39
27
29
38
54
69
99
163
188
140
93
49
988
H'
48
27
28
36
45
56
94
160
191
146
97
59
987
Tuberculosis
M
613
581
585
58C
646
641
713
701
692
669
640
590
7,651
If
527
484
489
514
511
512
533
590
558
585
496
516
6,315
Cancer
M
637
659
604
638
603
594
571
595
619
643
622
658
7,443
i!'
613
628
613
615
623
593
578
587
588
610
615
605
7,268
Diabetes
M
58
56
62
65
66
86
78
85
73
73
65
68
835
It
74
74
68
80
82
94
102
98
109
99
90
93
1,063
Organic Diseases of the . .
M
855
784
802
832
903
995
1,052
1,070
994
925
884
794
10,890
Heart
F
725
613
650
667
697
834
950
891
780
744
629
670
8,850
Diseases ol the Eespiratory
M
757
646
743
844
1,000
1,250
1,500
1,594
1,519
1,197
1,042
830
12,922
System
\f
519
472
471
581
723
895
1,083
1,217
1,088
895
726
609
9,279
Diarrhbea and Enteritis . .
M
1,021
941
866
764
503
265
203
166
185
338
782
1,069
7,103
V
894
820
787
678
457
264
164
127
137
309
644
895
6,176
Infancy
M
663
697
695
703
686
719
734
656
683
614
666
680
8,196
F
58C
543
571
608
562
579
616
561
504
528
521
532
6,705
Old Age
M
692
664
629
671
754
857
905
873
836
733
722
732
9,068
M
629
567
548
566
631
.697
748
726
688
628
581
570
7,579
Total all Causes
10,406
9,681
9,469
9,633
9,604
9,881
10,411
10,309
10,215
9,570
9,984
10,146
109309
i'
8,152
7,667
7,391
7,724
7,702
7,897
8,279
8,411
7,895
7,697
7,802
7,967
94,584
TABLE CLIV. — Shewing for each Equalised Month the Average Relative Frequency of Death due to
Various Causes, the Population being Constant throughout the Year. Based upon 9 Years'
Experience (1907-1915). AustraUa.
Cause of De.ath,
Sex
Jan.
Feb.
Mar.
AprU.
May.
June.
July.
Aug.
Sept.
Oct.
Nov.
Dec.
Year.
Typhoid Fever
M
1.589
1.657
1.773
1.599
1.199
.779
.449
.391
.357
.367
.663
1.179
12.000
H'
1.567
1.844
1.905
1.436
1.178
.979
.494
.320
.288
.304
.612
1.071
12.000
Whooping Cough . .
M
1.139
.942
.630
.664
.910
.930
.975
1.176
1.258
1.095
1.138
1.144
12.000
1''
1,185
.978
.835
.742
.761
.744
1.082
1.107
1.205
1.206
1.105
1.052
12.000
Diphtheria and Croup
M
.752
.706
.921
1.141
1.381
1.426
1.316
1.015
.948
.790
.776
.829
12.000
F
.81C
.752
.739
1.258
1.295
1.414
1.241
1.098
.994
.774
.914
.711
12.000
Influenza
M
.468
.332
.349
.463
.657
.840
1.199
1.982
2.281
1.707
1.134
.593
12.000
it
.58C
.332
.345
.434
.550
.676
1.140
1.942
2 325
1.779
1.182
.714
12.000
Tuberculosis
M
.961
.912
.917
.910
1.012
1.006
1.118
1.100
1.086
1.049
1.003
.926
12.000
F
1.001
.919
.929
.976
.971
.973
1.013
1.121
1.061
1.112
.943
.981
12.000
Cancer
M
1.027
1.062
.975
1.029
.972
.958
.921
.959
.998
1.036
1.164
1.061
12.000
F
1.011
1.037
1.012
1.015
1.029
.979
.954
.969
.971
1.007
1.016
.998
12.000
Diabetes
M
.839
.811
.888
.927
.952
1.235
1123
1.224
' 1.047
1.041
.938
.975
12.000
F
.834
.837
.767
.901
.929
1.056
1.148
1.102
1.232
1.124
1.012
1.052
12.000
Organic Diseases of the
M
.942
.864
.884
.917
.995
1.096
1.159
1.179
1.095
1.O20
.974
.875
12.000
Heart
F
.983
.831
.882
.904
.945
1.130
1.288
1.208
1.058
1.008
.853
.909
12.000
Diseases of the Eespiratory
M
.703
.599
.690
.784
.929
1.161
1.392
1.480
1.411
1.112
.968
.771
12.000
System
F
.671
.611
.610
.752
.935
1.157
1.400
1.573
1.407
1.157
.939
.788
12.000
Diarrhoea and Enteritis
M
1.725
1.590
1.463
1.290
.850
.448
.344
.281
.312
.570
1.321
1.805
12.000
F
1.737
1.594
1.530
1.317
.889
.512
.319
.246
.266
.600
1.252
1.739
12.000
Infancy
M
.971
1.020
1.017
1.030
1.004
1,053
1.075
.961
.999
.899
.975
.996
12.000
F
1.038
.971
1.002
1.089
1.005
1.037
1.102
1.003
.902
.945
.933
.951
12.000
Old Age
M
.916
.878
.833
.888
.997
1.134
1.197
1.155
1.106
.970
.955
.969
12.000
F
M
.996
.897
.868
.896
.999
1.103
1.184
1.150
1.089
.995
.920
.903
12.000
Total all Causes . .
1.047
.974
.952
.969
.966
.994
1.047
1.037
1.027
.963
1.004
1.020
12.000
F
1.034
.973
.938
.980
.977
1.002
1.050
1.067
1.002
.977
.990
1.011
12.000
426
APPENDIX A.
Fig. 105.
The distances from the centres of the circles shew the average ratios of the death-
rate per month to the average rate for the entire year, the ratios for males being
denoted by firm lines, and those for females by dotted lines, the succession of months
being clockwise. In the case of absence of fluctuation the sector-boundaries would
all be on the circle marked " 1," e.g., " Cancer." In the case of " Influenza" it will
be seen that the September rate is more than double the average for the year.
33. Studies of particular causes of death : voluntary death. — ^Although
the study of particular causes of death might appear not to belong
to the general theory of population, it is really an essential. For example,
if diseases, the incidence of which is characteristic of earlier life, be com-
batted, the consequence will be an increase in deaths from those which
MORTALITY.
427
characterise later years {e.g., tuberculosis and cancer). Again statistics
of voluntary death or suicide, are of special importance, inasmuch as
they disclose the regularity of human conduct even in matters which
might be thought to be peculiarly under individual control, and be
imagined to lie outside regular law. But suicide follows well-defined laws,
and even as regards the mode of death the regularity is remarkable, as
the following table shews : —
TABLE CLV.—
Mode
Of
Voluntary
Death. Australia 1907-1
5.
J
Number of Suicides.
*
o
EH
Range.
Mode of Death.
1907.
1908.
1909.
1910.
1911.
1912.
1913.
1914.
1915.
1 •
1
r
Poison
Asphyxia
Hanging and Stiangulation
■ Drowning
Firearms
Cutting Instruments
Precipitation from Height
Crushing
Other
57
2
71
37
129
61
6
3
19
88
1
68
31
146
54
4
6
15
70
2
67
24
138
74
7
5
11
79
72
42
134
79
3
8
15
93
2
69
43
133
65
2
6
33
128
4
79
34
168
76
8
17
127
2
79
25
163
88
6
10
16
121
2
72
30
201
76
4
2
26
105
84
38
196
89
4
8
13
868
15
661
304
1,408
662
36
56
165
.2079
.0036
.1583
.0728
.3373
.1586
.0086
.0134
.0395
57
67
24
129
54
2
11
96.4
1.7
73.4
33.8
156.4
73.6
4.0
6.2
18.3
128
4
84
38
201
89
7
10
33
92.5
2.0
75.5
31.0
165.0
71.6
3.5
6.0
22.0
Total, Males . .
385
413
398
432
446
514
516
534
537
4,175
1.0000
385
464
537
461
^Poison
Asphyxia
Hanging and Strangulation
Drowning
Firearms
Cutting Instruments
Precipitation from Height
Crushing
Other
32
12
19
3
5
1
2
2
35
15
14
7
6
2
2
3
54
9
19
6
5
1
3
34
10
19
6
13
2
52
1
10
13
9
9
2
2
70
12
11
10
8
1
6
76
1
22
14
9
4
2
1
2
61
15
17
4
3
4
2
3
64
1
18
21
5
6
3
2
2
478
3
123
147
59
59
14
11
25
.5201
.0033
.1338
.1600
.0642
.0642
.0152
.0120
.0272
32
9
11
3
3
2
53.1
0.3
13.7
16.3
6.6
6.6
1.6
1.2
2.8
76
1
22
21
10
13
4
2
6
54.0
0.5
15.5
16.0
6.5
8.0
2.0
2.0
4.0
Total, Females
76
84
97
84
98
118
131
109
122
919
1.0000
76
102
131
103
Ratio of Females to Males
Ratio of Males to Females
.197
5.07
.203
4.92
.244
4.10
.194
5.16
.220
4.56
.230
4.36
.254
3.94
.204
4.90
.227
4.40
.220
4.54
.194
3.94
.219
4.60
.254
5.16
2.37
4.55
* It is worthy of note that the mean of the highest and lowest number of suicides in any year is sensibly equal to
the arithmetic mean. The male population increased about 18.40 per cent, on the period covered, and the female
21.82 per cent.
The ratio of the total females of age 16 and above, to the total males
of 16 and above, was about 1.10904, and of 21 and above was 1.12391.
This would indicate a frequency of 4.097, or 4.042 to 1 for male, as com-
pared with female suicides. But this relative frequency is very variable.
On the whole it is rapidly increasing. The ratios of the death-rates of
males and females according to age are as follow, viz. : —
Age.
Ratio of
Death
Bates
Smoothed
Ratio
10-14
1.7
.74
15-19
0.97
1.37
20-24
2.15
25-29
3.01
2.62
30-34
3.25
3.25
35-39
4.32
3.88
40-44
5.09
4.51
I
45-49 50-54
6.17
3.93
5.13
5.76
55-59
5.14
6.39
60-64
7.86
7.30
65-69
10.67
8.80
70-74
9.07
11.50
75-79
14.27
15.60
80-84
28.79
28.80
85-89
These results shew that the ratio of the rate of suicide by men to
that of suicide by women increases about 0.125 per annum till about age
60, when it becomes more rapid. The general result is, that this rate
p can be expressed between the ages 10 and 57.5 as : —
(652).
.p = 0.1256 {X — 6.63)
428
APPENDIX A.
after which the points he upon the curve indicated by the numbers 6.39,
7.30, etc., in the preceding result as smoothed.
The annual fluctuation of suicide is fairly well-defined . By correcting
the results so as to make them represent what would have been furnished
by records of equal months, and a constant population ^ (as at the middle
of the period), the following values are obtained, viz. :—
TABLE CLVI. — Number of Suicides per diem in a Population of 1,000,000 Persons.
Australia, 1900 to 1915.
Period.
Jan.
Feb.
Mar.
April.
May.
June.
July.
Aug.
Sept.
Oct.
Nov.
Dec.
1900-1909'
1907-1915
.359
.376
.371
.356
.335
.326
.336
.335
.310
.306
.284
.262
.301
.346
.326
.295
.307
.351
.353
.358
.323
.356
.345
.381
Mean . .
.367
.364
.330
.335
.308
.273
.324
.310
.329
.3'56
.340
.363
These results are given by 0.3291 + 0.0354 sin (x + 72° 4') — 0.0117 sin 2 (x + 73°.22')
I sin 3 (x + 12° 49')— 0.0142 sin Hx + 40° 520—0.0131 sin 5 (x+fy.W) + 0.0104 sin
urn. Boy. Soo. N.S.W., xlv., p. 99.
+ 0.0031 am a va: -f- la as ;— u.ux*a sin
6x : Journ. Boy. Soo. N.S.W., xlv., p. 99.
The final mean results probably do not define the curve representing
an indefinitely large number of cases. The results given are based upon
only about 10,000 oases', and at least 10 times this number would be
necessary to get satisfactory results. The distribution is more likely to
be of the form. *
(653). .y= A+ B sin x-\- G cos x= A+b am {x+P)+c COS {x+ y)
(654) . . .4=(2'i» y)/n ; B=b cos ^— c sin y ; 0=b sin ^+c cos y.
'■ The population records give for the population at the middle of each month
the following results, 00 omitted : —
Jan. Feb. Mar. April. May. June. July. Aug. Sept. Oct. Nov. Dec.
209,686 210,012 210,338 210,662 210,983 211,305 211,657 212,039 212,421 212,834 213,278 213,723
Females —
194,153 194,513 194,873 195,054 195,055 195,056 195,442 196,211 196,981 197,766 198,567 199,369
' See " Studies in Statistical Representation" (Statistical Applications of the
Fourier series), by G. H. Knibbs, Journ. Roy. Soc, N.S.W., xlv., pp. 76-110, IQll-
in particular pp. 97-110.
XVI.— raGRATION.
1. Migration. — The effect of immigration, and indeed of migration
generally, is to modify the age, sex, and race constitution of a community,
and these facts are well illustrated in the statistics of any new country
(e.g., the Commonwealth of AustraUa). Concentrations of population
due to seasonable or similar influences, or from other causes, may also
become a factor of importance from particular points of view. Por ex-
ample, statistics of morbidity or of mortahly, the object of which is to
differentiate between urban and country hygienic conditions, may be
materially affected even by temporary concentrations of populations in
cities ; for example, by the fact that serious impairments of health may
lead to transfer to the cities for special treatment, with a consequent
increase of the mortality and morbidity rates ; and so on. Certain
obvious economic consequences may, too, arise from such concentrations.
For these reasons statistics for particular purposes are often Hmited as
regards precision.
In countries where the migration of adults is a striking characteristic,
the constitution of the population according to age ceases to be normal ;
but the aggregates obtained by inclusion of the group of countries between
which the migration takes place, tend to restore the normality. In
AustraJia financial arrangements between the component States have,
among other things, led to records being kept (a) of oversea migration,
(6) of interstate migration by sea, and to a partial record (c) of overland
migration. All of these shew fluctuations of annual period.
Records of overland migration by road are not kept, but such
migration is assumed to be in balance, that is to say, the immigration
and emigration are supposed to be equal. It will be seen later that over-
land immigration by rail virtually balances the overland emigration.
2. Proportion born in a country.- — The correlation of birth-place and
age in any population is of sociologic importance. ^ In the following
results, from the 1911 Austrahan Census, the " unspecified" cases (as to
whether the birth-place was Australia or outside of Austraha) have, for
each age-group, been distributed in the proportion of the numbers given
as born in and out of Australia, respectively. The results are as shewn
in Table CLVII. hereunder and in Fig. 106. These disclose the fact that
the initial preponderance of persons born in Australia diminishes very
rapidly with age ; this of course being due to the fact that the commence-
ment of colonisation was at a point of time nearly identical with the birth
of the present oldest inhabitants.
1 An analysis of the Australian population will be found in the Census Report,
Vol. I., pp. 120-125.
430
APPENDIX A.
TABLE CLVn. — Shewing according to Age and Sez the Proportion of Persons Living
in hut not Born in Australia.^
Proportion not
Born in
Proportion not
Proportion not
Proportion not
Born in
Age
Age
Bom in
Age
Born in
Age
last
Australia.
last
Australia.
last
Australia.
last
Australia.
Birth-
Birth-
Birth-
Birth-
day.
day.
1
day.
day.
Males.
Females
Males. ^Females
Males.
Females
Males.
Females
.0036 .0036
15-19
.0403
.0243
50-54
.4536
.3244
85-89
.9792
.9756
■ 1
.0106 1 .0103
20-24
.0699
.0513
55-59
.5875
.4915
90-94
.9781
.9886
"2
.0160 .0166
25-29
.1866
.1100
60-64
.7014
.6485
95-99
.9569
.9449
3
.0202 1 .0193
30-34
.2290
.1531
65-69
.7572
.7181
00 and
.9143
.8966
i
.0215 .0207
35-39
.2538
.1806
70-74
.8880
.8653
over
5-9
.0249 1 .0242
40-44
.3007
.2083
75-79
.8952
.9318
10-14
.0239 .0232
45-49
.3834
•2673
80-84
.9731
.9637
The results in the table are graphed in Fig. 106, the Curves M, M'
and F, F' denoting respectively the results for males and females. The
Proportions bom in Australia.
l-(ll j 1 1 ) 1 [ II 1 )
Mil ij^^fij 1°
z Vi
*
: V
a
J - 2 „
t- 3
H
V* ^1
.3"' _i:?_ :
t- '2
-<c -«
A *J
g-e X
-tt *1
,.s
= t J
12 e
■"•i iUu
t -.«|
«1* J I* ^ z
», tla / I
'" + ' ^
g' .,/ - ^ /.
■3.,^'^^ieZ" ^
: '1
l.] f - « , ^ •- -
nfi
s -it
0=:="**
~0 K) 20 50 40 50
60 » 80 90 100
Fig. 106.
Curves M and I'" shew respectively the
proportion of males and of females born in
Australia. Curves M' and F' are plotted
on ten times the vertical scale of Curves M
andF.
irregular form of the curves is due to
the age-pecuUarities of the migration.
As the population develops by natural
increase the curves will tend to become
similar to the dotted forms, the F and
M curves to become identical, and both
will approach more and more the base-
line.
It might be supposed that by com-
paring the Census results with the birth-
registration results, reduced according
to the mortality, so as to shew the
number of survivors, the excess of
immigrants over emigrants wou(d
appear. Such is not the case, how-
ever, notwithstanding the striking
regularity of the results : see Census Report, Vol. I., pp. 93-94.
1 See Census Report, Vol. II., pp. 130-1 for males, and pp. 132-3 for females.
The vmspeoified according to age have been ignored. Let the total T of either sex
be made up of / those bom in Australia, O those born outside, and V the un-
specified. Then the adjusted numbers /' and 0' will be respectively, I' =
/ . T/(T - U); 0' = O . T/{T - U).
Item.
No. of Males
No. of Females
Ratio of excess
of reduced
registration
Nos.
N.S.W. Vict. Q'land. S. Aus. W. Aua. Tas.
M
F
P
22,957
22,136
.0282
.0321
.0301
15,869
15,089
.0183
.0323
,0251
8,329
7,967
.0346
.0336
,0341
5,378
5,124
.0444
.0357
•04Q2
3,808
3,684
.0373
.0231
,0303
2,761
2,584
.0120
.0240
,0178
Total.
(Cwlth.)
59,102
56,584
.02774
.03174
.02970
MIGRATION.
431
These excesses, ranging from about IJ to about 4| per cent., are only
in part accounted for by the migration of infants (see p. 94 above referred
to), in fact only one-tenth may be referred to migration. They disclose
the necessity of fixing the age exactly by recording the date of birth.
The practice of accepting loose statements as regards age is from every
point of view most unsatisfactory for the purposes of accurate tabulation.
3. Correlation, owing to migration, between age and length of resi-
dence. — The length of residence of the proportion of persons not born in
Austraha, shewn in Table CLVII. and Fig. 106, is furnished by data given
in the Report of the Census, Vol. II., pp. 392-393. The middle of the age
and length-of-residence groups may be regarded as a sufficiently accurate
indication of the average value in both cases. It will then be seen that,
for any given length of residence, there is an age at which the numbers
are a maximum.
The maximum values are — for males — about as follows : —
MAJ.ES.
Average leugth of
residence . . (yra.)
0-1
1-2
2-3
3-4
4-5
5-10
10-15
15-20
20-25
25-30
30-35
35-40
Age giving maximum
numbers . . (yrs.)
24.2
24;7
26.3
27.0
28.6
30.3
34.6
39.7
44.0
(28.6)
49.0
(31.6)
53.8
(37.0)
58.6
(42.5)
Approx. numbers
9,590
4,420
3,180
2,430
1,460
868t
l,148t
l,040t
3,000
(904)-
3,600t
Wt
l,120t
Total (and average) in
(1210)+
(446)t
age group
39.228
17,571
12,760
9.347
5,816
3,644
4,440
4,315
11,478
13,946
8,386
4,563
t Tliese maximum numbers correspond to the maximum ages on tlie two lines above.
The above results shew that the curves are dimorphic and con-
sequently that the relation between the frequency of migration and the
age of the migrants is not simple. This is to be expected in a new country,
where special tendencies in migration are likely to be in evidence from time
to time. The complete record of migration requires that migration
should be tabulated according to age, and for some purposes a Census is
preferably tabulated under the headings " age at entry into the Common-
wealth," and "duration of residence."^ Only in this way can the
relation between age and frequency be accurately and conveniently
ascertained.
Before deaUng in detail with the several classes of migration, the
general theory may be indicated.
4. The theory of miration. — Migration, either into or out of any
territory, varies according to age and sex. The character of these
variations change greatly from time to time, when long periods are con-
sidered, but are ordinarily fairly constant for short periods. As between
place and place, it is, no doubt, other things being equal, also a function of
their distance apart.
1 Thia is analogous to the tabulating with the headings " (Juration of marriage"
with " age at marriage" instead of " age at the Census,"
432 APPENDIX A.
Let the ratio of the number migrating (T) in a unit of time (1 year)
to the population (P) in which it occurs, be called the migration ratio
(t) ; then we shall have : —
(655) T = T/P = Z^^T / SI P
the last expression giving the weighted average over the n years included.
That is, it is the ratio of the sums of the migrants to the aggregate of the
mean populations of the same years.
Let the number of migrants consist of T,n males and Tf females,
then the ratios of each of these to their sum may be called the sex-ratios
a and y respectively, ^ of the migration. Thus : —
(656) a = T^/ (T^ + Tf ) ; y = Tf / {T^ + Tf ).
Thus we shall have for the annual number of male and female
migrants, respectively : —
(657) T„ = Pra; Tf = Pry.
The male and female " migration ratios," however, are given by
(658) T„= TJM; r^= Tf / F ;
and if the number of males and females in the population be equal, we
shall have : —
(659) T„ =2a.T; t = 2y . t.
The components of the fluctuation of annual period are distinctly
traceable — ^in many cases the causes can be assigned.
The " migration ratio" r. is not a population-ratio indicating the
number of different persons migrating : it merely represents the relative
quantity of migration independent of the individuals. It has & fluctuation
of annual period, and minor periods within that, and these can be repre-
sented and dealt with as already indicated, see Part III., § 5, pp. 39, 40
in particular formulae (90) to (101), and Part XI., §§ 16, 17, pp. 169-172.
^ The first letters of avjip and yvvq.
MIGRATION.
433
The " migration-ratio" is a function of age and of time, and is pro-
bably in all cases polymorphic, that is : —
(660).
.r =(f> (x) = SiA'e'^), or = 2'(4a;»»e-"
in other words, it may be regarded as the sum of a series of curves of one
or both of the types shewn, see formulae (23) to (39a), pp. 22 to 24, and
formulse (.147) to (156), pp. 52 to 55. Like nearly all statistical curves it
will probably not conform exactly to any simple expression. The
variation with time will ordinarily be considerable in new countries.
The characteristics of the annual fluctuations are not quite identical
for the sexes : hence each of the components (T^ and Tf ) may be
analysed separately, or the total ( T^, + T/ ) may be analysed, and the
fluctuation of the sex-ratio, determined for individual months, may be
analysed.
5. Migration-ratios for Australia. — ^The migration-ratios for Aus-
traha, determined as indicated by formulae (655) to (659), are as follow :
TABLE CLVin. — Shewing the Migration-ratios for Australia and the Sex-ratios of
the Migration for Oversea and Interstate Sea Migration and for Migration by
Railway.
Oversea Migration,
1909-1913.
Interstate Sea Migkaiion,
1909-1913.
Interstate Migration by
Railway, 1914-1916.
To (I)
or
from (B)
Males.
Fe-
males.
Per-
sons.
Eatio
Males
to
Total.
Males.
Fe-
males.
Per-
sons.
Ratio
Males
to
Total.
Males.
Fe-
males.
Per-
sons.
Ratio
Males
to
Total.
N.S.W. I
B
Vic. I
-B
Qld. I
B
S. Aus. I
B
W. Aus. I
B
Tas. I
B
.05237
.03654
.02195
.01376
.02284
.00928
.02199
.01130
.05561
.02308
.02129
.00910
.02644
.01751
.01336
.00788
.01472
.00267
.01048
.00372
.03502
.00963
.01104
.00566
.04003
.02748
.01763
.01080
.01912
.00626
.01632
•00757
.04676
.01730
.01630
.00742
.68549
.69666
.61875
.63298
.64953
.80570
.68336
.75729
.67788
.76057
.66986
.62842
.04557
.04394
.07226
.07313
.05516
.05028
.05873
.05502
.07460
.08202
.22592
.24873
.03251
.03095
.04807
.04766
.03251
.03137
.03534
.03463
.07288
.07593
.16258
.18082
.03935
.03775
.06009
.06032
.04482
.04165
.04720
.04497
.07386
.07940
.19505
.21564
.60666
.60976
.59749
.60242
.66902
.65623
.63097
.62041
.57566
.58876
.59379
.59163
.18966
.19104
.19580
.19071
.16804
.16970
.22646
.23406
.09635
.09447
.07766
.08047
.11287
.11476
.10576
.10170
.14426
.14406
.13582
.13474
.14238
.14413
.16490
.16655
.67506
.68094
.70974
.69681
.63129
.62984
.67288
.68857
The table shews that as regards oversea migration, immigration is
preponderant : in interstate sea migration it is also generally preponder-
ant, the exceptions being— Victoria, " males" and " persons" ; Western
Austraha, " males," " females" and " persons." Interstate migration
by railway shews an approximate equaUty between immigration and
emigration, the balance on either side being variable.
That these results have very accordant values from year to year will
appear from the following table ; — ■
434
APPENDIX A.
TABLE CLIX.— Interstate Imm^ation by Sea, 1909-1913.
1
Migration-ratios.
PjitioofMale
Migrants to
Total Migr'nts
s
Mlgration-iatios.
1
Mlgration-iatlos.
1^1
Year
^^^'■' mills} Z^:
Males.
Fe-
males.
Per-
sons.
Males.
Fe-
males.
Per-
sons.
1909
1910
1911
1912
1913
•
CO
.0401
.0408
.0487
.0488
.0454
.0278
.0294
.0347
.0359
.0320
.0343
.0354
.0420
.0427
.0390
.6134
.6026
6066
.6008
.6110
1
>
.0651
.0670
.0747
.0763
.0714
.0407
.0442
.0504
.0526
.0491
.0528
.0556
.0625
.0644
.0603
.6099
.5994
.5967
.5915
.5931
■6
a?
.0575
.0521
.0559
.0522
.0557
.0323
.0323
.0323
.0311
.0320
.0460
.0430
.0452
.0425
.0448
.6801
.6571
.6714
.6641
.6717
1909
1910
1911
1912
1913
1
.0536
.0555
.0605
.0610
.0578
.0315
.0349
.0371
.0362
.0337
.0427
.0453
.0490
.0488
.0458
.6356
.6209
.6278
.6352
.6351
1
.0693
.0790
.0783
.0709
.0698
.0691
.0757
.0753
.0707
.0664
.0692
.0776
.0770
.0708
.0683
■5705
.5808
.5811
.5694
.5756
1
.1886
.2017
.2210
.2465
.2464
.1331
.1396
.1606
.1792
.1829
.1614
.1713
.1862
.2139
.2158
.5961
.6007
.5886
.5935
.5915
Excluding Federal Territory.
TABLE CLX. — Shewing ior the Years 1909 and 1913*, the Ratio of Male Migration
to the Total Migrationt, and the Proportion of Males, Females and Persons, under
12 Tears of Age, to the Total Number of Emigrants. Australian Interstate
Migration by Sea.t
states from
N.S. Wales.
Victoria.
Queensland.
S. Australia.
W. Australia.
Tasmania.
Masc.M..066
Masc. M. .061
Masc. M. .086
Masc. M.
.161
Masc. M. .096
» .059
„ .063
„ .070
.126
J, .087
To
.607 F. .095
.665 F. .112
.591 F. .109
.573 #.
.212
.518 F. .079
N.S. Wales.
.600 „ .085
.667 „ .118
.599 „ .103
P. .095
.598 ^
.163
.499 ., .098
P. .078
P. .078
.188
P. .087
„ .069
„ .082
„ .083
'•
.141
Masc.M..061
Masc. M. .084
Masc. M. .065
Masc.M
.182
Masc. M. .061
J, .063
,. .085
„ .063
.155
„ .059
To
.612 F. .086
.649 F. .119
.647 F. a22
.539 #.
.196
.618 F. .088
Victoria.
.578 ., .082
.600 „ .122
.677 „ .099
if. .085
.550 ,
.182
.600 ,, .096
P. .071
^. .096
P.
.189
P. .071
„ .071
„ .100
„ .075
»
.167
Masc.M..059
Masc. M. .085
Masc. M. .033
Masc. M.
.106
„ .063
„• .081
„ .109
.270
To
.687 F. .126
.649 F. .136
.831 If. .160
.610 f.
.067
Nil.
Queensland.
.678 „ .116
P. .080
.642 „ .130
!^. .103
.567 „ .155
¥. .054
.525 ^,
.263
.001
„ .080
„ .098
„ .129
»
.267
Masc.M..095
Masc. M. .063
Masc. M. .018
Masc. M.
.112
Masc. M. .024
„ .078
„ .052
., .156
,106
,, .000
To
.601 F. .054
.659 F. .113
.829 F. .089
.629 #.
.189
.971 F. .053
S. Australia.
.588 „ .128
.674 „ .109
P. .080
.427 „ .070
P. .030
.635 ,,
161
.348 ,, .000
P. .111
i.
.140
P. .029
„ .098
„ .070
„ .107
..
.126
„ .000
Masc.M..162
Masc. M. .178
Masc. M. .058
Masc. M. .120
Masc. M. .097
„ .143
„ .148
.543 F. .184
„ .103
.340 F. .121
„ .112
„ .417
To
.556 F. .193
.607 iF. .191
.633 F. .056
W. Australia.
.566 „ .174
„ .170
!&. .181
.439 „ .181
if. .100
.620 „ .158
.343 ., .087
¥. .176
P. .148
P. .082
„ 156
„ .157
„ .091
„ .130
„ .200
Masc.M..080
Masc. M. .054
Masc. M. .100
„ .081
„ .053
„ .111
To
.483 F. .072
.620 F .082
.409 F. .IOC
Nil.
NU.
Tasmania.
.494 „ .087
.612 „ .096
P. .065
„ .154
P. .076
P. .IOC
„ .084
„ 069
„ .136
• The upper figures are for the year 1909, the lower for the year 1913. t The masculinity
of the migration in the table is the ratio of males to persons. J Based upon the departures front
and arrivals in the States indicated,
MIGRATION.
435
6. Periodic fluctuations in migration. — Periodic fluctuations of
migration are exhibited alike by oversea migration, by interstate migra-
tion by sea, and by migration overland. The following tables give the
variations for the first and second for Australia. Table CXLI. shews also
the monthly variations of the sex-ratio (or masculinity) of the migration.
To express these results by Fourier series, see Part III., § 5, pp. 38-40,
and also Part XI., § 16, pp. 169-171.
TABLE CLXI.^Shewing Oversea Migration into and from Australia during the
period 1909-1913, and its Fluctuations for " Persons" during the Year.
(For equalised months and a constant population).
I
or
B
I
£
I
B
Totals tor 1909
-13.
Jan.
Feb.
Mar.
AprU.
May.
June.
July.
Ajg.
Sept.
Oct.
Not.
state.
Persons.
Males.
Females
Dec.
N.S.W.
337,997
232,056
.6856
.6967
231,634
161,666
106,303
70,390
8.^9
1.002
.697
.692
1.110
1.101
.665
.670
1.180
1.096
1.227
.679
.680
.989
1.101
.708
.712
.992
.884
.708
.707
.792
m
.848
.723
.721
.876
.806
.689
.735
.920
.761
m
.694
.709
.985
.896
.679
.716
1.091
.976
.659
m
.712
1.070
1.316
1.082
Masc.
.698
.642
m
.689
.705
Vict.
I
B
I
B
116,603
71,425
.6187
.6330
72,148
45,211
44,455
26,214
.883
1.189
.632
.663
1.040
1.288
.612
.620
.990
1.529
.914
1.386
.637
.601
1.105
.974
.652
.631
.856
.801
.659
.652
.756
.766
.645
.658
.832
.627
.636
.689
.994
.602
m
.628
.664
1.024
.720
.613
.658
1234
.877
.594
.649
1.372
1.241
Masc.
.593
.580
m
.565
m
.626
Qld.
I
B
I
B
58,507
19,161
.6495
.8057
38,002
15,438
20,505
3,723
.850
1.125
.650
.890
.717
m
1.244
.670
.854
1.114
1.434
.743
1.243
.677
.702
m
.963
1.022
.672
.748
1.165
.801
.688
.777
.855
.697
m
.637
.775
.989
.709
.628
.812
1.172
.777
.711
1.264
.895
.626
.848
.938
1.062
.670
.863
1.230
.991
Masc.
.595
.840
.592
.841
.865
S. Aust.
I
E
I
E
33,496
15,529
.6834
.7573
22,890
11,760
10,606
3,769
.902
.939
.733
.833
1.004
1.305
.713
.769
1.037
1.699
.918
1.480
.721
.691
1.038
1.084
.654
.761
.854
.808
.652
.811
.840
.719
.698
.791
.731
m
.723
.762
,826
.816
.684
m
.747
.789
1.050
.726
.708
.802
1.796
.880
.568
m
.744
1.014
.953
Masc.
.714
.680
m
.659
.737
W. Anst.
I
B
I
E
7
B
I
E
67,168
24,846
.6779
.7606
45,532
18,897
21,636
5,949
1.389
.932
.715
.793
1.169
1.058
:703
.750
.702
1.436
.847
1.298
.724
.709
1.059
1.198
.682
.718
.860
.895
.662
.758
1.095
.796
.654
.772
.594
m
.797
.708
.801
.842
.641
m
.612
..806
.811
.679
.653
.788
1.556
1.076
.897
.610
m
.781
1,873
Masc.
.746
.683
.707
.838
Tas.
15,633
7,121
.6698
.6284
10,472
4,475
5,161
2,646
.841
1.518
.622
m
.685
1.105
1.582
.641
.580
1.279
1.169
1.427
.705
.583
1.029
.755
.704
.612
1.036
.639
.696
.652
1.002
.499
m
.689
.628
.840
.561
.671
.637
1.040
.590
.675
.671
.838
.952
.814
.722
.821
m'
.625
.624
.607
1.005
1.732
1.120
Masc.
.686
.553
m
.648
.690
The quantity underlined is the greatest, and that marked m the least during the year.
The two upper figures in each section are the relative average magnitudes of the migration for the
month, the monthly average for the year being unity.
The two lower figures are the migration-ratios tor the correspondlng'months, viz. the ratio o£ the
migrants to the population of the State,
436
APPENDIX A.
In Table CLXII. hereunder the fluctuations of interstate migration
by sea are shewn, and the " migration-ratios" are also shewn.
TABLE CLXn. — Shewing the Fluctuations for " Persons" in the Interstate Migration
by Sea in Australia for the Period 1909-1913.
(ForequaUsed months and aconstant population and the migration ratios xl.OOO.OOO.)
State.
FLUCinATION RATIO (TOIAI =
12.000) AND MlGRATIOK-KATIOS FOB PERSONS.
Mi-
grants.
Jau.
Feb.
Mar.
April.
May.
June.
Jjly.
Aue.
Sept.
Oct.
Nov.
Dec.
To—
' Victoria
Q'land
S. Aust
W. Aust.
Tas.
N. Terr.
137,916
16,344
109,542
12,982
20,788
2,464
16,218
1,922
33,617
3,972
825
98
1.645
1.254
1,707
1.011
1,094
1.194
245
1.263
200
1.868
615
.60
5
1.045
1,428
1.059
1,146
1.485
1.157
1,575
1.225
1,325
1.325
272
1.366
.891
1,214
1.318
.663
903
1.256
1,359
.776
159
.809
129
.437
j»
145
.108
9
.595
m
810
.952
1,030
.622
128
.683
109
.460
152
.102
8
.641
873
.883
955
.592
m
122
.636
m
101
.448
148
.87
7
.797
1.085
.782
846
.744
153
.763
122
.661
219
.69
m
5
1.164
1,586
.706
m
764
.860
176
.858
137
.730
242
.93
8
.965
1,314
.711
769
.879
180
.871
139
.827
274
.78
6
1.283
i
00
2,105
1.195
1,293
1.243
255
1.068
170
1.991
1,748
.902
CO
n
1,425
1.096
225
1.333
212
.498
165
.128
10
976
1.184
305
1.270
202
1.200
397
.97
8
243
1.160
^
218
.907
300
.217
18
185
1.983
659
.83
7
656
.93
8
Total ..
318,806
37,781
1.425
1.228
3,866
1.106
3,482
1.178
3,708
1.032
3,251
.859
2,704
.710
2,237
.701
m
2,207
.772
2,429
.925
2,911
.852
2,682
1.212
4,489
3,816
■■N.S.W.
145,326
21,973
1.354
1.274
2,333
1.343
2,458
1.297
2,376
.958
1,755
.788
1,352
.629
1,151
.610
m
1,117
.743
1,361
.709
1,298
1.088
1,992
1.267
2,479
8,302
t^
Q'land.
25,828
3,905
.935
304
.909
296
.987
321
1.187
387
1.467
477
1.515
1.272
414
1.031
336
.760
247
.606
m
197
.662
212
.679
493
221
to
S. Aust.
28.006
4,235
1.266
1.231
435
1.233
435
1.212
428
1.083
382
.787
278
.701
247
.631
m
323
.779
275
.775
274
1.060
371
1.262
II
ft,-
447
442
W. Aus.
46,031
6,960
1.082
627
1.339
777
1.438
1.307
758
1.220
708
.926
537
.716
415
.729
422
.710
412
.703
m
407
.864
501
.966
f
835
560
1
Tas.
163,688
1.614
1.263
1.041
1.009
.683
.694
.663
.602
.748
.786
1.112
1.905
^
23,220
3,126
2,425
2,013
1,952
1,322
1,149
1,264
1,164
1,447
1,520
2,151
3,686
£
N. Terr.
166
.92
.10
m
1
1.46
.39
2.46
1.16
.77
.92
1.38
.31
1.60
.64
-
24
2
3
1
5
2
1
2
3
.6
3
1
Total
398,915
1.390
1.246
1.207
1.174
.925
.758
.695
.649
.745
.736
1.041
1.434
60,317
6,986
6,266
6,066
5,901
4,649
3,810
3,493
3,264
3,745
3,697
5,230
7,211
MIGRATION.
437
TABLE CLXII.^Shewing the Fluctuations for "Persons" in the Interstate Migration
by Sea in Australia for the period 1909-13 — continued.
FLtrCTUATION RATIO (TOTAT. =■ 12.000) AND MIOBATION-RATIOS FOK PEMOlfS.
State.
Mi-
grants.
Jan.
Feb.
Mar.
April.
May.
June.
July.
Aug.
Sept.
Oct.
Nov.
Dec.
'N.9.W.
106,280
1.140
.957
.976
.890
.811
.781
.788
.905
1.036
1.057
1.037
1.622
U3
03
34,731
3,300
2,769
2,825
2,578
2,347
m
2,261
2,282
2,619
2,998
3,058
3,001
4,693
i"
Vict.
19,664
1.088
.914
.818
.959
.721
.773
.785
1.039
1.131
1.205
1.071
1.496
6,426
582
490
438
514
m
386
414
420
556
606
645
573
801
II
9. Aust.
593
1.23
.69
.93
.92
1.78
.63
1.11
.97
.69
1.03
1.06
.97
ft.
194
20
10
20
14
29
10
17
36
11
16
17
16
g
W. Aust.
325
1.11
1.22
.85
2.14
1.85
.85
.52
.59
.92
.89
.29
.77
1
116
10
11
8
19
16
8
5
5
8
8
3
7
J
Tas.
62
0.0
4.8
4.7
2.5
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
20
8
8
4
N. Terr.
629
.79
.93
.95
1.47
.89
1.09
.77
1.16
1.11
1.04
1.18
.67
L
173
11
13
14
21
13
16
11
17
16
15
16
9
Total
127,463
1.130
.952
.953
.908
.804
, .780
.788
.926
1.049
1.078
1.040
1.692
41,651
3,924
3,302
3,307
3,151
2,791
m
2,708
2,736
3,213
3,640
3,743
3,610
5,526
N.S.W.
22,116
10,774
1.285
1,153
1.422
1,277
1.494
1.231
1,105
.956
858
.694
623
.591
531
.560
m
503
.772
693
.870
781
.866
768
1.270
00
1,341
740
i
Vict.
25,828
12,583
1.222
1,281
1.445
1.188
1,245
1.069
1,121
.770
807
.793
831
.673
706
.626
m
657
.749
786
1.105
1,160
1.027
1,077
1.333
II
ft,
1,515
1,397
Q'land.
1,019
.77
1.21
.84
1.07
1.17
1.25
.98
.84
1.18
1.03
.66
m
27
1.00
i
496
32
50
35
44
48
52
40
35
49
43
42
1
W, Aust.
43,341
1.128
1.275
1.317
1.185
1.049
.869
.800
.783
m
1,377
.786
.911
.896
1.001
Total
21,115
1,984
2,243
2,318
2,086
1,845
1,529
1,408
1,383
1,603
1,577
1,760
a
&
92.303
44,909
1.188
7,941
1.357
1.318
7,501
1.163
6,681
.950
5,568
.810
4,617
.717
4,331
.686
m
4,455
.777
5,614
.956
6,862
.920
6,954
1.158
8,348
10,528
'N.S.W.
20,370
1.292
1.350
1.232
.967
.884
.804
.685
.579
.830
.930
1.062
1.385
CO
14,180
1,527
1,595
1,456
1,142
1,045
949
809
684
981
1,100
1,255
1,637
to
CO
Vict.
45,690
1.283
1.310
1.135
.978
.828
.609
.519
.608
.796
1.077
1.093
1.763
iH
31,806
3,400
3,476
3,007
2,592
2,195
1,615
1,375
1,612
2,110
2,852
2,898
4,674
ft,
Q'land.
531
1.08
.79
1.22
.93
1.45
1.22
.68
.86
.77
1.02
.95
104
.X
370
33
24
38
28
45
38
21
26
24
31
29
32
4
S. Aust.
47,205
32,860
1.088
2,980
1.182
3,236
1.096
3,000
1.050
2,877
.833
2,281
729
m
1,996
.775
2,122
.776
2,125
.913
2,499
1.032
2,827
1.012
2,772
1.514
4,145
&=■
N. Terr.
266
0.0
1.17
0.0
.267
.13
1.22
27
.45
0.0
3.39
.00
2.71
1
185
18
41
2
26
4
7
52
42
u
^
Total
114,062
79,401
1.200
7,941
1.262
8,348
1.134
7,501
1.010
6,681
.841
5,568
.698
4,617
.655
m
4,331
.673
4,455
.848
5,614
1.037
6,862
1.061
6,954
1.691
10,528
438
APPENDIX A.
TABLE CLXn. — Shewing the Fluctuations for "Persons" in the Interstate Migration
by Sea in Australia for the Period 1909-lZ— continued.
Fluotuation Eatio (Total = 12.000) ahd Mioeation-eatios foe Peesons.
State.
Mi-
grants.
Jan.
Feb.
Mar.
April.
May.
June.
July.
Aug.
Sept.
Oct.
Nov.
Dec.
00
l-H
N.S.W.
37,786
3,939
1.566
1.474
4,837
1.482
4,865
1.442
4,735
1018
3,343
1.044
3,427
.826
2,708
.553
1,815
.735
2,411
.520
m
1,709
.613
2,013
.728
to
5,140
2,388
II
Vict.
168,563
17,572
1.606
23,512
1.619
1.432
20,970
1.223
17,911
.821
12,014
.686
10,053
.693
10,149
.680
m
9,952
.709
10,389
.876
12,831
784
11,485
.871
fX,
23,700
12,756
rt
S. Alls.
339
.57
.67
4.01
3.33
.81
.78
0.0
.07
.28
.71
.60
.17
s
a
35
17
20
118
98
24
23
2
8
21
18
5
i
W. Aust.
1S8
0.0
.46
3.49
6.68
1.06
.31
•■
1
1.
16
6
48
92
15
4
Total
206,846
21,564
1.695
1.690
28,563
1.447
26,001
1.271
22,839
.867
15,396
.752
13,506
.716
12,857
.656
m
11,768
.718
12,809
.810
14,560
.762
13,^516
.843
28,669
15,150
N.S.W.
412
.87
.64
.61
.68
.75
1.11
.93
.41
1.72
1.81
1.49
1.08
M
CO
o
39,800
2,900
2,100
2,000
1,900
2,500
3,700
3,100
1,300
5,700
6,000
4,900
3,600
II
Vict.
210
.67
1.66
1.37
.51
.40
.40
.67
.91
.80
1.49
1.26
2.06
B,
20,300
9,700
2,800
2,300
8,700
6,800
6,800
9,700
1,500
1,300
2,500
2,100
3,500
^'
Q'land.
387
.75
.63
.44
m
1,300
.84
1.02
.93
1.12
1.39
.66
1.77
1.24
1.33
Tl
37,400
2,300
1,600
2,600
3,200
2,900
3,600
4,400
2,000
5,500
3,900
4,200
^
W. Aust
161
0.0
1.57
0.0
1.49
1.49
.75
1.12
.30
.37
2.01
.62
2.38
s
15,600
2,000
1,900
1,900
9,700
1,400
3,900
4,800
2,600
6,800
3,100
S Total
1,170
.66
.91
.61
.78
.88
.87
.96
.81
1.02
1.76
1.23
1.62
113,200
6,200
8,600
5,700
7,400
8,300
8,200
9,000
7,600
9,600
1,600
1,200
1,400
The upper figures are the relative average magnitudes of the migration for the month, the monthly
average for the year being unity. Those underlined are the maximum-values and those marked
" m" the minimum values during the year.
The small figures are the number of migrants (" persons") per 1,000,000 population in the State
from which the migration takes place.
That Land Migration also shews marked periodicity is evident from
Table CLXIII. It is worthy of notice that the total immigration for a
year is sensibly equal to the total emigration for the same period
though the want of balance for individual months may be considerable.
TABLE CLXIII. — Shewing the Periodic Fluctuation of Overland Migration (by
Railway) for equalised months and a Constant Population.
Australia, 1914-1916. ("Persons.")
Month.
AEKIVAIS or iMMIQEATIOIf.
DBPAETUEES OE EMir.EATION.
N.S.W.
Vic.
Q'land.
S. Aust.
N.S.W.
Vic.
Q'land.
S. Aust.
January
February
March
AprU
May
June
July
August
September . .
October
November . .
December
1.1517
.d824
1.1091
1.2085
.9584
.8034m
.8389
.8734
-.9394
.8700
.9091
1.3549M
1.0855
.9812
.9427
1.1153
.9353
.8360
.8269 m
.8985
1.0494
1.1441
.9884
1.1967 M
1.5297M
1.0387
1.2059
1.3487
1.0189
.7474
- .7472
.8454
.7266
.7610
.7117 m
1.3188
1.1928
.9737
.9944
1.1649
.9761
.8007 m
.8528
.9445
.9602
.9047
.9350
1.3002 M
1.2452
1.0044
1.0339
1.1811
.94.18
.7888 m
.7999
.8938
.9432
1.0000
.8809
1.2790 M
1.0973
.9437
1.0848
1,1745M
.9928
.8533 m
.8921
.9179
1.0177
.9115
.9970
1.1174
1.2142
.9738
1.1270
1.2167
.8907
.7072
.6995 m
.8231
.8349
.8703
.8219
1.8207 M
1.2352
1.0580
1.0250
1.2827M
1.0278
.8643
.8904
.8488 m
.8893
.8970
.8725
1.1091
Mean No. for
equalised mnth.
Aggr. Popn. . .
67,102
48,188
24,278
18,063
67,007
47,804
24,516
18,244
M denotes the maximum and m minimum value.
M [ORATION. 439
7. Migration and Age.— If the ages of migrants of each sex are re-
corded at the moment of entry into or exit from any community, it is
possible to know continuously the constitution of the population accord-
ing to sex and age, once a population Census has been taken. Results
forwarded to the compiling authority only at long intervals require cor-
rections, of the type referred to in Part XI., §§ 7-9, pp. 152-160. The
deduction of ages is best effectuated by referring all the results to the one
point of time, say the end of the calendar year.
8. Defects in migration records and the closure of results.— Not-
withstanding that elaborate care was taken as regards the record of
emigration, it has been found in Australia that errors occur therein of
considerable magnitude. From the 1901 Census and the intercensal
records up to the Census of 1911, it appeared that, if the discrepancy were
attributed wholly to this source of error, it would amount, in the case of
males, to 0.1459 of the whole recorded male migrants outward (de-
partures) and in the case of females to 0.0995 of the whole recorded female
migrants outward. A still more extraordinary result was that apparently
the island-continent of Australia was rapidly losing females.^
Suppose that a statistical element Eq is accurately ascertained at
anypoint of time {e.g., as at a Census) and after n years is again accurately
ascertained and found to ba En ; and further that the intervening changes
are e^, 62 , ... en. Then : —
(661).. En=S!„+k (61+62 H-. . ..+en); or k={E„~E„)/{ei+ez+ . .+e„)
The quantity k may be called the coefficient of proportional linear adjust-
ment, and El, E^, etc., may be found by the successive additions, viz., of
kei, ke^, etc., instead of the unadjusted change. We may, however,
correct the results as indicated in (662), that is : —
(662) . . En= ^o+ei+e2+ • • +e«+€ =Eo+{ei+K)+{es+K)+ . . +(e„-f/c).
in which last expression K = e/n, the total defect of closure, e, being
divided equpUy among the changes. Thus in this case Ei, E^, etc.,
may be found by successive additions, viz., of bi+k, e^+K, etc. This
may be called simple linear adjustment. The question as to whether
one or the other or either is legitimate, must always be decided by the
nature of the case, and obviously no general rule can apply.
1 Upon a change being made on the system as between State and State, such
that the aggregate of the State -increments of population gave the increment of
population to the Commonwealth, this peculiarity vanished.
XVn.— raSCELLANEOUS.
1. General.' — It is proposed in this part to refer to a number of
miscellaneous matters, which have not been included in previous parts,
and which either do not fall under any particular heading, or have been
omitted from ear her consideration.
2. Subdivision of population and other groups.^ — The values of
group-subdivisions, which are obtained by dividing groups bj'^ the middle
ordinate, are given earUer, see Part VI., § 4, pp. 80-81. These formulae
are not always applicable. Two questions often arise, viz. (i.) the
value of the subdivisions or (ii.) of their ratios to each other.
Considering firsfc the subdivision of a group g into two parts, let it
be supposed that the function, representing a series of groups, viz.,
g^i, g_i, g,gi. . . .g^, is a-\-bx-\-cx^-\- etc., then we shall have^ :—
(663)
g^i =ig - gig [61(sri-g_i)- 44:(g^-g-2)+ 19(g3-?-3)-3M9'4-?-4)+etc.]
gr_j denoting the portion of the group g on the negative side of the middle
ordinate of that group. This formula is in general suitable about maxima
and .minima values, but may, of course, be inappropriate ii g^. — g./^
increase more rapidly than the coefficients diminish. It may often be
employed, however, when pairs of terms in the square brackets are sharply
convergent.
Another process of arriving at values for the subdivision of groups
into halves is the following :■ — Let the values of the successive groups be
C, B, A,M, A',B' and C", and M, the group to be divided. Then the
portion next to .4 is ^ :—
(664)..Af'=JM-2^[201(^'-.4)-44(£'-£)+5(C"-C)— ..]
which in many cases gives substantially the same result as (663), though
it is not an identical formula, and apparently might be regarded as not in
agreement therewith.
1 This is deduced by finding, in terms of the groups themselves, the values of the
constants a, 6, etc., of the curve : and then integrating between the limits which
give the first half of the group to be subdivided.
2 This is easily derived from the usual formula for interpolation into the middle,
viz., F(ii = 'F -\- \ a' — \b„ -\- ,-§g d„ — xi^jj/o + etc., by regarding the aggregates
G,0-\-B, G-\-B -^ A,0+B-\- A-'rM, etc., as successive totals represented by
ordinates represented by a -f px -{- yx^ -\- etc. ; finding the value to the middle
ordinate of group M and subtracting C-\- B-\- A.
MISCELLANEOUS. 441
In the case of groups rapidly increasing or rapidly dimiaishing in
amount — as for example the numbers dying at the beginning of life in
0-1, 1 to 2, etc., days, weeks, months or years, the following method of
subdivision may be followed : —
Let it be required to divide each of a series of larger groups A, B, C,
etc., for equal limits of a variable into s smaller groups, viz., aj, «£ > • • • ■
as ; 6i, . . . 6s ; Cj, . . . Cg ; etc., and suppose that £=m^ ; C=m'B
=min' A ; etc. Then if m'=m, etc., it is 6bvious that the successive
values of the smaller groups will be : —
(665) . . (oi + noi + n^Oi +...) + (6i + w6i + w^&i + • • •) + etc. =
in which n is the sth root of m and m'. The brackets shew the groups,
the sum of which give the original values A, B, G, etc. Since from each of
any three adjoining groups an equal quantity Q may be cut off or added,
so that the altered values A', B', C will be A', m^ A', ml A', we can
constitute the group-divisions by adding a common value Q/s to each of a
series of quantities of the type of (665) above, n^ in this case being the sth
root of Too . Hence we have : —
(666) a {I + w + w2 + ... n»-i) = A' = A — Q ;
from which, since n is known, a can consequently be determined, and the
series o, na, n ^a, etc., to which, if a comrnon quantity q= Q/s is added we
obtain ax, «£ . ©tc. Thus :■ —
(667) ai = A'/{l + n + n^ + ... + n'-^) + q = a + q ;
02= na -\- q; a^ = n^a + q ; etc.
In applying this method practically, any group may be subdivided
by treating it as B, and dividing it according to the indications of the
groups on either side A and G.
3. The measure of precision in statistical results.— Statistical results,
expressed without regard to their possible or probable error, often suggest
the attainment of a precision far beyond that which the data can furnish.
For example, if the ratio of the survivors after one year be given (as in
life-tables) to 5 decimals, the results imply for Australian data an average
precision of og-e for the first year of 1.1 hour, or at its terminal of 0.4 hour.
For other countries it will be much the same. Again, in the case of the
instantaneous rate of mortality at the end of the first year, the expression
to 5 places of decimals implies a precision, in the time or epoch to which it
may be deemed to apply, of 8 days. In both cases the apparent precision
is illusory, 1 forasmuch as the recording of the facts and their actual
1 See Census Report, Vol. III., p. 1215, and also p. 1212.
442 APPENDIX A.
variableness does not conform to this order of precision. For example,
births and deaths are not recorded as regards age to 0.4 hour per annum
even on the average : nor can the point of time to which they may ap-
propriately be referred be deemed to be ascertained to 8 days or its
equivalent in a decade. Actuarial tables are often carried to 7 places
of decimals. A unit in the last place is (on the average) for ages 1 to 2
about equivalent to an age-difference of 2 miuutes, and, owing to the
diminution of death-rate with the lapse of time, also to about the same
as to the poiat of time to which the result is presumed to apply.
Let u and y denote respectively fuifctions of time (t) and of age (x),
then if : —
(668) Au = Idt; Ay = JAx; or I = du/dt ; J = dy/dx
in the Umit, / and J are the ratios of relative importance — as compared with
the units of u and y — of precision in the units respectively of t and x.
These ratios serve as guides in fixing the relative accuracy required in
the data giving the two co-ordinates. If iu graphing results, the units
on the axis of abscissae are, respectively, I and J times the units on the
axis of ordiaates, then the curve wiU make an angle of 45° with either axis,
and this, in so far as it is practicable to foUow it, is the best scale-relation
between ordinate and abscissa for any graph intended to be used for
analysis.
The life-tables published in connection with the Australian Census of
3rd April, 1911, foUow the usual tradition as regards the number of
figures to which the results are expressed. It is not, however, implied
that the precision indicated is realised, they merely are followed for the
sake of oonsistencj^ in the results.
By suitable combinations of arithmetical and graphical methods
results can be obtained to any required degree of practical precision. ^
4. Indirect relations. — ^It is often necessary to establish statistical
relations which reaUy depend upon some intermediary statistical relation.
For example, the average num.ber of children bom to an individual, or
" average issue" may be related to age of " mothers," and such a relation
would, of course, be a direct one. For certain purposes, however, (e.g.,
social insurance) the average issue may be required as related to the age of
fathers. The later relation, though physically indirect, is a regular and
important one. Nevertheless, it is one which may be deduced by means
of certain data from direct relations ; at the same time it is not prefer-
able to obtain it in this way. The relations according to " wives"
and " husbands " are both given immediately by the Census, and the
relation so given is, in general, to be preferred to the deduced relation :
see Fig. 107.
1 If the value of / or J is not between the Umits J to 4, the natural scale for
both co-ordinates is not ordinarily satisfactory in graphing a function ; however the
mode of variation of the greater co-ordinates will assist in the determination of a
truly smoothed curve.
MISCELLANEOUS.
443
Fertility Curves.
A B
=s.=rs-iu^ = s 40Averag6s
SO 60
C F
Fig. 107.
Curve A shews the ratio, according to age, of first bkths to all births.
Curve B shews the probability, according to age, of a nuptial birth ; see also p. 242 and p. 243.
Curve Ca shews the probability, according to age, of an ex-nuptial birth on the assumption
(1.) that they are attributable wholly to the never-married.
Curve Cb shews the probability, according to age, of an ex-nuptial birth on the assumption (ii.)
that they are attributable equally to the never-married, widowed, and divorced.
Our-e D shews the average issue, according to age, of wives at the Census of 1911.
Curve E shews the average issue, according to age, a&related to husbands at the Census of 1911.
Curve ITB ' shews the average interval, according to age, between marriage and first-births.
Curve F6 shews the average interval, according to age.between marriage and first-births, occurring
within 1 year of marriage.
5. Limits of uncertainty. — The limits of an uncertainty in any
deduced quantity may be due to possible errors in the numbers upon
which it is founded, or upon an uncertainty as to the particular quantity
which should be employed. The first cause of uncertainty is sufficiently
illustrated by the ratio of, say, first births to all births : for prediction
purposes the smoothed numbers in Table CLXIV. are really more
probable than the crude numbers : see Fig. 107.
The second cause of uncertainty is illustrated in the following
example : — If the " never married," the " widowed" and the " divorced"
are regarded as a homogeneous class, the probability of a case of
ex-nuptial maternity during one year is found by dividing the number
of births in one year by the sum of the average numbers in the three
444
APPENDIX A.
classes. If, however, they are not homogeneous as regards this proba-
biUty, a more accurate result might be obtained by dividing by the never
married. The general probabiUty must lie between the two results : see
the curves marked Ca and Cb on Pig 107, and the results in columns
marked I. and II. respectively in Table CLXIV.
It may be noted that the characteristics of a variation may be
wholly changed by restriction within limits. This is seen by taking the
interval according to age between marriage and a first birth, when the
consideration is restricted to the lapse of 12 months, or is indefinite :
see the curves FG and PE' respectively.
TABLf
CLXIV
—Shewing Rates of First to All Births,
and Probabilities of
Ex-nuptial Maternity
. AustraUa. 1907-14.
ProbahUity Ex-
Probability Bx-
Ratio oJ First
nuptial Maternity
Ratio of First
nuptial Maternity
to all Births.
hased on
to all Births.
based on
Age.
assumption
Age.
assumption
Crude.
Smooth-
I.
n.
Crude.
Smooth-
I.
II.
ed.
ed.
12
1.0000
.000015
.000015
13
1.0000
.9970
.000062
.000062
34
.0994
.1040
.01325
.0118
14
1.0000
.9930
.00037
.00037
35
.0923
.0920
.0130
.0115
15
.9404
.9715
.0016
.0015
36
.0817
.0825
.0127
.0110
16
.9407
.9430
.0042
.0042
37
.0703
.0730
.0122
.0103
17
.9130
.9035
.0085
.0085
38
.0640
.0640
.0110
.0094
18
.8602
.8450
.0131
.0131
39
.0583
.0560
.01045
.0084
19
.7627
.7627
.0157
.0162
40
.0524
.0485
.0095
.0072
20
.6594
.66j4
.0172
.0173
41
.0437
.0425
.0076
.0058
21
.6912
.5912
.0180
.0179
42
.0381
.0370
.0059
.0044
22
.5285
.5170
.01835
.0181
43
.0352
.0338
.0043
.0030
23
.4534
.4485
.0181
.0179
44
.0351
.0310
.0030
.0018
24
.3360
.3960
.0176
.0174
45
.0349
.0285
.0020
.00123
25
.3482
.3482
.0169
.0161
46
.0244
.0295
.0012
.00080
26
.3098
.3080
.0160
.0154
47
.0360
.0338
.00085
.00040
27
.2722
.2710
.0154
.0147
48
.0255
.0428
.00055
.00022
28
.2352
.2370
.0149
.0141
49
.0769
.0555
.00036
.00013
29
.1)87
.2065
.0146
.0135
60
.1333
.0790
.00022
,00009
30
.1800
.1795
.0143
.0130
51
.0909
.1230
.00012
.00006
31
.1555
.1560
.0140
.0126
52
.1429
.2500
.00004
.00004
32
.1324
.1355
.0137
.0128
53
.00002
.00002
33
.1160
.1182
.0135
.0120
54
.00001
34
.0994
.1040
.01325
.0118
55
I. denotes the ratio of birtlis to the never -married ; 11., the ratio of births to the aggregate of
the never-married, widowed and divorced.
6. |The theory of " happenings" or " occurrence frequencies." — ^Ih
order to establish a rational theory of, and to completely interpret, the
frequency curves met with in the various elements of the statistics of
population, a theory of the frequency of occurrences of various kinds is a
first requisite, and the type-forms of distribution estabhshed by Prof. K.
Pearson and his co-workers are a contribution thereto, based upon the
application of the theory of probabihty, plus certain empirical assumptions
by means of which assymetrical forms of various kinds are deduced.
Recently a foundation has been laid of a perfectly general theory of the
frequency of occurrences, by Prof. Sir Ronald Ross. This latter seems to
have had its birth in an attempt made in 1866 by Dr. Parr to develop a
definite theory of an epidemic (cattle plague) i. In 1873-5 Dr. G. H.
' Dr. William Farr, "On the Cattle Plague," Journ. Soc. Sci., 20th Mar., 1866.
MISCELLANEOUS. 445
Evans endeavoured to extend Fair's theory to other epidemics, i The
subject was again reopened by Dr. J. Brownlee^ in a series of very
significant contributions, and later, by Ross. Quite recently the last-
named has put forward a definite theory, the fundamental elements of
which are outlined in this section.^ Although the main object was
initially the determination of a basis for a theory of epidemics, the results
are entirely general, and may be called the theory of " occurrences" or
" happenings."
The, differential equation of independent occurrences, reduced to its
simplest expression, may be deduced as follows : —
Suppose a population P to consist of two parts, viz., A a part which
is unaffected, and Q a part which* is affected*, by any " happening, "°
so that P= A-\-Q. Suppose also that some portion, viz., hdt, of the
unaffected part becomes affected in the time dt, and also that a portion
rdt of the affected part reverts to the unaffected part in the same element
of time, so that the element of increase of the affected part is (h — r) dt ;
and finally let bdt, mdt, idt and edt denote in the unaffected part, the rates
of birth, death (or mortaUty), immigration and emigration respectively ;
and Bdt, Mdt, Idt and Edt denote the similar rates in the affected part.
Obviously therefore :• —
(669). .dP= A (b—m+i—e)dt + Q{B—M+ I~E)dt = ( Av+ Q V)dt ;
{Wl{i)..dA=A{b-m-\-i-e-h)dt-^Q(B-\-r)dt={A[v-h)+Q{B+r)]dt;
(611).. dQ = Ahdt+ Q(—M+ I~E—r)dt = \ Ah+Q{ V—B-r)}dt;
^Dr. G. H. Evans, " Some arithmetical considerations on the progress of
epidemics," Trans. Epidemiol. Soc. London, Vol. 3, Pt. III., p. 551, 1873-5.
2 Dr. J. Brownlee (i.) Theory of an Epidemic," Proc. Roy. Soc. Edin., Vol.
26, Pt. IV., p. 484, 1906; (ii.) "Certain considerations on the causation and course
of epidemics," Proc. Roy. Soc. Med., Lond., June 1909 ; (iii.) " The mathematical
theory of random migration and epidemic distribution," Proc. Roy. Soc, Edin.,
Vol. 31, Pt. II., p. 261, 1910 ; (iv.) " Periodicity in infectious disease," Proc. Roy.
Phil. Soc, Glasgow, 1914; (v.) " Investigations into the theory of infectious
diseases, etc.. Public Health, Lond., Vol. 28, No. 6, 1915 ; (vi.) " On the curve of
the epidemic," Brit. Med. Journ., May 8, 1915.
' Lieut. -Col. Prof. Sir Ronald Ross, (i.) " The logical basis of the sanitary policy
of mosquito reduction." Cong. Arts and Sci., St. Louis, U.S.A., Vol. 6, p. 89, 1904,
and Brit. Med. Jouin., May 13, 1905 ; (ii.) The prevention of malaria in Mauritius,"
Waterlow and Sons, Lond., 1908, p. 29-40 ; (iii.) The prevention of malaria, J.
Murray, Lond., 1910 ; 2nd Edit., 1911 ; Addendum on" the theory of happenings,"
1911 ; (iv.) Some quantitative studies in epidemiology. Nature, Lond., Oct. 5, 1911 ;
(v.) " Some a priori pathometric equations," Brit. Med. Journ., Mar. 27, 1915 ;
(vi.) " An application of the theory of probabilities to the study of apriori pathome-
try" ; Proc Roy. Soc, Lond., Vol. 92, ser. A„ July 14, 1915, pp. 204-230. See also
H. Warte, " Mosquitoes and Malaria," Biometrika, Lond., Oct. 1910, Vol. 7, No. 4,
p. 421.
* The affection may be of any nature, such as a disease, etc., and the supposition
is quite general.
"■ The " happening" is the becoming affected, and is equally general with th?
preceding supposition.
446 APPENDIX A.
and writing v and F for the algebraic sum of the quantities in the brackets
in (669), the final forms of the preceding equations are given as is necessary
of course, dP=d A-\-dQ. It may be noted that only a A and d Q contain
terms representing the happening (h) and reverting elements (r), and that
QBdt appears in (670) but not in (671), because, in general at least, the
progeny of the affected part are not affected at the instant of birth.
Although the variation elements b, m, i, e and B, M, I, E will, if long
periods are considered, generally be functions of time, they may be re-
garded as constant when short periods only are under review. Consequently
for elementary cases mean values may be taken without sensible error, ^
similarly in regard to the reverting elemeni.^
The most important element is the happening element, h, which it is
to be clearly understood ordinarily falls on both groups ( A and Q) alike.
Should, however, it fall upon individuals already affected, it merely
reaffects them and does not cause them to pass from one group to the
other. Hence, though the total number of " happenings" is P.hdt=
{A-\-Q) hdt, the number Qhdt are already affected and must not be taken
account of. The actual number of new cases Gdt, say, is thus only Ahdt.
Thus :—
(672) Odt/Phdt =^ ; or G = hA = h{P ~ Q)
" Happenings" may be divided into two classes, viz. : — (a) those in
which the frequency of the happening is independent of — ^and (b) those on
which is dependent upon — ^the number of individuals already affected.^
In independent happenings h and G are constants, in dependent happeninge
they are functions of Q.
* If , as is often the case, the " happenings" have no effect on the birth, death
and migration rates, then we may have b= B, m=M, i= I, e= E, and consequently
v= V, which may also occur fortuitously though the several terms differ. In general
6 is less than B in marriages, m than M in accidents, while in certain alarming
epidemics (e.g., cholera, plague, malaria) i is greater than 7, and e less than E, in
which case v is greater than V. In fatal accidents M= 1, and B, I and E are all 0,
which value may also be assigned when considering happening among the same
individuals. If a surrounding population be not affected 1=0 ; if affected indivi-
duals cannot move S=0.
" In the case of " independent happenings" — see later, rdt denotes merely the
proportion of affected individuals who may become reaffected in the time dt. {e.g., by
divorce in marriage). In " dependent happenings" it implies loss of capacity for
affecting others (e.g., in infectious disease it implies both immunity and loss of
infectivity). In some diseases r may be zero (e.g., leprosy and organic diseases, fatal
accidents) ; it may be of small value (e.g., many zymotic diseases) ; it may be of
high value (e.g., snake-bite, heat-stroke, etc.), and it may be imity (e.g., alight
accidents).
» To the former belong cases which are attributable to what may be called
external causes (e.g., accidents, non-infectious diseases, etc.) ; to the latter belong
all cases attributable to propagation from individual to individual (e.g., infectious
diseases, etc.).
MISCELLANEOUS. • 447
In independent happenings, therefore, the happening falls upon the
same proportion Qidt) of the population in every element of time. Put
x= Q/P and P— Q for A, then equations (669) and (671) give :—
(673) dP/dt = vP — {v — V) xP
(674) d {xP)/dt = hP {1 — x) + (V — B- r) xP
and by difEerentiation : —
(675) d (xP)/dt = xdP/dt + Pdx/dt.
From these three last equations, we have after dividing by P, and
eliminating d {xP)/dt and dP/dt : —
(676) dx/dt = h -{h + v — V +B + r)x+{v — V)x'^
which gives one form of integral if v— V=0, and a different one if v and V
are unequal.
When the sum of the variation elements of the affected group is
constant the case may be called the equivariant case, the total population
is unaltered. 1 Putting :-^
(677) K = h+ B+ r; L = h/K ; y=L—x; hence
(678) dx/dt= — dy/dt = K {L—x)=Ky ; dy/y= — Kdt ;
which gives on integrating :■ —
(679) log y=~Kt+C,OTy = y„ e-^\
yo being the value of y at the beginning of the " happening." Con-
sequently, since y„ = L — Xq : —
(680) X = L — {L — Xo)e-^t
viz., the proportion of the total population affected at the time t, the pro-
portion being Xq when ( = 0. *
When V is not equal to V, we have the general case of independent
happenin^gs which involves the integration of (676). This may be written
in the form : —
(681). . . .dx/dt = K (L—x) L'—x) = K (a—^—x) (a+^—x)
1 An example would be the occurrenee of slight accidents in which case r=l,
.or the attainment of a oertaia standard of wealth tending to diminish simultaneously
the birth, death, and migration rates of the affected by an equal decrement. If the
progeny of the affected are also aSected B should be omitted from (670), and in-
serted in (671), and will disappear in. (674) and (676).
' Obviously in (673) if v — F=0, a differential equation of the sajne form as
(678) is obtained, hence P= P|,e"', formula (2), p. IQ herein.
448 APPENDIX A
in which a={h+B+r+ K)/2Ka,nd j8 = ^(a^ — h/K), the roots
L= a — j8 and L'= a+jS, being always real and positive when v > V.
This gives : —
(682).... x=L-(L'^L) (I^z,)/l(L'-x„) e^^P'~{L-x„)\
which simphfies sUghtly if x^=Q. The relative number of the affected
depends upon whether K, that is whether v— V is positive or negative,
the former being usually the case in injurious happenings and the latter
the case in beneficial ones. This gives :■ —
(683). .P=P„e'". Le-^i'^/(L'—L) ; or Poe"". ~Le-^^'''/(L'—L)
the former expression being appropriate when K is positive, the latter
when it is negative.
Among dependent happenings the case of proportional happenings is
important as a first approximation to the study of the infection of a com-
munity. In this instance A is a function of Q and consequently of t.
If each affected individual affects c others in a unit of time the total
happenings in the time dt will be cQdt. The number of new cases per
element of time may be taken as probably : —
(684) Gdt/cQdt = A/P ; or G = cQ {I — x) ; h = ex.
This gives : —
(685).. dx/dt =Za;(i;-a;), in which K=o—v+ V; L=l-{B+r)/K,
from which may be obtained : —
(686) x=L/{ 1+ {L/xo- 1) e-K^"f.
This gives regular bell-shaped curves : x^ and Qg can never be zero.
Sufficient has been indicated to shew the value and reach of Prof.
Ross's analysis of the question, and to render evident the fact that it is
the foundation of a rational theory of "occurrences" of any kind, which
can be numerically defined, in a population.
7. Actual statistical curves do not coincide with elementary type
loims. — The importance of a rational theory of " happenings" does not
consist in the fact that the curves deduced from elementary suppositions,
meticulously correspond to actual statistical frequencies, but in the fact
that deduced types give the general configuration. Since in actual
cases what may be called the frequency of initiation is variable, the deduced
forms of frequency at any given moment are only partially applicable to
actual cases. Moreover any assymetrical and polymorphic curve, and
indeed even any regular curve, can be built up in an infinite number of
ways. The dissection of a, curve into additive components is therefore,
MISCELLANEOUS. 449
in general, purely empirical. Although this is so, when extra-mathe-
matical reasons exist for the acceptance of an hypothesis of constituent
elements, whose origins, and general characters, are known, it may be
possible to effect an analysis into components which yields a real and not
merely a formal interpretation.^
In general, type-curves, the interpretation of which is impossible and
is ignored outside certain selected points (e.g., the points where they meet
the axis of abscissae) are logically unsatisfactory.
The function of a " theory of happenings" and of the " theory of
probability," is therefore one of guidance ia interpretation, and of
deciding as to the applioabihty or otherwise of particular types of mathe-
matical expression for the representation of the change of frequency with
change of the variable. Mere arithmetical tests of the " goodness of
fit" of particular mathematical expressions are significant or otherwise
according as they conform to what is known a priori, or is deducible from
a priori considerations, and these must certainly be taken in conjunction
with the observations over the whole range of experience. *
8. International norm-giaphs and type-curves. — The function
strved by the creation of norms has been indicated in Part VIII., § 6,
p. 102. When norms for every important population-character
have been computed, it is desirabk that they should be graphed and used
internationaUy. This could be done by printing squared graphed paper,
with the norm shewn thereon, say in pale colour (or by a very fine line).
The graphing of the same character on such paper for any particular
population, would then immediately disclose the nature of its deviation
from the normal. In this way the population phenomena could be
graphically studied in their comparative relationships.
An extension of the system would be for each country to shew by
pale tint not only the international norm, but also its own norm for (say)
the previous decade.
Type-curves for international use would also greatly assist in the
work of a better technical reduction of statistical results. The forms
desirable or necessary would doubtless be more readily recognised when
the international norms had been obtained.
^ For example in the harmonio analysis of tides, the forma and periods of the
components are determined by celestial positions (i.e., of the sun, moon, " anti-sun,"
" anti-moon," etc.), and the elements to be ascertained are merely the epoch of each
component and its amplitude.
* For example, to systematically vary the representation of facts in order to
agree with some adopted mathematical expression to which it is thought they oitght
to conform, is only to delude oneself. The character of terminal conditions is often
known a priori, and the mathematical expression representing the facts should not be
merely one in substantial arithmetical agreement with the frequency, but one which
expresses as accurately as may be the law of its change. Similarly, the adoption of
an expression which disturbs the observed critical values of the frequency, vitiates
tjie results,
450 APPENDIX A.
9. Derivative elements from population-theory — It is beyond the
purpose of this monograph to discuss the various derivative branches of
the theory of population ; such, for example, as the estimation from
probate-records of the aggregate of private wealth ; of the economic
value of an average man or woman ; of the economic value of different
classes of persons ; the cost of; and economic value of, education, etc.
The present increasing length of life tends to give a higher average
economic value other things being equal — to an individual : the average
wealth possessed per individual is probably also increasing.
Although all that relates to population may, in a comprehensive
view, be regarded as belonging to its theory, it is quite appropriate that
purely economic questions should be separated out. Therefore, while
results obtained by means of the development of the population-theory
are essential and are of the first order of importance, in any attempt,
for example, to reach decisions as to the economic aspects of population,
the questions that arise are so extensive that they must be treated
independently. Nevertheless, the value of a suitably developed theory
of population is not seen until it is viewed in the light of all its applica-
tions among which the economic is but one.
Similar observations apply to the anthropometric elements of the
population. These are probably correlated with elements treated here-
inbefore ; nevertheless, it is preferable to deal with them independently.
10. Tables for facilitating statistical computations Mathematical
tables of various kinds have been prepared for faciUtating statistical
computations, among which may be specially mentioned "Tables for
Statisticians and Biometricians," by Prof. Karl Pearson, F.R.S., etc.
In this monograph the following tables are solely for facUitating the
computation or illustrating the mode of deducing quantities which
enable required quantities to be found by inspection : —
Tables I. VI. XVII. & XVIII. XXXVI. & XXX\^I. XL. LXV. LXVI
Pages 20 77 123 159 163 217 219.220
Tables LXVII. LXXV. LXXXI. CXLIV.
Pages 221-222 247 266 398
11. Statistical integrations and general formulae.— Reference has
already been made in Part VI., §§ 6-8, pp. 82-84, to statistical integra-
tions, and references were given to various tables, see p. 82. The integ-
ration of functions of a single variable is the subject of one of the
Cambridge Tracts in Mathematics and Mathematical Physics, No. 2.
This and the works previously mentioned will enable most integrations
occurring in practical cases to be effected. For convenience the following
are given ;—
MISCELLANEOUS. 451
Table of Integrals and Limits.
/^ .log^. lx>9(i + f )•.y^^=ilog(a.±6):/.^|^ = ^:.^-~?logC«^S):
2a. "'^ 20.^^9 9 2axt6v3 2a 'i'^ cu^Fp ^Tp at^o^+o; «-^.jj^^/o,
>/ — oicTS 2a. a.' ^ J? ''=''■ ' "^ " c^
/x'dc.^'.-^{^'.az^f)(^'-a^^f), etc. . f^^a^^m^) = /^,imt/3.6^-4«4
/^.iloa.^'-A /:^,s«eabove! /^ =i_ A^]oq#V %2«?/"-# , see above :
fiisn{ajc*h)da- — i-]ogcos(aa;+8J :ycot(aj5+S)<ie-^]ogsia(aa:t*):
J "■ m.loga "^
452 APPENDIX A.
Table of Integrals and Limits.
fjlLH-r =- —Sl a-' log a-' a'Claqa)'' a'QBqaT" , Ooaar ^ fa'^r. ■
yV-'"V-r Y-li n^^ n-^' ^^o;''^ •> . Tu^'l oq j: ) i nx. n'sc' i . n.'j'QoqaO 'iJ. Tut n-'j:' K.
^"'■'"•cte-yV'^il*'urloga;ti,(7vx3oga:)V^,(.7ijr:log«)'+- } d» :
•av5iA.= e(cos?*l^isia«) : ^i-logx cte- ^(Jogx-i^g) . true aUo if^O.
I 1 f dx- ■ Joa a 1 i 1 . 1 1 i.
■'l7H.-l)e/x(at»x;'^' (m-i)B0"'-' Xm-i)a4l(pi-Z)/>"'"' (m-3)ajzi"'-" ^ (m.-V^a'^^'^-*' tar- '■fi'^
'5^>-(^9-)"-^ = ^{Oo9xr-^,(togxr.|^aio3x)-- . -^^^^^ ) :
^3^4^,1^ = -. i'„^^-^'^'^"'''''.I'J^-l = J)enofc]ogCl.j:c)Tori,g'x;
1 (l+a)» = e : I tlva:)- =1:1 (l-^f- I Cl**y)' - e*j 1 «Ji -Jcqa. .-. X £^=i
I(log*Ca>fl) -log*a:)= : I (H-|+|*- ti -logx) - y- 0-5772156G490--- -EuIex's canst.
1. K l*i+A--4Vlo9 =c} = 1:1 cC/x. -oo : I a7x-"'= oo if»77T. lie + :I e^. S ;
LxaoQJ:-0: I. Jogx/x»-=€x. : X«»loqj:--oo : I ^. : I "t(TO-i^..-l7n--7t+l)
Iix*'=l,.-. 0°'l (nDtiroanafi^r) r l^'^'^l :i a:"""^ 0: I x.^i.e^-^ = e -. 114"= e^ "'""^
«-+" ""^o '-+0 *-*' J£-flie espisaeteani
I(l+3o9»)==-^ = e: I Ci'"<-2'V-'».')/'a.'"-^: L {(a-i)'+(a:+2)%- +{a.-v770'}/77i.'*H.i
3IaillBCrum.~^Qu£S. T(x=X-4€16521-j»0'8556O32 ■• : a:* fonar- 0-3678794- =0G922OO7.
1 (l+^i.)''=I [U-+^x)'tI ;if j^<i-0,J^a.oo,jma.;2<a.;^a..77j^,11ifin.I $(;c).e'?'
In the above Tables the sign "=" merely denotes that a following
quantity is also a value of the Lategral, not that it is necessarily equal,
as will be obvious in the first few examples. The general formula
facilitate the integration of many types of expression occurring in
statistics.
XVra. CONCLUSION.
1. The larger aim o£ population statistic. — ^Inasmuch as population
is the foundation-element of all branches of social statistics, its complete
study is both of practical value and general interest. For this reason an
adequate "theory of population" has become a necessity. Moreover,
international relationships have made it evident that the proper co-ordina-
tion of the whole world's statistical method and effort has become an im-
perious need of civihsation. ^ This emphasises the importance of the for-
mation of a basis for international comparisons, and is a desideratum yearly
becoming more urgent. Only by a sufl&ciently wide survey of human
facts can the required norms of all sorts be estabhshed, norms which
represent the characters of the great unit constituted by the aggregation
of all the nations. It is only in the comparatively slow secular changes
of these norms, that the drift of mankind in the gross can be unequivocally
revealed ; when that drift is ascertained, the quicker and more marked
variations of individual nations and populations can then be forced to
disclose the real significance of their differentiating tendencies. The
limits of human expansion are much nearer than popular opinion imagines;
the difficulty of future food supplies will soon be of the gravest
character ; the exhaustion of sources of energy necessary for any notable
increase of population or advance in the standards of living, or both
combined, is perilously near. Within periods of time, insignificant com-
pared with geologic ages, the multiplying force of Uving things, man
included, must receive a tremendous check. The present rate of increase
in the world's population cannot continue for four centuries,* and the
extraordinary increase in the standard of Hving which has characterised
the last few decades must quickly be brought to a standstill, or be deter-
mined by the destructive forces of human extravagance. Very soon
world-politic will have to face the question whether it is better that there
should be larger numbers and more modest Uving, or fewer numbers and
lavish living ; whether world-moraUty should aim at the enjoyment of
life by a great multitude, or aim at the restriction of life-experience to a
few, that they may Uve in relative opulence. The statistician of the
1 This is the raison d'etre of the " Institute International de Statistique," and
the ideal aim which its activities have in view.
• If we take the present population as about 1,700,000,090, and the annual rate
of increase as 0.01159 the increase doubles the population in 60.15 years, and gives
a population 3.16 times as great in 100 years ; thus in 200 years the population
will have increased 10-fold, and in 400 years 100-fold. We thus get, at the end of
successive centuries, the following populations in round numbers : — 100 years,
5,380,000,000; 200 years, 17,040,000,000 ; 300 years, 53,930,000,000 ; 400 years,
170,710,000,000.
454 APPENDIX A.
futiire will utilize all discovery of the mysterious play, and no less
cryptic, limitation of life-force to make prediction sure. Given co-
ordinated international effort, there would be no difficulty in so directing
future statistical technique that all countries and all analysts could add
their quota in a form suitable for the wider study of the drift of man-
kind in the more important relations of civic, national, and international
life.
In earlier days monarchs utilized statistic as a basis for judging the
probability of success in operations of war and plunder. That use has
not disappeared, but the plexus of relations, which, through the fructi-
fying power of science, the modern world has seen established, particularly
in the realms of industry and commerce, has shewn a growing measure of
economic solidarity in the affairs of mankind. The modem world
responds to everything that profoundly touches any one nation. By
the conditions of modern hfe mankind tends to be welded into a unit.
By the magic of invention, humanity has been quickened ; distance — ^if
not annihilated — ^has been immensely shortened ; life has been enriched
in the potentialities of material and psychical enjoyment, and be it said
also in the plane of its possible intellectual and moral effort. The destiny
of mankind will therefore be the supreme problem of those statisticians
of the future, who have an adequate outlook on that science and art
with which it is their privilege to concern themselves. For the craftsman
with acute and microscopic vision there are a multitude of analyses to be
made ; for one with the capacity for reaching wide generalisations there
is no end of larger work, while for him who is happily able to see both
the trees and the forest of the statistical landscape, there is the most far-
reaching task of all, the creation of a statistical world-picture, which
shall reveal the secrets of man's place in the many-sided world of social-
economics, using that word in its fullest and most ideal sense.
2. The impossibility of any long-continued increase o! population
at the present rate. — An increase of population at the rate of 1 per cent,
per annum is often regarded as unduly slow ; the increase for the United
States between 1790 and 1860 was nearly 3 per cent, per annum, a rate
which has recently also been attained in AustraHa. That this rate cannot
possibly last even five centuries is a fact, however, that, though immensely
important, is not realised.
It has been contended in reply to Malthus that experience has shewn
that food-production will advance even more rapidly than the growth of
population. It can do so for only a very limited time. The false infer-
ence has been drawn from this fact that therefore ahnost any population
can be provided for. The point demands attention, for the argument
is a plausible one. Notwithstanding this it is invalid, as can easily be
shewn . ^
1 See L. Hirsch, La th^orie de la population de Th. R. Malthus, Biblioth^aue
Universelle, Deo. 1916, No. 252, pp. 553-567, and Jan. 1917, No. 253, pp. 141-154.
CONCLUSION. 456
If the earth's present population be taken as low even as 1,500,000,000
persons (which is, of course, an underestimate), and its land area, exclud-
ing the Arctic and Antarctic continents, be assumed to be, say,
33,000,000,000 acres ; and if further it is supposed that by some means it
is possible to make the whole of this land-area yield an average of as much
as 22.8 bushels of food-corn per acre, per annum, the total yield would be
only 752,400,000,000 bushels.
In Australia, and in fact generally, the food-corn consumption is
on the average, about equivalent to 5.7 bushels per annum, viz., one-
fourth of the amount above assumed, which means that the total popula-
tion which could be fed with 5.7 bushels of food-corn per annum together
with other foods in like proportion, would be only 132,000,000,000. At
a rate of increase of population of 0.01 per annum,^ somewhat less than
the rate for all countries which have accurate statistics, it would require
only 450 years to exhaust the food requirement mentioned (more exactly
449.96 years). That no possible increase of the earth's reproductiveness
can materially affect the question can also be readily shewn. For — to
postulate the impossible — let it be supposed that every acre of area on the
earth's entire surface could produce as much as 228 bushels, that is, ten
times the above amount, with other foods in Like proportion : this being
done, it would take less than 700 years (681.37 years) for the population
to exhaust the food supply. " The fundamental element in Malthus' con-
tention is thus seen to be completely established. Even a low rate of
increase must soon exhaust the possibilities of food-supply, and as we
have seen already the material of the earth is inadequate to provide
bodies for any long-continued increase quite apart from the food question.
It is quite clear therefore that statistical analyses of the world's progress
in various ways will soon become of the highest order of importance.
3. Need for analysis of existing statistical material. — ^At present there
exists a large and accumulating mass of unanalysed material. Numerical
data have in many instances already become a burden, and in other cases
threaten to become one. But when their significance has been pene-
trated they seem no longer tedious ; they have been transformed into
illuminating and interesting facts.
Here, however, we need a word of warning. The problem of all so-
called knowledge is to subsume what we know — or think we know — ^under
suitable elementary conceptions, conceptions, in fact, that are within our
intellectual grasp, and that we can mentally handle. As in physios the
Boyle-Charles gaseous laws, the molecular law of equal numbers in equal
volumes at equal pressures and temperatures, and the conception of mass
1 The number of years in which a population is doubled is given by the
following quantities divided by the increase. When the increase or
divisor is .000 .010 .020 .030 .040 .050
the numerator is .6931 .6966 .7001 .7035 .7069 .7103
( See also footnote p. 31.)
436 APPENDIX jA.
as independent of velocity,^ are but crude statements of the actual facts,
so crude that their elementary simplicity entirely disappears when neces-
sary qualifications are made, so likewise does a deeper knowledge of
statistic reveal that relations subsisting among crude data are subject
to corrections that, not infrequently, are very elaborate. The more simple
and obvious of these relations constitute a kind of rough frame-work
about which more subtle and accurate conceptions may cluster,* or, to
change the figure, they are a skeletal foundation on which the body of
justly conceived statistic is to be built up.
4. The trend of destiny. — ^To the extent man is ignorant, he is both
the puppet of Fortune, and the victim of Desire. Anyone who has
seriously refiected upon the facts of the last ten decades must realise that,
within the next ten, tremendous problems will arise for solution and these
wUl touch fundamentally the following matters, viz. : —
(i.) The multiplying power of the human race ;
(ii.) The organic constitution of Nature and the means at human
disposal for avoiding the incidence of its unfavourable aspects ;
i.e., eugenics .LQ its wider sense ;
(iii.) The enhancing of the productivity of Nature, and the limits of
its exploitation ;
(iv.) The mechanism of the social organism, and the scheme of its
control ;
(v.) Internationalism and the solidarity of humanity.
For the adequate study of these matters, not only will the mere
technique of the collection and analysis of statistic require to be much
advanced, but the popular opinion as to the value of the effort will also
have to progress. Given, however, an intelligent public opinion, as to the
utility of statistical inquiries, there would be some ground for hope that
the great questions, the analysis of which would throw Ught upon human
destiny, could be properly attacked. It is for educational departments,
worthy of the name, to create such opinion by the mechanism of their
systems, in order that each human being should be sufficiently interested
to cordially co-operate, by accurately furnishing the necessary data in the
taking of a census of population or wealth. Census-taking is a costly
operation, but it is the foundation of all branches of statistic that have a
direct human interest. Its value and the facihty of using it would be
immensely increased if it were meticulously accurate. The importance
of technique and of precision, matters apparently of httle moment, can
be rightly estimated only when the ultimate aim of all statistical inquiry
is realised to be " the study of man's destiny" as the denizen of a world
of limitations.
1 According to modem views " mass" in matter becomes infinite when its
velocity equals that of light.
• Thus, for example, the crude mortality of one popiilation may agree with that of
another, but when corrected may seriously differ, shewing either a better or worse
state of things regarding the conditions of human life.
INDEX.
APPENDIX A.
A Page.
AbsciasBB, oentroid verticals bounded
by curve. Table LXXXI. . . 266
Abscissa, exact value of, corresponding
to quotient of two groups . . 395
Actuarial population, theory of . . 407
Age at marriage, average differences 226
Beginning and end of fertility . . 238
Mother, effect on total issue all
durations marriage, Table XCII. 281
Belationshi^s, conjugal . . . . 224
Age-fecundity distribution . . . . 334
Surface, Fig. 91 335
Age-genesio distribution . . . . 333
Ageneeio surface defined . . . . 265
Age-polyphorou3 distribution, Table
CXXII 336
Surface, Fig. 92 335
Ages at marriage, error corrected.
Table LVI 194
At marriage, errors in . . . . 193
Exact, least mortality . . . . 401
Aggregate mortality, composite char-
acter of 392
Aggregates, areal and volumetric for-
mulae . . . . . . . . 75
Statistical, group-heights, values
of. Table VIII. .. •.-.■■ 80
Statistical, group-subdivisions,
value of 80
Summation, and integration of . . 75
AhUeld 307
Zeit f. Geb. u. Gyn., 1902, p. 230.
Allen, J. M., F.I.A., " On the relation
between the theories of com-
pound interest and life contin-
gencies." Jour. Inst. Aot., Vol.
XII"; p.p. 305-307 .. ..409
Anamorphosis . . . . . . . . 297
Projective . . . . . . . . 45
Annual rate increase, various popula-
tions. Table III. . . . . 30
Ansell, Sterility Estimates . . . . 327
Arithm.etioal mean, error of, rate con-
stant . . . . . . . . 12
Not constant . . . . . . 13
Assymetrioal curve . . . . . . 448
Auerbach, Felix : Graphischen Dar-
stellungen . . . ■ ■ ■ 9
Average age, quinquennial age-groups
primiparae . . . . . . 257
Issue, non-linear according to dura-
tion of marriage . . . . 282
Life, children dying before 1 year 151
Number •children born, varying
intervals after marriage, 1908-14,
Table XC 280
o Page.
Barford, F. W., " Studies in Statistical
Representation" . . . . . . 44
Beltiotio coefficient . . . . 388
Bernheim, A., Philadelphia . . . . 306
Deutsche med.Wochenschrift 1899,
quintuplets, p. 274.
Betterment, Footnote 2 . . . . 385
Footnote 3 388
Birth and immigration, non-uniform 26
In early age. Table LXXII. . . 239
In old-age, probabiUty, Table
LXXI 238
Marriage and divorce rates. Fig. 56 177
To registration, interval .. .. 151
Birth-rate, effect of marriage rate upon 166
Influence of infantile mortality . . 146
Physiological annual fiuctuation . . 172
Birth-rates, Australia, 1860-1914,
Table XXXVIII 160
Crude 143
1860-1914, Table XXXIX. . . 161
B«sidual . . . . . . . . 160
Secular fluctuation . . . . 160
Various countries, 1860-1913, Fig. 63 168
Births, annual periodic fluctuation of 166
Influence on birth-rate . . . . 144
Proportion due to pre-nuptial
insemination . . . . . . 278
Registered, Australia, 1907-14,
Table XLII 168
Seasonal fluctuations, according to
sex, Table XLIII 168
Various intervals after marriage 276
Bivitellins (see corrigenda) . . . . 307
Blaschke, E., Ph. D. Calc. of probabih-
ties, Wien 1893 362
Born in country, proportion . . . . 429
Brides and bridegroonas, nuptial and
ex-nuptialmaternity, etc.. Table
LXXIII 242
Brownlee, Dr. J. 445
1 ' Theory of an epidemic ' . . 445
2 ' Certain considerations on the
causation and course of
epidemics ' etc. etc, . . 445
458
APPENDIX A.
C Page.
Censua, piogresaive irnprovement in
results 108
Change of coefficients expressing rate 13
Of rates of mortality. Fig. 99 . . 381
Changes, constitutive, organic . . 7
In ratio female to male mortality 375
Changing ratios various age-groups,
Table CXXXVIII 379
Children borne 332, 343
Age and duration fecundity distri-
bution (VIII.) 340
Age and duration polyphorous dis-
tribution (X.) .. ... ..340
Age-fecundity distribution (III.) 334
Age-genesic distribution (I.) . . 333
Age -polyphorous distribution (V.) 335
Characteristic scheme compilation
fecundity by ages, etc. Table
CXIX 332
Durational fecundity distribution
(IV.) 335
Durational genesic distribution
(II.) 333
Durational polyphorous distribu-
tion (VI.) 336
Duration and age-fecundity distri-
bution (VII.) 340
Duration and age-polyphorous dis-
tribution (IX.) 340
Ratios, married mothers by age
groups, and durations marriage
to totals, same nimiber. Table
CXXIV 341
Women by age-groups, durations
marriage. Table CXXV. . . 344
Women who bore 'k' children, by
ages durations, marriage. Table
CXXIII 338
Climcwiterics in mortaUty, absence of . . 399
Coghlan, T. A., " Child-birth in New
South Wales" 272
Complex Elements ; Fertility and
Fecundity 297
Component-elements of force of mor-
tality. Table CXLVUI. . . 413
Conjugal conditions, features of fre-
quencies. Table LI. . . . . 186
Num.bers at each age, Table L. . . 183
Batios, ciu:ves of . . . . 185
Conjugahty and nuptiahty norms . . 232
Constants, exponential curves . . 40
For periodic fluctuations . . . . 38
Constitution, conjugal, of populatioj;! 180
Conjugal, of population, Austraha,
3rd April, 1911, Table XLIX. . . 182
Of popijlation . . . . 2
Continuous interest, development of
theory (Allen, J. M.) . . . . 10
Contours diisogenic . . . . . . 349
Correction, computed average interval
marriage to first birth, popula-
tion increasing. Table LXXXVI. 275
Protogenesio interval population,
characters not constant . . 274
Page.
Corrections, fertility-ratio, (13-23) for
previous maternity, Table XCV. 291
Necessary in statistics involving
duration . . . . . . . . 298
Correlation, owing to migration between
age and length of residence 43 1
Crude death-rate . . . . . . 370
Curve-constants, determination inter-
mediate from instantaneous
values . . . . . . . . 34
Curve, assymetrical . . . . 448
Exponential, for variation of rate.
Fig. 2 23
Of organic increase or decrease . . 394
Polymorphic 448
Curves, actual statistical, do not
coincide with type forms . . 448
FJexible 62
Generalised probability, projec-
tions of normal curve . . . . 57
Of probabiUty derived by projec-
tion 250
Prof. Pearson's type-forms . . 49
Special types, their characteristics.
Fig. 8 47
Special types of . . . . . . 47
Curve-tracing, Frost's, footnote ^ . . 9
Data, soheme of examination . . 395
Davenport, C. B., statistical methods. . 298
Death-rates, annual fluctuation . . 424
Crude . . . . . . . . 370
Secular changes . . . . 372
Curves of improvement 20 years. . 385
First 12 months of ]ife,Fig. 104 . . 423
From particular diseases aaoord-
ing to age and sex . . . . 415
In age -groups, their secular
ohanges. Table CXXXV. . . 376
According to cause, age, and
sex. Table CLl 420
Males and females, also married
and unmarried males and
females. Table CXLVI 401
Per diem at beginning of life.
Table CXLII 390
Ratio of improvement 20 years . . 385
Deaths, actual number in Austraha
according to cavise, age and sex.
Table CXLIX 416
Each equalised month from various
causes all ages. Table CLIII. . . 425
From particular diseases accord-
ing to age and sex . . . . 414
From particular causes . . . . 414
Mode of voluntary. Table CLV. 427
Defects in migration records and
closure of results . . . . 439
Deformation, systematic . . . . 297
INDEX.
459
De Morgan, Prof., " On a Property of
Mr. Gompertz's law of mortal-
ity." Joum. Inst. Act., Vol.
VIII., July 1859, p.p. 181-184,
also Phil. Mag., Nov. 1839
Dependent happenings . .
De Vries. statistical methods . .
Difference, age, husbands and wives
: ' ; at census . .
.J ■ Age, husbands. Table LXIX
Differences, E^ges brides and bride-
grooms. Table LXX. . .
Evaluation of, from coefficients . .
Ilusbands any age and age of wives,
Fig. 64
Leading formulae (54-68a)
Digenous fertility and fecundity
Digenesio surfaces, & diisogenio
contours . .
Diisogenic graphs, their signifioance . .
Contours & digenesic surfaces
Surface, Fig. 97
Diisogens, their trajectories and tan-
gents
Diminution average issue by recent
maternity
Diovular and uniovular multiple births
Triplets, theoretical distribution,
Table CIV
Twins, theoretical distribution,
Table CIII
Dissection of multimodal curves
Distance from initial value any range
on axis of abscissse to the
ordinate to which ratio any two
groups should be ascribed, whole
range being unity, Table CXLIV.
Distribution unspecified quantities,
double-entry tables
Divorce acts, influence on divorces.
Table LIII
Curve, abnormality of
Frequencies of, Table LII. . .
Its secular increase
Marriage and birth-rates. Fig. 56
Statistics, desirable form of . .
Double -entry tabulations unspecified
data. Table XCVII
Duncan, , J. Matthews, term of
" Fecundity," etc
Sterility, Glasgow & Edinburgh . .
E
Easter, periodicities due to . .
Position of, for 200 years. Fig. 55 . .
Table XLIV
Economics, purpose of ....
Effect of recent maternity on issue,
various durations of marriage.
Table XCIII
Elderton, W. PaUn, statistical methods
Empirical expressions for population-
fluctuations . .
Equalization, irregular periods
Page.
405
446
297
225
226
228
37
227
36
233
349
350
349
361
352
286
306
308
307
63
398
300
188
188
187
186
177
189
300
234
327
173
173
174
6
288
297
26
171
Evaluation constants, curves various
types of fluctuation
Evans, Dr. G. H., " Some arithmetical
considerations on the progress
of epidemios " . .
Examination of data, scheme of
Excess, multiple births by ages in
number per confinement. Table
XCIX
Ex-nuptial protogenesis
Exponential curve, for migrations,
utility of . .
Curves, evaluation of constants . .
Expressing variations of rate,
examined
Page.
40
44S
395
304
257
25
21
19
Factor, survival . . . . . . 295
Factors correcting fertility -ratio for
previous maternity. Table XCIV. 290
Farr, Dr. Wm., " On the cattle plague"
Jour. Soc. Soi., 20th Mar. 1866 444
Fecundity, actual . . . . . . 235
By ages, duratiojis marriage . . 331
Correction for infantile mortality 291
Characteristics, types distribution.
Table CXX 333
Complete tables . . . . 349
Definition of 234
Distributions by ages, durations
marriage . . • • ■ ■ 337
and ages at marriage . . . . 345
During given period- . . . . 324
Existing 324
Fertility and sterility, theory of . . 319
Physiological or potential . . 235
Polygenesic and gamogenesic dis-
tributions . . . . . . ■ • 285
Tables, previous issue ignored . . 320
Femininity, definitions of . . . . 13 1
Fertility, age of beginning and end . . 238
Age of greatest . . . . . . 290
And fecundity, correspondence,
correlation . . . . . . 297
And fecundity, digenous . . . . 233
Crude, corrected for previous
maternity . . . . . • 289
Curves, Fig. 107 443
Definition of . . ■ ■ ■ • 234
Fecundity, derivation of words,
iootnote . . . . ■ • • • 234
Reproductive efficiency . • 233
Monogenous . . . . . • 233
Sterility and fecundity, theory . . 319
Tables, previous issue ignored 320
Fertility-ratio, crude, factors correct-
ing for previous maternity,
Table XCIV ..290
Fertility-ratios, nuptial, exnuptial,
compared by ages. Table CXI. 318
First-birth, according to age and dura-
tion of marriage . . . . 261
Probability various intervals after
marriage . . . . . . . . 245
460
APPENDIX A.
Page. '
Flexible curve, evaluation of constants,
Fig. 21-4 56
Fluctuation, annual, in frequency of
. ^ IKniarriage . . . . . . ■ ■ 180
IS Of births, annual periodic 166
^Secular, in birth-rates . . 160
Fluctuations, continuous, finite . . 7
Curves for and their constants . . 40
Dissection multimodal, into uni-
modal elements ■ ■ . ■ ■ . 63
For persons interstate migration
by sea in Australia, Table
CLXII 436
In frequency of births. Fig. 54. . 167
Of rate, secular, empirical formulae
for 26
Of ratio female to male death-
rates, according to age . . 399
Periodic, evaluation constants . . 38
In migration . . . . . . 435
Overland migration by rail.
Table CLXIII 438
Polymorphic and other . . . . 42
Fluent Life tables 380'
Frequencies of conjugal conditions,
critical features in. Table LI. . . 186
Of fertiUty, terminal. Figs. 66-70 244
Frequency, births after diSerent periods
between menstruation and par-
turition. Table LXXXVIII. . . 277
Births between 240 and 332 days
after menstruation. Table
LXXXIX 278
Of births, corrected, periodic
fluctuations. Table XLI. . . 167
Of death, various causes each
equalised month, Table CLTV. . . 425
Of deaths from particular diseases 414
Of initiation 448
Twins, various countries. Table CI. 306
Q
Galton, F., graphic method, Proe. B.S.
Lend 298
Francis, on Isogens . . . . 350
Qamic surface . . . . . . . . 228
Surface, theory of .. ..201
Surface, curves equal, oonj ugal fre-
quency. Fig. 66 . . . . 229
Gramogenesic, polygenesio and fecund-
ity distributions . . . . 285
General trend, modification of . . .. 382
Genesic and gestate elements in mor-
tality 413
And gestate elements in mortaUty,
Fig. 103 413
Distribution, durational . . . . 333
Geometrical forms and graphs, curves
representing . . . . . . 8
Gestate element in mortality . . . . 413
Force of mortality 376
Gestation period, range of . . . . 276
Gini, Statistical methods . . . . 297
Page.
Giompertz theory, its limitations and
developments . . . . . . 410
" On the Nature of the Function
expressing the Law of Human
Mortality " 405
Gompertz-Makeham-Lazarus theory
of mortality . . . . . . 405
Graph, polymorphic fluctuations,
simple cases. Figs. 6 and 7 . . 44
Graphs, AustraUan population, accord-
ing to age and sex . . . . 125
AustraUan population according to
age and sex. Figs. 43-44 . . 126
Diisogenic, their significance .. 350
Graphics and smoothing in population
analysis . . . . . . . . 85
Graphic smoothing, advantages over
others 124
Group-heights, formulae depending on 67
for diiierent ranges of the variable 78
Grouping repeated, coeflacients for.
Table XIV. 119
Groupings of data, non-homogeneous 224
Group-intervals, evaluation of, from
extended groups . . . . 262
Groups, average value of . . . . 73
Group sub -divisions . . . . . . 80
Group-totals, curve of for equal inter-
vals
Group-values, Adjustment of . .
Determination of, constants being
known . . . . . . . . 72
Ideal distribution . . . . . . 65
Representation by equations . . 65
Their limitations . . . . . . 64
Growth, various populations . . . . 26
Of population, rate of . . . . 31
Gyration, radius of . . . . . . 273
72
64
Handbuch. d. Med. Statistik., Fr.
Prinzing, 1906, pp. 381-2 . . 390
Happenings, theory of . . . . . . 444
Independent . . . . . . 446
Dependent . . . . . . . . 446
Herschel, Sir John, Logic of graphic
smoothing. Trans. Astr. Soc,
Vol.V 124
Hirsch, L., " La theorie de la population
de Th. Malthus, Biblioth^que
Universelle " Dec. 1916, No. 252
pp. 553-567 and Jan. 1917, No.
253 pp. 141-154 454
Homogeneity as regards populations 103
Horlacher, Wurtt, Korr. Bl. 1840,
quintuplets . . . . . . 306
Human mortality . . . . . . 370
Race, its multiplying power . . 456
INDEX.
461
I Page.
Immigration and birth, non-uniform 26
Interstate by sea, Table CLLX. . . 434
Increase, annual relative, various
countries, 1906-11, Table V. . . 31
Annual, various populations.
Table III. 30
Ot population, present rate im-
possible for long duration . . 454
Resulting from non-periodic migra-
tion 24
Various populations, rates, 1790-
1910, Fig. 4 29
Indirect relations . . . . . . 442
Independent happenings . . . . 446
Infantile —
Deaths, proportion born in year
recorded. Table XXXVII. . . 159
Proportion of, births oonstant 162
Proportion in year of record . . 158
1909-12, Table XXXVI. . . 169
Mortality andbirth-rate, relations of
about 1900, Table XXXIII. . . 147
And birth-rate, world-relation
between . . . . 147
Influence on birth-rate . . 146
Table XXXII 146
Relative frequency of, Fig. 48 160
Initial frequency twins by intervals
after marriage . . . . . . 312
Instantaneous rate increase, relation
to period-increase . . . . 11
Integrals and limits, table of . . . . 451
Indefinite and definite, Table of . . 84
Integrations, important statistical . . 82
Interval between marriage and first-
birth 257
Birth and registration . . . . 151
Evaluation of, from limited group-
values . . . . . . . . 26 1
For exponential curves . . . . 264
Marriage and first-births later than
9 months after marriage. Table
LXXXIII ..269
Intervals and groups, subdivision of 37
Average groups all first-births . . 267
In months, first-births. Table
LXXXII 267
Interstate immigration by sea, Table
CLIX . . . . 434
Internationalism and solidarity of
humanity . . . . . . 456
Isogeny, initial, or isoprotogeny . . 234
Isoprotogens and isogens . . . . 234
Isoprotogamy, Footnote 1 . . ■ . 202
Isogamy, Footnote 1 . . . ■ • . 202
Issue, according to age and duration of
marriage
279
According to age. Table CXIV. . . 322
Average and protogenesic indices.
Figs. 74 and 75 ... . . . 268
By durations of marriage. Table
CXIII ..322
Diroinution by recent maternity 286
Kiaer, A. N., Isogens, etc 350
Knibbs, G. H., " Determination " and
Uses, Population Norms, etc." . . 106
Improvement in Infantile mortality;
annual fluctuations and age-
frequency . . . . . . 294
" Nature of the Flexible Curve" . . 44,55
" Studies Statistical Representa-
tion" 42
" Studies in Statistical Represen-
tation Joum., Roy. Soc, N.S.W.
XLV. pp. 76-110, 1911, in
particular pp. 97-110 .. .. 428
" The Flexible Curve"; footnote 1 19
" The Improvement in Infantile
Mortality, its Annual Fluctua-
tions and Frequency according
to Age in Australia." Journ.
Aust. Med. Cong., Sept., 1911,
pp. 670-679 389
" Volumes of Solids Related to
Transverse Sections" . . . . 76
KOrOsi, Joseph, Phil. Trans. Lend., 1896
232, 240
Estimate as to legitimate natality 350
Land migration . . . . . . . . 438
Laska, Dr. W., Collection of Formulse 9
Least mortality, exact ages of . . . . 40 1
Lewis, J. N. and C. J., Variations of
Maeoulinity, 1906 .. ..136
Life, children dying before 1 year . . 151
Life-tables, fluent . . . . . . 380
Limits of uncertainty . . . . . . 443
Linear adjustment, co-efficient . . 439
Adjustment, simple . . . . 439
Grouping, error of . . . . . . 117
Lommatzsch, G., Zeit. f. saehs. stat.
Bureau, 1897, Bd., XLIII., p, 1 390
Loria, Dr. Gino, Algebraic and Trans-
cendental Plane Curves . . 9
M
Malthusian coefficient, the . . . . 164
Equivalent interval . . . . 163
Equivalent intervals different rates
increase. Table XL. . . . . 163
Law, the . . . . . . . • 162
Malthus, T. R., Essay on Principle of
Population . . . . . . . • 164
Male nuptial ratio . . . . . ■ 241
Marriage and birth-rates, means, 1860-
1909, Table XL VII 179
Birth and divorce rates, Fig. 56 . . 177
Fluctuation of annual period . . 180
Frequency, according to age . . 199
Curves of. Fig. 61 . . . . 209
In age-groups . . - . 211
Table LXIII 211
ProbabiUty at any age . . . . 198
In pairs of ages . . . . 224
Theory probabiUty, in age-groups 214
462
APPENDIX A.
Page.
Marriage-rate, crude . . . . . . 176
Reaction upon birth-rate . . . . 166
Marriage-rates, Australia, 1907-14,
Table LV. 193
Secular fluctuation of . . 179
Various, 1860-1913, Table XLVI. 178
Marriage-ratios of unmarried . . 232
Marriages according to ages brides.
Table CXXVl 345
According to durations marriage,
Table CXXVIX 346
At given ages, Australia, 1907-14,
Table LIX 197
Distributions according ages, dura-
tions marriage. Table CXXVIII. 347
Frequency of, in pairs of ages 189
Number, according to age, 5-year
groups. Table LX 199
Number according to age. Table
LIV. 190,191
Number in different months, Aus-
traUa, 1908-14, Table XLVIH. 180
Numbers and difierences of age . . 192
Tabulation in 5-year groups . . 198
Married jJersons together. Census, 3rd
April, 1911, 5-year groups. Table
LXVIII 224
Women bearing n children, age-
polyphorous distribution. Table
CXXII 336
Women childless. Table CXVI. . . 326
Masculinity, age-groups, censuses Com-
monwealth and England, 1881-
1911, Table XXX 140
All births 136
And femininity, definitions of, for-
mula, 333-335 131
Change with age. Table XX. . . 130
Table XXn 133
Coefficients, ex-nuptial and still-
births 137
Definitions of . . . . . . 131
Femininity, relations between,
Table XXI 132
France, 1865-1876, Table XXXI. 141
In Australia, Table XXV 134
Intensification Coefficients, W.
Aust., 1897-1913, Table XXVIII. 138
In Victoria, Table XXIV 134
Its secular fluctuations . . . . 139
Of First-bom 138
Of populations, 1900, Table XIX. 130
Of unmarried. Fig. 63 . . . . 213
Table LXIV 212
Batios, all births, W. Aust., Table
XXVII 136
Batio, still to live-births. Table
XXIX 138
Tljeories of 140
Masculinity —
Unmarried, in age-groups . .
In 5-year groups, Table LXVn.
In 2-y6ar age-groups. Table
LXVI
Variations of, according to age.
Fig. 47
Various countries. Table XXVI. . .
Maternity -ratios, nuptial & exnuptial.
Fig. 80
Maternity-frequency, nuptial and ex-
nuptial
Mathematical Analysis, its value
Conception, rate of increase
Conceptions, importance of
Mayr, Dr. Georg von. Gender and birth,
footnote . .
Mean age of population
Arithmetical, error of, rate constant
Bate not constant
Mortality, improvement in mor-
tality and ratio, relative im-
provement. Table CXL.
Population, determined, rate con-
stant
Bate not constant
Measure of precision in statistical
results
Migration and age
Effects of
Effects of. Fig. 1
Exponential curve representing . .
Interstate . . . . ~
Non-periodic, effect of
Exponential curve for
Oversea
Overland
Periodic fluctuations of . .
Batio
Batios for AustraUa, Table
CLVIII
Beeords, defects in . .
Theory of . .
Misstatement, accidental and their
fluctuations
Ages at marriage. Figs. 60, 60a . .
Age, correction-factors. Table
LVII
Distribution according to age and
magnitude
Of age, Australia, 1911, Figs. 37,
38
Of age in years. Fig. 41 . .
Of age, ratio, censuses 1891, 1901,
1911, AustraUa, Table Xn. . .
Of ages, analysis 1660 cases,
census, 1911, Table XIU.
Of age, theory of correction
Belative frequency of. Fig. 40
Smoothing of populations in age-
groups
Systematic, characteristics of
Elimination of
Modification of general trend . .
Moments, approxim.ate computation of
Monogenous fertility and fecundity
Page.
218
221
219
139
135
292
240
2
10
4
7
106
12
13
386
11
12
441
439
18
19
25
435
18
22
435
436
435
433
433
439
431
111
194
195
114
110
114
111
113
109
113
116
112
119
382
81
233
INDEX.
463
Page.
Monogenous natality . . . . . . 233
Mortality, class . . . . . . . . 370
Composite character of aggregate,
according to age . . . . . . 392
Cxirve and probability of death,
relation between . . . . 40S
Curves, variation with time, Fig.
100 387
General 370
Gestate force of . . . . . . 376
Human . . . . . . . • 370
Improvement ratio . . . . 386
Improvement ratio, significance
of variations . . . . . . 387
In earlier childhood . . . . 412
Least, exact ages of . . . . 401
Bate of, beginning of life . . . . 389
Rates as related to conjugal
condition . . . . • . . ■ 401
Changes of. Fig. 99 . . . . 381
Fig. 101 391
In childhood. Table CXLVII. 412
Norm of 413
Secular changes according to age 374
Theory of variation with age . . 402
Moser, Ludwig, ' Die Gesetze der Leb-
ensdauer ' . . . . . • 405
Mothers, average age, first-births, 5-
year groups, Table LXXVIII. . . 257
Multimodal and unimodal fluctuations 63
Mutiple births, Australia 1881-1915,
Table C 305
Frequency (Germany) 1906-11,
Table Cn 307
Frequency by previous issues.
Table CIX 315
Relative frequency . . . . 305
Secular fluctuations . . 316
N
Natality, general
Index
Monogenous
Tables
Natural resources, effect of
E fleet of increased knowledge . .
Efiect on population
Nature's resources, exploitation of . .
Non -homogeneous groupings of data . .
Non-hnear average issue according to
duration of marriage . .
Non-periodic migration, exponential
curve
Norm, for comparative purposes
Of mortalily -rates . .
Population, reproductive efficiency
and genetic index
Representing constitution popu-
lation according to age . .
142
237
233
236
1
17
16
1
224
282
22
6
413
237
105
Page.
Norms, Dr. Ogle's proposals, 1891 .. 105
For masculinity and persons . . 132
Importance of oreating . . . . 103
Nuptiality and conjugality . . 232
Of conjugal ratios .. .. .. 186
Of population . . . . . . 104
Of masculinity and femininity . . 131
Population for 1900, Table XI. . . 106
Variations of . . . . . . 104
Norm-graphs and type-curves . . . . 449
Number at confinement, function of age 303
Bom in country . . . . . . 430
Deaths in Australia according
to cause, age and sex. Table
CXLIX 416
Dying per month first 4 weeks
of Ufe, Table CLII 422
Nuptial and ex-nuptial maternity . . 243
And ex-nuptial maternity, fre-
quency, Figs. 66-70 . . . . 244
Exnuptial fertility compared . . 317
Fertility-ratios, Fig. 87 317
Fertility-ratios compared by ages.
Table CXI 318
First-births, all durations marriage,
all ages. Table LXXVII. . . 262
First-births, proportion, various
intervals after marriage. Table
LXXXVII 276
Protogenesic maxima, curve . . 266
Surface . . . . . . 265
Ratio, male.. .. .. .. 241
Nuptiality and conjugality norms . . 232
Nuptial-ratio, defined . . . . . . 176
Nyhoff., Groningen, Zeitschr f. Geb. u
Gyn 306
Occurrence frequencies theory of . . 444
Ogle, Dr., Proposed Norma for various
Rates 106
Organic increase or decrease, curve of 394
Orthogonal trajectories . . . . 203
Oversea migration into and from Aus-
tralia, 1909-19 13, Table CLXI. . . 435
PaUn Blderton, W 62
Pearson, Prof. K., Type forms for statis-
tical curves . . . . . .49, 63
" The chances of death " Vol. I.,
1897, p.p. 1-41 402
Peithogamio infiuence . . . . 313
Periodic ohajiges, minor, elimination of 7
Elements, non-periodic represent-
ation of . . . . . . 17
Persons living in but not born in Aus-
tralia, according age and sex,
Table CLVII 430
Physical and psychical characters of
population . . . . . . 102
physiological or potential fecundity. . 236
464
APPENDIX A.
Plasticity curve . .
Polymorphic curve
Fluctuations
Polygenesic fecundity and gamogenesic
distributions
Surface
Population, characteristics of increase,
secular
Conjugal constitution of ..
"de facto," "de jure"
Fluctuations, nature of . .
During given period . .
Of, through births,
and migration
Growth of, rate identical all ages
Varying rates . .
Increase birth and immigration,
non-uniform
Masculinity of
N. S. Wales, Table XXIII. . .
Mean age of . .
Norms for 1900, Table XI.
Numerical constitution, at given
moment . .
Oscillatory fluctuations of
Physical and psychical characters
of
Prediction of future
Proportion contributed, various
age-groups, etc.. Table CXXI.
Proportion sterile. Fig. 88.
Bange of the wider theory . .
Statistic, larger aim
Theory necessary . .
Population-characters, oonspectus of . .
Populations, various countries. Fig. 3,
and Table II
Prediction of population (Watson) . .
Prinzing,H.,HandbuohdMed. Statistik
Probabilities first-birth to 6 years from
marriage. Fig. 71
Marriage and maternity, maximum
Maximum of first-birth . .
Probability, birth at early ages. Table
LXXH
By age, nuptial exnuptial twins,
triplets. Table CV
Curves, projection on various sur-
faces. Figs. 30-33
First-birth various intervals after
marriage . .
Marriage, in age-groups . .
Marriage in pairs of ages . .
Nuptial first-birth to 6 years from
Marriage, Table LXXV.
Nuptial, exnuptial, confinements
their ratio, 5-year age-groups.
Table CXII
Of Birth in old-age. Table LXXI.
Of death and mortality curve,
relation between
Of first-birth, maximum. Table
LXXVI
Of twins, according to age, nuptial,
exnuptial , , , , , ,
Page.
389
448
42
285
331
6
.. 180
6
6
.. 99
deaths
99-100
127
128
26
130
133
106
105
98
5
102
129
334
321
102
453
1
96
27-8
1
306
249
245
248
239
310
61
245
214
223
247
319
238
408
248
309
Page
Productivity of nature . . . . . . 456
Projection, oblique, of probability
curves, on plane . . . . 60
Proportion bom in Australia, Fig. 106 430
Nuptial first-births, various inter-
vals after marriage. Table
LXXXVn 276
Protogamic frequency, apparent pecu-
liarities 208
Surface, oharacteristics of. Fig. 62 210
Characters on . . . . 203
Contours 208
Positions for 5-year groups.
Table LXII 207
Positions for year-groups.
Table LXI 205
Theory of 201
Protogenesio Index for Australia, Table
LXXXV. 270
Index, from age at, and duration
of, marriage . . . . 271
Indices, according to age, Table
LXXX 259
And average issue. Figs. 74
and 75 268
Interval, first-births, not earlier
than 9 months after marriage.
Table LXXXIV ^70
Unprejudiced . . . . 268
Quadratic indices and intervals . . 272
Surface, Fig. 72 255
Profiles, Fig. 73 . . . . 256
Q
272
Quadratic intervals and indices
Quinquennial age-groups, primiparse,
average age . . . . . . 257
Quintuplets . . . . 306
Bemheim, A., Philadelphia . . 306
Horlacher, Wiirttemberg . . . . 306
R
Radius of gyration . . . . . . 273
Rates, as related to conjugal condition 401
Beginning of life . . ' '. . 389
Changes of. Fig. 99 . . 381
First twelve months of life . . 415
Kg. 101 391
Mortality in childhood. Table
CXLVII .. ..412
Of first to all births and proba-
bility exnuptal birth. Table
CLXIV 444
Secular improvement. Table
CXXXIX 383
Ratio, female to male mortality, Table
CXXXVII 378
Fluctuations of female to male
death-rates according to age . . 399
INDEX
465
Page.
Ratio in age-groups of deaths from
particular causes to total deaths
from all causes, Table CL. . . 418
Infantile to total deaths, Table
CXXXIV. (A) 374
Of male migration to total migra-
tion, proportion Males Females
and persons under 12 years to
total emigrants, Table CLX. . . 434
Variation of female to male mor-
tality-rates by age. Table CXLV. 400
Ratios between mean mortality and
mortality - improvement - ratios.
Table CXLI 388
Changing, for different age-groups.
Table CXXXVIII 379
Conjugal, Australia, 1911, Fig. 59. . 185
Norms of . . . . . . 186
Curves of conjugal . . . . 185
Female to male death-rates and
rates infantile mortaUty, Table
CXXXVI 376
Married to unmarried wonxen.
Table XLV 175
Regularity of unspecified to speci-
fied cases. Table XCVIIl. .'. 302
Regularity, ratios unspecified to speci-
fied cases, Table XCVIIl. . . 302
Relation between infantile mortahty
and birth-rate, Table XXXIV. 149
Reproductive efficiency, measurement 235
ficproductivity, crude and corrected . . 293
Nuptial, exnuptial, secular changes.
Table XCVI 293
Secular trend . . . • . ■ 292
Residual birth-rates, Australia, 1904-14
Table XXXV 152
Results, subdivision of, for equahsed
quarters . . . . • • • • 169
Resources, dependent on human inter-
vention, infiuence of . . . . 17
S
Savorgnan, statistical methods 297
Sohroeder (Lehrb. d. Geburt.) twins,
triplets, quadruplets . . . 308
Secular changes in crude death-rates 372
Changes, in mortality . . . . 374
In mortality, determination of
general trend . . . . 382
Of death rates, Table CXXXIV. 373
Vary with age . . .. ■• 378
Fluctuations of rates, empirical
expression of .... ■ • 26
Infiuences on rates of increase . . 14
Improvement, mortality rates.
Table CXXXIX. .. ..383
Trend of reproduotivity . . 292
Senile element in force of mortality . . 411
Element in mortality. Fig. 102 . . 412
Sheppard W. F., statistical methods . . 297
Significance variations in mortality
improvement ratio . . . . 387
Page.
Smoothed or graphic results, testing of 94
Smoothing coefficients, table of. Table
XVIII 123
Of sxufaoes 229
Processes, characters of . . . . 88
Solidarity of humanity and Interna-
tionalism . . . . . . . . 456
Solution exponential curves, values
t log. t, etc., Tablo f 20
Spencer, Wells, & Sims, sterilities . . 327
Sprague, T. B., M.A. Translation
Lazarus' paper Jour. Inst. Act.,
Vol. XVm., pp. 54-61, 212-213 406
Statistical data, elements of original,
Table X 96
Data, justification for smoothing 87
Smoothing, graphic methods 88
Object of smoothing . . . . 87
Theory of smoothing . . . . 86
Integrations and general formulae 450
Material, need for analysis . . 455
Results, measure of precision . . 441
Standard of Uving, effect of . . . . 1
Sterility -ratios, according to age, Fig.
89 329
By ages, durations marriage, 327
Table CXVII. . . . . 328
Curves by durations marriage . . 331
Curves of equal. Fig. 90 . . . . 329
Degree, all ages, durations marri-
age. Table CXVIII 330
Fertility and feoimdity theory . . 319
Proportion sterile. Fig. 88 . . 321
Still-births, masculinity, ooefficients 137
Sub-division of groups . . . . . . 80
Of groups, population and other . . 440
Suicides per diem population 1,000,000,
Table CLVl 428
Summation-formula-coefficients, Table
XVEI 123
Methods, defect of . . . . 121
Eliminating error, weighted
mean . . . . - . 120
Papers on, various authorities 122
Processes, smoothing ooefficients.
Table XVIII 123
Surfaces digenesic . . . . . . 349
Survival coefficients, progressive changes 296
Factor .. .. .. ..295
SX^stematic error, elimination of. Table
XV 119
Error, elimination of. Table XVI. 120
Table of integrals and limits . . . . 451
Theory, niigration . . • • . . 431
Occurrence frequencies . . . . 444
Of happenings 444
Of mortality, Gompertz-Makeham-
Lazarus . . . . • • • • 405
Variation of mortality with age . . 402
466
APPENDIX A.
Total issue mothers, various age-
groups, 1908-14, Table XCI. . .
Trend o£ Destiny
Of population changes, and analysis
Triovulation, small frequency . .
Triplets, frequency each month after
marriage (first births). Table
cvin
Probability by ages
Probability by durations marriage
Twins, frequency according order con-
finement . .
Frequency by months after marri-
age, Table CVIII.
For 24 months after marriage.
Table CVII
With age of husband . .
Probability according to
nuptial, exnuptial
By durations marriage
Ratios, Table CXXXIII.
Relative frequency various coun-
tries. Table CI
Triplets, nuptial, exnuptial, proba-
bility by ages, Table CV.
Probability by durations
marriage. Table CVI.
Secular fluctuations frequency
Fig. 86
Secular variation frequency.
Table CX
Type-curves and norm-graphs . .
Development of
Evaluation of constants . .
Page.
281
456
2
309
313
310
311
314
313
313
367
309
311
366
306
310
312
316
316
449
61
62
Unimodal and multimodal fluctuations 63
Uniovular & diovular, multiple births 306
Univitellins (see corrigenda) . . •. . 307
Unmarried, masouUnity, Table LXIV. 212
Unspecifled cases, regularity of . . 302
Data, double entry tabulation.
Table XCVII 300
Validity of curve, how tested . . 24
Value of abscissa corresponding to
quotient of two groups . . . . 395
Values of E. Table CXLIV . . . . 398
xit^/li^, that is 1 + / (ii) in 611,
Table CXLIII 391
Variation in ratio, female to male mor-
tality-rates by age. Table CXLV 400
Of rate, simple . . . . . . 18
Variations of population dependent
on natural resources . . . . 19
Of rate, disoontinaous, periodic . . 25
Simple, forms of. Fig. 2 . . 23
Vassali, .^natom. Anzeiger. Bd. X No.
10, sextuplets . . . . . . 306
Voluntary deaths follow regular law . . 427
Studies of particular causes . . 426
Mode of 427
W
Waite, H., " Mosquitoes and Malaria "
Biometrika, Lond., Oct. 1910,
Vol. VII., No. 4, p. 421 . . . . 445
Weinberg, Phys u Path. d. Mehrlings g. 307
Westergaard, Prof. Harald, "Scope and
Method of Statistics " Journ.
.Amer. Stat. Assoc, Vol. XV.,
Sept. 1916, p. 254 . . . . 402
Whewell, Novum Organon Renovatum,
Bk. Ill 125
Wiokens, C. H., " Investigations con-
cerning a law of mortality "
Journ. Aust. Assoc. Adv. Soi.
XIV., p.p. 526-536 . . . . 406
Women bearing more than 10 children.
Table CXV 325
Bearing nth child by age -groups
etc., Table CXV. .. ..325
Married bearing 'n' children age-
polyphorous distribution. Table
OXXII 336
Childless 326
Table CXVI 326
World's population, estimates of, 1806-
1914, Fig- 5 33
Population, rate of increase . - 30
Limited . . . . . . 454
Populations, estimates various
authorities. Table IV 30
World-norms, creation of . . . . 103
Relation between infantile mortality
and birth-rate . . . . . 147
Yule, G. U. statistical methods
297
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