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PUBLISHED BY 

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CHAPMAN & HALL, Limited, LONDON 






MATHEMATICAL MONOGRAPHS 

EDITED BY 

MANSFIELD MERRIMAN AND ROBERT S. WOODWARD 



No. 15 

MORTALITY 

LAWS AND STATISTICS 



BY 



ROBERT JIENDERSON, 

ACTUARY OF THE EQUITABLE LIFE ASSURANCE SOCIETY OF THE UNITED STATES 



FIRST EDITION 
FIRST THOUSAND 



NEW YORK 

JOHN WILEY & SONS, INC. 
LONDON: CHAPMAN & HALL, LIMITED f> 

1915 




COPYRIGHT. 1915, 

BY 
ROBERT HENDERSON 



THE SCIENTIFIC PRESS 
BROOKLYN. N. Y. 



PREFACE 



THE present work is designed to set forth in concise form the 
essential facts and theoretical relations with reference to the 
duration of human life. A description is first given of those 
mortality tables which have had the greatest influence on the 
development of the science of life contingencies or on its appli- 
cation in this country. A few chapters are then devoted to 
the mathematical relations between the various functions 
connected with human mortality, to the analysis of probabilities 
of death or survival, so as to lead to their simplest form of 
expression in terms of the mortality table, and to the general 
mathematical laws which have been proposed to express the 
facts of human mortality. The connection is then established 
between the mortality table and mortality statistics and some 
investigation made of the corrections which must be allowed 
for in interpreting such statistics. 

The methods of constructing mortality tables from census 
and death returns and from insurance experience are then 
taken up. The methods adopted for the purpose of adjusting 
the rough data derived from experience are next described 
and their theoretical basis investigated. Some of these methods 
of construction and graduation are then illustrated by a new 
mortality table now first published. In the Appendix ten 
useful tables are given. 

The scope of the treatise is confined to life contingencies 
excluding all monetary applications, so that the combination 
of the theory of compound interest with that of life contin- 
gencies is not touched upon. A warning may not, however, 
be amiss that the present value of a sum of money payable 

iii 



IV PREFACE 

at death cannot properly be calculated by assuming it to be 
payable at the end of a definite period equal to the expectation 
of life, nor can the present value of a life annuity be calculated 
by assuming it to be certainly payable for that period. 

R. HENDERSON. 

NEW YORK, March i, 1915 



CONTENTS 



PAGE 

CHAPTER I. MORTALITY TABLES i 

II. THE MORTALITY TABLE AND PROBABILITIES INVOLVING 

ONE LIFE 17 

III. FORMULAS FOR THE LAW OF MORTALITY 26 

IV. PROBABILITIES INVOLVING MORE THAN ONE LIFE 34 

V. STATISTICAL APPLICATIONS 45 

VI. CONSTRUCTION OF MORTALITY TABLES 51 

VII. GRADUATION OF MORTALITY TABLES 68 

VIII. NORTHEASTERN STATES MORTALITY TABLE 95 

APPENDIX. DATA FROM VARIOUS MORTALITY TABLES 100 

MORTALITY TABLES 

DR. HALLEY'S BRESLAU TABLE 3 

DEATHS AND POPULATION, NORTHEASTERN STATES, 1908-1912 96 

THE NORTHAMPTON TABLE 100 

THE CARLISLE TABLE 101 

ACTUARIES', OR COMBINED EXPERIENCE, TABLE 102 

AMERICAN EXPERIENCE TABLE 103 

INSTITUTE OF ACTUARIES' HEALTHY MALE (H M ) TABLE 104 

BRITISH OFFICES O M[51 TABLE 105 

NATIONAL FRATERNAL CONGRESS TABLE 106 

NORTHEASTERN STATES MORTALITY TABLE, 1908-1912 107 

RATES OF MORTALITY PER THOUSAND ACCORDING TO TWELVE TABLES. . . . 109 

DEATH RATES PER THOUSAND ACCORDING TO VARIOUS TABLES no 

DIAGRAMS 

1. COMPARISON OF AGGREGATE AND ANALYZED RATES OF MORTALITY .... 67 

2. COMPARISON OF GRADUATED AND UNGRADUATED RATES OF MORTALITY. 93 

3. RATES OF MORTALITY BY VARIOUS AMERICAN TABLES 99 

v 



MORTALITY LAWS AND STATISTICS 



CHAPTER I 
MORTALITY TABLES 

1. THE subject of human mortality is one which, from its 
nature, is of widespread interest to mankind. It has always 
been recognized that it is impossible to predict the duration 
of any individual life and that the only thing that could be 
taken as certain on the subject was that death would come 
sometime to each one. In other words it has been recognized 
that the date of death of any individual is subject to chance. 
The scientific study of the subject of chance is, however, a 
comparatively modern development of mathematics and con- 
sequently the science of life contingencies is also comparatively 
modern. 

2. Like all other events, whether considered as chance events 
or as certainties, the death of any individual or his survival 
to any specified date is the necessary result of those forces 
which have been operating upon him. The cause of our ig- 
norance regarding the result in the case of an individual is the 
limitation of our knowledge regarding the forces operating 
and their effects. We do know, however, that among the 
important ones affecting the result are climate, sanitary 
conditions, medical attendance and habits of life. These vary 
in their tendency and effectiveness as we pass from one 
locality to another or at different times in the same locality 
and may even differ in a recognizable way as between different 
individuals in the same locality and at the same time. The 
results of the observations made with respect to human mor- 



2 MORTALITY LAWS AND STATISTICS 

tality under any given set of circumstances are frequently set 
forth concisely in the form of a Mortality Table showing the 
number surviving to each age out of a given number living at 
some selected initial age. A brief description is here given of 
some of the Mortality Tables which have had a relatively im- 
portant part in the history of the science of life contingencies. 

THE BRESLAU TABLE 

3. This table is of importance because it represents the first 
attempt to construct a mortality table from which to deduce 
the probabilities of survival and the values of life annuities. 
It was formed by the celebrated astronomer, Dr. E. Halley, 
from returns of the deaths in the City of Breslau, Silesia, 
during the five years 1687 to 1691 inclusive. Owing to the 
fact that the births during the five years only slightly exceeded 
the deaths (the numbers being 6193 and 5869 respectively) 
he assumed that the population might be considered a sta- 
tionary one. He therefore appears to have graduated by 
inspection the average number dying per annum at the various 
ages and assumed that this gave the decrements of the table. 
Dr. Halley published his table in two columns, the first headed 
" Age Current," and the second " Persons," and it appears 

' from the explanation given to have been in modern notation 
equivalent to a table of values of L x -i, the population at age 
x next birthday. 

4. On the basis of this table Dr. Halley solved various prob- 
lems regarding survival and calculated annuity values, thus 
laying the foundations of the science of life contingencies and 
preparing the way for the transaction, on a scientific basis, 
of the important business of life insurance, although it was 

/not until nearly seventy years had elapsed after the publi- 
cation in 1693 of this table that the first company to operate 
on a scientific basis was established. Mr. E. J. Farren, writing 
in 1850, said regarding this table: 

" With respect to its form, as has already been stated, 
no improvement has as yet been adopted, beyond inserting 



MORTALITY TABLES 



the column of differences or deaths, and choosing higher num- 
bers for exemplification. Of its two principles of construction, 
viz., as to the number of living being deducible from the num- 
ber of deaths, by aid of the assumption of a stationary pop- 
ulation; and as to the number of deaths at contiguous ages 
after childhood being allied in number; the former principle 
was generally prevalent in the construction of such tables, 
until the appearance of Mr. Milne's Carlisle Table in 1815, 
but is now as generally abandoned; the latter characteristic 
is still operative and considered as valid in all the best tables." 

The mortality indicated by this table was considerably higher 
than that shown by more modern tables. 

5. The table as given by Dr. Halley is as follows: 



Age 
Cur- 
rent. 


Per 

sons. 


Age 
Cur- 
rent. 


Per 

sons. 


Age 
Cur- 
rent. 


Per 

sons. 


Ag, 
Cur- 
rent. 


Per- 
sons. 


Age 
Cur- 
rent. 


Per- 
sons. 


Age 
Cur- 
rent. 


Per- 
sons. 




I 


1000 


8 


680 


IS 


628 


22 


586 


29 


539 


36 


481 


Age. Persons 


2 


855 


9 


670 


16 


622 


23 


579 


30 


531 


37 


472 


7 5547 


3 


798 


10 


661 


17 


616 


24 


573 


31 


523 


38 


463 


14 4 584 


4 


760 


ii 


653 


18 


610 


25 


567 


32 


515 


39 


454 


21 4 270 


5 


732 


12 


646 


19 


604 


26 


560 


33 


57 


40 


445 


28 3 964 


6 


710 


13 


640 


20 


598 


27 


553 


34 


499 


4i 


436 


35 3 604 


7 


692 


14 


634 


21 


592 


28 


546 


35 


490 


42 


427 


42 3 178 


























49 2 709 


























56 2 194 


Age 
Cur- 
rent. 


Per- 
sons. 


Age 
Cur- 
rent. 


Per- 
sons. 


Ae 
Cur- 
rent. 


Per- 
sons. 


Age 
Cur- 
rent. 


Per- 
sons. 


Age 
Cur- 
rent. 


Per- 
sons. 


Age 
Cur- 
rent. 


Per- 
sons. 


63 I 694 
70 I 204 


























77 692 


43 


417 


50 


346 


57 


272 


64 


2O2 


71 


131 


78 


58 


84 253 


44 


407 


51 


335 


58 


262 


65 


192 


72 


120 


79 


49 


100 107 


45 


397 


52 


324 


59 


252 


66 


182 


73 


109 


80 


4^ 


S., , 


46 
47 


387 
377 


53 
54 


313 
302 


60 
61 


242 
232 


67 
68 


172 
l62 


74 
75 


98 

88 


81 
82 


34 
28 


Total 34 ooo 


48 


367 


55 


292 


62 


222 


69 


152 


76 


78 


83 


23 




49 


357 


56 


282 


63 


212 


70 


142 


77 


68 


84 


20 





The column on the right hand is evidently a summary 
of the preceding figures in groups of seven years; with an 
additional item giving 107 as the population at ages 85 to 100 
inclusive. 



MORTALITY LAWS AND STATISTICS 



THE NORTHAMPTON TABLE 

6. This table was first published in 1783 in the fourth 
edition of Dr. Price's work on Reversionary Payments. It 
was constructed by Dr. Price on the basis of a return of the 
deaths in the parish of All Saints, Northampton, England, 
during the forty-six years from 1735 to 1780 inclusive. These 
deaths by ages were as follows: 



Ages. 


Number of Deaths. 


0-2 
2-5 


1529 
362 


5-10 

IO-2O 


2OI 
189 


20-30 


373 


30-40 
40-50 
50-60 
6O-7O 
70-80 


329 
365 
384 
378 
358 


8o-QO 


199 


QO-IOO 

Total 


22 
4689 







7. Owing to the fact that the total number of baptisms 
during the same period was only 4220, or almost exactly ten 
per cent less than the number of deaths, Dr. Price, apparently 
believing the baptisms to correctly represent the births, as- 
sumed a stationary population supported by births and immi- 
gration at age 20. He supposed that the immigration was 
sufficient to supply thirteen per cent of the total deaths. The 
actual process followed appears to have been to transfer a 
sufficient number of deaths from age group 20 to 30 to age 
group 30 to 40 to equalize the numbers in the two groups. 
The number of deaths in the various groups was then pro- 
portionately increased so as to make the total number 10 ooo. 
Thirteen per cent of this number or 1300 in all were then de- 
ducted pro rata from the groups above age 20. All the groups 
both below and above age 20 were then increased pro rata 
so as to bring the deaths above age 20 up to the same figure 



MORTALITY TABLES 5 

as before. This increased the total deaths to n 650, which 
was taken as the value of IQ. A subtraction of the deaths in 
the successive groups gave the values of lz, Is, ho, ho, etc., 
the intermediate values being afterwards inserted. 

8. From the above explanation it will be seen that in 
this case, as in the case of the Breslau table, only the death 
returns were available, without any enumeration of the pop- 
ulation, and the resulting table indicated rates of mortality, 
especially at the younger ages, which, in the light of subsequent 
experience, appear unduly high. In fact, instead of being 
stationary, the population and the births had been regularly 
increasing and the population and deaths at the young ages 
were consequently proportionately higher than would have 
been the case in a stationary population. Thus when it was 
assumed that the number attaining any given age was equal 
to the number dying above that age the result was an under- 
statement of the former number by the amount of the total 
increase during that period in the population above that age, 
subject to adjustment for immigration or emigration. This 
understatement of the denominator of the fraction determining 
the rate of mortality, of course, overstated that rate. 

9. The Northampton Table was adopted by the Equitable 
Society as a basis for its calculations immediately after its 
construction. This fact, combined with the success of that 
Society, caused its adoption for many purposes for which it 
was not suitable. An outstanding illustration of this is the 
fact that the British Government based their rates for the sale 
of annuities upon it and consequently sustained a serious loss 
because the longevity of the annuitants proved much greater 
than was indicated by the table. Until within a few years 
the Northampton Table with five per cent interest was the 
basis prescribed by the court rules in New York State for 
the valuation of life interests and dower rights, but the Carlisle 
Table has now been adopted instead. 



MORTALITY LAWS AND STATISTICS 



THE CARLISLE TABLE 

10. This table was constructed in 1815 by Dr. Milne and 
was the first table to take both the deaths and the corre- 
sponding population into account. It was based on two 
censuses of the population of the parishes of St. Mary and 
St. Cuthbert, Carlisle, taken January i, 1780, and December 
31, 1787, or an interval of eight years, and on the deaths for 
the nine years 1779 to 1787. 

The following schedule shows the data: 



Age Last Birthday. 


Population. 


Deaths 1779 to 1787. 


Jan. 1780. 


Dec. 1787. 


O 








39 


I 








173 


2 




1029 


1164 


128 


3 








70 


4^ 








I 51 


5- 


9 


908 


1026 


89 


10- 


14 


715 


808 


34 


15- 


*9 


675 


763 


44 


20- 


29 


1328 


1501 


96 


30- 


39 


877 


991 


89 


40- 


49 


858 


970 


118 


5- 


59 


588 


665 


103 


60- 


69 


438 


494 


173 


70- 


79 


191 


216 


152 


80- 


89 


58 


66 


98 


90- 


99 


IO 


ii 


28 


100-104 


2 


2 


4 


Tola 


1 


7677 


8677 


1840 





ii. The figures for the deaths are derived from a record 
kept by Dr. J. Heysham, and the population as of January, 
1780, is also derived from an enumeration by him, taking account 
of the ages. The population as of December, 1787, appears 
to have been merely enumerated in gross and then distributed 
by ages in the same proportions as had been found in 1780. 
It will be seen on examination that the proportions on the 
two dates are the same, which is a condition scarcely likely 
to be realized in two actual enumerations. It was assumed 



MORTALITY TABLES 7 

that the average population during the period covered by the 
observations could be represented by the mean of the two 
censuses. The method followed in deducing the rates of 
mortality will be described in Chapter VI. 

This table presented a more accurate statement of the 
probabilities of death at various ages than any preceding table 
and was widely used for insurance calculations. It has now 
been largely superseded for this purpose by tables more recently 
constructed from the experience of insured lives. It is, however, 
still in use for special purposes. 

Owing to the small extent of the data on which it was based 
and the graphic method adopted in redistributing the pop- 
ulation and deaths into individual ages, the rates of mortality 
were somewhat irregular, particularly at the older ages, and 
various regraduations of the table have been made with the 
idea of removing the irregularities. 

THE ENGLISH LIFE TABLES 

12. At various times tables of mortality have been con- 
structed on the basis of the census returns and registration 
of deaths in England. On account of the fact that an exten- 
sive series of monetary tables was based on it the most widely 
known of these tables is the English Life Table No. 3, which 
was constructed by Dr. Farr on the basis of the censuses of 
1841 and 1851 and the deaths of the seventeen years 1838-54. 
Separate mortality tables were constructed for male and female 
lives starting with radices of 511 745 and 488 255 respectively 
or for the combined table, i ooo ooo at age zero. The method 
followed in constructing these tables will be described in 
Chapter VI. 

13. At about the same time as these tables were constructed 
other tables, known as the Healthy Districts Life Tables, were 
also constructed from the census returns for 1851 of the sixty- 
four English districts having at that time an average death 
rate below 17 per thousand, and the deaths in the same dis- 
tricts during the five years 1849-53. These tables were pre- 



8 MORTALITY LAWS AND STATISTICS 

sen ted in threefold form, the radix at age zero for the male 
table being 51 125 and for the female 48875, the two added 
together constituting a mixed or combined table with a radix 
100 ooo. The Healthy Districts Male Table with certain 
modifications was used by the Committee of the Actuarial 
Society of America in charge of the Specialized Mortality 
Investigation as a basis for the comparison of the mortality 
in the different classes. 

14. The more recent of the series of English Life Tables 
are designated as Nos. 6, 7, and 8. The English Life Tables 
No. 6 were based on the census returns of 1891 and 1901 and 
the deaths of the ten years 1891 to 1900 inclusive and, while 
not originally prepared by Mr. Geo. King's method described 
in Chapter VII, have been readjusted by that method. The 
English Life Tables Nos. 7 and 8 have just been published and 
were prepared by Mr. Geo. King by his method. The former 
set are based on the census returns of 1901 and 1911 and the 
deaths of the ten years 1901 to 1910 inclusive, while the latter 
are based on the census of 1911 adjusted for increase to the 
middle of that year and on the death returns of the three 
years 1910 to 1912 inclusive. The special feature of these tables 
is that not only do they indicate an improvement in mortality 
as compared with the earlier tables of the series, but, when 
compared with one another they indicate that the improvement 
was still progressing. The No. 7 Tables show a lower mor- 
tality throughout than the No. 6 and the No. 8 Tables a 
lower mortality at practically all ages than the No. 7. 

THE ACTUARIES', OR COMBINED EXPERIENCE, TABLE 

15. This table, also known as the Seventeen Offices' Ex- 
perience Table, was prepared in 1841 by combining the experi- 
ence, by lives, of the Equitable and Amicable Societies with 
the experience, by policies, of fifteen other companies as con- 
tributed in 1838 to a committee of actuaries. It was thus 
the first example of a mortality table formed by combining 
the experiences of different insurance companies into one 



MORTALITY TABLES \J 

general average. It appears to have covered in all 83 905 
policies or lives of which 13 781 were terminated by death, 
25 247 were terminated otherwise, and 44 877 were in existence 
and under observation when the observations closed. The 
total of the numbers exposed to risk, for one year at each age, 
was 712 163 indicating an average duration of 8.5 years. 

1 6. Probably owing to the mixed nature of the data, which 
as above stated, was partly by lives and partly by policies, 
and to the fact that the average duration of the experience 
contributed by the companies other than the Equitable and 
the Amicable was only 5.5 years, this table was never widely 
used in Great Britain for insurance purposes. It was, however, 
prescribed by the State of Massachusetts as the basis for 
the valuation of the reserve liabilities of life insurance companies. 
The example of Massachusetts was later followed by New York 
and other states with the result that for many years the 
Actuaries' Table with four per cent interest was the accepted 
valuation standard in the United States, although the pre- 
miums actually charged by the companies were as a rule based 
on a different table. 

THE HEALTHY MALE (H M ) TABLE 

17. This is the most important of the group of tables 
published in 1869 and representing the results of the Insti- 
tute of Actuaries' Mortality Experience, 1863. They were 
based upon data contributed by twenty British life insurance 
companies regarding their experience up to 1863 on insured 
lives. The H M table was based on the experience of male 
lives insured at regular premium rates, and duplicate policies 
on the same life, whether in the same or in different companies, 
were carefully eliminated. This table represented a much 
broader experience than that upon which the Actuaries' Table 
had been based, confirmed in a general way the results of 
that experience and obtained immediate acceptance as a fair 
representation of the average mortality of insured lives. 
The official H M Table was graduated by Woolhouse's formula 



10 MORTALITY LAWS AND STATISTICS 

but it was subsequently regraduated by King and Hardy 
according to Makeham's formula, with a modification at the 
younger ages, and extended down to age zero by means of 
rates of mortality taken from the Healthy Districts Male 
Table. This graduation of the table is published in Part II 
of the Text Book of the Institute of Actuaries. 

18. In the construction of this table all lives of the same 
attained age were included together without regard to the period 
elapsed since medical examination. But an analysis of the 
experience indicated that the rate of mortality among lives 
recently insured was much less than among lives of the same 
attained age who had been insured for a longer period. 
Accordingly a second table, known as the H M (5) Table, was 
formed by omitting the experience during the calendar year 
of issue and the next four calendar years. This table was 
taken as representing the ultimate rate of mortality after the 
effects of selection had worn off. The rates of mortality at 
the young ages are considerably higher in the H M (5> table than 
in the H M , but the two rates gradually approach one another 
and coincide at the extreme old ages where there are no recently 
selected lives. 

19. These two tables used together were adopted by many 
British companies for the valuation of their liabilities, and the 
H M Table was for many years prescribed for that purpose 
by the laws of Canada. It will be noticed that the rates of 
mortality according to the H M Table are lower than those 
for the same ages in the Actuaries' Table except for ages 46 
to 50 and ages 73 to 85 inclusive and ages 95 and over. The 
difference is not, however, important except at the young ages, 
where it is considerable. 

THE BRITISH OFFICES' LIFE TABLES, 1893 

20. These tables represent the experience on insured lives 
of sixty British life insurance companies during the thirty 
years from the policy anniversaries in 1863 to those in 1893. 
The data were compiled under the joint supervision of the 



MORTALITY TABLES 11 

Institute of Actuaries and the Faculty of Actuaries in Scot- 
land and was classified into male and female lives and accord- 
ing to the plan of insurance issued. The O M Table represents 
the experience of male lives insured on the Ordinary Life plan 
with participation in profits. The total number of lives under 
observation was 551 838, of whom 149 566 were insured prior 
to 1863. Of these 140 889 died, 148 392 withdrew and 262 557 
remained insured in 1893, the total number of years of risk 
being 7 056 863. The O M ( ' Table represents the same ex- 
perience, omitting the first five policy years and covers 5 324 862 
years of risk and 129 ooi deaths. These tables were grad- 
uated by Mr. G. F. Hardy. The O M (5) table was first grad- 
uated by the application of Makeham's formula, the dif- 
ferences in the values of log p x by the two tables being then 
graduated by the use of a double-frequency curve. A select 
or analyzed table was also prepared from the same data and 
is known as the O [M1 Table. In this table separate rates of 
mortality are indicated for each age at entry for the first ten 
policy years, merging into an ultimate table at the end of that 
time. This select table was also graduated by Makeham's 
formula, different constants being used for the different policy 
years. The rates of mortality by the O M table are lower 
throughout than those in the H M table as graduated by Make- 
ham's formula and also, with unimportant exceptions, than 
those in the official H M table. The O M (5) Table is the basis 
at present prescribed for the valuation of policies in Canada. 

THE AMERICAN EXPERIENCE TABLE 

21. This table was constructed by Mr. Sheppard Homans 
and was first published in its present form in 1868. No com- 
plete record has ever been made public of the method adopted 
in its construction, but it has always been understood that 
the mortality experience of the Mutual Life Insurance Company 
of New York was used as a basis. As that experience covered 
only a few years and therefore did not include any exposures 
or deaths at extreme old ages it must have been supplemented 



12 MORTALITY LAWS AND STATISTICS 

from other sources. The table is a very smoothly graduated 
one and evidence has been discovered which seems to indi- 
cate that the author first constructed a table of values of the 
reciprocal of the rate of mortality showing the number of 
lives out of which one death would be expected at each age. 
From these values the usual columns of the mortality table 
were then formed. 

22. The first publication of the table was in the schedule 
of an act prescribing it as a basis of valuation in the State of 
New York and although it was temporarily abandoned in 
that state for the sake of uniformity it is now the legal standard 
in practically every state of the Union. The table as originally 
published was found to conform very nearly to Makeham's 
law, and was subsequently regraduated in accordance with that 
law for use in connection with joint life calculations. 

The American Experience Table has been widely used in 
America as a basis for insurance premiums even when another 
table was prescribed as a legal basis of valuation, as it presented 
a conservative view of the mortality after the effect of selec- 
tion had worn off. The rates of mortality shown were higher 
than those in the Actuaries' table for ages 30 and under and 
for ages 78 and over, but lower between 30 and 78. Com- 
pared with the H M Table, which was published about the 
same time, it gave higher rates of mortality for ages under 
36 and over 80 and slightly lower for the intervening years. 
Compared with the O M(6) it shows higher rates of mortality 
for ages 40 and under and for ages over 70 and lower values 
for the intermediate ages. It will thus be seen that in general 
the American Experience Table seems to give relatively low 
rates of mortality for the central ages and high rates for the 
young and old ages. 

THE NATIONAL FRATERNAL CONGRESS TABLE 

23. This table was constructed by the Committee on Rates 
of the National Fraternal Congress, an association of Fraternal 
Societies in the United States of America, and was presented 



MORTALITY TABLES 13 

in its original form at the annual meeting of that association 
in 1898. It was based on the experience up to that time of 
the societies connected with the Congress. It was subse- 
quently regraduated by Mr. Abb Landis and reported in its 
amended form the next year. Compared with the American 
Experience Table the rates of mortality are lower throughout, 
although the difference is proportionately smaller at the older 
ages than at the younger. Compared with the M Table 
the rates of mortality are higher at ages 20 to 27 inclusive 
and for ages 81 and over and lower at the intervening ages. 

24. This table of mortality with interest at four per cent 
is prescribed as a basis for minimum rates of contribution in 
fraternal orders by the laws of several States. It is worthy 
of note that a subsequent investigation was made of the experi- 
ence during the year 1904 of 43 societies. This experience 
covered 2 880 166.5 years of exposure and 19 414 deaths, and 
the rates of mortality in the resulting table were lower than 
those in the National Fraternal Congress Table for ages up to 
52 inclusive and for ages 79 and over, but higher for the inter- 
vening ages. 

THE M. A. TABLE OF THE MEDICO-ACTUARIAL MORTALITY 

INVESTIGATION 

25. This table was constructed in 1912 by the joint committee 
of the Medical Directors' Association and the Actuarial Society 
of America in charge of the Medico-Actuarial Mortality 
Investigation into the relative mortality of special classes of 
risks. It was intended for use as a standard with which to 
compare the mortality of the special classes. It was there- 
fore based on the experience of the same companies as con- 
tributed to the special class experience on policies issued during 
the same period and observed up to the same date. The data 
used were based on the experience of the companies on policies 
issued during the month of January in odd years and July in 
even years from 1885 to 1908 inclusive, observed to the anni- 
versaries in 1909. The total number of policies was 500 375, 



14 MORTALITY LAWS AND STATISTICS 

the total years of exposure 2 814 276 and the number of policies 
terminated by death 20 222. The table is shown in the form 
of analyzed rates of mortality for the first four policy years 
with an ultimate table for the fifth and subsequent years. 

A special feature of this table is that the difference between 
the rates of mortality in the early policy years and those shown 
for the same attained ages in the ultimate table is relatively 
small. This has been explained on the theory that an improve- 
ment in general mortality conditions was going on during 
the time of the observations and that, owing to the fact that 
the observations in the early policy years were on an average 
made at an earlier date than those for the longer durations, 
this partly concealed the true effect of selection. This theory 
was confirmed by investigating separately the experience on 
policies issued in the years 1885 to 1892 inclusive, those issued 
in 1893 to 1900 inclusive, and those issued in 1901 to 1908 
inclusive. A progressive improvement was shown in passing 
from one group to the next. In the ultimate part of the M. 
A. Table the rates of mortality are throughout lower than those 
for the same age in the American Experience Table, but prac- 
tically equal at age 69. Compared with the National Fraternal 
Congress Table, they are lower at ages under 55 and over 80 
but higher between those ages. For ages under 70 the rates 
of mortality are lower than in the ultimate part of the British 
Offices' O (M1 Table, but after that age they agree exactly with 
that table. 

26. This table was constructed in a special way for the 
special purpose above indicated and is not recommended by 
its authors for any other purpose. The experiences, however, 
of some individual companies which have since been investi- 
gated appear to confirm substantially the ultimate part of 
the table as a -fair representation of ultimate mortality of in- 
sured lives in American and Canadian Companies transacting 
a normal business. 



MORTALITY TABLES 15 

MCCLINTOCK'S ANNUITANTS' MORTALITY TABLES 

27. These tables were constructed in 1899 by Dr. McClin- 
tock on the basis of experience of fifteen American companies, 
collected and analyzed by Mr. Weeks. The data comprised 
the entire experience of the companies on annuities up to the 
anniversaries of the contracts in 1892. Separate tables were 
constructed for male and female lives, the number of lives 
taken into consideration being 4365 males and 4821 females. 
Although this was an experience of American companies only 
about one-fourth of the number of annuitants were actually 
American lives, the remaining three-fourths representing an- 
nuities granted abroad by the companies. The experience 
was taken out strictly by lives, all duplicates being carefully 
eliminated, and in the case of deferred annuities only the ex- 
perience after the annuity became payable was considered, 
owing to the uncertainty with regard to the date of death 
during the deferred period. 

28. Each table was graduated by Makeham's formula, 
(Art. 50), the same value of c being used for the two tables 
and in consequence of this fact the principle of uniform seniority 
may be used, although in a modified form, even where the 
lives are not all of the same sex. The formula adopted was 
colog />z = log b+c x log h, where Iogc = .o4 and for the male 
table log b = .003 2 and log log h = 5.5 5; for the female table 
log b = .0015 and log log ^ = 5.43. The rate of mortality is 
higher throughout the male table than for the same age in the 
female table, the difference being proportionately greatest at 
the young ages. The rate of mortality in the male is higher 
than in the American Experience Table up to age 62 and 
lower above that age. In connection with these tables it 
should be remembered that at the young ages they are purely 
theoretical, there being only two actual deaths at ages under 
40 in the male experience and three in the female. These 
tables are now prescribed by the law of New York State as the 
basis for the valuation of annuity contracts issued by life 
insurance companies. 



16 MORTALITY LAWS AND STATISTICS 

THE BRITISH OFFICES' LIFE ANNUITY TABLES, 1893 

29. These tables are derived from the experience of British 
Offices in respect of life annuitants, male and female, during 
the period 1863 to 1893, including the British Annuity experi- 
ence of three American companies. Both select and aggregate 
unadjusted tables were constructed, duplicates being separately 
eliminated for each. After the final elimination of duplicates 
for the aggregate tables the total number of male lives involved 
was 6728, the number of years of risk 53 599 and the number 
of deaths 3503. For the female table the number of lives was 
1 8 951, the number of years of risk 173 519 and the number of 
deaths 9107. 

30. The graduated tables constructed from these data were 
shown in the select or analyzed form with separate rates of 
mortality for each of the first five contract years, merging 
into an ultimate table at the end of the fifth year. The male 
table was graduated by Makeham's formula (Art. 50), mod- 
ified for duration, the value of logioc being .038. The female 
table could not be graduated as a single series by that law. 
A second series was therefore introduced and it was assumed 
that' / w +i = /iS.f+C+i> where /$+, and /$+, each con- 
formed to Makeham's formula modified for duration. The 
rates of mortality in the ultimate part of the male table 
are lower than in McClintock's table for ages under 50 and 
over 82 and higher for the intervening ages. In the ultimate 
female table the rates of mortality are higher throughout 
than hi McClintock's table. The value at 3^ per cent interest 
of an annuity at date of issue is somewhat higher by the 
British Offices' Male Table throughout than by McClintock's 
Table. By the female table the value at date of issue is lower 
than by McClintock's table for ages under 62 and from age 
69 to age 75 inclusive and higher for ages 62 to 68 inclusive 
and for ages over 75. 



CHAPTER II 

THE MORTALITY TABLE AND PROBABILITIES INVOLVING 

ONE LIFE 

31. The mortality table has been defined as " the instru- 
ment by means of which are measured the probabilities of 
life and the probabilities of death." It may be considered as 
primarily a table showing how many on an average survive 
to each attained age out of a given number living at some 
selected initial age. The symbol l x is used to denote the number 
surviving to age x and if a be the initial age selected it is evident 
that l a represents the given number observed, since they are 
all living at that time. The mortality table, therefore, asserts 
that on the average out of /<, persons living at age a, l x will 
survive to age x, where x is any higher age. But if on the 
average in each N out of a series of cases in which an event 
A is in question A happens on pN occasions, the probability 
of the event A is said to be p. The probability, therefore, 
that a life aged a will survive to age x is l x /l a . 

32. This is a property, however, which is not confined to 
the initial age a. Consider any third age y greater than x. 
The probability, then, of a life aged a surviving to age y will 
be ly/la- But this event may be considered as a compound 
event, being composed of a life aged a surviving to age x and 
a life aged x surviving to age y. The probability of the first 
is lx/ l a ', therefore, by division, the probability of the second 
is ly/lx- This may also be demonstrated from the considera- 
tion that the l v survivors at age y are the survivors out of l x 
living at age x, because they are all included among the l x , 
and there are none included in l x who, if surviving at age y, 
would not be included in /. Therefore, again the probability 
of a life aged x surviving to age y is l v /l x . A single table, 

17 



18 MORTALITY LAWS AND STATISTICS 

therefore, of the values of l x gives by a single division the 
probability of survival for any age and period included in its 
range. 

33. The probability of a life aged x surviving n years is 
designated by n p x , and since the attained age at the end of 
the period is x+n we have 

npx=l x +n/lx ........ (l) 

The probability of surviving one year is denoted by p x , 
so that 

px = lx+i/lx ........ (2) 

From these equations it is evident that we have the gen- 
eral relation 

np X = px-px+l'Px+2 Px+n-l- ... (3) 

We thus see that from a complete table of the values of p x 
the probabilities over longer periods can be calculated. Eq. 
(2) can in fact be stated in the form l s +i=l x px by the appli- 
cation of which the successive values of l x can be calculated, 
starting from any given value. 

34. Hitherto we have dealt with the probability of sur- 
vival over a specified period. The complementary probability 
is that of death within the period. The probability of a life 
aged x dying within one year is denoted by q x . Since the life 
must either survive one year or die within the year we have 



whence we obtain the following 

(4) 



It is usual to designate the function (l x l x +i) by d x , so that 
dx denotes the number dying between the ages x and x+i, 
or at age x last birthday, out of l x living at age x; or out of 
l a living at age a. 

35. Suppose that w i is the highest age at which any 
survivors are recorded in the mortality table, so that l w -\=d w -i 
and /. = o, then 



MORTALITY TABLE AND PROBABILITIES INVOLVING ONE LIFE 19 



or 



, 

' 



In fact, if a frequency 'curve is supposed to be drawn suchV_^ 
that the height of the ordinate corresponding to any age x 
is proportionate to the number dying at that age, then d x will 
be the area included between the ordinates for ages x and #+i, 
and l x will be the entire area between the ordinate for age x 
and the limit of the curve. 

36. The probability of a life aged x dying within n years 
is denoted by \ n q x and we have, since \ n qx-\- n px= i, 



(6) 



37. The probability that a life aged x will die in the nth 
year from the present time is evidently compounded of the 
probability that it will survive n i years and that having done 
so it will then die within one year, and is denoted by n -\\q x . 
Hence we have n -i\q I = n -ipx'q x +n-i- 

From this equation or from the consideration that the 
number, out of l x living at age x, who die in the wth year there- 
after is d x+n -i we have 

i _ 

n-i\Qx- 



, 

'X v'x4-n 1 vi 



38. We have heretofore considered only the values of /* 
for integral values of x, but it is evident that deaths occur at 
all times throughout each year of age so that l x may be con- 
sidered as a continuously varying function. Let us investigate 
its rate of decrease at any particular age. This rate is the 
limit when n vanishes of the function (l x l x+n )/n which rep- 
resents the average number dying per annum over a period 
of n years. This limit may be expressed in the language of 

the infinitesimal calculus as ^. The ratio of this instan- 

dx 

taneous rate of decrease of l x to the corresponding value of l x 



20 MORTALITY LAWS AND STATISTICS 

is called the force of mortality and is denoted by /u*, so that 
we have 



Unless, however, it is possible to express l s as an algebraic 
function of x we cannot determine exactly the value of -, 

08 

and consequently of n z - Certain approximate expressions 
can, however, be determined on the assumption that differential 
coefficients of a high order may be neglected. We have 



and 



/ =/ ++ + + . etc 

dx 2 dx 2 6 dx 3 24 dx* 



I =i -h _i_j+ 

dx 2 dx 2 6 dx 3 24 dx 4 



by Taylor's theorem, whence, 



x . x . 
- | -+etc. 
dx 3 dx 3 



d 3 l 
If we assume that -r-| and higher orders may be neglected, 

(LOO 

and put h = i, we have as a first approximation 

=m -i } 

dx 2 
whence 

J7 

.... (9) 



d 5 l 

39. Next let us assume that - \ and higher orders may be 

dx- } 

neglected, and substitute for h successively i and 2. Then 
we have 

, , dl x . i d 3 l x 



dl x . 8 d 3 l x 
^--\ y-r. 
dx dx ' 



MORTALITY TABLE AND PROBABILITIES INVOLVING ONE LIFE 21 

whence 



and 



40. It will be noted that by the principles of the infinitesimal 
calculus we have 

d log e l x _dlx/j 

/I 'V* /T^ 

U/-k U/* 

whence we have, 

_ d \0ge lx / v 

41. We have already pointed out that trie distribution 
of the lives according to duration may be represented by a 
frequency curve, and in this case as in the case of frequency 
curves in general, we may seek for some typical value to rep- 
resent the curve as a whole. There are three quantities some- 
times used for this purpose. The first is the mode, or that 
value of the variable for which the probability is the greatest. 
In the mortality table this corresponds to the age at death for 
which l x n x is the greatest. The most probable duration of life 
is therefore the difference between the present age and this 
definite age provided the present age is less than the age of 
maximum deaths. 

42. The second quantity corresponds to the median in the 
theory of frequency curves and is the duration which the life 
has an even chance of surviving. It is known as the vie probable 
sometimes translated into probable lifetime. Its value for any 
age x is determined by solving for n the equation 

npx = \ Or l x +n=%lx- 

43. The function most commonly used, however, for the 
purpose of summarizing the probabilities of survival of a 
given life is the expectation of life which corresponds to the 
mean value in the theory of frequency curves. The curtate 
expectation of life is the average or expected number of com- 



22 MORTALITY LAWS AND STATISTICS 

plete years survived by lives of a given age. Its value may 
be calculated as follows: 

Out of l t lives at age x, d z die in the first year without 
completing a year of life after age x, d x+ i die in the second 
year after completing one year, d x+2 after completing two 
years and so on. Therefore if we designate the curtate dura- 
tion of life at age x by e z we have, 

lie* = d x+ 1 + 24 +2 +3^+3 +etc. . . . (12) 



Substituting now for d x +i, etc., their values in terms of 1 X 
we have 

s -^+4) +etc., 



or 

_r + l+r + 2+r + 3"~. _ n~is+ n / \ 

*~ ~~ '' 



(14) 



This same expression for the curtate expectation may 
also be obtained by considering separately the number com- 
pleting each year. The number completing the first year 
is / I+ i, those completing the second year l x+2 , the third year 
/z+3, and so on. Therefore the total number of years com- 
pleted by the l x lives is /z+i-Hz+a+^+s+etc., and the average 
is obtained as above. 

44. The calculation of the curtate expectation takes account 
only of years of life entirely completed and omits the fraction 
of a year survived in the year in which death occurs. The 
complete expectation of life, denoted by e x , includes this 
fraction of a year. In arriving at a first approximation to 
the value of e x , it is usual to assume that this fraction, which 
may have any value from zero to one year, averages half a 
year, so that we have 

(15) 



MORTALITY TABLE AND PROBABILITIES INVOLVING ONE LIFE 23 

Or substituting for e x its value in terms of l x , we have 

o _ \lx + /X + 1 +lx + 2 + lz + 3 +etC. 



_Z, J +L g +i+L r +2+etc. 

; > I 10 / 

*z 

where generally, 

.-i(4+4+i}- 

45. The exact expression for the complete expectation of 
life is evidently 



Integrating by parts in the regular way this becomes 

l*e* = - [&+!" + f "/,+<& = rl x+t dt, . . . (18) 

/0 ^/O 



since tl x+t vanishes at both limits. This integral cannot be 
evaluated exactly unless an algebraic expression in integrable 
form can be substituted for l x+t . An approximate expres- 
sion can, however, be obtained by dividing the integration 
into yearly intervals. We have 

/*< + ! /"I 

/ ,//_ I / .Al, 

ll-^-lUl I I'x-^l-l-Jl lilt 

J* Js*, 

I 1 / /// /f2 fJ2J W /73/ IA .J4/ 

1/7 I L****+* I n ** l X + t i '* t* t-I + t , '* C* *3; + / 

= ' [L Yt + h 1 \r r 

dt 2 dt 2 6 d/ 3 24 t/^ 4 



-I 



4- 



etc.dA. 



_. i dl s+t , i </ 2 /z+ t , i cPl x+t . i d 4 l x+t 

>-H"T T] "TT -- Tn ' 775 ' TTt -- r"tC. 

2 dt 6 dt 2 24 dt 3 120 J/ 4 
from Taylor's theorem; also, 

i, ,, v , ^i dl x+t i d 2 l x+t ^ i <P/ g+< i d 4 l x+t 

-- l*+t-i ---- -7: I --- T^ I --- ^s~~l o . rt +etc. 

2 a/ 4 rf/ 2 12 rf/ 3 48 fi/ 4 



/dlg+t+i _ dfx+t\ _ 
\ dt dt / 



d 2 l x+t . i d?l x+t 
dt 2 2 dP 6 



APfc-M+i d*l x+t \_d*l x+t 
\ dt 3 ' dt 3 ) dfi 



24 MORTALITY LAWS AND STATISTICS 

Hence we have 



/ 



I2\ dt dt 



i APU-M d?l x+t \ 



720\ 

Putting then / successively equal to o, i, 2, 3, etc., we get 

XI / 

i2\dx dx/ 

+-L 

7 

rv^-Ku*+^) T ( dii+z dix+i ] 

./ i2\dx dx / 



i d*l x+1 



s t+l 

, ., , o etc. 

72o\ dx* dx* 



etc. 



Whence, summing and remembering that at the upper limit 
l x+t and all its differential coefficients may be assumed to vanish, 
we have 

Ci rH-*i j-v 00 / i 1 dl x i d 3 l x 

l x+ tdt = ?lx-rZi l x + t -\ - --- ^-r+etc., 
Jo 12 dx j2odx- i 

17 1 v7 J 7 I d l x . 

= 2/ z +2 x l z+t -- l x n x -- -+etc. 
12 720 dx 6 

i d*l 

Hence, if we neglect the term - f and all higher differ- 

"j2odx 3 

ential coefficients, we have 

l x e x = 5^ + 21 l x+t - 



or 

ex = ?+ex-^Mx ........ (19) 

This shows that the correction to the first approximation 
is approximately faux- As the value of this correction is 
very small except at extreme old age it is usually neglected 
and the first approximation used for e x . 



MORTALITY TABLE AND PROBABILITIES INVOLVING ONE LIFE 25 

46. From a table of the expectation of life it is possible to 
derive directly the corresponding rates of mortality, for we 
have 



so that we have 
or 

*X-{-\ f \ \ jk / * I \ f \ 

x ~ \ ^ "T~ @x + 1 / /'z \ ^ ~i ^c -f-l/j" \ 2O J 

whence, 

<?* 

#*=- , 



and 



Eq. (21) gives the rate of mortality in terms of expectations 
and Eq. (20) gives a rule for determining e x from p x and e x+ i. 
By the successive application of this formula, beginning at" - 
the oldest age, the expectation of life for all ages may be 
computed from the rates of mortality without constructing 
the l x column. 

47. The value of n x may also be expressed in terms of 

complete expectations of life, for we have l x e x = I l x dx, from 

Eq. (18). Differentiating, then, with respect to x we have, 
after changing sign, 

7 _ / dtx _ j 

* dx 
or 

de 
HxCx = i +-r = I \ (r- ie x +i], approximately 

(LX 

whence 

4 



CHAPTER HI 
FORMULAS FOR THE LAW OF MORTALITY 

48. Before the various labor-saving devices now in use 
in connection with the calculation of monetary values from 
the mortality table had been invented, the desirability of 
reducing, if possible, the mortality table to a mathematical 
law in order to facilitate such calculations was especially 
evident. The first attempt of this kind was made by 
DeMoivre in his " Treatise of Annuities on Lives," for the 
purpose of passing from the expectation of life to the value of 
a life annuity. His assumption was the very simple one that 
the value of d x was the same for all ages, or in other words, 
that l x decreased uniformly up to the limiting age. The equa- 
tion for l x in terms of x can therefore be written 

l x = a(wx), (i) 

where x is less than w. 
Whence 

dl x , 

= -a=d x , 
dx 

and therefore, 

a i i ^ 

q x = n* = r- i ( 2 ) 

a(w x) wx 

also 

/**%/ /*"** 

\ I e I = \ l x dx= ( a(w x)dx= [(wx) 2 ] v =-(ivx) 2 

J* J* 

whence 

O J/ \ / \ 

This equation may also be stated in the form w = x-}-2e x , 
which shows that, if the formula applied, the function x+2c x 
would be a constant. The calculation of this function for 

26 



FORMULAS FOR THE LAW OF MORTALITY 27 

two or three ages at intervals, or the examination of the d* 
columns of any mortality table based on actual experience 
will show that DeMoivre's hypothesis is only a rough ap- 
proximation to the truth. While it accomplished its -purpose 
of enabling approximate life annuity values to be calculated 
from the expectation it cannot be accepted as a statement of 
the true law of mortah'ty. 

49. In the next attempt the problem was approached directly 
by an investigation of the causes of death. It was made by 
Benjamin Gompertz, who assumed that the force of mortality 
increases in geometrical progression with the age. This may 
be written as follows: 

M. = 5c*, (4) 

where B and c are constants for a given mortah'ty table, but 
may have different values in different tables. From this 
equation we have 

d log* l x _ _ E * 

UL . 

dx 
Whence, integrating with respect to x, we have 

D 

log /z = loge &- r C*, 
logeC 

or 

/* = */, (5) 

TO 

where \og e g=-- , or B = - log e g log e c. We may thus 

lOge C 

express \i x in terms of the constants of the mortality table by 
substituting this value for B, so that we have 

x =- (loge g' loge C]C X (6) 

50. Gompertz's formula constituted a genuine approxima- 
tion to the law of mortah'ty, but it was found that it did not 
apply to the period of childhood, and that even at adult ages 
it would not cover the complete range without a change of 
constant at an age in the neighborhood of 50 or 60. To remedy 
this Mr. Makeham proposed to modify Gompertz's formula 



28 MORTALITY LAWS AND STATISTICS 

in a way actually suggested by the reasoning of Gompertz 
himself, who had stated that "It is possible that death may 
be the consequence of two generally coexisting causes; the 
one chance, without previous disposition to death or deteri- 
oration; the other a deterioration, or increased inability to 
withstand destruction." The modification consisted in adding 
a constant to the expression for the force of mortality, which 
became 

x = A+Bc? ........... (7) 

= - k)g e S- (loge ' loge C)C* ..... (8) 

by putting A = log fl 5 

Substituting then - , f r M* and integrating, we get 

CLOC 
loge l x =log e k+X loge S+C* log, g, 

where log e k is the constant of integration, or 



This formula has been applied to various mortality tables 
with considerable success, reproducing them very closely from 
about age 20 to the end, but not covering the period of infancy. 
Certain other tables, however, cannot be reproduced by this 
formula. 

51. The simplest way of determining the constants in 
Makeham's formula is from four equidistant values of log/*. 
We have. 

log l x = log k +x log 5 +c* log g, 

log /,+, =log k + (x+t) log s+c* + ' log g, 
log l x+2t = log k + (* 4- 2/) log s +c* +2t log g, 
log / z+3t = log k+(x+st) log s+c x+3t log g. 
Taking now the differences, we have 

log l x+t -log /, =/ log *+<?(<? -i) log g, 
log /x+2 log l g+t =t log 5+c* +l (c l - 1) log g, 
log /*+; - log l x+2t = Mog *+c* +2 V - 1) log g. 



FORMULAS FOR THE LAW OF MORTALITY 29 

Taking differences again, we have 

log l x+ 2t - 2 log l t+t +log l x = c*(c l - i) 2 log g, 

log l x+3t ~ 2 log /*+2* + log l x+t = *+'(<? - l) 2 log g. 

Dividing the second by the first, we have 

^ ^log I***-* log k+2i+log lx+t 

log /,+ - 2 log /x+t + log /, 

From this equation c is determined and then in succession 
log g, log s, and log k. 

52. For example, according to the Makehamized American 
Experience Table, we have, 

log k + 20 log 5 +C 20 log g = log /20 = 4.96668 

log &+40 log S + C 40 log g = log /4Q = 4.89286 

log + 60 log J + C 60 log g = log /60= 4.76202 

log k +80 log 5 + C 80 k)gg = log/80= 4.l6l22 

20 log S+C 20 (C 20 i) log g = .07382 

201og5+C 40 (c 20 l) logg= .13084 

20 log + C 60 (c 20 - l) log g = .60080 

c 20 (c' 20 i) 2 log g = .05702 

. ^(c 20 ~i) 2 log g--. 46996 
Taking logarithms 

20 log c+ 2 log (c 20 -i)+loglogg = "2.756o3W 

40 log f + 2 log (C 20 l) +log log g = 1.67206^ 

20 log ^ = .91603 

log c = . 045801 5 . . (a) 
Hence 

c 20 = 8.242 

log (c 20 -i)=. 85986 
20 log c+ 2 log (c 20 -i) = 2.63575 
log log g=^.756o3- 2.63575 =4.12028^ 

log 8= -.00013191 . (b) 



30 MORTALITY LAWS AND STATISTICS 

20 log C+log (C 20 -l)-flog log g = 2. 75603**-. 85986 



c 20 (c 20 - 1) log g=-. 00787 
but 

20 log S+C 20 (c~ l) log g = .07382 
.'. 20 log 5 = - .06595 

log 5 =-.003 297 5 . . (c) 
20 log c+log log g = 3. 03 631 n 
c 20 logg = .00109 

20 log 5 + C 20 log g= .06704 

but 

log&-f2ologS + C 20 k>gg = 4.96668 

/. logk = 5-0337 2 (<0 

53. It is interesting to compare the values thus calculated 
with those from which the table was constructed. The dif- 
ferences arise from the fact that in the mortality table the 
value of l x is expressed to the nearest unit and that we have 
used five-figure logarithms throughout. The comparison is as 
follows: 

Exact Value. Calculated Value. 

logc .04579609 .045802 

logs .003296862 .003298 

log g .00013205 .00013191 

log& 5.03370116 5-337 2 

The value of log c in the Makehamized American Experi- 
ence Table is one of the highest which it has been found nec- 
essary to use. The range of values lies between .036 and .046 
with a general average close to .04. 

54. Unfortunately for the widest usefulness of Makeham's 

formula it is not possible to evaluate the integral j ks'g^dx 

otherwise than approximately, so that it does not serve the 
original purpose for which a mathematical law was sought. 
Any mathematical law, however, gives a very smooth series 



FORMULAS FOR THE LAW OF MORTALITY 31 

which enables formulas of approximate summation or inte- 
gration to be used which greatly reduce the labor of cal- 
culating the values of complicated benefits, and Makeham's 
law in particular offers great advantage in the calculation 
of probabilities of survival involving more than one life, on 
account of the form of the expression for log n p x . This point 
will be discussed further in connection with those probabilities. 
The advantage thus secured is so great, however, that it is 
considered that a mortality table which is to be used for 
monetary calculations should be adjusted so as to conform to 
Makeham's law if it can be accomplished without departing 
too seriously from the facts upon which it is based. The 
general subject of adjustment or graduation will be taken 
up in a later chapter. Modifications have been proposed to 
Makeham's formula for the purpose of making it fit certain 
tables more closely. These modifications consist in adding 
terms to the expression for // z . One modification assumes 
n x =A-}-Hx-\-Bc x , adding the term Hx to Makeham's expres- 
sion, and another takes the form n x = ma x -\-nb x . 

Both of these modifications sacrifice a considerable por- 
tion of the advantage which can be secured from the use of 
Makeham's formula in its original form. 

55. Another formula has been proposed by Wittstein, 
which is intended to cover the entire range of life from infancy 
to extreme old age. He assumes that the values of q x may 
be expressed in terms of x as follows : 



From the form of this expression it is evident that the 
first term becomes equal to unity when x = M and that where 
a is greater than unity and n is positive the value of this term 
increases regularly up to that value. It appears therefore that 
M+i must equal the limiting age in the table. Also for 

infant mortality we have q = a~ M ^-\ , showing that the 



32 MORTALITY LAWS AND STATISTICS 

probability of death during the first year after birth is slightly 

greater than . Also 
m 



ax 

Now it is evident that this vanishes when 

M 



= (Mx) or x = 



w + T 



and it will be found that for all cases arising in practice this 
represents a minimum value of q x . In applying this formula 
to mortality tables it is found that for normal mortality in 
temperate climates a is approximately 1.42, n is approximately 
.63, M is between 95 and 100 and m is not less than 6. With 
these values it is evident that the second part of the expression 
for q x decreases rapidly as x increases and becomes negligible 
at about age 25, so that the first term may be taken to rep- 
resent adult mortality and the second term to represent the 
additional mortality of infancy. This formula does not possess 
the practical advantages of Makeham's and consequently has 
not been much used in practice. 

56. Still another method was adopted by Prof. Karl Pearson, 
who took the numbers dying at the various ages and analyzed 
the series into the sum of five frequency curves typical respect- 
ively of old age, middle life, youth, childhood, and infancy. 
The table selected was that known as the English Life Table 
No. 4 (males) and the expression which he deduced for 
was as follows: 



,.22l5(z-71.5) 



if 4.g-[.Q5524U-41.6)] 
_j_ 2.6e~ I- 09092 **" 22.5)]* 

+8.50-2)- 3271 <r- 3271(z - 3) 



FORMULAS FOR THE LAW OF MORTALITY 33 

In the first four curves the maximum values are at ages 
71.5, 41.5, 22.5, and 3 respectively, while the fifth theoretically 
extends below age zero, the ordinate becoming infinite at age 
.75. The method has not, however, been applied to other 
tables and it is difficult to lay a firm foundation for it, because 
no analysis of the deaths into natural divisions by causes or 
otherwise has yet been made such that the totals in the various 
groups would conform to these frequency curves. 



CHAPTER IV 
PROBABILITIES INVOLVING MORE THAN ONE LIFE 

57. IN calculating probabilities involving more than one 
life it is usual to assume that the probabilities of survival or 
death of the various lives involved are independent of one 
another, so that the probability of a compound event is found 
by simply multiplying together the elementary probabilities 
of which it is composed. For example, the probability that 
two lives now aged x and y respectively will both be alive 
at the end of n years is found by multiplying the probability 
that the life aged x will be alive, n px, by the probability that 
the life aged y will be alive, />. For the sake of brevity it 
is usual to write (x) for a life aged x. If then the probability 
that both (x) and (y) will survive n years be denoted by n p xv , 
we have 



'x+n' 



/ \ 
(l) 



Similarly, where more than two lives are involved, we have 

npxyz ~ nfx' npy' nPz .... ^2^ 

58. These probabilities of joint survival are the elementary 
forms to which other probabilities are usually reduced. It 
is interesting to investigate the form which they take when 
Makeham's law applies. We have 



\ogs+c x+n 
= log s+<f(c n -i) logg. 
Similarly, 



log nPv = n\ogs +c v (c* - 1) log g. 

34 



PROBABILITIES INVOLVING MORE THAN ONE LIFE 35 

Therefore, 



}(c n -i} logg 
Let us now take two lives of equal age w. Then we have 

log npww = 2n log S+ 2C w (c n - l) log g. 

If, therefore, 2c v> = c x +c v , the value of n pww will be the same 
as that of n p xv for all values of n. Thus in all questions 
relating to the joint continuance of two lives aged x and y 
we may substitute two lives of equal ages w. The relation 
2c v> =c t +c v may be expressed in another form by dividing 
through by c* when we have ic w ~ x = i +c v ~ x , from which it 
follows that the value of w x depends only on that of y x 
and is independent of the actual values of x and y. Similarly 
for any number m of lives (x}, (y), (z), etc., we have 

log npxvt . . . (m) = 2 log n p x = mn log s+ (c n - 1) Zc* log g, 
= m\n\ogs+(c n -i) c"\ogg\=m\og n p w , 

provided mc" ! = 'Lc z , so that m lives of equal ages may be sub- 
stituted for any m lives. 

59. Under Gompertz's law this relation takes a simple form 
because the term involving log s disappears and we have 

log npw ...<> = (c n - 1) Zc* log g = (c n - 1) c w log g = log n p w , 

provided c w = Zc*, so that here a single life may be substituted 
for any number of lives. In this case, too, the addition of the 
same number of years to each of the ages x, y, z, etc., will add 
the same number of years to w. This property of Gompertz's 
and Makeham's laws is known as the property of uniform 
seniority. 

60. Another way in which the principle can be applied to 
Makeham's law is by constructing a hypothetical mortality 
table such that l' x = ks mx g^, so that we have 

log np'x = mn log S+C*(C B - 1) log g. 
We have, therefore, 



ic = nP 



xvz . . . (m) 



36 MORTALITY LAWS AND STATISTICS 

for all values of n provided c v> = I,c f as in Gompertz's law. 
It would thus be necessary to construct a special table for each 
value of m, but once constructed it would apply to all combina- 
tions of m lives. 

61. Hardy's modification of Makeham's law may be written 



or 

log l x = log k+x log s+x 2 log r+c* log g, 

log npx = log l t +m ~ ^g l*=n log S + (2WZ+W 2 ) log T +C*(c" - l) log g, 

= (w log s+n 2 log r) + 2nx log r 
-H?(*- 1) log g. 

log n p xvt ...() = w(w log s+w 2 log r) + 2n 2x log r 



w log /> = w(w log s+w 2 log r) + 2www log r+mc u (c n - 1) log g. 

The first term in each of these expressions is the same and 
the last terms can be made equal by putting, as in Makeham's 
law, mc w = '2c t . This will leave an outstanding difference in 
the second term of 2W log r(2x mw). Since an addition of 
/ years to each of the ages will add the same number of years 
to w and leave this expression unchanged, it follows that its 
value depends only on the differences of the ages. Since the 
variable n enters in the same way into this expression as into 
mn log s, we may consider it as a modification to be applied to 
the value of s. In fact, if we put 

/ i Zx \ 

log s =log s +2 log r{ -- w ), 

\m ) 
and 

log n p'w = (n log s' +n 2 log r} + inw log r + (c n - 1) c w log g 

then we have 

log npxvz ...(m)=m log n p' w . (3) 

It will be seen, however, that this modified value of s depends 
not only on m, but also on the differences of the ages, so that 
the complications are considerably increased. 



PROBABILITIES INVOLVING MORE THAN ONE LIFE 37 

62. If we assume 

k-fr^y, 

or 

log /, = log k+a* log r+tf log s, 
we have, 

log n px = (a n i) a x log r + (b* i) b x log 5. 

If, therefore, w is determined by the equation 

tf/V-2 
and / is determined so that 



then we have 
log np xv * ...()= (<? ~ i) to" log r+ 0" - 1) /&" log $ = / log .. 

From this it follows that for the m lives (x), (y), (z), etc., we 
may substitute / lives of equal ages w. The difficulty is that 
/ is not usually integral and it would, in practice, be found 
necessary to determine any required value by a double inter- 
polation because w also is usually not integral. 

63. We have seen that for a single life (#) we have the 
relation e x = 2 n p x . Similarly out of a large number N of groups 
of m lives aged respectively x, y, z, etc., we find Nip xyz . . . (m) 
complete the first year, Nzpxyz ...(*> complete the second, 
and so on, so that if we denote by e xyz . . . (m) the average 
number of years completed during the joint continuance of 
the m lives, we have 

Cxyz . . . (m)= ^nPxyz . . . (m) , ..... (4) 

Consequently where any expression occurs involving a sum- 
mation with respect to n of the probabilities of joint survival, 
we may substitute a joint expectation. 

64. Heretofore we have dealt with the probability that 
all of the lives involved shall survive. Similar reasoning 
will, however, show that the probability that every one of 
the m lives will be dead at the end of n years is obtained by 
multiplying together individual probabilities of death. This 
probability is expressed by 

\nOxi/z . . . (m) Or I npxvt ...() 



38 MORTALITY LAWS AND STATISTICS 

We have therefore, 

|9**... (>= In?* -Uv |?*... = (l- n />z)(l ~nA,)(l ~npz}- (5) 

In this symbol the bar over the letters denoting the ages 
of the lives involved signifies that the last survivor of the 
lives is in question, the probability designated by n q xyt . , . (w) 
being that the last survivor of the m lives shall have died before 
the end of the nth year. 

65. The complementary probability is 

npxvz . . . (m) I ~ ( I ~ npx ) ( I ~ npy) , 

and is evidently the probability that at least one of the lives 
will survive n years. By expanding the product and reducing 
the equation may be written as follows: 

nPxyz . . . On) = ^npx ~ ^npxy + ^npxyz ~ etc. . . (6) 

In this equation the summation extends over all probabilities 
similar to the one under the 2, that is, involving the same 
number of lives. 

66. Let us now investigate the probability that exactly 
r out of the m lives will be alive at the end of n years. This 
probability is designated by n p M The probability 

xyz . . . (m) 

that r particular lives, (#), fy), etc., are alive and the remain- 
ing (mr) lives (z), (w), etc., all dead is evidently 

nPxv . . . (T)-\ n (fzw . . . (m-r) Or n p xv . . . (r) ( I n p z ) ( I n /> w ) . . . 

and the total probability sought is the sum of these probabilities 
for all the combinations r at a time of the m lives, or 

pxt . . . <r)(l -/>)(! ~npw) ... (7) 



(m) 



From the form of the expression it is evident that it may be 
expanded in a series of probabilities of joint survival involving 
from r up to w lives; also that each probability involving more 
than r lives will appear more than once in the expression because 
it will appear once for each combination r at a time of the 
lives involved in it; also that the sign of any probability in- 



PROBABILITIES INVOLVING MORE THAN ONE LIFE 39 

volving r-\-t lives is positive or negative according as t is even 
or odd. Thus we have 



v . . . (r+1) r+2Crnxy . . . (r+2) 

etc. 

. . . (r) Bw . . . (r+1) 



67. This may be verified by supposing all the ages x, y, 
z, etc., to be equal, in which case 2 B /> Z1 , 2 . . . (r+0 becomes equal 
to m c T+ tnpx T+t , because m c r+t is the number of terms included 
in the summation and each term becomes equal to n p x r+t . 
The whole expression therefore reduces to 

m^T npx T+lCr'mCr+1 ' npx \r+2Cr'm.Cr+2nPx etc., 

= mCr'nPx mfr'm-r^l 'npx imCr'm-rCz'nPx CtC. 



This is evidently the proper expression for the probability 
in question, because the probability that any particular r of 
the m lives aged x are all alive and the remaining (mr) all 
dead is n px(i n px} m ~ T , and there are m c r different groups of 
r lives included among the m. 

68. A very convenient symbolic notation is sometimes used 
to condense the form of Eq. (8) by substituting Z l for 
2 n pxvz ...> when the equation takes the form 

m p in =Z r - 

zyz . . . (m) I ' 2 

r + 2 ) (r + 3 ) zr+3 

1-2-3 



In this connection it is to be remembered that the expres- 
sion is purely symbolic and that no operations can be performed 
upon it which in any way disturb the meaning of Z'. 



40 MORTALITY LAWS AND STATISTICS 

69. Let us now investigate the probability that at least 
r out of m lives will survive n years. This is denoted by 
,/> r _ and it is evident that we have 

xyt . . . (m) 



xt/z . . . (m) xyz . . . (m) xyz . . . (m) 

From this we see that the expression may be written in the 
form 

nPxyt . . . (r+2)+etC. 



xvz . . . (m) 

where a\, az, etc., remain to be determined. But from Eqs. 
(8) and (10) we have 

-\-' +(-!) r+fCf, 
r+t- id +r+t- 1^2) ~ (r+t- 1 



Therefore, we have 

A *V A i 

Pxyz . . . (r) ~"^nPxyz . . . (r+1) T *>*& . . . (r+2) 



. . . (m) 

etc., 
-Z* ' 



(n) 



where the same meaning is assigned to Z r as before. 

70. It is to be noted that although the relation is purely 
symbolic and the function of Z has no meaning except as 
expanded in ascending powers and then interpreted, we have 
the following relation: 

n p 



. . . (m) 



=Z r (i+Z)- (r+1) +Z r+1 (i+Z)- (r+2) +Z r+2 (i-f-Z)- (r + 3) 

+ . . . etc., 
=/> M +/> fr+ir +/> fr+2] +etc.. 

xyz . . . (m) xyt ... (m) xyz . . . (m) 

as in Eq. (10). 



PROBABILITIES INVOLVING MORE THAN ONE LIFE 41 

71. Also if we have m lives aged respectively x, y, z, etc., 
the expected number of survivors at the end of n years is 
Srft iri , where the summation extends over all values 

xyz . . . (m) 

of r from unity to m. Expressed symbolically this becomes, 
from Eq. (9), 

Z(i+Z)- 2 +2Z 2 (i+z)- 3 + 3 Z 3 (i+Z)- 4 +etc., 

{i + 2 Z(i+Z)- 1 + 3 Z 2 (i+Z)- 2 +etc.}, 



(12) 



This may be also verified by reasoning similar to that by which 
Eq. (n) was deduced. We thus see that the expected number 
of survivors out of any group of lives is found by adding together 
the individual probabilities of survival. 

72. Another class of probabilities involving more than 
one life relates to the order in which the deaths occur. The 
probability that (x) will die in the nth year from the present 
time is 

l . 

~npx, 



- 
lx lx Px- 

and the probability that (y) will be alive at the end of the 
wth year is n p v . The probability therefore that (x) will die 
in the wth year and (y) will be alive at the end of that year is 



Summing this function for all values of n from unity up, we 
get the total probability that (y) will be alive at the end of the 
year in which the death of (x) occurs This sum is 



Similarly, the probability that (x) will be alive at the end 

of the year in which the death of (y) occurs is - e x -^\ e^, 

Pv-i 



42 MORTALITY LAWS AND STATISTICS 

and the probability that both deaths will occur in the same year 
is the complement of the sum of these probabilities and is there- 
fore, 

i i 



, *< * . y , *t Y M. 

Px-l py-l 

It may be assumed that where both deaths occur in the same 
year the chances are even that the death (x) will occur before 
that of (y) . The total chance therefore that (x) will die before 
(y) denoted by Q\ u is 

0, f i _ e \ + lL + 2e i c _ i 1 

\Px-l X Z j 2\ p x _i X p y _i X " l j 

= l| I+ _l_ e x . l e x .\ . (13) 

21 px-l Py-1 } 

73. The same probability may be otherwise expressed in 
terms of the infinitesimal calculus by indefinitely reducing 
the intervals considered. The probability that the death of 
(x) will occur in the interval of time between t and t-\-dt will 

d I 4. 

be dt/lg. and the probability that (y) will be alive 

at 

I 
at that time is . The total probability therefore that 

ly 

(y) will be alive when the death of (x) occurs is 



dl x 



7 /" 

// fl f i 
X l -y U'V^y0 

~T ;" * ~T \*X*V^Xy ) y 

Ix'v dx 



i d 



PROBABILITIES INVOLVING MORE THAN ONE LIFE 43 

It> i ii / 
dx r 



dx ' 

K^z-i :v~ 6 Wi :v) approximately. (14) 

74. Similarly, where m lives, (x), (y), (2), etc., are involved, 
the probability that (x) will die first is 

dlx+t* i 7, 



(TO) 



I ( 
~Y~J 

**'v ... 70 



7, 
at 



. . . (m) 



: yz . . . (m))- 



75. Also from the fact that the probability that (x) will 
die rth in order out of the group of m lives may also be stated 
as the probability that, when (x) dies, there will be exactly 
m r survivors of the m i lives other than (x), we may 
express this probability by the use of Eq. (9) in terms of 
probabilities of dying first. In fact, if we denote by Y l the 
sum of the values of Ql vz ... for (x) along with all groups 
/ at a time of the (m i) lives (y), (z), etc., we have 



where the summation included in Z covers only the m i 
lives (y), (z), etc. 

X=o 
// z+l ,p x Z l dt = Y 1 for all values of /. Therefore 

expanding, integrating, and condensing, we have 



For m = 3 we have 

Qxvz = ^xxvz-\-^(t-x-l:vz~ e x+i:vz), - - (i?) 

^ z = F(i + F)- 2 = F- 2 F 2 , 

(18) 

..... (19) 



44 MORTALITY LAWS AND STATISTICS 

76. Where Makeham's law is assumed to hold, the prob- 
ability Ql v takes a special form. We have generally 

Qxy~ 77" I ly+t ,. dt = I Iv+Jx+tVx+tdt, 
IxlyjO WvJo 



c' t p xv dt. 
Similarly, 



Therefore, 



or 

B I c' t p xy dt = * y (i 2 A e xv }. 
Jo c -\-c 

Substituting this in the expression for Ql v , we have 

^\\ A O C/ / 



A I - C 

-A- -e xv (20) 



77. For Gompertz's law we have A=o, and this equation 
takes the form 



We thus see that according to this law the probability of 
survivorship depends only on the difference of ages. 



CHAPTER V 
STATISTICAL APPLICATIONS 

78. SUPPOSE that in a certain country there is neither 
emigration nor immigration and that the number of births 
occurring each year is uniform and equal to IQ. Then it is 
evident that on the assumption that the law of mortality 
remains unchanged the number of inhabitants attaining age x in 
each year is l x , being the survivors out of the IQ who were born 
x years before. Therefore, at any moment the number in the 
existing population whose age is between x and x+dx will 
be l x dx, being the survivors out of the I dx who were born in the 
interval dx years x years earlier. The total population at age 
x last birthday or between ages x and x+i is, therefore, 



If now we assume that the deaths are evenly distributed, so that 



we have for the population at age x last birthday denoted by 
L x the following 

L x = f Q \l x -td x )dt = l x -%d x = $(l x +l x+1 '). . . (i) 
Also the deaths occurring per annum between the ages 

x and x+dx will be l x n x dx = -~ dx. Therefore, the total deaths 

dx 

per annum at age x last birthday or between ages x and x+i 

rx+i $ 

will be -j^dx = l x l x +i=d x . Summing this for all ages 

J z dx 

x and over we see that the total deaths for those ages is l x , 
so that the aggregate number of deaths per annum at all ages 

45 



46 MORTALITY LAWS AND STATISTICS 

will be /o, which is also the number of births. The population 
is therefore constant in total number and also in age com- 
position. 

79. The total population at age x and over will be 2 X L X , 
which is usually denoted by T x , so that we have 



The total population at all ages will be 

T Q = l<>eo, ....... (3) 

The general average death rate is obtained by dividing 
the total number of deaths per annum by the total population, 
its value is therefore equal to lo/T = lo/loe = i/ei or the re- 
ciprocal of the complete expectation of life at birth. Similarly, 
the average death rate at ages x and over will be 

lx/T x = l x /l x e x = i/e x . 

The total number of deaths per annum between ages x 
and x-\-n will be l x l x + n , and the total population at the same 
ages will be T x T x+n , therefore, the average death rate between 

those ages will be ~ * +n . When n is equal to i , this be- 

* x 1 x+n 

comes the central death rate for age x last birthday and is 
denoted by m x , so that we have 

l x l x +i _d x _ d x d x / v 

mx- - i/;,/ \~j_ij' ' ' w 

JL x -I x+1 J^x 2 WE "T Ix+l) h -2 a x 

If we divide both numerator and denominator by l x , we 
have m x in terms of q x , as follows: 

*, =-V or ---i ..... (s) 
i-rffc m * Q* 

So. Assuming uniform distribution of deaths within each 
year of age, the sum of the ages at death of all those dying 
in a year at ages x and over will be 



= (x+e x )l x . 



STATISTICAL APPLICATIONS 47 

Since the total number of deaths at those ages is l x it follows 
that the average age is x+e s . Putting x equal to zero, we 
see that the general average age at death is e . 

The aggregate of the ages at death of those dying between 
ages x and x-\-n is evidently 



and the number of deaths is l x l x + n , so that the average age is 

(T x - T x+n ) +xl x -(x+n)l x+n _ T x -T x+n -nl x+n 
l 1 / / 

l x t x +n "X "x+n 

81. We have seen that the population at age x last birthday 
is L x , and the total population is T , so that the proportion 
of the total population at age x last birthday is L X /T and the 
proportion between ages x and x-\-n is (T x T x+n )/T . Sup- 
pose, for example, that all young men are required to serve in 
the army from age 18 to 21, then, assuming a stationary pop- 
ulation, the proportion of the total male population so serving 
will be (Tig 2n)/2V 

82. We have hitherto assumed that we are dealing with a 
stationary population. A consideration, however, of the ques- 
tion leads to the conclusion that such a condition never exists, 
but that, owing to various disturbing factors, the percentages 
of the total population at the various ages will not be exactly 
the same as in the assumed stationary population derived 
from the mortality table representing the actual death rates 
experienced. It is evident that, if for any reason we have in 
one community more than the normal percentage of the pop- 
ulation at those ages where the death rate is low and in another 
community less than the normal percentage at those ages, 
then, even though the death rate at every individual age might 
be the same in the two communities, the general average 
death rate in the first will be less than in the second. It cannot 
be assumed, therefore, that a higher average death rate nec- 
essarily means a more unfavorable mortality experience. A 
correction must first be made for the difference in age dis- 
tribution. 



48 MORTALITY LAWS AND STATISTICS 

83. One method of making this correction is to construct 
the mortality tables representing the observed death rates, 
analyzed by ages, in the two communities and calculate from 
such tables the complete expectation of life at birth. From 
the fact that in a stationary .population the general average 
death rate is the reciprocal of this expectation it is readily 
seen that this amounts, in effect, to substituting for each actual 
population the stationary population corresponding to its 
actual mortality. It is readily seen that this method may be 
applied to the death rates for ages above any assigned age, 
or within given limits, by constructing the corresponding por- 
tion of the mortality table and calculating the average death 
rate in the stationary population. For example, it might 
be desired to compare the mortality in two communities for 
ages 15 and over or for ages 15 to 64 last birthday inclusive. 
The corrected death rate for the former would be i/Ji 5 , and 



_ j 
* 15 5 



for the latter =^&f-. The labor of constructing a 

il5~-*65 2i 5 L z 

mortality table is, however, considerable and other methods 
of correction are usually followed. 

84. Although the stationary population is largely of theo- 
retical interest the notation derived from it is useful with cer- 
tain modifications in connection with actual population statistics. 
For this purpose 6 X represents the deaths between age x and 
age x+i, and X z = ^+0 z+ i+etc., is the total number of deaths 
at age x and over, but is not equal to the number attaining 
age x. For the population between ages x and x+i the symbol 
L x is retained. The symbol T x is also used to denote the 
total population at ages x and over, so that we have 



as before. The general average death rate is then \o/T , but 
is not equal to i/Jo except for a stationary population. Simi- 
larly for the average death rate at age x and over we have 
\x/T x but not i/l z , and for ages between x and x+n we have 

\c~ \c+n/*x~ 1 x+n- 



STATISTICAL APPLICATIONS 49 

85. One method of correcting the death rates of different 
communities is to analyze each into certain age groups, usually 
quinquennial up to age 15, then decennial up to age 85, with 
a final group for ages 85 or more last birthday, the average 
death rate for each group being used. These death rates are 
then applied to a standard proportionate distribution of the 
population into these age groups. One standard which has 
been used is the age distribution of the population of England 
and Wales according to the Census of 1801. The general 
average death rate for the standard population on the basis of 
the observed group rates for each community is thus calculated 
and this is considered as the corrected death rate for the com- 
munity. In this way all communities entering into the com- 
parison are placed on the same footing with respect to age 
distribution. The same method may be extended to cover 
varying proportions of the two sexes by analyzing the statistics 
for the different communities and also the standard popu- 
lation in this way. It may, in fact, be extended to cover any 
factor, such as occupations, considered as having an impor- 
tant bearing on the mortality to be expected and for which the 
necessary data can be obtained. 

86. Another method of comparison is to use a standard 
scale of death rates for the different groups into which the 
actual populations are analyzed. The actual population in 
each group is then multiplied by the standard death rate and 
the expected deaths according to the standard are thus cal- 
culated. The total of the actual deaths in the community 
is then expressed as a percentage of the expected and these 
percentages for the different communities are compared. 

87. These two methods have been described as applying 
to a whole community, but it is evident that they apply also 
to a part, such as those aged x and over, or those whose ages 
lie between x and x+n, or those who are engaged in a certain 
occupation. In fact, what may be considered as mortality 
index numbers for various occupations have been formed 
from the census and death returns (in England. A standard 
population is taken, analyzed into the five decennial age groups 



50 MORTALITY LAWS AND STATISTICS 

between 15 and 65, the aggregate population being such that 
the expected deaths according to the general average death 
rates for occupied males in the various age groups will total 
up to 1000. The actual death rates for the various age groups in 
each occupation are then applied to this standard population 
and the t resulting total of expected deaths gives a number 
whose r^tio to 1000 measures the general mortality of the 
occupation. This is in effect the standard population method 
above described with the addition that instead of recording 
the corrected average death rate we record its ratio to an 
average death rate based on the same standard population 
combined with standard group death rates. 

88. The standard population method is the one most used 
for the comparison of general population mortality statistics, 
while the standard death rate method is most used in con- 
nection with the mortality of insured lives. In connection 
with such insurance statistics three modifications are made. 
The first is that the actual experience is usually analyzed into 
individual years of age and sometimes also into years elapsed 
since medical examination. The second is that the rate of 
mortality or probability of dying within one year is usually 
used instead of the death rate or average force of mortality, 
and that along with it the exposed to risk of death, which is dis- 
cussed under the head of construction of mortality tables, must 
be used instead of the population. The third is that amounts 
insured or amounts at risk are frequently taken into account 
instead of lives, so that we compare actual losses with expected 
losses rather than actual deaths with expected deaths. 

89. In this chapter it has been assumed that the period 
covered by the statistics is one year. Where a period other 
than one year is dealt with, we must take the average deaths 
per annum, and in any event whether for a period of exactly 
one year or otherwise the average population during the period 
must be taken. The ratio will, of course, be the same if both 
of these are multiplied by the period, so that we have on the one 
hand the total deaths and on the other the aggregate number 
of years of life during the period. 



CHAPTER VI 
CONSTRUCTION OF MORTALITY TABLES 

90. In the second chapter it was shown that in any mor- 
tality table we have the relation l x+ il x p x for all values of 
x and that consequently if we have a complete table of the 
values of p x we can, by starting at the initial age and working 
forward progressively, construct a complete mortality table. 
A little consideration also shows us that there is an insuperable 
practical difficulty in the way of constructing the l x column 
of a mortality table by taking a large group of lives of a given 
age and following them throughout the balance of their lives, 
observing the number surviving to each age. This difficulty 
arises not only from the length of time that would necessarily 
be consumed in waiting for the last one to die, but also from 
the fact that out of any large number some are certain to pass 
out of the knowledge of the observers and from the moment 
that any do so disappear the further observations are nullified 
by our ignorance of the time of their death. A correction 
is therefore necessary and this correction can be most con- 
veniently applied by a method which also obviates the neces- 
sity of waiting until some particular group of lives selected 
at a young age have all died. This method is to use the rela- 
tion already quoted and to determine separately the values 
of pi for each year of age. By this method the observations 
do not necessarily extend over a longer period than one year, 
although a longer period is usually taken in order to eliminate 
the effect of special conditions. In that event the observations 
at different times for the same year of age are combined. 

91. The observations are not, in fact, made directly on the 
value of p x , but rather on that of m x determined by the relation 

(i) 

51 



52 MORTALITY LAWS AND STATISTICS 

where 6 X represents the deaths observed at age x, last birthday, 
and L x is the corresponding population. 

But we have in terms of the mortality table 

m x _ d z _l x l x +i i Px 

from which we have 

_2m x 
2+m x 
and 

*"*>WI 

q* = T--px = 



92. In connection with population statistics it has been 
usual to calculate m x from the data and then to pass to q x 
and p x . In connection with observations on insured lives, 
on the other hand, the practice has been to determine the 
value of L X +%6 X denoted by E x for each age and so to pro- 
ceed directly to q x by the equation q x = 6 x /E x . The problem 
therefore reduces *to the determination of the values of B x for 
each value of x, and of the corresponding values of L x or E x . 
The methods followed vary, of course, with the form in which 
the facts are presented, and the conditions in connection with 
general population statistics differ so much from those in 
connection with insured lives that it is well to take up the 
two cases separately. 

93. In the case of general population statistics the in- 
formation regarding the deaths is usually derived from the 
registration returns and it is a necessary condition, for their 
use in the determination of death rates, that the registration 
should include all the deaths coming within the scope of the 
investigation. It is evident that, to the extent that the returns 
are incomplete, the numerator of the fraction determining the 
death rate is understated and consequently the death rate 
itself is also understated. For this reason the statistics can 
be used of only those countries, states or municipalities in 
which the laws and their enforcement are such as to secure 



CONSTRUCTION OF MORTALITY TABLES 53 

substantial accuracy in the death returns. In the United States 
those states and parts of states which, in the opinion of the 
Federal Census Bureau, comply with this requirement con- 
stitute the registration district. The area included in this 
district is extended from time to time as the registration becomes 
more complete. Particulars of the deaths in the various com- 
ponent parts of the registration area are published annually by 
the Census Bureau. 

94. For information regarding the population corresponding 
to the deaths reported we must depend upon the census results. 
As a census is made only periodically, some means must be 
devised of passing from these figures at periodical intervals 
to the average population by age groups during the interval 
covered by the observed deaths. The census returns and the 
death returns are also frequently given only for groups of 
ages, and we have therefore an additional problem to solve, 
namely, that of passing from age groups to individual years of 
age. 

95. Let us take first the problem of finding the average 
population in a certain age group during a specified period. 
For the sake of simplicity we will first suppose that the total 
population analyzed by age groups is known for the beginning 
and end of the period and that the whole period may be con- 
sidered for this purpose as a unit of time. Let the total pop- 
ulation at the beginning be PO and at the end PI, also let the 
population in any particular age group at the beginning be 
aP and at the end (a +b) PI. Then, evidently, the sum of 
the values of a for all age groups must be equal to unity and 
this is also true for the values of (a +6), so that the sum of 
the values of b must be zero. Also suppose the ratio of increase 
of the total population during the period is r, so that we have 
Pi=rPo. Then it is assumed that at any time t during the 
interval the population in the age group is P (a-{-bt)r t , the 
sum of the values of which for all age groups is evidently PQ^. 
In other words the total population is supposed to vary in 
geometrical progression, while the percentage of that total 
in the particular age group is supposed to vary in arithmetical 



54 MORTALITY LAWS AND STATISTICS 

progression. On these assumptions the average population 
during the period is 



r i 



P 

= -M 



log e r [log e r (log e r) 2 ]' 

r ] 

'loge 



96. It is evident that if the period covered by the obser- 
vations were the interval between two censuses, the census 
returns would give directly the values of P . P\, a and b. 
But the dates upon which the census is taken do not usually 
coincide with the limits of the period of observations. Suppose, 
therefore, that we have the results of two censuses taken at 
the times t\ and fa counting from the beginning of the period 
and that the corresponding total populations are PS and P 4 , 
also that the populations in the age group are A PS and BP 4 . 
Then, according to the assumptions already made, we have 



or 
P 4 = r"P or 

(h - /i) log r = log P 4 - log P 3 , 
logr = (logP4-logP 8 )/(/2-/i), .... (5) 
log P = log Ps-h Iogr = (t 2 log Ps-h log P 4 )/(/ 2 -/i). (6) 
a+bfa =A, 
a+bt 2 = B, 



(7) 
(8) 



97. The expression for the average population in the age 
group, when we substitute in Eq. (4) these values of a and 
b, takes the form 



CONSTRUCTION OF MORTALITY TABLES 55 

B-AI r i 



T> ' I f* "1 i 

*<N \. : r- 7-1 7 

log* r I /2 *i fe <i\f I loge ry 

_ r-ii t 2 A A t r i \ 
= -^oi ii r~] rl 1 ) 

log e r[/2 h t 2 ti\ri log e r) 

B / r i 



tz ti t 2 ti\ri loge r 
ri\t 2 i / r i 



= A P 

og,r 

p r-i i / r i \ _ ti ,v 

lo ge rl/ 2 -/iV-i log e r/ h-til' 

Since A For 11 and J5P ^ 2 are the numbers shown in the 
two censuses for the age group in question, it follows that we 
obtain the average population for any age group by multi- 
plying the numbers shown in the two censuses by 



/I fe ^l\r I loge P. 

and 

..r i 



Iog e r[t 2 -ti\r-i \og e r 

respectively, and adding together the products. These factors 
are the same for all age groups and may be calculated once 
for all. 

98. The average deaths per annum may be obtained by 
dividing the total deaths during the period by the number of 
years included, or the same object can be accomplished by 
multiplying up the average population by the number of years 
to get the aggregate population or years of life corresponding 
to the total number of deaths. The latter is the course usually 
followed. 

99. Having, then, the total deaths and the corresponding 
population by groups of ages the remaining problem is to as- 
certain the death rates for individual ages. An approximation 
which was formerly used was to divide the total deaths by the 
total population, and assume that this represented the force 
of mortality at the middle of the interval, or, in terms of the 



50 MORTALITY LAWS AND STATISTICS 

notation explained in Chapter V, n x+ ? = ~ ^f^_ Where 

2 T x T x+n 

n is odd this gives directly the value of m x +j(-i), the two 
functions being approximately equal and each equal to 
d t+i(n _ !)//*+, Thus &+}(,_!> is obtained by Eq. (3) of Chapter 
VI. Where n is even, however, x-\-\n is an integer. The value 
of q x+ m is then determined on the assumption that during the 
year n x increases in geometrical progression at the ratio r 
determined from the values of /JL X for the neighboring groups. 

We have then since n x + t = - , x+t 

at 



From these values of q x the intermediate values are found 
by a formula of interpolation. 

100. It was always recognized that the quinquennial age 
group from 10 to 15 required special treatment, and it has 
recently been shown by Mr. Geo. King that the method under- 
states the death rate at the older ages. This can be seen 
by taking any mortality table and, assuming a stationary pop- 

ulation, comparing the values of ^ ~* with those of m x+2 . 

T- x~ 1 x+5 

In view of this fact some more accurate method is desirable. 
Greater accuracy has been attained by distributing the total 
deaths and population of each age group into individual years 
of age. 

101. In the construction of the Carlisle table this distribu- 
tion was effected by a graphic method. On a base line dis- 
tances were laid off consecutively representing the number 
of years included in the successive age groups. On these 
bases rectangles were constructed whose area represented the 
total number (of deaths or of population as the case may be) 
in the age group. The heights therefore represented the 
average number per year of age in each group. A continuous 



CONSTRUCTION OF MORTALITY TABLES 57 

line with continuous curvature was then drawn through the 
tops of these rectangles such that the area included between 
it and the base was the same in each interval as that of the 
corresponding rectangle. The base was then subdivided to 
represent individual years and ordinates erected to the curve 
so drawn. The area between the base and the curve in the 
interval representing each year of age then represents the 
number assigned to that year. And the total should agree 
with the total for the group. 

102. Under this method, however, it was found difficult 
to read off the diagram with sufficient accuracy and an analyt- 
ical method of redistribution has been devised. If we take 
the population for the various age groups and sum from the 
oldest group downwards we obtain a series of numbers rep- 
resenting the total population older than the respective ages 
which are the points of division between the groups. In 
other words the values of T x are given for a series of values 
of x. If then the values of T x can be interpolated for unit 
intervals we can calculate the values of L x because we have 
L X = T X T X+ I. Also, if the deaths are similarly treated, we 
have a series of values of \ X = I,6 X , from which, by interpolation 
and differencing, the successive values of 6 X may be deter- 
mined. The successive values of m x , q x , or p x may then be 
determined from the relations already given. 

103. The intervals used in tabulating death returns and 
population statistics in the publications of the United States 
Census Bureau are individual years from birth to age 4 last 
birthday, inclusive, and five-year intervals thereafter. The 
returns of some countries, however, give only ten-year intervals, 
beginning with age 15. This grouping is adopted in order to 
avoid a transfer of lives from one group to another arising from 
a tendency to state ages at a multiple of ten years. Where 
the interval is ten years it is readily subdivided into five-year 
intervals by the finite difference formula: 



. (n) 
This may be easily demonstrated by expanding by Taylor's 



68 MORTALITY LAWS AND STATISTICS 

theorem each of the functions on the right and assuming that 
fourth and higher differential cofficients vanish. 

This formula does not apply to the last interval, where 
we use instead the equation 

4 (f( x )+f(x-2t)\=6f(x-t)+f(x-$t)+f( X +t), (") 

which may be similarly demonstrated. 

104. For the sub-division of the five-year intervals a special 
interpolation formula is used which ensures a continuous series. 
The first and second differential coefficients are determined 
at each point of junction by the formulas 



I2fif"(x) = 

(14) 

These formulas may be obtained by expansion as above, 
except that the fourth differential coefficient is not neglected. 

For each interval a function is then found such that the 
values of the function itself and of its first and second dif- 
ferential coefficient at the beginning and end of the interval 
will be equal to those so determined for those points. As there 
are six conditions to be satisfied it follows that if a rational 
algebraic function is to be used it must be of the fifth degree. 
It may readily be demonstrated that the following function 
satisfies the conditions for the interval from x to x+t: 



t 

/"(*) V(t-hY , 
~ 



fi 
f(r I fi ,'(* + *) 

~~1*~~ ~T~ ~p ' ' ' 

This may be seen by differentiating with respect to h and 
then putting h equal to o and / successively. 



CONSTRUCTION OF MORTALITY TABLES 59 

105. This equation takes a simpler form when expressed 
in terms of central differences as follows: Let 5 denote an 
operation such that 



8*f(x)=f(x+2f)-4f(x+t)+6f(x)-4f(x-t)+f(x-2t), 
etc. 

Then, from Eq. (13) we have 



or 

(x)}. (16) 



Similarly, 

(f (*+<) = tA*+/) H-WCs+O -iWGH-0} - 1/0) -5 2 /(*)Ki7) 
And from (14) we have similarly, 

(*), .... (18) 



Substituting, then, these values in Eq. (15) and collecting 
like terms, we have, after reduction, 



-4' 

106. This method cannot be applied in the above form 
below age 15 because f(x 2t) enters into the formula and the 
mortality differs so much at infantile ages from that at other 
ages that it is not safe to assume that /(o) can be determined 
from the same rational algebraic function as the values of f(x) 
above age 5. Having determined, however, /(i6) the first 



60 MORTALITY LAWS AND STATISTICS 

and second differential coefficients at age 10 may be determined 
by the equations 

. . . (21) 



These values enable us to interpolate the values of f(x) 
for ages n to 14 inclusive by the use of Eq. (15). The values 
for ages 6 to 9 inclusive may then be filled in by determining 
values for/'(5) and/"(5) such that if 5 is put for x in Eq. (15) 
the values of 7(3) and 7(4) will be determined by putting h 
successively equal to 2 and i . 

107. Sometimes it is found preferable to interpolate by the 
above methods values for log T x and log X^ instead of those 
of T x and X z , but the principle is the same. In fact, any single 
valued reversible function of T x or \ x can be used if it is found 
to furnish a series more appropriate for interpolation. One 
of the most valuable suggestions in this line is probably the 
use of the ratios of the values of T x and X^ to their values in 
a stationary population derived from some standard mortality 
table from which all minor irregularities have been removed. 

The further treatment of mortality statistics of the general 
population will be considered under the heading of graduation, 
in the next chapter. 

1 08. When dealing with the mortality experience of a 
life insurance company or group of such companies the prob- 
lem is a different one, because in this case information is usually 
available as to the exact date when each life first came under 
observation, so that the death if it had occurred would have 
been included, as well as the exact date when it passed out 
from observation. The problem in this case is to determine 
the most convenient way in which the data can be analyzed 
and how labor can be saved without sacrificing accuracy. 

It is not proposed to describe how the data can best be 
collected as that will depend upon various circumstances, 
particularly as to the mechanical sorting and tabulating devices 
which may be available and as to the nature of the records 
from which the information is to be extracted. Attention 



CONSTRUCTION OF MORTALITY TABLES 61 

will be confined to the principles to be adopted in classification. 
Aggregate mortality tables will first be dealt with, because, even 
where tables are constructed that are analyzed both by age 
and by policy duration, the differences in duration are usually 
neglected when the duration is in excess of some assigned 
limit and an aggregate table used for all longer durations. 

109. The first point to which attention is directed is the 
analysis of the deaths, three essentially different methods 
having been used. The first method is known as the age year 
method and might otherwise be described as the exact method. 
Under this method the date of birth is noted and the deaths 
are classified precisely according to age last birthday at the 
time of death. In this case it is necessary to determine the 
number observed within each year of age and as the same 
life is usually observed through a series of ages the calculation 
can usually be most conveniently made by a continuous process, 
the value of E x+i being determined from that of E x by the 
proper modification. The particular form which the modi- 
fication will take will depend on the treatment of the new 
entrants and withdrawals. The deaths at age x last birthday 
are always treated as included in E x for the full year or as 
included in L x for an average of half a year. In the case of 
new entrants and withdrawals four different methods are now 
available. Under the first method the exact age at entry or 
withdrawal may be noted and the new entrant treated as 
exposed for the fraction of a year of age after entry, the with- 
drawals being similarly treated as exposed for the fraction of 
a year elapsed since birthday at the time of withdrawal. Let 
us denote by n x the number of new entrants at age x last 
birthday and by f x the aggregate of the fractions of a year 
since last birthday at time of entry. Also, let w x denote the 
number of withdrawals and g x the aggregate of the fractions 
at withdrawal. Then we have, evidently, 



E x+ i = E x +(n x+l -e x -w x+1 )-(f x+l -f x ) + (g x+1 -g I ). (23) 



62 MORTALITY LAWS AND STATISTICS 

no. Under the second method of treating new entrants 
and withdrawals the exact fraction in each case is not cal- 
culated, but a general relation is assumed such a.sf x =fn x or 
gx = gw x , based on an examination of part of the data taken 
at random. In this case we would have 

E,+ 1 = E x + (n x+ i-e x -w x+ i) -f(n x+ i - n x ) +g(w x+l - w x ) 

= E x -e x +{fn x +(i-f)n x+1 }-\gw x +(i-g)w x+l \. . (24) 

Sometimes it is assumed that/ and g are each equal to \. 

In this and the preceding section those under observation 
at the commencement of the observations are treated as entrants 
at that time and those under observation at the close as 
withdrawals. 

in. Under the third method the new entrants and with- 
drawals are classified according to mean age at entry or with- 
drawal, the mean age being calculated by deducting the calendar 
year of birth from the calendar year of entry or withdrawal. 
On the assumption that birthdays and dates of entry and 
withdrawal are evenly distributed over each calendar year 
this will give approximately correct results, the cases in which 
the age is overstated balancing those in which it is under- 
stated. The maximum difference between the exact age and 
the mean age is one year. Where the observations are closed 
at the end of a calendar year with a number still under obser- 
vation, or started in the same way with a number already 
under observation, these cases must be specially treated, as 
the assumption of distribution of entry or exit over the cal- 
endar year does not apply. It may, however, be assumed that 
on an average half a year has elapsed since the last birthday. 
If then n x be the new entrants at mean age x, w z the withdrawals 
at the same age, a x those under observation at age x last birth- 
day when the observations began, and e x the corresponding 
number at the close of the observations, we have 



\. (25) 



CONSTRUCTION OF MORTALITY TABLES 63 

112. Under the fourth method the age nearest birthday 
at entry or exit is taken instead of the mean age, the average 
being again correct on the assumption of uniform distribution. 
In this case those under observation at the opening and 
closing of the observations do not require special treatment, 
but may be grouped at age nearest birthday. If then <r x and 
e x be the numbers of such cases at age x nearest birthday, 
we have 

113. The second method of analyzing the deaths is by the 
policy year method under which the completed age at death 
is determined by adding the curtate duration (or integral part 
of the duration) to the age at entry determined according 
to whatever rule may be adopted for that purpose. Under 
this method the age at entry is usually taken as the age nearest 
birthday, but may be taken as the mean age. The age at 
withdrawal and of the existing at beginning or end of obser- 
vations is determined by adding the duration to the age at 
entry. The duration may be calculated exactly in each case 
for the withdrawals, but it is usual to open and close the obser- 
vations on policy anniversaries in chosen calendar years when 
the policy year method is adopted, so that there will be no 
fractional exposures in the case of the existing. We have, 
therefore, 



e x + l . . (27) 

where w x is the number of withdrawals at completed age x 
determined as for the deaths, and g x is the aggregate of the 
fractional durations. Approximate methods of calculating the 
values of g x are sometimes used. 

The duration of the withdrawals may also be taken as an 
exact number of years in each case, the mean duration or the 
nearest duration being used. When this is done fractional 
durations disappear and we see that in this case Eq. (26) holds. 



64 MORTALITY LAWS AND STATISTICS 

114. The third method of analyzing the deaths is by cal- 
endar years, the completed age at death being the age nearest 
birthday at the beginning of the calendar year of death. Under 
this method the ages at entry and withdrawal may be deter. 
mined by any of the methods outlined for the policy year 
method and the equations of relation will be the same as for 
that method. In many cases where the calendar year method 
is applied, the age at the beginning of the calendar year is not 
taken as the age nearest birthday, but is calculated by adding 
the curtate duration plus half a year to the age nearest birth- 
day at entry or by adding the curtate duration to the age next 
birthday at entry. In the former case the ages for which the rates 
of mortality are determined will not be integral, but will be of 
the form x+%, where x is integral. 

115. To summarize, let x be the exact age at entry, x\ 
the age last birthday, so that the age next birthday is x\ + i, 
(x) the age nearest birthday, \x the mean age, and [x] the 
the assumed age at entry. Also let / be the exact duration, 
t\ the curtate duration, (/) the nearest integral duration, and 
{/| the mean duration. Then, under the age year methods 
the completed age at death is taken as x+t\, under the policy 
year methods it is taken as [#]+/|, and under the calendar 
year methods it is taken as either [#] + |/| , or |ff-H| | or 
\x !| + |^|. I n the last expression x \ \ is used to designate 
the mean age six months before entry. In connection with the 
age at entry we may have [x] taken as x, x\+f, (x) or x\, and 
in connection with the withdrawals the age at withdrawal 
may be taken as x+t, (x-\-t], \x-\-t\, x+t\+g, [x]+t, 



116. The census methods may also be conveniently applied 
to a life insurance company's experience, a classification of 
the lives insured being made at the close of each calendar 
year, giving the population in each year of age at that time. 
The completed age may be taken as x+t\ or [#]-H|, usually 
the latter. If, then, the deaths of a given calendar year be 
analyzed by completed age at death, the average population 
during that year for any age year can be determined by taking 



CONSTRUCTION OF MORTALITY TABLES 65 

the mean of the populations at the beginning and end of the 
year. The exposed to risk will then be determined by adding 
to this mean population one-half of the deaths. Each success- 
ive calendar year is treated in the same way and the experi- 
ence gradually accumulated, the experience of as many years 
as desired being combined. 

By a modification of this method the deaths are observed 
from the middle of one calendar year to the middle of the 
next, or from the policy anniversaries in one calendar year to 
those in the next, and the population at the end of the first 
year is taken to represent the corresponding average pop- 
ulatfon. 

117. Where an analyzed mortality table is to be constructed 
the policy year method is usually adopted and the experience 
during each year of duration is treated separately so far as 
is considered necessary. This amounts to the same thing 
as treating each age at entry separately for as many years 
of duration as is decided upon. This latter method is the 
more convenient for descriptive purposes, because new entrants 
are thereby eliminated after the start and we need consider 
only deaths, withdrawals, existing at the beginning of obser- 
vations and existing at end. Assuming that experience before 
the policy anniversary in the first calendar year or after it in 
the last calendar year of the observations is neglected and 
that nearest durations are taken for withdrawals, we have 



(20) 
(29) 



118. Having obtained the values of qw+t, those of l [x]+t , 
and d lx]+t are obtained as follows: The ultimate mortality 
table representing the mortality after the effect of selection 
is assumed to have disappeared is first constructed in the 
way indicated in Chapter II by taking an initial value of l x 
and multiplying successively by the successive values of p x . 
The values of l lx]+t are then constructed in reverse order, 



66 MORTALITY LAWS AND STATISTICS 

beginning with the value of t at which the analyzed mortality 
merges into the ultimate. The working formula is 

(3) 



The values of d lx]+l are then derived by differencing from 
the relation 



119. Diagram No. i is shown to illustrate the relation 
between the rates of mortality in an aggregate table and in 
a select or analyzed table based on the same data. In this 
diagram, in order to avoid the necessity of a change of scale, 
the ordinates are made proportionate to log (1 + 100^), instead 
of to q x . A scale is given on the margin, however, showing 
the values of q x corresponding to different lengths of ordinates. 
The tables selected to illustrate this point are the O M and O IM1 
tables. It will be noted that at the young ages the rates of 
mortality in the aggregate table approach those shown in 
the analyzed table for the first year, whereas at the older ages 
they approach and become indistinguishable from those in 
the ultimate part of the analyzed table. 



CONSTRUCTION OF MORTALITY TABLES 



67 



\ x 



3 a 

O O 

I I 

3 I 




2 



S -9 



-C 

& 



8 1 

+ 1 



-2 . 



'S tn 

g S3 

"2 
2 6 
o, 2 



9 H 



h o 

S to 

S ^ 

M <U 



CHAPTER VII 
GRADUATION OF MORTALITY TABLES 

1 20. WHEN the probability of dying within a year is q x 
and n lives are exposed to the risk of death, the expected 
number of deaths is nq x . This means that in a large number 
of such instances, where n lives are observed in each instance, 
the average number of deaths occurring will be nq x . It does 
not mean, however, that in every instance exactly nq x deaths 
will occur. In fact, unless the values of n and q x happen to 
be so related that nq x is an integer, the exact relation cannot 
hold. Any number of deaths from none up to n is theoretically 
possible. The probability of exactly r deaths is shown by 

In 
the principles of the theory of probability to be , <f x p x n ~ 



rn r 



and in that case the deviation of the actual number of deaths 
from the expected will be rnq x . The mean value of the 
square of this deviation may be shown to be np x q x . Where 
n is large this deviation is equally likely to be positive or neg- 
ative and there is approximately an even chance that it will 
exceed \^npxq x in absolute magnitude. If, then, a value 

7* 

q' x = be determined by dividing the observed number of 
n 

deaths by the number exposed, there is an even chance that 
this value will differ from the true probability q x by at least 



as much as -*/, and the deviation is equally likely to 

3\ n 

be positive or negative. We have no means, however, of deter- 
mining from the observations themselves at age x whether 
q' x is greater or less than q x and in the absence of further 

68 



GRADUATION OF MORTALITY TABLES 69 

information the hypothesis is adopted that q x = q' x , because 

\n 
that is the value of q x which makes the probability = p x n ~ r q x r 

\r\n-r 

of the observed facts a maximum. 

121. If, now, we have made similar observations at a series 
of consecutive ages for which the true probabilities of death 
are q x , q x+l , etc., the principle of continuity would lead us to 
expect that the successive values of q x for the different ages 
would form a smooth series. Where the values of q' x are 
calculated each will differ from the corresponding value of 
q x by a quantity which may be large or small, positive or 
negative, a positive deviation being as likely to be followed 
by a negative one as by a positive. The theory, however, 
indicates that the probable values of these deviations in the 
value of q x decrease as n increases and that positive and neg- 
ative values tend to counterbalance one another. It follows 
from the above that we must expect the values of q' x to form 
a somewhat irregular series, the amount of the irregularity 
depending on the numbers under observation. The problem 
of graduation is to remove the irregularities from this series 
and to approximate as closely as possible to the true values 
of q x . 

122. The methods which have been adopted for this purpose 
come under four general classes. Under the first method a 
diagram is made to represent graphically the observed facts and 
a continuous curve is then drawn as a basis for the graduated 
series. Under the second method the graduated series is 
formed by interpolation on the basis of values determined 
for fixed intervals, these values being so determined as to give 
an interpolated series fitting as closely as possible to the ob- 
served facts. Under the third method the individual terms 
of the graduated series are each determined by a summation 
of adjacent terms of the original series, a correction being 
introduced to allow for the curvature. Under the fourth 
method a mathematical formula containing arbitrary constants 
is used to express the series and the constants are determined 
so as to agree in certain respects with the observed facts. 



70 MORTALITY LAWS AND STATISTICS 

123. After a graduated series has been constructed it is 
usually tested with respect to the two points of smoothness 
and closeness to the observed facts. With respect to smooth- 
ness the fact that a series is determined by a mathematical 
formula is usually taken as sufficient, but when it is not so 
determined the criterion usually adopted is the smallness of 
the third differences in the graduated series. This smallness 
is sometimes tested by inspection of the differences after they 
have been taken out, but in comparing two different graduations 
of the same series, if it is desired to have a numerical measure 
of their departure from absolute smoothness, the sum of the 
squares of the third differences or the sum of the absolute 
values irrespective of sign of such differences may be taken 
as such measure. 

124. With respect to closeness to the observed facts the 
requirements usually made are (i) that the total number of 
expected deaths and their first and second moments about any 
assigned age shall be the same as for the actual deaths, and (2) 
that the departures in individual groups shall not, on the 
average, materially exceed in magnitude those expected in 
accordance with the theory of probability. This comparison 
is usually made by recording the difference between the actual 
and expected deaths at each age. A continuous summation 
of these deviations is then made with due regard to sign. The 
smallness of the numbers in this column of accumulated devia- 
tions, the frequency of changes of sign and the extent to which 
positive and negative terms balance one another indicate the 
extent to which the first requirement is complied with. The 
sum of the deviations without regard to sign tests directly 
the second requirement if the average deviation which is 
approximately equal to %^/np x q x be calculated for each age 
for the purpose of comparison. The comparison may also be 
based on the sum of the squares of the departures or on the 
sum of those squares each divided by its mean value np x q x . 
The test in this last form is supported logically by the fact 
that if the number observed is large the quantity so arrived 
at is proportional to the logarithm of the ratio between the 



GRADUATION OF MORTALITY TABLES 71 

probability of the observed facts and that of the expected 
according to the graduated table. 

125. The graphic method of graduating mortality tables 
arises naturally from the graphic method of representing 
them. Under this method the values of q x or of m x are rep- 
resented by points in a diagram. For convenience in plotting 
the diagram accurately ruled section paper is ordinarily used, 
the years of age being represented by equal intervals along 
the base line and the rate being represented on a suitable scale 
by the distance of the point from the base. When the points 
corresponding to the successive ages are plotted and joined 
by straight lines it is found in an ungraduated table that the 
result is a zigzag line full of minor irregularities, but showing 
indications, the strength of which depends on the volume of 
the observations, of an underlying regular law. The graduation 
of the table is effected by drawing among these points, but 
not necessarily through any of them, a regular curve to rep- 
resent this law. Preliminary groupings not covering equal 
intervals, but so arranged as to produce the greatest attainable 
regularity are made in order to bring out this law. After the 
curve is drawn, the values of the ordinates are read off and 
the results corrected to remove any irregularities due to errors 
in reading. A comparison is then made between the expected 
and actual deaths on the lines indicated above and, if a rel- 
atively large and persistent deviation in either direction is 
accumulated in any section of the table, the curve is amended 
to reduce or eliminate it. 

126. In applying this method the difficulty is found that, 
if the scale of* the diagram is sufficiently large to permit of 
accurate reading in one part of the curve, it will be too large 
in another. This difficulty is met by plotting not the actual 
value of q x but some more slowly varying quantity from which 
it can be determined. One method is to take as a basis some 
mortality table already constructed from a mathematical 
formula such as Makeham's and to plot either the ratio of the 
observed q x to the rate at the same age in the table, or the 
difference of the rates. Another method is to use log (i + 100^) 



72 MORTALITY LAWS AND STATISTICS 



or log(i + io<fc) instead of q t . Specially ruled section paper 
has been prepared with the vertical spacing so arranged that 
if the values of q z are plotted according to the ruling the actual 
distances from the base will be log (1 + 1009*), so that, if this 
paper is used, the whole diagram is reduced to practicable 
dimensions without altering the scale where q x is small, and 
after the curve is drawn the values of q x may be read off directly. 

127. In the interpolation method the problem may be 
divided into three parts. The first is to determine which func- 
tion of the mortality to interpolate, the second is to deter- 
mine the values for the selected ages of that function and the 
third part is to interpolate the intermediate values. Various func- 
tions have been used as a basis for interpolation and the method 
of determining the values at the selected ages will depend on 
the function selected. Where l x or log l x is used we must have 
an ungraduated mortality table as a basis and we obtain our 
points for interpolation by simply taking the values at the 
selected intervals from this ungraduated table. Where, how- 
ever, q x or some function of q x , such as log q x , m x , or log m x , 
is selected for interpolation we must group the data so as to 
determine with as much precision as possible the value of the 
function for the selected ages. For this purpose the exposed 
to risk, or the population, and the deaths are usually com- 
bined into age groups. In the case of the earlier English 
Life Tables the simple but somewhat inaccurate assumption 
was made that the total deaths in an age group of five or ten 
years divided by the total population corresponding to them 
would give the force of mortality at the central age of the 
group as explained in Chapter VI. 

128. Where more accurate values are desired the redis- 
tributed values of the deaths and exposures or population for 
some one year of age in each group may be calculated by 
methods similar to those described for census statistics in the 
preceding chapter. For this purpose quinquennial age groups 
are usually used, and in the case of general population statistics 
these groupings are given to us ready made or if not can only 
be formed by subdividing decennial groups. In the case, how- 



GRADUATION OF MORTALITY TABLES 73 

ever, of the experience of insurance companies the groups are 
formed by combining the figures for individual ages and we 
have freedom of choice as to the limits of the groups. 

129. If the central year of each quinquennial group is 
taken, it is usual to assume that the population and the deaths 
for the successive years may be expressed by a rational 
algebraic function of the third degree, so that we have, for each 
function, an equation of the form 



Summing this from h= 2 to h=-\-2 inclusive we get for the 
total number included in the group of which x is the central 
age, 5/00 +5/" GO- This may be expressed by the equation 
Wr = 5/00+5/"00> if we use w x to denote the total of the group. 
Similarly, summing from h= r j to h=+ f j inclusive we 
have 



Eliminating/" 00 between these two equations, we have 



or 

5f(x}=iv x -^8 2 w x . ....... (i) 

It thus appears that the adjustment to be made to the total 
of the groups to obtain the number for the central year is the 
same in form for the deaths, the population, or the exposed 
to risk. 

130. Where the first year of each group is taken, it is necessary 
to use four age groups in order to determine the value of 
We have 

' '" 



= S f(x) - 40/(*) + i6 5 r 00 - 



74 MORTALITY LAWS AND STATISTICS 



TOO. 

~6~ J 



- 2 



.*. 6257(0;) = 25( 

f ( ^_2w a; _3 g+2 -3 +a ( . 

J()= 25 625 

131. On account of the lack of symmetry of the above 
formula, it may be considered desirable to select a year of 
age, one-half of which will be in one age group and the other 
half in the next group. In this case and in cases where the 
force of mortality is to be determined it is more convenient 
to work with a function <f>(x) representing the number per 
unit of age at exact age x, so that 



and 

T 5 
Jo 
or 

/*+2i 

Wx _ =J +(x+h)dk. 



GRADUATION OF MORTALITY TABLES 75 

Expanding <j>(x+h) and integrating, we have 
J-io 2 6 



24 



24 



!= ( <t>(* 

Jo 



Wx+2' 

2 4 



24 

j 2 C 

-- " 



But 

/(*-*)= r + w-f-*)^= s ^(*)+- L 

./-i 24 

and 



12 



2 ). (3) 

This covers the case where half of the year of age is in each 
group. 

132. Where the force of mortality is required at the age 
corresponding to the point of division we have 



or 

IO<j>(x) = (w t -3-%8 2 'W I -3) + (Wx+2-%& 2 Wx+2), . (4) 

which is the formula to be used for the deaths and population. 



76 MORTALITY LAWS AND STATISTCIS 

133. For the age corresponding to the center of the group 
we have 



X + 2J 
*t* 
2J 



24 



24 

whence 



or 

T 

<Z<b(x) =Wr- I 5 2 W-r ( $) 

J-r\ / X J I ) \J/ 

2 4 

134. These adjustments enable us to calculate the values 
of q x , m X) or n x for values of x separated by quinquennial ages, 
and the remainder of the problem consists in interpolating 
intermediate values. For this purpose a formula of osculatory 
interpolation will be found most satisfactory. Three such 
formulas have been proposed. The first, which is the basis 
of the other two, is the one described in Chapter VI in con- 
nection with the redistribution of population and deaths, 
the simplest form for practical application being that given in 
Eq. (20) of that chapter. 



The other two formulas are simplifications of this by 
omitting some of the conditions, the first differential coeffi- 
cient only being determined for the points of junction. 

135. In Karup's form, which is the simpler, this first dif- 
ferential coefficient is determined on the assumption of a 
curve of the second degree for/(#), the value so derived being 
2 lj'( x ')j(x-\-f}f(x t}. A curve of the third degree is then 



GRADUATION OF MORTALITY TABLES 77 

determined so as to have the required values and first dif- 
ferential coefficients at the points of junction. The equation 
of this curve is found to be 



(7) 



136. Greater accuracy without material increase of work 
can be obtained by determining the first differential coefficient 
on the assumption of a fourth difference curve by Eq. (13) 
of Chapter VI, namely, 



This condition is then found to be satisfied by the equation 

H^yV/C*) -***/(*))] } 



137. When making this interpolation it is usually necessary 
to assume the age at which q x becomes equal to unity and the 
age chosen should be consistent with the data at the old ages. 
When it has been chosen it will be advisable to arrange the 
division into groups, if possible, in such a way that the ages 
for which the function is determined will form with the lim- 
iting age a regular series of differences. For example, if it is 
assumed that (7102 = 1 or w 10 2 = 2, then we should so arrange 
the groupings as to determine the rates for ages ending in 
2 and 7, starting the groups at those ages if it is intended to 
use the first year of each group and starting the groups at ages 
ending in o or 5 if the central age of each group is to be used. 

138. If we have interpolated a series of values of M* for each 
age, then we can pass to values of p x by the approximate 
formula 






78 MORTALITY LAWS AND STATISTICS 

+l 



fx 

colog e />* = I 

Jx 



2 as 6 ax 2 24 



24 



/ v 



139. The summation methods of graduation can be applied 
only when we have constructed a complete table of the ungrad- 
uated values of the function to be graduated. In investigating 
the effect of a graduation formula we may consider the un- 
graduated value of a function as consisting of two parts, one 
the true value V x of the function which would result from 
an indefinitely extended experience and the other the error 
or deviation E x of the observed value from the true value. 
The fundamental assumption which is the basis of all gradua- 
tion is that the values of V x form a regular or smooth series 
and that the values of E x form an irregular series, the fluctua- 
tions in the value of any one term being independent of those 
of the neighboring terms. It is also assumed that the mean 
value of each error, E x , is zero when the sign of the error is 
taken into account. 

140. Let us first assume that over a short range of values 
of x, the values of V x may be considered as forming an arith- 
metic series. Then it is evident that if we take the arithmetic 
mean of an odd number of terms of the series of values of V x 
it will be exactly equal to the middle term. Consequently, 
if we take a similar average of the ungraduated values of the 
function it will differ from the true value V x corresponding 
to the middle term by the average of the values of E x , and 
it follows from principles of averages that the mean absolute 
value of this resultant deviation will be much smaller than that 
of the individual ungraduated values. In this case, therefore, 
a simple average of a number of terms will constitute a grad- 
uation of the series. It is evident that the same process may 
be repeated without disturbing the value of V x , but the effect 
on the values of E x diminishes with the successive summation 



GRADUATION OF MORTALITY TABLES 79 

because the neighboring values are no longer independent, 
each original deviation now affecting, although to a smaller 
extent, a number of successive values. For example, suppose 
we take the average of five successive terms, then the error 
in the resulting average will be %(E x -2+E x -i+E x +E x+ i+E x+2 ). 
If we assume that the mean value of the square of each of 
these primary errors is jU2, then it is evident that the mean 
value of the square of the error of the averages will be |/*2. 
If we repeat the process a second time, the expression for the 
error will be 



the mean value of the square of which is 
85 17 17 i 

,V2 = - M2 = --- M2. 

25- 125 25 5 
If it is repeated a third time the error is 



the mean value of the square of which is 

I75 1 I0 3 J 7 i 
- J - i ViU2 = - ---- M2- 
1252 125 25 5 

141. Let us, however, investigate the effect on the third 
differences of the series. On the assumption made, that the 
values of V x form an arithmetic series, the third differences 
will arise entirely from the errors, the expression for the third 
differences in the ungraduated series being 



the mean value of the square of which is 20/12- When the 
first average is taken the third difference becomes 



the mean value of the square of which is 



12 ^ 

M2= -- 20/*2. 

25 125 



80 MORTALITY LAWS AND STATISTICS 

For the second average the third difference is 



the mean value of the square of which is 
12 13 

M2 = --- -20M2- 
625 25 125 

For the third average the third difference is 



the mean value of the square of which is 
20 iii 



125 15 25 125 

It will be noted that for successive summations the effect 
on the smoothness of the series of errors does not diminish 
as rapidly as the effect on the absolute values. 

142. The summations do not necessarily all extend over 
the same number of terms nor is the number of terms in each 
summation necessarily odd because, while an average over 
an even number of terms of a series does not give a term of 
the series but a term midway between two of them, a second 
average over the same or a different even number of terms 
will bring us back to the original series. An even number 
of summations over an even number of terms each may there- 
fore be introduced and the resulting averages will correspond 
to terms of the original series. For example, suppose, instead 
of taking the average in fives three times we take the average 
in fours, fives and sixes. Then the expression for the resulting 
error will be 



the mean value of the square of which is -^-=-M2i r a little 

I20 2 

less than for the third average in fives. The expression for 
the third difference is 



-\-E x +E x - 1 +E x - 2 E x -6, 



GRADUATION OF MORTALITY TABLES 81 

Q 

the mean value of the square of which is -^2- We thus 

I20 2 

see that by making the periods unequal a slight increase in 
weight is obtained on the individual term and a great increase 
in weight in the third difference. In other words, with unequal 
summations a much smoother series is obtained and it is 
at the same time a little more accurate. 

143. Thus far we have assumed that the series of true 
values is an arithmetic series, the general term of which may 
be expressed by a linear function. It is necessary, however, 
to take into account the second and higher differential coeffi- 
cients. The summation graduation formulas ordinarily used 
contain a correction so that a series the general term of which 
is of the third degree will be reproduced. Where the function 
is of the third degree, we have, generally, 



If, then, we add together n terms of this series, we have 



- lt/ (s) + ^y^)V t( *- l)( ^^^ 

-L ~i 

But 



n(n 2 
24 



If, then, we denote 

V / \ / 

n 



82 MORTALITY LAWS AND STATISTICS 

by (n]/(rc), we have, generally, 



or 



n 24 

144. The operation of taking the average of n successive 

n 2 i 
terms has therefore introduced an error / (*) into the value 

24 

of f(x). If, then, we repeat the process with the same or a 
different number of terms, an additional error of the same form 
is introduced. We also see that, since according to our assump- 
tions /"(x) vanishes for all values of x and consequently J"(x] 
is linear in form, the error introduced by the first average is 
carried forward unchanged. If, then, three successive averages 
cover p, q, and r terms, we have, 



pqr 24 24 24 



24 

Where the number of terms in each average is the same and 
each equal to n this may be expressed as follows: 



145. A correction must accordingly be introduced to com- 
pensate for this error. If, then, we use y n f(x) as a short expres- 
sion iorf(x-\-n)+f(x n) we have 



or 



we have, therefore, 



or 

( i + 2a + 2b + 2c)f(x) - (ayi + by 



GRADUATION OF MORTALITY TABLES 83 

If, therefore, we substitute, 

(i + 2a + 2b + 2c)f(x)-(ayi+by 2 +cy3)f(x) for/(X), 
before averaging, we have 



pqr 



If then a, b, and c are so determined that 

/> 2 +g 2 +r 2 -3 
= ^ -^ 



24 

a series of the third degree will be exactly reproduced. We 
have here three unknowns and only one condition, so that the 
equation is an indeterminate one and the various formulas 
result from the use of different values of p, q, and r and of 
a, b, and c, the condition in some cases not being exactly satisfied. 

146. The first summation formula correct to third differences 
was that devised by Woolhouse. It was not originally con- 
structed as a summation formula, but it was afterwards found 
to take that form. In this formula p = q = r = 5, so that 



= 

24 

Also a = 3, and b = c = o, so that the formula takes the form 

M 3 

-{7 37ii/0*0- Woolhouse's formula when expanded as a 

function of the terms of the series becomes 

li7( 2 5 + 2 47l + 2172 + 773+374 -276-377)/(*). 

147. J. A. Higham, who first showed that Woolhouse's 
formula might be applied by the summation method, suggested 
an alternative compensating adjustment which was equiv- 



84 MORTALITY LAWS AND STATISTICS 

alent to putting a=-i, b = i and c-o, thus still keeping 
= 3. His formula therefore took the form 



which expands into 

7^(25 +2471 + 1872 +1073 +374-270- 277 -ys)f(x). 

148. Karup's graduation formula uses summations in fives 
with the compensating adjustment a= f, 6 = 0, c = f, so that 
the formula becomes, 



149. G. F. Hardy used the same compensating adjustment 
as Higham, but used successive summations in fours, fives, and 



_____ 

sixes, so that for his formula ^3iV> an ^ ft is 

24 

not fully compensated, the remaining second difference error 
being T V /"(#). This is, however, small and is approximately 
counterbalanced for some functions such as q x and m x by the 
fourth difference error. A lack of compensation to this extent 
is therefore considered admissible. This formula becomes 



150. The most powerful summation formula which has been 
put to practical use is probably that of Spencer, which includes 
when expanded 21 terms of the original series. In this formula 
two summations in fives and one in sevens are used, so that 

p -rq -rr 3 = 4 ^y so a= , 6=0, and c = \, so that the 

24 
formula reduces to 



175 35 



151. A still more powerful formula would be given by 



__ 

putting p = 5, q = 7, and r=n, so that - ^ = 8, and 

24 



GRADUATION OF MORTALITY TABLES 



85 



in the compensating adjustment we have a= i, 6=0, and 
c = i , so that the formula becomes, 









152. The weight of a graduation formula is found by expand- 
ing the formula, adding together the squares of the coefficients 
and taking the reciprocal, that being the average proportion 
in which the mean square of the errors is reduced by the gradua- 
tion. The smoothing coefficient is found by similarly expanding 
the third difference of the graduated terms, adding together the 
squares of the coefficients, dividing by twenty and extracting 
the square root. It measures the effect of the graduation on 
the mean absolute value of the third differences. 

153. The following table shows a number of summation 
formulas and their weights and smoothing coefficients: 



Author. 


No. of 
Terms. 


Formula. 


Weight. 


Smoothing 
Coefficient 


Error. 


Finlaison 
Woolhouse. . . 
J. Spencer . . 
Higham 
G. F. Hardy. 
J. Spencer . . . 
Karup 


13 
IS 

15 
17 
17 
19 
19 

21 
21 
27 


[sP 

125 

[slMV 


8.92 
5-50 
5-4 
5-87 
6.07 

6-73 
6. 14 
6.98 
6.70 
9.11 


I 
125 

I 

15 

I 
60 
I 

56 
I 

95 

i 

85 
i 

105 

i 


!2.6/ ly (x) 


320 !4l3M 3*1 

WtsH^l 


1 20 

tfizL i M "- t 


/J. Spencer (a) 
J. Spencer (b) 
Henderson. . . 


UH5H6], 


1 60 
i 
141 

i 
326 


600 l3 " 

ISluHUjir I > 


12 


g 1131 Taj 



154. One difficulty in connection with the application of 
summation formulas to the graduation of tables is that in 



86 MORTALITY LAWS AND STATISTICS 

order to determine any graduated value of the function it is 
necessary to know a number of ungraduated values above 
and below it and, unless the function is such that it disappears 
and the ungraduated values beyond a certain limit may be 
assumed to be zero, there will be a portion at either end of the 
table for which values will not be obtained, and some sup- 
plementary means must be adopted of completing the table 
if it is necessary to have it complete. In the case also of 
insurance companies' experiences, the values of the functions 
near the limits are usually derived from very limited data 
and are consequently very irregular. Both of these difficulties 
may be overcome by taking the three last graduated values 
of the function that are considered as determined with sufficient 
accuracy as a basis and determining a function of x of the 
third degree which will reproduce these three values and will 
also make die total expected deaths for ages beyond equal 
to the {expected)? The general expression for a function of 
the third degreein x such that f(a) = u a , /(&) = u b , and f(c) = u c is 



_i_ 

i 



__ . 

/ T\/ \~*a i /-L \/L \*& ' / \ / lA** 

(a b)(a c) (b a)(b-c) (c-a)(c b) 



where k is an arbitrary constant. 

155. Taking up next the application of a general law with 
arbitrary constants to the graduation of tables the law which is 
most frequently used is that of Makeham, according to which 
n x = A -i-Bc 1 , or l x = ks x g cx . The constant k is merely in the nature 
of a radix and does not affect the rates of mortality. This 
formula may be applied in two general ways: first, to construct a 
graduated mortality table from the original data of the exposed 
to risk (or population) and deaths without the explicit deter- 
mination of the ungraduated rates of mortality; and second, 
to graduate a rough table without direct reference to the 
original data. 

156. Probably the simplest method of determining the 
Makeham constants from the original data is to group the 
exposures (or population) and the deaths into quinquennial 



GRADUATION OF MORTALITY TABLES 87 

age groups and then, by the process already described in con- 
nection with interpolation methods, determine the values of 
q x , m x or n x at quinquennial intervals. Where q x or m x is 
determined, we can proceed immediately to colog p x from 
the known relations. Now we have 

colog p x = log l x - log 1 I+ 1 , 

= (log k+x log s+c x log g) 



It is therefore of the same form as n x . If, then, we neglect 
the values at young ages and at extreme old ages as derived 
from insufficient data and start with some age y in the neigh- 
borhood of age 30, we have 



colog />j,+3 colog A+5 + 5 colog /Wio+5 col g / 
+3 colog 



= i8a+/3c"(i 
Similarly 

colog A/+15+3 colog A+20 + 5 col S ^+25 + 5 colog p v+30 

+3 colog />+36+colog ^+40 = S 2 

= i8a+/3c" +15 (i +C 5 )(i +c 5 +c 10 ) 2 , 
and 
colog &+ 30 +3 col S ^+35 + 5 colog p v+4 o + 5 colog ^ 

+ 3 COlog p y + 50 + COlog />, + 56 = 



From these equations we see that 



88 MORTALITY LAWS AND STATISTICS 
l8 = TT " = _ 1 3 . 



Thus, c, a, and are determined. Similarly, the values 
of c, A, and B may be determined from the values of Hx and 
from either a and /3 or A and 5 we may proceed to the values 
of s and g. 

157. If a further refinement is required, we may assume 
that the values determined as above are only approximate 
and that 

colog p x = (a+Mh) + (p+Mk)c x (i 

\ 

where h, k, and / are small quantities and M is the modulus 
of common logarithms, or logio e. Then approximately 

q x = q' x +hp' x +kc x p f x +lxc x p f x , 

where q' x and p' x are derived from the constants a, /3, and c. 
We have, then, three unknowns, h, k, and /, to determine and 
three equations are required. These equations may be ob- 
tained by making the total number of expected deaths and 
the first and second moments of the expected deaths equal 
to those of the actual deaths. 
The equations so obtained are: 



' ' x +k2xc*E x p' ' x +l?x 2 c x E I p ' x = Zx(6 x -E x q' x ), 
' x +k2x 2 c x E x p' x +l2x?c I E x p' x = '2x 
These equations are seen to be equivalent to 



= Z(x-a)(6 x -E x q' x ), 
x - d) 2 E x p' x + (k +al)Z(x - a) 2 c z E x p' x +l2(x - a 



GRADUATION OF MORTALITY TABLES 89 

where a is any suitable quantity used for the purpose of re- 
ducing the numbers involved. From these three equations the 
values of h, k, and / are determined, and the values of a, 0, 
and c are corrected accordingly. 

158. A shorter process is to assume that the value of c is 
accurate, and consequently l = o, and to determine h and k 
from the first two equations or to determine A and B directly 
from two equations depending on the relation 



The two equations are 



159. Where a mortality table has been constructed and it 
is desired to graduate it by Makeham's formula, the simplest 
method is to determine the constants from the values of log l x 
at four equidistant ages by the method described in Chapter 
III. Unless, however, the ungraduated table is already com- 
paratively smooth the constants so determined will depend 
to too great an extent on the particular ages selected. To 
minimize the irregularity we may take, instead of individual 
values of log l x , the sums of a number of consecutive values. 
Then we have: 



C I 

log l x = n log * + W (*+/)+-- log s 

-i). 
- log g, 

C 1 



"~ 1 



_ | _ | 
+ -'logg, 

C ~ ~ A 



90 MORTALITY LAWS AND STATISTICS 



I 



_ L , 
-S 2 = nt log 5-f - - log g, 

c i 



. , 

4-03 = / log sH -- - - log g, 

C ~"~ X 



I 



ci 

= c t . 

160. This method does not, however, entirely eliminate 
the objection *hat special importance is given to special points 
of division. To obviate this it has been suggested to use all 
the values of log/ z except those at extreme old ages and at 
young ages and to so determine the constants that the sum 
of the values and the first, second, and third moments will 
be reproduced. This can best be expressed in terms of sum- 
mations as follows: 

Suppose a is the youngest age to be included and let n 
be the total number of ages to be included. Then 

= (log k -{-a log s) +x log 5 +cV log g, 
= log k'+x log s+c* log g'. 

Where log k' = log k -{-a log s and log g' = c log g. Also 

(2) C 1 I 

i5 2 = 2S~ 1 log a+z = x\ogk'-i logs-| -log/, 

2 C I 

,.(2) ^.(3) ( r x t v 1 , 

lg 



24 

( C* I X X (2) 

~M 7 vi~7I rv5~T7I T\ I S S i 



GRADUATION OF MORTALITY TABLES 91 





' log k'-\ --- log s 

24 120 



Where # (f>) =#(# i)(# 2) . . . (# r+i). 

From these four equations, after putting x equal to w in 
each, log k', log 5, and log g' may be eliminated, leaving an 
equation in c for solution. This equation will be of the (n i)th 
degree and the numerical solution may be obtained to any 
required degree of accuracy. After the value of c is obtained 
those of log k', log s, and log g' follow readily. 

161. In the preceding discussion it has been assumed that 
the mortality table is an aggregate one, or in other words 
that it is analyzed only according to attained . age, and where 
select or analyzed tables are required some. , modification of 
the method is necessary. This usually consists in making 
a and ft, in the equation 



functions of the duration, so that we have as the general 
expression 



The constants for each year of duration may be deter- 
mined by any of the methods already described, the data for 
each year of duration being treated as representing a mortality 
table complete in itself but the same value of c being used 
for all. 

162. The values of f\(f) and/2(/) so derived will, however, 
be somewhat irregular, so that they themselves require further 
graduation. It is usually assumed that they become con- 
stant after some definite duration, such as five years or ten 
years, the constant values being determined from the aggre- 
gate experience for all longer durations. In selecting formulas 
for graduating the values of these functions during the period 
of selection, the following conditions should be satisfied: (i) 
a smooth junction between the curves representing the select 



92 MORTALITY LAWS AND STATISTICS 

and ultimate tables; (2) an agreement between the graduated 
and ungraduated values in the first year, as special importance 
Js attached to the rate of mortality in that year; (3) an agree- 
ment between the aggregate graduated and ungraduated 
values of these functions during the period between the date 
of entry and the ultimate table. Considerable experimenting 
will usually be necessary to determine the function complying 
with these conditions. 

163. In considering the method of graduation to be adopted 
in any particular case it is evident that a graduation by Make- 
ham's formula possesses an advantage over the others on the 
scale of smoothness and since three arbitrary constants are 
available the sum of the deviations and of the accumulated 
deviations can be made to vanish. In view of these advan- 
tages and of its other advantages in connection with the cal- 
culation of joint contingencies, a graduation by that formula 
will be the best, provided the absolute deviations in groups 
do not materially exceed the expected and provided there is 
no characteristic feature of the experience which will not be 
reproduced by the formula. 

164. Where a mathematical law cannot be applied it will 
usually be found that where the data is very scanty the 
graphic method will produce the best results as irregularities 
will occur of wide range, such as neither the interpolation nor 
the summation method is competent to remove. Where, how- 
ever, the data is more extensive, so as to give a satisfactory 
degree of regularity under the operation of the interpolation 
or of the summation method, those methods will be the more 
satisfactory as the values derived do not depend on the judg- 
ment of the operator except as exercised in the selection of the 
particular graduation formula to be used, and they can be 
obtained to a greater degree of accuracy than is possible under 
the graphic method. 

165. Diagram No. 2 illustrates the relation between an 
ungraduated series of rates of mortality and a graduated series. 
The irregular line represents the rate of mortality shown in 
the ungraduated experience, during the ten-year period from 




s 



I 111 1 |1 slill 111 



94 MORTALITY LAWS AND STATISTICS 

the policy anniversaries in 1899 to those in 1909, of a large 
American life insurance company on policies issued in the 
Northern States and in force more than five years at the time of 
observation. The ordinates are proportionate to log (i + ioo^ z ) . 
The regular lines represent the same rates graduated by Spencer's 
formula with a preliminary adjustment at the extreme ages 
and the rates in the M(5> table which was graduated by Make- 
ham's formula. The rates by the M. A. table are not shown, 
partly-because they . jKOiild .be practically indistinguishable in 
the diagram from those by the graduated experience of the 
company from age 35 to age 65 and from those by the M(5) 
table from age 70 to age 100. The comparatively wide fluc- 
tuations in the rates by the ungraduated table at the extreme 
ages should be noted. The irregularity in the M(5) table 
at the extreme old ages is due to the fact that the values of 
l x and d x were tabulated to integers only and the values of 
q x recalculated from them instead of being calculated directly 
from the constants in the formula. 



CHAPTER VIII 
NORTHEASTERN STATES MORTALITY TABLE 

1 66. Some of the methods described in the preceding chap- 
ters will be illustrated by the construction of a mortality table 
for the Northeastern States. The data used will be the 
death returns for the five calendar years 1908 to 1912 inclusive 
and the census returns as of June i, 1900, and April 15, 1910, 
for the New England States and the three Middle Atlantic 
States, New York, New Jersey, and Pennsylvania. Table I 
shows the total deaths in these states for the five-year interval 
arranged by age groups and the total population at the two 
dates arranged in the same way. The average population by 
age groups for the five-year interval is also shown. 

167. In this table the average population in each group 
is determined by means of Eqs. (5) and (9) of Chapter VI, 
only the population for which the ages are stated being taken 
into account. The cases where the age is returned as unknown 
are an extremely small percentage of the total and the effect 
of their omission is negligible. June i, 1900, is 7 T 7 T years before 
the beginning of the observations, which cover five years, there- 
fore, /i = |V April 15, 1910, is 2-sir years after the beginning 
therefore, /2 = ii- Also ^3 = 21004724 and P = 25 836 088, 
so that we have log r = . 0455239 and the two factors entering 
into the determination of the average population are .03154363 
and 1.0304817 for June i, 1900, and April 15, 1910, respectively. 
The total years of life are then obtained by multiplying the 
average population by five. 

168. We then apply Eq. (3) of Chapter VII and obtain the 
values of Lg k , Z, 141 , L m , etc., and of 14J , 6 m , 24i , etc. The 
value of 6 9 j is determined by leaving out of account the deaths 

95 



96 



MORTALITY LAWS AND STATISTICS 



TABLE I 
DEATHS AND POPULATION NORTHEASTERN STATES. 1908-1912 



Age Last 
Birthday. 


Deaths, 
1908-1912. 


Population. 


Total 
Years of 
Life. 


April 15. 1910 


June i, 1900. 


Average 
1908-1912. 





397 985 


574480 


476 810 


576951 


2 884 755 


I 


84939 


55 632 


426 773 


507 583 


2537915 


2 


357S7 


553 698 


448816 


556 419 


2 782 095 


3 


22 I ID 


541 178 


449855 


543 484 


2 717 420 


4 


15694 


515976 


442067 


517 760 


2588800 


0-4 


556 491 


2 690 964 


2 244321 


2 702 197 


13510985 


5-9 


42475 


2 400 180 


2 IIO 213 


2 406 778 


12033 890 


10-14 


25885 


2 285 642 


I 908 183 


2 295 121 


11475605 


iS-iQ 


42677 


2 385 256 


i 888 668 


2 398 388 


II 991 940 


20-24 


64 604 


2 554 686 


2 O24 318 


2 568 703 


12 843 515 


25-29 


71 700 


2 405 723 


i 977 342 


2 416 681 


I 2 083 405 


30-34 


75273 


2 121 420 


i 738 577 


2 131 243 


10656 215 


35-39 


86752 


I 984 723 


i 562 115 


i 995 946 


9 979 730 


40-44 


86342 


I 671 571 


i 305 952 


i 681 329 


8 406 645 


45-49 


90895 


i 399 363 


i 053 884 


i 408 775 


7 043 875 


5>-54 


99493 


i 174 250 


899 808 


i 181 660 


5 908 300 


55-59 


103 030 


840 368 


697 132 


843 994 


4 219970 


60-64 


118 590 


686 755 


575 880 


689 523 


3 447 615 


65-69 


128 504 


5H970 


418332 


5I747I 


2 587 355 


70-74 


128 074 


355 427 


292 946 


357 020 


i 785 loo 


75-79 


"3 343 


2IO 122 


177814 


2IO 918 


i 054 590 


80-84 


82 412 


IO2 741 


88019 


103097 


5I548S 


85-89 


45 174 


39617 


3i 504 


39831 


I99I55 


90-94 


15993 


10 198 


7923 


10259 


Si 295 


95-99 


3497 


I 851 


i 523 


1859 


9 295 


zoo and over 


678 


26l 


270 


260 


i 300 


All known 


I 981 882 


25 836 088 


21 004 724 


25 961 053 


129 805 265 


Unknown 


1038 


32485 


4I97I 






All 


I 982 920 


25 868 573 


21 046 695 







at ages o to 4 and assuming that d 2 wr is equal to 8 2 w\2 in the 
formula. The exposed to risk at each of these ages is then 
determined by the formula E x L x -\-\d x . The same formula 
also applies at ages i to 4 inclusive, but for age zero it is 
found that the average age at death of those dying within 
one year of birth is only three-tenths of a year, so that we 
use instead E Q = L Q -}-^QQ. From the values of log 6 X and 



NORTHEASTERN STATES MORTALITY TABLE 



97 



log E x the values of log q x are then determined. These figures 
are shown in Table II. 

TABLE II 

CALCULATION OF VALUES OF log q x FOR INFANTILE AND QUINQUENNIAL AGES 



X 


ioL x 


IO8 X 


ioR x 


log g z 


O 


28 847 550 


3 979 850 


31 633 445 


1.09972 


I 


25 379 ISO 


849 390 


25 803 845 


2.51742 


2 


27 820950 


357 570 


27 999 735 


. 10621 


3 


27 174 200 


221 l6o 


27 284 780 


3.90879 


4 


25 888 ooo 


156 940 


25 966 470 


-78i34 


9i 


23 180 579 


57344 


23 209 251 


39283 


145 


23 234 918 


62 207 


23 266 02 1 


.42712 


195 


25 046 068 


109 046 


25 zoo 591 


63793 


24* 


25 302 916 


138 167 


25372000 


73605 


295 


22 725 822 


145 085 


22 798 364 


.80372 


345 


20 660 018 


162 848 


20 741 442 


.89494 


395 


18 499 612 


174 237 


18 586 731 


97194 


445 


I537833I 


175 75i 


15 466 206 


2-0555I 


495 


13 005 892 


19 556 


13 101 170 


.16271 


545 


10 068 338 


201 374 


10 169 025 


. 29672 


595 


7 53 953 


220 568 


7 641 237 


.46038 


645 


6 039 903 


249 733 


6 164 770 


.60756 


695 


4 35i 046 


260 645 


4 481 368 


76464 


74i 


2 796 270 


246 450 


2 919 495 


.92642 


795 


i SGI 735 


199 302 


i 601 386 


i. 09501 


84* 


650 085 


127 131 


713651 


25077 


895 


205 286 


57085 


233 828 


38763 


945 


37512 


15 140 


45082 


.52612 


99 


3879 


2 225 


4992 


. 64906 



169. The value of log <7 4i to serve as an initial term in a 
systematic interpolation is calculated from those of log <7s, 
log 5-4, and log<7 9J , on the assumption that for the interval 
in question log q x may be considered as a rational algebraic 
function of the second degree in x, the resulting value being 
3.72417. An additional term is supplied at the end by assuming 
<7io4i = i or log <7io4i = .00000. Eq. (8) of Chapter VII is then 
applied to the interpolation. In this interpolation t is given 
the value 5 and h takes successively the values, \, f , -f , -J, and f . 
From these values of log<7 2 , the values of l x and d x are then 
derived. The following table for the first five ages shows the 
working process: 







2 



3 8i 

3 



+ 



.2 a 

H > *S 



8 S| 

hie 
-IM 

U2 Ti c 

WOO 
| K G 



CO c3 

6 <3 

Z oj 



X! 



NORTHEASTERN STATES NORTALITY TABLE 



99 



(I) 


(2) 


(3) =log (2) 


(4) 


(5) =(3) +(4) 


(6) =antilog (5) 


Age. 





log*,. 


log9 z 


\ogd x 


*i 





IOOOOO 


5.00000 


i .09972 


4.09972 


12 581 


i 


87419 


4.94161 


2.51742 


3-45903 


2878 


2 


84541 


.92707 


. 10621 


.03328 


I O8O 


3 


83461 


.92418 


3.90879 


2.83027 


6 77 


4 


82 784 


91795 


78134 


.69929 


500 


5 


82 284 











170. In accordance with the usual custom the values of 
q x shown in the table have been adjusted to agree exactly 
with the values of l x and d x and do not agree with the values 
of log q x used in constructing the table. In the accompanying 
Diagram, No. 3, the values of log (i + ioo^) are plotted in 
comparison with the similar functions according to three tables 
representing mortality among American insured lives. 



APPENDIX 



FUNDAMENTAL COLUMNS AND OTHER DATA FROM VARIOUS 
MORTALITY TABLES 

x = age ; l s = number living at age x ; 
d x = number dying at age x last birthday. 

NORTHAMPTON TABLE 



X 


/x 


dx 


x 


/* 


d z 


x 


/* 


d x 


o 


ii 650 


3000 


35 


4 oio 


75 


70 


i 232 


80 


I 


8650 


i 367 


36 


3935 


75 


7i 


i 152 


80 


2 


7283 


502 


37 


3860 


75 


72 


i 072 


80 


3 


6781 


335 


38 


3785 


75 


73 


992 


80 


4 


6 446 


197 


39 


3 7io 


75 


74 


912 


80 


5 


6249 


184 


40 


3635 


76 


75 


832 


80 


6 


6 065 


140 


4i 


3 559 


77 


76 


752 


77 


7 


5925 


no 


42 


3482 


78 


77 


675 


73 


8 


S8iS 


80 


43 


3404 


78 


78 


602 


68 


9 


5735 


60 


44 


3326 


78 


79 


534 


65 


10 


5675 


52 


45 


3248 


78 


80 


469 


63 


ii 


5623 


So 


46 


3 J 7 


78 


81 


406 


60 


12 


5573 


50 


47 


3092 


78 


82 


346 


57 


13 


5523 


5 


48 


3014 


78 


83 


289 


55 


14 


5473 


5 


49 


2936 


79 


84 


234 


48 


IS 


5423 


50 


5 


2857 


81 


85 


1 86 


4i 


16 


5373 


53 


5i 


2 776 


82 


86 


145 


34 


17 


5320 


58 


52 


2694 


82 


87 


in 


28 


18 


5 262 


63 


53 


2 6l2 


82 


88 


83 


21 


iQ 


5 199 


67 


54 


2530 


82 


89 


62 


16 


20 


5 132 


72 


55 


2448 


82 


90 


46 


12 


21 


5 060 


75 


56 


2 366 


82 


9i 


34 


IO 


22 


4985 


75 


57 


2 284 


82 


92 


24 


8 


23 


4910 


75 


58 


2 2O2 


82 


93 


16 


7. 


24 


4835 


75 


59 


2 I 2O 


82 


94 


9 


5 


25 


4 760 


75 


60 


2038 


82 


95 


4 


3 


26 


4685 


75 


61 


1956 


82 


96 


i 


i 


27 


4 610 


75 


62 


874 


81 








28 


4535 


75 


63 


793 


81 








29 


4 460 


75 


64 


712 


80 








30 


4385 


75 


65 


632 


80 








31 


431 


75 


66 


552 


80 








32 


4 235 


75 


67 


472 


80 








33 


4 160 


75 


68 


392 


80 








34 


4085 


75 


69 


3" 


80 









100 



APPENDIX 



101 



CARLISLE TABLE 



X 


lz 


d x 


X 


/. 


d x 


X 


h 


d x 





1OOOO 


1539 


35 


5362 


55 


70 


2 4OI 


124 


I 


8461 


682 


36 


537 


56 


71 


2 277 


134 


2 


7779 


505 


37 


5251 


57 


72 


2 143 


146 


3 


7274 


276 


38 


5194 


58 


73 


i 997 


156 


4 


6998 


2OI 


39 


5136 


61 


74 


I 841 


166 


5 


6797 


121 


40 


575 


66 


75 


I 675 


160 


6 


6676 


82 


4i 


5009 


69 


76 


1 515 


156 


7 


6594 


58 


42 


4940 


7i 


77 


i 359 


146 


8 


6536 


43 


43 


4869 


7i 


78 


i 213 


132 


9 


6493 


33 


44 


4798 


7i 


79 


i 081 


128 


10 


6 460 


29 


45 


4727 


70 


80 


953 


116 


ii 


6431 


3i 


46 


4657 


69 


81 


837 


112 


12 


6 400 


32 


47 


4588 


67 


82 


725 


IO2 


13 


6368 


33 


48 


452i 


63 


83 


623 


94 


14 


6335 


35 


49 


4458 


61 


84 


529 


84 


IS 


6300 


39 


5 


4397 


59 


85 


445 


78 


16 


6 261 


42 


Si 


4338 


62 


86 


367 


71 


17 


6 219 


43 


52 


4276 


65 


87 


296 


64 


18 


6 176 


43 


53 


4 211 


68 


88 


232 


51 


19 


6i33 


43 


54 


4143 


70 


89 


181 


39 


20 


6090 


43 


55 


4073 


73 


90 


142 


37 


21 


6047 


42 


56 


4000 


76 


9i 


i5 


3 


22 


6005 


42 


57 


3924 


82 


92 


75 


21 


23 


5963 


42 


58 


3842 


93 


93 


54 


14 


24 


5921 


42 


59 


3749 


106 


94 


40 


10 


25 


5879 


43 


60 


3643 


122 


95 


3 


7 


26 


5836 


43 


61 


35 2 i 


126 


96 


23 


5 


27 


5793 


45 


62 


3395 


127 


97 


18 


4 


28 


5748 


5 


63 


3268 


125 


98 


14 


3 


20 


5698 


56 


64 


3 143 


125 


99 


ii 


2 


3 


5642 


57 


65 


3018 


124 


IOO 


9 


2 


31 


5585 


57 


66 


2894 


123 


IOI 


7 


2 


32 


5528 


56 


67 


2771 


123 


102 


5 


2 


33 


5472 


55 


68 


2648 


123 


I3 


3 


2 


34 


5417 


55 


69 


2525 


124 


IO4 


i 


I 



102 



MORTALITY LAWS AND STATISTICS 



ACTUARIES', OR COMBINED EXPERIENCE, TABLE 



X 


I* 


d t 


X 


l 


d x 


X 


lx 


d x 


IO 


100 000 


676 


40 


78653 


815 


7 


35837 


2327 


II 


99324 


674 


4i 


77838 


826 


71 


33510 


2 351 


12 


98650 


672 


42 


77 012 


839 


72 


3i 159 


2 362 


13 


97978 


671 


43 


76i73 


857 


73 


28797 


2358 


14 


97307 


671 


44 


753i6 


881 


74 


26439 


2339 


15 


96636 


671 


45 


74435 


909 


75 


24 100 


2303 


16 


95965 


672 


46 


73526 


944 


76 


21 797 


2 249 


17 


95 293 


673 


47 


72582 


981 


77 


19548 


2 179 


18 


94 620 


675 


48 


71 601 


I O2I 


78 


17369 


2 092 


19 


93945 


677 


49 


70580 


i 063 


79 


15277 


1987 


20 


93 268 


680 


5 


69517 


i 108 


80 


13 290 


i 866 


21 


92 588 


683 


5i 


68 409 


i 156 


81 


11424 


i 730 


22 


9i 95 


686 


52 


67253 


i 207 


82 


9694 


i 582 


23 


91 219 


690 


53 


66 046 


i 261 


83 


8 112 


i 427 


24 


90529 


694 


54 


64785 


i 316 


84 


6685 


i 268 


25 


89835 


698 


55 


63469 


i 375 


85 


5417 


i in 


26 


89137 


703 


56 


62 094 


i 436 


86 


4306 


958 


27 


88434 


708 


57 


60658 


i 497 


87 


3348 


811 


28 


87726 


7H 


58 


59 161 


i 561 


88 


2537 


673 


29 


87 012 


720 


59 


57 600 


i 627 


89 


i 864 


545 


30 


86 292 


727 


60 


55973 


i 698 


90 


1319 


427 


31 


85565 


734 


61 


54275 


i 770 


9i 


892 


322 


32 


84831 


742 


62 


52505 


i 844 


92 


570 


231 


33 


84089 


75 


63 


50 66 1 


i 917 


93 


339 


155 


34 


83339 


758 


64 


48744 


i 990 


94 


184 


95 


35 


82581 


767 


65 


46754 


2 061 


95 


89 


52 


36 


81 814 


776 


66 


44693 


2 128 


96 


37 


24 


37 


8 1 038 


785 


67 


42565 


2 191 


97 


13 


9 


38 


80253 


795 


68 


40374 


2 246 


98 


4 


3 


39 


79458 


805 


69 


38128 


2 291 


99 


i 


i 



APPENDIX 



103 



AMERICAN EXPERIENCE TABLE 



X 


lx 


d x 


X 


l x 


d x 


X 


lx 


d x 


IO 


IOOOOO 


749 


40 


78106 


765 


70 


38569 


2391 


II 


99 251 


746 


4i 


77341 


774 


71 


36178 


2 448 


12 


98 505 


743 


42 


76567 


785 


72 


33730 


2487 


13 


97 762 


740 


43 


75 782 


797 


73 


3i 243 


2 505 


14 


97 022 


737 


44 


74985 


812 


74 


28738 


2 501 


15 


96 285 


735 


45 


74173 


828 


75 


26 237 


2 476 


16 


95 550 


732 


46 


73345 


848 


76 


23 761 


2 431 


17 


94818 


729 


47 


72497 


870 


77 


21330 


2369 


18 


94 089 


727 


48 


71 627 


896 


78 


18961 


2 291 


19 


93362 


725 


49 


70731 


927 


79* 


16 670 


2 196 


20 


92637 


723 


5 


69 804 


962 


80 


14474 


2 091 


21 


91 914 


722 


5i 


68842 


I OOI 


81 


12383 


I 964 


22 


91 192 


721 


52 


67841 


i 044 


82 


10419 


I 8l6 


23 


90471 


720 


53 


66797 


i 091 


83 


8603 


I 648 


24 


89 75i 


719 


54 


65 706 


i 143 


84 


6955 


I 470 


25 


89032 


718 


55 


64563 


i 199 


85 


5485 


I 292 


26 


88314 


718 


56 


63364 


i 260 


86 


4193 


I 114 


27 


87596 


7i8 


57 


62 104 


i 325 


87 


3079 


933 


28 


86878 


7i8 


58 


60779 


i 394 


88 


2 146 


744 


29 


86 1 60 


719 


59 


59385 


i 468 


89 


I 4O2 


555 


3 


85441 


720 


60 


57917 


i 546 


90 


847 


385 


3i 


84721 


721 


61 


S637I 


i 628 


9i 


462 


246 


32 


84 ooo 


723 


62 


54743 


i 713 


92 


216 


C37 


33 


83 277 


726 


63 


53 030 


i 800 


93 


79 


58 


34 


82551 


729 


64 


5i 230 


i 889 


94 


21 


18 


35 


81 822 


732 


65 


49341 


i 980 


95 


3 


3 


36 


81 090 


737 


66 


4736i 


2 070 








37 


80353 


742 


67 


45 291 


2 I S 8 








38 


79611 


749 


68 


43 133 


2 243 








39 


78862 


756 


69 


40 890 


2 321 









104 



MORTALITY LAWS AND STATISTICS 



INSTITUTE OF ACTUARIES HEALTHY MALE (H M ) TABLE 



X 


lx 


d x 


X 


lx 


dx 


X 


/* 


in 


10 


IOOOOO 


490 


40 


82 284 


848 


70 


38 124 


2371 


II 


99510 


397 


4i 


81 436 


854 


71 


35753 


2433 


12 


99 "3 


329 


42 


80582 


865 


72 


33320 


2497 


13 


98784 


288 


43 


79717 


887 


73 


30823 


2554 


14 


98 496 


272 


44 


78830 


911 


74 


28 269 


2578 


15 


98 224 


282 


45 


77919 


950 


75 


25691 


2527 


16 


97942 


3i8 


46 


76969 


996 


76 


23 164 


2464 


17 


97624 


379 


47 


75973 


i 041 


77 


20 700 


2374 


18 


97 245 


466 


48 


74932 


1082 


78 


18326 


2 258 


19 


96779 


556 


49 


73850 


i 124 


79 


16068 


2138 


20 


96 223 


609 


50 


72 726 


i 160 


80 


13930 


2015 


21 


95614 


643 


Si 


71 566 


i 193 


81 


"915 


1883 


22 


94971 


650 


52 


70373 


i 235 


82 


10032 


I 719 


23 


94 3 2 i 


638 


53 


69138 


i 286 


83 


8313 


i 545 


24 


93683 


622 


54 


67852 


1339 


84 


6768 


1346 


25 


93061 


617 


55 


66513 


1399 


85 


5422 


i 138 


26 


92444 


618 


56 


65 "4 


i 462 


86 


4 284 


941 


27 


91 826 


634 


57 


63 652 


1527 


87 


3343 


773 


28 


91 192 


654 


58 


62 125 


i 592 


88 


2570 


6i5 


29 


90538 


673 


59 


60533 


1667 


89 


1955 


495 


3 


89865 


694 


60 


58866 


i 747 


90 


i 460 


408 


31 


89171 


706 


61 


S7"9 


1830 


9i 


i 052 


329 


32 


88465 


717 


62 


55 289 


1915 


92 


723 


254 


33 


87748 


727 


63 


53374 


2001 


93 


469 


I9S 


34 


87 021 


740 


64 


5 1 373 


2076 


94 


274 


139 


35 


86281 


757 


65 


49 297 


2 141 


95 


135 


86 


36 


85524 


779 


66 


47 156 


2 196 


96 


49 


40 


37 


84745 


802 


67 


44 960 


2243 


97 


9 


9 


38 


83943 


821 


68 


42 717 


2274 








39 


83122 


838 


69 


40443 


2319 









APPENDIX 



105 



BRITISH OFFICES' O M < 5 > TABLE 



X 


(. 


dx 


| 

X 


lx 


d x 


X 


lx 


d x 


IO 


107 324 


658 


45 


82 oio 


984 


80 


15531 


2 151 


II 


106 666 


658 


46 


81 026 


1018 


81 


13380 


2 OO7 


12 


1 06 008 


656 


47 


80008 


i 056 


82 


"373 


1847 


13 


i5 352 


655 


48 


78952 


i 096 


83 


9526 


1674 


14 


104 697 


654 


49 


77856 


H39 


84 


7852 


1493 


IS 


104 043 


654 


50 


76 717 


1185 


85 


6359 


1308 


16 


103 389 


654 


Si 


75532 


i 234 


86 


5051 


I 122 


17 


102 735 


655 


52 


74298 


1286 


87 


3929 


943 


18 


102 080 


655 


53 


73012 


1343 


88 


2986 


773 


19 


101 425 


655 


54 


71 669 


i 402 


89 


2 213 


617 


20 


ioo 770 


657 


55 


70 267 


1464 


90 


i 596 


480 


21 


100113 


660 


56 


68803 


1529 


9i 


i 116 


360 


22 


99453 


661 


57 


67 274 


1598 


92 


756 


263 


23 


98792 


664 


58 


65676 


i 669 


93 


493 


183 


24 


98128 


667 


59 


64 007 


i 742 


94 


310 


124 


25 


97461 


672 


60 


62 265 


1819 


95 


1 86 


79 


26 


96789 


676 


61 


60 446 


1897 


96 


107 


49 


27 


96113 


68 1 


62 


58549 


1975 


97 


58 


28 


28 


95432 


688 


63 


56574 


2055 


98 


30 


IS 


29 


94744 


694 


64 


54519 


2i33 


99 


15 


8 


30 


94050 


703 


65 


52386 


2 211 


IOO 


7 


4 


31 


93347 


711 


66 


50175 


2285 


IOI 


3 


2 


32 


92 636 


720 


67 


47890 


2355 


IO2 


i 


I 


33 


91 916 


732 


68 


45535 


2 421 








34 


91 184 


744 


69 


43 "4 


2478 








35 


90440 


757 


70 


40 636 


2527 








36 


89683 


771 


7i 


38 109 


2565 








37 


88912 


788 


72 


35544 


2 591 








38 


88 124 


806 


73 


32953 


2 6O2 








39 


87318 


825 


74 


30351 


2 596 








40 


86493 


846 


75 


27755 


2 572 








4i 


85647 


869 


76 


25183 


2 529 








42 


84778 


895 


77 


22654 


2 466 








43 


83883 


922 


78 


20 188 


238l 








44 


82961 


95i 


79 


17807 


2 276 









106 



MORTALITY LAWS AND STATISTICS 



NATIONAL FRATERNAL CONGRESS TABLE 



X 


In 


d t 


X 


/ 


d x 


X 


/* 


d x 


20 


100 000 


500 


So 


81 702 


935 


80 


20 270 


2799 


21 


99 5o 


SGI 


Si 


80767 


981 


81 


I747I 


2659 


22 


98999 


502 


52 


79786 


i 029 


82 


14 812 


2485 


23 


98497 


503 


53 


78757 


1083 


83 


12327 


2 280 


24 


97994 


505 


54 


77674 


i 140 


84 


10047 


2050 


25 


97489 


507 


55 


76534 


i 202 


85 


7997 


I800 


26 


96 982 


5io 


56 


75332 


i 270 


86 


6 197 


1539 


2? 


96472 


5i3 


57 


74 062 


1342 


87 


4658 


I 277 


28 


95 959 


5i7 


58 


72 720 


1418 


88 


338i 


1023 


29 


95442 


522 


59 


71302 


i 501 


89 


2358 


788 


30 


94920 


527 


60 


69801 


1588 


90 


i 570 


579 


3* 


94393 


533 


61 


68213 


1681 


9i 


991 


404 


32 


9386o 


540 


62 


66532 


1778 


92 


587 


264 


33 


93 320 


548 


63 


64754 


i 880 


93 


323 


161 


34 


92 772 


557 


64 


62874 


1985 


94 


162 


89 


35 


92 215 


567 


65 


60889 


2094 


95 


73 


44 


36 


91 648 


578 


66 


58795 


2 2O6 


96 


29 


19 


37 


91 070 


59i 


67 


56589 


2318 


97 


10 


7 


38 


90479 


606 


68 


54 271 


2430 


98 


3 


3 


39 


89873 


622 


69 


51841 


2539 








40 


89 251 


640 


70 


49 302 


2645 








4i 


88611 


660 


7i 


46657 


2744 








42 


87951 


683 


72 


43913 


2832 








43 


87 268 


708 


73 


41 081 


2909 








44 


86560 


734 


74 


38172 


2969 








45 


85826 


761 


75 


35203 


3009 








46 


85065 


790 


76 


32 194 


3026 








47 


84275 


822 


77 


29 1 68 


30l6 








48 


83453 


857 


78 


26 152 


2977 








49 


82 596 


894 


79 


23175 


2905 









APPENDIX 



107 



NORTHEASTERN STATES MORTALITY TABLE (1908-12) 



Age. 

X 


Number Living. 
/* 


Number Dying. 
d x 


Rate of Mortality 
per Thousand. 

looo q x 


Expectation 
of Life. 

e t 


o 


IOOOOO 


12 581 


125.81 


50.41 


I 


. 87 419 


2878 


32.92 


56.59 


2 


84541 


I 080 


12.77 


57-50 


3 


83461 


677 


8. II 


57-24 


4 


82 784 


500 


6.04 


56.70 


S 


82 284 


386 


4.69 


56.04 


6 


8 1 898 


3" 


3-8o 


55-31 


7 


81587 


261 


3-20 


54-Si 


8 


81326 


228 


2.80 


53-69 


9 


81098 


207 


2-55 


52.84 


10 


80891 


195 


2.41 


Si-97 


ii 


80696 


190 


2-35 


51.10 


12 


80 506 


190 


2.36 


50.21 


13 


80316 


196 


2.44 


49-33 


14 


80 1 20 


207 


2-58 


48.45 


IS 


79913 


222 


2.78 


47-58 


16 


79691 


244 


3-o6 


46.71 


17 


79447 


27O 


3-40 


45-85 


18 


79 177 


299 


3.78 


45-oo 


19 


78878 


329 


4.17 


44.17 


20 


78549 


354 


4-Si 


43.36 


21 


78i95 


374 


4.78 


42-55 


22 


77821 


390 


S- 01 


41-75 


23 


77431 


402 


5-i9 


40.96 


24 


77029 


4i3 


5-36 


40.17 


25 


76616 


424 


5-53 


39-38 


26 


76 192 


435 


5-71 


38.60 


27 


75757 


445 


5-87 


37.82 


28 


7S3I2 


456 


6.05 


37 05 


29 


74856 


468 


6.25 


36.26 


30 


74388 


482 


6.48 


35-49 


31 


73906 


499 


K75 


34.72 


32 


73407 


5-i8 


7.06 


33-95 


33 


72889 


537 


7-37 


33-19 


34 


7235 2 


557 


7.70 


32.43 


35 


7i 795 


575 


8.01 


3 J -68 


36 


71 220 


59i 


8.30 


3 -93 


37 


70 629 


607 


8-59 


3-i8 


38 


70022 


623 


8.90 


29.44 


39 


69399 


639 


9.21 


28.70 


40 


68760 


656 


9-54 


27.96 


4i 


68 104 


674 


9.90 


27.23 


42 


67430 


693 


10.28 


26.49 


43 


66737 


7i3 


10.68 


25-76 


44 


66024 


734 


II . 12 


25.04 


45 


65 290 


758 


ii. 61 


24-31 


46. 


64 532 


785 


12. 16 


23-59 


47 


63747 


813 


12.75 


22.88 


48 


62934 


845 


13-43 


22. 17 


49 


62 089 


879 


14.16 


21.46 


5 


61 210 


915 


14-95 


20.76 


Si 


60 295 


955 


15-84 


2O.O7 



108 



MORTALITY LAWS AND STATISTICS 



NORTHEASTERN STATES MORTALITY TABLE (1908-12) Continued 



Age. 

X 


Number Living. 
lx 


Number Dying. 
d z 


Rate of Mortality 
per Thousand. 

looog* 


Expectation 
of Life. 

e x 


52 


59340 


998 


16.82 


19.38 


53 


58342 


1045 


17.91 


18.71 


54 


57 297 


I 096 


19 13 


18.04 


55 


56 201 


I 153 


20.52 


17.38 


56 


55 48 


I 217 


22.11 


16.73 


57 


53831 


1285 


23.87 


16.10 


58 


52546 


1355 


25-79 


I5-48 


59 


5i 191 


1424 


27.82 


14-88 


60 


49767 


1489 


29.92 


14.29 


61 


48 278 


1547 


32.04 


13.72 


62 


46731 


i 601 


34.26 


13 IS 


63 


45 130 


1652 


36.61 


12.60 


64 


43478 


i 702 


39 IS 


12.06 


65 


4i 776 


1752 


41.94 


n-54 


66 


40024 


1802 


45-02 


ii .02 


67 


38 222 


1850 


48.40 


10.51 


68 


36372 


1894 


52.07 


10.02 


69 


34478 


1933 


56.06 


9-55 


70 


32 545 


1964 


60.35 


9.08 


?i 


30581 


2986 


64-94 


8.64 


72 


28595 


2 OOO 


69.94 


8.20 


73 


26595 


2004 


75-35 


7.78 


74 


24591 


1998 


81.25 


7-37 


75 


22593 


1982 


87 73 


6.98 


76 


20 611 


1955 


94-85 


6.60 


77 


18656 


1914 


102.59 


6. 24 


78 


1 6 742 


1857 


110.92 


5-90 


79 


14885 


1783 


119.78 


5-57 


80 


13 102 


1693 


129.22 


5.26 


81 


ii 409 


1588 


I39-I9 


4-97 


82 


9821 


1470 


149.68 


4-69 


83 


835i 


1342 


160.70 


4-43 


84 


7009 


I 2O7 


172.21 


4.18 


85 


5802 


1068 


184.07 


3 95 


86 


4734 


929 


196.24 


3-73 


87 


3805 


795 


208 . 94 


3-52 


88 


3010 


669 


222. 26 


3-32 


89 


2 .341 


554 


236.65 


3-12 


90 


1787 


451 


252.38 


2-93 


9i 


1336 


360 


269 . 46 


2-75 


92 


976 


282 


288.93 


2-59 


93 


694 


214 


308.35 


2-44 


94 


480 


157 


327.08 


2.29 


95 


323 


in 


343-65 


2.16 


96 


212 


77 


363 20 


2.04 


97 


135 


Si 


377-78 


1.91 


98 


8 4 


34 


404.76 


1.77 


99 


50 


21 


42O.OO 


i .64 


100 


2Q 


13 


448.27 


1-47 


101 


16 


8 


5OO.OO 


1-25 


102 


8 


5 


625.00 


1. 00 


103 


3 


2 


666.67 


83 


104 


i 


I 


IOOO.OO 


So 



APPENDIX 



109 



RATES OF MORTALITY PER THOUSAND ACCORDING TO VARIOUS 

TABLES 



Age. 


Northamp- 


Carlisle. 


English Life 
No. 3. 


Healthy 
Districts, 


English Life 
No. 6, 


North- 
eastern 




ton. 




Mixed. 


Mixed. 


Mixed. 


States. 


o 


257.5I 


I53-90 


149.49 


102.95 


156.53 


125.81 


5 


29.44 


17.80 


13-43 


10.27 


7-43 


4.69 


10 


9.16 


4-49 


5-73 


4-36 


2.50 


2.41 


IS 


9.22 


6.19 


5-36 


4.87 


3.11 


2.78 


20 


I4-03 


7.06 


8.42 


7-30 


4-36 


4-51 


25 


I5-76 


7-31 


9.38 


8.08 


5-36 


5-53 


30 


17.10 


IO.IO 


10.32 


8-57 


6.48 


6.48 


35 


18.70 


10.26 


ii .42 


9-03 


8.36 


8.01 


40 


20.91 


13.00 


12.87 


9.69 


10.84 


9-54 


45 


24.02 


14.81 


14.84 


10.82 


13.17 


ii .61 


5 


28.35 


13-42 


17-53 


12.62 


17.12 


14-95 


55 


33-50 


17.92 


22.76 


15-35 


22.77 


20.52 


60 


40.24 


33-49 


30.66 


22.99 


32.32 


29.92 


65 


49.02 


41.09 


43.46 


35-35 


45-44 


41.94 


70 


64.94 


51.64 


63.80 


53-24 


67.20 


60.35 


75 


96.15 


95-52 


93-94 


80.54 


95-59 


87.73 


80 


134-33 


121.72 


I35-5I 


120.43 


146.00 


129. 22 


85 


220.43 


175.28 


189 . 29 


174.95 


215.41 


184.07 


90 


260.87 


260 . 56 


254.84 


245-03 


290.56 


252.38 


95 


750.00 


233-33 


33I-I7 


325-11 


396.91 


343-65 


IOO 




222. 22 


412 .56 


413 .04 


600.00 


448 27 














T~T V / 


Age. 


Actuaries'. 


American 
Experience. 


Healthy 
Male. 


British 
Offices, 
OM(5). 


National 
Fraternal 
Congress. 


Medico- 
Actuarial. 


o 














5 














10 


6.76 


7-49 


4.90 


6.13 






15 


6-94 


7-63 


2.8 7 


6.29 






20 


7.29 


7.80 


6-33 


6.52 


S-oo 




25 


7-77 


8.06 


6.63 


6.89 


5-20 


4-7 


30 


8.42 


8-43 


7.72 


7-47 


5-55 


49 


35 


9-29 


8-95 


8-77 


8-37 


6.15 


5.1 


40 


10.36 


9-79 


10.31 


9.78 


7.17 


5-7 


45 


12.21 


ii .16 


12.19 


12. OO 


8.87 


7-5 


50 


15-94 


13-78 


15-95 


15-45 


11.44 


10.6 


55 


21 .66 


i8.57 


21.03 


20.83 


I5-7I 


15-8 


60 


30.34 


26.69 


29.68 


29.21 


22.75 


24.0 


65 


44.08 


40.13 


43-43 


42.21 


34-39 


39-0 


70 


64-93 


61.99 


62.19 


62.19 


53-65 


61.7 


75 


95-56 


94-37 


98-36 


92.67 


85.48 


91.9 


80 


140.41 


144-47 


144-65 


138.50 


138.09 


137-2 


85 


205 . 10 


235-55 


209.89 


205 . 69 


225.08 


203.7 


90 


323-73 


454-55 


279-45 


300.75 


368.79 


297-8 


95 

IOO 


584.27 


IOOO.OO 


637-04 


424.73 


602 . 74 


423-3 
576 -4 

















110 



MORTALITY LAWS AND STATISTICS 



DEATH RATES PER THOUSAND ACCORDING TO VARIOUS TABLES 



Table. 


Infancy. 
Ages 0-2 


Child- 
hood. 
Ages 

3-IO. 


Youth. 

ARCS 
10-25. 


Maturity 

Ages 
3S-ds. 


Old Age. 

ARCS 
over 65. 


MIXED POPULATION TABLES: 
Northampton 


249-3I 
I30-33 
118.19 
73-48 
116.71 
87.25 

128.24 
80.28 
128.07 

107-95 
66.51 
105.30 


32.49 
24. 21 
16.07 
10.93 

8-57 
5.56 

16.10 
10.85 
8.60 

16.04 
ii .00 

8.54 


II .60 
6.26 
6.79 
5.89 
3-65 
3.60 

6.62 
5-43 
3-74 

6.97 
6-37 
3-56 

7.14 
4-77 
5-92 
3-84 
6.42 

7 99 


24.23 
15-80 
16.47 
12.66 
14.96 
13-74 

17.03 
12.76 
16.24 

15-90 
12.57 
13-74 

14.86 
14.44 
15-70 
13-36 
14.14 

13-65 
10.92 


91.87 
89.89 
89.45 
81.37 
92.35 
86.69 

92.35 
83-31 
96.79 

86.86 
79-47 
88.88 

91 .12 
90.8l 

91-57 
89.16 
89.25 
90.12 
84.50 


Carlisle 


English Life No. 3 


Healthy Districts 


English Life No. 6 


Northeastern States 


MALE POPULATION TABLES: 
English Life No. 3 


Healthy Districts 


English Life No. 6 


FEMALE POPULATION TABLES: 
English Life No. 3 


Healthy Districts 


English Life No. 6. 


INSURANCE EXPERIENCE: 
Actuaries' . . 


Healthy Male H M 






Healthy Male H M(5) 






British Offices' O M 






British Offices' O M(5) 






American Experience 






National Fraternal Congress 















INDEX 



Actuaries' Table, 8 
Age Year Method, 61 
American Experience Table, n 
Analyzed Mortality Table, Construc- 
tion of, 65 

Breslau Table, 2 

British Offices' Life Annuity Tables, 

1893, 16 
British Offices' Life Tables, 1893, 10 

Calendar Year Method, 64 
Carlisle Table, 6 

Carlisle Table, Method of Construct- 
ing, 56 
Census Returns, Mortality Tables from, 

S3 

Central Death Rate, 46 
Combined Experience Table, 8 
Construction of Mortality Tables, 51 

Death Rate, Central, 46 
Death Rate for Communities, Cor- 
rected, 47 
DeMoivre's Formula, 26 

English Life Tables, 7 
Expectation of Life, 21 

Force of Mortality, 19 

Gompertz's Formula, 27 
Graduation of Mortality Tables, 68 
Graphic Method of Graduation, 71 

Hardy's Graduation Formula, 84 
Healthy Male Table, 9 
Higham's Graduation Formula, 83 

Insurance Experience, Mortality Tables 

from, 60 

Interpolation Formulas, 76 
Interpolation Method of Graduation, 72 



Joint Survival, Probabilities of, 34 

Karup's Graduation Formula, 84 
Karup's Interpolation Formula, 76 

Makeham's Formula, 27 

Makeham's Formula, Graduation by ; 
86 

M. A. Table, 13 

McClintock's Annuitants' Mortality 
Tables, 15 

Medico-Actuarial Mortality Investi- 
gation, 13 

Mortality, Force of, 19 

Mortality Tables, i 

Mortality Tables, Construction of, 51 

Mortality Tables, Meaning of, 17 

National Fraternal Congress Table, 12 
Northampton Table, 4 
Northeastern States Mortality Table. 
95 

Pearson's Analysis of Mortality Table 

32 

Policy Year Methods, 63 
Population Statistics, 52 

Seventeen Offices Table, 8 
Spencer's Graduation Formula, 84 
Stationary Population, 45 
Statistical Applications, 45 
Summation Methods of Graduation, 73 
Survivorship, Probabilities of, 41 

Tests of a Good Graduation, 70 

Uniform Seniority under Makeham's 
Law, 34 

Wittstein's Formula, 31 
Woolhouse's Graduation Formula, 83 
111 



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