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A JOURNAL FOR THE STATISTICAL STUDY OF
BIOLOGICAL PROBLEMS
FOUNDED BY
‘W. F. R. WELDON, FRANCIS GALTON anp KARL PEARSON
KDITED BY
KARL PEARSON
VOLUME X
APRIL 1914 TO May 1915
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VIL
VIII.
ail
XIII.
CONTENTS OF VOL. X.
Memoirs.
Congenital Anomalies in a Native African Race. By HuGH STANNUS
STANNUS
Tables of Poisson’s Exponential Binomial Limit. . By H. E. Soper
On the Poisson Law of Small Numbers. By Lucy Wairaker
The Relationship between Weight of the Seed Planted and the
Characteristics of the Plant Produced. By J. ARTHUR HARRIS
On the Probability that two Independent Distributions of Frequency
are really Samples of the same Population, with special Reference
to Recent Work on the Identity of Trypanosome Strains. By
KARL PEARSON
On Homotyposis and Allied Characters in Eggs of the Common
Tern. By WitittAm Rowan, K. M. Parker and JULIA BELL
A Piebald Family. By E. A. CocKAYNE
Clypeal Markings of Queens, Drones and Workers of Vespa ae Ws.
By Oswatp H. Larrer : : :
Table of the Gaussian “Tail” Functions; when the “Tail” is
larger than the Body. By Atice LEE
Contribution to a Statistical Study of the Cruciferae. Variation
in the Flowers of Lepidiwm draba Linnaeus. By James J.
SIMPSON
Nochmals iiber “The Elimination of Spurious Correlation due to
Position in Time or Space.” Von O. ANDERSON
Statistical Notes on the Influence of Education in Egypt. By
M. Hosny
Height and Weight of School Children in Glasgow. By Ernen M.
ELDERTON .
PAGE
269
280
288
1V
XIV.
XV.
XVI.
Contents
Numerical Illustrations of the Variate Difference Correlation
Method. By Bearrice M. Cave and Kari PEARSON
An Examination of some Recent Studies of the Inheritance Factor
in Insanity. By Davip HERON
On the Probable Error of the Bi-Serial Expression for the Corre-
lation Coefficient. By H. E. Soper
XVII. On the Partial Correlation Ratio. Part I, Theoretical. By L.
ISSERLIS
XVIII. Association of Finger-Prints. By H. WaItre
XIX.
XX.
XXI.
On the Problem of Sexing Osteometric Material. By KARL PEARSON
Further Evidence of Natural Selection in Man. By Eruet M.
ELDERTON and KARL PEARSON
Frequency Distribution of the Values of the Correlation Coefficient
in Samples from an indefinitely large Population. By R. A.
FISHER :
XXII. Appendix to Papers by “Student” and R. A. Fisher. HprIroRra
XXIII. Tuberculosis and Segregation. By Atice LEE : :
XXIV. The Influence of Isolation on the Diphtheria Attack- and Death-
(1)
(11)
(iii)
(iv)
(Vv)
(vi)
(vil) -
rates. By Eraet M. ELDERTON and KARL PEARSON
Miscellanea.
The Statistical Study of Dietaries, a Reply to Professor Karl Pearson.
By D. Noet Paton
The Statistical Study of Dietaries, a Rejoinder. By Kart PEARSON
Note on the Essential Conditions that a Population breeding at
random should be in a Stable State. By Kart PEARSON
The Elimination of Spurious Correlation due to Position in Time or
Space. By “Srupentr”
On certain Errors with regard to Multiple Correlation occasionally
made by those who have not adequately studied the Subject.
By Kart PEARSON
Formulae for the Determination of the Capacity of the Negro Skull
from External Measurements. By L. IsseRLIs
Note on a Negro Piebald. By C. D. Maynarp
169
72
175
179
181
188
193
(viii)
(1x)
(x)
(x1)
(x11)
(xiii)
(xiv)
(xv)
Contents
Note on Infantile Mortality and Employment of Women. By ETHEL
M. ELDERTON
Announcement of Prize Essay by Professor F. M. URBAN
Corrigendum, Congo Female Crania in Vol. vu. p. 307
On Spurious Values of Intra-class Correlation Coefficients arising from
Disorderly Differentiation within the Classes. By J. ARTHUR
HARRIS
Correction of a Teena made by Mr MAJOR GREENWOOD,
Junior. By K. P.
Note on Reproductive Selection. by Dain HERON
On the Probable Error of a aot Coefficient. By Karn
PEARSON
On Medieval English Femora, A Reply to Professor ee By
KARL PEARSON . ; ;
Plates.
Plate I. (1) Samuti, an Ateliotic Dwarf. (2) Subgiant, Height
1:92 m., with Wife and Albinotic Child. (8)—(4) Etimu,
aged 25, an Achondroplasic Dwarf .
Plate II. (5)—(6) Masimosya, aged 19, Gynaecomastos, cae Shek
features which were formerly described as those of Partial
Hermaphroditism. (7) oe ee Idiot. (8) Case of
Hydrocele testis
Plate III. (9)—(11) Boy, Heol 15 years, showing Siac only
Plate IV. (12)—(13) Son of Matikwiri, aged 7, a case of Scapho-
cephaly. (14) Cases of Umbilical Hernia. (15) poet Hand
and Foot in a Child, aged 5
Plate V. (16) Case showing faint es nepresion of 1 upper
lip. (17) Blantyre boy, aged 10, with Hare-lip. (18) Young
Woman with two nipples on left breast. (19) Gobedi, aged 22,
with congenital Humeral Micromely .
Plate VI. (20) Ndala, Split Hand, left only. (21) Chibisa, aged
30, elongation of all segments of middle finger and its meta-
carpal bone. (22) Shortening of the fourth metatarsal bone.
(23) Case of Se Oe Shortening of the left ce
toe
Plate VII. Transition of Poisson’s disor Limit of the Bi-
nomial Series into the Gaussian
. to face p. 24
24
24
24
24
24
vi Contents
Plate VIII. Sample Eggs, Common Tern, Natural Size (In colours) to face p. 146
Plate IX. Types of Mottling of Eggs of Common Tern . i . 146
Plate X. Dr Maynard’s Piebald Negro. i : ; : a 193
Plate XI. Pedigree of Cockayne’s Piebald Family . : >» 200
Plate XII. | Photographs of members of Cockayne’s Piebald Family Fr ,
Plate XIII. .) ip ‘
Plate XIV. $ 3 5 9 % » »
Plate XV. ” ” » »” ” ” 7)
Plate XVI. " 3 3 5) 9 » »
Plate XVII. Right forearm of a member of Cockayne’s Piebald
Family to show leucotic patches. : » »
Plate XVIII. © Leucotic forehead, blaze and white ca of ae
in a member of Cockayne’s Piebald Family f i 4
Plate XIX. Model of Skew Regression Surface giving mean Weight
of Girls of Class B in Glasgow Schools for a given Height and
Age. : ; ; : ; ee 49)
Plate XX. Types of Finger-Prints : : > eee
Corrigenda and Addenda.
The following corrections and additions to the memoir on the Partial Correlation Ratio,
pp. 391—411, have been received from Mr L. Isserlis :
Page 396. Eqn (28) for = read “; Eqn (30) for 7yy= read ry,=; Eqn (35) for me
Oz Oz
a z
oa
read 4
Oz
Page 397. Eqn (87) for ry= read 1,=; Eqn (38) for cdz, read cqy2; Line following
Eqn (45) for d=—Cyx, read d=—Cry; Eqn (46) for ys day read -ys'Gn2; Line following
Eqn (46) for wf,? read ,,R,
Page 402. Line 8 from top: after,
c, \2 av
melee =Isy {(a+24) (1- yn) a
insert “provided the SDs of w for constant y are homoscedastic, or sufficiently so, for this
to be an approximation.”
Line 6 from bottom for y,p—736 read ys h— 306.
Expression at foot of page, factor (7zyQx2—9.%y) of first term in large curled bracket
should read (Tay Qa?y — Yay?) and factor (7x29x%—Ty2Fay) Of expression in numerator of last line
of same bracket should read (12Qx2y —TyzJxy2)-
Page 403. In denominator of second term of expression in first line for 1- yn? read 1—,n,?.
Page 405. In denominator of left-hand side of Eqn (70) for g,22—7? read 9422-7? xy.
me read 14 rey
Page 406. Line below Eqn (71) and in Eqn (72) for - oe
L474, QP vy
Vole X. Parte“ | April, 1914
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VoLuME X APRIL, 1914 No. 1
BIOMETRIKA
CONGENITAL ANOMALIES IN A NATIVE
AFRICAN RACE .
By HUGH STANNUS STANNUS, M.D. Lond, Medical Officer, Nyasaland.
(1) I HAVE thought it would be of interest to put on record some observations
made by myself in Nyasaland during the past seven years, on the subject which
appears as the title of this paper.
These observations relate to members of a native population of Bantu stock,
belonging to several main tribes, namely, Mananja, Yao, Ngoni and Tumbuka,
with a few references to the Nkonde in the north and the Nguru from the south-
east.
My interest in the subject was aroused by the frequency with which some
abnormalities were seen and I think the facts I bring forward will go to shew that
this unusual incidence is real and not only the result of the ease with which
observations may be made among a partially clothed community.
Statistics dealing with the subject, to be of value, must treat of large numbers,
such have however only been possible in a few instances to be referred to later.
I speak therefore largely from impressions in appraising the rarity or otherwise of
any particular condition. It should be remembered in this direction that the
cases now to be reported have been met with more or less casually, most of them
while travelling on the path or in some village, few in the course of Native
Hospital work and none in any Special Department.
Classification is a matter of some difficulty for many reasons and as the number
of anomalies to be described is not very large it is perhaps more convenient to
consider the various conditions according to the anatomical part affected.
One large section of congenital anomalies, Anomalies of Pigmentation, I have
already dealt with (Biometrika, Vol. 1x. pp. 333—365), and they will not be
touched on in the present paper.
Biometrika x 1
2 Congenital Anomalies in a Native African Race
(2) Dealing with those deviations from the normal in’which there is a change
of a more or less general nature, I refer firstly to Infantilism, at the same time
recognising that such a condition may not constitute a truly congenital anomaly.
To the class designated Idiopathetic Infantilism I should. relegate a woman
aged 22 years seen in 1911 at Zomba who presented the figure and development
of a girl of 13. There was no breast development, no pubic or axillary hair and
the rounded contours of the body and limbs usually associated with this age in a
woman were wanting; menstruation had not commenced. In other respects she
appeared normal and her mental development was but little if at all below the
average.
(3) In W. Nyasa I encountered a very excellent example of the Ateliotic Dwarf,
a perfect “little man,’ a man in miniature 1:25 metres in height. Another case
which I think must be considered as one of simple dwarfism is here reproduced :—
Samuti, aged 35, a Yao, 1:42 metres high. He is shewn together with a man of
1:85 metres. Samuti shews no other abnormality (Plate IJ, (1)).
No case of Cretinism or Myxoedematous Dwarfism has been seen. I may here
mention that Cachetic Infantilism is well seen in some cases of spinal caries
among Natives just as among HKuropeans.
A paper on “Congenital Humeral Micromely” in the Nouvelle Iconographie
de la Salpétriere, T. xxiv. pp. 463—471, Paris 1911, by Dr S. A. Kinnier Wilson
and myself, contains references to two cases of Achondroplasia in Nyasaland.
Since then I have heard of two other cases and seen a fifth :—Etimu, male, aged
25 years, a Yao, son of Masinjiri of Ndindi’s near Chipoli, Dedza District. The
subject stated that he had no children and that no member of the family was known
to have been similarly affected. He is a perfect example of the condition as the
photographs will attest, and further remarks are unnecessary (Plate I, (3) and (4)).
The following measurements were made and tracings of his hands are here
depicted (Fig. 1):
(1) Head: maximum length. : : : ; . ,20:1>em:
(2) - breadth . : : : : 2 elas
(3) circumference . 5 : : F : . 600
(4) Nose: length, base to root . : ; : 3°6
(5) breadth, across nostrils. : : : : 45
(6) Face: bizygomatic breadth . : : Become 0)
(7) length, nasion to chin 4 3 : 5 ae lala
(8) F. to commissure of lips : ; 6'7
(9) Standing height . : ‘ ; LS 2
(10) Span of arms. ; : : . diss
(11) Arm: acromion to external condyle of humerus. Bi a4)
H. S. Srannus 3
(12) Forearm: external humeral condyle to tip of ulnar
tubercle. : : ; : : 3 : Se aliecm:
(13) Forearm to tip of middle finger . ‘ : : oe Oe
(14) Leg: top of iliac crest to head of fibula. ; 5 SS
(15) 5 - - to external malleolus — . eS
(16) . . z to sole of foot i : ~ ol
(17) Trunk: upper border of sternum to umbilicus . . 3&4
(18) ‘ symphysis pubis 45
BY:
Left. Fig. 1. Etimu. Right.
(4) No case of actual Gigantism has been seen. Tallness or shortness often
runs in families. The tallest man I have ever seen measured 1°92 metres. He
was the father of an albinotic child and had internal strabismus but no signs of
acromegaly (see Plate I, (2)). Another man who I have not seen but who was
measured by Dr Davey at Kota Kota was 2:0 metres in height. No case of
Acromegaly has been seen by myself.
(5) The following case in the want of development of the lower jaw and
zygomatic arches might be considered as the converse to acromegaly (Fig. 2).
From the sketch the subject will at once be recognised as a type of Congenital
Idiot, the above-mentioned features and ill-formed pinnae together with the rather
bird-like appearance being characteristic.
1—2
4 Congenital Anomalies in a Native African Race
Jaidi, male, aged 20 years, a Yao of Chumbosa, Bursali, is the second child of
a family of three, the elder brother being dead and the younger sister normal.
No family history was elicited.
Fig. 2. Jaidi,
The growth of the face is defective as before noted, the zygomatic arches are
so little developed that there are practically no cheeks. The descending rami of
the jaws converge very considerably so that the floor of the mouth is very narrow
and the horizontal rami are so short that the symphysis is situated mid-way be-
tween the lower lip and the neck as they lie on one horizontal plane. The palate
is high and narrow.
The following measurements were made:
Maximum occipito frontal . : : ~ 19:tsem:
e bi-parietal . : : ; : . 138
Bizygomatic at junction of zygoma with temporal . > 123
Nose: length. : : ; : : S47
breadth . ; : : ; : ; : : 318
Face: nasion to commissure of lips . : oleae
» 5» symphysis of chin . : | lisa
Right external strabismus is present and vision defective.
Though mentally an imbecile with an impaired speech he is an excellent field
labourer. He states that no woman would marry him but that he has had sexual
intercourse and that he is capable of the act.
H. S. Srannus 5
A few other cases of Congenital Idiocy have been seen and include an example
of Spastic Diplegia, a Mongol Idiot aged 4 years in W. Nyasa district and two
microcephalic idiots met with in adjacent villages in Chikala district, in neither of
which were factors of etiological interest elicited.
(a) Aged 22, male, looked like a boy of 12 in physical development, the head
was very stnall but no measurements were made; the palpebral fissures were
markedly slanting downwards and inwards and an internal strabismus was present ;
the ears and palate were normal; the hands large and like those of a man.
(b) A male infant aged one year with so marked a degree of microcephaly as
to approach in type anencephaly, the resemblance being the more marked as the
protuberant eyes and lips were like those characteristically found in anencephalic
monsters (Plate IT, (7)).
(6) The following case is given at length (Plate II, (5) and (6)).
Masimosya, aged 19 years (1911), a Yao of Chipi’s village Zomba, exhibits a
marked want of development of sexual organs (male) associated with large breasts.
The general form of the body is that of a woman; the attitude, voice, laugh,
facial aspect and expression resemble those of a woman rather than of a man.
The teeth are good, the body and limbs well developed and there is a fair deposit
of subcutaneous fat. The breasts (see photo) are remarkable, being large, with
large well-formed nipples and well-marked areolae, dark in colour. They have
started to beome pendulous and resemble exactly those of a nulliparous woman
of the same age. The abdomen is well formed and round the umbilicus there
is a deposit of fat such as is commonly seen in women; the pelvis appears large.
There is some hair in the axillae but none on the face or body. The pubes is
rather prominent resembling the female mons veneris and there is some develop-
ment of hair upon it. The penis is very small, only two inches in length and of
infantile type, the glans is covered by a prepuce and there is no deformity. The
scrotum is very small indeed and only contains one testicle, the left, which can be
felt as a small body about the size of a bean, three-eighths of an inch long. The
right testicle is not apparently present in the scrotum or inguinal canal. The
scrotum shews no tendency to be divided nor is there anything in the arrangement
of the skin to suggest labia. No rectal examination was made. .
The subject is insane. He is fairly tractable and good-natured. He has
delusions and hallucinations, it is reported, with various phases of the moon, when he
is said to travel 15 miles to bathe in a certain stream, etc. He has tried to burn
down some houses. I could get very little of his history. The mother and father
are said to have been normal; the only other child, a girl, was insane and died in
the Central Asylum. The subject once cohabited with a woman who was to have been
his wife, but she ran away the next day and I was unable to find out from him if
he had any sexual desire. Such is a case which would have been called one of
Partial Hermaphroditism but in the absence of further data I shall not discuss it.
6 Congenital Anomalies in a Native African Race
(7) Obesity. No-cases of general obesity outside normal limits with possibly
a congenital origin have been seen. Steatopygy does not occur.
(8) Symmetrical Lipomatosis is conveniently considered here though perhaps
not strictly within the subject. Three old women have been seen all presenting
the same abnormal feature, namely, the presence of symmetrical lipomata in both
axillae, each about the size of a small orange. In a fourth case the affection was
one-sided, the subject giving a history of the gradual descent of the tumour from
the upper aspect of the shoulder into the arm-pit.
That these tumours were lipomata I can only support by clinical examination,
they certainly were not of the nature of the pads seen in myxoedema and no
signs of that disease were present. There is the possibility that they were acces-
sory breasts but they did not present the characters found in undoubted cases of
this condition. These tumours may have a similar pathogeny to the masses seen
on either side of the back of the neck of men and specially described by
Sir Jonathan Hutchinson; on account of their possible paleogenetic significance
I have included notes on these cases here.
(9) Lymphatism, Post-mortem examination on a boy 10 years of age who died
after receiving a blow on the head revealed a thymus gland of considerable bulk,
4 inches long. The blow had not severed the soft tissues over the skull and in
the absence of any other evidence of injury or disease one might suspect the case
to be one of lymphatism, an inherent disorder which had predisposed to death.
In a second case, that of a woman aged 40 years who died after moderately severe
burns, a body 44 inches long of yellow colour and firm consistency was found lying
on the anterior surface of the heart, the apex of this body being at a level with
the 2nd costal cartilage.
(10) Coming now to Malformations, there is a well-defined deformation of the
skull of which I have seen several examples, the main points of which are well
shewn in the photographs. The extreme height of the cranium and marked
dolicocephaly without bossing of the forehead, while the sides of the vault of
the skull are flattened, are characteristic. The photographs depict a boy aged 7,
son of Matikwiri, headman of Mlanje, whose two younger sisters are said to
resemble him exactly in the deformity present (Plate IV, (12) and (13)).
The second case is a boy aged 15 years, the head measured 21°5 cm. long and
12°5 cm. broad (Plate III, (9)—(11)).
(11) Congenital Ptosis is not uncommon and is associated with the typical
expression due to this disability. A slight degree of Epicanthus may be fairly
often observed; more marked, it is sometimes seen associated with obliquity
of the palpebral fissures giving a regular mongolian character to the face
(Fig. 3).
Buphthalmos has been seen on two occasions in young adults with a history of
its congenital nature but nothing else of note; tension normal and vision appa-
rently good.
H. S. Srannus T
Microphthalmos was once seen associated with coloboma of the iris and choroid
(see below).
Coloboma. This defect was met with in two brothers aged about 18 and 17
years, but neither parent nor, as far as 1 could ascertain, any other member of the
family was similarly affected. Bwanali the elder presented a coloboma of the iris
and choroid of the left eye; there was also a small opacity on the posterior surface
of the lens, which however could not be traced more deeply but which suggested a
remnant of an “arteria centralis.” .
2 CLs
Fig. 3. Epicanthus.
The right cornea shewed some superficial opacities, the iris appeared normal,
but examination of the fundus revealed a large white triangular area with the
apex near the disc with here and there small masses of pigment. The middle
portion of the white area was on a much deeper plane than the rest of the fundus,
forming a posterior staphyloma, the whole composing a kind of posterior coloboma
(Fig. 4).
Right eye. Left eye.
Fig. 4. Bwanali Coloboma.
This boy also had an accessory nipple.
The younger brother Pete presented on the right side a microphthalmic eye
with coloboma of iris and choroid resembling the condition in his brother, with an
8 Congenital Anomalies in a Native African Race
opaque spot on the posterior surface of the lens. The eye is convergent and
vision poor; he counts fingers at one yard. The left eye is normal.
Dermoid Cysts of the Face have been seen in the situations shewn in the sketch
(Fig. 5). One of these was excised and found to contain the usual pultaceous
Fig. 5. Dermoid cysts of face.
mass mixed with hairs. These hairs examined microscopically were found to be
spindle shaped, tapering at each end, brown diffuse and granular pigment was
present in them.
A relic of the cleft between the median and upper external processes of the
foetal face was on one occasion seen as a small pit at the lower extremity of and
just external to an epicanthal fold.
(12) Congenital Naevus. Only two cases of naevus have been seen. One a
woman presented a small naevus just to the left of the middle line on the forehead
at the margin of the hairy scalp, 1 cm. in diameter. The second was a man with a
similar growth 1 cm. in diameter on the lower lip just to the right of the middle
line (Ching’waya of Zomba).
(13) Har. The general conformation of the ear varies a good deal; some of the
types are shewn in the sketches (Fig. 6) but all these must be considered as coming
J III
Fig. 6.
within the limits of normal variation. In one case a kind of Accessory Lobule was
noted ; the subject was an albino. A number of persons with Accessory Auricles
H. S. Srannus 9
have been seen. These consist of little subcutaneous nodules of cartilage forming
tubercles one to four in number situated just in front of the tragus, the affection
being usually bilateral.
An abnormality seen affecting a woman in N. Nyasa consisted in the direct
prolongation of the skin from the side of the head on to the outer surface of the
pinna so that the upper margin of the ear was hidden, though easily felt beneath
the skin.
Helical fistula. Under this name have been described the remains of the first
branchial cleft found as little pits on the helix. The condition is certainly rare
in England and persons exhibiting the anomaly are sometimes shewn as interesting
cases at medical societies. That heredity plays a part in its incidence is well
known as illustrated by a case shewn by Dr Prichard at the Royal Society of
Medicine, an infant with symmetrical helical fistulae, whose mother, four siblings,
maternal grandmother and two great-aunts all exhibited the same defect.
Having noted this same anomaly in quite a number of natives I became interested
to ascertain the actual incidence. The statistics given below embody the results
of my observations covering nearly 6500 individuals of all tribes. The popula-
tions of whole villages were taken so that consecutive unselected persons were dealt
with,
Tribe core Right Left pou
. { Males 416 7 6 3
No gis { Females | 612 13 Soni bas
eS Males ... | 100 | — 4 1
Females... 136 — 3 —
Wionsa J Males... 1941 | 34 22 12
6” | Females... 2576 | 69 53 23
Wankonde* ... as 455 | 4 8 5
Awemba a ace 48 1 — 1
Anyanja ee oa 65! 1 1 -—
Ahenga sb aah | 142 3 5 —
Totals... .. | 6491 132 110 50
Thus among 6491 individuals of all ages and both sexes a total of 292 were
found to have helical fistula (4°5 °/,). It was more commonly unilateral, affecting
the right side a little more often than the left, giving percentages of 2°08 and 1°69
respectively and for bilateral cases 0°77°/,. Taking each sex we see that the
proportions between the three numbers are almost the same.
2457 males 3 41 By) 16
3324 females... 82 64 28
* These figures were kindly supplied by Dr Davey.
Biometrika x
10 Congenital Anomalies in a Native African Race
The actual incidence in the two sexes is however greater among females than
males in the proportion of 5:2°/, to 3°6°/,. An abnormality occurring so fre-
quently as 45 per mille might almost be considered to be a variation within the
limits of the normal. The fact remains, however, that it is the persistence of
a foetal character and abnormal, if the whole of mankind be taken into con-
sideration.
Dealing more in detail with this defect, there is some variation in the exact
site of the fistula; the sketches (Fig. 7) serve to illustrate the extremes of posi-
tion in three directions.
JIIVG
Fig. 7.
Three cases presented two pits on the same side, one each in positions A
and B. In these three cases the affection was bilateral and symmetrical. The
common position at which the pit is found is in D. In another case not included
in the series a pit was observed resembling those above mentioned but situated at
the junction of the tragus and lobule as in £.
These helical fistulae, which I have described as little pits, consist of a small
opening on the skin 1 or 2 mm. in diameter leading into a blind sac 1 or 2 mm.
deep; often this sac opens out into a little ampulla which can be seen and felt
under the skin. The ampulla and canal are generally filled with a little plug of
sebaceous matter.
In three cases the skin in this situation looked like scar tissue and presented
a honey-combed appearance, there being several openings into the ampulla giving
the impression that an abscess had formed at some past date in the ampulla with
consequent loss of tissue.
The fistula is so common and so unremarkable that most tribes have no name
for it and one cannot elicit long pedigrees to shew its incidence in families. Cases
of heredity were common enough but the type was not necessarily the same in
members of the same family; thus a mother with Left Fistula had a child with
Right and Left, or again, three brothers were seen two with the Left side affected,
the third with Right Fistula.
No malformations in connection with other branchial clefts have been seen,
H. S. Srannus 11
(14) Lips, Mouth and Palate. Most natives shew a well-marked tubercle in
the median line on the “red” margin of the upper lip; in a few however this is
replaced by a distinct groove which involves the red margin of the lip or only the
subjacent fold of mucous membrane (see sketch Fig. 8 and photo, Plate V, (16)).
Jae ‘ =
Fig. 8.
These cases resemble one of a Hindu (recorded in the Lancet, Oct. 2, 1909, by
Thurston), who besides having the median hare-lip was the subject of poly-
dactylism. In one of my cases there was a considerable gap between the upper
central incisors but no further abnormalities were present.
In a single case notching of the upper lip was found to the left of the middle
line with a mark running up to the nostril which looked like a scar. There was
no question of any operation having been performed, though the condition
resembled exactly an artificial repair of a lateral hare-lip (Fig. 9). A similar
Bigs 9:
case has been shewn at the W. Lond. Med. Chir. Society in which there was,
besides, a deformity of the nose and a family history of hare-lip. I have only seen
one case of ordinary Hare-Lip, a Blantyre boy aged 10 years (1909), the affection
being left-sided and unassociated with any cleft of the palate (Plate V, (17)).
Among 30,000 natives examined in the northern districts of this country no
case was seen.
No case of typical Cleft Palate has come to my notice; on the other hand I
have seen three cases which owing to their non-association with defects in the
upper lip are of great interest. All three cases, one a boy aged 10 years (1906),
the other two adult males, presented complete Absence of the Premasilla and
attached teeth. In the boy there was also a Median Perforation in the hard
Palate. Congenital perforations of the palate apart from clefts are apparently
rare in Europe. Dundas Grant (Roy. Soc. Med. April 1910) has recorded the case
of a girl aged 16 years with a perforation above and to the right of the base of
2—2
12 Congenital Anomalies in a Native African Race
the uvula with no history of trauma or syphilis. Prof. Karl Pearson has drawn
my attention to a skull which was brought by Du Chaillu from Fernand Vas in
the Congo (see Biometrika, Vol. vit, Plate XXVI); this shews congenital .
absence of the premaxilla, but the two maxillae have not approximated in the
mid-line in front as in my own cases, and we do not know the condition of the
soft parts, but it is interesting to see this anomaly from another part of Africa.
(15) Teeth. Native children are said to be born sometimes with teeth; it is
possible that this is not very rare as there is a common superstition regarding
them. I have seen one case with this history, to be mentioned later, as having
deformities of the lower extremities. A gap of as much as 4 of an inch between
the lower central incisors has been noticed a number of times, the other teeth
all being regular and touching one another. A similar condition may be seen also
affecting the upper pair of incisors, one that I am not conversant with among
Europeans. Among 1500 natives examined for statistical purposes in regard to
caries the following numerical abnormalities were noted:
(a) Complete reduplication of the set of teeth in an adult, the second set
lying on the palatal side of what appeared to be the normal set. I have every
reason to believe that this was a case of true reduplication, that is to say, the
result of growth from doubled enamel organs and not of retention of the deciduous
teeth.
(b) Reduplication of upper incisors.
(c) Reduplication of right lower bicuspid.
(d) Reduplication of both bicuspids in the lower jaw on each side and in the
upper jaw on the right side in a woman aged 24 years.
A single case of a Bifid Eatrenuty to the Tongue was seen in an albino
child.
(16) Polymazia and Polythelia. 14 cases of these anomalies have been met
with casually, so that I imagine this anomaly by excess is comparatively not
uncommon. Short notes of these cases are given below for purposes of com-
parison :
(a) Male adult, accessory nipple springing from the skin at the right sternal
edge opposite the 38rd intercostal space, it was large and well formed like a
woman’s but there was nothing resembling an accessory mamma beneath it.
(b) Male aged 45. Insane and suffering from spinal caries. There was a
rudimentary accessory nipple in Scarpa’s triangle on the right side 14” below
Poupart’s ligament.
(c) Adult female, an accessory nipple on the right breast, small but well
formed and lying above the one proper to the breast; both are patent and milk
can be drawn through both.
H. S. Srannus 13
(d) Adult male, the accessory nipple is situated in a line with the left nipple
below it and half-way between it and the costal margin.
(e) and (f) Two women each had two nipples to the right breast.
(g) A young woman was found to have two nipples on the left breast
(Plate V, (18)).
(h) Male with congenital coloboma iridis mentioned above has an accessory
nipple just above and to the inner side of the right nipple.
(«) Young male adult has just at the outer edge of the areola of the left
breast a very small accessory nipple, and beyond this and above it over the third
intercostal space another flat nipple with areola and hairs.
(j) Male, presents a rudimentary nipple in the left groin just below the
middle of Poupart’s ligament.
(k) Young adult male shews a small accessory nipple just below and internal
to the right nipple; his brother, father, and grandfather are all possessed of the
same identical anomaly. The subject has no children, no nephews or nieces.
(1) Female in hospital with syphilis has a small accessory nipple springing
from the skin of the chest wall just internal to the point of the left pendant
breast.
(m) A woman with well-formed accessory breast in the right axilla. It is
breast-shaped and pendant though there is no nipple. The woman volunteered
the fact that it was a breast and said it swelled with pregnancy. The right breast
was twice as big as the left.
(x) An old woman with symmetrical masses in each axilla resembling rather
the symmetrical lipomata mentioned elsewhere: see p. 6. She states that they
appeared at puberty and thinks them to be breasts but denies that they enlarged
with pregnancy.
In the Japanese this condition has been shewn to be not unrare, and among
them tuberculosis has been found to be more frequent than among the normal
population. I can only support the idea with one case (No. b).
(17) Meningocoele and Spina Bifida. No typical case has been noted. A man
was seen with a little dipple of the skin over the lower part of the sacrum in the
median line having a little fold of skin on either side forming two small vertical
lips.
(18) Penis, Testicle; Hernia.
Epispadias, hypospadias and extroversion of the bladder have never been seen.
T have seen a boy aged 18 years with a short penis enclosed in a fold of skin
from the upper surface of the scrotum (Fig. 10).. The boy had other deformities
which are described later. When examining a number of recruits I was surprised
to find in a large proportion the right testicle hanging lower than the left, the
14 Congenital Anomalies in a Native African Race
reverse of what is known to occur among Europeans. On examination of 400
consecutive men, adults, between the ages of 30 and 40 years, I found in 166 or
41°5°/, the right testicle lower than the left. In the remainder or 58°5°/, the
right testicle was on a level with the left, or rather higher in the scrotum. I also
got the impression that, associated with right lower testicles, the testicles and
penis were large. In another series of 280 men, the left was lower than or on the
Fig. 10. Boy aged 18.
same level as the right in 185; the right lower in 88. There were two cases of
left cryptorchidism, one of right cryptorchidism; one each left and mght hydro-
coeles and two right inguinal bubonocoeles. ites
I have come across a number of cases of undescended testis among other
natives, in some associated with a swelling in the inguinal canal, in others there
was complete cryptorchidism. Inguinal hernia is not infrequent in adult males
but I can give no figures relating to a large number of persons. In a single man
it was associated with umbilical hernia. I have never seen a femoral hernia.
Umbilical hernia is common enough especially in children. The following figures
though small in number give some idea of the frequent incidence of the condition.
They refer to all the children in a single village and may therefore be said to be
unselected in any way.
Age = P oe aE
O— 1 year 18 13 6
1— 2 years 44 27 12
3—10_,, 102 44 10
Totals 164 84 28 | =276,
giving roughly COM. 0k Gis
_—_—Yr—
40 °/,
That the hernia diminishes in size even to disappearance after childhood, as
indicated by these figures, is certainly true as the same incidence is undoubtedly
not found among adults. The protrusion is sometimes very marked and takes the
————— eee ar wv ~
a
H. S. Stannus 15
shape of the finger of a glove, some several inches long and curving downwards
(Plate IV, (14)). Writing recently E. M. Corner in doubting the commonness of
congenital sacs in hernia in general, as insisted on by some writers, has shewn in a
series extending to between two and three thousand observations that herniae in
children are often multiple and associated particularly with a ventral hernia, a
diastema, which though very rare at birth 1s common in young children and of the
nature of a true hernia. He believes that this ventral protrusion, which is certainly
not congenital, is caused by increased abdominal pressure due to gaseous distension
of the bowels the result of fermentative processes, and that other herniae are due
to the same cause. Among native children abdominal distension is almost the
rule, “ pot-bellied” is an expression always used in speaking of them. This dis-
ténsion is due largely, I believe, to fermentative processes, and also a second factor,
absent in European children, namely enlarged spleen. Of 50 children under the
age of 5 years taken from among those with umbilical hernia, 43 or 86 °/, were
found to have the ventral protrusion as described by Corner. 18 of these had
enlarged spleens and 20 shewed a considerable abdominal distension. In none
was any other hernia found. In these cases we see par emcellence the effect of
intra-abdominal pressure, in producing first ventral hernia and ‘secondly umbilical
hernia. The weakness of the umbilical scar is due, I have little doubt, to the
method of treating the cord at birth. The custom prevailing among many is to
bind the whole cord and placenta on to the child’s abdomen till it separates ;
with others the greater part of the cord is so treated after severing the placenta ;
in any case there must be considerable tension, I think, at the umbilicus and
sepsis is more likely to occur. Cursham Corner has said that the size of the
bulging is proportional to the length of cord left proximal to the ligature, and
the same principle adapted to natives who use no ligature may be true, and
thus account for the very “long” umbilical hernias.
I am therefore inclined to agree with Corner that the umbilical and the ventral
herniae of children are due largely to intra-abdominal pressure, but though my
numbers are small, the absence of any other hernia among my cases must be
taken to mean that for their production there is another factor to be taken into
account, and that is, I believe, in Corner’s cases some congenital structural
anomaly, namely a congenital sac, and, conversely, I think congenital sacs are
uncommon among natives of this country.
(19) Malformations of the Extremities. Various forms of Congenital Talipes
are met with which call for no special comment.
A peculiar condition characterised by symmetrical shortening of the humeri
has been observed and forms the subject of a paper by Dr 8. A. Kinnier Wilson
and myself referred to above; certain deformities of the hands and feet are also
therein dealt with.
Since this paper was written I have seen three other cases of Congenital
Humeral Micromely, one of which I mention here as there is a family history of
16 Congenital Anomalies in a Native African Race
the defect, a point of some interest and one which I had not elicited in previous
cases.
Gobedi, male, aged 22 years, a Yao employed as a machila carrier in Zomba,
exhibits the deformity in typical form well represented in the photograph
(Plate V, (19)).
The head of each humerus appears to be poorly developed and though move-
ment at the shoulder joint is free, a certain amount of fine crepitus is elicited,
such as was found in several of the other cases.
The point of interest however is the fact that the maternal aunt is stated to
have had the same congenital anomaly.
The subject has no brothers or sisters and his own two young children are
stated to be normals, his mother and father and more remote relations are not
known to be affected.
Besides these the following cases deserve mention.
A boy was seen, 18 years of age, with a peculiar deformation of the hands,
stated to be congenital; the fingers and thumb shewed considerable thickening
about the Ist interphalangeal joints with marked ulnar deflection; the bridge of
the nose was depressed, the lips very thick, and epicanthus present. There was
also the penile deformity above mentioned, except for which I should have
doubted the statement in regard to the congenital nature of the hand deformity
(Fig. 11).
Fig. 11, Boy aged 18,
H. S. StTannus 17
I saw at Bandawe a female infant aged 14 years presenting multiple defor-
mities. The astragalus of the left foot was apparently implanted in a cup-
shaped depression on the lower end of a very much shortened thigh. The femur
of this leg was short but around it there was an abnormal amount of muscle as
if the usual amount of muscle for a normal had been cramped up into the
shortened limb ; the foot could be freely moved by the child. The left foot had only
a hallux and two toes with a partial cleft between the hallux and the adjacent
toe, but I think four metatarsal bones. The right thigh was also somewhat
shortened but the bones of the leg apparently both present, the knee-joint could
not be distinctly made out and was flail. Right talipes equinovarus present, also
right internal strabismus. No history of similar deformity in family, a brother a
year older was born with two upper incisors. Father and mother normal. The
father has two other wives with six and ten children respectively, all normal.
Such gross congenital deformities are from time to time recorded in Europe, thus
Lockart Mummery described a case of congenital absence of the femur in a male
child, etc., in the Brit. Med. Jour. for November 5, 1910.
In a male 35 years of age I found Congenital Absence of the Right Fibula, the
tibia being bowed forward with 8 inches shortening of the limb, the foot on the
same side had only three metatarsal bones and three digits including the hallux.
A woman was seen with congenital shortening of one leg to the extent of four
inches.
A single case of unilateral Congenital Dislocation of the Hip has been met
with.
(20) Split Hand and Split Foot Deformities. The photograph, Plate IV, (15),
serves to shew moderately well the deformities met with in a male child aged
5 years (1905): in the absence of a skiagram it is impossible to go into the detail
of the bony conditions present. There was no admitted history of similar or
other deformity in the family.
A second case, Ndala of Njalusi’s Mangoche, shewed a similar deformity of the
left hand but in a less degree; he was otherwise normal and stated that no other
members of his family were similarly affected (Plate VI, (20)).
These cases are interesting to compare with those collected and classified by
Lewis and Embleton in Biometrika, Vol. v1, 1908.
(21) Shortening of the Fourth Metatarsal Bone. When first I entered the
country my attention was attracted by a number of natives who presented a
shortening of the fourth toe.
Since then Captain Hughes has noticed the condition in Egypt. The descrip-
tion he gives is as follows (Lancet, July 16, 1910):—“ The fourth toe is markedly
retracted usually behind the level of the fifth toe. The phalanges are not appa-
rently abnormally short, and the metatarsal bone can be felt unfractured but with
the head very much farther back than usual. Commonly the digit is pushed
Biometrika x 3
18 Congenital Anomalies in a Native African Race
upwards by the pressure inwards of the fifth toe. The condition is sometimes
unilateral sometimes bilateral.” He adds that in one case the second metatarsal
and, in another, the third metatarsal were also shortened. In a single case he
saw a similar condition in the hand, shortening of the second and fifth meta-
carpals.
The above description corresponds exactly with the condition seen in this
country. I have also seen other toes than the fourth affected, and I shew a photo-
graph of a man’s feet with involvement of the metatarsal of the hallux; in-
another case the fifth was affected; in another case, a woman, the common variety
was associated with shortening of the third metatarsal of the left foot (see Plate VI,
(22), (24) and Fig. 12).
/
. /
=.
Fig. 12. Fig. 13.
OM 2
(22) Syndactyly of various degrees has been observed; sketches of two
examples are given in Fig. 138.
(23) Polydactyly is not at all uncommon. I have casually come across some
dozen cases in five years.
In the majority the supernumerary digit consists of a miniature phalanx
attached to the skin of the hand or foot at the level of the head of the fifth meta-
carpal or -tarsal bone. Such digits are often removed in childhood, leaving
a small cartilaginous nodule at the seat of removal. Most commonly it is a
symmetrical affection of both hands and feet; in other cases hands or feet alone
(Plate VJ, (23)), or one extremity only, present the deformity. In some the
accessory digit is well formed and an accessory metatarsal or metacarpal bone
more or less complete is present. In one case it was the hallux which was
reduplicated, the two digits being partially fused. In another, reported to me,
the supernumerary digit in each hand was situated on the radial side of the
first finger with probably an accessory metacarpal bone in connection with it.
The feet bad extra digits beyond the fifth toes.
(24) The following case is of some interest:
Chibisa, male, an Angoni of Kawenga’s, aged 30 years. The deformities in-
volve all segments of the right upper and lower limbs and to a minor extent the
OO
H. S. Srannus 19
left limbs (Plate VI, (21) and Figs. 15—18). On the right side there is shortening
of the humerus and forearm (10 cm. difference between the two sides), but
elongation of all the segments of the middle finger and its metacarpal bone; the
middle finger itself measures 10} cm. The metacarpal bones and phalanges of
the other fingers are, I think, absolutely a little shortened. The left arm and
hand are normal, except that this hand as also the right hand shew a little
nodule at the base of the little finger where a supernumerary digit was removed,
®
Fig. 15. Fig. 16.
O
Qinee
E™
Fig, 17. Fig. 18.
The right foot presents a similar condition to the right hand, elongation of
phalanges and metatarsal affecting the second toe, the toe itself being 7 em.
long. The tibia is somewhat bowed outwards. The left foot presents shortening
of the metatarsal bone of the hallux. .
3—2
20 Congenital Anomalies in a Native African Race
The photograph and sketches illustrate some of these points. Other measure-
ments were as follows:
Height 166°5 cm.; span of arms 164°5 cm.;
Maximum fronto-occipital 18°0; maximum biparietal 13'8; -
Nose length and width 4:4.
(25) Congenital Anomalies of the Kidney. Post-mortem examination on a
native prisoner who died of pellagra revealed the presence of a double kidney
on the left side and none on the right.
From the sketches (Fig. 19) it will be seen that the upper part was the one
proper to the side while the lower half was the abnormal portion.
L.Suparenal-|--
Fig. 19. Churinigu. Kidney of Left side double.
The two parts were really very distinct, partly separated by a groove and
cleft.
The lower viscus had been felt during life as a tumour in the abdomen of
unknown nature as it lay along the left side of the vertebral column. The kidney
was unfortunately removed before dissection of vessels, etc. was made, but the
sketch shews the arrangement of these at the hilum of the kidney. .
The two ureters united below the lower pole of the double organ, the distal
ureter being nearly twice the normal size.
Se a Se ee
HLS. Sraynvus 21
The bladder was normal; there was no right ureter. The suprarenal body
of the right side was in its normal position and appeared normal. No other
abnormalities were remarked.
(26) Some suggestive observations have been made by Dr Ewald Stier,
published in the Deutsche Zeitschrift fiir Nervenheilkunde (Band xiv, Heft 1-2,
S. 21), from which the generalisation is made that in all anomalies of overgrowth the
right side of the body is much more frequently involved than the left, whereas in
anomalies of undergrowth the left is more commonly the site of the condition
than the right, this distribution being the result of a preponderance of persons
with a leading or superior left cerebral hemisphere, as with left-handed persons
the converse was found to be true. In other words, the plus anomalies occur on
the right side in right-handed people and the minus anomalies on the left side,
the left hemisphere being the superior hemisphere, the converse being true.
I have therefore tabulated my observations, and though small in number they
tend to confirm the idea assuming that the African native is right-handed. This
remains unproved and a less marked superiority of the left hemisphere may
account for non-conformity of my few cases to Stier’s rule.
| |
Right | Bilateral) Left
Plus Anomalies: ° |
Reduplication of teeth 2 3 0)
Polymazia : : ae eel 1 0
Polythelia ... soe She i0 | O 5
| Polydactyly ... 1 1 2
Minus Anomalies:
Hare-lip ; | @ 2
Cryptor chidism 2 0 2
Absence of Fibula : 1 0 0
~ and Tibia... 0 0) 1
| Split "hand, foot 0 1 1
| Shortened metacar pal, tarsal 0 1 3
Syndactyly : 1 0 0
Coloboma iris = 1 1 0
Plus and Minus together : |
Chibisa 1 (+) — | 1(-)
In considering these cases it should be remembered that the majority of my
observations have been made casually among natives met in the bush, in villages,
etc., others in the course of routine work among troops, prisoners, etc., the few
were the result of special investigation.
(27) Concluding Remarks. The notes of cases which I have thus collected
together form rather a medley of facts but I think certain deductions may be
made from them.
22 Congenital Anomalies in a Native African Race
It would appear that
(1) The slighter the anomaly the greater the frequency with which it may
be observed.
(2) The more marked degrees of deformity are only seen in children and
those in places where European influence is felt.
(3) Cases of heredity are only seen among the lesser anomalies.
(4) The least obvious congenital anomaly is a helical fistula, and this is
found in 46 °/, of the population and is frequently inherited.
The difference in the observed incidence between the minor anomalies and
those of more marked proportions may be real or only apparent. I think the
latter supposition is true for reasons which can be deduced from the facts given
above.
It is the custom among all the tribes of this country to destroy all deformed
children at birth. Any minimal deformity such as a helical fistula is of course
unrecognised, an accessory nipple is probably hardly noticeable, accessory digits
which can be removed by a nick with a knife are matters of no import, while a
foot with six well-formed toes would hardly be considered worthy of note. These
abnormalities are therefore comparatively common, but hare-lip, cleft-palate,
deformities common enough in Europe, are among the rarest in this country; a
child with a hare-lip would be seen to resemble a hare and would be immediately
destroyed. Children with the greater deformities would certainly be destroyed.
In recent years under European influence native customs fall into abeyance and so
we see my single case of hare-lip in a boy aged 10 at Blantyre, a township of 25
years standing, a child with gross deformities of the lower extremities born prac-
tically on a mission station; or, to quote another example, an albino reported by
myself was the fifth albino child born, the first four having been killed at birth
by order of a chief, who in later years came under the influence of an up-country
mission station, for which the living albino has to thank his survival. The gross
abnormality of absence of premaxilla would pass unnoticed as the deformity is
slight. History relates that in the case of the child with lobster claw deformity of
hands and feet, it was only saved from a summary death by the efforts of the
mother.
I think with the evidence as it stands one may with fairness say that con-
genital anomalies are common among the natives of this country. Secondly, I
think one may also deduce from the facts stated that abnormalities of all kinds
are at least not uncommon. In the few cases in which I have adduced statistics
there can be no doubt, in other cases it is rather a matter of one’s impression.
I have shewn that certain congenital anomalies among natives of Nyasaland
are common and have attempted to argue that probably many of them are
common.
ae
H. S. Srannus 23
(28) Very few statistics are available for comparison, but I should hke to refer
to some by writers in Egypt. Prof. Madden cites in a letter to the Lancet a case
of cleft-palate which he operated on as the first in 11 years during surgical work
at the Kasr-el-ainy Hospital, and assigns as the cause of the lack of such cases
the “truly awful struggle for existence” which would eliminate infants so handi-
capped. The Lancet remarked (Lancet, July 3, 1909), in an annotation upon this
letter, that Prof. Elliot Smith considers it to be impossible to endeavour to explain
this rarity of congenital defects in Egypt, unless the time-honoured scapegoat of
our too modern civilisation be invoked to account for their frequency in other
countries. Statistics of the Kasr-el-ainy Hospital compiled by Dr Day are quoted
in 1907; among 2630 total surgical admissions the only congenital deformities
were 5 hare-lips, 2 talipes, 2 imperforate anus, 1 extroversion of bladder; in 1908,
2702 admissions, 3 hare-lips, 2 imperforate anus, 1 hypospadias, 1 undescended
testicle, 1 meningo-encephalocoele. Capt. G. W. G. Hughes, R.A.M.C., in a paper
to the Lancet, July 16, 1910, referring to this annotation, remarks “ Readers will
be interested to hear that our too modern civilisation is innocent of this slur,”
and goes on to shew that many congenital defects are by no means uncommon.
Dealing with males between the ages of 14 and 21 years he gives the following
figures :
Hare-lip in 0°041 7%.
Cleft-palate 0016.
Polydactylism 0-058 '/ and 0:04 °%, in two series.
Shortened metatarsal 0°37 7% and 0:23 %
Other deformities of fingers and toes 0:22 7.
Talipes 016 %.
Among the thousands of ancient Egyptian bodies which Prof. G. Elliot Smith
has unearthed and examined, a single case of cleft-palate was met with, a female
of 20 years of age with a skull of negroid type, of between the 4th and 6th
century B.c.; only one case of talipes (T. equinovarus) was recorded.
It is obvious that in Egypt surgical treatment is not sought in cases of
cleft-palate and rarely for other congenital defects but many of them are common
enough.
May the rarity of defects among the ancient peoples of Egypt be due to the
same cause that acts in Nyasaland to-day? Were the children affected with
deformities killed at birth and “thrown onto the dust-heap” where their remains
were soon lost trace of? Of chief interest to me are the figures published by
Captain Hughes. He shews that a shortening of the 4th metatarsal bone occurs
in percentages rising to 0°37 of males examined. This defect is peculiarly common
in this country. Again, polydactylism occurs in 0°05 °/, and other deformities of
fingers and toes in 0:22 °/, of Egyptians, both deformities very frequently met
with by myself in Nyasaland.
24 Congenital Anomalies in a Native African Race
He however does not mention polythelia and polymazia nor helical fistula.
I should be very interested to learn if this last insignificant anomaly was looked
for. There is no doubt that one of them, helical fistula, occurs with a frequency
in Nyasaland which cannot be rivalled by any other among peoples of any race.
I think one may also say with certainty that the incidence of others (shortened
4th metatarsal and polydactylism) in this country is far in excess of that among
Europeans, though probably much about the same as in Egypt.
Upon what hypothesis can these facts be explained? Is there a single cause
or are there many at work? These are questions which I shall not attempt to
enter into, but by simply recording my observations I shall hope to stimulate
others to do the same, for only by accumulating facts can it be hoped that such
problems will ever be solved.
Biometrika, Vol. X, Part | Plate |
(1) (2)
Samuti, an Ateliotic Dwarf. Subgiant, Height 1:92 metres, with Wife and
Albinotic Child.
Etimu, aged 25, an Achondroplasic Dwart.
Biometrika, Vol. X, Part | Plate Il
(5) (6)
Masimosya, aged 19, Gynaecomastos, with other features which were formerly described as those of
Partial Hermaphroditism.
(8)
Microcephalic Infant. [Case of Hydrocele testis included by an over-
sight of Dr Stannus in the photographs, and
engraved in consequence, Discovered too late
to rearrange plates.]
4
Plate Ill
Biometrika, Vol. X, Part |
‘ATeydaooydvog SurAoys
(01)
‘supad GL pase ‘sog
/
Biometrika, Vol. X, Part | Plate IV
i
5
(13)
Son of Matikwiri, aged 7, a case of Scaphocephaly.
(14) (15)
Cases of Umbilical Hernia, Split Hand and Foot in a child, aged 5,
be
Biometrika, Vol. X, Part | Plate V
(16) (17)
Case showing faint medium depression Blantyre boy, aged 10, with Hare-lip.
of upper lip.
(18) (19)
Young woman with two nipples on left Gobedi, ayed 32, with congenital Humeral
breast. Micromely.
i
. i ' i
fi ‘ Sat hte . ‘
. i 2 : ,
. i 7
- a is eo '
- a —s ~ % 7 ”
i =, 4 i - 7
1 3h whe = : - : ue
- ~ 4 bo ge A?
- f 4
7 7 7 | .
sd : i . 7
, Fey , et
Biometrika, Vol. X, Part | Plate VI
(20) Ndala, Split Hand, left only. (21) Chibisa, aged 30, elongation of all segments of middle finger,
; and its metacarpal bone,
Zanes Ss. es EE
(22) Shortening of the fourth metatarsal bone.
(24) Shortening of the left great toe.
C 7 P a
7 . A - 5
—
‘
mie 22 in: Jase S >= + : - PK} : :
¥ 5 e 7
. ’ a i
= i j
ae : (,
2 ae O24 \, = :
e 2
bd ry
TABLES OF POISSON’S EXPONENTIAL
BINOMIAL LIMIT.
By Ee it SOPER, “McA:
In his treatise, Recherches sur la Probabilité des SJugements, Paris, 1837,
Poisson* shows that the series of frequencies
(Dean am Oct n(n =i) pg + + ee (DEK ip +
2 atin nih Lr
given by the expanded terms of the binomial
(p aL qs
becomes in the limit, when gq is diminished, and n increased, indefinitely, but so
that nq remains finite and equal to m, the exponential series
S me m
e™(l+_m+o——+...+—+4...]3
2! r!
and he points out that the terms of this series will give the proportional
frequencies of the occurrences
OF pie ee ney
times, in any sample, of an event, every occurrence of which is equally likely in
the sample and independent of the other occurrences, and which is of such
frequency that m events occur in the sample on an average.
The series is arrived at by “Studentt,” when considering the theoretical
frequencies in sample drops of a liquid of minute corpuscles supposed distributed
at random throughout the mass of the liquid.
The event may also occur in time, each occurrence being supposed to take
place with equal probability in any finite period taken as the sample, and to act
independently of the occurrences of all the other events. A physical example,
which appears by the closeness of the observed to the theoretical frequencies to
* pp. 205 et seq.
+ Biometrika, Vol. v. p. 351, ‘‘On the Error of counting with a Haemacytometer,”
Biometrika x 4
26 Poissows Eaponential Binomial Limit
satisfy these conditions, is the number of a-particles discharged per {-minute or
}-minute interval from a film of polonium *.
In vital statistics the sample may be an individual or house or community and
the event an accident or disease and so on. But it must be borne in mind that
for such series as the above to be applicable the occurrence of one event in the
sample must not preclude or influence in any way the occurrence of a second.
The probability of « occurrences, m being the mean number, in a sample, is
e-™ m*/x |
and in the tables which follow this is evaluated for m= 0'1, 0°2... to 15:0 and for
x =0,1, 2... up to such an integer as gives a figure in the sixth place of decimals,
the number of places tabulated.
The terms of the series were calculated, each by a fractional operation upon
the preceding, beginning with the modal term and going both forward and back.
Thus if m=7'6 the term e~*® x (7'6)'/7 ! was first calculated by tables of logarithms,
and the succeeding terms were then obtained seriatim by the operations
16 76 76
meee ie eI)
and the preceding ones by the operations
ag eae etc
06 5G" AEG ge
done with a mechanical calculator, first a multiplication and then a division.
tc.,
Seven places of decimals were thus calculated and the series is checked by the
total, which differs from unity by the remainder (a figure in the eighth or later
place of decimals in all the present cases) and the algebraical sum of the errors of
seventh figure approximations.
Poisson’s exponential series has been previously calculated to four places of
decimals by L. von Bortkewitsch+ for values of m from 0:1 to 10:0.
The present tables give the probability of each number of times of occurrence
of the event. For the sums of these values, that is, the probability of occur-
rence of the event, a given number of times or greater, or a given number of times
or less, reference must be made to a second paper in this issue of Biometrikat,
where such probabilities are calculated for integral values of m from 1 to 30.
* See Rutherford and Geiger: ‘‘The Probability Variations in the Distribution of a-Particles,”
Philosophical Magazine, Vol. xx. p. 700, 1910. See also EK. C. Snow, ‘‘ Note on the Probability Varia-
tions, &c.,” Vol. xxi. p. 198, 1911, who finds the variance of experiment from theory to be such
as would occur once in six experiments and once in three experiments respectively of the limited time
taken, were theory exact. In a note to the first paper H. Bateman gives a proof of the exponential series
of probabilities arrived at from considerations of this problem.
+ Das Gesetz der kleinen Zahlen, 1898. A comparison of the table printed therein with the present
table shows agreement except as to the fourth figure; the nearest fourth figure is not given, in rather
many instances, in the tables of Bortkewitsch.
+ Lucy Whitaker, B.Sc. ‘On the Poisson Law of Small Numbers,’ Vol. x. p. 37 et seq.
H. E. Soper
TABLE of e™m*/«!: General Term of Poisson's Exponential Expansion
(“Law of Small Numbers”).
m
# | : | a ca : af
0-1 o2. |, 03 04 | O08 0-6 0-7 0:8 09 | 1-0
| | |
0 | 904837 | ‘818731 °740818 | 670320 | °606531 | 548812 | -496585 | -449329 | 406570 | 367879 | 0
1 | 090484 | "163746 °222245 | -268128 °303265 | -329287 | °347610 | -359463 | °365913 -367879| 1
2 | 004524 | -016375 °033337 | 053626 | 075816 | -098786 | °121663 | 143785 | "164661 | -183940; 2
8 | 000151 | -001092 003334 | -007150 °012636 | -019757 | -028388 | 038343 | 049398 -061313| 3
4, | 000004 | -000055 = =*000250 | ‘000715 *O01580 | -002964 | 004968 | -007669 | °011115 | 015328 | 4
i) — 000002 *000015 | 000057 | 000158 | -000356 | ‘000696 | -001227 | ‘002001 003066} 5
6 — — ‘000001 | -000004 000013 | *000036 | *000081 | *000164 | 000800, -000511] 6
ih — = ~ — | 000001 | 000003 | -000008 | 000019 | *000039 | 000073 | 7
8 as = = | -- “000001 | *000002 | ‘000004 | “000009 | 8&
9; ~ = - — _ — _ — — | 000001] 9
| : -
x Hes 1°2 1°3 Lp SN 1 1-6 HUSH 1°8 1:9 20 zy
0 | 332871 | 301194 | -272532 | -246597 | -223130 | .201897 | 182684 | -165299 | °149569 | -135335| 0
1 | 366158 | -361433 | -354291 | °345236 | °334695 | 3823034 310562 | -297538 | -284180 | -270671| 1
2 | °201387 | -216860 | *230289 | *241665 | *251021 | .258428 | °263978 | -267784 | °269971 | *270671 | 2
3 | 073842 | -086744 | 099792 | 112777 | *125510 | -137828 | 149587 | -160671 | 170982 | 180447 | 3
4 | 020307 | -026023 | -032432 | 039472 | ‘047067 | 055131 | 063575 | 072302 | ‘081216 | 090224 | 4
5 | 004467 | 006246 | -008432 | 011052 | °014120 | -017642 | ‘0216154 -026029 | 030862 | (036089 | 5
3 | 000819 | °001249 | ‘001827 | 002579 | °003530 | -004705 | ‘006124 | -007809 | *009773 | ‘012030 | 6
7 | 7000129 | :000214 | 000339 | °000516 | ‘000756 | ‘001075 | °001487 | -002008 | -002653 | 003437 | 7
8 | 000018 | 000032 | -000055 | ‘000090 | ‘000142 | -000215 | 000316 | -000452 | -000630 000859 | 8
9 | -000002 | :000004 | ‘000008 | :000014 | ‘000024 | -000038 | “000060 | -O00090 | -000133 | 000191 | 9
10 = “000001 | ‘000001 | ‘000002 | ‘000004 | -O00006 | 000010 | -000016 | -000025 | -000038 | 10
11 — — = — = ‘000001 | ‘000002 | 000003 | “000004 | 000007 | 11
12 ae ee — — | — {| 000001 | 000001 | 12
u 21 22 23 24 2:5 2°6 2:7 28 2:9 30 x
| _ ie
0 | °122456 | *110803 | *100259 | -090718 | :082085 | ‘074274 067206 -060810 | °055023 049787 | 0
1 | :257159 | °243767 | 230595 | -217728 | -205212 | -193111 | 181455 | °170268 | 159567 | 149361] J |
2 | 270016 | 268144 | -265185 | -261268 | 256516 | -251045 | -244964 | 238375 | 231373 224042 | 2
3 | 189012 | °196639 | *203308 | 209014 | °213763 | °217572 | *220468 | -222484 | *223660 +224042| 3
4 | 099231 | -108151 | *116902 | *125409 | -133602 | °141422 | 148816 | -155739 | 162154 -168031| 4
5 | 041677 | -047587 | -053775 | 060196 -066801 | -073539 -080360 | 087214 | 094049 -100819| 4
G | 014587 | 017448 | 020614 | ‘024078 | 027834 | 031867 | -036162 | -040700 | 045457 -050409 | G
7 | 004376 | 005484 -006773 | -008255 | 009941 | 011836 | 013948 -016280 -018832 -021604| 7
8 | 001149 | ‘001508 :001947 | 002477 | 003106 | 003847 | ‘004708 -005698 | ‘006827 -008102!) 8
9 | 000268 | 000369 °000498 | ‘000660 | -000863 | ‘001111 | 001412 | -001773 | *002200 002701, 9
10 | :000056 | 000081 | *G09114 | -000158 | :000216 -000289 | °000381 | -000496 | 000638 -000810 | 10
11 | 000011 | 000016 | -000024 , -000035 | -000049 -000068 | -000094 | -000126 | ‘000168 -000221 | 11
12 | :000002 | °000003 | -000005 | ‘000007 | 000010 | -000015 | ‘000021 | °000029 | ‘000041 -000055 | 12
13 — 000001 | 000001 | ‘000001 | >000002 | ‘000003 | -O00004 | *000006 | 000009 = -000013 | 13
14 — — | = | -- _- ‘QON0OT | *000001 | *OO00001 | 000002 -000003 | 14
fon — —-| — | — — = = _ — 000001 | 15
|
28 Poisson's Exponential Binomial Limit
TABLE—(continued).
| 11 | *000287 | -000368 | *000467 | 000587 | -000730 | -00090) | °001102 | -001337 | -001610 | 001925
12 | ‘000074 | 000098 | °000128 | 000166 | -000213 | -090270 | 000340 | -000423 | -000523 -000642
3% | ‘000018 | -000024 | *000033 | -000043 , -000057 | -000075 | ‘000097 | 000124 | -000157 | -000197
14 | -000004 | -000006 | “000008 | -000011 000014 | 000019 | -000026 | -000034 | “000044 | -000056
| 15 | 000001 | *O00001 | “000002 | -000002 | -000093 | -000005 | 000006 | *CO0009 | -000011 | -000015
MM
v ie eer ae — as
81 32 3-8 ow) ass 36 Pil 3:8 39 40
0 | 045049 | -040762 | -036883 | -033373 | -030197 | 027324 | -024724 | 022371 | 020242 | -018316| 0
1 | *139653 | -130439 | 121714 | -113469 | -105691 | -098365 | -091477 | -085009 | -078943 | -073263| 1
2 | 216461 | -208702 | *200829 | -192898 | -184959 | -177058 | -169233 | “161517 | 153940 | -146525| 2
3 | 223677 | -222616 | -220912 | -218617 | -215785 | -212469 | 208720 | -204588 | -200122 | -195367| 3
4 | 173350 | -178093 | 182252 | -185825 | -188812 | -191222 | -193066 | -194359 | -195119 | -195367| 4
5 | ‘107477 | -113979 | *120286 | -126361 | -132169 | -137680 | 142869 | -147713 | 152193 | 156293) 5
6 | 055530 -060789 | 066158 | 071604 | -077098 | -082608 | -088102 | 093551 | -098925 | 104196 6
7 | 024592 | -027789 | -031189 | -034779 | 038549 | -042484 | 046568 | -050785 | -055115 | -059540| 7
8 | 009529 | 011116 | 012865 | -014781 | -016865 | -019118 | -021538 | -024123 | -026869 029770! 8
9 | 003282 | -003952 | °004717 | -005584 | -006559 | -007647 | -008854 | 010185 | -011643 | -013231| 9
10 | -001018 | -001265 | ‘001557 | -001899 | -002296 | -002753 | 003276 | 003870 | 004541 | 005292 | 10
1
2Q
me
Soe
s
5
16¢)| = : — 000001 -000001 | -000001 | -000001 | -000002 , -000003 | -000004 | 16
Ly =) ieee ee or ee = a “000001 | 000001 | 17
al at | ‘49 43 Fy NS 46 Te 48 yo G0 ae
0 | 016573 | -014996 | °013569 | -012277 | -011109 | -010052 | ‘009095 | :008280 , -007447 | :006738
1 | 067948 | -062981 | 058345 | °054020 | 049990 | -046238 042748 | °039503 | -036488 | -033690
2 | *139293 | +132261 | 125441 | +118845 | -112479 | 106348 | -100457 | ‘094807 | -089396 | -084224
3 | 190368 | +185165 | 179799 *174305 | -168718 | -163068 | °157383 | °151691 | -146014 | -140374
4 | 195127 | 194424 | -193284 | 191736 | :189808 | -187528 | 184925 | +182029 | *178867 | °175467
5 | 160004 | +163316 | °166224 | +168728 | 170827 | °172525 | -173830 | °174748 | -175290 | 175467
G | *109336 | -114321 | +119127 | 7123734 | -128120 | -132270 | 136167 | 1139798 | 143153 | 146223
7 | (064040 | -068593 | ‘073178 | ‘077775 | 082363 | 086920 | 091426 | 095862 | -100207 | *104445
032820 | -036011 | 039333 | 042776 | 046329 | 049979 | -053713 | 057517 | ‘061377 | -065278
9 | -O14951 | -016805 | 018793 °020913 | -023165 | 025545 | 028050 | °030676 | -033416 | -036266
10 | -006130 | 007058 | 008081 | °009202 | -010424 | -011751 | -013184 | 014724 -016374 | 018133
11 | -002285 | 002695 | -003159 | -008681 | -004264 | -004914 | -005633 , 006425 007294 | 008242
12 | 000781 | -000943 | -001132 | 001350 | -001599 | 001884 | :002206 | 002570 -002978 | :003434
3 | -000246 | -000305 | 000374 | 000457 | 000554 | -000667 | 000798 | -000949 001123 | -001321
14 | 000072 | 000091 | -000115 | 000144 | -000178 | -000219 | 000268 | 000325 | 000393 | -000472
15 | 000020 | -000026 | -000033 | -000042 | -000053 000067 | -009084 | -000104 -000128 | 000157
16 -Q00005 | 000007 | 000009 | ‘000012 | -000015 -000019 | -000025 | 000031 -000039 000049
17 | 000001 | -000902 | *000002 | *000003 | -000004 -000005 | -000007 | *000009 | *000011 | :000014
7)
WOANAAK OHS
iS |) Ga — | 000001 | *000001 | -000001 | -000001 | -000002 *000002 - -000003 | -000004
1S) = _— ces — = — — | 000001 | -000001 | *000001
| |
ole of Se Nord a4 55 56 a7 eee se ye) i) OO) x
0 006097 | °005517 | 004992 | -004517 | -004087 | -003698 | 003346 | -003028 | 002739 | 002479
031093 | ‘028686 | -026455 | °024390 | 022477 | ‘020708 | 019072 | -017560 | 016163 | -014873
2 | -079288 | 074584 | 070107 | -065852 | -061812 | -057982 | :054355 | -050923 | -047680 | -044618
3 134790 | 129279 | 123856 | 118533 113323 | *108234 | +103275 | 098452 | ‘093771 | 089235 |
ES
Ce e®R >
H. E. Soper 29
TA BLE—(continued).
m
qr
Le
Or
©
nn
aS)
N
s
aT
ial
OU
fo
~t
~
oa)
Ne)
QD
S
171857 | *168063 164109 | -160020 | *155819 | *151528 | 147167 | °142755 | 138312 | 133853) 4
175294 | 174785 173955 | -172821 | 171401) -169711 | 167770 | -165596 | *163208 | *160623 5
*149000 | °151480 | -153660 | 155539 | °157117 ) °158397 | *159382 | -160076 | 160488 | 160623) 6
“108557 | °112528 | -116343 | -119987 | °123449 126717 | 129782 | +132635 | °135268 1387677) 7 |
069205 | 073143 -077077 | -080991 | ‘084871 -088702 | 092470 -096160 | ‘099760 *103258 8
039216 | ‘042261 -045390 | 048595 | °051866 | -055192 ‘058564 | -061970 | °065398 | 06838 | 9
“020000 | 021976 024057 | -026241 | °028526 -030908 | °033382 | -035943 | 038585 | 041303, 10
-009273 | °010388 -011591 | -012882 | °014263 -015735 017298 | 018952 | -020696 | 022529 | 71
003941 | ‘004502 -005119 | -005797 | °006537 -007343 | -008216 | 009160 | 010175 | ‘011264 | 12
001546 | 001801 | 002087 | 002408 | °002766 -003163 -003603 | -004087 | 004618 | °005199 | 13
‘000563 | -000669 | -000790 | -000929 -001087 -001265 | 001467 | -001693 | °001946 | "002228 | 14
‘000191 | 000232 | -000279 | -000334 000398 000472 | 000557 -000655 | ‘000766 000891 15
000061 | ‘000075 | -000092 | -000113 | *000137 | -000165 | 000199 | -000237 | *000282 | ‘000334 16
000018 | -000023 | -000029 | -000086 | ‘000044 | -000054 | ‘000067 | 000081 | ‘000098 | ‘000118 | 17
“000005 | ‘000007 | ‘000008 | :000011 | ‘000014 | -G00017 | 000021 | -000026 | -000032 | ‘000039 | 18
000001 | 000002 -000002 | -000003 | -000004 | -000005 | -000006 | -000008 | -000010 -000012 | 19
= — 000001 000001 | 000001 | -000001 -000002 | -000002 | 000003 | -000004 | 20
= = —- ; — ar — -= | :Q00001 | *O00001 | *OO0001 | 21
6'l 6°2 63 Or4 6'5 O°6 oy 6°8 Org 70 we
002243 | -002029 | -001836 | °001662 | ‘001508 | -001360 | -001231 | -001114 | -001008 | ‘000912 | 0
013682 | °012582 | -011569 | -010634 | 009772 | ‘008978 | °008247 | 007574 | °006954 | ‘006383 | 1
041729 | :039006 | :036441 | °034029 | 031760 | 029629 | 027628 | -025751 | -023990 | 022341 | 2
084848 | ‘080612 | -076527 | °072595 | ‘068814 | 065183 | 061702 | -058368 | -055178 | 052129 | 3
129393 | °124948 | :120530 | -116151 | °111822 | -107553 | :103351 | (099225 | -095182 | ‘091226 | 4
"157860 | °154936 | *151868 | -148674 | °145369 +141969 | 138490 | 134946 | -131351 | 127717) 6
"160491 | °160100 | -159461 | +158585 | °157483 | -156166 | °154648 | 152939 | 151053 | -149003 | 6
139856 | *141803 | *143515 | 144992 *146234 | -147243 | :148020 | 148569 | °148895 | *149008 | 7
"106640 | -109897 | :113018 | °115994 | ‘118815 | °121475 | -123967 | °126284 | -128422 | 130377] $&
072278 | ‘075707 | :079113 | -082484 | 085811 | 089082 | 092286 | -095415 | 098457 | 101405 | 9
044090 | 046938 | -049841 | -052790 | 055777 058794 | 061832 | 064882 | 067935 | ‘070983 | 10
024450 | 026456 028545 | :030714 | °032959 | 035276 | -037661 | -040109 | -042614 | “045171 | 11
012429 | ‘013669 | 014986 | :016381 | °017853 | -019402 | -021028 | -022728 | -024503 | 026350 | 12
005832 | 006519 | -007263 | ‘008064 | °008926 | -009850 | :010837 | -011889 | -013005 | 014188 | 13
002541 | ‘002887 003268 003687 | (004144 -004644 | -005186 | 005774 | ‘006410 ‘007094 | 14
001033 | °0011938 -001373 | -001573 | 001796 -002043 | -002317 | 002618 | -002949 003311 | 14
‘000394 | :000462 | -000540 | -000629 | 000730 | -000843 | -000970 | 001113 | 001272 *001448 | 10
000141 | ‘000169 | 000299 | 000237 | ‘000279 *000327 | *000382 | -000445 | ‘000516 | ‘000596 | 17
000048 | -000058 | 000070 | ‘000084 | ‘000101 000120 | :000142 | ‘000168 | 000198 -000232 | 1S
‘000015 | *000019 | -000023 | -000028 | -000034 -000042 | -000050 | -000060 | -000072 ‘000085 | 19
000005 | ‘000006 | -000007 | -000009 | °000011 -000014 | ‘000017 | -000020 | 000025 000030 | 20
000001 | *090002 | -000002 | -000003 | *000003 | -000004 | 000005 | -000007 | ‘000008 -000010 | 271
= -— | -000001 | 000001 | -000001 ‘000001 | *000002 | -000002 | -000003 -000003 | 2.2
= = | -—- —- | — | = — 000001 | 000001 000001 | 22
q
q
30 Poisson's Exponential Binomial Limit
TABLE—(continued).
™m
av ax
Hi | 7:2 13 Th 1s} v6 at 78 79 8-0
0 | :000825 | 000747 | -000676 | -000611 | 7000553 | -000500 000453 | 000410 | -000371 | 000335 | 0
1 | :005858 | ‘005375 | 004931 | 004523 | 004148 -003803 | (003487 | -003196 | 002929 | 002684] 1 |
2 | 020797 | 7019352 | -018000 | 016736 | ‘015555 | 014453 | (013424 | -012464 | 011569 | 010735 | 2
3 | 049219 | 046444 | 043799 | 041282 | 038889 036614 | 084455 032407 -030465 | 7028626] 3
4 | (087364 | -083598 | -079934 | 076372 | (072916 069567 | 066326 -063193 060169 | 057252 | 4
5 | 124057 | 120382 | +116703 | -113031 | °109375 105742 | -102142 -098581 ‘095067 | 091604} 5
6 | 146800 | "144458 | 141989 | :139405 | 136718 °133940 | "131082 | -128156 125171 | *122138| 6
7 | *148897 | *148586 | *148074 | -147371 | °146484 | °145421 | 144191 | -142802 | -141264 | 139587] 7
8 | 132146 | -183727 | °135118 | +136318 | ‘137329 | -138150 | *138783 | -139232 | -139499 | -139587| 8
9 | *104249 | *106982 | 109596 | *112084 | *114440 | °116660 | 118737 | -120668 | -122449 | 124077] 9
10 | ‘074017 | 077027 | 080005 | -082942 | 085830 | ‘088661 | °091427 -094121 | 096735 | -099262 | 10
11 | ‘047774 050418 | °053094 -055797 | °058521 | °061257 063999 -066740 | 069473 | -072190) 11
12 | 028267 | -030251 | ‘032299 | -034408 | 036575 | 038796 041066 | -043381 | 045736 | -048127 | 12
3 | :015438 | 016754 018137 019586 | ‘021101 | -022681 | °024324 -026029 | 027794 | 029616 | 13
14 | 007829 | 008616 009457 | -010353 | 011304 | -012312 | 013378 -014502 | -015684 | 016924 | 14
5 | 003706 -004136 004603 -005107 | *005652 006238 ‘006867 -007541 | 008260 | -009026 | 15
16 | 001644 001861 002100 002362 | 002649 002963 | °003305 -003676 | -004078 | 004513 | 16
17 | -000687 | -000788 | ‘000902 | °001028 | 7001169 | ‘001325 | 001497 | -001687 | -001895 | 002124 | 17
18 | -000271 | 000315 | -000366 | -000423 | 000487 | *000559 | 000640 | 000731 | *000832 | 000944 | 18
19 | -000101 | °000119 | *000141 -000165 | 000192 °000224 | *000259 -000300 *000346 | 000397 | 19
20 | 000036 -000043 | 000051 | 000061 | 000072 | 000085 | 000100 | -000117 | 000137 | 000159 | 20
21 | 000012 -000015 *000018 | 000021 | °000026 | -000031.| *000037 000043 | -000051 | -000061 | 21
22 | 000004 | -000005 -000006 | -000007 | 000009 -000011 | *000013 -000015 | -000018 | -000022 | 22
23 | 000001, -000002 *000002 | -000002 | “000003 , 000004 | *000004 | -000005 ‘000006 | 000008 | 23
24 = = ‘000001 | -000001 | 000001 *000001 | 000001 -000002 | -000002 | *00C003 | 2
25 = 23 = ae = mn — 000001 | 000001 | -000001 | 25
= | ae raw
x | 8 8-2 8:3 Sy 85 SiG) ile oa 88 | 89 90 | «x
0 | 000304 | -000275 | 000249 | *000225 | 000203 | 000184 | -000167 | 000151 | 000136 | -000123 | 0
1 | -002459 | ‘002252 ‘002063 | ‘001889 | -001729 | -001583 | -001449 | 001326 | 001214 | OOL111) 1
2 | 009958 -009234 008560 | ‘007933 | 007350 006808 -006304 -005836 -005402 | 004998 | 2
3 | 026885 | 025239 | 023683 | 022213 | 020826 | 019517 | ‘018283 | 017120 | -016025 | ‘014994 | 3
4 | 054443 | 051740 049142 046648 | 044255 -041961 039765 | 037664 035656 | -033737 | 4
5 | 088198 | 084854 081576 | 078368 | ‘075233 | 072174 069192 | -066289 | 063467 | -060727 | 5
6 | 119067 | "115967 | "112847 | °109716 | "106581 *103449 | -100328 | 097224 | 094143 | 091090 | 6
7 | 137778 snes 1305 “a105) "129419 | -127094 | °124693 | -129224 | 119696 | °117116| 7
8 | -139500 | *139244 | -138823 | "138242 | 137508 136626 | 135604 | -134446 | 133161 | 131756 | 8
9 | 125550 | *126866 | 128025 | "129026 | *129869 | -130554 *131084 | "131459 | "131682 | -1381756| 9
10 | -101696 | *104031 | 106261 | *108382 | °110388 | -112277 *114043 | -115684 | *117197 | 118580) 10
11 | :074885 | ‘077550 | ‘080179 | ‘082764 | ‘085300 | -087780 | °090197 | 092547 | °094823 | 097020 | 11
12 | :050547 | -052993 | -055457 | °057935 | 060421 062909 | -065393 | -067868 | °070327 | °072765 | 12
13 | 031495 | ‘033426 | 035407 | ‘037435 | °039506 -041617 °043763 , 045941 | -048147 | -050376 | 13
14 | -018222 | 019578 | 020991 | (022461 | ‘023986 | 025565 -027196 | -028877 | 030608 | 032384 | 14
15 | -009840 | 010703 | 011615 | °012578 | 013592 | -014657 ‘015773 | 016941 | -018161 | ‘019431 | 15
16 | -004981 | 005485 -006025 | °006604 | °007221 | -007878 | 008577 | -009318 | 010102 | °010930 | 16
17 | -002373 | 002646 002942 | 003263 | 003610 -003985 004389 | -004823 | ‘005289 | ‘005786 | 17
18 | -001068 | 001205 -001356 | °001523 | 001705 | -001904 | 002121 | -002358 | 002615 | -002893 | 18
19 | -000455 | -000520 | -000593 | °000673 | 000763 | -000862 | 000971 | -001092 | °001225 | 001370) 19
20 | 000184 ee ia ‘000283 | -000324 | -000371 | 000423 | 000481 | -000545 | ‘000617 | 20
H. KE. Soper 31
TABLE—(continued).
me
xv rs Bo
Sl 82 S38 Sh Sb S'6 Ont ss SD | 9:0
21 | 000071 | ‘000083 | ‘000097 | ‘000113 -000131 | °000152 | ‘O00L75 | *000201 | *000231 | ‘000264 | 27
22 | -000026 | ‘000031 | -000037 | ‘000043 | -000051 | ‘000059 | 000069 | ‘000081 | 000093 | *000108 | 22
28 | 000009 | -000011 | ‘000013 | ‘000016 | 000019 | 000022 | *000026 -000031 | -000036 | 000042 | 23
24 | 000003 | -000004 | 000005 | ‘000006 | -000007 | ‘000008 -009009 | ‘000011 000013 | ‘000016 | 24
25 | -000001 | 000001 | ‘000002 | :000002 ‘000002 | *000003 | -000003 | °000004 | 000005 | ‘000006 | 25
| 26 — — —- 000001 | 000001 | *O00001 | “000001 | *QOQ0OL1 | *O00002 | -000002 | 26
2Y = — — a = — — ‘OOO0O1 | ‘OOO0001 | 27
Da Oot 9°2 Fs a4 hes) IO Oey 9°8 99 10°0 4B
| sons.
0 | 000112 | 000101 | ‘OO0091 | -000083 -000075 | ‘000068 | *OO0061 | *000055 | *000050 | ‘000045 | U
1 | 001016 | -000930 | ‘000850 | :000778 | ‘000711 | 000650 | ‘000594 | -000543 | 000497 | :000454 I
2 | 004624 | 004276 | 003954 | 003655 | -003378 | 003121 | (002883 -002663 | 002459 | 002270 g
3 | 014025 | 013113 | °012256 | 011452 | ‘010696 | °009987 | *009322 | ‘008698 | ‘008114 | ‘007567 3
4, | 031906 | 030160 | 028496 | -026911 | -025403 | :023969 | ‘022606 | °021311 | 020082 | 0189174
5 | 058069 | -055494 | -053002 | 050593 | -048266 | 046020 | 043855 °041770 | 039763 | °037833 | 5
6 | °088072 | -085091 | °082154 | °079262 | ‘076421 | -073632 ‘070899 | °068224 | *065609 | °063055 6
7 | 114493 | -111834 | *109147 | +106438 | °103714 | *100981 | ‘098246 | -095514 | °092790 | ‘090079 7
8 | 130236 "128609 | °126883 | °125065 | 123160 | °121178 | °119123 | *117004 | *114827 | *112599 8
9 | 131683 | °131467 | *131113 | 130623 | -130003 | 129256 | °128388 | *127405 | °126310] 1251109
10 | *119832 | *120950 | °121935 | *122786 | °123502 | °124086 | °124537 | °124857 | -°125047 | -125110 10
11 | :099188 | -101158 | *103090 | *104926 | -106661 | *108293 | ‘109819 | *111236 *112542 | 113736 11
12 | ‘075176 | ‘077555 | ‘079895 | -082192 | -084440 | 086634 | ‘088770 | -090848 | °092847 | 094780 12
18 | 052628 | :054885 | ‘057156 | 059431 | -061706 | (063976 | ‘066236 | 068481 | ‘070707 | (072908 13
14 | °034205 | :036067 | °037968 -039904 | ‘041872 | °043869 | °045892 | 047937 | “050000 | °052077 14
15 | 020751 | 022121 | °023540 *025006 | 026519 | 028076 | ‘029677 | ‘031319 | ‘033000 | ‘034718 15
16 | °011802 | 012720 | ‘013683 -014691 | -015746 | -016846 | ‘017992 -019183 | °020419 | -021699 16
17 | (006318 | -006884 | 007485 | 008123 | :008799 | 009513 | -010266 | -011058 | °011891 | ‘012764 17
18 | 003194 | 003518 | -003867 004242 | 004644 | 005074 | 005532 | ‘006021 | ‘006540 | ‘007091 18
19 | (001530 | -001704 | 001893 ‘002099 | 002322 | -002563 | °002824 | °003105 | °003408 | 003732 | 19
20 | :000696 | 000784 | ‘000880 | 000986 | -001103 | -001230 | *001370 | °001522 | *001687 | 001866 | 20
21 | 000302 | 000343 | ‘000390 000442 -000499 | 000563 | 000633 | 000710 | *000795 | ‘000889 | 21
22 | 000125 | 000144 | °000165 *O000189 | 000215 | 000245 000279 | *000316 | ‘000358 | -000404 | 22
23 | 000049 | -000057 | ‘000067 = “000077 000089 | -000102 *000118 | *000135 | 000154 | -000176 | 23
24 | 000019 | -000022 | ‘000026 -000030 | 000035 | ‘000041 | *000048 | *000055 | ‘000064 | -000073 | 24
| 25 | ‘000007 | -000008 | 000010 | -O00011 | *000013 | ‘000016 | “000018 | ‘000022 *000025 | -000029 | 25
| 26 | 000002 | -000003 | *000003 | :000004 | ‘000005 | ‘000006 | *O00007 | *O00008 000010 | 000011 | 26
27 | "000001 | 000001 | *OOO001 | *O00001 | *000002 | -000002 | *O00002 | "000008 *000004 | 000004 | 27
28 — —- | = — 000001 | ‘000001 | ‘000001 | 000001 000001, -000001 | 28
QO — = —- | — ee 000001 p22
@ 10:1 10°2 L0°3 L104 L0°5 106 L0°7 L0°8 KORY) | 110 7
0 | -000041 | :000037 | -000034 -000030 -000028 | -000-25 | ‘000023 | -000020 | -000018 | *OO0017 0
1 | 000415 | *000379 | 000346 | 000317. °000289 | -000264 | -000241 | :000220 | 000201 | ‘000184 7
2 | 002095 | 001934 | 001784 -001646 | -001518 | :001400 | 001291 | -001190 | -001097 ; 001010 | 2
3 | 007054 | 006574 | °006125 | -005705 | 005313 | 004946 | 004603 | 004283 | -003984 | *008705 | 3
1 | | |
32 Powssovs Exponential Binomial Limit
TABLE—(continued).
UD
x = ine ey ] BL
10°1 10:2 10°3 10"4 | 10°5 | 10°6 L0°7 10°S 10:9 | 11:0
|
—|- ——|——|——— | _——
4 | O17811 -016764 | -015773 | 014834 | -013946 | -013107 | -012313 | -011564 | 010856 | -010189 |] 4
5 | 085979 | °084199 | °032492 | 030855 | *029287 | ‘027786 | °026350 | 024978 | 023667 | 022415 | 5
6 | °060565 | -058139 | (055777 | 053482 -051252 049089 | 046991 | -044960 | 042995 041095 | 6
7 | 0873887 | 084716 | 082072 | 079458 °076878 | (074334 | 071830 | 069367 | 066949 064577, 7
8 | °110326 | *108013 | *105668 | 103296 *100902 | -098493 | ‘096072 -093646 | (091218 | ‘088794 | 8
9 | °123810 | 122415 | °120931 | -119364 “117720 | *116003 | °114219 -112375 | -110475 °108526; 9
10 | 125048 | °124863 | °124559 | 124139 +123606 +122963 | -122215 | -121365 | -120418 | 119378 | 70
17 | 114817 | *115782 | °116633 | °117368 | -117987 | *118492 | -118882 | *119159 | -119323 | -119378 | 71
12 | 096637 | 098415 | *100110 | °101719 | *103239 | :104667 | *106003 | +107243 | :108386 | 109430 | 12
13 | ‘075080 | ‘077218 | ‘079318 | -081375 | ‘083385 | °085344 | ‘087248 | -089094 | 090877 | °092595 | 13
14 | °054165 | °056259 | 058355 | *060450 | -062539 | °064618 |} 066683 | °068730 | ‘070754 | 072753 | 14
15 | °0B6471 | °038256 | 040071 | 041912 | °043777 | °045663 | 047567 | -049485 | 051415 053352 | 15
16 | °023022 | 024388 | 025795 | °027243 | °028729 | °030252 | -031810 | 033403 | 035026 ‘036680 | 76
17 | (013678 | 014633 | °015629 | 016666 | ‘017744 | °018863 | 020022 | -021220 | 022458 | 023734 | 17
18 | :007675 | *008292 | ‘008943 | -009629 | -010351 | °011108 | -011902 | :012732 ‘013600 | 014504 | 78
19 | ‘004080 | 004451 | *004848 | *0054%71 | -005720 | -006197 | °006703 | -007237 | 007802 | -008397 | 19
| 20 | °002060 | -002270 :002497 | -002741 003003 | 003285 | -003586 | 003908 | -004252 | 004618 | 20
21 | 000991 | °001103 | *001225 | ‘001357 | *001502 | °001658 | -001827 | °002010 | 002207 | ‘002419 | 21
22 | 000455 | -000511 | 000573 | 000642 | 000717 | 000799 | :000889 | -000987 | -001093 | °001210 | 22
23 | -000200 | -000227 | -000257 | 000290 -000327 | -000368 | 000413 | 000463 | -000518 | ‘000578 | 23
24 | 000084 -000096 | -000110 | 000126 -000143 | 000163 | -000184 000208 | -000235 -000265 | 2
25 | 000034 | 000039 | *000045 | *000052 , 000060 | *000069 | ‘000079 | -000090 | 000103 | ‘000117 | 25
26 000013 | 000015 *000018 | -000021 | 000024 -000028 | -000032 000037 | -000043 | 000049 | 26
27 | 000005 | 000006 | 000007 | *000008 | 000009 | :000011 | *000013 | :000015 | 000017 | *000020 | 27
28 000002 | -000002 | *0000038 | *000003 | -000004 | *000004 | :000005 | *000006 | -000007 | -000008 | 28
29 | 000001 | ‘000001 | -000001 | °000001 | 000001 | 000002 | *000002 | -000002 | -000003 | 000003 | 29
3O — _ — a — 000001 | ‘000001 | -000001 | *000001 | ‘000001 | 30
a let ED 113 114 TAG 5 = eto Mie 11°8 11:9 12°0 iz.
0 | 000015 | 000014 | ‘000012 | ‘000011 | ‘000010 | *000009 | °000008 | °000008 | *000007 ‘000006 |} 0
Z | -000168 | °000153 | 000140 | 000128 | ‘000116 | *000106 | ‘000097 | -000089 | ‘000081 | ‘000074 | 7
2 | 000931 | -000858 | 000790 | 000727 | 000670 | ‘000617 | 000568 | 000522 | 000481 | 000442 | 2
3 | 003445 | -003202 | 002976 | 002764 | ‘002568 | 002385 | -002214 | -002055 | 001907 | ‘001770 | 3
4 | 009559 | 008965 | -008406 | 007879 | 007382 | 006915 | 006476 | 006062 | -005674 | 005309 | 4
5 | °021221 | 020082 | -018997 | 017963 016979 | °016043 | 015153 | ‘014307 | -013504 | ‘012741 | 5
G | ‘039259 | °037487 | -035778 | 7034130 | 032544 | -031017 | °029549 | 028137 | -026782 | 025481 6
7 | 062253 | -059979 | -057755 | 055584 | 053465 | 051400 | 049388 | -047432 | 045530 | 043682 | 7
8 | 086376 | °083970 | 081579 | -079206 | ‘076856 | 074529 | 072231 | -069962 | 067725 | ‘065523 | 8
9 | 106531 | -104496 | 102427 | -100328 | 098204 | 096060 | -093900 | °091728 | 089548 | ‘087364 | 9
10 | “118249 | -117036 | 115743 | 114374 | +112935 | *111430 | *109863 | 108239 "106562 | °104837 | 10
11 | 119324 | +119164 | °118899 | °118533 | -118068 | *117508 | °116854 | *116110 | -115281 | 114368 | 11
12 | *110375 | 111220 | +111964 | *112607 | *113149 | °113591 | +113933 | -114175 | 114320 | *114363 | 72
13 | °094243 | 095820 | -097322 | 098747 | -100093 | °101358 | -102539 | -103636 | -104647 | 105570 | 13
14 | 074721 | (076656 | -078553 | -080409 | -082219 | 083982 | 085694 | :087350 | -088950 | 090489 | 14
15 | 055294 | 057236 | 059177 | 061110 | -063035 | -064946 | 066841 | -068716 070567 | 072891 | 15
16 | -038360 | ‘040065 | -041793 | 043541 | 045306 047086 | 048877 | ‘050678 | 052484 | -054293 | 16
17 | 025047 | -026396 | -027780 | -029198 | 030648 | °032129 | :033639 | 035176 | 036739 | ‘0388325 | 17
18 | 015446 | 016424 | -017440 | -018492 | 019581 | -020706 | 021865 | ‘023060 | -024288 | 025550 | 18
19 | -009023 | 009682 | -010372 | ‘011095 | 011852 | :012641 | 013465 | -014322 | -015212 | ‘016137 | 19
Se ee ee ee
ay
ee ee ee ee
Po pain hs:
H. E. Soper
TA BLE—(continued).
Vit
Lv = = = =F = —
111 11-2 113 [14 HN, | LTO 9) eat te?
| | | |
20 | 005008 | -005422 | 005860 | -006324 -006815 | 007332 | ‘007877
21 | :002647 | 002892 -003153 | :003433 -003732 | ‘004050 | 004388 |
22 | 001336 | -001472 001620 001779 | -001941 | ‘002136 | -00233
23 | 000645 | -000717 | -000796 | 000882 | -000975 -001077 -001187
24 | °000298 | -000335 | 000375 -000419 | -000467 | 000521 | -000579
25 | 000132 | -000150 000169 | 000191 | 000215 | 000242 | -000271
26 | °000057 | 000065 000074 -000084 | 000095 | 000108 000122
| 27 | 000023 | 000027 | 000031 —-000035 “000041 "000046 | -000053
28 | ‘000009 | -OVDOL1 | 000012 -000014 | ‘000017 | -000019 | 000022
| 29 | -Q00004 “O00004 000005 *OO0006 | *OO0007 | *O00008 | -~OO0009
| 30 | -000001 | -000002 ‘000002 | -000002 | -000003 | -000003 | -000003
| el *YODVO1 | -000001 | *000001 | 000001 | *OO0001 | ‘000001 |
OY | — -- — — -—
| F =
i || StF 12:2 12°3 124 io 12°6 12°7
| r | “aanel|
0 000006 | -000005 | *000005 | 000004 -000004 | *000003 000008 |
1 000067 -000061 | *000056 | 000051, 000047. *000042 -000039
2 000407 000374 | 000344 -000317 000291 -000268 | -000246
3 001641 | -001522 | °001412 | -001309 | -001213 | 001124 | -001042
4 | 004966 | -004643 | °004841 | 004057 | -003791 | 003541 | 003307
5 | 012017 | -011330 | 010679 | -010062 | -009477 | ‘008924 | 008400 |
6 | 024233 | -023037 | °021892 | 020794 | -019744 | (018740 | -017781
7 | 041889 | 040151 | 038467 | 036836 | -035258 | ‘033733 | -032259 |
8 | 063358 | -061230 | °059142 | -057095 | 055091 | °053129 | 051212
9 | -085181 | -083000 | ‘080828 | -078665 | 076515 074381 | (072266,
10 | *103069 | 101261 | °099418 | ‘097544 | -095644 °093720 -091777
11 | 113876 | °112308 | 111168 | +109959 | -108686 | 107352 | °105961
12 | *114321-| -114180 | 113947 | -113624 | -113215 | °112720 | -:112142
13 | °106406 | -107153 | °107811 | -108380 | -108860 | °109251 | *109554
14 | 091965 | :093376 094720 095994 | -097197 | ‘098326 | -099381
15 | 074185 | ‘075946 | °077670 | 079355 | -080997 | °082594 | 084143
16 | 056103 | -057909 | °059709 /-061500 | (063279 ‘065043 | 066788 |
17 | :039932 | -041558 | °043201 | 044859 | -046529 ‘048208 | -049895 |
18 | ‘026843 | 028167 | ‘029521 | -030903 | -032312 °033746 | °035204
19 | -017095 | -018086 | ‘019111 | -020168 | -021258 | °022379 | 023531
20 | -010342 | -011033 | ‘011753 | -012504 | -013286 | 014099 | 014942
21 | 005959 | -006409 | ‘006884 | -007383 | 007908 | ‘008459 | -009036 |
22 | -003278 | 003554 | 003849 | 004162 | 004493 | ‘004845 | -005216
23 | 001724 | -001885 | ‘002058 | -002244 | -002442 | ‘002654 | 002880
24 | 000869 | 000958 | -001055 001159 | 001272 | *001393 001524 |
25 | 000421 | -000468 | 000519 | -000575 | 000636 | ‘000702 | -000774
26 | :000196 | :000219 | -000246 | -000274 | -000306 | °000340 | -000378 |
27 | -000088 | -000099 | ‘000112 | 000126 | -000142 | ‘000159 | 000178 |
28 | -000038 | 000043 -000049 | -000056 | -000063 | ‘000071 | -000081 |
29 | 000016 -000018 000021 | -000024 | -000027 | -000031 | -000035
30 | 000006 | -000007 000009 -000010 | -000011 | (000013 | ‘000015
31 | -000002 | -000003 -000003 000004 | 000005 | -000005 | 000006
32 | 000001 | -000001 000001 -000002 ‘000002 000002 | 000002 |
Bo = | = — “000001 | 000001 | “000001 | *O00001
BY ee ee = == i) = ae
Biometrika x
ao
ints ———— rs
it} || rile) | BO)
008450 | -009051 | -009682. 20
004748 | 005129 -005533. 21
| -002547 | 002774 | 003018 | 22
001307 | 001435 | 001575. 23
000642 °000712 | ‘000787 24
“000303 | 000339 | ‘000378 25
000138 | -000155 *OOO1L74 26
“000060 | *O00068 | 000078 | 27
‘000025 | -000029 | ‘000083 | “5
© 000010 000012 | -O00014 | 79
“000004 | 000005 -000005 | 3
‘000002 | -000002 - *000002 3
*000D01L | 000001 | “000001 | 37
I
12'8 12°9 13:0 xv
000003 | 000002 | 000002} 0
000035 | -000032 | *000029 | 1
“000226 *000208 | ‘000191 2
000965 | ‘000894 | *000828 | 6
003088 | °002882 | *002690 | 4
007905 | 007436 | 006994 | 5
016864 | -015988 | °015153 | 6
030837 | -029464 | ‘028141 | 7
049339 | °047511 | 045730 | 8
‘O70171 068100 -066054) 9
089819 | -087849 | ‘085870 | LO
104516 | *103023 | *101483 | 11
-111484 | -110749 | :109940 | 12
-109769 | -109897 | (109940 | 13
100360 | -101263 | "102087 |) 14
085641 | ‘087086 | 088475 15
(068513 | ‘070213 -071886 16
‘051586 | 053279 | -054972 | 17
036683 | 038183 039702 | 18
024713 | (025925 -027164 | 19
015816 | 016721 °017657 | 20
009640 | 010272 °010930 ) 2
‘005609 | -006023 | ‘006459 | 22
003122 | :003378 ‘003651 | 25
001665 | ‘001816 | 001977 | 24
000852 | ‘000937 | °001028 | 25
000420 000465 | 000514 | 26
“000199 | -000222 | 000248 | 27
000091 , 000102 ‘000115 | 28
000040. -000046 | *000052 | 29
000017 | -000020 | *000022 30
“000007 ‘000008 | 000009 31
“000003 -000003 | ‘000004 52
“000001 | -OO0001 | -000002 | 3:
SS = 0000014173
5
34 Poissons Eaponential Binonial Limit
TABLE—(continued).
m
| | | \
13°2 LO258 Wy lovin A elo olen elo G 137. | 13:8 13°9 1L‘0 |
|
> St Ss INO
© © % Ww & 7
9 | *000568 | -000626 | ‘000689 | :000758
000002 ‘000002 *000002 -000002 | 000001 ‘000001 *000001 | *000001 000001 ‘000001
000027 | ‘000024 | *000022 -000020 | -000019 | ‘000017 000015 | 000014 | -000013 | -000012
000175 ‘00016 | *000148 000136 -000125 | ‘000115 *000105 | -000097 | -000089 | *000081 |
‘000766 | 000709 | °000657 -000608 000562 | -000520 000481 | -000445 000411 | 000380 |
‘002510 | *002341 | 002183 | -002035 -001897 | ‘001768 °001648 | 001535 -001429 | ‘001331 |
"006575 006180 °005807 | 005455 005123 ‘004810 004514 | -004236 ‘003974 | 003727
"014356 013596 | *012872 | -012183 | 011526 | 010902 *010308 009743 | 009206 | :008696
‘026867 | 025639 | "024458 | -023322 022230 | ‘021181 | 020173 | 019207 | -018280 | 017392
"043994 -042304 -040661 -039064 -037512 | -036007 084547 -033132 031762 | 030435
‘064036 | °062046 *060088 | 058161 | -056269 | -054410 | 052588 | -050802 | -049054 | 047344
| 083887 | ‘081901 | ‘079916 | -077936 075963 | -073998 | -072046 | ‘070107 -068185 | -066282 | 10
‘099901 098281 | 096626 | -094940 | 093227 -091489 -089730 | -087953 ‘086162 | -084359 | 11
ee HOBLOO fine a aon ue 104880 | *103687 102441 | -101146 | ‘099804 | -098418 | 1?
| "109898 | *109773 | *109566 | -109279 -108914 | *108473 | °107957 | -107370 | *106713 | *105989 | Lo
"102833 | *103500 “104087 | 104595 *105024 | -105373 +105644 105836 | 105951 | *105989 | 14
_ 089807 | °091080 | °092291 | -093439 | -094522 | 095539 , 096488 | ‘097369 | 098181 | 098923 | 15
| 073530 075141 | ‘076717 | -078255 079753 | -081208 | -082618 | 083981 | -085295 | 086558
WH OHMWAAK Cs WHS
a
D
‘056661 | °058345 | -060019 | 061683 -063333 | -064966 ‘066580 | -068173 | 069741 | 071283 | 17
041237 | *042786 | °044348 | 045920 | -047500 | -049086 | (050675 | °052266 | 053856 | 055442 | 18
039400 | :040852 | 19
027383 | ‘028597 | 20
018125 | 019064 | 1
011452 | :012132 | 22
028432 029725 | 031043 | -032385 | 033750 | 035135 °036539 -037962
| 018623 | 019619 | 020644 | -021698 | -022781 | °023892 -025030 | -026193
‘O11617 | °012332 | ‘013074 | 013846 -014645 | -015473 016329 | -017213
‘006917 | °007399 | 007904 | -008433 -008987 | 009565 -010168 | -010797
003940 | -004246 | ‘004571 | 004913 -005275 | ‘005656 006057 , ‘006478 | :006921 | :007385 | 25 |
“002151 | 002336 | ‘002533 | :002743 | 002967 | 003205 | 003457 | -003725 | 004008 | 004308 | 24 |
001127 | ‘001233 | 001348 | 001470 -001602 | 001744 | 001895 | ‘002056 | 002229 | -002412
000832 -000912 | 000998 | -001091 | ‘001191 | :001299 | 26 |
000275 | ‘000306 | °000340 | 000376 -000416 | 000459 | -000507 | 000558 | (000613 | :000674 | 27
‘000129 | -000144 | ‘000161 | 000180 -000201 | :000223 | :000248 | 000275 ‘000305 | 000337 | 28 |
000058 | “OO0066 ‘000074 | 000083 | -000093 | 000105 | -000117 | -000131 -000146 | :000163 | 29 |
000025 | -000029 | 000033 | -000037 | 000042 | 000047 ‘000053 “000060 -000068 | ‘000076 | 30 |
000011 -000012 | :000014 | -000016 | 000018 | °000021 | -000024 | °000027 -000030 | :000034 | 31 |
"000004 | 000005 | -000006 | -000007 | ‘000008 | -000009 | 000010 | -000012 | 000013 | ‘000015 | 22 |
“000002 | -000002 -000002 | -000003 | ‘000003 | 000004 | 000004 | *000005 | ‘000006 | ‘000006 3S |
| 000001 | ‘OO0001 | ‘000001 | -000001 | -000001 | -000001 | -000002 | ‘000002 -O00002 | 000003 of
D —- |; — a — | :000001 | ‘000001 | ‘000001 | ‘OO000I | ‘000001 | 35
141 | Ife | 14°38 4 | 145 146 et 14-8 | 14-9 15°0 “
| 4
0 |} *Q00001 | 000001 | 000001 , 000001 000001 | = — | —- —— a ei =. | © 5
1 | 000011 | -000010 | 000009 | -000008 — -000007 | 000007 | ‘000006 | ‘000006 | -000005 | *000005 1 :
2 | 000075 | ‘000069 | -O00063 » 000058 | *000053 | 000049 | -000045 000041 | ‘000088 | *0000384 | 2
3 | *0G0352 000325 | 000300 | ‘000277 :000256 | ‘000237 | 000219 | -000202 | ‘000186 | ‘000172 | 3
4 | 001239 -001153 -001073 | 000999 -000929 | -O00864 | 000803 -000747 -000694 | 000645 | 4
5 | 003494 -003275 -003070 | -002876 -002694 -002523 -002362 002211 | ‘002069 | °0019386 | 3 | g
6 | 008212 -007752 | ‘007316 | “006902 -006510 | ‘006139 | 005787 °005454 | 005138 | (004839; 6 ¢
7 | (016541 | -015726 | 014946 -014199 -013486 | -012804 | -012152. -011530 -010937 | 010370) 7 | ;
& | 029153 | 027913 | 026715 | 025559 | -024443 | -023367 | -022330 | 021331 | -020370 | 019444 | 8 2
9 | 045673 | 044040 | 042447 | -040894 | 039380 | ‘037907 | (086472 | "035078 | 033723 | ‘032407 9
10 | 064399 | -062537 | 060700 | -058887 057101 | 055343 | 053614 | -051915 050247 | 048611 10
11 | :082547 | 080730 | -078910 | 077089 | -075270 | 073456 | ‘071648 :069850 -068062 | ‘066287 | 17
\ |
7
Ds
er
H. E. Soper
TABLE—(continued).
m
35
Lyl | 142 Lyd | Lh 4 1d | 146 TH 148 19 15-0
s | = | Rae ae - ; a
§-096993 095530 | -094034 | -092507 -090951 089371 ‘087769 | 086148 | 084510 | 082859 12
105200 | -104349 | 103437 -102469 | -101446 | *100371 -099247 | 098076 | -096862 | 095607 13
105951 *105839 | 105654 -105396 | -105069 | -104672 *104209 103681 | 103089 | 102436 14
§ -099594 | -100195 | 100723 -101181 | *101567 101881 *102125 | -102298 | 102402 | -102436 15
‘087768 088923 | -090021 091063. -092045 092967 -093827 | -094626 095361 | 096034 16
072795 | -074277 | 075724 077135 | ‘078509 | -079842 | -081133 | 082380 | -083581 | 084736 17
(057023 *058596 | -060158 -061708 063243. ‘064761 -066259 067735 069187 | 070613 18
042317 043793 | 045277 -046768 | 048264 -049763 -051263 | 052762 | 054257 | 055747 19
029834 031093 | 032373 ‘033673 034992 -036327 037678 -039044 | 040422 | ‘041810 20
020031 | -021025 | 022045 -023090 | ‘024161 -025256 026375 027517 028680 | -029865 27 |
012838 -013570 | 014329 | 015114 -015924 -016761 -017623 | 018511 | 019424 | 020362 22
007870 008378 | ‘008909 | 009462 | 010039 ‘010640 -011264 011911 | -012584 | 013280 23
(004624 -004957 | 005308 | 005677 | 006065 006472 -006899 “007345 °007812 | 008300 24
002608 -002816 | 003036 | 003270 ‘003518 -003780 -004057 °004348 -004656 | 004980 25
001414 001538 | 001670 “001811 -001962 “002123 -002294 002475 | -002668 | -002873 | 26
“000739 ‘000809 | -000884 -000966. ‘001054 -001148 | -001249 “001357 001473 | -001596 27
000372 -000410 000452 | 000497 | 000546 000598 -000656 -000717 “000784 | 000855 28
‘000181 -000201 | -000223 | -000247 -000273 000301 -000332 000366 “000403 | 000442 29
‘000085 “000095 | “000106 “000118 | °000132 “000147 | -000163 -000181 | -000200 | 000221 40
000039 “000044 000049 | -000055 000062 | 000069 -000077 000086 -000096 | -000107 4
“000017 {000019 | 000022 | 000025 | -000028 | -000032 -000035 | -000040 | “000045 000050. 32
000007 -000008 | 000009 000011 | -000012 “000014 -000016 000018 -000020 | -000023 33
§ 000003 -000003 | -000004 -000005 -000005 | 000006 000007 000008 | 000009 | 000010 34
000001 “000001 | -000002 000002 000002 “000002 “000003 000003 000004 | 000004 35
— 900001 | 000001 -000001 | -000001 000001 -000001 | 000001 | «000002 | -000002 36
=e aa eae ap as ae = 000001 | -000001 ‘000001 37
ON THE POISSON LAW OF SMALL NUMBERS.
By LUCY WHITAKER, BSc.
PART I. THEORY AND APPLICATION TO CELL-FREQUENCIES.
(1) Introductory.
Let p denote the probability of the happening of a certain event A, and
q = 1-—p, the probability of its failure in one trial. Then it is well known that
the distribution of the frequencies of occurrence n, n — 1, n — 2... times in a series
N of n trials is given by the terms of the point binomial
VG @ eo a/) MMH rMn Raa hone yoAusdcbbooodote. (1).
The fitting of point-binomials plotted on an elementary base c to observed
frequency distributions has been discussed by Pearson*, and he has indicated that,
if c be unknown, the problem can be solved in terms of the three moment coefficients
Hy x, M4 required to find c, p and n. In actual practice but few cases of frequency
ean be found which are describable in terms of a point-binomial, and of these few
a considerable section have n negative, p greater than unity and q negative; thus
defying at present interpretation, however well they may serve as an analytical
expression of the frequency. .
The hypothesis made in deducing the binomial (p+ q)” as a description of
frequency is clearly that each trial shall be absolutely independent of those which
precede it. In this respect it may be said that binomial frequencies belong to the
teetotum class of chances, and not to those of card-drawings, when each drawing
is unreplaced. In the latter case the “contributory cause groups are not inde-
pendent,” and our series corresponds to the hypergeometrical rather than to the
binomial type of progression ft.
Using the customary notation 8, = p;°/"s°, Bo = M4/M, the binomial is determined
from :
n=2/{8—B.+B}, c=ov6—28,432,
Pq =$ 8B — B+ B,)/(6 — 28, + 38;)
* «Skew Variation in Homogeneous Material,” Phil. Trans. Vol. 186, A, p. 347, 1895.
+ Phil. Trans. Vol. 186, A, p. 381, 1895.
Lucy WHITAKER 37
In order that n should be positive, it is needful that
3—-B,+ 8, =4(6—28,+ 28),
should be positive. If this is satisfied clearly c will be real because 8, is always
positive. Further then
1 6— 28,4 28,
Ree) eS Goa eae)
is always less than a quarter and p and q will therefore be real. If the reader
will turn to Rhind’s diagram, Biometrika, Vol. vit. p. 131, he will see that the line
3—6.+ 8,=0 cuts off all curves of Types III, IV, V and VI, and includes a
portion only of Type I, with a part of its U and J varieties. The binomial
description of frequency, therefore, is not—considering our experience of frequency
distributions—likely to be of very universal application.
(2) Further Linutations.
Now let us still further limit our binomial by supposing :
(i) that the unit of grouping of the observed frequencies corresponds to the
actual binomial base unit c and (ii) that the first of the observed frequencies
corresponds to the term Np” of the binomial*.
In this case the mean m of the observed frequency measured from the first
term of the frequency will be equal to the nq of the binomial and the standard
deviation of the observed distribution will be equal to Vnpg. We have thus:
Doo KG — LC, TN —O) sc cn cda ete nscrwes: (111)
and n and q will both be negative, if m be less than o% The condition for a
positive binomial is therefore that o be less than 4/m.
(3) Probable errors of the constants of a Binonual Frequency.
It is desirable to find the probable errors of p and n as determined by these
formulae. We have:
pa = nq, M2 = MPs
buy’ = Gqdn+ nog, Sp. = pon + ngdp + npog,
assuming deviations may be represented by differentials.
Hence, since dp = — dq:
Ou. —(p — q) du, = Gon and pdm,’ — bu. = nqdq.
Square each of these results, sum for all samples and divide by the number of
samples, and we have:
Oe at qy Cie 2(p—-q) C5 Oi) City iy Gon
2 oot Vi ,=Nn20"%o,2.
Oe HD fay! 270 4% p, a Hy fg
* The exact nature of these limitations must be fully appreciated. The best fitting binomial to
a given frequency distribution will usually be far from one in which the first term of the binomial
corresponds to the first observed frequency. The modes of the binomial and the observed frequency
will closely correspond, but the ‘‘tails” of the binomial may be quite insignificant and correspond to no
observed frequencies.
38 On the Poisson Law of Small Numbers
Now (ai is the standard deviation of variations in ~. and therefore
oO", = (fa — pe’)/N.
Similarly oy! is the standard deviation of variations in the mean and therefore
Lo Heo sures t f 1 ween
o*,/=Ms/N. Lastly the product o,,0,/7,,,,' measures the correlation betwe
deviations in #, and yp,’ and is known to be p,/N*.
Thus we have:
1
eer ee Bo pay q)? fa = 2(p—@) pst,
wePay = + {a — Pa? + Pe — pps}.
Butt fy = npg {1 +3(n—2) pg}, |
een Arrs bh 00 (iv).
= mpg(p-G, He = npg |
Whence after some purely algebraical reductions we deduce:
n p ad
r= >= Pe el aN | erm Aan (bet |) | adodoccdcous00s '
° VNY (1- 4) - me A) aes 2(1- *) ©)
we pasl E noe Sp .
Op) = Oy vv 2 ae ip Cae aaa (vi).
Formulae (v) and (vi) are very important; they enable us to obtain the
probable errors for x and p when a binomial limited in the present manner is
fitted to a frequency distribution.
We see at once, that as n grows large and q grows small
0, =o, approaches the limit V2/N,
or the probable error, 67449 /2/N, of p and q is finite. But o® being finite op
becomes infinitely great, or the probable error of n indefinitely large. Thus when
the n of the binomial is very large, g being very small, the probable error of its
determination is so great that its actual value is not capable of being found
accurately. Again, suppose V embraced 200 observations, the probable error of q
would be of the order ‘07; if NV corresponded to only eighteen observations, then
the probable error of q would be of the order ‘22. It is clearly wholly impossible
* Biometrika, Vol. 11. ‘‘On the Probable errors of Frequency Consiants,” see p. 275 (iv), p. 276 (vii),
and p. 279 (xii).
_ + Phil. Trans. Vol. 186, A, p. 347, 1895.
+ There is no difficulty in obtaining the probable errors of n and p from the more general values
in (ii). In this case
t= 0No%g +o Be 208, 78,7B,p,»
Cp=oq= es Ve? ,$42B,"0°2, — 2B, og %R,7,2,°
The values of TB.» Fg, and "g,p, tor different values of 8, and B, have been tabled by Rhind, Biometrika,
Vol, vit. pp. 136—141.,
Lucy WHITAKER 39
from series of observations even of the order 200, much less of order 18, to assert
that g is or is not really a “small quantity.” Thus the observed value of q corre-
sponding to a population of extremely small q might easily show g=°15 to °50!.
(4) Poisson—Law of Small Numbers.
A last limitation of the point-binomial is made by supposing the mean m = nq
to remain finite, but g to be indefinitely small. We write :
We
N(p+ qr =N(Q-g4qr=N-gt (1+ 4)
me me
=N(1-q)? (14+)! nearly
Ne aa (1 +m So + ei + a) ;
Here the successive terms give the frequency of occurrence of 0, 1, 2, 3...
successes on the basis of each success not being prejudiced by what has previously
occurred. This is the Law of Small Numbers. It was first published by Poisson
in 1837*. It was adopted later by Bortkewitsch, who published a small treatise
expanding by illustrations Poisson’s work+. The same series was deduced later
by “Student” in ignorance of both Poisson and Bortkewitsch’s papers, when
dealing with the counts made with a haemacytometert.
The mean is at m from the first group, the other moments as “Student” has
shewn § are:
f,.=™M, pg=mM, py =3m?4+m.
Hence B,=1/m, B,.-—3=1/m.
When the mean value is large, 8,, 8, and the higher §’s approach the values
given by the Gaussian curve.
Clearly the Poisson-Exponential formula contains only the single constant
m = p, and its probable error is therefore °674490//N = 67449 m This will,
if V be reasonably large and m not too big, be a small or at any rate a finite
quantity (i.e. not like o, for g very small). Hence it might be supposed, although
erroneously, that the Poisson-Exponential formula was capable of great accuracy
in addition to its great simplicity. But this is to neglect the fundamental
assumptions on which it is based, namely:
(i) that the data actually correspond to a binomial,
(ii) that in that binomial g is small and n large.
Clearly (i) shows us that, if we can find the binomial, it will actually be closer
to the observed frequency than Poisson’s merely approximate formula.
* Recherches sur la Probabilité des Jugements. Paris, 1837, pp. 205 et seq.
+ Das Gesetz der kleinen Zahlen, Leipzig, 1898.
£ “On the Error of Counting with a Haemacytometer,” Biometrika, Vol. v, pp, 351—5, 1907.
§ They may be deduced at once from (iv).
40 On the Poisson Law of Small Numbers
Secondly (11) can only be justified as an assumption by actually ascertaining
the form of the binomial from the data and testing whether n is large and q small
and positive. It appears absurd to base our formula on an approximation to
a binomial of a particular kind when, on testing in the actual problem, such a
binomial does not describe the results. As a merely empirical formula, the
Poisson-Exponential of course can be tested by the usual processes for measuring
goodness of fit, but no such test nor any discussion of the probable errors of their
results have been provided by Bortkewitsch himself nor by Mortara, who has
followed recently his lines in a work to be considered Jater. Asa matter of fact in
the cases dealt with by Bortkewitsch, by Mortara and by “Student,” n will be found
almost as frequently small and negative as large and positive, and q takes a great
variety of values large and negative and large and positive, as well as small
and positive. Thus the initial assumptions made from which the “law of small
numbers” is deduced are by no means justified on the material to which it has so
far been applied.
(5) Application of the Law of Small Numbers to determine the Probable Errors
of Small Frequencies. Given a distribution of frequency for a population NW let ny
be the frequency in the cell of the sth row and éth column of a contingency table
(or if we drop t, 7, would stand for the frequency of any class). Then if we take
a random sample of WV individuals from this population, the chance that an indivi-
dual is taken out of the 7, cell is fis/|N, and that it is not is 1— 7: Therefore if
the original population be so large that the withdrawal of an individual does not
affect the next draw, the frequency of individuals in M random samples of WV will
be given by the terms of the binomial :
M {( 1") + ke
Now, if jig/N be very small, and WN large this will approximate to the
Poisson series :
Me (1 +m+ stat =) :
where m == x V. But 7y/N will approximately be the mean proportion of the
whole in the st cell of the sample itself =ny/N, or m= ny. Thus if in any cell of
a contingency table, or in any sub-class of a frequency whatsoever, we have a
frequency ny small as compared to the population V, then in sampling, this small
frequency will have a distribution approximating to the Poisson Law, and tending
as my, becomes larger to approach the Gaussian distribution*. It would appear,
* Such approach is usually asswned when we speak of
67449 Alf nae ( e “)
as the probable error of the frequency n,. But such a ‘probable error’’ has really no meaning if 7,
be very small and the exponentiai law be applied.
Lucy WHITAKER 41
therefore, that the Poisson Law of Small Numbers should be applied in order to
deal with the errors of random sampling in any small frequency, and an appeal
should not be made—as is usually the case—to Sheppard’s Tables on the assump-
tion that the frequency is Gaussian.
The following Table I illustrates the results obtained (a) from the Binomial,
(b) from the Poisson-Exponential and (c) from the normal curve on the two
hypotheses that (1) the frequency is 10 in the 1000 and (6) is 30 in the 1000.
But here a word must be said as to which Gaussian is to be compared with the
Binomial or the Poisson-Exponential. The usual method of fitting a Gaussian is
to give it the same mean and standard-deviation as the material to which we are
fitting it. For example, we should compare the Poisson exponential with a Gaussian
at mean m and with standard-deviation ym, or the point binomial with mean ng
TABLE IL.
Comparison of Binomial, Poisson-Exponential and Gaussian for cell-frequency
variations in samples for case of 10 and 30 in a total population of 1000
Percentage Frequency
|
10 in 1000 30 in 1000
; = | a a =|
Binomial | Reese Gaussian Binomial | feo Gaussian
0 00004. | = =-OU0005 ‘00132 19 | -00848 | ‘(00894 = ‘01100
1 ‘00044. (00045 =|) 00327 20 ‘01287 ‘01341 01553 CO
2 ‘00020 ‘00227 | ‘007385 Bil ‘01857 sO1OIG 9) <O2718" 9)
3 00739 ‘00757 ‘01491 22 ‘02556 02613 =| © :02792 =|
4 01861 | -01892 | -02736 28) ‘03362 08408 | *08544 |
5 | -03745 03783 | -04539 | 2, | -04233 | -04260 | -04373
6 ‘06274 ‘06306 ‘06806 25 05110 | 05112 =| «=-05198
or 7 “08999 “09080 ‘09224 26 05927 =| ‘05898 ‘05970
S 8 "11282 *11260 “11300 27 | ~ 06613 ‘06553 ‘06625
Es 9 “12561 Sb abet 28 ‘O7107 07021 | ‘07104
oo | 29 ‘07367 | ‘07263 =| ‘07360
oH |
8 |10| -12574 ‘12511 "12526
E | 30 | -07375 ‘07263 ‘07367
eS 11 114381 11374 | 11334
12 (09516 | °09478 | :09271 | 31 07137 ‘07029 ‘07126
13 ‘07305 ‘07291 ‘06854 | 32 “06684 06590 =| ~=-06659
14 "05202 705208 | 04580 | 33 ‘06064 05991 =| -06013
15 03454 | 03472 = :02°767 34 05334 05286 =| 05246
16 °02148 ‘02170 ‘O1511 Bo. 04553 04531 04423
wy *01256 ‘01276 ‘00746 36 ‘03775 ‘08776 =| *03602
18 ‘00693 ‘00709 00333 =| 37 ‘03042 ‘(03061 | *02835
19 ‘00362 ‘00373 00134 38 *02384 02417 *02156
20 ‘00179 | = ‘00187 “00049 389 ‘01819 ‘01859 01584
21 ‘00085 ‘00089 ‘00016 40 01351 ‘01394 ‘01125
22 ‘00038 00040 = 00005 41 ‘00979 (01020 ‘00771
ASS 00016 | -00018 ~~ -0001 42 ‘00691 ‘00729 “00511
Biometrika x 6
42 On the Poisson Law of Small Numbers
and standard-deviation “npg. These will, however, not be identical standard
deviations as p is not truly unity. In ordinary practice, in testing for example the
30 in 1000 frequency, we should put the centre of our Gaussian at our 30 group,
and use a standard deviation = V30 (1—30/1000) = /30 x ‘97 = 5°39444 to enter
the table of the probability integral. This is, of course, the Gaussian we obtain
by the method of least squares, but to assume that it is “the best” is to argue in
a circle, because we then take least squares as a test of what is best*. It is
not the Gaussian which is directly reached by proceeding either to a limit of the
Binomial or to the Exponential, for example, by applying Stirling’s Theorem. It
will be seen by examining Table II that the Gaussian curve develops out of the
exponential by a mode at the point midway between the two equal terms, rather
than by a mode at the mean, which coincides with the centre of the second of
them. If we apply Stirling’s Theorem to the term+
in
N La nr gr
|n —r |r eve
of the binomial NV (p+q)”" it becomes
No g-kr- mgt 4 (p- 9-9),
U, e
V 2a Vinpq
i.e. the ordinate of a Gaussian curve of Standard Deviation Vnpq and mean at
ng—4(p—q). These give for the Poisson-Exponential the Gaussian with standard-
deviation ,/m and mean m—- }. The above type of curve which gives frequencies
by coordinates and not by areas has been termed by Sheppard a ‘spurious curve
of frequency’; at the same time it is the method by which Laplace and Poisson
first reached the normal curve, and the real point at issue is whether we shall get
better approximations to the discontinuous frequencies of the binomials by using
Gaussian ordinates than by using the areas of a Gaussian curve. At the same
time it has been shewnt that if a Gaussian curve gives a series of frequencies by
its areas, then if its standard-deviation be o?, a spurious Gaussian frequency curve
with standard deviation given by o,?= o°? + +;h’, h being the sub-range, will closely
give the frequencies by its ordinates. It seems probable therefore that the
Gaussian curve with mean at ng—4(p—q) and standard deviation Vnpq— 5
will more closely represent the binomial for cell frequency variation by its areas,
* There is a further flaw in this treatment—the Gaussian is continuous, the Binomial and the
Poisson-Exponential are not. If t, be the rth term of either of the latter series, we ought really to
make
r+1-m N = a 9
So” [je- | ——eé Fash alee
r—m N2ro
a minimum by the conditions du/dm=du/dc=0. No complete solution of this problem has hitherto
been determined.
Die
+ The final form for wu, may be obtained by neglecting the terms in n in the formula given by
Pearson, Phil. Trans. Vol. 186, A, p. 347, footnote. i
+ Biometrika, Vol. ur. p. 311,
*
7
rz,*
et)
-
a
t - 7
2 “7 :
7 =
>
ic
a
. -
a
=
a
Frequencies {total 1000)
150
100
10)
Biometrika, Vol. X, Part I.
150
LO
oO
Catisstarn.
Plate VII
35
: Gatisstan.
ied
Frequencies (total 1000)
Biometrika, Vol. X, Part I. Plate VII
10\0- aes Mee
=|
1 4 —_
350, 150 | 1 aoe
— 5]O
Mp9 1
300 100;
|
250 Tegal 501
200; OG 5 10
De
1501
Caussian
4 \
“TIE 7.2/0 |
100) 100) J , we
| pa r]
| } | b 15 20 25 30 35
ete
| oak yp Gausstan.
50 5@ oF Pht = 30 i >
| 14 -
| |
Oa Daa ae 6 OF 8 DION MSD IGT BIOs
| 35 40
Ww
Lucy WHITAKER 43
than if we apply the ordinary process of mean ng, standard deviation Vnpq, and
Sheppard’s table for areas to the frequencies. It will be noted that this amounts
to using Sheppard’s correction on the crude second-moment and slightly shifting
the central ordinate towards the side of greater frequency. This is the Gaussian
curve used in Table I.
The object of the present section of our work is to indicate how far it is
legitimate to use the Poisson-Exponential up to cell frequencies of the order 30
in a population of about 1000* and how far we then reach a state of affairs, which
for practical purposes may be described by ordinary tables of the Gaussian. It
will be seen from Table I that the Poisson-Exponential even for ny =10 and 30 is
not extremely divergent from the Binomial.
In Plate VII the transition of the exponential histograms of frequency towards
the Gaussian form is indicated for cell-frequency = 1, 5, 10, 15, 20, 25 and 30; in
the cases of 10 and 30 the corresponding Gaussian curves are drawn.
It will be seen that with due caution the Poisson-Exponential may be reason-
ably used up to frequencies of about 30 in the 1000, and that after that it would
be fairly satisfactory to use the areas of the Gaussian curve as provided in the usual
tables.
(6) In order to table the results of the Poisson-Exponential for easy use, it
seemed desirable to turn them into percentages of excess and defect. For example
take the distribution for a frequency 5. It is:
Per cent. of Cases in which:
0 006,737,945 a defect of 5 occurs : 0674
1 033,689,725 3 4, oy more Pe: 4043
Dy, "084,224,310 > 3 or more vor: 12°465 ,
3 140,373,850 P 2 or more an 26°503
4, 175,467,310 as 1 or more ee 44-049
5 175,467,310 the true value ar 17°547
6 146,222,755 an excess of 1 or more ~. S 38°404
a 104,444,825 ms 2 or more . 23°782
8 065,278,015 - 3 or more ‘ 13337
9 036,265,564 * 4 or more s 6809
10 (018,132,782 P 5 or more *. 3183
Tt ‘008,242,178 _ 6 or more ' 1370
12 003,434,238 " 7 or more a 0°545
13 001,320,860 _ 8 or more . 0202
* Of course in the Poisson-Exponential itself the total frequency plays no part; it is only useful in
testing the validity of the approximation.
6—2
44 On the Poisson Law of Small Numbers
Thus we see that if the true value of the frequency be 5 for the average sample,
it will only lie outside the tange 1 to 10 in 674 + 1:370 = 2044 cases per cent., or
the odds are 49 to 1 that the value found will be from 1 to 10.
On the other hand it will lie outside the range 2 to 8 in 4043+ 6°809 =10°852 °/,
of cases, or once in about 9 trials the frequency will lie outside this range. Or,
again, once in about every four trials (25°8°/,) the result will fall outside the
range 3 to 7.
On the other hand if we write «= 5 (1 — 005) = 223047, we have — 4°5
and +55 as the deviations from a mean 5 of all beyond 0°5 and above 105,
giving w/o =—2:0175 and + 24658 respectively. These cut off tail areas of
02181 and ‘00684, respectively. Thus in 2°865—not 2°044—per cent. of cases
we should assert that the frequency would he outside the range 1 to 10, or the
odds that it would lie inside this range are now only about 34 to 1, not 49 to 1.
Calculated from the Gaussian the frequencies outside ranges 2 to 8 and 3 to 7
correspond to 10°1°/, and 26:2°/, of the trials instead of 10°9°/, and 25°8°/,. If
we take for the standard-deviation of our Gaussian Vnpgq — yy = 2°21171, we find
that the odds in the first case are still only 35 to 1, but the percentages in the
other two cases are 11°3 and 25'8.
It will be clear that near the centre of the curve—especially when we equalise
the excess and defect of the Gaussian by taking equal ranges on both sides—it
does not give bad percentages of frequency, but that it does not lend itself to
the accurate determination of the range for reasonable working odds such as
50 to 1.
It will be noted that the total area in excess and defect of 2 and more
= 23°782 + 26°503 = 50285, or corresponds very nearly to the “probable error.”
Actually the Gaussians with standard deviations of 2°23047 and 2:21171 give
probable errors of 1:504 and 1-492 respectively, so that the Gaussian with 1°5 as
the probable error is very nearly accurate.
Table II gives the Poisson-Exponential; it will enable the reader to appreciate
the range of probable variation in small frequencies. Thus we realise that in
37°/, of cases in which the true frequency is 1, the cell will be found empty ;
in 13°5 per cent. of cases it will be empty when the actual frequency is 2, and in
5 °/, of cases when the frequency is 8 and in 1°8 °/, when the frequency is 4. These
results indicate how rash it is to assume that a sample 4-fold table with one zero
quadrant signifies perfect dependence or association in the attributes of the
material sampled. The second line below gives the percentages of cases that 0
would appear in a cell when the actual number to be expected is that in the first
line calculated from Table IT on the usual theory of a priort probabilities :
Actual ih 1 4 5 6 | ren aes | 9 &over
116 | 0-43 0-16 | 0-06 0°02
| Percentage
63°21 eee || 8°55 | 3°15
0°01
Lucy WHITAKER 45
TABLE II.
Table of Poisson-Exponential for Cell Frequencies 1 to 30.
Cell Frequencies
Per cent occurrence of values differing by x or
Per cent. occurrence of values differing by « or more in excess
Fy 1 2 8 J 5 6 ¢3 8 9 10
22
al
20
19
18
a 17
elen 16
lp ae
= |
Soi
sia! 12 |
oo ii |
‘6 10 005
i 9 ae 012 ‘050
ce 8 034 123 yi |
5 if 091 302 623 | 1:033
S 6 248 7730) |) e tea7d |) 222123) |e 25925
5 ——=——| -674 | 1°735 -| 2:964 | 4:238 | 5-496 | 6:708 |
4 1°832 | 4:043 | 6-197 | 8-177 | 9-963 | 11°569 | 13-014
3B ———_| 4:979 | 9:158 | 12°465 | 15:120 | 17-299 | 19-124 | 20-678 | 22-022 |
2 |__| 13-534 | 19:915 | 23-810 | 26-503 | 28°506 | 30-071 | 31°337 | 32-390 | 33-282 |
1 | 36-788 | 40-601 | 42°319 | 43°347 | 44-049 | 44:568 | 44°971 | 45-296 | 45°565 | 45°793
Actual] 36°788 | 27-067 | 22°404 | 19°537 | 17-547 | 16-062 | 14°900 | 13-959 | 13°176 | 12°511
1 | 26:424 | 32°332 | 35-277 | 37:116 | 38-404 | 39°370 | 40-129 | 40-745 | 41-259 | 41°696
2 8030 | 14-288 | 18°474 | 21-487 | 23-782 | 25-602 | 27°091 | 28:°338 | 29-401 | 30°323
B 1°899 | 5-265 | 8:392 | 11-067 | 13°337 | 15-276 | 16-950 | 18°411 | 19°699 | 20°845
4 366 | 1°656 | 3°351 | 5113 | 6:809 | 8-392 | 9852 | 11°192 | 12-422 | 13-554
5 ‘059 453 | 1191 2°136 | 3:183 | 4:262 | 5:335 | 6°380 | 7°385 | 8°346
6 ‘008 ‘110 “380 813 | 1°370 | 2:009 | 2°700 | 3°418 | 4:146 | 4:875
i 001 024 ‘110 284 +545 883 | 1:281 1726 | 2°403 | 2°705
8 000, -005 029 092 | 202 363 “572 ‘823 | 1:110 | 1°428
9 =e ‘001 ‘007 027 ‘070 140 ‘Q4] 372 | 532 ‘719
10 = 000 002 008 | 023 ‘051 096 159 | 242 346
11 is = “O00 “002 ‘007 ‘O18 ‘036 ‘065 “105 “160
a | 12 as = : ‘001 002 006 013 025 | 044 | ‘071
E 13 — ; = _ “000,001 002 ‘005 “009 ‘O17 “030
Se eR hs = i000 ‘001 002 | 003.) 007 ‘O13
15 == weet) t= See S25 ‘000 ‘001 ‘001 | 002 ‘006
EB | 16 = ae a = 000 ‘000 ‘001 002
eel 17% = = — a = = - “000 001
18 _ | = = ‘001
19 = — — —_— | — — -—— a — “O00
20 = a = 224 (ies = = = = —
21 = = aa == hie = = a a —_
22 = = -_ = = = = = = =
23 = = _ = = _ = == = —
2 imate ee os — = os ee ee le = _
Ne = —_ = ao Se dig ee See =
26 — = — — = a= = = = oe
at | — —— == = =
46 On the Poisson Law of Small Numbers
TABLE I1—(continued).
Cell Frequencies
av 11 12 | 18 Uy 15 16 iif 18 19 20
| |
Ly ed 22 |
hes 21
= 20 a
a 19 sik eS
=p 18 —
a | 17 = — ‘000 “000
es | 16 | | — | »:000'| 000) | COM mmoue
ea | 15 | | 000 000 | ‘001 002 | 004 | ‘007
Be | Ly ‘000 | -001 002 | 004 | ‘008 | ‘015 | -026
eB ole ite ‘000 | 001 004 | 009 | 018 | 032] -052 | -078
eed He 001 003 | 009 | 021 040 | -067 | ‘104 | -151 209
8 || any 002 008 | 022 | -047.| -086 | +138 | -206 | ~-289"\") -3Sia/Nae500
es | 10 020 052 105 }_ 181 279 | 401 543 | -706 | 886 | 1-081
a | 129 121 229 | 374 | 553 | -763 | 1:000 | 1:60 | 17538 | 1-832 | 2-139
peril 8 ‘492 | -760 | 1-073 | 1:423 | 1:800 | 2-199 | 2-612 | 3-037 | 3-467 | 3-901
-£] 7 | 1510 | 2-034 | 2589 | 3-162 | 3-745 | 4-330] 4-912 | 5-489 | 6-056 | 6-613
ae 5 | 3-752 | 4582 | 5-403 | 6-206 | 6:985 | 7:740 | 8-467 | 9°167 | 9-840 | 10-486
is 5 | 7861 | 8-950 | 9-976 | 10-940 | 11-846 | 12-699 | 13°502 | 14-260 | 14-975 | 15-651
8 4 | 14319 | 15-503 | 16°581 | 17°568 | 18-475 | 19-312 | 20-087 | 20-808 | 21-479 | 22-107
2 3 | 23-198 | 24-239 | 25-168 | 26-004 | 26-761 | 27-451 | 28-084 | 28°665 | 29-203 | 29-703
2 2 | 34-051 | 34-723 | 35°317 | 35-846 | 36-322 | 36-753 | 37-146 | 37-505 | 37-836 | 38-142
1 | 45-989 | 46-150 | 46°31] | 46-445 | 46°565 | 46-674 | 46°774 | 46-865 | 46-948 | 47-026
| |
|Actual, 11-938 | 11-437 | 10-994 | 10°599 | 10-244 | 9-922 | 9-629 | 9°360 | 9-112 | 8:884
“A 1 | 42-073 | 42-404 | 42-695 | 42-956 | 43-191 | 43-404 | 43-597 | 43-776 | 43-939 | 44-091
8 2 | 31:130 | 31-846 | 32:486 | 33-064 | 33-588 | 34-066 | 34°503 | 34-909 | 35-283 | 35-630
z 3 | 21-871 | 22-798 | 23°639 | 24-408 | 25-114 | 25-765 | 26-367 | 26-928 | 27-451 | 27-939
= 4 | 14596 15-559 | 16-450 | 17-280 | 18-053 | 18-776 | 19-451 | 20-088 | 20-686 | 21-251
2 5 | 9-261 | 10°129 | 10-953 | 11°736 | 12°478 | 13-184 | 13-852 | 14-491 | 15-099 | 15-677
= 6 | 5593 | 6-297 | 6-983 | 7°650 | 8-297 | 8-923 | 9-526 | 10-111 | 10-675 | 11-219
s | v7 | 3219 | 3742 | 4266 | 4-791 | 5-311 | 5-825 | 6-399 | 6-826 | 7:313 | 7-789
|= | 8 | 1769 | 2-198 | 2-501 | 2°884 | 3-275 | 3-669 | 4-064 | 4-461 | 4-856 | 5-248
© 929 | 1160 | 1:407 | 1-671 ; 1°947 | 9-932 | 9-593 | 9:824 || 3107 \\iardae
| > 10 467 | -607 762 | . 933° | 1117 | 1-312 | 1-516 | 1732" |S kobaeeoaiep
foes ana) Ble 225 305 | 396 | 502 | 619 | -746 | -882'| 1-030 | 1:185 | 1:348
aa | 12 104 | +148 | = -201 261 331 ‘All ‘497 | +595 | -699 | - -809
58 | 13 047 | 069 | 097 | -131 172 | -219 | -272 | 333 | 400 | “478
Seca, 020} ‘031 046 | -063-| 086 |" 1140) “14d | 1625) eos mmmoes
ea | Is 008 ‘O14 ‘O21 ‘030 042 ‘057 074 ‘096 121 149
295 | 16 003 | -006 | 009 | -013 | 020 | 028 | 036 | 050 | 064 | -081
cali | ll 001 002 | 004 | 006 | ‘009 | 014] 017 025 | 033 | 042
i 18 001 000 | 002 | -002 | 004 | 006 | 008 | ‘O12 | “O17, |) =e-0z2
2 19 000 | 000-001 001 002 | 003 | 003 | -006 | 008] O11
8 20 ae ae 001 ‘000 | -001 ‘002 | 002 | 003 | -004 | -005
5 Of |, was = 000 | 000 | -000 | 001 001 | 002 | -002 | -003
E Bea = 000 | 000-001 001 | ‘001
So eee dine = = = ze = an = 000 | 000 | 001
° 24 = = = = = = = = = 000
= 25 at em | ees = = = = = = ==
8 26 = = ke i oi
R a7 _ — — — — — —
aw 28 Es at, ae ee Sn (ure — = = =
Lucy WHITAKER
TABLE Il—(continued).
Cell Frequencies
4
=
v 21 22 23 24 25 26 or 28 29 30
Al ,
° Op | ae es ee = = = = = = = ‘000
5 21 = ral -_ oe = =e ‘000 000 ‘000 ‘001
8 20 —_ 2 — — 000 000 ‘O01 001 ‘001 “002
| 19 ae aa -000 -000 ‘001 ‘001 002 | = -008 004 006
2p 18 -000 000 001 001 002 004 006-009 012 ‘O17
|-Ba | 17 ‘001 002 003 ‘005 008 ‘O11 016 | 023 ‘O31 ‘O41
some 16 003 ‘006 ‘010 O15 022 ‘031 043 | 056 073 092
| alee eee 012 020 -030 043 O59 ‘078 “102 “129 160 195
An Mew 039 058 ‘081 “109 142 “180 224 273 328 387 |
ES 1 ‘ll “150 ‘198 252 314 384 “460 543 632 727 |
Se | 19 277 355 “443 540 ‘647 762 884 | 1:014 1-151 1-293 |
pon lett “625 763 912 | 1:072 | 1°240 | 1-417 | 1-601 | 1°791 1987 | 2-187
een | 10 1-290 | 1°512 | 1°743 | 1:983 | 2:229 | 2-482 | 2-739 | 3:000 | 3-263 | 3:528
ze 9 2-455 | 2°778 | 3°107 | 3:440 | 3°775 | 4111 | 4:446 | 4°781 | 5-114 | 5-444
ee 8 4-336 | 4°769 | 5-200 | 5-626 | 6:048 | 6-463 | 6°872 | 7-274 | 7-669 | 8:057
zg Hi 7°157 | 7°689 | 8-208 | 8-713 | 9-204 | 9-682 | 10-147 | 10-599 | 11-038 | 11°465
38 6 | 11°107 | 11:704 | 12-277 | 12:827 | 13°358 | 13°867 | 14°357 | 14°830 | 15-285 | 15°724
aa 5 | 16292 | 16-900 | 17-477 | 18-025 | 18°549 19-048 | 19°525 | 19°981 | 20-417 | 20-836
= 4 | 22°696 | 23-250 | 23°771 | 24:263 | 24°730 | 25-172 | 25°591 | 25-990 | 26-371 | 26-734
© 3 | 30-168 | 30:603 | 31:010 | 31°391 | 31°753 | 32-094 | 32-416 | 32-721 | 33-011 | 33-287
2 2 | 38-426 | 38-691 | 38:938 | 39°168 | 39-387 | 39°593 | 39°786 | 39:970 | 40143 | 40-308
1 | 47-097 | 47:164 | 47-226 | 47-283 | 47-340 | 47-392 | 47-440 | 47-486 | 47-530 | 47-572
Actual] 8°671 | 8:473 | 8-288 | 8-115 | 7:°952 | 7:799 | 7°654 | 7-517 7°387 | 7:264
2 1 | 44-232 | 44:363 | 44-485 | 44-603 | 44°708 | 44°810 | 44:906 | 44:997 -| 45-083 | 45°165
8 2 | 35-955 | 36-258 | 36-542 | 36-812 | 37-062 | 37-299 | 37-525 | 37-739 | 37-942 | 38-135
2 3 | 28-397 | 28:828 | 29-235 | 29-620 | 29-982 | 30:326 | 30°653 | 30-965 | 31-262 | 31-546
5 4 | 21-785 | 22-290 | 22-770 | 23-227 | 23-660 | 24-074 | 24:469 | 24°847 | 25-208 | 25-555
a 5 | 16-230 | 16°758 | 17-264 | 17-748 | 18-211 | 18-655 | 19-083 | 19-493 | 19-888 | 20-269
# 6 | 11-744 | 12-251 | 12-740 | 13-213 | 13-669 | 14:110 | 14°538 | 14°951 | 15-351 | 15°738
a ” 8254 | 8709 | 9°153 | 9:585 | 10-007 | 10-418 | 10°819 | 11°210 | 11-591 11-962 |
B 8 5-637 | 6:022 | 6-402 | 6-777 | 7:146 | 7:509 | 7:866 | 8-218 | 8°562 | 8-901
& 9 3°742 | 4:052 | 4:362 | 4:670 | 4:978 | 5-284 | 5°588 | 5-890 | 6-188 | 6-484 |
ES 10 2-415 | 2-654 | 2°895 | 3:188 | 3°385 | 3-632 | 3°880 | 4:129 | 4:377. 4°625
a 11 1517 | 1:692 | 1°873 | 2-057 | 2-246 | 2-438 | 2-633 | 2°831 | 3-030 | 3-230 |
tai | 812 927 | 1°051 | 1°18] 1315 | 1°456 | 1:599 | 1:°747 | 1:899 | 2-053 | 2-210
isso 13 “552 637 727 “821 92] 1025 | 1°134 | 1:247 | 1°362 | 1:481
HS 14 320 376 437 “500 570 643 720 ‘801 885 | 973
4 15 ‘181 217 256 298 “345 394 ‘448 “504 564 626
2 | 16 ‘100 -122 147 ‘173 204. 237 272 311 352 395
‘ee | 17 054 -067 082 ‘098 ‘118 139 162 “188 ‘215 | +245
ig 18 -028 036 ‘045 ‘055 ‘067 ‘080 095 ‘111 129 | +149
° 19 O15 ‘019 024 -030 ‘037 ‘045 054. ‘065 076 | 089
S) 20 007 010 013 016 ‘020 ‘025 ‘03 ‘037 044 052
B 21 004 -005 ‘007 009 ‘Oll | 014 ‘O17 021 025 ‘030
= 22 002 ‘002 004 ‘005 006-007 009 ‘O11 O14 ‘O17
8 23 ‘001 ‘001 002 ‘003 003-004 ‘005 006 ‘007 ‘010
S Qh 001 001 ‘001 002 002 002 003 003 004 006
= 25 ‘000 000 001 ‘001 ‘001 001 ‘001 002 002 003
8 26 =s ee -000 ‘000 000 ‘001 001 ‘001 ‘O01 002 —
x or es = : ‘000 -000 000 -000 ‘001
a 28 Bs = = -_ _ : |
“O00
48 On the Poisson Law of Small Numbers
PART II. CRITICISMS OF PREVIOUS APPLICATIONS OF
POISSON’S LAW OF SMALL NUMBERS.
(7) We now turn to the illustrations which various authors have given of
the Law of Small Numbers.
“Student's” Cases. We take first the series given by “Student” in his memoir
on counting with a Haemacytometer*. They are of special importance because
the series at first appear of fairly adequate size, namely consisting of 400
individuals, and further we should anticipate that the Law of Small Numbers
would hold in his cases. He obtains better fits with the binomial than with the
exponential but, as he remarks, he has one more constant at his disposal. On the
other hand, if the exponential be a true approximation, the binomial ought to come
out with a large n and a small but positive g. “Student” finds for his four
series :
L400 x (111893 — 1893)-2™,
Il. 400 x (97051 + 02949 624,
III. 400 x (1:0889 — -0889)-2"™,
IV. 400 x (9525 + 0475989",
{I. and IV. may, perhaps, be held fairly to satisfy the conditions, although it
is not certain if 46 is to be considered a large n or ‘05 a very small q.
I. and III. fail to satisfy the conditions at all, unless the probable errors of q¢
and n are such that g might really be a small positive quantity and n really large
and positive. The following are the values for the four series of n and q and their
probable errors :
I g=— 1893 +0647, n=— 3°6054+4 1:2209.
Il. qg=+°0295 +0457, n= 46°2084 + 71°7373.
III. g=— 0889 + 0534, nm = — 202473 + 12°1165.
IV. qg=+ 0475 +0452, n= 985263 + 93°7494.
Now while these results are very satisfactory for II. and IV., they are not
wholly conclusive for I. and III. We can approach the matter from another
standpoint; the probable error of g for p=1 is
il .
67449 Ta V2 = 67449 x 0707
in “Student’s” cases. Thus the deviation of q from q a very small quantity is for
I. 2°68 times the 8. D., and for III. 1:26 times the 8. D. Since g may be either
positive or negative, we may reasonably apply the probability tables and the odds
against deviations occurring as great as these are in one trial about 250 to 1 and
9 to 1 respectively. Hence in four trials we should still have large odds against
their combined appearance.
* Biometrika, Vol. v. p. 356.
Lucy WHITAKER 49
We have said that the results for Il. and IV. are fairly satisfactory, ie. we
mean that they are consistent with g being small and positive and n being large ;
but of course they are also consistent with g being negative and n being small and
negative.
It will be obvious from these results for “Student’s” data that it is extremely
difficult to test the legitimacy of the bypothesis on which the “Law of Small
Numbers” is based. In none of the cases dealt with by Bortkewitsch, much less
in those dealt with by Mortara, are the populations (V) anything like as extensive
as those considered by “Student.” But populations of even 400 give, as we see, too
large values of the probable errors of g and w for us to be certain of our conclusions.
(8) Bortkewitsch’s Cases. Taking Bortkewitsch next, he deals with the
following cases :
I. Suicides of Children in Prussia for 25 years: (a) Boys, (b) Girls, 25 cases.
II. Suicides of Women in eight German States for 14 years: 112 cases or
8 subseries of 14.
III. Accidental Deaths in 11 Trade Societics in 9 years: 99 cases, or 11 sub-
series of 9,
IV. Deaths from the Kick of a Horse in 14 Prussian Army Corps for 20 years:
280, or, as Bortkewitsch, 200 cases.
It will be noted at once that Bortkewitsch’s populations (1) are far too small
for any effective determination of the legitimacy of his application of Poisson’s
formula to his data.
We take his cases in order:
I. (a) Suicides of Boys.
TABLE IIL.
| Number of Suicides _... ON LZ AS We 5 6" | 7andiover
| | |
| -——-—-— cae at Se }
| Number of Years So 4 | 8 | Des) 4s 108 1 | 0
The binomial is:
25 [1:2033 — -2033]-°™,
Mean 1:9600 and yw, = 3:2584.
We have g= — 2033 +°2421, n= — 96425 + 109416.
If y were really zero its probable error would be +1908. Clearly 25 cases are
wholly inadequate to test the legitimacy of applying the Poisson-Exponential to the
frequency*. But to what extent is the reader made conscious by Bortkewitsch
that his cases fail entirely to demonstrate the legitimacy of applying his hypotheses ?
* The x2 for the binomial is 2°379 and for the exponential 2°836, showing a somewhat better
result for the binomial.
Biometrika x 7
50 On the Poisson Law of Small Numbers
I. (6) Suicides of Girls.
TABLE IV. E
Rees ae | |
Number of Suicides... 0 vi | 2 3 |
| — ———
Number of Years a 15) | oan | 0)
The binomial is: :
25 ['7418 + °2582 7,
Mean = 4400 and yu, = 3264.
We find g ='2582 +1012, n=1°7041 +°7850.
As in the case of the boys’ suicides, if g were practically zero its probable error
would be + ‘1908, and there is nothing in this result again to justify us in asserting
that q is indefinitely small and n indefinitely large.
Actually we have:
TABLE V.
Number of Suicides per Year.
0 1 aye mans
x 7 |
Actual ... ee 15 9 1 ure
Bortkewitsch 16-1 71 [cg | ees
Binomial (a)... 15:0 8°9 11 —
Binomial (0) 15:2 8°7 11 —
(a) is the binomial considered above, (b) is the binomial obtained by taking
n a whole number = 2, and g= mean/2 = ‘22, we. 25 (78 + °22).
It is clear that either (a) or (b) gives better results than the Poisson-Expo-
nential. Applying the test of goodness to fit, we have
x? = ‘007 for the binomial (qa),
x? = 610 for Bortkewitsch’s solution.
Both give P > ‘60 but the first is much better than the second.
If both boys and girls are taken together, we find the binomial
25 (9333 + (0667).
This is the nearest approach to a small q and big 7 we have so far found—ze. the
nearest approach so far to an exponential, but it is reached by a process, «.e. that of
adding together two series of entirely different means and variabilities in a manner
which cannot be justified, for Bortkewitsch’s hypothesis depends essentially on the
homogeneity of his material. Even here the fit of the point binomial is slightly ;
better than that of the exponential.
Lucy WHITAKER 51
II. Suicides of Women in hight German States. Bortkewitsch gives the
following table :
TABLE VI.
Number of Suicides of Women per Year
State - - Totals
Ore iee ee eel Sachse he Sl o.8| Fo
| |
| (a) Schaumbureg- tnppe: 4) 4) 2) 4)/—]--|]— | 14
(b) Waldeck... Mee huer dl 8 A he | 14
(ce) Liitbeck ee ves elle i geale cele a eee os 14
(d) Reuss a. L. . ce soni | oil | Sl soulmate 2a Ue Pe ms 14
_(e) Lippe PR Gy OU) BE ay a | | 14
(f) Schwarzburg- Rudolstadt ... | — ; 1}—}; 2)/—) 56]; 8) 2) 1)—)— 14
(g) Mecklenbure- Strelitz soe |e MBL) PE ee ea eT So eee eae 14
| (A) Schwarzburg-Sonderhausen SN a ed Sc he 14
Totals 112
The resulting binomials are :
(a) 14( 9714 + 0286)",
(b) 14( ‘8571 +°1429)9%,
(c) 14( 5819 +4181),
(d) 14 (1:0058 — -0058)-##24,
(e) 14(1°3929 — -3929)-77,
(f) 14( 6071 + :3929)3%,
(g) 14(1°5792 — 5792)-91"7,
(h) 14 (16609 — -6609)-3,
Thus it will be seen that of the eight binomials only four have a positive q,
and of these only one can be said to have a very smajJl g, and even in this case the
n is not indefinitely large. Of the four negative binomials three have quite
substantial q’s, and the fourth with its small negative qg corresponds most closely
to the Poisson-Exponential. The probable error of g for g=0 is +:2549. The
number, 14, of cases taken is therefore wholly inadequate to test whether the
Poisson-Exponential may be applied to these data. The mean value of q is
negative and = — ‘0820 + ‘0901, and the standard deviation of g=:3928 + -0637,
which are within the limits of random sampling of g =0 with a standard deviation
of 3779. We shall return to a different manner of considering the point later.
At present we wish only to indicate that the hypothesis is that q is a very small
positive quantity and that data which give ga standard deviation of ‘3928, or in
the next example of 4714 are really inadequate to test such a hypothesis ; for in
the resulting binomials g may easily lie anywhere between +°8 and —°8, and it
is not possible to demonstrate that its real value is practically an exceeding small
positive quantity.
7—2
52 On the Poisson Law of Small Numbers
III. Accidental Deaths in 11 Trade Societies. Bortkewitsch provides data
from which the following table is deduced:
TABLE VII.
Accidental Deaths
Index Number
of Society Totals
1 ] | ;
0 BNO NG We | eae | 10 al
13 S Syl 9
1h ; 2 3 9
12 fq 3 9
20 =a a 9
23 ei 9
QF AG pas 9
29 Ne ie 9
Al Te hee 9
40 1 2 9
42 }— | — 9
55 — 2 9
Totals ... | 16 | 7
The resulting binomials are:
(18) 9( 4914+ -5086)5°8,
(14) 9( 61844 -3816)"%,
(12) 9:(1:9227 = 9227), 27,
(20) 9 (11282 — +1282)-s2"e00,
(23) “9° 9921 2.0079) eenes
(27) 9( 52294-4771),
(29) 9 (14130 — -4130)72™,
(41) 9( 8454 + +1546)9°66,
(40)° 9 (2:0342 — 1:0842)-27™4,
(42) 9( 9822+ -0678)72”,
(55) 9( 6154+ °3846)n2,
Of these eleven binomials seven have a positive g; only one of these (23)
actually corresponds to a really small q and large n, although a second, (42),
approximates to this condition. In the five other cases the q’s are quite sub-
stantial; in (13) the q is larger than p. Of the four negative q’s none can be said
to be so small and the » so large as to suggest that they really correspond to the
Poisson-Exponential. The probable error of q for q=0 is, however, + ‘3180, and
thus for such small series, no test whatever can be really reached of the legitimacy of
applying the Poisson-Exponential to such data. We may note, indeed, that seven
of the eleven values of g exceed the probable error and two of these are more than
three times the probable error. We should only expect two negative values of ¢
as great or greater than ‘9227 in 80 trials, whereas two have occurred in 9 trials,
Lucy WHITAKER 53
so that the odds are considerably against such an experience.
g is — 0469 +0959 and the standard deviation of g is ‘5127 + ‘0678, both results
compatible with g indefinitely small and a standard deviation = “4714. The main
problem, however, of the legitimacy of applying the Poisson-Exponential to such
series cannot be answered by data involving only total frequencies of 9 to 14
cases in the individual series.
He clubs the
results given for each application of the Poisson-Exponential together and
examines the observed totals against the sums of the calculated totals. Thus
calculating the 11 Poisson-Exponential series* and adding them together
Bortkewitsch finds for observed and calculated deaths:
TABLE VIII.
Accidental Deaths in 11 Trade-Societies.
Bortkewitsch examines the matter from another standpoint.
The mean value of
Number of Deaths
Observed Frequencies
Sums of 11 Exponentials
Single Binomial .. | 3°8 | 95
0
10 | 11 | 22
oe O) Te 13 4: 16 7
3-7 15°2 | 14-3 | 12°3 | 9°8
|
20|12)o7| 0-7
13°9 | 15°6
14-8 | 124 9°6
138 & over} Totals
If we attempt to fit a single binomial to the observed line of totals, we obtain:
m= 43636, o2=7'°5849
leading to the negative binomial :
O97 382 — 7802). 7s,
g=— 1382, + 18297,- n=— 59111 + 1391,
or the constants are significantly substantial with regard to their probable errors.
The resulting frequencies are given in the last line of the table above. The reader
Here:
* The values of the means and standard deviations for the eleven societies are :
m | o m o m o
13 7889 1:969 23 6°222 2-485 || 40 2°889 2°424
14 2-556 1:343 27 1'889 0-994 || 42 | 4-556 2-061
12 2556 | 2217 29 5889 2-885 || 55 4°333 1°633
20 4:333 | 2-211 41 5111 2079 || |
All these means are less than 10, which is the limit reached by Bortkewitsch’s Tables for the Poisson-
Exponential. Bortkewitsch says he has taken the societies for which ‘the statistics indicated the
smallest numbers of such accidents.’’ This is not very clear. It is certain that a society with a mean
number of accidents =100, if it consisted of 200,000 members, would be more suitable for application
of the exponential, than one with a mean of 8 if it only contained 10,000 members. Both Bortkewitsch
and Mortara confine their results to means less than 10, and seem to indicate that ‘‘ smallness” has
been determined by the absolute frequencies, but clearly it is relative frequency with which we have to
deal. The use of such a term as Das Gesetz der kleinen Zahlen for the Poisson-Exponential seems open
to serious objection, if it be associated with ‘‘m” an absolutely small number, and not with smallness
of ‘¢q.”
+ For q=0, the probable error would be +°0959 and accordingly q is very divergent from the
Poisson-Exponential value of zero.
54 On the Poisson Law of Small Numbers
will be surprised to see how closely the single negative binomial determined by
two constants gives the same result as the sum of the eleven Poisson-Exponentials
determined by eleven constants, no one of which is really of any significance for its
own exponential*. If we apply the condition for “goodness of fit,” »?= 5°83 for
the single binomial and y?= 5°88 for the sum of the eleven Poisson exponentials,
leading to P='950 and P= ‘951 respectively, or the fit with a-single negative
binomial is slightly better than that with eleven exponentials. The two constants
are significant, the eleven constants have no real significance for their individual
series, as is demonstrated by the fact that the binomials for these series do not
approximate to the Poisson-Exponential type.
We may now consider the previous case of suicides of women from the same
standpointt. The following are the data as given by Bortkewitsch :
TABLE IX.
Suicides of Women in Hight German States.
| Number of Suicides 0 | 1 | 2 | Beall oA | 8 | 9 | 10 & over Totals
Or
fo)
Se
| Observed Frequencies 9? S198 O05 115 Uae ee eee ate 3 112
Sum of 8 Exponentials | 8°0 | 16°9 | 20°3) 18°7 | 15-1 | 11°4 | 8°3 | 5°6 | 3°6 | 21 | 2°0 112
| |
|
Single Binomial ... | 12°6 | 18:4 | 18°8 | 16°4 | 13-2 | 9°9 | 7:2 | 5-1 | 35 | 24 | 4°5 112
i |
For the single binomial we have :
m = 3'°4732, o7? =8:2312,
leading to: 112 (2:3699 — 1°3699)- 25354,
where q=— 13699 +°1490, n= — 25354 + 8076.
If q were very small its probable error would be +0901. The values of g and n
are quite significant, g is large and negative and n is small and negative. The
resulting frequencies are given in the last line of the table as “Single Binomial.”
Turning now to the test of “goodness of fit,” we have for the sum of the 8 ex-
ponentials y?= 7:957, and for the single binomial y?= 7°740, leading to P= 633
* If the reader will turn to the first footnote on p. 53 he will note that for nine cases, the standard
deviations of the means (o//9) are roughly about -7 or errors of +1 to +1:5 may easily occur in the
means. Hence with the possible exception of (13) and (27) the m’s have not significant differences, and
are not typical of the individual societies.
+ The values of the means and standard deviations are: :
| m o | m o |
Schaumburg-Lippe | 1:°429 1-178 Lippe Are ag on 2°857 1-995
Waldeck eligi) poco 1:378 | Schwarzburg-Rudolstadt ... 5143 1:767
Liibeck ... ais 2-571 1:223 | Mecklenburg-Strelitz or 5°286 2°889
Reuss a. L. we =| 2648 1:631 Schwarzburg-Sonderhausen | 5°642 3-061
The standard deviation of the mean is here o/V14, or, say, 5. Thus errors of 1 might easily occur
in the values of m. There are probably significant differences between the first five and the last three
states, but not between the first five among themselves or the last three among themselves. Thus the
Poisson-Exponentials, if correct in theory, are not significant for the individual states,
Lucy WHITAKER 55
and ‘654 respectively. Thus again the single binomial with only two constants
give a fit slightly better, than the sum of eight exponentials with eight constants.
Bortkewitsch looking at the observed frequencies and the sum of 8 or 11
exponentials—without using any satisfactory test for “goodness of fit ”—assumes
that the coincidence is so good as to justify his hypothesis. But a better fit can
be obtained with two instead of 8 or 11 constants by simply using a negative
binomial. We must note here that Bortkewitsch is using the final coincidence
merely as justification of the Poisson-Exponential; the total frequency is not
describable in terms of the 8 or 11 constants as it is in terms of the two, for
these eight constants are not really significant for his individual eleven trade
societies or for the suicides in the individual eight states. If he wants to describe
the total, he has no constants by which he can do it. If, on the other hand, he
wishes to describe what has occurred in the individual societies or states, we have
seen that their binomials differ very widely from Poisson-Exponentials. If, lastly,
no stress be laid on the individual cases as having too large probable errors, but
only on the general coincidence with total frequencies, then the same coincidence
would justify us in using a single binomial with two constants only*. It appears
to us that to properly test the Poisson-Exponential, we need not 9 or 14 instances
in the individual case, but several hundred instances,—more, indeed, than “Student”
has taken—and that no proof of the “Law of Small Numbers” can be obtained
on data such as those of Bortkewitsch or Mortara.
IV. Deaths from the Kick of a Horse in Prussian Army Corps, omitting four
Corps with Bortkewitsch.
Here the results are:
TABLE X.,
| Number of Deaths ... 0 1 Z 3 | 4 Totals
Number of Corps —... 109 65 22 3 1 | 200
Whence :
m='61, p.='6079
and the binomial is:
200 (996,557 + 003,443 771707,
This is the first of Bortkewitsch’s illustrations for which his hypothesis that q is
small and n large is really justified by his data. For:
q = 0034 + 0670,
n= 1771711 + 3449:108.
The probable error of g for q really zero is + ‘0674.
* Of course immensely better general total fits are obtained by using the sums of the actual 8 or 11
binomials than by the Poisson-Exponential sum or the single binomial, but the results in that case
involve 16 or 22 non-significant constants.
56 On the Poisson Law of Small Numbers
The actual results as given by the binomial and the Poisson-Exponential are:
TABLE XI.
| i
Number of Deaths ... | 0 eae ae 2 3 4, and over |
= we |I— | | | |
Observed Hae as 109° =4)), Ob" 22 3 1
Binomial a ax 108°6 | 66°4 | 20°2 Ail 0-7
Exponential ... “3 108°7 | 66°3 20°22 | 4:1 O'7 |
Actually if we work to two decimal places in the frequencies we have y? = ‘61
for both binomial and exponential, or the goodness of fit is practically identical.
In this case it seemed worth discussing the binomial fit more at length.
Taking the moment coefficients about the mean we have:
(1) Mean =ng="6100.
(11) bs = npg = 6079.
(111) Hs = npg (p — g) = 590,562.
(iv) fy = npg (1 + 8npgq — Opq) = 1:°643,373.
We have already discussed the binomial from (i) and (ii), giving x” for goodness
of fit ="6096. Using (11) and (111) we have for the binomial
200 (985,739 + 014,261),
giving x? = 665.
Using (111) and (iv) we have:
200 (979,524 + °020,057 303,
giving x? = 707.
Putting : B.= Me | pe? and Bi = M3/ ps?,
we have: B,—-3 =(1—6pq)/npq, Bi. = —4pq)/npy,
and working from 8, and £, we find:
200 (969,150 + :030,850)89™,
and in this case x? = 1:1286.
This of course does not give a bad fit, but it is clear that working from the
lowest moment coefficients, as we might anticipate, gives the best results.
But if q be the chance of death from the kick of a horse, and n the number of
men in an army corps, then the binomial should be
200 (p +q)”
Now it is obvious that none of the binomials give, by their value of n any
approach to the real number of men in an army corps. If we start with the
Lucy WHITAKER 57
number of men 7 in an army corps as 50,000*, we have ng ="61 and g=:000,0122,
thus reaching the binomial
200 (-999,9878 + 0000122),
giving as compared against Bortkewitsch :
Binomial Bortkewitsch
0 108°6876 108°6703
1 66°3002 66°2889
2 20°2213 20°2181
3 41115 41110
4 and over °*7035 ‘7034
and y? = 608,298 608,318
or, the slight advantage to the binomial exists but is of no significance.
Now it seems to us that in this case the use of the exponential is justified for
the total frequencies, but as far as describing those frequencies is concerned, it
gives no better result than the binomial. But as in the other five of Bortke-
witsch’s cases the Exponential is not justified by the individual series themselves.
It is perfectly true that the exponential has a definite theory behind it, and
is interpretable in terms of that theory, i.e. we must suppose the probability of an
occurrence very small and the chance of its repetition absolutely identical. But
is the second of these conditions ever likely to be demonstrable a priori, or must
* This supposes that every man in the army corps is equally liable to death from the kick of
a horse; of course a very arbitrary assumption.
+ To illustrate the idleness of the application of the Poisson-Exponential even to these data for the
Prussian Army Corps, we give here the binomials for the whole of the 14 corps.
Index Number
of Corps Binomial
G 20 (-95 + -05)16-0000
: 20 (1325 — +325)-2-4615
ul 20 (1:5667 — 5667) —1-0585
Tl 20 (-9 + +1)6-0000
IV 20 (-6 + -4)1-0000
W 20 (-6318 + +3682) 1-4938
Vi 20 (1:0912 — :0912)—9-3202
VII 20 (-9 + +1)6-0000
VI 20 (-65 + +35) 1-000
Ix 20 (‘8115 + *1885)3-4483
x 20 (1°05 — -05)-15-0000
XI 20 (1-11 — -11)-11-3036
XIV 20 (1:05 — -05)—24-0000
XV 20 (1:1 — *1)—4-0000
One seeks in vain through these binomials for any approach to q very small and positive and very
large and positive. In no case does n approach the number of men in an army corps, say 50,000,
or q equal the chance of a death from the kick of a horse, say, ‘0000122! It seems impossible by
clubbing such equations together to give any satisfactory proof that the Poisson-Exponential really does
apply to individual cases. In the 20 years involved, there were doubtless great changes in both
the training and the personnel of each army corps, and the results obtained may be just as much due to
such causes as to the errors of small samples.
Biometrika x
58 On the Poisson Law of Small Numbers
not we a posteriori demonstrate it from the data themselves? Child suicide may
be influenced by example, by environmental conditions in different districts,
possibly even by meteorological conditions in different years. Again, even in
different army corps the conditions may be far from uniform, the spirit of the
corps, the teaching with regard to the handling of horses, the experience of past
life according to whether the corps is raised in town or rural districts may all tell.
Even Bortkewitsch before he gets his best fit removes four corps or 80 observations
from his data. We do not criticise this removal, but even unremoved he says the
fit of theory with experience leaves “wie man sieht, nichts zu wiinschen iibrig”
(p. 25). But the binomial is before removal:
280 (1:085,714 — 085,714)-8 155
in which q is not very small and is negative, and n is not very large and is not
positive. It is true that the probable error of g for q insignificant is in this case
+0570, but this only shows that the data were insufficient in quantity to
determine whether the exponential could be applied or not.
(9) Mortara’s Cases.
Mortara* in an interesting paper has realised the possibility of repetitions not
being independent and has discussed a constant @’, by which he proposes to test
such influence. This quantity Q should be unity, if the Bortkewitschian hypo-
thesis can be applied. He then takes 16 or 17 districts with records of 10 years,
and calculates the mean number of deaths from some special cause per year, say,
for each district for those years. If this mean number exceeds 10, he casts out
that district, presumably on the ground either (1) that such a number is no
longer small, or (ii) that it differentiates the district from those with lower
numbers. Thus Bologna with 10°9 deaths by murder is excluded and Bergamo
with 84 is included, although Q’=1 for both. Bologna with 7:1 deaths from
smallpox is included, but Pavia with 12°3 is excluded although the Q’ of the
former is 2°5 and that of the latter 1:7. What method should be employed in
dealing with the frequency of the excluded districts which may amount to 50 °/,
of all districts is not discussed. Having thus reduced his available districts,
Mortara proceeds to apply the exponential to each individual district ; he adds up
the results for each district and compares his totals with the observed totals. It
will thus be observed that he fits his exponential to ten observations, and then adds
together five or more districts to get his totals. We can equally well apply this
process by fitting a binomial to each 10 observations and then adding up such
results. But it is quite clear that on the basis of ten observations, it is, owing to the
large probable errors, wholly impossible to assert, whether a binomial of the kind
required by the Bortkewitsch-Mortara hypothesis,—i.e. one of very small positive q
and very large positive n—really is justified. We can illustrate this at once from
Mortara’s Tables (see his pp. 42 and 45) for deaths from Chronic Alcoholism. The
* « Sulle variazioni di frequenza di aleuni fenomeni demografici rari,” Annali di Statistica, Serie v.
Vol. 1v. pp. 5—81. Roma, 1912.
Lucy WHITAKER 59
observed numbers, and those deduced from the binomials are given in the
accompanying table. At the foot are the observed totals, Mortara’s exponential
totals and the binomial totals.
TABLE XII. Deaths from Chronic Alcoholism.
Oe Cason aber) |) ling | 9 | io | ar | a2 las |e |
| | | |
| | | i. i st | |
Calabria 1 ® 4 — 2 = |) = | | — | — | Observed
1:49] 2°84) 2°70) 1°71 81]; -31}) -10] -03)| -01 = | —- | Mortara
118} 2°85} 3:06} 1:91 "76 | °20} -03 | - | = | | Binomial
Foggia 1 2 4 — | 2 oS || | | | O.
1:00} 2°30] 2°65] 2:03) 1:17 54 | “21 07 | °02] -O1 | M.
96) 2°29) 2°70 | 20S eelaliSae <b3 19-06 Ol; — | B.
Siracusa 2 1 3 — 2 2) | - —;/;—|— | 0.
*82| 2°05] 2°56] 2°14 34 67 | 23) “10 | -03)| “Ol — — |/—}]}—|— |M.
112) 2:16). 2°33} 1°85] 1:21] ‘69 S4ay | kr | S07) 03 Ol B
|
Potenza 2 — 2 2 1 1 ea ee | = /O
4 1°30) 2°09) 2°23) 1:78) 1:14 Gil 288 O04 ol; -- |—/;—}|]— |M
78| 1:6 1:95} 1°80] 1°41] -98 63} °38 21 12} ‘06; °03; -01} 01] — | B
Catanzaro | 1 1 Smee eee = 1 1 1 — | — Re pel |= Peony 6)
15 s63= l32.|- 1-85)| 1:95.| 1:63) 1-14) 369 | +36 17 ‘OT (0953 |) COE |) |) Se 1 AE
Solmelecon LAG (0 1363) Peli | 95 75] 57 | +43 31 23) °16/ 12) °08 7/B
Salerno 1 1 il _— 2 — il 1 2 — | — ae Oz
06 31 e19ee eso) 72") 75a 1491-09) -69)) <39 20 9} 04] °0 “Ol | M.
40 86) 1:18} 1°31} 1°27)1°14] °95 lide | s09 45 33 | °24 | ‘17 22 1B
| |
Cosenza 2 = I = l Sy 8) 1 a el =| 1}/—|— |0O. |
06 29 foe 29) |e e1e68i\ dere Io 112 73| °42] °22 0) | 05.) -02)| 01) M
43 88} 1 ele 7d D30) TiO 93 75 59 | °44 33 24/)°17|°13| °33)B.
Bologna = | = 3 ea al i || ea ae een eee 1 S| 0)
‘Ol 06 21 49) +88] 1:24] 1°47] 1°49 | 1°32 | 1:04 74| °48) °28/°15|) °14) M.
‘40 46 97) 1:06] 1:05] -98 87 76) ‘64 | 53 43°35 8] “21 7 133
: | | | if |
| |
Totals 10 8 Qi alin 4: WM BY i | 8} il il 2 1 | — 2 | O}
. 4:00 | 9°78 | 13:07 | 13-09 | 11-33 | 9:03 | 6-81 | 4°87 | 3:27 | 2°08 | 1:24} -70| -38|/-19] -16]M.
6°15 | 12°75 | 14°82 | 12°64 | 9°28 | 6-57 | 4°70 | 3°46 | 2°54 | 1-88 | 1:38 | 1°02 | °75 | 55 | 1°43 | B.
| | aera bao Z =
The following are the binomials for the 8 districts out of 16 which Mortara
has selected.
Reggio Calabria 10 ( *7842 + -+2158)ts8so9
Foggia 10 ( 9609 + -0391)87"
Siracusa 10 (1°3000 — °3000)-3#
Potenza 10 (15500 — °5500)3!2
Catanzaro 10 (2°7524 — 1°7524) 229%
Salerno 10 (2:3510 — 1:3510)-27
Cosenza 10 (25308 — 1:5308)-*#
Bologna 10 (33161 — 23161)"
60 * On the Poisson Law of Small Numbers
Examining these we see that there are only two in which g and n are positive
and only one in q is small and positive and n moderately large. The probable
error of g for 10 observations on the assumption that n is very large and q very
small is + ‘3016 and is quite inconsistent with the last four districts being samples
from exponentially distributed frequencies. The other four districts may or may
not belong to such frequencies—the data are wholly madequate to determine
whether they do or not. Reggio Calabria and Foggia have the lowest Q's,
ie. 09 and 1:0. But that six districts out of an already selected eight give
negative q and a seventh a relative large g and small n suggests the inapplicability
of the hypothesis adopted. If we seek for “ goodness of fit” of the totals, we find:
Binomial Exponential
y? = 2512 47-92
P0336 ‘0000
Thus the odds against the binomial system are 28 to 1, but the odds against
the exponential are enormous. It does not seem possible to justify the treatment
of such data by the use of the Poisson-Exponential.
Let us turn to a second of Mortara’s illustrations, that of deaths from small-
pox. He rejects first six out of the 17 districts, the remaining ten are given in
Table XIII. The districts give the following binomials:
Venezia 10( 9500+ 0500)
Bologna 10( 9889+ 0111)"
Treviso 10 ( 2:2000 — 1:2000)-*8
Pavia 10( 1:8000 — 8000)
Caghari 10( 45190 — 3:5190)->
Padova 10 ( 36833 — 2°6833)-%#
Verona 10( 56000 — 4:6000)-™"
Brescia 10 (O97 20 — B8i0i 2h)
Bergamo 10 ( 23821 — 1:3821)-7=9
Catanzaro 10 (156128 — 14-6128)~ 76
Vicenza 10 ( 34854 — 2°4854)-Ve97
Out of the eleven cases only two give g small and positive; not a single one
gives for g anything like the chance of a death from small-pox in the district, nor
for n anything like the population of the district. There is an increasing divergence
from the positive binomial as Mortara’s Q’ increases in value. We see that in nine
cases, however, a negative binomial not the exponential is required to describe the
frequencies. The probable error of gq, for insignificant q is as before + 3016, and
therefore it is improbable that g is zero in at least 9 out of these 11 districts.
Examining the totals we find
Binomial Exponential
v2 = 9°64 * 570°79
Oi ‘000,000
Lucy WHITAKER 61
TABLE XIII.
Deaths from Small-powx (1900—1909).
: fy | 12 or
0 1 2 8 4 5 6 : g er ea lee | more
Venezia 4 5 = 1 7 || le Observed |
4:49 | 3°60] 1°44 38 08 01) -- — = — —|— — | Mortara
4°40} 3°71} 1°46 36) 06) ‘01 -—— | Binomial
Bologna 4 4 1 1 — —|/|— —}|} — |90.
4:07 | 3°66} 1°65| 49 ear Nfs ee: ©) een |r| ean cn | i ee IVT
4:04} 3°68} 1°65 “49 ‘ll 02 Ol; — |) — B.
Treviso 5 3 7 pe == 1 | | 0!
3°68 | 3°68} 1°84 61 15| :03 ‘OL| - - M.
Halls 236) 1-18 | ‘61 By ON ‘09 ‘05 03) ‘Ol; — | — = B.
Pavia 4 3 el eee fee lL catty tea 0.
SO) 382624) 2:17 87 26 ‘06 Ol = | = M.
4:14] 2°76 e538) 79 40 19 09 O05 02 01} ‘01 |) — | = B.
Cagliari 5 1 1 1 SV es eel aes peo ee eee | a ea ee eS
OS Ocou eouiOln woo “99 42 IL) 04 OL y= —}— = M.
4:07] 1°89} 1°17 79| 55! -39| -28| -21| -15| -11/:08|-°06/ -25 |B.
Padova 3 3 _ 2 = 1 a |e == = 1/— = 7 |KO):
‘91 | 2°18) 2°61} 2°09] 1°25 ‘60 24 08 03 Ol}; — | — — M.
o2'| 2°03) 1°40 98 MON 250 36 26 19 13°] “10 7 16 | B.
Verona 4 3 — 1 = = 1 ome ee oe |e | i O.
OM Ql Sei :6la|) 22098) 125 ‘60 | +24) -08 03 ol) — J] — — M.
4:07 | 1°74] 1:09 ‘75 4 40 | °31 733 18 14 | -11 | ‘09 Fata) || 15%,
Brescia 2 3 2 2) = == = = a ee | 13 0)
ES alan le QO 2 Oil) || a2, 1°82 | 1°20] ‘66 31 33 05 | 02 | — — M.
4°99) 1°42 87 6 47 coll 30 24 20 17 | *14 | °12 79 |B.
Bergamo 2 — 2 2 — | 1 — 1 1 1 }/—/—] — |9.
*20 79)| 1°54) 2:00)) 1°95 1°52 99 5)5) 27 12 | 04! -02 ‘Ol | M.
Hotere dol 1:57) 1:46) 1723 98 74 54 38 he | Altsy | aly 24 |B.
|
Catanzaro 3 3 1 1 1 |g en |e | 1* | 0.
*20 “ON 1254.1 2:00)) 5951) 1:52 "99 YD) ||) 47/ 12 | :04 | 02 ‘O1 M.
4°80 |} 1:20 el 50 38 ill 25 21 18 LG 4 | el2s 04s 1B:
Vicenza 3 = 1 1 1 1 = 1 1 = — | — 1 O.
‘17 GSae eso alee 1:95 | 1°60 | 1:09 BS} 15 | 706 | °02) -O1 M.
Qe SOs da? 1eQBnle LeODie 282, 65 51 39 30 | SBA PI | °48 B.
lias ae |
Totals Somos, ie i.| 2 2 CRden No Wo sor ery) wae. Os
19°24 | 24°97 | 21°50 | 16°54 | 11°76 | 7°58 | 4°38 | 2:25 | 1°07 ‘46 | °16 | -06 03. «| M.
40°25 | 23°70 | 14:05 | 8°58] 5°78 | 4°16 | 3:08 | 2 L254 ed 59 Olles7ios eo soln b
|
Brescia, and 1 at 27 in case of Catanzaro, if the means were to agree with those given by Mortara.
* 1 at ‘12 or more’ in cases of Brescia and Catanzaro was found to signify 1 at 20 in the case of
62 On the Poisson Law of Small Numbers
In other words the binomials give a reasonable total fit, the exponentials a
practically impossible one.
But there is another question to be asked in such series as those of Mortara:
What justification is there in cutting off at 10 cases, say of murder? <A province
may have a million inhabitants and, perhaps, 40 murders occur in a year*. Hence
the binomial is for ten year returns
10 24,999 il 1,000,000
x eecce a saat)
but this is as close as anything can be desired to the exponential series. It may
be reasonable to apply a separate series to districts giving 4°2 and 36°6 murders
per annum respectively, but it is difficult to see why the latter district should be
altogether excluded from treatment. If the theory of the binomial be applicable
at all, then it applies practically as well to districts with 40 murders as to districts
with 4; for, we need no indefinitely small g to get a closely exponential series.
If we take the case of deaths by murder, Mortara has retained only 6 out of 16
provinces, yet his criterion @’ (see his Table, p. 51) is not more divergent from
unity for the rejected provinces than for those retained ; the binomials are indeed
Reggio Treviso 10( ‘7000 +3000)"
Venezia 10( °5619 + °4381)s"
Vicenza 10( 9571 + 0429)842!
Padova 10 ( 4774 + 5226)»
Pavia 10 (1'8162 — -8162) °°
Bergamo 10( ‘8857 +1143)
only one of which gives qg small and positive and n large.
The mean Q for the retained provinces is ‘967 with a range from ‘7 to 14 and
for the rejected 1:03 with a range from ‘8 to 1:4. Even if—which is not the case
—the probability of an individual being murdered were too great for the ex-
ponential, it ought to follow the binomial, but this, as a rule, it does not do, unless
we give some wholly new interpretations to g and 7; the actual values render the
theory of the binomial as stated inapplicable.
(10) Mortara’s Criterion.
As a matter of fact the only test of whether an exponential will legitimately
fit a given series or not is to determine the binomial (p+q)”" and ascertain
whether p is slightly less than unity. But:
p = npgq/ng
_ (Standard Deviation)
7 Mean
* We assume that each individual is equally likely to be murdered. But if there be a graduated
probability for murder throughout the community, what right have we to apply Poisson’s series at all?
The essential basis of the application—equal chance of each individual—is wanting.
Lucy WHITAKER 63
Now if m, be the number of deaths, say, occurring in any year and there be
1 years under consideration, then:
S? (ms — nq
l : >
(Standard Deviation)? =
or, if we use the form preferred by Bortkewitsch*
8S? (ms — nq)?
eet ee
S? (ms, — nq)
(l—1)nq ©
This in other notation is Mortara’s Q”, the only criterion he’ actually uses
provided by his equation (17 ter), p. 18. Thus his Q’, which he says must not
differ much from 1, is only /p, and it would be better to use p—which has a
direct physical meaning—than Mortara’s Q =,/p. Clearly Mortara’s somewhat
elaborate process of deducing Q’, does not amount to more than saying: Fit a point
binomial and test if p is slightly less than unity. We contend that it is best
straight otf to fit the binomial.
Hence : p=
It is true that Mortara does not reach his Q”, our p, by the simple process of
asking whether the binomial is one with a positive probability less than unity.
He endeavours to obtain it by considering whether there is “lumpiness” in the
observations. But it seems to us clearer and briefer to ask: Are the contributory
cause-groups independent as in teetotum spinning? If so, the data will fit a true
binomial and p will of necessity be a positive quantity less than unity. If they
are not of this character then p must of necessity be greater than unity. It is of
interest to see how Mortara’s test of dependence of contributory cause groups
2
leads to a criterion, but he actually only gets his Q”, Le. our binomial p after
2
a series of hypotheses which much limit, and that in no very obvious manner,
* The use of NE or en in the value of the standard deviation when l is small has been several
times discussed. It may be dealt with as follows: The probable errors of a mean as deduced by the
two processes are
E=-67449 . o/a/1,
and E’=-67449 .o/,/t-1,
now B= 61449 6] JT(1 +5) +-~- ]
1
= -67449 +. (« eee ot. )
wl Not OE
1
may and —— is less and often much less than ‘67449.
wel /21
Hence if we only know o from the observations themselves, and this is the usual case, we have:
1 /
2s gh
Jt
where o’ differ from o by a quantity usually far less than the probable error of ¢. In other words the
refinement of using H’ for F is idle having regard to the accuracy of our observations; and the form
used by Bortkewitsch and Mortara with ,/1—1 for nibs of no importance.
Now the probable error of o is °67449
E’ = -67449
64 On the Poisson Law of Small Numbers
the nature of those contributory causes groups. Of course if their dependence
were of the nature of successive draws from a pack, then the result would be
a hypergeometrical series and Q? would have no physical meaning for the series
at all.
(11) We will deal with one further illustration out of many considered by
Mortara which are of like character. In the case of Marriages of Uncle and Niece
(see Table XIV, p. 65), where the distribution of Q’s is the most favourable
for his theory, the binomials are
Reggio Marche 10( "7000 + -3000)'°
Umbria 10( :9000 + -+1000)°°
Basilicata 10 (14000 — -4000)7*
Sardegna 10( °44545 + °55455)198%
Emilia 10( 9818 + -0182)2020
Abruzzi 10( 8429 + °1571)788
Lazio 10 (12548 — -2548)—12 1646
Puglie 1OGeS =o ace
Veneto 10 (1:34.44 — +34.44.)~1'S064
Toscana 10:(2:2667 A266 7) 28"
Calabria 10 (13584 — -8584)—2#°88
of which only one (Emilia) approaches the conditions for an exponential distribu-
tion. If we test the totals at the foot of Table XIV, we find the result much to the
advantage of the binomial, for which P = ‘902 as against ‘714 for the exponential.
(12) On Mortara’s own showing nearly all the Qs of his numerous series are
greater than unity, and very few of the binomials are positive. If we consider the
distribution of Q's, given in his work omitting Table 13 (Deaths from Malaria) we
find a range from ‘5 to 3°6 with a mean Q at
1:2565 + 0847,
while for the distribution of all the p’s in the binomials we have determined, we
find a range from ‘4 to 15°6 with a mean p at 2°5655 + ‘3817.
These results are sufficient to show that there is no real distribution of p round
the value unity but the binomials have a distinct tendency to be negative.
(13) But the whole theory of Poisson’s exponential law in the hands of Bortke-
witsch and Mortara appears essentially vague. The binomial is built up on the
assumption of the repetition m times of a number of independent events, of which
the chance of occurrence is identical and equal to g. The population is n and the
chance of occurrence q in the case of each individual. The mean frequency of
occurrence is ng. But if g be very small we have seen that the series 1s
—m [1 , m me
e ppnow nt soy v8 ;
i ——
Lucy WHITAKER 65
TABLE XIV.
Marriages of Uncle and Niece (1900—1909).
: as
0 1 ze | 8 pe Nae |e oer |e om ip it | 22 | 23 | 14 15 | 18%
| | over
er
Marche O.| 7 3 = = | |
M.| 7-41] 2-22] -33] -04 | | | |
Bale iz 2 |
Umbria 0. 6 3 1 — — | |
M 6:06 | 3°03 s(oa wld 02
B. | 5°90} 3:28 ‘73 08 | -—
BasiicamOnime. | o3 | — | 1 | — | ; | | | | |
M 5°49 | 3:29 99 ‘20 03 | |
B. | 6°04] 2°59 92 31 10 03 Ol. |
Sardegna O. 2 5 3 = -
M.| 3°33] 3°66) 2°01 74 20) °05 O1
Pelpeccores-o6 | 3:03) = | — | — | —
| |
Emilia =O. 1 3 2 Zee 1 —
M Il-l1l| 2°44) 2°68} 1:97] 1°08} °48]) ‘18 05) -Ol
B 1:09] 2°48} 2°70] 1:98 | 1:08 47| ‘17 05 Ol
Abruzzi ©, || — SUF whee ail 3 | 2 _- 1 — | — |
M 61) 1°70) 2°38) 2:23] 1°56 8 Se G 06 02
B Se elon mcd S|) 2°43 | 1°68 87 34} *11 03) —
isazion ©. | 1 eal a2 3) — | 2}/—) 1] —) —
M ADM eA One Dele 224 173!) 1:07 5D 25; 10} ‘03 OL
B 63} 1°56! 2°09} 2:00} 1°54) 1°01 59 31 14} -06 03 O01} ‘O01
Puglie O.| — 30 al 2 nh peal 1 72 ae bbe |e fee ae |
M 27 FOS) wake) Loi U38 | “835 427) =19) 08 03 01 | — |
B Sone lesOny We77 | 1-80 | 1538 |1:14| -77| °49) 29] +16) +10) :05] :03 | ‘Ol | ‘OL
|
Veneto O. 1 = 1 l 3 1 i a 2 | |
M. 11 SHOP aoa OOM le OO! Ts fn) 28 82) “46 23 10 OA 025-01)
Bs 21 ‘70 | 1°26] 1°62] 1°67 46 | 1°13 TS Mh > aap 30) 17 09 | 05 | °02 | -O1 | O01
Toscana O. _ =e 1 2 2 2 1 Le — | — 1
M. O04 24 66) 1:19) 1:60] 1°73 / 1°56) 1:20} -81 49} °26 13 | 06} ‘02 | 01 | —
B. 31 SM lsOvaleelaton al 2ifal elie) 1-00 83 | °65 5O 37 27 | 19 | 13} 09 | 06] :10
|
Calabria O = - 2 2 1 1 1 ented tg | —| 1
M — Ol 05:; “16. 36 4) 94] 1:20 | 1°33] 1:32] 1:17 | :95) -70| 48 | -31 | -18] -20 |
B 00 O03 wi; 26 48 73 96 | 1:16 Deal OL C7 6Ou plaleon I 2a. a6
: | ea |
Totals O. 24 24 1} || ile 8 9 5 5 3 it 1 it = |) SE ee al
SS M. | 24°88 | 19°47 | 14:93 | 12°72 | 10°39 | 7:93 | 5°76 | 4°10 | 2°96 | 2-17] 1°57] 1°13 | -78| -51 | -32 | -18] -20
\ B. | 24°22 | 22°16 | 16°46 | 11°73 | 9°35 | 6°88 | 4°98 | 3°74 | 2°84 | 2°19 | 1°71 | 1:29 | -97 67 | 48 | °32] -26
|
}
J
As Biometrika x 9
66 On the Poisson Law of Small Numbers
from which x has disappeared, and in this exponential we have seen that
Bortkewitsch and Mortara suppose m small, ic. 10 or under. We have seen
that there is no reason why m should be absolutely small, and that the name
given by Bortkewitsch to the Poisson-Exponential—ie. the “Law of Small
Numbers ”—is misleading. But supposing the mean occurrence m to be small,
it by no means follows that g need be small and n finite. For if g="2 and n=4,
m would be “small ”—and the sort of small number with which our authors deal,
but the mere fact that the mean frequency of occurrence was 2 would not justify
our using the Poisson-Exponential for
(Golam ey)
The fact is that when our authors speak, of the deaths in a Prussian Army
corps from the kick of a horse, or the suicides of schoolgirls, or the deaths from
chronic alcoholism as being “small,” they really mean small as compared with the
number of persons exposed to risk. They had probably in mind all the men in
the army corps, all school-girls or all individuals liable to death in the towns
considered. But are all men in the army corps,—or only the cavalry, the artillery,
etc..—equally liable to death from the kick of a horse? Is every school-girl equally
hable to commit suicide or only a very few morbid and unhealthy minded girls?
Is every individual equally liable to die of chronic alcoholism, or only perhaps the
10 or 12 confirmed and aged drunkards in a town? The moment we realise these
doubts, what is the population n to be considered? It is not m being small, but
the smallness of m/n that leads us to believe that the binomial may have passed into
an exponential. But if only six school-girls per year in a community are in the
least likely to commit suicide, what is the justification for the “law of small
numbers,” if the average number of suicides be 65? Further, if we pass to even
a large community in which the tendency to commit suicide is graded—a very
probable state of affairs—m might be small and n large, and yet since q is not
constant, the binomial and its exponential limit would not be applicable ; and this
non-applicability would not depend on “lumpiness””—i.e. contagion or example in
occurrence. Thus the probability might be:
(Pit h) (P+ G2) (Ps + Gs) +++ (Put Yn)
with all the p’s independent (as in spinning differently divided teetotums) and not
correlated (as they would be in drawing successive non-returned cards from a pack).
It would seem therefore that a priort we should not expect the conditions for the
exponential to be fulfilled in most of the cases selected by Bortkewitsch and
Mortara, although with perfect mixing we might expect it in the cases cited
by “Student.”
(14) In order to test this point on adequate numbers, the ages at death of all
persons dying over 70 years of age were extracted for a period of three complete
years from the notices of death in the 7imes newspaper for the years 1910—1912:
see Table XV. These announcements of death are those of individuals in a fairly
limited class, which may be considered stable in numbers for these three years.
67
Lucy WHITAKER
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68 On the Poisson Law of Small Numbers
Table XVI shows that the announcements of deaths over 70 years of age only
amount to 3°74 per day for males and 3°52 for females. These are certainly “small
numbers,” but “small” with regard to what? Are we to consider n as the number
of the population which embraces, (i) all the individuals of the limited classes of
the same range of ages as the defunct, (11) all the individuals announced as dead
on the same day, (iii) all the individuals of whatever ages of the class which
announces deaths in the Times? Or, should we refer to all the individuals in the
community of that range of ages, or the whole community at large, 1e. the chance
that in a population of so many millions an individual over 70 or 80 as the case
may be will die and have their death announced in the Times newspaper? Well,
it really does not matter, because if for any one or all of these populations the
binomial (p + q)” applied, we should get if g were small and n large, the Poisson
series
Cm @ +m+ ae nag + 4
Ziel ;
and this quite regardless of the size of n. If therefore we did find a series in
which g was very small and n large, we might not be able to say to which, if any
of the above populations n applied. On the other hand the mere fact that m is
small is no justification for the use of the “law of small numbers” as is sometimes
implied. If it be argued that the small number of people who die over 80 and
have their names recorded in the Times are drawn from a small population, we
reply so it may be argued are the school children who commit suicide, the uncles
who feel any inclination to marry their nieces, or the men liable to die of chronic
alcoholism ; and we can in the case of the announcement of deaths test the values
of g and n on fairly adequate numbers. As a matter of fact we do not know, in
attempting to apply the Poisson formula, what is the population from which we
are drawing our individuals, and the justification of the Poisson formula lies only
in showing that there actually does exist a binomial for which qg is small and
n large. We might imagine that as we got to the higher ages practically every
person of that age would die, or that in our notation q would be 1 nearly and p be
a very small quantity ; thus an approach might be made to the Poisson-Exponential.
But the approach to the Poisson-Exponential arises not through q approaching
unity but from q becoming very small. Nor again in the lower age groups do we
find ourselves left with a positive binomial.
In all cases except women over 90 years of age, we find that a negative
binomial best fits the observations. Even in the case of the announcements of
deaths of women over 90 years, we find that the approach of the binomial to the
Poisson exponential depends on
/ i: 53°3333
(1 a 553555)
being measured with sufficient approximation by e = 2°71828. But
(1:01875)?°8 = 269323,
Lucy WHITAKER 69
and is therefore not a very close approximation, a result shown when we use
a binomial by the substantial improvement in the measure P of “goodness of
fit.” Even in this case we are not prepared to say what is the population for
which the g = ‘01875 in the case of these announcements of deaths of women over
90 years of age. It can scarcely be that there are only 29 women over 90 years
TABLE XVI.
Constants for Deaths of Aged.
Men.
| | |
Probable _ Probable lee omiaal Expo-
Age over | p q Error n Error m , P a | nential
| of q | of n P
| 70 years... | 112965 | — 12965 | + 03314 |—28°8747 + 7°3734| 3°7436| °1355 0045
| 80 years... | 1°12152 | — 12152 | + 03349 |—14:0703 + 3°8704 | 1°7099 | °9358 | *1129
| 85 years ... | 1:01903 | — 01903 | + :02902 | —43:2996 + 67°5797 | °8289) “9737 | “9715
90 years... | 1°00654 | — :00654 | + 02934 | — 42°8498 |+192°3069 | -2801 6741 | 6672 |
| | . | |
Women.
= , = |
| Probable Probable | | Bi al Expo- |
Age over Dp qd | Hrror n Error m eee nential
of q | of n | ee
|
ei. Paice:
70 years ... | 1°34012 | -- 34012 | + 04161 |—10°3522 | + 1°2307 | 35210) -8084 | -0000
80 years... 1°20770 | - -20770 | + °03294 |—10°4400] + 1°8309 | 2°1569 -9686 70018
85 years ... | 1°14507 | — °14507 | + 03077 |— 8°1447) + 1°9627 | 1:1816| ‘9860 | -1062
90 years ...| *98125 | +°01875 | + ‘02779 | + 29-0573 +43:0634 | 5447| ‘9848 | -8116
| | |
of age living in the country, whose deaths are likely to be announced in the Times
when they occur. Further the probable error of qg is such that actually this case
might equally well be a random sample from material following a negative
binomial. Analysing our material we see that our first two cases of males and
the first three of females are such that they could not possibly be random samples
from positive binomials, the probable errors of q are too small. Next, seven cases
out of the eight do give actually negative binomials and the eighth might, having
regard to its probable errors, well be a negative binomial. Thus although our
daily occurrences are certainly in Bortkewitsch and Mortara’s sense “small numbers,”
they give no support to the use of a Poisson-Exponential.
If it be said that these “small numbers” differ in character from those used
by our authors, the reply must be: we know in none of these cases the real
population from which deaths are to be considered as drawn. The chances of
death are certainly graduated with age, but the chances of suicide are graduated
with temperament, and the same is true of alcoholism, or again the chance of
70 On the Poisson Law of Small Numbers
death by accident is graduated with occupation. At any rate until those who
support the use of the “law of small numbers” demonstrate its application on
material, where the probable errors are sufficiently small for us to measure the true
value of gq and n, no advance can be made. Nor until we have clear ideas of the
population in which the chance is q, is it possible to assert that it may be used
for the suicides of school children, and the marriage of uncle and niece, and must
not be used for the deaths of aged people, which certainly occur in “smaller”
numbers.
In the illustrations of deaths we have taken, certainly the Poisson-Exponential
is not the rule, although the distributions appear to approach it, as towards a limit,
when the number of deaths approach zero. But our data which show the rule of
the negative binomial appear to show it in no more marked manner than much of
the data selected by Mortara himself indicate the negative binomial, although owing
to the sparsity of his material his results are far more erratic and unreliable. Nor
is Bortkewitsch much behind Mortara in the evidence he produces for a negative
binomial being as reasonable a description—possibly owing to inherent lumpiness—
as a positive binomial of these “small number” frequencies.
(15) Conclusions.
(a) The Poisson-Exponential gives a fairly reasonable method of dealing with
the probable deviations of small sub-frequencies in the case of random sampling.
When the average value of a sub-frequency is not more than 3°/, of a population,
then Poisson’s formula suffices in most practical cases to determine the range of
error likely to be made. Tables are given to assist its use.
(b) The application of the Poisson-Exponential to various data by Bortkewitsch
and Mortara has hardly been justified by those writers, for they have not tested
whether the probability q is small and positive and the power n large and positive
in the cases considered by them. When this is actually done, it is found that
their hypotheses, having regard to the probable errors of q and n, are largely
unjustified in the case of their illustrations. Even in such cases where it is
justified, a binomial gives a better result as measured by the test for goodness
of fit.
(c) Negative binomials repeatedly occur and give just as good fits, where
they occur, as positive binomials. In the illustrations taken by Mortara, the
frequency 10 used is so small that it is not possible to assert that either positive
or negative binomials are demanded by the data. Still the average p of his results
is very significantly in excess of unity.
(d) Mortara like Bortkewitsch cuts out of his data straight off all districts
with, on the average, more than 10 cases in the year. But the g obtained from
20, 40, or even 100 cases in a population of 100,000 is a small g in the sense that
the resulting binomial is adequately expressed by a Poisson-Exponential. There
Lucy WHITAKER 71
appears to be no valid reason for such a procedure, except the experience that
many such cases actually give negative binomials*. It seems to us theoretically
unjustifiable to apply the exponential to 8 cases say in a district of 100,000, and
not apply it to 12 cases in a district of 200,000. Actually p may be 1:4 in the
first case and only 09 in the second.
(e) We consider that the reasonable method in every case is not to start with
the Poisson-Exponential, which screens the truth or falsity of the a@ prior
hypotheses, but to fit a binomial regardless of the magnitude of p. The fact that
quite as good fits are obtained with negative as with positive binomials suggests
that a new interpretation of these cases of “negative probability” is requisite.
Several cases of the interrelation of “contributory cause groups” which provide
a series represented by a negative binomial (p—q)~” have been recognised f.
A general interpretation based on a very simple conception seems needed for
these demographic cases in which the law of small numbers appears far more often
to correspond to a negative than to a positive binomial.
This paper was worked out in the Biometric Laboratory, and I have to thank
Professor Karl Pearson for his aid at various stages.
* Can we cite in addition perhaps, the fact that existing tables of m*e~”"/x! do not extend beyond
m=10?
+ Pearson, Biometrika, Vol. tv. p. 208.
THE RELATIONSHIP BETWEEN THE WEIGHT OF THE
SEED PLANTED AND THE CHARACTERISTICS OF
THE PLANT PRODUCED air
2
By J. ARTHUR HARRIS, Ph.D., Carnegie Institution of Washington, U.S.A.
I. Inrropucrory REMARKS.
1. In Biometrika, Vol. rx. pp. 11—21, March 1913, were published constants
showing the relationship between the weight of the seed planted and the number
of pods on the plants produced in twenty experimentally grown series of Phaseolus
vulgaris. From the economic view point, number of pods is the most important
character which could have been chosen, total weight of seed matured only
excepted. But to the student of morphogenesis, or of the physiology of seed
production, other characters are of equal interest, while the comparison of the
correlations for various features must yield results of significance.
The purpose of the present communication is the presentation of the constants
measuring the influence of the weight of the seed planted upon the number of
ovules formed and the number of seeds developing in the pods of the matured
plant.
These various relationships have now been worked out for a relatively large
bulk of material. Altogether there are 29 individual series belonging to 5
varieties, involving 17,953 plants, from which 119,192 determinations of the
number of ovules and seeds per pod have been made. The reply to the possible
suggestion that the expenditure of effort in the collection-and analysis of such
masses of data is quite unjustifiable is twofold. First, a major portion of the
labour involved was necessary for investigations not touched upon here. Secondly,
there are many problems of morphogenesis and physiology which can only be
solved by the amassing of large series of accurately determined biometric constants
which when sufficiently numerous may themselves be the materials for statistical
analysis. The data here contained are recorded in partial fulfilment of such
requirements for certain definite morphological and physiological problems.
The present paper is limited strictly to matters of fact; general discussions are
reserved until further data—much of which is already available in a raw state—
are reduced.
— Te eee ee ee le
——_
J. Av HRARRIS
TT.
MATERIALS.
The first paper may be consulted for details not entered here.
analysed are drawn in part from the series already considered for the relationship
between weight planted and number of pods produced.
73
The data
In addition to the White
Flageolet, Navy and Ne Plus Ultra varieties already treated, several lots of
Burpee’s Stringless and two of Golden Wax are available.
III.
ANALYSIS OF DATA.
2. Data for Number of Ovules and Seeds per Pod.
Tables III—VI, similar to those of the preceding paper, give in a condensed
form the data for the correlations discussed.
Table I* gives the correlations
TABLE I. Correlation and Partial Correlation Coefficients.
| | |
Namber coro: Nummberc! Correlation, Partial Correlation, Partial
F eight Weight | pare Weight | :
Series of end Pos of Pods | eadiOvdlese| Correlation, and, Seeds Correlation,
Plants Twp Examined | no mane ng pws
| | |
LL 1141 |--008+°020; 8043 026+°008; ‘027+°008 —°013 +4008 | — 013 + :008 |
LG 182 066 + 050 806 1534023 | +140+°023 —-100+°024 |—:103 + 024 |
GG 750 |-°368+°021) 6310 018+°008' -029+'008 ‘004+°008} ‘0164-008
GGH 583 208 + 027 5251 045+°010) 01947009 024+ °010 | — 004+ ‘009
GGH2 499 176+ 029 3502 0934°011;) -083+°011 063+°011} 049+-011
GGHH 396 193 + 033 2656 — 022+ °013 | — -042+°013 —:-029+-013 |— 048+ 013
GGD 514 "159 + 039 1438 ‘1074018; ‘089+4°'018 ‘O71+°018} -068+°018
| GGD2 449 215 + 030 1227 0444°019;) °018+°019 ‘079+°019] ‘0624-019
| GGDD 342 137+ 036 | 807 1014023} -092+°024, -089+°024} -076+°024
A 1484 177+:017| 14029 010 + :006 | —"039+°006 = 007 + :006 | — 054 + 006
| HHA 1271 1454-019) 11230 — ‘000 + 006 | — -030+°006 = 016 + 006 | —-014 + 006
HD 1416 "129+°018| 5581 — 044+ °009 | — 067+ :009 — 049+ 009 | — 052 +:009 |
| HDD 1204 121+°019;} 5449 — 029 +009 | — 065 +4°009 —-010+:009 |— 030 + ‘009
DD 513 282 + 027 1827 098+ °016) -009+°016 0504-016} -008+-016
| DDD 459 215 +°030 2018 0444°015|} ‘0004°015) 0464°015} ‘006+4°015
| DH 670 258 + 024 5955 075 +009 | — ‘005+°009 = ‘076 +009 | — (013 + 009
| DHH 565 152+°028, 5019 0454°010} *008+°010, ‘011 +°010 | —-025+°010
OSC 530 150+°029| 2569 059+ °013} 032+ °013 | 031+:°013} -024+°013
USS 680 155+:025| 6605 023 +008 |—-000+°008) -041+°008| -024+ 008
OSH 361 129+ °035 3.406 ‘032+ :012| 0014-012} 0374-012] -020+-012
| USHH 224 1434-044 1743 112+°016} ‘098+°016 ‘011 +016 |— -004+-016
USD 312 195 +037 | 802 127+:023/ °098+:024; -071+:024| -067+-024
| USDD 237 241 +-041 | 851 2384022} +175+°023|) °1314°023] -090+:023
FSC 586 147 +027 2876 047+°013] -017+°013|} :089+°012} -073+°013
FSS 868 098 + 023 7809 021+°008) *001+'008) -026+4-:008; -004+4-008
FSH 475 "100 +031 4541 049 +°010} -018+°010 —-045+°010 |— -073 +:010
FSHH 427 121 + 032 3837 015+°011 /—-013+°011 -0404°011) ‘0174-011
FSD 428 130 + ‘032 1449 "060 + 018 | — 027+°018 —-019+-018 = +036 +018
FSDD 387 144+ 034 1556 0B7+°017) 013 +017 | ‘047+ °017) 024+ 017
|
* The weight of the seed planted was weighted with the number of pods counted.
Sheppard’s correction was used for seed
weight, but not for the integral variates ovules per pod or seeds per pod,
differ slightly from those of Table II of the first paper.
Biometrika x
Thus w and o,
10
74 Weight of Seed and Characteristics of Plant
between weight of seed planted and ovules per pod, 7,., and between weight
8 Pp per } 8
planted and number of seeds matured per pod, 7s. The partial correlation
coefficients, 3
i= Li wo me wp Pr. po ae Tws — Trp pss
p' wo P = ) DWS: = 5 2 5)
v1- “wp Vl=r r ‘po V1 — Pop V1 — ns
showing the correlation for weight (w) and ovules (0) and weight and seeds (s) for
constant numbers of pods (p) per plant are also given. These require in addition
to the correlations here given 7), 7p). and r,s, the correlations between the number
of pods per plant and the number of ovules and seeds in these pods. Values
of 7», are available from the preceding paper (Biometrika, Vol. 1x. p. 21, Table
VII) and from a supplementary table giving nine additional constants*. For the
reader’s convenience these are reprinted in this table. The values of r,, and ry,
will be published in connection with another problem.
The probable errors have all been calculated on the basis of the number of
pods examined as V. There is considerable question whether the actual number
of seeds planted should not have been used instead; the degree of trustworthiness
of a constant is perhaps not greater than is indicated by the lowest number of
actual measurements (irrespective of the number of associated measures taken).
The point is not of the greatest practical importance for the present case, since the
number of series is so large that conclusions can be drawn from the run of the
constants as a whole and too much weight need not be given to individual series.
A glance at the table shows that the correlations are low throughout. The
suggestion naturally arises that some of the extremely low values may be due to
non-linear regression. The regression straight line equations and the results of
Blakeman’s test} are given in Table II. Here 7, 7 and the straight line equation
for the regression of ovules and sceds per pod on weight planted (in working units)
are determined by the conventional formulae. The final two columns give the
values of
vil se Mee
ue: Miele =i"
when €=7?— 7? and xy, = 67449//N.
All the straight lines are shown in Diagram 1. The empirical means are
indicated in all of the cases where it can be done without confusion. The slope is
very slight and the agreement of observed and predicted means not very close,
especially near the ends of the range, where the number of observations is small.
There is, however, no clear indication that a curve of a higher order would describe
the results better than a straight line. This irregularity is precisely what is to be
expected in cases of low correlation.
* Harris, J. Arthur, ‘“‘An Illustration of the Influence of Substratum Heterogeneity upon Experi-
mental Results.” Science, N. 8. Vol. xxxvi. pp. 345—346, 1913.
| Blakeman, J., Biometrika, Vol. 1v. pp. 332—350, 1905,
~T
Gr
J. A. Harris
TABLE II.
Tests for Linearity of Regression.
| | _ |
Correlation, r, | Correlation-Ratio, | Regression | Blakeman’s | Blakeman’s
Series and n, and Straight Line —_ Criterion, Criterion,
Probable Error | Probable Error Equation Test Ad Test B
x |
a eel |
For Ovules: | | |
USS 0232 + °0083 0657 £0083 | -54230+4-0074 wv 3°720 | 1°688
DHH 70445 + *0095 0788+ 0095 | 4:°9385+4°0257 w 3°431 1151
USDD 2381 + 0218 "2978+ 0211 3°6886 + “LOOL w 11°096 2°102
GG D2 0442 + 0192 1276+ ‘0189 4:7224+4 0137 w 3°152 1:967
FSS 0209 + ‘0076 0403+ °0076 =| 5560640153 w 2°263 “754
HH “0098 + :0057 "0661 + °0057 5°3600 + 0056 w 5°159 1°678
For Seeds: |
USS ‘0407 + ‘0083 0946+ 0082 | 3:5870+°0206w | 5"182 2351
DHH 0111 + :0095 05414 °0095 | 4°1521+4°0106 wv 5573 1°869
USDD 1313 + :0227 1932+ °0223 | 2°1840+-0940 w 8712 | 1°650
GG D2 0794+ 0191 "1760 + ‘0187 2°4735 + 0846 w | 4181 | 2°529
FSS ‘0261 + 0076 0499+ :°0076 =| -3°0712+ "0269 w | 2°793 931
HH ‘0068 + *0057 0953+ 0057-42119 + 0058 w 8421 2°739
Blakeman’s criterion has been applied in two ways, A and Bb. In the first the
actual number of pods examined has been taken as V. In test B the number of
seeds planted (not the weighted number) has been used in obtaining x,. If the
first test be accepted as the proper one, it follows that regression cannot safely be
regarded as linear. But there are two important points to be taken into account.
The correlation ratio 7 depends upon the squares of the differences in means, hence
it has always a positive value, which may be very substantial because of the errors
of sampling when the number of individuals per array is small. Thus when r
approaches zero 7 is limited by 7, the mean values of 7 for zero correlation*.
Hence a test for linearity based on a comparison of 7 with a very low value of r
may be misleading, Again, as pointed out above, the significance of both r and 7
should perhaps be tested on the basis of the lowest number of measurements. If
this be done, as it is in test B, there is found very little evidence for non-linear
regression. Certainly, one. cannot possibly assert that the low values of 7, which
is seen throughout these experiments, is due to the number of ovules (seeds) per
pod at first becoming larger and then decreasing after a maximum is reached as
one passes from the lowest to the highest grade of seed weight.
The results of Table I are also shown graphically in Diagram 2. Here the
relationships for weight of seed planted and number of pods on the plant developing
are also indicated as a basis of comparison. The values of both ry. and r,s are in
general conspicuously lower than the low values of 7,,. But very few of them
drop below the zero bar; one is forced to the conclusion that there is a distinct
though very slight correlation between weight and ovules and between weight
and seeds.
* See K. Pearson, Biometrika, Vol. vit. pp. 254—256, 1911.
. 10—2
76 Weight of Secd and Characteristics of Plant
Consider in somewhat greater detail the signs and magnitudes of these
correlations*,
Of the 26 values of 7, only 4 are negative. The mean value of the 22 positive
coefficients is +°0673; the mean of the 4 negative is — 0236; the mean of all
(regarding signs) is +°0533.
For the relationship between weights of seed planted and number of seed
matured per pod, 7s, 21 constants are positive and 5 are negative. The mean of
the positive coefficients is + 0502; the mean of the negative values is — ‘0303 ;
for all 26 correlations the mean (regarding signs) is + ‘0348.
Thus both correlations are (as is clear from the diagrams) unquestionably
positive but very low.
Apparently the relationship for weight and ovules is slightly closer than that
for weight and seeds per pod, but the difference is too slight to justify any final
conclusion.
Consider now the question whether the observed correlations ry, Ts are to be
regarded as direct biological relationships between the two variables w and o or w
and s, or whether they are to be looked upon as merely necessary resultants of
other interdependences. At present, the only other demonstrated correlation
which might tend to bring about sensible values of 7», and rys is that between
number of pods per plant and number of ovules formed and number of seeds
developing per pod. Since number of pods per plant is known to be correlated
with weight of seed planted, while both number of ovules and number of seeds per
pod are correlated with number of pods per. plant, some correlation must be
expected between weight planted and number of ovules and seeds per pod. If
now the observed values of 7%. and r,s which are always small, are merely the
necessary resultant of the relationships 7), 7po, ps, one would expect the partial
correlation coefficients, ,?0, pws, to be sensibly zero. If these partial correlations
are not sensibly zero, it can only mean that there is a direct (causal) relationship
other than the one just considered between number of ovules (or seeds) and the
weight of the seed planted.
The partial correlations and the correlations are shown side by side in
Diagrams 3 and 4. The lowering of the degree of interdependence between both
weight and ovules and weight and seeds by the correction for number of pods per
plant is clearly marked. In a number of cases in which the correlation coefficient
is positive the partial correlation coefficient is negative.
Thus only 4 of the 26 values of 7. are negative, while 9 of the partial cor-
relation coefficients have the minus sign. In only 5 cases is ry, negative, but in
11 of the series, the sign of 7s is negative. The mean values of the partial cor-
relations are very close indeed to zero. Thus j7y.='0186 as compared with
Two = 0533; pFws= "0099 as against 7, = 0348.
* T have already shown (Science, N. 8. Vol. xxxvu1. pp. 345—346, 1913) that the LL, LG and GG
series are open to question because of the lack of certain precautions in the cultures; while they are
included in the table of fundamental constants to avoid any possible criticism of selection of series they
will be left out of account in the following disctssions.
Ci
J. A. Harris
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Weight of Seed and Characteristics of Plant
78
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1
Biometrika x
Weight of Seed and Characteristics of Plant
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83
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‘A ATEV
84 Weight of Seed and Characteristics of Plant
IV. RECAPITULATION.
The facts presented in this paper and in the preceding studies justify the
following conclusions.
1. In Phaseolus vulgaris there is a sensible relationship between the weight
of the seed planted and the number of pods on the plant developing from it. The
correlation is always low, averaging only about °166, but under proper experimental
conditions the coefficients have always been found to be positive. When experi-
ments are not made with all necessary precautions substratum heterogeneity may
completely obscure the influence of seed weight, reducing the correlation to
practically zero or even bringing about a substantial negative correlation.
2. There is also a significant positive correlation between the weight of the
seed planted and the number of ovules and the number of seeds in the pods pro-
duced by the plant developing from it. These correlations are so low that on
relatively small samples negative values may be found. They average only about
one-fifth to one-third the magnitude of the correlation for weight planted and pods
per plant.
The relationship for weight and ovules is numerically higher than that for
weight and seed, but on the basis of the number of series now available the
difference cannot be asserted to be significant.
3. Morphogenetically and physiologically, the observed correlations between
weight and ovules and weight and seeds are to be regarded as the resultant of two
other correlations, namely, that between the weight of the seed planted and the
number of pods per plant and that between the number of pods on the plant and
the characteristics of these pods. This conclusion is based’on the fact that the
partial correlation coefficient for weight of seed planted and number of ovules or
seeds per pod for constant number of pods per plant is practically zero.
CoLtp Sprina Harpor, N.Y.
August 20, 1913.
ON THE PROBABILITY THAT TWO INDEPENDENT DIS-
TRIBUTIONS OF FREQUENCY ARE REALLY SAMPLES
OF THE SAME POPULATION, WITH SPECIAL REFER-
ENCE TO RECENT WORK ON THE IDENTITY OF
TRYPANOSOME STRAINS
By KARL PEARSON, E.RS.
(1) In Biometrika, Vol. vin. p. 250, I discussed fully the mathematical
process requisite for measuring the probability that two independent distributions
of frequency are really samples of the same population. As far as I am aware this
is the only complete theory of the subject which has been published. I believe it
to be scientifically adequate, and it has already been applied to a large number of
problems*.
Before that paper was published, it had been usual to compare any constants of
two frequency distributions together, and by a due consideration of their difference
relative to the combination of their probable errors to determine the probability of
the identity of those constants. This could be repeated for any number of corre-
sponding constants, and if theoretical curves of frequency had been fitted, their
divergence or correspondence measured by the divergence or correspondence of
their complete series of constants. The method above referred to, however, as
based on the general theory of sampling, calls for no hypothesis as to the general
theory of frequency. It takes the observed distributions and measures the prob-
ability that both are samples from a large population. The population may be
homogeneous or heterogeneous; provided the samples are truly random samples
we obtain a measure of the probability of their common origin.
In the course of a long statistical experience I have learnt that it is wholly
impossible to reach any safe conclusions as to the identity or non-identity of
populations by any process of mere graphical comparison of frequency distributions.
* In actual practice the x? test of ‘‘ goodness of fit’’ should always be made with not too fine group-
ing at the terminals, especially when any group in the tails appears to be contributing largely to the
total of y?. This point was recognised ab initio (Phil. Mag. Vol. u. p. 164), and has recently been
re-emphasised by Edgworth, Journal R. Statistical Society, Vol. uxxvu. p. 198.
86 A Study of Trypanosome Strains
The distributions in appearance are wholly dependent on the choice cf scales
and the eye alone cannot possibly make any measure of the degree of accordance,
which will have scientific value.
In the accompanying Diagram I. for example, we have the frequency distri-
butions permille of two strains of trypanosomes, (aa) from a Donkey and (bb) from
a Hartebeeste. These we are told are identical. Below (cc) and (dd) are given the
frequency distributions of head-breadths for two races, Egyptian and English women,
separated by 7000 years interval. These strains we know to be different, but the
eye that judges (aa) and (bb) to be the “same” * might well suppose (cc) and (dd) to be
also the same. Actually when we come to the quantitative measure of divergence,
the probability that (aa) and (bb) are samples of the same thing is P < 000,000,1,
while the probability that (cc) and (dd) are the same is P=‘001. In other words it
is 10,000 times as likely that Egyptians of 6000 B.c. and the English of 1680 a.p.
are the same strain as that the trypanosomes from the Hartebeeste and those
from the Mzimba Donkey are of the same strain. Both may indeed be of the “same
strain” if a sufficiently wide meaning be given to the term. But is such a racial
resemblance as we find between the Prehistoric Egyptian woman and the English
woman diluted 10,000 times what we understand in ordinary language by the “same
strain”? All the mathematician can understand by “sameness of strain” is the
identity which corresponds to random samples of the same population. Ifthe identity
has been modified by a long evolutionary process, by markedly differential environ-
ment or treatment, is it not better to have some measure of a scientific nature of
the extent of the difference or of the sameness? The eye can never provide any
judgment of value on such a point. Especially is this the case if the graphs
represent percentages, as the degree of divergence is of course a function of the
number employed to determine the percentages. A deviation of frequency by per-
centages based upon samples of 200 might look to the eye absolutely like the
deviation of frequency due to samples of 2000, but the scientific measure of the
probability of sameness would be widely modified.
That the reader should have evidence how excellent is the test, I have taken
the cranial lengths (Flower’s measurement) of 67 female skulls dug up in Liverpool
Street and compared them with the like lengths of 142 female skulls dug up in
Church Lane, Whitechapel. It is possible that both these sets of crania formed
part of the contents of plague pits, or there may be an interval in date of a century
between them+. Diagram I bis shows the data arranged as percentage frequency
curves. The x? for 17 groups proceeding by 2 mm. ranges = 19°38, giving P = ‘250,
or once in four trials, if the material drawn from were the same, we should obtain
pairs of samples more divergent than the pair recorded. In other words we can
be confident that the Liverpoo! Street and Whitechapel crania represent persons
* An attempt to define the word ‘‘sameness” as used by writers on trypanosome strains would
doubtless serve a useful purpose, and emphasise the fact that we can only define ‘‘sameness” by appeal
to the theory of sampling, or by the adoption of some quantitative measure of the grade of likeness,
+ See Biometrika, Vol. 11. p. 191 and Vol. vy. p. 86.
Kari PEARSON
MICRONS.
15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
els SeReRa es
jHRUBERSAEEESRERRRREED
Sone oe e
\ \
aera HE THEE
aa, Mzimba Strain of Trypanosome,
bb. Strain frcm Hartebeeste.
MILLEMETRES.
c. Cranial Breadths of Egyptian Women 6000 B.c.
dd. Cranial Breadths of English Women 1680 a.p,
Dracram J.
88 A Study of Trypanosome Strains
MiLLEMETRES.
170-5 180-5 190-5
a
BEB Ae
ofA a FS
cc. Whitechapel ¢’s.
dd. Moorfields ?’s.
Diacram I bis.
of the same strain. That P= ‘25 and not, say, ‘85 may be merely a result of
random sampling, or it may arise from some difference of period or social class.
(2) In a long series of papers recently published in the Proceedings of the
Royal Society, Section B, conclusions are reached as to the identity of various
strains of trypanosomes. These conclusions are largely based on a comparison of
graphs of the frequency percentages obtained by measurement of hundreds of
trypanosomes.
To some extent mean values are given for the different strains, but no argu-
ments whatever can be based on them, for in no case has the probable error of the
difference been calculated. Even if it had been calculated, this constant alone
would not have sufficed to determine the sameness or difference of the strains.
Further, the percentages of various forms in the strains are sometimes given;
but again no attempt has been made to determine whether the differences of
these percentages are or are not significant. It seems sufficient here to consider
the far more valid test of the sameness or diversity of the frequency-distributions
as a whole.
I shall divide my investigation into four parts:
(i) The probability of identity of the strains on the evidence presented in the
reports of the Commission of the Royal Society, Nyasaland, 1912.
(ii) The probability that the host or the animal in which the trypanosome is
cultivated makes essential differences in the distributions of frequency.
Karu PEARSON 89
(iii) The probability that the strains are alike after allowance has been made
for the host.
(iv) The nature of the heterogeneity which is statistically demonstrable in
the bulk of trypanosome measurements.
I should like before considering the material to indicate one or two very
important points. I am not concerned here with the truth or error of the con-
clusions drawn by Sir David Bruce and his collaborators. I am only concerned
with the nature of the process by which they have drawn their inferences. That
process consists in a measurement of the individual trypanosomes and an appeal
to the statistics of these measurements—in short to what I should term biometric
reasoning. There may well be other means of discussing the resemblances of the
different strains of trypanosome,—either by microscopic examinations of diver-
gencies in the life history of the different strains or by differentiation in their
action on different hosts, or otherwise. But in the present case the appeal to statistics
of measurement has been made. Drs Stepbens and Fantham in their paper on
T. rhodesiense (R. S. Proc. Vol. 85, B. p. 227) actually term their work a “biometric
study,” and the later papers of Sir David Bruce and others are no less “ biometric.”
Now if an appeal be made to statistics, then by a statistical method alone can the
answer be given. Further, that method must be the analysis of the modern fully
equipped and highly trained statistician. Such a statistician, and he alone, can
assert or deny on the basis of statistics the probability of any of these strains
of trypanosomes being samples of the same population; he alone is in a position
to judge the value of the evidence provided by the frequency distributions. If he
finds substantial “divergence” where Sir David Bruce and _ his collaborators
assert “sameness,” then either statistical theory is wrong, or Sir David Bruce
understands by “sameness” something quite different from the “sameness” of the
statistician, and something which cannot be judged by the methods of statistics, to
which accordingly no appeal should have been made, or only an appeal after a long
series of control experiments. The “sameness” postulated by Sir David Bruce is
something quite incompatible with the “sameness” found by the statistician when
he investigates two samples of 100 crania of the same race or two samples of 1000
blood corpuscles of two series of frogs of the same race. It is what the statistician
calls marked divergence and not sameness. If it be asserted that the extreme
divergence actually existing between the strains of trypanosomes statistically dis-
cussed is due to difference of individual host and not to difference of strain, it will
be clear that the divergence and not the sameness ought to have come out of the
statistical investigation, and then control investigations ought to have been made to
explain that divergence by environmental or other differences. But this is & priori
to assume the identity of the strains and a posteriori to seek an explanation of
marked divergence deduced statistically, whereas in the actual papers this great
divergence is assumed to be statistical sameness and this sameness used as an
argument for identity of strains. The statistician coming to the data critically
Biometrika x 12
90 A Study of Trypanosome Strains
does not of course assert dogmatically that any two strains are not of identical
race. What he does assert is that no argument for the sameness of the strains can
be based on the statistics provided; for these actually show wide divergence, and
he asks if the strains are @ priori assumed to be “same,” for a full @ posteriort
examination of the sources of the divergence.
The scope of the present paper is not the complete investigation of all the data
of the Royal Society Commission, nor an endeavour to obtain from the published
data the full conclusions which may be legitimately drawn from them. Its purpose
is to illustrate the statistical methods which ought to be applied to such material
and to indicate the essential necessity of control experiments on strains known to
be the same or accepted as different. A point should be noted here, namely, that
I have only found two cases where the strains on the basis of the statistical
evidence are said to be different. The first is in the case of Trypanosoma evansi
and Trypanosoma brucei. Sir David Bruce* gives (1911) the frequency distri-
bution of lengths of 820 individuals of 7. evansi and compares it by means of a
graph of percentages with 7. brucei. The percentages of the latter appear to be
deduced from the lengths for two series of 160 trypanosomes and 200 trypanosomes
cultivated in a variety of animals (Uganda, 1909, and Zululand, 1894) and pub-
lished in the preceding yeart, but no reference is given in the paper to the
original of the percentages in the graph, nor is any demonstration given in the
paper of 1910 of the statistical sameness of the Uganda and Zululand strains—
there is merely said to be “ marked resemblance{” where the trained statistician
finds marked divergence§. Stephens and Fantham|| use the curve of 1911 to
assert that there is a “general resemblance between the curves representing
the measurements of these trypanosomes (7. gambiense, T. rhodesiense, T. brucet)”
and consider that this “general resemblance” shows that “the method is a
trustworthy one.” It is not clear what “the method” referred to really sig-
nifies. The statistical comparison of means and maximum and minimum
lengths without statement of probable errors, and the mere graphical exami-
nation of frequency curves are wholly inadequate to determine sameness or
R. S. Proc. Vol. 84, B, p. 186, 1911.
|+ R.S. Proc. Vol. 83, B, pp. 5 and 11, 1910.
R
|
+ R. S. Proc. Vol. 83, B, p. 12.
§ The two distributions are as follows:
oa | 5 fe
|} 13) 14|}15| 16) 17) 18 | 19) 20 | 21) 22) 23) 24| 25 | 26 | 27 | 28 | 29 | 50 | 31 | 82 | 83 | 34 | 85 | Totals
(hers jth, elie lice ee i
Uganda, 1969 |—|—] 1, 2} 4} 6/10/26 14/14/12} 9]12) 6} 6/12/10) 7] 1) 2/3) 3)—/ 160
| Zululand, eee a 3} | 11 | 11 | 20 | 32 ma 4} 4) 3] 5/3] 7] 7 |10/13/13] 8 |10} 8} 3]1] 3] 200
| | |
These give x2=101:18, leading to P<-000,001, or not once in a million trials would two so divergent
distributions be obtained by sampling the same population.
|| R. S. Proc. Vol. 85, B, p. 233, 1912,
KARL PEARSON 91
divergence. Only a month later than Stephen and Fantham’s paper appeared
another paper by Sir David Bruce and others* comparing human trypanosomes
from Nyasaland with 7. brucei and T. rhodesiense and T. gambiense. But the
curve for 7. brucei is wholly different from that of a year earlier. Instead of a
minimum at 24 microns there is now a maximum at 24 microns, and the “ general
resemblance” of 7. brucei to 7’. evanst is much increased. We are now told that
T. rhodesiense (Stephens and Fantham) is “a distinct species, nearly related to
T. brucei and T. gambiense,” and the conclusion drawn that “the human trypano-
some disease of North-east Rhodesia and Nyasaland is not the disease known as
Sleeping Sickness in Uganda and the West Coast of Africat.” But the divergence
between the frequency distributions of 7. brucei and the human trypanosome of
Nyasaland when accurately measured is of exactly the same order as that which
suffices to demonstrate the identity of the human Nyasaland trypanosome and
T. rhodesiense. Thus the two cases in which divergence is asserted, i.e. (i) 7. brucez
and 7. evanst, (11) T. brucei and 7. rhodesiense, seem to be differentiated largely on
the base of unanalysed statistical evidence of a nature precisely like that which in
other cases is interpreted to mean close “general resemblance” or “sameness.”
We do not feel that we are in the possession of independent evidence of differen-
tiation which would enable us to test how far statistical divergency corresponds to
recognised morphological differences of strain,—a fundamental requisite if we are
to interpret as “sameness ” a statistical divergence of an extremely high order.
In concluding these introductory remarks we must refer to the types of
trypanosome in Nyasaland recognised by Sir David Bruce and his colleagues as
distinct on other grounds than numerical measurements. They are:
(a) TZ. brucei vel rhodesiense. This is said to be the cause of the human
trypanosome disease of Nyasaland. The modal length appears to be 24 to 25
micronst. According to Bruce and colleagues 7’. gambiense appears to have a
mode of 20 microns, but there is evidence for a submode at 26.
(i) TZ. pecorum. This is said to be the cause of trypanosome diseases of
domestic animals in both Uganda and Nyasaland. The modal length varies from
13 to 14§. There is no statistical evidence of bimodality.
(ii) 2. stmiae. This attacks monkey, goat and warthog. Oxen, dogs, white
rats, etc., are said to be immune. The length distribution appears to be very
homogeneous and with a single mode at 18 microns]}.
* R.S. Proc. Vol. 85, B, p. 431, 1912.
+ R. S. Proc. Vol. 85, B, p. 433, 1912. In 1913, however, we find that ‘there is some reason for the
belief that T. rhodesiense and T. brucei are one and the same species,” see Sir David Bruce and others,
R, S. Proc. Vol. 86, B, p. 407.
+ R. 8. Proc. Vol. 84, B, p. 331. Stephens and Fantham’s measurements on 7’. rhodesiense
suggest modes at 20 and 26. Ibid. Vol. 85, B, p. 231. The double mode—roughly 18 to 20 and
28 to 29—appears in the Zululand (1894) and Uganda (1909) strains of T. brucei. Ibid. Vol. 83,
B, p. 12.
§ R. S. Proc. Vol. 82, B, p. 468, and Vol. 87, B, p. 14.
|| R. S. Proc, Vol. 85, B, p. 477, and Vol. 87, B, p. 48.
92 A Study of Trypanosome Strains
Gv) ZL. caprae. This is found in waterbuck, ox, goat and sheep. The dis-
tribution of length is apparently homogeneous and the mode at 25 microns*.
I leave out of account several forms of trypanosome referred to by Sir David
Bruce and colleagues, e.g. 7. vwaa, T. uniforme, T. ingens, etc., of which no large
series of measurements were at my disposal.
With the exception of 7. simiae, which occurs in the warthog, the above
trypanosomes appear to be found generally in the wild game and all of them are
found in the Glossina morsitans. Sir David Bruce and his colleagues suppose the
differentiation into these classes to precede the consideration of individual strain,
but the exact modus differentrationis is not clear from the memoirs.
(3) Method of Investigation. The actual formula employed in the present
investigation is very simple and can be applied by anyone able to do ordinary
arithmetic. If N and N’ be the sizes of two samples and the corresponding
frequencies :
Joy Jas ise aise Tas eae Ss:
as Us ws; wise) VEN ile. fess
where fp, f, are the frequencies falling in the p™ category, then if
(Io _ tv \?
cou CEB
| et Te,
be calculated, the probability P that the observed or a greater divergence between
the two series would arise from sampling the same population is obtained by
determining P from y? by my method of testing “goodness of fit.” This method
was first published in the Phil. Mag., Vol. 50, p. 157, 1900. The shortest method
of actually determining P is by aid of Palin Elderton’s tables for P with argument
x’ issued in Biometrika, Vol. 1. p. 155, 1902. This is the process used in the
measurements of sameness and divergence provided below.
(4) On the Probability of the identity of the Strains discussed by Sir David
Bruce and others.
(a) I take first the question of the “sameness” of the Wild-game strains of
trypanosomes as isolated from five antelopes—reedbuck, waterbuck, oribi, and two
hartebeeste. Sir David Bruce and others discuss these strains in a papert of
February, 1912, and conclude, apparently from the statistical data, that ‘the five
Wild-game strains resemble each other closely and all belong to the same species.”
Now these Wild-game strains have a distinct advantage for they are all
obtained from the trypanosomes ultimately taken from the rat as host; they were
passed from the infected antelope through healthy goat, monkey or dog, which
* R. S. Proc. Vol. 86, B, p. 278.
+ R. S. Proc. Vol. 86, B, p. 407, 1913. In the Table p. 405 for 2500 trypanosomes under the
heading 31 microns read a frequency of 33 not 53.
Karu PEARSON 93
became infected, to the rat. The frequencies of lengths of the trypanosomes in
microns were as follows :
From Rat 15|16|17|18|19| 20
21 | 22| 23 | 24\ 25 | 26 | 27 | 28 | 29 | 30 | 31| 382 \ 33 | 384| 85
Hartebeeste (1) ...
Hartebeeste (2) ...
Oribi 2 ee
Waterbuck
Reedbuck
Mzimba (Donkey)|
Strain i
I questioned first whether the strains found in the two Hartebeeste were the
same; they give
x? = 108'69, and therefore P < ‘000,000,1.
In other words not once in 10,000,000 trials would two such divergent samples
arise if the Hartebeeste strains were samples of the same population. I now
compare the Waterbuck and the Oribi; these provide y? = 109°25 and P <-000,000,1,
and again the ewtraordinary divergence, not the sameness, is the statistical
feature. The reader may rest assured that equally incompatible results arise
when we compare the other antelopes. Statistically we are compelled to assert
either that the trypanosome strains in these different antelopes were different
species, or that, not only the infected species of antelope, but the individual
antelope of the same species (as in the case of the two Hartebeeste) immensely
modifies the strain of trypanosome. In short not the “sameness” of the strains,
but their great statistical divergence is the fact which impresses itself on the
biometrician. No biometrician could possibly accept the view of Sir David Bruce
and his colleagues that* :
“Tt is evident from these tables and charts that the various strains of. this
trypanosome, as they occur in wild game are remarkably alike. This is what
might be expected. Here the trypanosome is at home; it is leading a natural
life. It may be supposed to be saved from variation by constantly passing and
repassing between the antelope and the tsetse fly.”
Our authors, it will be noted, directly appeal for “likeness” of strains to the
tables and charts.
With these immense measures of statistical differentiation, we ask : what would
be the values of x? and P, if examples of differentiated strains of trypanosomes could
be found? If differences of host or treatment can produce these wide divergences,
how without a preliminary study of the same strain in different hosts and under
different treatments can we be certain whether these large divergences mean the
same strain differently treated, or different species of trypanosomes ?
* R. S. Proc. Vol. 86, B, p. 406.
94 A Study of Trypanosome Strains
(b) The next comparison I make is between the Mzimba (Donkey) Strain
taken through rats and the above wild-game strains. I have added the data for
the Mzimba Strain to the last table (p. 93): it is given by Sir David Bruce and
others in a paper on the Mzimba Strain*. I compare the Reedbuck and the
Mzimba (Donkey) strains first. We find:
xy? = 53°37, P =-000,05.
Thus only once in 20,000 trials would a divergence as great as this arise, if the
two strains were samples from the same population.
The results of comparing the Mzimba strain with Waterbuck and Hartebeeste (1)
- give respectively
x= 11423, P= <-000,000,1,
and x = 00, = = 000000R-
These give for practical purposes impossibility of a common source, thus still
further demonstrating that the marked feature of the wild-game and Mzimba
strains is divergence, not sameness.
Sir David Bruce and his colleagues writet : “The trypanosome of the Mzimba
strain is the same species as that occurring in the wild-game inhabiting the
Proclaimed Area, Nyasaland.” In an earlier paper a diagramt is given of the
frequency distribution of 3600 trypanosomes of Human strain taken from the rat
alone. These are drawn from four native cases of sleeping sickness in Nyasaland
and from one European case from Portuguese East Africa. As the individual
cases for the rats alone are not given, they have had to be read off the per-
centage diagram, but the frequencies must be very nearly correct. This Human
strain may be compared with the 7. rhodesiense, the T. brucei, the Mzimba
(Donkey) strain and a strain obtained from a native woman suffering from
“ Kaodzera,” the so-called sleeping sickness of Nyasaland. The frequencies of
these five strains are given in the following table. I first compare the trypano-
somes of Nyasaland given as (b) above with 7. bruce: and T. rhodesiense, for this
is the comparison made by the authors themselves§.
Taking the trypanosomes of Nyasaland (b) and the 7. brucei as figured in
percentage curves by Bruce and others, we have
y2=7217, P< -600,000,1,
or it is impossible to ascribe any degree of sameness to these two strains. We
now compare the Nyasaland strain (b) with 7. rhodesiense, and find
x? = 69:95, P=-000,01 ;
* R.S. Proc. Vol. 87, B, p. 31, 1913.
+ R. S. Proc. Vol. 87, B, p. 34.
+ R. S. Proc. Vol. 86, B, p. 301.
§ R. S. Proc. Vol. 85, B, pp. 431 and 433,
Karu PEARSON 95
thus once in 100,000 trials two such divergent samples might be drawn. Although
there is less divergence than in the case of 7. brucei and Nyasaland (5), it is idle
to speak of such a degree of divergence as sameness.
Length in Microns.
LONI | TAN 1S VLGNLT)| LS) | 19 | 20°) 21.) 22) 23 | 2h |.25 | 26 | 27 | 28 | 29
| Mzimba (Donkey) (4) ~... | —|—|— = | 2}14| 41] 91] 79| 56] 53] 38] 39] 22} 19] 16} 15] 9
Human, Native Woman (b)/—|—| 1] 4/19) 42) 63] 81] 75| 91) 65; 66; 93] 91) 107/}110/104| 87}
Human, mixed (c) ...|/—!—}|—]| 1] 4/46] 111] 159 | 219 | 288 | 312 | 365 | 359 | 314 | 314 | 231 | 218 | 198
T. brucei... eae ..{-—!| 5 | 8/14/17) 40] 63] 55) 66] 63) 75} 87} 93} 80) 82} 72) 50] 38
| T. rhodesiense >... ...| 1 | 3 |10]19|29/35| 67) 54] 92] 51) 74] 56] 68] 59} 85] 61] 72] 50
| |
| a c eee Se el
Length in Microns.—(continued).
| 380 | 31 | 82) 33 | 34) 85 | 36:| 37 | 38 | 39 | Totals Remarks
a — sz
Mzimba (Donkey) (a) ...| 2] 2/ 2 (=| - 500 | &. S. Proc. Vol. 87, B, p. 31.
eer? | | Rats only.
Human, Native Woman (b)| 49} 27/23/13] 7| 1) 1 |—|—|—| 1220 | &. S. Proc. Vol. 85, B, p. 427. |
| Various hosts.
Human, mixed (c) ... | 132} 125]}90/59/30/13] 8 | 2 | 2 | —} 3600 | &.S. Proc. Vol. 86, B, p. 301. Read |
; | from diagram. Rats.
TT. bruces... oe | 27] 26/18/11} 4) 4; —|—| 2 |-—| 1000 | &. S. Proc. Vol. 84, B, p. 331.
| | Read from diagram
T. rhodesiense sen Peamozipe2o (lid. 13) Oat | Lo) — a 1 | 1000 | & S. Proc. Vol: 85, B, p. 227. |
| Various hosts.
To further establish our point let us compare the Human strain (c) for 3600
trypanosomes with the 7’. rhodesiense. Here y? = 325'47 leading to P < ‘000,000,01.
In other words the great degree of divergence for the case of the Nyasaland native
woman is exceeded at least a thousand times, when we take the big example of
four natives and one European.
Sir David Bruce and his colleagues write of these strains :
“(1) The trypanosome of the human trypanosome disease of Nyasaland is
T. rhodesiense (Stephens and Fantham).” In other words the P =-000,01 is inter-
preted as sameness.
“(2) This is a distinct species, nearly related to 7. brucei and 7. gambiense,
but more closely resembling the former than the latter.” In other words they at
this date distinguished between 7’. brucei and T. rhodesiense*, and as a result of
this distinction proposed to call the human trypanosome disease of North-east
Rhodesia and Nyasaland by the name “ Kaodzera” as not being identical with the
sleeping sickness of Uganda and the West Coast of Africa. If we, however,
compare 7’. brucer and T. rhodesiense we find y? = 46°83 and P=-019. In other
* R.S. Proc. Vol. 85, B, p. 433, 1912.
96 A Study of Trypanosome Strains
words once in about 50 trials we might expect to get two samples from the same
population as divergent or more divergent than the distributions found for
T. brucei and T. rhodesiense. We have in fact in the cases of these two trypano-
somes reached our first instance of comparative sameness, and the statistics should
have shown Sir David Bruce and his colleagues that 7’. brucei and T. rhodesiense
were relatively the same, and though both differed from the human trypanosome
of Nyasaland widely, the approach to 7. rhodesiense was only slightly closer.
The accordance—speaking in a relative sense—of 7’. rhodesiense and T. brucei
was asserted by Stephens and Fantham in March, 1912*. In May, 1912, Bruce
and others, speaking of the 7. rhodesiense, term it a distinct species; in February,
1918, they say—although without publishing further frequency distributions—
that “There is some reason for the belief that 7. rhodesiense and T. brucei
(Plimmer and Bradford) are one and the same species,’ + and in a further paper of
the same month, “Evidence is accumulating than 7. rhodesiense and T. brucei
(Plimmer and Bradford) are identical{.” In May, 1913 (R. S. Proc. Vol. 87, B,
p. 34), we are told that the Mzimba strain is identical with the wild-game strain
and that “it has already been concluded that this species is 7. brucei vel T. rhode-
stense.” As far as the statistics of the subject go the only really weighty evidence
for the identity is that of 1912, on which, without statistical analysis, the
distinction between the two species was asserted.
(c) We will next consider the possible identification of 7. gambiense with
T. rhodesiense and with T. brucev.
The second identification is seggested by Sir D. Bruce and others in the words§:
“Whether these slight differences are fundamental or only accidental it is
impossible at present to say, but enough has been written to show that Trypano-
soma gambiense and Trypanosoma brucet approach each other very closely in
shape and size.”
The following table|| provides the data for 7. gambiense to be compared with
the distribution of 7. rhodesiense ranging from 12 to 39 in the last table.
Microns.
ont | | | | |
T5416 ee | 18| 19 ie 2) | 21 | 22 23 | 24| 25 26 | 27 | 28 | 29| 80 | 31 | 82 | 33 | 34| 35 | 86 | 37 | 88 | 39 | Totals
| | | | al |
| | | | | |
9 | 21. m6 9 114 aie 85 | 61 | 47 a ie Sila Opiellen | Aaa eet = ze - | 1000
lie =) I
The Sear adie are ran a ate of hosts.
For the 28 classes we have, y?= 140°27 and P<-000,000,1. The chief point
therefore is the complete divergence, not the resemblance of the two series.
* R. S. Proc. Vol. 85, p. 238, 1912. § R. S. Proc. Vol. 84, B, p. 332.
+ R. S. Proc. Vol. 86, B, p. 407. || R. S. Proc. Vol. 84, B, p. 330.
+ R.S. Proc. Vol. 86, B, p. 302.
KARL PEARSON 97
Stephens and Fantham, who term their work a “biometric study,” speak of
“the general resemblance between the curves representing the measurements of
these three trypanosomes (7. gambiense, T. rhodesiense, T. brucet).’ They con-
tinue: “We do not consider, however, that identity of measurement would
necessarily imply identity of species. We still believe that the difference in
internal morphology, namely the presence of the posterior nucleus, is sufficient to
separate 7’. rhodesiense both from 7. gambiense and T. brucei*.’ As a matter of
fact the “ biometric study ” of the data does not indicate identity in the measure-
ments, but confirms the result of internal morphology by proclaiming wide
differentiation +.
(d) We can now compare 7. brucei and T. gambiense. Of these Sir David
Bruce writes: “Whether these slight differences are fundamental or only acci-
dental it is impossible at present to say, but enough has been written to show
that Trypanosoma gambiense and Trypanosoma brucei approach each other very
closely in size and shapet.” The biometric commentary on this is that for length
of the two series yx? = 126°52, giving P< ‘000,000,1 and that as far as size is
concerned the samples ditfer immeasurably, ie. far beyond the limits of the
calculated tables of P.
We should thus conclude, merely from the statistical evidence, for close same-
ness in 7. brucer and T. rhodesiense but for marked divergence of both from
T. gambiense.
* R. S. Proc. Vol. 85, B, p. 233.
+ In a later section of this memoir I show that Stephens and Fantham have been markedly biased
in their judgment of even and odd units of measurement (p. 129 below), and that the recognition of
this makes a wide difference in the goodness of fit of my resolution into components to their data for
T. rhodesiense. It seems desirable therefore to inquire whether this bias affects the test of ‘‘sameness”’
of T. rhodesiense with T. gambiense, T. brucei, and the Human strains (b) and (c), see the Tables
pp. 95—6. The data were accordingly classified into groups of two microns, starting with 12 and 13,
14 and 15, etc., so as to get rid of the even bias as far as possible, and we find :
Old Unit Ranges New Two Unit Ranges
Strains compared
| on x? Ps n x? 1p)
|
TL. rhodesiense and T’. gambiense | 28 140°27 <'000,000,1 14 118°73 | < :000,000,1
T. rhodesiense and T. brucei ... | 28 46°83 019 | 14 25°76 ‘018
T. vrhodesiense and Human
strain (b) a ae reales 28 69°95 000,01 14 45°92 000,06
T. rhodesiense and Human |
strain (c) | 28 325°47 < ‘000,000, 01 14 253°37 | < -000,000,01
The bias towards even numbers of Stephens and Fantham has thus not substantially influenced our
results, which still show the relative likeness of 7’. rhodesiense and T. brucei, and the marked divergence
of the former from 7. gambiense and the human strains.
{ R.S. Proc. Vol. 84, B, p. 332,
Biometrika x 13
98 A Study of Trypanosome Strains
(e) It seemed well worth while to investigate how far the two Nyasaland
strains of Human Trypanosomes given in the table on p. 95 agree or differ. The
first (b) of these strains from a native woman of Nyasaland may be compared with
(c) a compound strain from four natives and a European. We find
x? = 172°36
giving P < :000,000,1.
In other words, the two Nyasaland strains from human beings are indefinitely
differentiated. I now compare the Mzimba (Donkey) strain* (a) with human
strains (b) and (c), we find:
for (a) and (b) x? = 22316 giving P certainly < :000,000,01 ;
for (a) and (c) x? = 348°55 . < 000,000,01.
Thus the trypanosome strain found in the donkey appears to be absolutely
incomparable with that found in man in Nyasaland, just as the strain found
in the donkey differed from that found in wild-game.
(f) We may now turn to a memoirt by Sir David Bruce and others com-
paring the Mvera cattle strain, the wild-game strain, and the wild Glossina
morsitans strain. They give on p. 18 of that paper the graphs for 500 specimens
of T. pecorum, the wild-game strain, and of the wild Glossina morsitans strain taken
from a variety of hosts. The following are the frequencies:
Microns.
_— * aa
Strain 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | Totals
== |. a
Mvera Cattle Strain eee Met bana a 15 | 64 | 101) 186) 114) 59 Seal 500
Wild-Game Strain ... we | —| — 2) 34 | 85) 172) 119 | 63 | 292) 3 | — 500
Wild G. morsitans Strain ...| 1 | 4 | 16 | 42 |129) 147/103) 42 | 15} 1 | — 500
We compare first Mvera cattle strain with the wild-game strain and find for
our 10 categories
x7 = 34554, P= 000,243.
This is a relatively low degree of divergence considering that P has been running
into 1 in 10,000,000! But it means that if these two strains were samples of one
and the same population, we should only expect two such divergent samples to
occur 1 in 4000 trials.
* This Mzimba strain of trypanosome is discussed in a paper headed: ‘Morphology of the various
strains of Trypanosome causing Disease in Man in Nyasaland.—The Mzimba Strain’ (R. S. Proc.
Vol. 87, B, p. 26); it is said to be of the Nagana type and is identified by Sir David Bruce and
colleagues with 7’. brucei vel rhodesiense, the source of the human trypanosome disease.
t+ R. S. Proc. Vol. 87, B, p. 4.
KARL PEARSON 99
Next we find for Mvera cattle strain and the wild Glossina morsitans strain,
x? = 40°508, or P=-000,008,
or only once in 125,000 trials would a pair of samples so divergent arise when
testing the same material.
Lastly, testing the resemblances of wild-game strain and wild G. morsitans
strain, we find
x? = 35°41, or P =°000,2,
not such a gigantic divergency as we have found in many cases, but a difference
so great that it only occurs once in 5000 trials requires explanation as divergency
and cannot be used as an argument for “sameness.”
It will thus be quite clear that as far as the measurements of length go, there
is wide divergence to be accounted for between the trypanosomes found in the
cattle, the wild-game and the tsetse fly, and that statistically this divergence is the
remarkable feature. Yet the conclusion of Sir David Bruce and his colleagues,
arguing very largely from the frequency distributions, is that “The Mvera cattle
strain, the wild-game strain and the wild G. morsitans strain belong to the same
species of trypanosome, 7. pecorum*.”
d
It will be seen that actual statistical analysis does not in any way confirm the
bulk of the conclusions reached by Sir David Bruce and his collaborators. The
strains may or may not be ultimately of like origin, but what is quite clear from
the analysis is that, if we are to rely on the measurements, then it is the diver-
gence, not the sameness of these strains, which should have been emphasised.
No stronger evidence could be deduced of the danger of appeal to statistics
when the statistics are not handled by the trained statistician. The mere appeal
to the resemblance of frequency curves given in the form of percentages, often
based on widely different totals, is an only too common error of medical investi-
gations ; it is by no means confined to the Scientific Commission of the Royal
Society, Nyasaland. But it has recently become so marked a feature of Series B
of the Proceedings of the Royal Society, that a vigorous protest is really needful.
Thus in the very last part issued (Vol. 87, B, p. 89) occurs a paper on “ The
Trypanosomes causing Dourine.” In this paper there may be microscopic evidence
to differentiate the strains A, B and C dealt with; on that I cannot express an
* A further conclusion is also reached (Ibid. p. 26) ‘‘7'. pecorum, Nyasaland, is identical with the
species found and described in Uganda.” Unfortunately the species found in Uganda is dealt with
in a paper (R. S. Proc. Vol. 82, B, p. 468) which provides no frequency distributions, and does not tell
us the total number on which the mean length—13°3 microns—is based. The mean value of the
T. pecorum, Nyasaland is 13-954 (R. S. Proc. Vol. 87, B, p. 3) and the standard deviation is 1-393 in
microns, thus the probable error of the mean is °67449 x 0623. Assuming the Uganda trypanosome to
be the same strain and to have the same variability as the 7’. pecorum, Nyasaland, the difference of the
means = ‘654, with a probable error of *67449 x /2 x 0623 =-67449 x -088, thus the deviation of the means
is 7:73 times its standard deviation. A deviation so great would only occur about once in 4 x 10" trials,
i.e., would be practically impossible if the two strains were identical. Here again it is excessive
divergence not sameness which the statistics indicate.
13—2
100 A Study of Trypanosome Strains
opinion. But on pp. 92—3 percentage frequency curves are drawn for the three
strains, and the following remark is made :
“A survey of the curves obtained by plotting out in percentages the various
lengths of trypanosomes encountered in each of the three strains is of interest.
It will be observed that in the case of rats the curves of each of the strains corre-
spond fairly closely.”
Now what do the authors mean by “fairly closely”? In their conclusions
they identify B and C and differentiate A. Unfortunately they have not given
their actual frequencies, and I have had to endeavour to reconstruct them from
the percentage curves. There results for the rat-data:
Microns.
| i eel ier ae | |
| 16|17|18| 19 | 20| 21 | 22| 23\ 24| 25 | 26 | 27] 28| 29 | 30| 31 | 82| 83 | 84| 85 | 36 | Totals
| | | | | | |
oo | | | | | cae | | | z
| [oe es te | ae ea | |
Berlin Strain A . | 1 | 1 {10] 9 |12/17|17)22|28]48 |47 |57/55|42 |39!37|/28/13| 81 6 | 3 | 500
Frankfurt Strain B...]—|—| 1] 3 | 5) 1] 4/10] 20/29*|18* | 25 | 24 | 35* | 23} 15)18}15] 8 | 3
East Prussian Strain C}—-| 1 | 4 3 | 6|12)15)22|24)27 | 28-|37|31])16 |10; 7) 5; 2);-—|—
ee 7 . : | | | | | |
We obtain the following results:
Strains A and Bs) ¥7=3111, P= 0627,
Strains A and C: y?=43°37, P= -0034,
Strains B and @: 4°=72:72, P=<:000,001
Thus to judge from rats only, Bb and C are far more divergent from each
other than either is from A; in other words the strain A is intermediate between
B and C and closer to 5, from which it is not immensely divergent; two such
samples as A and B might, as far as the length distributions go, be drawn from
common material once in 16 trials.
Now of course no one suggests that a conclusion drawn from this rat-material
is to replace one drawn from guinea-pig material, but the statistician cannot agree
that for rats “the strains correspond very closely”; and he finds it illogical to place
the evidence of the rat-data on one side and proceed to draw conclusions from the
ocular inspection of the guinea-pig curves, without noticing that the conclusion is
markedly opposed to the proper deduction from rat-data. Indeed while the guinea-
pig-datat give a relatively high degree of relationship between B and C (P =:0157)
it is not as high as the rats give between A and B (P=-0627); and while the
* The values given by the percentage graphs in these cases are respectively 21, 17 and 34, and
the total appears to be 247 and not 250 as stated. Hither 247 were used or the graph is in error.
The three individuals were introduced in a way calculated not to increase divergence.
+ The frequency distributions for the guinea-pigs have had to be reconstructed from the percentage
curves, the necessary data not being published by the authors.
KARL PEARSON 101
relationships of A and B (P< -000,000,1) and A and C (P< 000,001) are very
low, the origin of the second hump in the guinea-pig distribution for A requires
much more analysis and the certainty by control experiments, that it always
repeats itself, and is not the result of hitting a “ pocket.”
¢
It seems to me that any statistical analysis by modern methods of the trypano-
some data compels us to confess that either statistical methods must be discarded
entirely in these trypanosome investigations, or they must be pushed to their
logical conclusion, and used as the fundamental instrument of research which can
guide our enquiries by inference and suggestion when, and when only, it is handled
by the trained craftsman. Thus far the use made of statistical methods seems
merely to have confused the issues, and brave would be the man who would venture
to say after reading this section of our present paper that any two strains discussed
by the commission are definitely “same” or certainly differentiated.
(5) On the Probability that the Animal in which the Trypanosome vs cultivated
makes essential Differences in the Distributions of Frequency.
But the very method which casts apparent discredit on the results at present
reached seems able to lead us to definite conclusions provided we start with it as
the fundamental mode of investigation. Really very little inspection seems to indi-
cate that not only the host but the period of infection materially influences the
frequency distribution. These points have not been wholly disregarded by the in-
vestigators in this field, but they have had no quantitative measure by which they
could appreciate the relative influence of the various environmental factors. Nor
indeed could the method be fully applied without experimental observations on
trypanosomes of the same strain subjected to differential treatment. Knowing in
such cases the quantitative divergence produced, we should be in a position to infer
whether two strains from different sources were separate species or merely modified
by differential environment. Until we have such quantitative measure no hypothesis
of sameness or difference can flow from statistical treatment; nobody as yet knows
how much to attribute to environment, how much to attribute to individuality
of strain.
In endeavouring to throw light on this matter we are, however, checked at
the very start by the absence of effective material. In some cases the period of
infectivity is not given; in others we are not always able to break up the total
frequency by reference to the host, or to a single host. And even when we merely
classify by one type of animal as host, we may have reduced our material to such
small numbers that samples may be “same,” which on larger numbers would
show the marked divergence due to the emphasis of smaller differences*. Some
suggestive points can, however, be effectively dealt with and they are treated in
the following paragraphs.
* It may not be possible to differentiate Bavarian from Wiirtemberger on samples of 50 crania,
although quite possible on samples of 400.
102 A Study of Trypanosome Strains
(a) I ask what difference is made when a strain is passed through various
animals (goat, monkey, dog, rat) or through a single animal alone. Taking the
wild-game strain discussed by Sir David Bruce and others*, we have:
Microns.
10 | 11| 12 | 18 | 14 | 15 | 16 | 17 | 28 | Totals
Wild-Game Strain
(from various ae
Wild-Game Strain
(from a single rat 510)
Here we find y? = 65°37 and P < ‘000,000,1. In other words the distribution of
lengths of the trypanosomes of the wild-game strain obtained from various animals
differs so enormously from that obtained from a single rat that the two cannot be
looked upon as samples of the same population. The moment this result is realised
we appreciate that (1) it is impossible to compare two strains developed in a variety
of animals unless we have previously tested on the same strain the equal valency
of these animals, (11) a series of animals of even the same species may quite
possibly give widely divergent results from those obtained for a single animal.
Thus passing from a variety of animals in wild-game strain to a variety in wild
G. morsitans strain makes less difference (P = ‘000,008)—although great enough—
than passing from a variety of hosts to a single rat in the wild-game strain.
This rule is not universal, but it illustrates the absolutely essential need for
testing the effect of change of host before questioning the identity or non-identity
of two strains.
(b) I now turn to the Mvera cattle strain, and ask what differentiation is
produced by the dog and goat as hosts. The data are very sparse and unless we
get a high degree of resemblance may be worth little. They run+:
* R. S. Proc. Vol. 87, B, pp. 6 and 8.
+ R. S. Proc. Vol. 87, B, p. 3. I tested the relative interchangeability of goat and sheep in the case
of T. caprae. The data are as follows: (R. S. Proc. Vol. 86, B, p. 280)
Microns.
eee ie | | Wi tiee pee a |
T. caprae | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 82 | Totals
—— | |
Goat .. |—|—] 8 | 7 | 11] 85 | 43 | 50 | 88 | 28) 27; 17) 5 | 1 | —} 260
Sheep... | — | — J 1 | 10 | 12 | 29 | 39 | 31 | 28 | 20 5 | 3 id 1 180 |
leading to y?=18-088 and P=-1133 or the resemblance is considerable although not so great as we find
between goat and dog for the Mvera cattle strain.
KARL PEARSON 103
Microns.
| > | ]
| DON Le | 12" TS tp \\ 15 | 16 | 17 | Totals
|
fs | | |
Mvera Cattle, Goat ... | 1 Wed ala 225 26°) 19 is.) i 100
” ” Dog |
a eon ot lala ag ZI 8, | =— | 100
We have x? = 5°396 leading to P= "714, or in 71 pairs of samples out of 100
from a homogeneous population, we should get more divergent results. It follows
therefore that, as far as these small series of this strain go, goat and dog are
interchangeable as _ hosts.
Let us go a stage further and ask whether ox is interchangeable with goat and
dog. The following is the frequency distribution for the trypanosomes through
the ox:
Microns.
9 10 ae ei 13 || Lo |G NLT | is Totals |
, a\2 _|
| |
Mvera Cattle, Ox ... | — | A Pis | 33 | 44 | 49|91| 7 | 1 | 180 |
|
Compared with the goat strain, this gives
‘x? = 9559 and P =°3888,
and compared with the dog strain
xy? =9:461 and P= ‘3973.
Thus in about two out of five trials from a same population we should get
pairs of samples differing more than the dog and goat strains do from the ox
strain. We conclude that while for practical purposes dog, goat and ox strains in
the Mvera cattle trypanosomes are interchangeable, yet the dog and goat strain
are nearly twice as much alike as the ox strain is to either. Lastly—although it
is rather a rash proceeding—I compare rat with goat and dog. It is rash because
only 40 trypanosomes through the rat were measured, and this is wholly inadequate
for real determination. The frequencies for the lengths are:
ae oe ee ee ae
| |
| 9 | LO tL | 12) 18 14 | 15 | 16 | 17 | Totals
| |
—| | are
Mvera Cattle, Rat ... sas fe NN I | alee oy 40
; » Dogand Goat | 1 | 1 | 6 | 25 | 49 | 56 | 40) 21) 1 | 200
We find y?=21'329 and P=-0064. The small series of rat trypanosomes
probably accounts for no smaller value of P, but the odds of 155 to 1 are
sufficient to show that rat series must not be mixed with series from the goat,
104 A Study of Trypanosome Strains
dog or ox. This confirms the view obtained for the wild-game strain, that a
strain taken through the rat as host is incomparable with strains from other
animals.
(c) The totals considered for one species of host in (a) and (6) are rather
small. Larger numbers are forthcoming for the so-called Mzimba strain of
trypanosomes taken from a donkey at Mzimba. The frequencies are here*:
| ” ”
Microns.
16 | 17 18/19 | 20 | 21 | 22| 23| 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | Totals
| ; | | } | | | | le 4 | | |
Re cal as
Mzimba Strain, pos | 3 17 | 56 69 | 67 | 47 | 27 | 22 | 10/12] 7 4| 4 | 4|2)1 | 360
e | 2 i 41 91 79 56) 53 | 38/39 22/19/16/15 | 9 | 2/2/21) 500
este
We find y? = 25499 and P=-0619. Thus only about once in 16 trials should
we get such a degree of divergence as the two samples present, drawing them from
the same population. This is very far from such a divergence as we have noted
in the rat and dog for the Mvera cattle strain, or in the case of rat against other _
animals in the wild-game strain, which was extremely large. The only expla-
nations that occur to me here are:
(i) In the case of the wild-game strain and the Mvera cattle strain a single
rat seems to have provided all the trypanosomes, while in the case of the Mzimba
strain two rats were used; this might lessen the influence of individuality.
(ii) In the case of the Mvera cattle strain and the wild-game strain the
trypanosomes were ultimately taken from a great number of individuals. In the
Mvera cattle case we are told that 32°/, of the herd were affected, and we have
some details of 16 head of cattle and 5 donkeys naturally infected+. In the wild-
game case, the wild game affected were very numerous, covering cases of eland,
reedbuck, waterbuck, bushbuck, oribi, koodoo, hartebeeste, buffalo and hyaena.
Now can we start with the hypothesis that all the individual cattle and all the
individual wild game were each bitten by a fly carrying the same strain of
trypanosome? Have we any more right to suppose @ priort that one wild-
game strain of trypanosome and one cattle strain of trypanosome exist, and ask
whether these two are identical, than to ask whether the strains carried by hyaena
and hartebeeste are the same? We have already (p. 93) seen that the strains
from two hartebeeste are extremely divergent. What right have we @ priort
to classify all wild-game trypanosomes together and call them a wild-game strain ?
And if two antelopes, whether of the same or of different species, give widely
different results, why are the trypanosomes of oxen of the same herd or donkeys
and oxen from the same neighbourhood to be classed @ priort as of one species ?
* R. S. Proc. Vol. 87, B, p. 31.
+ R. S. Proc. Vol. 87, B, p. 15.
KaruL PEARSON 105
If we turn to the Mvera cattle, we find there were four sources of trypanosomes
for the ox, two for the goat, and the same two for the dog—these two sources being
two of the four cattle sources. There was only one source for the rat, but I have
not discovered how far it was identical with one of those for ox or goat*. In the
Mzimba donkey strain there was one source for dog and rat. In the wild-game
strain there were, I make out, ecght sources of trypanosomes for the goat, four for
the dog, and only one for the rat.
Thus the individuality, which might be supposed to influence the result,
because we are treating of trypanosomes in this case from a single rat, in the
Mvera cattle case from a single rat, and in the Mzimba data from only two rats,
may really arise from the fact that the rat strains in each case are derived from a
single source, while the dog, goat and ox strains show a multiplicity of sources.
The troublesome point is that the experimental part of the work has not been
designed to answer what seem to me fundamental questions. We cannot directly
inquire what difference the host makes because different hosts have rarely been
treated with the strain from a unique source. We can say that dog and goat are
interchangeable for the Mvera cattle strain, because both drew trypanosomes from
the same two sources ; but we cannot determine whether the difference in the ox
is due to difference of the host, or to the introduction of two more sources. Simi-
larly the divergence between the trypanosomes from rat and from other animals for
the wild-game strain may be due to using one rat and therefore one source, and not
the many sources of the other animals, or it may really be due to the differentiation
of the host. In the same way the difference between the two hartebeeste may be
due to individuality in the same species, or to infection from different strains.
(d) To some slight extent we may appreciate the effect of individuality by
comparing the two rats 512 and 513 im the case of the single source, the Mzimba
strain f.
The frequencies are as follows :
Microns.
| | Neal ee lee |
| Mzimba Strain | 16 | 17 | 18 | 19| 20 | 21 | 22 | 23 | 24 | 25 | 26
| | | |
|
|
|
|
7/1/1/2] 240
2/1 He 260 |
Rate? °... |—| 5
(Rat 513. 2) 9
{
The numbers are not as large as we should like; but they give
y2=17'89, P ="3306.
* R.S. Proc. Vol. 87, B, pp. 2 and 15.
+ R. S. Proc. Vol. 87, B, pp. 6 and 8 compared with 5. Rat from p. 8.
t R.S. Proc. Vol. 87, B, pp. 29 and 31.
x
Biometrika 14
106 A Study of Trypanosome Strains
Clearly then two samples as divergent as those found wonld occur on the
average once in three trials. It follows that two individual rats are really inter-
changeable and we note that the extent to which ox is interchangeable with dog
or goat for the cattle strain is very much the degree in which two rats are inter-
changeable. To judge from this single instance, individuality within the same
species of host is not very important, and when we find two hartbeeste differing
as those considered on p. 93, it seems much more likely, with the information we
have at present got, that the hartebeeste were infected with different strains of
trypanosome than that their individuality produced the enormous divergence
noted. Again the sensible divergence between Mzimba strain in dog and rat on
p. 104 is probably due to difference of host, but the enormous difference in the
wild-game strain between a single rat and dog and goat on p. 103 is probably due
to differences in the strains of trypanosomes in the various types of wild game
dealt with. We may consider whether the dog and goat data for the wild-game
strain differ sensibly. We have*
Microns.
i oe : :
| | 11 | 12 | 18 | 14 | 15 | 16 | 17 | 18 | Totals
+ :
| Wild-Game Strain, Goat ... | 1 | 16] 37 | 73 | 38) 26| 8 | 1 | 200 |
| 3s = 5 Dogs... | — | 12) 31))957 | 50 | 24 | 6. | ==s\eas@ |
| eal a
Here y?= 6:04 and P ='5378. Thus in more than half the trials we should
obtain from homogeneous material pairs of samples more divergent than those for
dog and goat. This confirms the view formerly expressed that as far as trypano-
somes are concerned dog and goat are interchangeable. We cannot yet say that
they are not interchangeable with the rat, as the mixture of strains in dog and
goat and the uniqueness of strain in the rat may account for the marked
divergence of the latter. Sir David Bruce and his colleagues do not appear to
have noticed the wide divergence of the distribution of the rat from the dog and
goat either as indicating the heterogeneity of the wild-game and the cattle strains
of trypanosomes, or as suggesting such wide differentiation of strain by the host, that
rat-material cannot be mixed with that from dog and goat. They do, however,
remark of the wild-game strain: “In this the rat is not a suitable animal, since
many strains of 7’. pecorwm have no effect on it}.” This suggests that 7. pecorum
is not homogeneous and that the rat exercises a selective influence on its strains.
The suggested rejection of the rat data seems, however, to be based upon the in-
convenience of its non-infectivity, and not on what might turn out to be of great
importance a selective influence on wild-game or cattle strains. It is not possible
to test this selective power in the present mstance, as we do not actually know
how heterogeneous either the cattle or wild-game material used really was.
* R.S. Proc. Vol. 87, B, p. 7.
+ R. S. Proc. Vol. 87, B, p. 7.
Karu PEARSON » 107
(e) If we turn to the 7. pecorum strain as actually found in the tsetse fly, we
see that Sir David Bruce and his colleagues deal with these trypanosomes passed
through a variety of animals, of which only goat and dog supply sufficient numbers
for any even approximately accurate treatment. The data are as follows*:
Microns.
beg. zo") 27 es 14| 15 | 16 | 17 | 18 | Totals
= rales ; | |
Wild G. morsitans strain: Goat | 1 | 3 | 12 | 21 | 55 | 60 | 32 | 12) 4 | —} 200
|
. ¥¢ ” Doe |=") == 13) 144) 34. 41, | 40 | 19:1 9 | — | 60
i | Ea
Wild G. morsitans strain: Rat | — | 1 | — | 3) 22 | 28/19] 6 | 1 | — 80
For goat and dog we find y?= 19°518, which give P=-0125. The resemblance
is therefore far less than we have found for goat and dog in other strains, only
once in 80 trials from homogeneous material would two samples of such divergent
character arise. Before we comment on this it seems desirable to compare the
very inadequate rat data.
For rat and goat we have
x? = 12201, P='1434.
For rat and dog we have
x?=11:370, P =-1245,
Accordingly we see that for this material the rat strain (i) lies between the
dog and goat strains, and (11) is definitely interchangeable with dog and with
goat, while the dog and goat are much more divergent. Now the sparsity here of
all the data must prevent any dogmatism; all we can reach is suggestion for
further investigation. But the following points should be notedt. The trypano-
somes through the goats were obtained from sva different goats, infected directly
from the wild fly; the trypanosomes from the dogs were obtained from only four
different sources, namely from a monkey directly infected by the wild fly, from a
dog directly infected, and from two goats (89 and 125), the former only of which
is identical with one of the former six goat sources. Lastly, the rats were infected
from one dog alone, upon which the tsetse flies had directly fed. This dog is not
identical with one of the dog sources. Now unless we assume that all the strains
of the trypanosome found in the tsetse fly are identical—which is certainly not in
accordance with the differences found in the strains of wild game from the “ fly-
country ’—it is by no means certain that the trypanosomes obtained from wild
G. morsitans, through goat, dog and rat as above noted came from anything like
the same sources. Further, the closer resemblance between rat and dog strains
* R. S. Proc. Vol. 87, B, p. 11.
+ R. S. Proc, Vol. 87, B, pp: 10, 11, and 19 to 22,
14—2
108 A Study of Trypanosome Strains
may simply be the result of the rat strain having been developed in the dog as
host. The divergence between the dog and goat strain may again be solely due to
the greater variety of sources in the goat. The data from the wild G. morsitans
experiments seem to indicate that the observed divergences between the strain
from rat and the strain from goat or dog may not be due to difference of host;
but to difference of source from which the material was drawn, and to difference of
treatment of the individual stock of trypanosomes, e.g. the number of hosts, ete.,
through which it has passed.
It seems absolutely certain that at the present time most light would be
thrown on the conditions for asserting sameness or diversity of strains, by well
devised experiments on strains from single sources passed through different species
of hosts in different manners, in order to determine the exact measure of divergence
produced by host and by treatment, and ultimately to devise a standard treatment
for all strains which we desire to compare.
The exact nature not only of host, but of standard treatment is most vital. We
can demonstrate the influence of treatment at once by considering the “ percentages
of posterior nuclear forms among short and stumpy forms” recorded by Sir David
Bruce and his colleagues for the wild-game strain*. All the trypanosomes were
from rats, and although the date of infection of the rat is, I think, not stated, the
dates of first extraction will be after much the same interval, and we can therefore
classify by date from first extraction. We find the following table:
Wild-Game Strains.
Percentage of Posterior-Nuclear Forms among
Short and Stumpy Forms.
From first Extraction 21°/, and under 22°/, and over | Totals |
|
6 days and under 18 6 24
| 7 days and over 6 18 24
| Totals 48
Using Sheppard’s formula for the four-fold table, we have for tetrachoric r
Oe
or, the correlation between this character of the trypanosome and the time
after infection of extraction is very considerable. It will be obvious that in a
standardised treatment this time of extraction will play a most important part.
But it again is not independent of the species of trypanosome, for if we take the
wild Glossina morsitans strainst, we find :
* R.S. Proc. Vol. 86, pp. 396—404, Tables III, VI, IX, XII and XV.
+ R. S. Proc. Vol. 86, B, pp. 410—418, Tables III, VI, IX, XII and XV. I have added one percentage
by random selection from the complete table by lot in order to give 60 cases, and save labour in
fractionising.
Kart PEARSON 109
Percentage of Posterior-Nuclear Forms among
Short and Stumpy Forms.
From first Extraction 7 °/, and under 8 °/, and over Totals
6 days and under 18 30, |
12
7 days and over
Totals 30
leading to r = — ‘309.
In other words using tsetse fly strains and not wild-game strains, but the same
host, we find that now the correlation is negative or the longer the infection the
smaller the percentage. Actually the five G. morsitans strains show remarkably
irregular results compared with the results for the wild-game strains; the ex-
tractions were spread over much the same period, 13 to 14 days on the average,
but were somewhat more numerous for the G. morsitans. Thus even the same
method of extraction may give widely varying results according to the nature
of the strain producing the infection, although the host be the same.
To the statistician who examines the frequency distributions provided by
Sir David Bruce and his colleagues for both wild-game strains and Glossina
morsitans strains, there can hardly remain a doubt about the heterogeneity of
the material in each case. We have already demonstrated this statistically for
the wild-game strains. These strains not only differ by immense differences
inter se, but intra se they are clearly heterogeneous. Whether this heterogeneity
is due to the mixture of separate strains, to dimorphism within the strain, or to
the combination of material drawn from the rat at various stages of infection, it is
not possible on the material at present available to determine finally. The same
remarks apply with even greater certitude to the wild G. morsitans strains than to
the wild-game strains. But we shall return to this point in the last section of this
paper. We have already noted that Sir David Bruce and his colleagues identify—
against the weight of the statistical evidence—the Mvera cattle strain, the wild-game
strain and the wild G. morsitans strain as belonging to the same species 7’. pecorum*.
They had previously identified other strains in wild game, G. morsitans and human
beings} with 7. rhodesiense which they elsewhere describe as vel brucet{. This is
again, I hold, against the weight of statistical evidence. But it is not clear from
the memoirs themselves what is the exact process by which an individual fly, an
individual human being, or the blood from a specimen of wild game is credited
with carrying a homogeneous strain. The sizes are so different in the cases of
T. pecorum and T. simiae that there may be no difficulty in distinction, but the
range is so great and to the statistician the material seems so heterogeneous in the
ease of T. brucei vel rhodesiense that, perhaps, a fuller description by the authors
* R. S. Proc. Vol. 87, B, p. 26.
+ R. S. Proc. Vol. 86, B, p. 42.
+ R. 8. Proc. Vol. 86, B, p. 426.
110 A Study of Trypanosome Strains
of the process of differentiation would aid him. This is of especial importance
if it should turn out, as I suspect, that the trypanosomes classed as T. brucei are
either dimorphic, or belong to two different species.
In another paper* we find the trypanosomes from G. morsitans, on the basis of
their infective powers on monkey, goat and dog, resolved into 7. brucei vel rhode-
stense, T. pecorum, T. simiae and T. caprae. But it is clear that the differentiation
was not done solely by infectivity, or there would have been no means of dis-
tinguishing 7. bruce: and T. pecorwm which attack all three—monkey, dog and
goat. The question arises, whether 7. pecorum, T. simiae and T. caprae being
readily identified by microscopic examination or size, the remainder was classed as
T. brucei, in which case the question of the heterogeneity of this group, which
appears to attack all animals, is rather supported than otherwise by this paper.
Frequencies of the Various Strains for Length.
Length in Microns.
| -.| | | | | |
Strain 9 |10)11| 12 | 28 °\ Th \eioy| 6 ety 78 | 19 20 21 | 22 | 28 | 24 | 26 | 26
| Ae aaa .
T. pecorum 2 | 6 | 42/193 | 452] 618/ 453/178] 51] 5 mee
T. simiae —{|—|—| — | — | 7) 28) 76) 93) 126) “92)/0 47i\" 221) eG io) ae ee
T. caprae —;/—-—|;-|]—- 1}— | 3] 8) 28) 49) 79.) 95) 80
|
(i) 7. rhodesiense —i—|—]1 3 10} 19] 29] 35! 67 | 54) 92 | 51| 74}> 56! 68 59| 85
Gal Ze bruce, 62-9) — ao 8| 14] 17} 40] 63) 55 66 63)| 75) 87) 93] 80] 82
(ii) 7. gambiense... | —|—|—]— |} 1 | — 9| 21| 56| 79|114/122 110] 85] 85] 61] 47] 49
(iv) Mzimba Strain | 8| 27) 791175 189 139/109} 72) 66) 36) 32
(v) G. morsitans ... | = 7| 3L| 148 | 230 | 326 252 237 | 184) 143 | 115 | 130 | 110
(vi) Wild Game ... | 1 8} 53/118 | 252 381 | 348 | 285 | 200 | 162 | 149 | 135
(vii) Human Strain | — | —|—}— | — 1} 10/ 41/154) 325 | 494 | 528 577 | 512 | 525 | 511 | 464 | 425
| (viii) Chituluka... | |= | - 1 8} 48} 81, 78) 71) 44, 46] 56) 53) 98) 120
| | | | |
Length in Microns—(continued).
| | jie | | | |
Strain 27 | 28 | 29 | 30 | 31 | 32 | 33 84 | 85 | 36 | 37 | 38 | 39 | Totals} Source
- | ee ae | aes
T. pecorum sti - | | |—|—| 2000 | R. S. Proc. 87, B, p. 13
7. simiae | esa ea 500 | Ibid. 85, B, p. 477
T. caprae 68| 57; 24) 9) 2) 2)/—|—|} - 500 | Zbid. 86, B, p. 278
(i) TZ. rhodesiense | 61| 72| 50] 52| 28| 13/13|-5| 1| 1|--|—| 1] 1000 | Jéid. 85, B, p. 227
(i) 7. brucei 72| 50) 38| 27) 26) 18)11} 4) 4)/—|—j| 2 |—] 1000 | Jbed. 84, B, p. 331
Gii) ZT. gambiense 47| 44) 31] 20] 11] 4] 4 - -—|—-| 1000 | Ldzd. 84, B, p. 330
(iv) Mzimba Strain | 24] 22) 16) 7) 4] 4)—| | —|—| 1000 | Zbcd. 87, B, p. 31
(v) G.morsitans ... |127|133/113} 96; 54; 44/11) 7; 2;—|—|—]/— | 2500 Lbid. 86, B, p. 419
(vi) Wild Game ... | 125/110] 62} 55} 33] 12] 7/ 3} 1|/—|--|—|—| 2500 | Jbid. 86, B, p. 405
(vii) Human Strain | 372] 347 307 | 198 | 167 | 123 | 77 | 36} 12/11} 2 | 1 |-—| 6220 | Zoid. 86, B, p. 330
(vill) Chituluka 111 | 128/138} 99/117} 91/63/27/11| 9) 1 | 1 |—J| 1500 | Zbed. 86, B, p. 291
* R.S. Proc. Vol. 86, B, p. 422.
KARL PEARSON 111
At any rate the exact method of differentiation adopted would be of interest
to the statistician. The result of the paper is that the four species of trypanosomes
occur in quite comparable permilles of tsetse flies caught in the sleeping sickness
area of Nyasaland, and there is no evidence to show that they or other strains also
may not occur side by side in the same fly or in the same specimen of wild game.
Further, these compound strains would then appear in different proportions in the
host. Some such hypothesis seems very needful to account for the extreme
heterogeneity of the wild game, wild G. morsitans, and human strains as recorded
by Sir David Bruce and his colleagues. The following table gives a comparison of
what appear to be homogeneous strains—T. pecorum, T. simiae and T. caprae—
with what appear statistically to be heterogeneous strains, ie. 7. brucer,
T. rhodesiense, T. gambiense, the Mzimba strain, the wild-game and wild G.
morsitans strains of human type, and the human strains themselves. The table
Means, Standard Deviations and Coefficients of Variation of eleven Strains
of Trypanosomes.
Seri M Standard Coefficient
Bs ean Deviation of Variation
T. pecorum 13°992 + :019 1°2816 +014 9°16 + ‘099
T. simiae 17°870 + 050 1°6558 + ‘035 9°27 +°199
T. caprae 25508 + :063 2°1011 +045 8°58 +184
() T. rhodesiense ... 23°577 +°100 4°6764+ 071 19°83 +°311
Qi) ZT. brucei 23529 + 094 4°3938 + ‘066 18°67 +°291
Qui) Z. gambiense 22°113 + ‘081 3°7867 + ‘057 17°12 + °266
(iv) Mzimba Strain... 217413 +:063 2°9586 + 045 13°82 + 212
(v) G. morsitans 22°695 + 058 4°3002 + 041 18°95 + 187
(vi) Wild Game se 22622 + 047 3°4541 + 033 15°27 +:'174
(vii) Human Strain ... 23°796 + °035 4°1262 + 025 17°34+°108
(viii) Chituluka 26°172 + ‘084 4°8414+.060 18°50 + °235
above, gives the means, standard deviations and coefficients of variation of these
strains. It will be seen that the first three are of a very different character to the
last five. The variation of the latter is about double that of the admittedly pure
strains, and throughout the whole course of our further work this possibility of
heterogeneity, and the differential selection of the components by the host must
be borne carefully in mind. Great divergences do not discourage the use of
biometric methods, and we get occasionally identities of strains which are quite
beyond the limits of chance coincidence and which point to definite possibilities if
only host, environment, and treatment are once effectively standardised. I propose
to try to throw some light on these points in the remaining sections of this paper.
(6) On the Probability that Strains are alike after allowance for the Host.
(a) Luckily in certain cases the treatment has been more or less alike. Thus
in the wild Glossina morsitans strain, the tsetse flies brought to the Laboratory
112 A Study of Trypanosome Strains
from the “ fly-country” were in one strain (I) fed on a monkey and in the case of
four other strains (II to IV) fed on dogs. From these animals thus infected others
were inoculated, but in each case only the trypanosomes from a single rat were
used for purposes of measurement and comparison. ‘The following table gives the
frequency distributions of the five strains, and chiefly on the basis of these
distributions, Sir David Bruce and his colleagues conclude that:
“The five wild Glossina morsitans strains resemble each other closely, and all
belong to the same species of trypanosome.” (p. 421.)
Wild G. morsitans Strains*.
Microns.
Strain I
|]
Se Oe
Investigating the statistical measure of resemblance
230 | 326 | 252 | 237
| |
| 25
143115 130
7
the following series of results :
Strains I and IT: x? = 81°88, P < 000,000,1,
Strains I and III: Va aLoosil, P < :000,000,01,
Strains I and IV: ye OOS: P < :000,000,1,
Strains I and V: 2— 115°77, P < :000,000,1,
Strains II and III: x? = 32812, P < :000,000,01,
Strains II and IV: x? = 184°88, P< :000,000,01,
Strains II and V: x? = 208:79, P < :000,000,01,
Strains III and IV: x? = 122°79, P < :000,000,1,
Strains III and V: x? = 147-20, P < :000,000,1,
Strains IV and V: x? = 23°90, P =:2470.
in the usual way we have
Statistically therefore there is not the faintest resemblance whatever between
any pair of these strains except the IV and V. These strains are for practical
purposes interchangeable. In one out of every four trials two pairs of samples of
500 from the same trypanosome population would give results more divergent than
those observed. But what is the source of this resemblance? Why are these two
strains alike and all the others widely divergent? There is nothing whatever in
the paper to account for this agreement, and it is the more remarkable because
Strains IV and V are to the statistician the most compound looking of all the
strains. But some uniformity of origin or treatment has caused the two com-
ponents to appear in like proportions, and at the back of this resemblance there is
some vital point, if we could follow it up. Were the two dogs bitten by the same
* RS, Proc. Vol. 86, B, p. 409 et seq.
fly, or Rats 658 and 660 really inoculated from the same dog ?
Kart PEARSON
Clearly
113
there is a
point here which ought to be cleared up, for otherwise the statistician could only
conclude that the wild G. morsitans strains are widely divergent, and that their
compound nature suggests that the tsetse fly carries various types of trypanosomes
and these in varying proportions,
(b) I now turn to the five human strains dealt with by Sir David Bruce and
his colleagues.
animals.
Human Strains.
A: Compounds from Various Animals*.
Let us first consider the human strains compounded from various
The following table gives the length distributions :
Microns.
| 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26
Strain I, Mkanyanga ... 4] 19 | 42 | 63! 81) 75) 91) 65) 66} 93] 91/107
” ) 2 2 | 12 | 55 | 108] 159 210 | 188 | 215 | 177) 138) 83
» II], Chituluka 1 8 | 48 | 81 | 78| 71] 44] 46] 56) 53) 98] 120]
» LV, Chipochola ... 2] 4 | 32 | 68 |110/101/109|106) 95° 95] 74] 64|
» V, Chibibi 1 8 | 20 | 58 | 117(122|123,107| 93) 98] 63} 51
|
—
Sum 41 | 154 | 325 | 494 [oes 577 | 512 | 525 | 511 | 464 | 425
Human Strains. A: Compounds from Various Animals—(continued).
Microns.
38 | Totals
Strain. I, Mkanyanga . 1220
ee al B 1500
» Ill, Chituluka ... 1500
IV, Chipochola ... 1000
V, Chibibi 1000
6220
We may conipare the strains precisely as in the case of the wild G. morsitans
We find:
Strains I and II:
Strains I and III:
Strains I and IV:
Strains I and V:
strains.
Strains II and III:
Strains II and IV:
Strains II and V:
Strains III and IV:
Strains III and V:
Strains lV and V:
x? = 408°50,
x? = 204-99,
x? = 180°63,
x” = 20540,
x” = 923°62,
x foi 0k,
veh, 00,
x? = 53132,
x? = 563°82,
x? = 16°81,
* R. S. Proc. Vol. 86, B, pp. 287, 291, 295, and 297.
B, p. 423.
Biometrika x
P =< -000,000,01,
P =< :000,000,01,
P = < :000,000,01,
P =<:000,000,01,
P = <:000,000,001,
P =< -000,000,5,
P =< 000,000,5,
P = <:000,000,01,
P = <:000,000,01,
P =7733.
For Strain I see R. S. Proc. Vol. 85,
15
114 A Study of Trypanosome Strains
Again we have the remarkable result that all the human strains are statis-
tically divergent beyond any possible comparison, except those of Chipochola and
Chibibi which show a high degree of correspondence. Now is this result the
outcome of treatment? We note the following diversity of hosts :
Strain I. Strain II. Strain III. Strain IV, Strain V.
Gross | Percentage | Gross | Percentage | Gross Percentage | Gross | Percentage | Gross | Percentage
Men 60 4°9 — 0-0 - 0:0 — 0 — 0:0
Monkey ... 100 8:2 160 10°7 160 1KO}27/ 160 16°0 160 16:0
Goat 20 16 60 4:0 80 5°3 80 ‘0 80 8:0
Sheep 60 4°9 20 1:3 1 0-0 ‘0 — 0°0
Dog ss 260 21°3 260 17°3 260 17°3 260 26°0 260 26°0
Guinea Pig | 120 9°8 _ oOo | — 0-0 — ‘0 — 0-0 >
Rat 600 49°2 1000 66°7 1000 66°7 500 50°0 500 50°0
Totals 1220 — 1500 — 1500 — 1000 — 1000 —
Now it will be clear at once that the percentages of trypanosomes drawn from
various types of host are identical only in the case of Strains IV and V, which we
have found in close accordance. But there is not great divergence in source
between Strains II and III although Strain I shows fairly wide differences. We
find, however, that II and III are statistically very unlike, the next closest
resemblances, although very slight, being between II and IV and V. It would
not seem therefore that the degree of similarity is wholly determined by similarity
of hosts. I have accordingly reinvestigated the five human strains by taking rats
only. But, of course, even then it is of vital importance to be certain that the
process of transfer from man to rat was the same in all five cases, and of this no
evidence is provided.
Human Strains. B: From Rat only*.
Microns.
15 | 16 | 17 | 28 | 19 | 20 | 22 | 22 | 29 | 24 | 25 | 26 | 27
|
Strain I, Mkanyanga =e lie 1 | 21 | 40 | 52 | 49 | 80 | 31 | 36 | 33 | 48] 52
» LL, 5, Rat 728 .~f—|—| 2] 4] 15 | 30] 57 | 72 | 85 | 72 | 59 | 44 | 26
» I, E, Rat 796... --... | — |— | 2 24 | 30.) 42 |60))) 61 "87 | 78ul soulmoralnoo
» III, Chituluka, Rat 952 | 1 | 3 | 21 | 27 | 23] 15] 10] 15 | 19 | 21 | 34 | 44 | 36
3 ILI, Chituluka, Rat 953 — ih 17 | 26 | 20 | 19 | 15 | 14) 26 | 18 | 33 | 40 | 34
,» IV, Chipochola, Rat 1337] — | — | 4| 6] 16| 29 | 53] 61 | 59 | 69 | 56 | 51 | 36
.) V, Chibibi, Rat 1660... | —- | -— | — | 4 {17 | 29 | 46 32 | 69 | 73 | 52 | 40 | 31
Sum Ne Ete via 1 | 5 47 | 112! 161 | 216 | 290 | 316 | 376 | 362 | 322 | 294 | 235
* R. S. Proc. Vol. 86, B, pp. 288, 289, 292, 293, 295, and 298. For Strain I see R. S. Proc.
Vol. 85, B, p. 423.
Karu PEARSON 115
Human Strains. B: From Rat only—(continued).
Microns.
: Totals
Strain I, Mkanyanga 600
eT. E). Rat 728 500
55 II, E, Rat 726... dis 500
» II], Chituluka, Rat 952 500
» III, Chituluka, Rat 953 500
» LV, Chipochola, Rat 1337 500
i V, Chibibi, Rat 1660... 500
Sum... male ... [219 | 210] 134| 108} 88 | 57 | 28 | y) S| dl 1 } 3600
This table with its two pairs of rats inoculated from the same strains is
peculiarly instructive. We can compare II, Rat 726, with I, Rat 728.
We find: x? = 36195, giving P =-0048.
This is far from the high degree of divergence we have found between the com-
pound human strains, but it is not satisfactory as a measure of the agreement of
the same strain in two hosts of the same species.
Applying the same test to the two Rats 952 and 9538 of Strain III we have:
Vv? =14715, giving P="9038.
This is, of course, quite satisfactory. We should not hesitate to assert identity
of strains and of treatment in the case of the trypanosomes from these two rats.
The statistician will feel fairly confident that there is a factor of divergence
between the trypanosomes of the two rats in Strain IT, which does not occur in
the two rats of Strain III. He will be almost certain that the strain was not
conveyed through the same steps or at the same stage of the disease to the rats in
Strain II. Unfortunately dates and processes are not discussed. Sir David Bruce
and his colleagues say that it is remarkable how much alike these distributions for
Rats 726 and 728 are, and again for the distributions for Rats 952 and 953 that
they also closely resemble each other. “It is curious and striking that the same
strain of trypanosome growing in two different animals should show this remarkable
similarity*.” The interesting point is that the statistician would agree with the
remarkable similarity in the latter case, but the divergence not the remarkable
resemblance in the first case would force him to seek for some explanation in
treatment. It will, I think, be clear from these illustrations that a strain of
trypanosomes, even if obviously compound, can be taken from a single source and
after inoculation into two different individuals of the same species be identified
as same; but to insure this result on every repetition the greatest caution will
have to be exercised as to identity of process and treatment.
* R.S. Proc. Vol. 86, B, pp. 289 and 293.
116 A Study of Trypanosome Strains
There are still further results of importance to be ascertained, however, from
our table of human strains. Let us compare Strains IV and V, which we found
resembled each other closely even for compounded hosts. We now reach
x? = 14085 and P=:5229.
Or, the probability that these two strains are identical has been reduced by
selecting out the rat data only. But the result is still so high that no one would
hesitate to assert that Chipochola and Chibibi were suffering from a disease due
to the same strain of trypanosome. The correspondence is so close that we have
combined Strains III and V for all other comparisons. In the case of Strain ITI,
we have added together the results for Rats 952 and 953. Such addition is less
reasonable for Rats 726 and 728, but without doing this, it is impossible to decide
which rat is to represent the E strain. I have then made the following com-
parisons :
Strains IV and V with III: y*?=525-67, P <:000,000,01.
There is accordingly no similarity at all between the Chituluka strain and that
common to Chipochola and Chibibi.
Strains IV and V with IT: y? = 64°70, P < ‘000,001.
Thus the strain from the European E from Portuguese East Africa diverges
from the Nyasaland strain widely, but not as widely as that of Chituluka does
from those of Chipochola and Chibibi.
Strain I with Strain HI: y? = 12613, P <:000,000,1,
Strain I with IV and V: y? = 21782, P < :000,000,01.
Thus the trypanosomes from Mkanyanga are widely divergent from those of
the three other Nyasaland cases. Nor are they any closer to the European E:
Strain I with Strain Il: yx? = 331°37, P <:000,000,01.
Thus with the exception of the Chipochola and Chibibi strains, the trypanosome
distributions from human sources differ widely. Nor is this to be wondered at, if
the human beings owe their trypanosomes to Glossina morsitans, for in that case
we should expect the human strains to be as diverse as we have found those from
the tsetse fly itself. It would remain to explain the close similarity of the
Chipochola and Chibibi cases. It would be interesting to know the history of
these cases with regard to locality and to the possibility of a unique source
of infection.
(c) In the case last dealt with, namely that of Chipochola and Chibibi, we
have the remarkable feature that the strains although significantly identical,
whether treated in the rat alone or in compounded distributions from various hosts,
resemble each other somewhat less closely in the single host series. This is not
generally the rule. Some of the big divergencies we have already noticed become
far less appreciable, nay, even become resemblances when we confine our attention
to one species of host. The chief misfortune which then too often arises is the
Kart PEARSON 117
paucity of the total numbers that we have at our disposal. I will consider,
however, from this aspect the relations of the three strains wild G. morsitans,
wild game, and Mvera cattle.
I compare first the lengths of 200 trypanosomes from wild G. morsitans and
wild-game strains. These yield for the host, goat* :
Microns.
| Satie
From Goat 9 | 10 | TA) 12. | 13 | 1h | 15 | 16 | 17 | 18 | 19 Totals |
é | | Ps aa |
Wild G. morsitans Strain | 1 | 3 | 12 | 21) 55 | 60 | 32 | 12) 4 | —| — | 200
Wild-Game Strain Hae | —|}— | Wk | 37 | 73 | 38 | 26) 8 2 200 |
|
giving : x? = 26782 and P=-0015.
To further test this, I take the same two strains in the dog as host+:
Microns.
| ere ee |
From Dog D0 | IED RED HES NGI A GESY IRE TRS | is} | Totals
i onl | fe
| eas | ee ee |
Wild G. morsttans Strain 5) 3 . 34 41 | 40/19) 9 | — 160
Wild-Game Strain ae al a 31 | 57 | 50 | 24 | 6 | —| 180
|
| \ | |
Here v= (045 and P= 3171.
The value we had previously found for a mixture of all strains was P = -0002.
Thus the two strains may be considered as identical when we deal with the
trypanosomes from the dog, as showing considerable divergence when we take the
goat, and as showing marked divergence when we take a great variety of hosts.
The weight of evidence in favour of a standardised treatment thus becomes very
great.
Let us look at precisely the same material for the wild-game strain and for
the Mvera cattle strain, first for the goat and then for the dog as host+. The
grave difficulty is the paucity of measurements thus differentiated.
Microns.
i =a Eee aaa | | : ie yi eel | lo |
| From Goat 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | Totals |
| : SS | : |
Wild-Game Strain ...|— | — | 1 | 16 | 37 | 73 | 38] 26| 8 | 1 | 900
| Mvera Cattle Strain ... | 1 1 OF la 225260) TOE sa 1} ==) 100
| ed |
This gives x’ = 14670, leading to P =-'1013.
* R.S. Proc. Vol. 87, B, pp. 6 and 11.
+ R. S. Proc. Vol. 87, B, pp. 6 and 11.
by by
. S. Proc. Vol. 87, B, pp. 3 and 5
+
+
118 A Study of Trypanosome Strains
Microns.
| From Dog | 11 | 12 | 13 | 14 | 15 | 16 | 17 | Totals
eee | ps ee Me |
| : | |
| Wild-Game Strain soe fe IQ ASI D7 de bOM24alaG | 180
Mvera Cattle Strain | || Lia O eel 8 | — 100
|
|
I
This leads to x? = 15992, P="0138;
Previously (p. 98) on the total series of different hosts we had found
P=-000,243, Thus by referring our material to individual hosts, we have reduced
the degree of divergency between the wild-game and Mvera cattle strains, but it
would be still hazardous to state that these strains are identical.
Lastly, we turn to the Mvera cattle strain and the wild G. morsitans strain
dealing with dog and goat as hosts separately * :
Microns.
From Goat ako) |) ib ae 1h 15 | 16 | 17 | Totals
| |
Wild G. morsitans Strain ... | 1 Sale | 21 | 55 |.60 32 | 12 | 4 200
Mvyera Cattle Strain eeu Cal 1 3] 14 22 | 26 | 19°) 2372 100
| |
This gives x= 1-968, P = °4368.
And again :
Microns.
| |
From Dog 9 | 10| 21 | 12 | 13 | 24 | 15 | 26 | 17 Bas
|
ee eee ae eer
5 5 |
Wild G. morsitans Strain... | — | -— | ela: | 34 | 41 | 40 | 19 | 9 160
Mvera Cattle Strain | —|—| 3/11] 27) 30} 21-4 8 |= 19 ico
| | | | | | | | |
resulting in x? = 11:120, P=:0852.
The Mvera cattle strain and the Glossina morsttans strain had for all hosts a
divergence measured by P= 000,008. Thus the great bulk of this divergence
is due to multiplicity of hosts f.
To sum up the results obtained for 7. pecorum in Mvera cattle, wild G. morsitans
and wild-game strains, the identification of these strains was quite illegitimate on
the basis of the compound host frequencies. It is reasonable on the basis of
* R. S. Proc. Vol. 87, B, pp. 3, 10—11.
+ It is worthy of note that in comparisons with the cattle strain the goat appears to give closer
results than the dog, but the dog appears the better in the comparison of the G. morsitans and wild-
game strains,
Kart PEARSON 119
trypanosomes taken from a single species of host. But how far the resemblance
in these cases is produced by a selective influence of the host and not necessarily
by an identity of all the members of the strain before transference to the host is
not demonstrated.
On the other hand while divergence due to host will account for the divergences
which are so notable in 7. pecorum, it will not account for the divergences in the
human strains; these are startlingly conspicuous even if we confine our attention
to a single species of host. Precisely the same remarks apply to the trypanosomes
similar to those causing disease in human beings found in wild game and in the
tsetse fly itself. There must be another source for these divergences.
(7) Discussion of the Heterogeneity which is statistically demonstrable in the
bulk of the Trypanosome Measurements.
The reader who has attentively followed the course of the argument in the
previous sections will be prepared for the next step in this memoir, the attempt to
account for the large divergences between strains of trypanosomes in individuals
of the same species by the heterogeneity of those strains. My suggestion is that
the strain in one fly differs from that in another because the components do not
appear in the same proportion, the strain in one specimen of wild game from that
in another, or in one man from that in another because they have been bitten by
a fly containing the components in unlike proportions. The host does make some
difference, either by nutrition or selection of trypanosomes, but it is a minor differ-
ence. Thus consider what we may probably hold to be pure strains and observe
the average differences in length found by Sir David Bruce and his colleagues:
Microns.
T. pecorum
T. simiae* T. caprac |
Mvera Cattle + Wild G. morsitans §
Goat 17°3 | Waterbuck 26°8 | Donkey 13°5 | Goat 13°5
Monkey Loa Ox 25°7 | Ox 14:2 | Monkey 13°6
— Goat 25°3 | Goat 13°8 | Dog 14:2
— Sheep 25°6 | Dog 13°8 | Guinea Pig 14°6
— | — Rat 14°8 | Rat 14:0
Max. Difference 0°8 Max. Difference 1°5 | Max. Difference 1°3 | Max. Difference 1:1 |
We may thus anticipate that in a pure strain the change of host would hardly
make a difference of more than 2 microns in the average length. We must
* R. 8S. Proc. Vol. 85, B, p. 479. : t+ R. S. Proc. Vol. 87, B, p. 3.
+ R. S. Proc. Vol. 86, B, p. 279. § R. 8. Proc. Vol. 87, B, p. 10.
120 A Study of Trypanosome Strains
accordingly be prepared for some such change as this in the shifting of the mean
when the host is varied.
We have next to inquire what type of curve accurately describes the strains
which we are fairly certain are homogeneous.
If the reader will turn back to p. 110 he will note at once a marked difference
between the distributions for 7. caprae, 7. pecorum and T. simiae when compared
with those entitled Mzimba strain, human strain, wild-game strain, 7. brucei,
T. rhodesiense, T. gambiense and the wild G. morsitans strain. The coefficients of
variation of the former group are all under 9°5 (mean = 9:00), the coefficients of
variation of the latter group are all over 13°5 (mean = 17:29). We recognise
therefore a totally different order of variability. Even in absolute variation as
measured by the standard deviations we find the first group with its mean
S. D.= 1°68 and the second with its mean 3:96. An examination of the graphs
scattered through the trypanosome papers to which we have referred will, we think,
convince the statistician that we have to deal with heterogeneous and not skew
homogeneous material*. It becomes of course important to ascertain whether in
the pure strains a Gaussian curve will suffice to describe the frequency closely
enough for statistical purposes, for, if it does, the analysis into at any rate two
Gaussian components of the heterogeneous strains becomes relatively direct, if
laborious. I will consider the 7. pecorum, T. simiae, and T. caprae strains
from this standpoint.
(a) T. pecorum (see p. 110).
Mean = 13:992 microns. S.D.=1:2816 microns.
boa Observed | Calculated
Microns Values | Values
9 and under 2 0°46
10 6 | 5°98 -
11 42 45°41 x°=7°630
12 193 192°52 P= 572
138 452 456°70
Ls 618 607°12
15 453 | 452°49
16 178 188-98
LT 51 44°16
18 and over 5 | 6°20
Hence in 57 out of 100 trials from material following the Gaussian distribution
a more divergent sample than that observed would actually be obtained. We can
therefore conclude that a simple Gaussian frequency adequately describes the
distribution in size of 7. pecorwm. This is illustrated in Diagram II.
* Note especially the bimodal graphs in R. 8. Proc. Vol. 83, B, pp. 5 and 11, for both the Uganda
and Zululand strains of 7. brucei, in Vol. 86, B, pp. 291—293, for human strains, in Vol. 86, B,
pp. 395, 397 for wild-game strains and pp. 409, 411, 417 and 419 for G. morsitans strains,
KARL PEARSON 121
Total 2000.
Frequencies per Micron.
1
'
'
'
t
OOo OAS S16 MSem9
Microns.
Driacram II. Gaussian fitted to T. pecorum Frequency.
(b) T. simiae (see p. 110).
Mean = 17'870 microns. S.D.=1°6558 microns.
Microns | Observed Calculated
Values Values
14 and under ui 10°46
| 15 28 27°63
16 | 76 63°92 x?=8'149
17 93 103°78 P= +520
18 126 118°32
19 92 94°66
20 47 D878
21 22, 20°96
22 6 5°80
23 and over 3 1:29
150
140 ;
3 130
2 120
S$ 110
Ss
S 100+
Frequencies per Micron.
QO
fe)
(OMmmsMNaee Se diemly is) 19-9901 21 Of a5) oO
Microns.
Dracram III. Gaussian fitted to T. simiae Frequency.
Biometrika x 16
122 A Study of Trypunosome Strains
We conclude that the Gaussian adequately describes the distribution of
T. simiae. In more than half the trials we should get a worse sample. See for
graphical fit, Diagram ITI.
(c) T. caprae (see p. 110).
Mean = 25°508 microns. S.D.=2°1011 microns.
Microns Observed Caleulated
Values Values
| 20 and under Aas 4:98) |
21 8 9°82
22 23 23°95
23 49 46°74 x?=5°175
24 79 73°05 ae
25 95 91°38 val
26 80 91°54 |
27 | 68 73°45
28 57 47°16
29 24 | 24°26
30 9 9°98
| 31 and over 4 | 4°38
This is a still more excellent fit; if the Gaussian represented the population,
in 92 °/, of samples we should get a more divergent sample than that observed.
The curve is given in Diagram IV.
150
140-
2 120 te
: 110, =
4 10
é 90,
= 80
= 70
= 60
eS 50
S 40
S
al a
va
IN
és os
19° 90 QI 29)" 93) 94995" 96) 227) (2829 sOn sl asa
Microns.
Diacram IV. Gaussian fitted to 7. caprae Frequency.
It will be clear from the above three illustrations of what we may term
homogeneous trypanosome strains that the Gaussian curve of frequency suffices
to describe adequately such material. It is equally clear that no Gaussian can
Kart PEARSON 123
possibly describe such skew distributions as we get in the wild-game strain or wild
tsetse fly strain of the trypanosome species identified by Sir David Bruce and
colleagues as 7. rhodestense*. It is equally impossible in the case of the human
strains figured in the paper of February 1913+. I illustrate this on the frequency
distribution for 6220 trypanosomes of human strains}.
Observed Calculated Observed Calculated
14 and under 1 | 75°45 26 425 | 520°55
5 10 =| 62°51 I a7 | 372 444-42
16 | 41 101'51 —s| 28 347 || 35796
17 154 | 155°50 | 29 | 307 271°88 |
18 325 | 224°73 I 30 | 198 194-81 |
19 494 306-27 | 31 | 167 | 131°68
20 528 393-73 | 32 123 83°91
21 577 | 47739 83 77 50°44
22 512 545°93 | 34 36 28°61
33 525 588-91 | 35 | 12 15°30
24 511 599°17 | 36 and over | 14 | 14:18
25 464 | 575°04 | |
|
Here y?= 501 and P< ‘000,000,001. In other words description by a Gaussian
is absolutely impossible. The histogram of observations and the curve are shewn
on Diagram V.
Now the suggestion that flowed at once from these results was the compound
nature of all the material classed under the headings :
G) TZ. rhodesvense.
que) L brucer.
Gu) TZ. gambiense.
(iv) Mzimba Strain.
(v) Wild G. morsitans Strain.
(vi) Wild-Game Strain.
(vii) Human Strain.
With the experience of the Gaussian fitting the homogeneous strains, the direct
step was to investigate whether the above material could be analysed into two
Gaussian components and to determine how nearly these components were in
agreement. The method of carrying out this analysis was provided in the first
of my series of Contributions to the Mathematical Theory of Evolution§, There
was nothing to prevent the process being applied to every individual frequency
given by the trypanosome workers, except the very laborious arithmetic. The
method was applied to the above seven cases, and also (viii) for the purposes of
illustration to a single human case, that of Chituluka, a native of Nyasaland, who
* See R. 8. Proc. Vol. 86, B, pp. 407 and 419.
+ See R. S. Proc. Vol. 86, B, pp. 285 et seq.
See R. S. Proc. Vol. 86, B, p. 300.
-
a
§ Phil. Trans, Vol. 185, A, pp. 71—110, 1894.
16—2
A Study of Trypanosome Strains
124
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KARL PEARSON 125
died of sleeping sickness*. With the single exception of 7. brucei every one of
these distributions broke up into two components, and into two components with
strikingly close means. I propose to call these two components 7. minus and
T. majus. I do not assert that they are distinct species; they may be dimorphic
groups of one and the same trypanosome species. But the recognition of their
existence seems to bring some order at least into the chaos we have already noted
as existing in the trypanosome measurements. ‘Two human strains or two wild-
game strains differ from each other with such wide divergence in their frequencies
because these two groups 7. minus and T. majus are mixed in the individual
in different proportions.
| Standard Coefficients of Size of
MSs Deviations | Variation Populations
Strain = =o |e
T. minus | T. majus | T. minus! T. majus | T. minus | T. majus | TL. minus T. majus
T. rhodesiense ...| 18°7418| 26-1122 | 2°3184 | 3-4397 | 12-370 | 13-173 eae Bees
7. brucei... «|| 19°8244 261122 2-6439 | 34134 | 13°337 | 13-072 Peace eos |
“T. gambiense _...|19°8926) 26-2463 | 2-0566 | 26260 | 10-339 | 10-005 |) ORY. | | BNE
Feamailecus apr SOD Re Pad iee N GokE ( 634-96 | | 365-04
| Mzimba Strain ...| 19°8966 | 24:0508 | 1°3961 | 3:1028 7°017 | 12-901 ) 635°, |) 365 om
| @. morsitans Strain | 19°6475 | 27-1966 | 1°7503 | 2-70138 | 8-908 | 9-932 eae 7 aa ’
Wild-Game Strain | 20-4418 | 25-8263 | 16332 | 2-8799 | 7-990 | 11-151 ae ae es
OO"D eon ipa dO bet
i 9e . F .
Human Strain ...| 20-3687 | 26-2930 | 1-9444 | 3-4470 | 9°536 | 13-110. ercee “1 | woe 7
Chituluka ... ...| 19°8410 | 28-7875 | 1-9785 | 2:8823 | 9-972 | 10-012 ae hoe y
Means... _... | 19°8315 | 25-9542 | 1-8498 | 3-0328 | 9-360 | 11-712 | 9 —- ee
T. simiae ... —... | 17°870 Be wp OOS ae aS 2700 = 100° -
lo
T.caprae... ..{ — |25:508 | — |21011| — | 8-580 = rae
| | | 100 °/,
| i 2 |
The table below gives the chief biometric characters of 7’. minus and 7’. majus
as found from the seven resolutions. The mean values of the constants for
T. minus and for T. majus are placed at the foot; in calculating these mean values,
Chituluka’s data have been excluded as already included in the human strain, and
also those for 7. brucei not directly resolved.
At the foot of the table I have placed the constants for 7’. simiae and T. caprae,
the nearest pure strains to 7. minus and T. majus respectively. I do not in the
hase Proc: Vol.86, By ip. 290.
126 A Study of Trypanosome Strains
least suggest there is any identity, but comparison may bring home to the
trypanosome worker the average sizes of the two components*. The differences
of the variabilities are, however, much larger, and the influence of host on
variability as well as on mean ought to be studied.
It will be seen at once that the divergence in the individual means of 7. minus
from the general mean is very slight, at most a micron, and well within the limits
which arise, as we have seen, from difference of host. It is a most remarkable fact
that from six independent reductions the mean size of JZ’. minus should come
out so nearly 19°8 microns. In 7. majus the correspondence is not so good; the
average of about 26 microns falls to 24 in the Mzimba strain and rises to 28°8 in
the case of Chitulukat. Still it does not appear to me that these changes of
mean of the 7. majus component are absolutely beyond the variation due to differ-
ences of host and treatment. Another more serious matter is the comparatively
wide range found for the variabilities ; but even here it is impossible to assert that
such differences will not occur with difference of host. For example the Mvera —
cattle strain, a fair sample of the simple 7’. pecorwm, gives:
| Fost M | Standard Coefficients of
Ot tas | Deviation Variation
Goat 13°80 1°462 10°592
Rat 14°75 *839 5689
Dog 13°79 1:087 7°885
Here while the means are within one micron, the differences in variability are
of the same order as those found in 7. majus from different hosts.
Again, taking a pure homogeneous strain as 7. caprae with goat and sheep as
host, which are scarcely so differentiated as man and antelope, we find:
Hoc Mean Standard Coefficients of
Deviation Variation
Goat 25°31 2-187 8°642
Sheep 25°60 1°92¢ 7512
|
Lastly, taking 7. simiae for goat and monkey we have:
r Standard Coefficients of
| Eos lean Deviation Variation
|
(ea ae eee — - aa |
| Monkey ... | 17°26 1-403 8127 |
| Goat | 18°11 1°687 9°315 |
* The maximum average length of 7’. caprae is 26°8 in the waterbuck and of 7’. simiae 18:1.
+ It should be noted that with the whole of the human data the mean is 26°33 and that Chituluka’s
mean is very exceptional.
Kart PEARSON 127
I think we may conclude that, allowing for the errors of random sampling
and the errors arising from the resolving process, the deviations observed in the
variability of our two components do not invalidate the hypotheses :
(i) That the widely divergent results obtained from different strains are due
to the existence in the same individual of two types of trypanosome with very
varying percentages from individual to individual.
(ii) That one of these types has a mean length of about 19°8 microns and a
variability of about 1°8 microns, the other a mean of about 26°0 microns and
a variability of about 3:0 microns. The means may vary 1 or 2 microns with the
nature of the host and the variability 0°5 to 1 micron.
The large type predominates in the Nyasaland human strains*, on the average
in about the ratio of 3 to 2, but the smaller type predominates in the G. morsitans
and wild-game strains in about the same ratio; while in the trypanosomes classed
as T. rhodesiense, and 7. gambiense as well as in the strain from the Mzimba
donkey the preponderance is still cf the smaller type and the ratio approaches
13 to 7. Whether these ratios are peculiar to the host or due to the infecting fly,
it is not at present possible to determine. But the hypothesis of the existence of
these two types,—whether as a dimorphism of 7. rhodesiense or as independent
species seems to bring some order into the apparent chaos of recent trypanosome
measurements.
The following paragraphs give the calculated constants of the reductions, and
the numbers of the diagrams showing the nature of the compound frequencies:
Ga) T. rhodesiense.
Mean = 23°577,
fly = 21°86874, fs = 1079°10255,
Ms = + 401986, Ms = + 1105°74834.
Reducing nonic:
249° — 298-7232q' — 5817q° + 1114°7684q° + 34°7620¢'
— 117924954? + 12°9808q? + 0891g + 0001 = 0
wheret p,=—10q.
The root is p,=—12:2578. This leads to the two components in the Table
p. 125. The histogram of the observations and the two component Gaussian
curves with their compound are given in Diagram VI.
The resolution is not a very good one; for 24 groups y?= 37°48, and P =:05, or
once in 20 trials only we should get a worse result. But an examination of either
the graph or the original frequency shows at once the cause of this divergence.
In their measurements Drs Stephens and Fantham have had a strange bias in favour
* The European from Portuguese East Africa had predominance of T. minus. See R. S. Proc.
Vol. 86, B, p. 288.
+ Notation of the memoir Phil. Trans. Vol. 185, A, p- 84, Eqn. (29).
ins
y of Trypanosome Sti
e
A Studi
128
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arawan Dau
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' Kart PEARSON 129
| of even numbers. No curve whatever could fit the data satisfactorily under the
7 circumstances! Either they used a scale graduated to 2 microns only, and had a
7 prejudice in favour of the scale markings, or else their even numbers were in some
way more conspicuous than their odd. Whatever the source of this peculiarity
i” may be, there can be no doubt of the bias*.
~| The only way to obtain a reasonable measure of the goodness of fit in Stephens
| and Fantham’s results for 7. rhodesiense is to group from 10 to 12, 12 to 14 and so
on in comparing the observed and calculated frequencies. If this be done we find
x? = 5:03 for 13 groups and P=-957, a splendid fit. The frequencies are as
follows :
26-28 | 28-30 | 30-32
10-14 | 14-16 | 16-18 | 18-20
Observed | 9 | 38-5 | 93-0 | 133-5
Calculated | 7-17 | 34°67 | 92-99 | 132-91
20-22 | 22-24 | 24-26 }2-34,| 34-36 | 36-38 | Totals
|
134°0 | 127°0 | 1385°5 “1395 | 112°0 | 60°
6
5 15 1000
124°79 | 124°28 | 146°35 | 145°56 | 106°55 | 56°22 | 21°36 | 5° 11
"17 | 999°85
{
(i) TZ. brucez. The data for this trypanosome were taken from Sir David
Bruce and colleagues’ diagram+. I have not come across the original publication
with the measurements involved in this diagram. Describing this species in
: July 1910, the authors speak of its well-marked dimorphism. This is very
obvious in the graphs for length given for the Uganda 1909 and Zululand 1894
strains, but the numbers given are far too slender (160 and 200 respectively) to
justify any attempt at analytical resolution. Graphically we may take it that
roughly the following are the means of the components:
T. minus. T. majus.
Uganda 1909 20 microns 28 microns
Zululand 1894 18 microns 29 microns.
These are not very widely divergent from the values
19°8 microns 26°0 microns
we have found from the seven resolutions.
In May 1911§ the two curves for Uganda and Zululand appear to be added
together to give a 7. brucei curve of length distribution. This is again markedly
bimodal with one component mean at 18°75 microns and the other at 27°5 microns,
both approximative. Thus far 7. brucei appears quite well to fit in with our other
material. But in September 1911 appears the diagram of 7. brucei said to be
* Bias of this or of a similar character is not uncommon—even in the pages of this Journal.
I remember once pointing out to a Scotch anthropometer his prejudice in favour of whole centi-
metres. He looked at his results, recognised the bias, and then gravely told me that it was not
due to any personal bias, but that the Creator must have designed Scotsmen on the metric scale!
+ R. S. Proc. Vol. 84, B, p. 331.
+ R. S. Proc. Vol. 83, B, p. 2.
§ R. S. Proc. Vol. 84, B, p. 186.
Biometrika x 17
130 A Study of Trypanosome Strains
based on 1000 individuals, Here there is a mode about 240, with possibly a sub-
mode at 19 microns, but the evidence for dimorphism has largely disappeared.
It is very desirable that we should know the details of this curve, ie. the nature
of the hosts and so forth, for it apparently replaces the earlier data and remains
the standard 7. brucei distribution. It certainly shows nothing of the definite
heterogeneity (or dimorphism) of the previous Uganda material.
Its constants are as follows :
Mean 28°5290,
fy = 19°30583, fy = 996°87764,
Le; = 10°54837, Hs = 2146°37930.
249° — 10186189’ — 4°0057q° + 140°6937q° + 62:0835¢q'
— 29°39409? + 11:2371¢@ + 1.44139 + ‘0331 = 0.
No suitable root of this equation exists and accordingly it would appear that
this distribution is not rigidly reducible to Gaussian components. This result is
so remarkable in view of the obviously bi-modal character of the earlier 7. brucei
distribution, and the resolution into two components of all the other seven
distributions, said to be allied to 7. brucez, that I determined to consider the
matter further by fitting Gaussians to the ‘tails’ of the 7. brucei distribution*.
I chose as the right-hand ‘tail’ the frequency from 28 to 38 inclusive, and as the
left-hand ‘tail’ the frequency from 13 to 18 microns-inclusive. The two resulting
components were :
T. minus. T. majus.
m, = 20°0817 (19°83), My = 26-4359 (25-95),
o, = 2°8685 (1°85), oy = 36399 (3:03),
he O2o0, Ny = 467°52.
The totals populations for each component are clearly not very good and their
combination exceeds by 9°6 °/, the total observed population; but the means are
not widely divergent from the average values resulting from our six resolutions, as
the numbers given in brackets testify. Accordingly I determined to select the
means of the components at values near the mean values of six reductions, and
after one or two slight betterments, determine the sizes of the populations and
their standard deviations so as to give the mean, and second and third moments of
the observed population. These provided:
T. minus. T. majus.
iy = 198244, ms — 261122;
ao, = 2°6439, o, = 34154,
n, = 410°83, Nei Oooala.
* Biometrika, Vol. 11. p. 1 and Vol. vi. p. 65.
;
a
4
ea =
Karu PEARSON 131
The following table gives the observed and calculated values :
Microns Observed | Calculated | Microns Observed | Calculated
B) | 5 | 3°44 26 82 72°74
Ls 8 | 5°80 | Bie 2) 67°98
15 14 12°25 28 50 59°71
16 17 22°79 29 38 48°10
17 40 37°05 30 Dif 36°04
18 63 52°87 ruil 26 24°79
19 55 66°68 82 18 15°67
20 66 75°44 | Go iil 9°09
21 63 78°43 BY A 4°84
22 75 77°49 85 4 PET
23 87 75°61 36 — |
2) 93 74:71 BH = -1:75
25 80 74:36 ©6|| 38 2 J
From these results we find y?= 29°92 and P=:22. Thus more often than
once in five trials we should get a worse divergence than the observed, if the
sample were taken from the calculated population. Some endeavour was made to
better the fit by small variations from the above solution, discussed by least
squares, but no improvement was effected. The two components are represented
in Diagram VII (p. 132).
(i) =T. gambiense.
Mean = 22°1130,
fy = 14°3389, jis = 531°3585,
fs = 29°1104, Hs = 2429-0948.
Reducing” nonic :
24¢° — 7178109’ — 30°5070q' — 300:02609q? + 869°6372q!
— 278°8475¢q° — 270°9547¢? + 58:9108¢q + 146050 = 0.
This leads to p,=—10qg=—91777, and the components given in the Table
p. 125. The two Gaussians and their compound are given in Diagram VIII -
(p. 133). We find y?= 11:96, giving for n’ = 18, P=°80 a splendid fit.
Gv) Mzimba Strain (from Donkey).
Mean = 21°4130,
fg = 87531, pis = 2935629;
és = 26°6602, bs = 1926°7045.
The reducing nonic :
24¢° + 53°5186q" — 25°5876q° — 4157069? — 171:2637¢4!
+ 227°12119° — 37:3371¢ — 30°8995q + 86177 = 0.
The required root is p, = — 10g = — 40000, which leads to the two components
given in the Table on p. 125. The two components and their compound curve are
17—2
ins
ypanosome Sti
a)
of T)
e
U
{ Stud
ok:
132
. etn
+ Sem —
=
——
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"sUuo.oUyy
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~
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08
06
OO t
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13
Kart PEARSON
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*sUwoloryy
Ge ve ES oe 1€ OF 62 8G LO 96 GB VG SG BG IG oe 6 st Zt OL Gi VI Si al
+
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O€l
OvL
OSL
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“‘OOOT 2970.7
Total 1000.
Frequencies per Micron.
134 A Study of Trypanosome Strains
figured on Diagram IX on this page. We have y?= 19:28, giving for n’=17,
P=-26 a fairly reasonable fit.
(v) Wild G. morsitans Strain.
Mean = 22°6952,
fy = 184918, fy = 7584420,
fe; = 43°0246, Ps = 89548788.
210
200
190
180
Mean
170
160
150
140
130
120
110
100
+ i
10 fj
\
es a
Bi. =
21-229 23.24 05 -96 27 28° 29 30 (Sil sORscmes.
“Microns.
14 5) Ge 8! sg
Diacram IX. Resolution of the Frequency of the Mzimba Strain into 7’. minus and T. majus.
~~~
KARL PEARSON 135
Reducing nonic:
24¢° — 224°6115q’ — 66°6402q° — 595-9589 + 5079°3305q!
— 4500°7030q? — 1460°5459¢? + 879'6116g¢ + 1522340 = 0.
The required root is p,=— 10g =—4°75085, which leads to the components
given in Table on p. 125. These components with their compound curve are
drawn in Diagram X (p. 136). Here y?=92'75 which for 20 groups gives
P< :000,000,1. Thus although the G. morsitans strain breaks up into two com-
ponents the combined curve is not a probable description of the frequency. One
would like to test another sample of this strain, at present it tells against the
validity of our reduction.
(vi) Wild-Game Strain.
Mean = 22°6220,
fe ol; fog = 404°4932,
ty = 29°0514, by = 2247-6657.
Reducing nonic:
24q° — 18°94469' — 30°38349q° — 250°2869q° + 851°7475¢'
+ 118°6154q? — 212°3972¢? + 15°4222¢ + 144283 = 0.
The root required is p, = — 10g = — 6°9859. There result the two components
provided in the Table p. 125. The two components and their compound are
figured on Diagram XI. (p. 137). We find y?=12°61 giving for n’=19, P=°81,
an excellent fit.
(vu) Human Strain.
Mean = 23°7963,
by = 170252, jt, = 7131660,
fs = 27-1889, Ms = 80341222.
Reducing nonic:
24¢° — 1381°3796q' — 26°5147¢q' — 89°8059q@’ + 96441764!
— 67427559q3 — 114°7894q" + 8144929 + 95887 = 0.
The root is given by p,=—10qg=—8'5576, which leads to the components
given in the Table on p. 125. The two curves and their compound are figured
in Diagram XII (p. 138). Although the two components merely from the
graphical point of view do not give a bad fit, the number of trypanosomes in-
volved is so large that the deviations are not reconcileable with random sampling
trom two such components. We find y? = 79°67, giving P < ‘000,001.
‘snip *J, pue snuww “J OYUT UTeI4YG supzisLow “DH PTIM ey} Jo Aouanbarg oy Jo uorynjosey “yx NvYDVIG
“sUuoLorTy
eS LES OGr Gor SGN 2G V6e UGG S666 AGE OGsG I SiveZ LAO Gina St
% O
eS
RM
S OOL
S
3 ;
= |
S =
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S 00G
(= 1
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= '
> 008
nS |
YS
= '
B 'S
iS
=x | OO¥V
\
|
|
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OOG
136
uowoy ad hauanbawy
00G6 1PIOL
Total 2500.
Frequencies per Micron.
KARL PEARSON 137
In order to determine how far heterogeneity of treatment or material might be
responsible we took further frequencies. In the first place we dealt with the 3600
measurements for trypanosomes through the rat only. The frequencies are:
| | _ ;
| | | |
15 |16|17)| 18 | 19 | 20 | 21 | DODD 2 | 25 26 27 | Q8 "29 | 80 1 8L 132.83 34 | 35 | 86 | 387 | 38 Totals!
| Py
| | ms : a i) =| aa
iL 5 | 47 |} 112 | 161 | 216 | 290 | 316 | 376 | 362 | 322 | 294 | 235 | 219 zu al 108 | 88} 57 | 28] 9 | 8) 1 1 | 3600 |
| | ee Ie | |
400
350
300
250
200
150
100
50
1415 16 1718 19 20 21 29 93 24 25 96 27 28 29 30 31 32 33 34 35
Microns.
Dracram XI. Resolution of the Frequency of the Wild-Game Strain into T. minus and T. majus.
Biometrika x 18
‘ains
y of Trypanosome Sti
St ul d
A
138
‘snlpu *J, pues snuww *f OUI UNG UBUIN_ 94} Jo Aouonbeag oy} Jo uOoIyN[OSay
| at
IN
--
\
*suoLorTy
ge LE 96 GE ve FE GE IS OF 66 8G LG a GG ue €6 66 16 OG 6 Bi 2I GI Gi FI Et
— e— er er Orr rer hl
‘IIX WvuovIG
va
0
001
006
00€
0Ov
00S
009
‘wou sad fawanbay
0669 2Y707,
Kart PEARSON 139
These give :
Mean: 24°6175,
ple = 15°25897, Hy = 602°23008,
fs = 19°21542, fs = 2023°21556,
leading to the reducing nonic :
249° — 80°8739q' — 13:2924q" — 42°3159q? + 306°5227¢'
— 166°4257q? — 248654q? + 12°6008q + 1:2081 = 0
which gives po = — 10q = — 70031.
This provides the two components :
T. minus. T. majus.
m, = 21:6772, my, = 26°9993,
o, = 2°2404, T, =a 298),
nm, = 1611:18, Ny, = 198882.
The components and their compound are figured in Diagram XIII, p. 140,
and we find for n= 21, y?= 52°68 and P=-00016. There has thus been much
improvement of goodness of fit, although the result is still unsatisfactory.
It is impossible, however, to look through the graphs given by Sir David Bruce
and others for the human strains* without being convinced of their fundamentally
bimodal character, although there appears to be much evidence of its being
disguised by heterogeneity of host and treatment.
(viii) Diagram XIV (p. 141) gives the resolution for the human strain from
Chitulukat. The constants
Mean = 26:172,
fin = 2302260, Hy = 1179°30786,
fy = — 3718226, bs = — 3248°43805,
leading to the reducing nonic:
24q° — 393°8678q' — 49°6370¢q° + 520°2910q° + 8226:94354'
— 12493°5620q' — 101-101 7¢@? + 855°7520q + 63°2383 = 0.
The value of the root is p,=— 10g =—16:2295 and this leads to the com-
ponents given in the Table p. 125, and illustrated in the diagram. The graph
while giving broadly some of the features of the case is by no means a satisfactory
fit; for n= 21 groups, y?= 86 and P is < 000,000,1. The diagram suggests that
we are probably dealing with a mixture of three components with means about
18°5, 25°5 and 31-0, but at present we have no satisfactory method of performing
multiple resolutions of this character.
* R. S. Proc. Vol. 86, B, pp. 285—302.
+ R. S. Proc. Vol. 86, B, p. 291.
18—2
—————
wns
y of Trypanosome Stra
Studi
A A
140
1
‘snfpu “7 pure snuw *7, oyut ATUO syey Ysno1yy ‘ureyg weunZ, wos souosoueddry, jo Aouonberq oy} Jo uoNNosery
*SUudLIUTT
‘IITX NVUOVIG
)
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006
00S
‘uowary wad fauanbaasy
“0096 17207,
‘sn(vu “J, PUB sNUUW “7 OFT VYNTNIYO oaAeN oy} Woy soutosouvddry, yo Aouanbary oy} JO UOTN[osay “ATX NVUOVIG
"swouarTy
66 8c LE 9E GE VE SE GE IE OF 66 82 LG 9G GS VS GG GG WW OG GI Bl Al QI Gl vl El Gl
141
KARL PEARSON
os
OOL
OSL
uowny ad houanbawg
00ST 1279.7
142 A Study of Trypanosome Strains
It will be seen that the following strains, 7. rhodesiense, T. brucei, T. gambiense,
the Mzimba, and wild game, give either reasonable or excellent results as combined
frequencies of 7. minus and T. majus. On the other hand the G. morsitans and
the human strains break up into reasonable pairs of componeuts, but the goodness
of fit test is not fulfilled. In the case of the human strain, we better matters
somewhat by taking the strain through the rat only, but the fit is still bad. If we
confine our attention to a single human being, the case of Chituluka, we still do
not get a satisfactory fit, although few statisticians could look at the four diagrams
published by Sir David Bruce and others for Chituluka*, and not recognise the
character of the material as being at least bimodal. The same applies to the
Mkanyanga data of an earlier paper}, it is distinctly bimodal. But besides this
bimodal character there are certain other features in the human data, and to a
lesser extent in the G. morsitans, which appear to some extent to disguise the
bimodal features. I am not prepared to assert definitely that this is the appearance
of a third component. It is of course easy to improve the fit of the distribution
by the introduction of such a third component, but the remarkable excellence
of a bimodal resolution for 7. rhodesiense, T. gambiense, and the wild-game strain
makes me hesitate at present to adopt such an expedient.
Owing to the courtesy of Sir David Bruce (who heard from Sir John Rose
Bradford that I was much puzzled over the differentiation of strains) I have been able
to examine a series of drawings of the various strains of trypanosomes. There is
no other morphological differentiation which impresses itself a priori on the layman
and statistician, and which might serve as anew measure of the possibility of differen-
tiation into 7. minus and T.majus. But it occurs to me that an index of breadth to
length of the nucleus might just possibly serve as a differential character of even
more importance than the length. It is only a suggestion and considerable caution
would have to be used in selecting only nuclei not near the dividing stage. But
it would be of striking interest to see how far the resulting frequency distributions
for the nuclear indices were or were not bimodal. I think a classification according
to nuclear index might possibly—to judge from the drawings—cut across the
forms “intermediate ” in length. But this is only a suggestion which may appear
idle to the student of the subject?. Some difficulty might also arise from the
doubt as to whether the index was really greater than 100, or the nucleus as
a whole had set itself athwart the “length” of the trypanosome. This difficulty
would certainly have to be considered in the “stumpy” 7. brucei and T. gambiense
* R. S. Proc. Vol. 86, B, pp. 291 to 293.
+ BR. S. Proc. Vol. 85, B, p. 428.
+ Several students of the subject with whom I discussed the matter stated that they considered the
nucleus so mobile and so impermanent in form, that a ‘‘nuclear index’ would prove of little value.
I think much objection could a priori be raised to the use of the trypanosome ‘‘length” on the same
grounds. ‘The problem is rather, whether in dealing with large numbers we do reach an average type.
It would only be possible a posteriori to justify the use of a nuclear index, i.e. if it were found to differ
sensibly from one pure strain to a second, and if it confirmed in such cases as 7’. rhodesiense resolutions
based on length frequencies.
KARL PEARSON 143
forms, but I am inclined to think that the index really passes through the value
100. Undoubtedly this range of index, or possible athwartness of the nucleus is
not conspicuous in the simple strains like 7. pecorum, T. simiae and T. caprae.
Conclusions. (i) If appeal be made to statistical measurements, judgment
between identity and diversity of strain must be formed by means of accepted
statistical processes and not by mere comparison of graphs.
(11) Statistical processes show that the conclusions already formed as to the
identity of trypanosome strains from mere inspection of the graphs cannot be
confirmed.
(111) There must be some standardised process of treatment both in regard to
host, and to method of and stage of infectivity at extraction.
(iv) Even making allowance for differences due to host and treatment, we
find remarkable divergences in the very strains asserted to be identical.
(v) It would appear that some order would be brought into the chaos, if we
could consider the strains described as 7. brucei, T. rhodesiense, T. gambiense, the
wild-game, the Mzimba, and very probably the tsetse fly and the human strains
as really consisting of two components, which for the time I have termed
T. minus and T.majus. It is highly desirable that additional measurements should
be made (? a nuclear index ascertained) to determine whether these lead also
to similar components.
I do not assume that this is a final solution of the problem, nor do I assert that
T. minus and T. majus represent necessarily, although probably, distinct strains ;
they may be dimorphic forms of one and the same strain occurring in different pro-
portions. But, I believe, that the suggestion of their existence may help to explain
some anomalies of the present chaos. I ought also to state quite frankly that this
paper is not written in a merely critical spirit. I believe that the trypanosome
workers have undertaken in their elaborate systems of measurements most laborious
and most valuable work, but, I think, the time has now come when without
trained statistical aid, but little further progress will be made in a very important
and urgent matter.
The very large amount of arithmetical work in this paper would never have
‘got carried through had I not had the ever ready assistance of my colleague
Miss Julia Bell; to Mr H. E. Soper also I owe help in the arithmetical work, but
I have to thank him in particular for the careful preparation of the diagrams, and
the planimetric determination of their frequencies by aid of which the x? for all
but two of the compound curves was found. In the case of YZ. brucei and
T. rhodesiense actual calculation of the areas of the normal curves was used.
ON HOMOTYPOSIS AND ALLIED CHARACTERS
IN EGGS OF THE COMMON TERN
By WILLIAM ROWAN, K. M. PARKER, B.Sc., anp JULIA BELL, M.A.
(1) Origin of the material and method of measurement.
The settlement of Common Terns, which provided material for the present
work, is one of old establishment on Blakeney Point, Norfolk. This is a shingle
spit of some 8 miles in length on the north coast of Norfolk, about 12 miles
west of Cromer. The colony is situated on the very end of the point, with
water on three sides. Here the spit is a combination of dunes, salt marsh and
shingle, and for the most part the nests are found on the open shingle on the
seaward side of the dunes. Nests are plentiful in the embryo dunes in some
years, though this year (1913) none were found there. The colony was more
scattered than usual and covered the greater part of a mile of sea front. To
avoid missing any clutches, Miss K. M. Parker, B.Sc., and Mr William Rowan
divided the nesting area into suitable well marked plots and worked these one
after another. Each of these again were worked in strips, till a patch was com-
pleted, when the workers moved on to a remote one, to give the birds a chance of
settling down again. After measurement each egg was numbered with indelible
ink, so that any one egg was never measured twice. In all 203 clutches were
handled.
(2) Reduction of the material.
The principal part of the work of tabling and reduction was carried out by
Julia Bell*. The characters dealt with were:
(i) Length of Egg : : ; : : L
(ii) Breadth of Egg, maximum value. : : B
(ii) Lateral Girth at section with maximum procaine : Gr
(iv) Longitudinal Girth . 5 ; : : : : i Gy
(v) Length-Breadth Index. ; : B/L
(vi) Mottling, as determined from a Pas of one eggs. M
(vii) Ground Colour, as determined from a tint scale. ; C
* The authors have to thank Miss B. M. Cave for certain tables and their correlation coefficients.
The Editor is responsible for the actual wording of this paper.
W. Rowan, K. M. Parker ann J. BELL 145
The Leneth of eze LZ may be considered as the easiest character to determine
fo) {=}
and needs no further comment.
The Breadth of egg B should be closely related to the Lateral Girth G,, and
in most cases the relationship G,=7B is very closely satisfied. If we sum and
take the means we have
a = Mean Lateral Girth/Mean Breadth.
This gives in the present material :
mw = 3224 as against 3142,
which marks an error of about 2°6 wee rather larger than we might anticipate, and
possibly due to the inclusion of a certain number of slightly damaged eggs, and
the measurement of the eggs in the field and not in the laboratory. The relation
between G, and B is a useful test of accuracy and should be determined with a
slide rule before the egg is finally replaced in the nest, or lost sight of.
The Longitudinal Girth G; is somewhat more difficult to measure, and a rough
test of its accuracy not so easy to determine as in the case of G,. We have, how-
ever, developed a formula for determining Gin terms of B and L, and on testing
it we find that as a rule the differences are below 15mm. Such a formula may
be useful as emphasising the need for remeasurements, when the observed and
calculated girths have values much in excess of 15mm. We are not prepared
to say, however, that the coefficients in this formula can be extended beyond
the case of the Common Tern.
While the Length-Breadth Index is valuable as giving a measure of the
ellipticity of the egg, it is not of much influence on the apparent oval shape,
unless we suppose some theoretical geometrical construction for the egg. Hf we
suppose the blunt end of the egg to be approximately spherical, the hemisphere
ending with the maximum breadth, then the egg might be considered as divided
into two portions, the upper or hemispherical with radius $B and the lower with
length from the base of the hemisphere (or ‘ equator’) to the lower pole = L—4$B.
The ratio of these two segments of the length depends only on the index B/Z.
Thus it is conceivable that this index has actually as much association with
ovality as with ellipticity, although without some geometric theory of egg-shape,
we are not able to make any dogmatic assertion as to the value of B/Z. It
seems, however, a character of considerable interest as being free of absolute size
and also some measure of shape. If J =B/L and O be the ratio of $B to L—4B,
: B/L
Le. O= = BIL eit
have correlated O for eggs of the same clutch as well as J. Of course, since O 1s
a function of J, there will be relatively little difference in the results.
=I/(2—JI), we may consider O a measure of the ovality, and we
Biometrika x 19
146 On Homotyposis in Eggs of the Common Tern
The mottling is a far more difficult matter for determination. The points
which may be considered are:
(1) Size and shape of individual splodges.
(ii) Portion of the egg over which these splodges are distributed.
(iii) Area of mottled surface as compared with whole area of the egg.
The fieldworkers selected 9 typical mottlings (see Plate IX) and named these
a, b,c, d,e, f, g,h, i; they then compared each recorded egg with these and selected
the letter which marked the egg on the scale most resembling the egg to be
recorded. There is little doubt that in this manner they divided the whole series
of eggs into differentiated classes. But it may be doubted whether the judgment
made depended on one only of the above three characteristics. Hence when we
came to arrange the eggs a, b, c, d,...h, 7 on a scale of mottling, we found that the
order would not be the same when we classified in turn by each of the three
characteristics. We endeavoured to place the eggs in order by extent of mottling,
ie. by (iii), but we think that the relatively low value of the homotyposis which
has resulted is possibly due to size and shape of the mottlings, (i), having had
as much influence on the classification as the extent of area mottling. Even
position on the egg, (11), can influence judgment considerably. We believe that
in future work on eggs, it would be desirable to classify the mottling of each
by using the three characteristics independently. Even then an ocular appre-
ciation, as this must be, may fail to give a very close measure of the nature of the
mottling and thus weaken any homotypic correlation.
The Ground Colour of these eggs varies through all shades of brown to
brownish greens and blue-greens. The fieldworkers attempted to give the value or
depth of ground-colour pigmentation without regard to the brown or green shade
of colouring. The scale of values is given at the foot of Plate VIII.
A point seemed worth consideration: assuming the pigments to be deposited
on the egg in its passage through the oviduct, it was conceivable that greater
pressure might indicate greater intensity of pigmentation. We accordingly
selected the broader egg in each clutch and investigated for every pair of eggs
from the same clutch whether the broader or narrower egg had the larger mass
of mottling and greater density of ground colour. We reached the following
results :
The broader egg in every possible clutch-pair has
Greater mottling in 26 cases | More dense ground colour in 25 cases
a + « = .
The same ‘ BY op The same > $ 39
Less . 40s, | Less dense 5 5 BY a
Perhaps not very much stress is to be laid on these results, but they suggest
that the total amount of pigment deposited is less the broader the egg, i.e. for the
same bird a relatively smaller egg will be more pigmented. A solution of this
Biometrika, Vol. X, Part I. Plate VIII.
Colour Value Scale
Cambridge University Press
SAMPLE EGGS,COMMON TERN, NATURAL SIZE
Gs)
enh
i
fe
Biometrika, Vol. X, Part | Plate IX
g h i
TYPES OF MOTTLING OF EGGS OF COMMON TERN.
W. Rowan, K. M. Parker anv J. BELL 147
rather unexpected result may, perhaps, be found in the suggestion that the total
amount of pigment is the same in both eggs, but the mottling and ground colour
will appear denser on the smaller surface of the smaller egg. The point deserves
consideration on the basis of larger numbers and possibly better defined measures
of pigmentation.
(3) Means and Variability.
Table I gives the means, standard deviations and coefficients of variation of
the several characters studied. It will be seen that the tern’s egg has for
quantitative characters relatively small variation. The values of the coefficients
TABLE I.
Means and Variabilities (Absolute Measurements in Centimetres).
Character | Mean SU ease
| Deviation Variation
|
Length ZL ae .. | 4:14+:007 180 + -005 4°34 + "12
Breadth 2B ‘ cog 2°98 + -004 099 + :010 3°33 4°09
| Girth G... ae ae 11°39 + ‘015 376 + 010 3°30 + °09
_ Girth G, mbt .. | 9°59+-014 347 + 010 3°62 4°10
| Index B/L ~ Lad 72°04 4 136 3°449 + 096 [4°79+ °13]
| Index of Ovality, O* .... | 56°35+°171 4:334+°121 [7°69 + °22] |
of variation are less than many of those which we find for the human skull
(3 to 8), but greater than those we know for the wing of the wasp. It is very
doubtful whether the coefficients of variation of the indices should be included
in such considerations, for the object of the use of these coefficients is to get
rid of absolute lengths, and this is already done in the case of indicest. It is
noteworthy that the length of the egg is only slightly more variable than the
breadth and the breadth-girth is actually more variable than the length-girth.
(4) Correlations.
If we turn to the correlation of characters in the same egg, we note that
while the ordinary product-moment correlation * has been calculated for all
measurable pairs of characters, this is not possible for the ground colour or the
mottling. Where mottling has been used with a quantitative character there
has been calculated and both corrections used. Where mottling has been con-
sidered in conjunction with ground colour, there we have adopted: mean square
contingency correcting for both number of cells and for class-index correlations.
* O=(B/L)/{2-(B/L)}.
+ For example, if we take 1/0 for our index of ovality its mean =176-32, the standard deviation
=11-24 and the coefficient of variation =6°38. Is O or 1/O the more variable? It does not seem that
the coefficient of variation can help us in such a problem.
19—2
148 On Homotyposis in Eggs of the Common Tern
Certain facts are at once obvious from this Table, others are obscured. In
the first place length and breadth of the egg of the Common Tern have a rela-
tively small relationship, while the relationship between the two girths is between
TABLE II.
Correlations of Characters in the same Egg.
Characters | Symbols | Correlation Remarks
— — = = = ——=—
Length and Breadth L, B \+:2220+ 0374 =
Longitudinal and Equatorial Girths | Gy, Gy +°5297 + 0284 =
Length and Longitudinal Girth ...) Z , G, +°8804+-0088 =
| Breadth and Longitudinal Girth ...| B, Gy) |+°5216+ :0286 =
| Index and Longitudinz al Girth B/L, Gy | —'3832 + 0336 —
Index and Length *: B/L, L |—"7284+°0185 =
Index and Breadth es BIL, Bo | +5033 + 0294 =
Mottling and Ground Colour M,C + ‘2260 (corrected C2) | More mottling, deeper ground colour
Mottling and Index... | M, B/L |}—-1550 (corrected y) | Less mottling, higher index
Mottling and Breadth | U, B — 1803 (corrected 7) | Less mottling, oreater breadth
| Ground Colour and Index ... C, BIL ‘0000 (corrected n) | No relationship
Ground Colour and Breadth | C, B |—*1506 (corrected n) | Fainter ground colour, greater breadth
two and three times as great. This probably flows from the consideration that
the correlation of G, and G, arises from B being a factor in both and only
secondarily from the correlation between Z and B. The correlation of the
longitudinal girth with egg length is 60% higher than that of longitudinal
girth with egg breadth; both these correlations are more substantial than that
of the longitudinal girth, G,, with the egg index, B/Z. The egg index correlated
with length is large and negative, and with breadth considerable and _ positive,
precisely the results we should anticipate would appear if the correlation were
largely spurious *.
In order to ascertain how far it was possible to predict the longitudinal girth
from length and breadth, double (for Z and B) and triple (for Z, B and B/L)
regression formulae were ened out. The following equations resulted :
(i) G.— G,=1-2701 (B-— B)+1:6415(L—L),
or, G, = 12701 B +1°6415 L +8224,
and = (ii) Gy — G; = — 17-2930 (B — B) + 146374 (L — L) +7636 (I —T),
or, G, = — 17-2930 B + 146374 L +-7636 B/L — 527239.
The first seventeen eggs were taken as a random set to test these results upon
with the following values:
* As a matter of fact the correlation of index and length for a constant breadth is —-997 and
of index and breadth for a constant length is +-996 instead of unity. These values indicate how closely
the linearity of regression holds in these quantitative measurements.
W. Rowan, K. M. Parker and J. BELL 149
TABLE III.
Observed and Calculated Longitudinal Girths.
Calculated Girth Ditference |
Egg | Observed |
Number | Girth | pmer l :
Gy (ii) | (i) As | Ay
ie - = | | =s
i 11-40 11°14 | 11-20 + 26 +°20 |
2 11°65 11°83 11°74 — 18 — ‘09
3 12°10 12°24 12°07 | -— ‘14 } 4°03 |
y 10-80 11°46 | 10°84 — “66 --04
5 11°70 Deo ee aS + °47 +°39 |
6 11-20 We elt — ‘07 —:14 |
? 12°15 13°19 12:31 — 1°04 —16 |
8 (i) 11-20 11719 1-27 + 01 — 07 |
8 (ii) 11°30 11-09 WED =e. 2 +03
9 (i) 11-50 1144 | 11°61 | + 06 Sil
9 (it) 11-40 11°36 11°45 + -04 — 05
10 11°50 11°52 11°61 — 02 —1l
11 11°80 WES 5 iy ele 72 + 25 08
12 11-90 11°62 11-74 + 28 +716 |
13 -(@) 11°10 11°01 10°94 209: |. E16
13 (ii) 10°80 10°75 10°78 + 05 | +02
18 (iii) 11°70 1455 |) 11:55 + 25 e155
e o3 a Root mean 354 146 |
| square A |
To judge by this small sample we obtain only increased inaccuracy by taking
the more complicated formula. We shall only make an error of about 14 mm. if
we calculate the longitudinal girth from
G, = 12701 B +1:6415 E +8224,
and for the egg of the Common Tern at least this is a convenient formula for
verifying measurements in the field.
The remaining correlations indicate sensible correlations, but these correlations
might well be substantially higher had a better scale of mottling been adopted
ab initio. In the first place we see that the mottling and the ground colour
are sensibly correlated, and the deeper the ground colour the more intense is
the mottling*.
We have already seen (p. 146) that for eggs of the same clutch the broader
has less intensity of ground colour and more meagre mottling. This is true
for the eggs of the Common Tern in general, although it is probable that a better
classification of mottling would bring out more marked correlations. The
* This might probably be asserted interracially as well as intraracially, compare for example the
swallow with the skylark, the lapwing with the ringed plover, etc.
150 On Homotyposis in Eggs of the Common Tern
following are the orders (a) of mottling chosen, (b) of breadth classes, (c) of
index classes :
| |
| (a) () (c)
| ser sy: . Order of Breadth Order of Index
——— —_ {ics = —___— _ —
Class B Class BIL Class
gte+d a 3°00 a 72°64
a c 2°99 f+i 72°54
b gtetd 2°97 C 72°30
Cc ttt 2°96 gtetd | 72:27
h h 2°96 h 71°95
ft+é b 2°95 b 70°54
= Mean 2°98 Mean 72°30
The relationship is small, but exists. It seems reasonable to suppose that
the order of mottling classes as given by B or B/L, where there is only one
displacement, may be a better one than that we have selected. But if in the
mottling order b and c were interchanged, it would agree with the B classification,
in so far that the three classes of least and of most mottling in the two classi-
fications would be the same.
We now turn to the ground colour. We see that the ground colour is
fainter, when the egg has greater breadth, but that there is no relation of the
index to the intensity of ground colour. The results of p. 147 are thus confirmed
by the general correlation of ground colour and breadth. Although there is no
high-correlation, we may assert that it is probable that the intensity of pigment
dees not depend on the pressure during transit of the oviduct, but rather on
a constant amount of pigment being distributed over a larger surface.
(5) Homotyposis in Eggs of the same Clutch.
The homotyposis, or degree of resemblance in character between eggs of the
same clutch may be studied on the present material. The chief direct and cross
homotypic correlations are given in Table IV.
Pearson has shewn* that the degree of resemblance of undifferentiated ‘like
organs’ might be expected to be equal to that of pairs of brethren, i.e. about *50,
and proved that this is so for many homotypes in the vegetable kingdom, a result
which has been since confirmed by much as yet unpublished material from the
animal kingdom, including a number of series of birds’ eggs. Thus the mean
value of the homotyposis for eggs of the Common Tern could hardly be improved
upon. Only the colour characters show irregularity, especially the mottling, a
* «On Homotyposis in the Vegetable Kingdom,” Phil. Trans. Vol. 197, A, pp. 285—379, 1900.
ad hee
W. Rowan, K. M. Parker ann J. BELL fod
feature we have already indicated as difficult to measure. It will be seen that
the correlation of the ground colour of an egg with the mottling of a second
(3989) has come out greater than the organic correlation between mottling and
ground colour in the same egg (2260).
TABLE IV.
Homotypic Correlations.
| Symbols | Characters Correlation
L, L£ | Lengths of Eggs in same clutch ... oe oes su ... | 4643 + 0346
B, B | Breadths of Eggs in same clutch ... fee bet i ... | 5176+ 0326
G, G, | Longitudinal Girths of Eggs in same clutch... et ... | 0076 + °0327
G@,, @, | Equatorial Girths of Eggs in same clutch ae oss ... | 4621 + 0350 |
: | Mean value ae te 4879
Direct
M, M | Mottling of Eggs in same clutch ... mo te “ee ae| °3500
C, C | Ground colour of Eggs in same clutch ... es one oe “5709
| Mean of six characters ... isi eis "4788
| aan |
L,B | Length of one Egg with Breadth of a second ... ... | 0922+ 0441 |
C, M Ground colour of one Ege with Mottling of a second ... | 3989 + :0379
Cross | £, G, | Length of one Egg with “Longitudinal Girth of a second... 4229 + '0362
B, G | Breadth of one Ege with Longitudinal Girth of a second... | *2530+-0416 |
1, Gy | Longitudinal Girth of one Egg with Equatorial Girth of a second | +2603 4 -0413
| B/L, ie Indices of two Eggs of same clutch ie .. | 65374 0308
Index | 0, Indices of ovality ‘of two Eggs of same clutch ... ee ... | 5527 + °0309
| 1/0, Ho Inverse of indices of ov ality 5361+ :0317 |
| |
| |
Mean of three Index Correlations —... 5475
Mean of nine Homotypic Correlations 5017 |
We feel that the classification by mottling is at present too uncertain, and
that until the result cited has been confirmed with larger numbers and more
definite categories, it would be idle to consider whether, while a given bird has
usually highly or lowly pigmented eggs both as to ground colour and mottling,
yet when in the individual egg there is an excess of mottling pigment, there
may be some tendency to a relatively less increase of ground colour. Thus the
correlation in the individual egg might possibly be less than the correlation
between eggs of the same clutch. Such considerations must be postponed until
the fact itself is adequately demonstrated.
152 On Homotyposis in Eggs of the Common Tern
Another relation suggested by Pearson* is that the cross homotypic corre-
lation of the characters # and y should on the average equal } (correlation of
w and w + correlation of y and y) x (the organic correlation of # and y). It is
clearly impossible from what has just been said to apply this to the cross
homotyposis of ground colour and mottling. We can apply it to the five cases
in which quantitative measurements have been made. Table V gives the
requisite data, the last two columns giving respectively the calculated and
observed cross correlations.
TABLE V.
Cross Homotypic Correlations.
Characters Direct Correlations ' Cross Correlation
oe | Organic Correlation | Te
| eee (1) and (2) | |
| (1) (2) (1) and (1) | (2) and (2) | | Calculated | Observed
|
DL B | +4643 5176 22.20) +1090 0922
L G, | “4643 OTK | “8804 | °4278 "42299
ie G, | +5076 “4621 “5297 | +2568 2603 |
B Gree ee ou7G Oi 5216 | "2674 "2530 |
| eee
| Gy 3) BEE “5076 SDDoM — 3832 | — +2083 — 2007
When we compare the calculated and observed cross correlations, we see
a striking agreement, or the theory that cross homotyposis is the product of
direct homotyposis and the organic correlation of the characters under investi-
gation holds very closely for the egg of the Common Tern.
The general results obtained are in good accord with those reached by previous
observers, and the authors hope to investigate one or two doubtful points on
fuller material this year.
* Phil. Trans. Vol. 197, A, p. 290.
W. Rowan, K. M. Parker anv J. BELL 153
APPENDIX OF CORRELATION TABLE
TABLE A. Length and Breadth of Egg.
Breadth.
aS es
[ | | | Totals
Sikhs = {8
39 89 Sa) Sn)
| 1
0)
1
| l
Hee ee oat ae SS el Sy 5
ee ee ah | ome le 5
yas fake S33 sto Ie (eee rs 15
SO eee ah hse eal | | ee | Pe 10
on hes ee G6 Bal) A 25
= ho Dule 6) hese) bal 5 26
S05 09 ef —— |) — | 1 | 1 Dae ot lh gorleSa is) 2 27
3 | 410—-p14 ee Sai) toe eon ebul|l 2 45
415—h19 | Sea! OF Bi 2 8 24
4:20—4-24 Win Siles | 6) 2 31
425—429 Dae AN Ae AGB il 2 23
430— 34 ee eG 3 ak) a 20
4°35 439 Towser lute ero alo. loa 14
44O—I 4h Oalenaileccceit al eae if
44S 49 =| bl) tj |—)| 2] 5
L50—b 54 Tj} 1{]—/ 2] — ] 6 |
455—h59 | : | ; 0
460—h65 1
465—469 0
LIO—-4VS 2
Totals | ME Oe ileae os oa | 14 | 24 | 61 | 61 | 57 | 35 | 20| 10 | 2 | 294
H |
TABLE B. Garth L, and Girth B. Girth B.
; i la2leaje|
za . 9
S| 5
| ileal
| Sales
10°00—10:09
10°10—10°19
10:20—10°29
10:30—10°39. |
10-40—10°49 |
10.50 10:59 4 —— | — |_-1 4 1 4h - |
10°60—10°69 1 : = = | —
< 10'70—10°'79 | — |.— | — ! 1 1 } 2 .
10°80—10°89 | —- yl ya a
e090 1090 == |= 1 yet 42) els) 1 |
Se Om 1c 9 ean| ie et he Rs al | 97, We Se | ee
© | i10o-1119] —| —| —|—| 2] 5] 9| 5] 2/—J—J1]—
Mcgee Ve See ee) SINS lg 10}, 8} 5)/—]} 1}—)}—
HATED Sa 30) | | ee ene) a em Fe el a |e ea
eter .0— 7 1-7.) |) apace Webern @7 ==. ey
| 11:50—11°59 Tees eee ae Ila acelin
eel 60— 11-694) — ||. | —— iL a=} | AR i) 5 5 5 = x
11°70—11°79 Sele Toshio eGullet —
| 11:-80—11°89 > Teele ea Go Goe co)
11°90--11:99 TOW POF ie Assen, oe |
el? 00== 72-99 — = | — |) — = | | BE i)
12:10— 12:19 | 1| 2
12:20—12:29 et
Totals
Biometrika x WY
Eggs of the Common Tern
yposis in
On Homot
154
“we
a] ana
HMO MHA A SS
maf OD 19 19 190 |
AAD O MOM |
age ee pee es
Sey aetna eh aera ee
S|) Se ee ua
y |7
Cia lecue cee
CUeOl | Galee aa
SSM sleOs oleae ile | =
See OE Neh enl Oe sl peel its ela
Te ee ealeG tl aer eG eal) ilar = oe
eich e\eGen ois) ee sed =
= WEL Rowalntoe eee Nigel =
[eh al at eee ier — =
Ti lee ee
STPIO,
67-61—04-6T
68-61 —06 6T
66-61-—08.61
61-6I—O1L-6T
60-@1— 00-81
66-TI—06-1T
68-TI—08-1T
6h-TE—OL-TT
69-TI—09-1T
65-1I—0¢-IT |
67-II—OF-1T |
68:-TI—08-1T
66-TI—06-TT |
61-TI—OT-TT
60-TI—00.TT
66-01 —06-0T
68:01 —08:0T
64-01 —O4L-0T
69-01 —09-0T
6¢-01—0¢-0L |
64-01—0%.0L
68-01 —08-0T
62-01 —06-0L
61-0I1—OL-0L
Me bels®)
60-0I—00-0T |
"Tyne pup bby fo yybuayT
‘OD FTavVit
SiS Hie] eis SS ~~ | & | ~ S ~ Og Gg | te
~ lL Ql} Al alT ale] se] ow] eo] ws S/R} RY Ss o D | a
ae alle st SN > N S an S in S NY > N S | a
IAS a eRe aL acl OG Cd en al a Wa | | |
S|) see || Seale Se || elf Se alfeosarn te Set] Seal Sea ge Nosed | seta [ees <s ce oe
‘yysuarTy
Girth L.
W. Rowan, K. M. Parker and J. BELL 155
TABLE D.
Breadth of Egg and Girth L.
Breadth.
»ni{[slaflsilaltssijafjsiloaltlxsiolwsloalrs
Ma OU OES He WSO | FO. EO Oa 8S |S lo | | ee
i) NX X » RW | & NW | & » % | % 9 | Ss)
| | | | | | | | anlee| | eal | | Totals |
ip S wy oS iW Ss wD S wD Ss Xe) S wD iS
VOM Ore || Reale eco |G) | Ose Os |S S| a | ose |e
RiRIl R{/RIRQIRI RI RI Rl nls {_n]_o] Ss
10:00—10°09 1
10°10-—-10:19 O08
10°20—10°29 2 |
10-30 —10°39 0
10°40—10°49 2 3
10°50—10:°59 | — | — | —| 1 1 - 2
10°60—10°69 | — | — if eee ee eee | ee ee oe ee 1
10°70 —10°79 | ee fag Fe = 4
10°80—10°89 | | 5}. 1 a] |e 3 13
10:90—=10:99 | —|—|)—| 1)}—|].1] 4| 4] 38] 5 18
00 —thOO— |e | |=) 1 Ae ee Sale A eee ea Se | ees 16
neon —— aie | 2 IO A | 4 | oe |) 1 24
11:20—11'29 | — se Pe Te a eLO! Ch eed 2d 32
AC eou wee fe eT lO Bee | Be | 4 = 3
11-40—11-49 | eee ne oaelise| Cree. | Tk |p=—. lea
11-50—11°59 | Dele eG oe Ob etal Wee ety cae
11:60—11°69 | eye aliemee ie etal Pret can els aie oh |)
11-°70—11°'79 | — - eae wo Bal) Vel 2) |
11-80—11°89 | Aa 28) OC an a5 | al
11:90—-11:99 Ses Paleese coer Oras | Seb ape
12:00—12:09 | — | — | — | =i ee been. Sl at Weg i) =
12:10—12:19 | — | — | | — | 2/—]| 1}/—] 2) 1J—
12:20—12°29 | | | 1
12°30 —12°39 =| | | = ae ee
12:40—12°49 | | | | | — | - i
pe tn
i i}
Heer ete oe a ae 24 | 61 | 61 57 | 35 | 20 | 10} 2
TABLE E. Girth LE and Index 100 Breadth/Length.
Girth ZL.
156 On Homotyposis in Eggs of the Common Tern
2
| =) DOARNAMONHMAA tH
$ AMI O10 N z
i=
67.6I—04-6I a
i
| 66-61 —08.6T S
| 66-61 06-61 =
| i
61-61—O01-61T No)
| 60-61 —00-é1 Ne) eS eS
66-TI—06-TT oS
|
yeaa ellie ere 9
68-1T—08-IT is 3S
: a x Kt
6L-TI—-OL-TT sy] oN
nl ————|
ee = =
69-TI—09-T1 | = =
> °
0, TT— Ne. oo = a
69-LI—0¢-T] <e ma)
ea ays x
64-LI—0V-1I g = eal
| Ce,.7T—ne aA OONHN aa = o>
65-1 09:0E a S
a | ee ee = ST
6%-TI—0@-TI | Aamo a ac a
4 oD tS ap b
| S § |¢1-4—o1-4 s)
6L-TIL—OL-TT a i Ss : =
Pomerat ae 3 4 —C0.4 5
: A 60-%—GO.
| 60-1 00-1 7 cee =
oy — Tl) SS tn tf —pn.t Re)
66-01 —06-01 eet io) = LOT ROOT HY
— 4 ~ =
a = S
1 O-—CR.E Ye)
68-0I—08-01 29 = ee ms a
1 N te.0—nG.e oO
62-01 —0L-0T _ 16-806- oy
, .E—G9.E Ne)
69-01 —09.01 = = 68-8 98-8 i
— & is
.E—09-¢ 10
| 69-01—09-01 aa a] 18.E—08-E
:
; arom 64-E-—-GAE oo)
| 6-01-—04-01 Po = eee
ae /.e ay pia =
6801-08-01 ° ee Bice
ee 69-2 —O9-¢ =
66-01 —04-01 a 2 ‘ ?
[Sue e =n perom a Py Sava (a
, Pale 19-6€—09-§ °o
OL-01—OL-0T =) :
60-01 —00-0T A
PA BABAARSRAW
MMBIHRD HNWVIwD
Mm he 4
| ~
Seeeeseeseese {es
3S X Sos Xe) S
Q > & SSSSRRRKKLSVE
*
o
ire)
fo
La)
‘XOpuyTy
W. Rowan, K. M. Parker anv J. BELL 157
TABLE G.
Breadth of Egg and Index 100 Breadth/Length.
Breadth.
fon | & + | d + | > = | oO s+ | st | Pe
BS I Se Sh all Ps Te pe teeky tS ce eS) SS ee tt caainllE S |
Rl Rl} _ eX |_ Raia Ie2le sl a&laelse lala] |
| ee era Ne eh ee ay tb) oh ll a Botals}
1D S LD 2 Ro > WwW > Ww > WD S w S
eso ek | els pole TS |S Sis | 1S
Ri RIL RIRI RIL RIRIRIRARIni HISD | SH /1H
| 62:0—63'9
64:0—65°9
| 66-:0—67°9
| 68:0—69'9
70°0—71°9
72-0—73'9
ThO—75'9
%6-0—77'9
¥80—79'9
80°0—81'9
82:0—83°9
84:0—85°9
| | ~1m ts = conn
Index.
WoL PDE
> OW DO NT bo Ow Or
| aa)
oe OTN
Shiono kas |
bo OOD) ATW bo
on)
i]
Pe web
bo bo OAT bo
Totals
TABLE H.
Ground Colour and Mottling.
Ground Colour.
Totals
15 16 | ; 73
(+3-96) | (—5-28) | (+ °70)
5 | 4 4 26
(+107) | (+ :69) | (—1:45)
So 14 3 49
is (23°73)
~ |
18 ‘ 101
=
| 13
| 29
291
158 On Homotyposis in Eggs of the Common Tern
TABLE J.
Index 100 Breadth/Length and Mottling.
Index.
S| SD [SD [Sa] Seonloaian | Lom onltoraicay
SPU eet met comm ipo ian Umut WIL CAy mylene cco Ionian | be iassy [1 Wo)
| © Ks) Ne) S ~ 2S ro NS rs ra) ore) ey |
| | | anya | | | | | Patel Totals
| San Ease ec eae a ge ell eral ieka eager |S |
| Sy =“ xe} Soha | oe) RN => Ne} nH SS eR |=
| Ne) Ne) Ne) 3° NN reel) ES BS ~ | w Se) ies)
oc | 2
a g+d+te 15 | 7 —
D> a 6 l
6 b Sule
= c 92/4) 99
h 5
Fri 5
Totals
TABLE K,
Breadth of Egg and Mottling.
Breadth of Egg.
|
& : 3 a) 35 | 3
esl | | | Totals
2) i=) S
S = RN
on -
| gtdte — | Qa Galva Feo UO ote GP oy ts :
= a —|—;—]—j;—] 1) 1] 6] 4) 6] 3] 3] 2) — 26
2 b ee eee Mee kerala sey ymoy lee |) ay aloe on 49
aa ¢ 1. | 2) — | 8 3s Bl aa O28 19y) alize |e onli b: 1) Sie heaton
h ee ee es ee ee hea
fri | ie ae eA eal S| ef 29
NN Oa a a a UO Po
Totals 1 | 2{1|]2| 41/14/24] 59/611 56] 35| 20/10] 2 | on
momAl li As Ht i é | |
W. Rowan, K.-M. Parker
TABLE L.
Ground Colour and Index
Ground Colour.
AND J. BELL
100 B/L.
9 Seat canny |
a eel edlaa fa i
62:°0—62'9
63:0—63'9
64 O—C49
65°0—65'9
66°0—66'9
67:0—67'9
68-0—68'9
69:0—69'9
70:0—70°9
71:0—71°9
72:0—72:9
73:0—738°S
ThO—TH'9
| 75:0—75°9
760-769
TiO Ti
| %78:0—78-9
79:0 —79°9
Index,
| Reo akacwwe pr |
| Si paroo. aa en og eo<ar'| =i | ws | wre bp
OD
Hew bono e | les
me NM DO or
80°0—80'9
S1'0—81°9
| 82:0—82:9
S3-0—83'9
8h0—8 49
85:0—-85'9
Totals
‘ig | 18
TABLE M.
Ground Colour and Breadth.
Ground Colour.
{
t
Re fee
|
Oh 9'59
260—2-64
2-65—2°69
270-27 4 |
2-75—2°79 = 2
S| 280-284 1 anes 4
canis 2:85-9:39 f— | 3} 1) .3)| 81%
Ss | 290-294 | 4| 8/13] 6) 12| 7
oO 2:95—2:99 Sol e380) Bo 88 dill a5
S00 S:0he Nee) | Le 10) Ob 10
3:05 —3:09 Sf 6 Biaor| 8
310—3:14 | — Nala he oe eaO
PUB — PUG) | eh eT ih a
ais 20-8121 =| 11 -
|
159
TABLE N.
Breadth of Egg in Pairs of same Clutch.
On Homotyposis in Eggs of the Common Tern
F0I—3 09
3:10—8:14
I — ITH)
Totals
|
1
pi
a S oom = a
» X R cS) & x ) x» | 3 85 S95 85
Die Ue es ees ee Poet el Maley Walls ix]
9» > a) AD YS Ww i) Ww S wD S wD
ME Sigal SSF eet COT SCO SO OS A SNE) OPS a
2 il RI Riel RIRliaAgl Rial ala |] a
. | | ]
| 255—259 | — | 1
260—2°64 | 1 ; |
265—2°7), = | cae Es
275—2°79 ~ | 1 1
| 280—2-84 = - | | 2 2) 4 2) L|— Wh ||
2-85—2°89 | 2 1 4 i) 1} —}--}| —
290—2'94 1 4 4/10); 14 8 5 1
2:95—2:°99 1 2 3) 16 | 12 Oy) || —
| S:00—3 04 it 1 12 |) 12 8 8} 2
- 8 4 6) |5——
8-i\ 135) 74
2 ale
=
on)
ase
-I
ou
or
or
is)
Totals
TABLE O.
Length of Egg in Pairs of same Clutch.
L00—4 04
4O05—4'09
st D> Ty D>
~ ~ XN Se
at
a lh | eal
a[ mle] es
SS eashe | Sse
L55—43B9
44O—Y 4S
J
=
ol
Sd ate Sd be 9) Dd
XY fol lolals
Sn) 35 35 An Sn)
| | | | |
wW SO WwW > WwW
~~ | %© rs) Dd DS
& | S Sn) Sn) 89
Se ED |
BT5—B TI | |-- 1
3-80—3°8 4 |
Totals
20} 14] 1
3-85—3°89 | — Ze ey Male ale at al —
3:90-—3'94. 4 — | — 2) 2 3 a ea
3953-99} 1 }—} 2/ 8] 6] 6] 2} 2}—] 2}—}—];—}—}]—]—
4:00 —404 = 6 28 23 225) See es es
4:05—4-09 | — =) Se a ae es le Sie ee
410—414]—| 2 1 ee lee eta eS Ee ey MN cet Ea The hl
| £15—419 | — | — 1;—|;—] 8] 4] 6] 2] 5)| 1 2};—;—|]—]—
| 420—4241—|—! 1} 1] 2) 1) 1] 2] 5) 2) -2) 2) =|] 1 | —] 3
| 4:25—429 : ;}—/ 4] 1] 3] 1} 2/—}]—jf 1]— =
| 80-484 1— |—|—]| Ly—] 1] 1) 4) 2) a 4 be |e
435—439 3 Deh — = | Se 4e |)
Y4O--L hh Sl eee | 1 |
| #45—4'49 =| pl 1
450—h 54 | | 5
1 oa 28 | 24 | 36 | 24 | 23 | 12 W 8
Totals
10:20—10:29
10°30—10'39
10°40—10'49
10°50—10°59
10°60—10°69
10°70--10°79
10°80—10'89
10:90—10°99
11:00—11:09
11°10—11'19
W. Rowan, K. M. Parker anv J. BELL 161
TABLE P.
Girth B in Pairs of same Clutch.
| S(E(SiSisl/SyF/s{/Sl/sisys
| ine) ine) ioe) Dd o> Dp D len) Ss S S S
| s s 4
| asec eter te dea al ie he i otels
| S S S S S S S S S S S S
R/S (S/S PRPSsl/Sl sel slaelSis
~ ~ ~a Dd =) Dp ier) im) S S S S
4 n 4 a
8:20— 8°39 9
8-40— 8°79 0
&*80— 8:99 2
9:00— 9:19 16
9°20— 9°39 37
9-40— 9°59 68
9°60-— 9°79 57
9°80— 9:99 31
10°00—10°19 11
10°20—10°39 3
10°40—10°59 0
10°60—10°79 3
Totals 230
TABLE Q.
Girth L in Pairs of same Clutch.
Sn oylliFas) Ss > a lala sf eS eS ss) es ee es
S)/s/s is & = {| 2] 8 SiSi/E l/s lsl/ sls
S =) S S S ™ mt ~ a il a ™ ™m | & N
™ ™ iat = Oh] = =e ~ ™ = ™ ™ ™ | ™
ieee | sales | | Loiieede alae Pa | ok Pa) | totals!
Ss S S S S S S S S S i S S =) S |
S/S) s |S © =| 2/8 e/S}/eF;el/se]}s]s
S S SS S ™ ™ ™ ban i iH ~~ a RN R
SO 4 nN 1 mm | N sO | i
1
be |
11°20—11°29
11°30—11°39
I1:-40—11-49
11°50—11°59
11:60—11°69
11°70—11°79
11'80—11°89
11:90—11°99
12:00—12:09
12°10 —12'19
Totals
Oalla=.
— | — 3 Il 2
4 2) — 1
3 2 + 3)
1 | — 3 6 4
2 1}|— 4 4
— 2 2 2 6
1 5 1) — 3
1
2); —}]— Fs |) al
—/;|—|— il 1
1 2
rar [esrron Iker l. ‘a> no hoses: |
| | | woanad | roe
1 @) || al 1
=at= | 2
2
Fle tte it aie, ee
A esa ltota lt ested ao) a) ee ee
ea OuINe a eal ik hel ae
3| 3
eon eee a Sol ae a
Sp Gel eese |ulegte es eel
3 ee In oi) wou
yal ige |e | uae | a ees
Se Woh Bou ee ee
1 =
(| een ee
a1 | 20 | 18 | 11 | ose elle
Biometrika x
21
162 On Homotyposis in Eggs of the Common Tern
TABLE R.
Mottling in Pairs of Eggs of same Clutch.
Totals
a
OQ
h
fti Totals
Pairs of same Clutch.
TABLE &.
Ground Colour of one kgg with Mottling of the other Egg for
Ground Colour.
Totals
gtet+d 8 67
(-5°16)
al | 6 Wly/
| (+2°66)
bh b 31
=
S Cc 10 82
s (-1°71)
h 1 14
(1-00) | (— -94)
f+t 4 3 13
(+2°14) | (41:20)
Totals 32 31 224
iS
In Tables R—T, the contingency of each cell is given in brackets,
W. Rowan, K. M. Parker anp J BELL 163
TABLE T.
Ground Colour in Pairs of Eggs of same Clutch.
a+b
12
42) | (+ °58) | (4+10°47)) (41:43) | (—2°97)
f 3 4 4 12 20 8 51
(— 5-96)] (—3-12) | (—3-12) |(+ 1°43), (48-28) | (+2°49)
g—k 1 4 1 2 | 8 8 24
(- 3-22) (42°49) | (45-41)
Totals 222
TABLE U.
Length of one Egg with Breadth of the other Egg for Pairs of same Clutch.
Length of one Egg of Pair.
Breadth of Second Egg of Pair.
a = = = ms
2) S/S 2 |
i Totals
ese
2:55—2'59 | | 1 | — c= = 1
260—2°64 | — | — eee | 1
265—2°7 = | = 0)
25— 27 1 1 2
2°80—2°84 = 1 1 a 1 2 2 1] — 1 12
2°85—2:89 | — | — | — 1 3 oe 3 3 }|— 2 16
2:90—2:94 1 4 1 7 3 6 i 2 is) 2 5 tn eee ee | 47
2:95—2:99 | 1 | — 4 2 2 8 9 6 (ji) zt 83 5 Ab ee eel 55
3:00—3 04 il 2 2 2 i 2H a3 5 4 5 4 6;—|]—]|1 52
POSER) | | 4 (oe | 4 4 2 3 2)— 1 |= 1} — Zo)
SHOE | | 1 2 3 3 1 1 Ae | |) SS ee 17
3:15—3'19 == | 1} — — | — 2
Totals 230
164
Length Girth of Second Egg.
Length and
TABLE V.
Girth L. in Pairs of same Clutch.
On Homotyposis in Eggs of the Common Tern
10°20—10°39
10°40—10°59
10°60—10°79
| 10°SO—10°99
11°00—11°19
11°20—11°39
11°40—11°59
11°60—11°79
11°80—11°99
1200—12°19
Totals
Length of First Egg
& |os fe |aelalalyalals
| | | | | | | | | | Totals
Si S S => } S S S
= S/Sia|/RXl/ S/S] s
39 oi|(sei sisal arlasrls
| to bo no bo co bo |
pre
EPDprhe OQD-!
— i
BORO Or
TABLE W.
Breadth and Girth L. in Pairs of same Clutch.
Breadth of First Egg.
2:55—2°59
Length Girth of Second Egg.
Totals
10°20—10'39
| 10°40—10°59
10°60—10°79
10°50—10°99
11°00—11°19
11°20—11°39
| 11:40—11°59
11°60—-11°79
11°SO0O—11°99
12°00—12°19
2°65—2°69
270-274
280—2'84
rele! |
285—2°89
47
| Gees ioe |
2:95—2:99
Sora! | | | 200-304
bo oR HAT
on
bo
3°05—3'09
bo
or
3-10—3'1h
lromaale| | |
Length Girth of Second Egg.
W. Rowan, K. M. Parker anp J. BELL
TABLE X.
Girth L. and Girth B. in Pairs of same Clutch.
Girth of First Egg.
165
10°20—10°39
8:20— 8°89
Breadth
aD Sd
NN lon}
Sc | oo
| |
S >
& | &
9:20— 9°39
9-40— 9°59
9:00— 9:19
10:00 —10°19 |
10°40—10°59
Index 100 Breadth/Length in Pairs of same Clutch.
op Meros50 lst ms a ad Se |p |
10°60—10°75 etl Se ee | |
OOO OO A | 2) G18 eo a) Se |
11-00—11°19 Ae ol ae pie eb |= melt
11°20—11°39 }—| 83| 9/11 ]}12]} 8} 4)/—]—j 1
ie om 0 oOl a a 9 3a) SAN tA Ni | 6) B)|| 2el\e=9)
11°60—11°79 Sel eouee ied | sole ot tle =
11°80—11°99 ees alpaca ee 50m al wt j=l) Sete
12-00—12°19 | } 2] 1] |
Totals oF Oo iO | 8 | 16 | 37 | 68 | 59 | 31 | eS lewia| 13
TABLE Y.
65
69 70
| TL NG 72
failed
76
= — 2 1
15 3D) | ol 1
3 say |p aay |i) baa)
15 | — 3 2°5
15 | 3 tf 75
15) 2eon 1 :on Le
— 5 | 5 5
3 6
1 2 15] 6°5
5 °B) || —— 3
| 1
bo LY BO Ge OH
OV CH
5
O55 |
— Lat
== 4/4]
= |?
3 I 1:
6 |6:
25/1
10
il
2
5 | 2
Se OW HO
5 |
‘i |
3 I — {5 | —
B59 ea ebe
2 | 5/—|]—|—
3 2°5| — | —
i 62: — | — |
ee
25
3)
Totals |
HPHOmlprEwoH
HOR HR HH
at
|
|
166 On Homotyposis in Eggs of the Common Tern
MONK FOr DMDOAADIEOD
sl AAMAS
wd A
ie ics re)
58 | 59 | 60 | 61
in Pairs of same Clutch.
TABLE Z.
100 B/L
2 — B/L
6321 85h C66 56 | 57
Index
TABLE AA.
Clutch.
f same
a
2 L/B—1 in Pairs
W. Rowan, K. M. Parker anv J. BELL
60-6—00-6
66-I1—£6-T
96-I—¥6-1
86-I—16-1
[egal Sse Ge
06-I—88-T
lie Wlealer? eee ete
48-I—G8-T
| Ja aa | faa] | |
| [aaa ece |
ee a
mame HTN WOODY SS
|
nN |p ora 69 a1 |
[Egat tina SenlhON |
69-I—49-T
[| <2 JAG Sts A eSeSiaECY
99-I—"9.1
Anan Pao walion
&9-L—19.T
sow | a! |
09-I—8G-T
AG EGET
4G.I—@G-T
167
168
Index of Second Egg.
|
TABLE BB.
On Homotyposis in Eggs of the Common Tern
Girth L and Index 100 B/L in Pairs of same Clutch.
Le
ng
th—Guirth of First Egg.
3 | 26
(39 | 48
51 |
> |X |< Di oma) fies ol | al |iey
S571 BO Sa S> oH SORE |e S= IEC Salas
> > > SSF | sStlosh PS St | Ss
ee ee ere IL |
S = SIS S Ss S > S S
X Sy Se) se Sis pat | SO at) ACO SS
=) =) S =) ™ ™ = ™ ™ RX
~ ial ~ ~ m el al ~ ~
SS a a ee ee
| 62:0—62:9 | 14s ee
63:0—638'9 —
64:0—64'9 | 2 1} — |] —
69:0=—65°9 - 1
66°0—66°9} 1 1 =: 1)>—|;— 2/— 2
| 67:0—67'°9 3 1 2 he
68:0—68'9 Healy oll 3 9 el) =
69:0—69°9 Lie Q aoe
_ - . e | ¢
70°0--70'9] —|—}|1 | 1 | Ge) 2B a 22 ae
CL 07129: 3) LON WS 8 2 Sued
T20—72:9 6 3] 6 7 6 | 7 | —
73:0—73'9 Sle Bol al) e6a) ¢3a|ee3a es
TLO—7TL9t— |—]} 1 5 4 ul 6 4 Py i
75:0—75:9] — |— | l 2 4 2 3 Ba lp
76-0—76:9}| — | l = 4 2 2 1 1S
77:0—77'9 > 1 Vets 3
78°0—78°9 1
79:0—79°9 —}|—|—| 1
| | |
Totals
MISCELLANEA.
I. The Statistical Study of Dietaries, a reply to
Professor Karl Pearson.
By Proressor D. NOEL PATON, F.R.S.
PROFESSOR PrARSON’S criticism of Miss Lindsay’s Study of the Diets of the Labouring
Classes in the City of Glasgow (Biometrika, Vol. 1x. Oct. 1913) is a good example of the
danger of one who does not understand the problems involved and who is ignorant of the
work already done upon a subject attempting to discredit the results of an investigation by
the application of mathematics according to his own fancy and in, what seems to me, a
totally illegitimate manner.
Not appreciating the questions which were under investigation, he starts his criticism by
demanding that our studies should afford a solution of problems other than those we had
before us, and, because he does not find the solution of these problems, he proceeds to abuse
the work.
Apparently in his opinion the object of the studies should have been to determine what
effect the diets which the families were taking at the time of the study had upon the
physique of the various individuals. He states that, if adequate anthropometric observations
had been secured in such a study, it would have been at once possible to co-relate these
with the diets. It is unnecessary to point out, as was pointed out in the Report, that the
physique is determined by the whole previous condition of life and by the influence of
heredity, and that it is absurd to attempt to relate it solely to the diet (Report, pp. 3 and 4).
The objects of the studies are quite clearly stated on p. 4 of the Report: “Do the
working classes of this city get such a diet as will enable them to develop into strong,
healthy, energetic men, and, as men, will enable them to do a strenuous day’s work ; or are
the conditions of the labouring classes such that a suitable diet is not obtainable? Further,
if a suitable diet is obtainable, and is obtained, is it procured, or can it be procured, at a
cost low enough to leave a margin sufficient to cover the other necessary expenses of the
family life, with something over for those pleasures and amenities without which the very
continuance of life is of doubtful value?”
It was accepted as proved by previous work that for the labouring classes: “If a family
diet...... gives a yield of energy of less than 3500 Calories per man per day it is insufficient
for active work, and if less than 3000 it is quite inadequate for the proper maintenance of
growth and normal activity.”
The first question investigated was: “Did the families examined receive this supply of
energy?” As regards the poorest classes this was answered in the negative. The validity of
this conclusion has not been challenged by Professor Pearson,
Biometrika x 22
170 Miscellanea
The second question considered was whether the diets contained a sufficient supply of
protein. Previous work indicates that this is probably something above 110 grms. per man
per diem. It was shown that in families with regular incomes of over 20s. a week the
average protein intake was above 110 grms., and that in families with regular incomes and
in those with irregular incomes of under 20s. a week the average protein intake was under
110 grms. This conclusion has not been refuted.
Accepting our premises, the final conclusion was (p. 27) “that while the labouring classes
with a regular income of over 20s. a week generally manage to secure a diet approaching the
proper standard for active life, those with a smaller income and those with an irregular
income entirely fail to get a supply of food sufficient for the proper development and growth
of the body and for the maintenance of the capacity for active work.”
The main points proposed for the study were thus elucidated. :
The part of the Report to which Professor Pearson specially directs his criticism is not
the main problem, but that dealt with on pp. 30 and 31—The Physique of Children in
Relationship to Diet, a subject taken up at the suggestion of Dr Chalmers. Professor Pearson,
having declared the data totally insufficient, proceeds to apply his statistical methods not to
refute Miss Lindsay’s conclusion, but to demolish other conclusions upon the relationship of
physique to income which were never deduced by us.
The very guarded conclusion in the Report was: “These show very markedly the relation-
ship between the physique and the food. When the weight is much below the average for that
age almost without exception the diet is inadequate.”
Weights alone were considered. Thirty-six children, boys and girls, were dealt with. As
the relationship of weight to income was not under consideration, they were classified not
according to the income but according to the energy value of the family diet. Hence
Professor Pearson’s remarks upon this point are quite beside the mark.
I give below, in a re-arranged form, the Table from Appendix IV. The individuals are
placed in two groups according to the energy value of their diets, with, opposite each child,
the average weight for the age, taken from the Report of the Anthropometric Committee
published in the Transactions of the British Association for the Advancement of Science,
1883, and with the difference between the weight of the child and the average weight. The
differences between groups 1 and 2 are sufficiently marked and warrant the conclusion as
stated above.
That is, of the children in families the diets of which yielded more than 3000 Calories per
man per day:
10 were above the standard or not more than 5 lbs. below it,
8 were more than 5 lbs. below it,
while of the children in families in which the diet yielded less than 3000 Calories
3 were above the standard or not more than 5 lbs. below it,
15 were more than 5 lbs. below it.
It must be remembered that the ‘standard’ is for the children of all classes and not for
those of the poorer classes.
The fact that the average age of the children in the second group was about 1? years
greater than that of the children in the first group does not account for the marked
difference.
The last question which Miss Lindsay had to consider was, how the necessary supply of
energy and of protein might be supplied without increased expenditure, and she was right in
stating that these can be more cheaply purchased in vegetable than in animal foods. She
Miscellanea 171
TABLE A.
Family Diets above 3000 Calories per Man per Day.
_. | Age in Weight | enenray|
Number | Calories ve ; Sex nee OA Weight | Difference
ears in lbs. .
in lbs.
2 4003 a f°) 39 46°7 - 77
2 4003 10 3 63 67°5 — 45
2 4003 8 3 50 54:9 — 49
2 4003 5 3 35 39°9 -— 49
36 4091 3°25 3 35 5450 0)
4 3882 8 2) 45 52°2 | — 72
32 3822 6°25 2 39 42°4 — 34
4 3882 6 ie) 39 42°4 - 34
4 3882 10 3} 56 67°5 -11°5
39 3422 10°5 2 55 65 —10°0
50 3471 6°25 io) 37 42-4 | — 54
50 3215 6 9) 47 42-4 + 46
BS 3116 6 2 43 42-4 + 06
18 3248 55 @ 43 41-0 ap 2450)
54 | 3282 5 Q 33 39°6 — 66
58 | 3080 6 3 38 44-4 | — 6-4
30* | 3136 55 4 21 4] | ~—20-0
49 | — 3841 55 3 42 41 + 1
* Family with rickets.
TABLE B.
Family Diets below 3000 Calories per Man per Day.
. A | 3 Standard
Number Calories ' Age . Sex Weight Weight | Ditference
in Diet in years in Ibs. nba:
d4 2690 13 ie) 76 87 —11°0
14 2690 12 Q 60 76°4 —16°4
14 2690 10 ie) 45°5 62:0 —17°5
US 2936 10 o 56 62:0 — 6:0
7 2931 9°75 io) 44 62°0 —18°0
D0 2686 5°75 2 42 42°4 — o4
14 2690 9 3 45 60°4 — 15-4
4l 2723 6°75 3 53 49°77 + 3:3
14 2690 6 3 36 44-4 — 84
57 2974 5 3 37 39°9 — 29
3 2891 5 rey 37 39°9 — 29
2 2772 5eD 2 34 41°0 — 7:0
24 2412 eT: 2 39 68°1 —29°1
21 2329 9 g 37°5 55°5 —18°0
24 2412 6 g 28 42°4 —14°4
21 2329 1 3 60 72°0 —12°0
10 2435 8 3 43 54°9 —11°9
59 1978 5 4 26 39°9 —13°9
172 Miscellanea
undoubtedly starts with the well-known conclusion that a Calorie in the food absorbed in a
mixed diet from whatever source, protein, fat or carbohydrate is of equal dynamic value.
Previous work amply justifies this.
She was not foolish enough to attempt to draw any conclusion from her investigations as
to the relative value of animal and vegetable food in the diets on the physical development
of the individuals.
Professor Pearson seems entirely unable to grasp the fundamental fact that the physical
development of the individual depends largely upon his past conditions of life. To co-relate
it with the special constituents of the food which he habitually eats will require not only an
enormous series of studies, but a full investigation of the character of the various food stuffs
and of the mode of cooking.
These points I tried to explain to him when I wrote to him in summer. He did not
write to me as, in his criticism, he says he did. Miss Lindsay forwarded to me a letter
from him to her, and I wrote a reply to Professor Pearson which he did not acknowledge.
In conclusion I would say that before he expects his criticism of a physiological problem
to be taken seriously, he had better make some attempt to understand the nature of the
problem. Certainly it is not my intention to waste time in replying further to his criticism
unless in the future it is more pertinent than is his present contribution.
II. The Statistical Study of Dietaries. A Rejoinder.
By KARL PEARSON, F.R.S.
I puBLISH Professor Noel Paton’s reply because it is very typical of the type of difficulty
which we meet with at present, when we assert that what is really statistical work must be
undertaken only by the adequately trained statistician and that when it is not, then the
investigation cannot be considered as falling into the field of science.
Professor Paton states that the following question given on p. 4 of the Report formulated its
object: “Do the working classes of this city get such a diet as will enable them to develop into
strong, healthy, energetic men, and as men, will enable them to do a strenuous day’s work; or
are the conditions of the labouring classes such that a suitable diet is not obtainable?”...
Now Professor Paton either assumes that the sample taken of the diet of the individual
family was their customary diet, or he does not. If he does, then the question: Was the diet
such ‘as would enable the working classes “to develop into strong, healthy, energetic men”?
has meaning. If he does not, not only is it idle, but the section dealing with the physique of
the children on the basis of a sample diet taken as a rule for a week (occasionally for a fortnight),
is beside the point.
But anyhow, I ask how he can possibly ascertain how the working classes will “develop into
strong, healthy, energetic men,” if he does not take an adequate anthropometric survey of the
families subjected to the dietaries recorded? He says that it is accepted and proved that “ If
a family diet...gives a yield of energy of less than 3500 calories per man per day it is insufficient
for active work ; and if less than 3000, it is quite inadequate for the proper maintenance of
growth and normal activity.” He further assumes with Miss Lindsay that calories from animal
and vegetable foods have equal “dynamic value.” I assert that neither of these conclusions,
Miscellanea 173
which he accepts, are based on adequate research and they are in fact refuted by Miss Lindsay’s
own material. For, if it can be shown that animal and vegetable calories have different results
on the physical development of the children, it is clear that the first statement as to how many
calories are needful for the proper maintenance of growth has no significance until a statement
is made with regard to the source of the calories. Professor Paton cites no evidence for his
statements; from what I have read on the subject of calories, I feel convinced that most
of the data on the matter would not stand for five minutes any adequate statistical analysis.
The Report, Professor Paton tells us, shows “very markedly the relationship between the
physique and the food.” Yet in a previous paragraph he says ‘that the physique is determined
by the whole previous condition of life and by the influence of heredity, and that it is absurd to
attempt to relate it solely to the diet.”
Now the only way to ascertain whether there was a marked relationship between the food
and the physique of the children was to correlate the two for a constant age and investigate
whether the correlations were such, having regard to their probable errors, that they could be
considered significant. I did this with the result that the total calories in the food and the
girls’ weight for constant age was not definitely significant with regard to the probable error,
while in the case of the boys the probable error was so large that it was impossible to say
whether the relationship was really considerable or not. In fact no marked relationship could
be deduced from Miss Lindsay’s data, they were too inadequate. If Professor Paton’s statement
as to the influence of heredity is to be trusted, then even my correction for age was inadequate,
and the data ought to be corrected also for physique of parent! If so, why was the parent not
measured ?
Professor Paton places before the readers of Biometrika two tables on which this “marked”
relationship is asserted by him to rest. One of the cases in his Table A, No. 32, is erroneously
placed in this table; the details show that the number of calories was 2949 and not 3822* ;
it should be in Table B. These tables contain 16 boys’ weights and 20 girls’ weights. Professor
Paton takes the British Association measurements, which are, of course, wholly inadequate as a
test of Glasgow children, and making no real correction for age+ considers whether the children
in the two tables were or were not above the quite arbitrary limit of 5 lbs. below standard. He
gives us no measure at all of the significance of the result, which is based on the vagaries of
sampling 16 boys of ages from 3 to 11, and 20 girls from 5 to 13; and he supposes in some way
that this treatment can possibly refute the correlation coefficient, wo,» Of weight and food
calories for constant age with its probable error! I can, however, throw more light on the
matter. Owing to the great courtesy of Dr Chalmers, Medical Officer of Health for Glasgow,
I have been able to more than treble the number of weights of the boys and girls subjected
to the dietaries. The results for total calories in food, C;, now aret:
Girls, 69 Boys, 55
alec, = +21 £08, al uc, = +05 +09.
Thus the relation for boys is now quite insignificant, and for girls may well be insignificant
also. At any rate although both correlations are positive, there is no “marked” relationship
between the physique and the dietary. Of course, it may be said that these weights (w) have been
taken at some interval after the dietaries were recorded, but unless we assume the dietary to be
a rough measure of the permanent feeding of the family, whose physique has been gradually
built up for years before the dietaries were recorded, the observations must be discarded as of no
value at all for testing physique, or as Professor Paton phrases it “development.”
* In the Appendix V of Rickety Families, it is given again ; this time as 2329 calories,
+ The deviation at each age would have to be measured in terms of the standard-deviation of weight
at that age; naturally the deviations are larger for older children.
+ I have to thank Miss B, M. Cave for the present series of correlations,
Miscellanea
=
aT
i
But the most interesting point ascertained from the new material is the confirmation of the
result that the higher the proportion of animal to vegetable calories the greater the weight. In
Biometrika, Vol. 1x, p. 533, we had for 16 boys and 20 girls:
Boys: alws Cy/C4= — 23416,
Girls : alws Cy/C_= — 12°15.
We now have for 55 boys and 69 girls:
Boys: alws Cp/C4= ~ 380+ 08,
Girls: ailing, Cy [C4 =— — 94. + ‘08.
These results seem to indicate that Miss Lindsay and Professor Paton, who supports her view,
are in error when they consider a calory the same whether it be from animal or vegetable food.
On the other hand, our larger numbers now indicate that :
(i) For a constant age the expenditure on vegetable or on animal food has no sensible relation
to weight.
(ii) For a constant age the number of calories in vegetable food has no sensible relation to
weight.
(iii) For a constant age the number of calories in animal food has a positive correlation with
weight for both girls and boys, being definitely significant in the first case (+°32 4°07) and not
so in the second (+08 + ‘09).
(iv) For a constant age the correlations of weight with ratio of expenditure on vegetable and
animal foods are for both boys and girls quite insignificant as compared with their probable
errors.
I am extremely obliged to Dr Chalmers for doing his best to supply additional material. As
far as it goes, it tends to show that calories are of far more importance than expenditures, but
that calories from animal food are more closely related to physique than are calories from
vegetable food*. The new material supports my criticisms that the failure to distinguish
between animal and vegetable calories stultitied the advice given by Miss Lindsay, i.e. to spend
money on oatmeal rather than on eggs. It also indicates that no safe conclusions with regard
to dietaries can be drawn until a reasonable anthropometric survey accompanies the record
of dietaries, and the whole is reduced with adequate statistical knowledge.
One point I can allow Professor Paton. It was an oversight on my part, when I said that
I had written to both Miss Lindsay and to himself; the letters in which Miss Lindsay and he
stated that to follow up the families now would be impossible were both replies to one and the
same letter of mine addressed to Miss Lindsay. The additional facts I desired were in their
opinion unascertainable, and further correspondence did not seem to me likely to be of any
service in achieving the end I had in view, namely to render of real service to science a piece of
recording work from which in my opinion then and in my opinion still, very misleading conclu-
sions had been drawn, and which conclusions in their turn had been exaggerated in the press
résumés of the paper. I do not think any such work as that done on dietaries by Miss Lindsay
and Professor Noel Paton will be of real value until (i) these dietaries are accompanied by
a thorough anthropometric survey of the whole families of the dieted and (ii) the equality of
animal and vegetable food calories ceases to be considered as a dogmatic truth.
* Of course the results show that on such data as are available, the food has relatively little relation
to the weight, there is no ‘‘marked ” relationship.
Miscellanea 175
III. Note on the essential Conditions that a Population breeding at
random should be in a Stable State.
By K. PEARSON, F.R.S.
Let us deal with bi-parental inheritance in the first place. Let « be a character in the father,
mean 2, standard deviation o,; let y be the same character in the mother, 7 its mean, and gy» its
standard deviation. Let z be the character in offspring of one sex, o3 be the standard deviation of
all offspring of this sex and Z the mean. Let: pu’, jus’, pag’ 3 oe” os”) pea” 3 aN poe!” pag’, pa’”, be the
moment coefficients about the means respectively of father, mother and offspring frequency distri-
butions. Let 7,, be the mean of the offspring of those parents, who have characters w and y, and
let the array of frequency of such offspring be given by fs (wv) du about 2,,, i.e. the character of any
offspring in this array is 7,,+4, where uw is independent of the parental characters x and y, but
Z,, is a function of w and y the parental characters. Some writers have suggested that the
offspring character should be taken as a blend of the parental characters, i.e.
z=}(et+y),
understanding by blend the mean of the parental characters. This appears to be very unsatis-
factory for:
(a) It supposes the parental characters to fix absolutely the offspring characters which is far
from a result of experience.
(6) It supposes the mother to reproduce the female size of character in the male and the
female offspring alike, whereas she contributes to each the sex character of her own stock, ice. if
she is a tall woman, she would contribute absolutely more to a son than to a daughter. The late
Sir Francis Galton got over this difficulty by “reducing female measures to their male equiva-
lents.” This he did by altering absolute measurements in the ratio of male to female mean
measurements. Thus he would take for the mean of his array of offspring
a ih oa @
2ay =F ange
if he were dealing with male offspring. A more reasonable hypothesis is to assume that
This will practically agree with Sir Francis’s form, if the coefficients of variation in the two sexes
are the same, i.e. 01/%=09/Y.
If we measure wv from the mean of the array of offspring we have
ee eyed, Ls
Z=503 @ + a) etuehelesmmteh alo stasntiasielastioe seraranetnarins arene eb (11).
We shall now suppose the offspring to follow the law (i), or
poet = +4 =") 4 ee Gan
1 2
where w and y are uncorrelated (mating at random), and w represents other influences than the
parental, and is therefore uncorrelated with # and y*. The frequency distributions of # and y
* This assumes the homoscedasticity of the arrays of offspring due to pairs of fathers and mothers
with characters « and y,
176 Miscellanea
may be taken as given by fi (v—2#) and fy(y—y). Let V,x WV, be the total number of possible
matings ad w
=SJi(%-2)foy-Z) dedy
and the total number of offspring V3 in any array
=f fs; (u) du.
I now propose to give the expression for the zth moment coefficient about the mean, i.e. py”,
of the population of offspring of a given sex. We have
NV, x Nox N3x on”= |] [2 o3” co 2 =!) + uf ve (w— 2x) fo (y—Y) x fz (u) dadydu,
the integration being extended over the whole of the frequency distributions of father, mother
and oftspring. Thus
t=n=s Jn—s ‘L(y —%)}n-8— BOON us
pal” aa Hos res = = 8 iB feet ea oy" 8- tot os
x fi (v7 — 2) fo (y— ¥) fs (u) dedy du.
Now «, y and wu being independent we have
1 = ‘ =
iA | (w- EB) a (x az @) Au = pn —3- t
: Y ; ” ”
al Yy-¥)' hly-y) dy= pi
: 8 fi lv
N; | us fs (uw) du=pyg
= g t=n— n-s 4
is fey = 03" i. - 3 + | Pin-s—t Ht bet }] eer sonaes (iv).
s=0 gr-3 |n—s|s t=0 jn- s—t|toy"- St gat os8
Thus we reach, remembering that py! = py" =p," =0,
i. 1 at bs ”
Pe 4 om (, + Hs, + any Synveraisiesayefolstaseleleqelstctouncajatesalerecciotetenttats(s(e{etetctahsfa¥uialetstetalstetstetatetetl Rete ieeis (Vv),
Hs = 3 o3° (# aF e) ipeall “ad aantearearoealemenitete te coduchiae ua ee ss: ete ae meee eee eee (v1),
Hl d 2 is 5 if Des. !
Ha = 76 os! (A+0% et) +5 oy (PE +8 ) Hal ee oh heNeeeeere (vii)
But po’ =077, pe” =o", and py” =o". Hence we must have
pl? =4o52 a a'o{bisieiolelatafe, a¥a\s otesa(e ajalejs\etarele(aoleleiate,sie7= slelelnfsle(erafatsl stale (vill).
If as usual we take 8, = p37/u.? and B,=4/p? we find from (vi) and (vii), writing s?= po'¥
/B rl ) {Va —3 Ve, ne Seen ERR acke ta inaah is (ix),
4
Byit = {2 uy = 74 (8! 4+ 8,” +6) — 32} REE ERR EES ER 0c (x)
Whence by the use of (viii)
/By¥ =2,/2 va” -5 war-+V,} ve ae ted! ik eae (xi),
: Jer ac
gVv=4 {182” — 16 (Bo + By! + 6) SIO. ai sosine satin ss heater cee (xii).
Miscellanea 177
Hence in order that the offspring population should be stable, it is needful that in the array of
offspring for given parents :
1
(a) FTI) 03.
(6) Nama, (Bi - 5 (B+ /B7)| =2y2/8,” (1-7) = 3 Wa”
if 8,” =8,/=B,", i.e. the skewness be the same for fathers, mothers and offspring.
i 1 mr
B=; (78)" —15),
if pee = Bo! =f,"
Thus, we have for the array of offspring of given parents
1
i op 03
enor ae
B= 5B, tones outta sae esac e caaioe ees ceavenges (xili).
Peal,
Be” —8= 5 (Br — 3)
Accordingly the variability of the array is less than that of the population of offspring ; and
the array (unless B,;'”=0, 8;’”=3) is more skew and has greater kurtosis than the general
population.
If 712, 723, 73, be the three correlations of father, mother and offspring we know that the mean
standard-deviation of the offspring of arrays having the same parents is
we 1 = 743 — 793" — 149? + 2712713731
s'=05 tony cy Cee
AY
and this equals if there be no assortative mating
ae epee
(712=0), o3N 1 143? — 193°.
If we could assume this equal to s we must have, since
y 1
=—- OG:
a/ 2 39
1 ls prepa aa S
2 =N1—- 713° — 193"5
2
leading to 1132 +723°=4,
or if the two parental correlations are equal to
113 =123 = "9.
In other words, if the parental influences were equal and there were no assortative mating
and the character in the array of offspring had the mean value
then the population could only be stable if
713 =723=0°5.
But this apparently noteworthy result only begs the question. By the general theory of
correlation the mean of the array of offspring is
ame 713712723 U— & 4 V3 —N12713 YY
Lis? Vox 1-732 oy)”
if there be no assortative mating,
Cn? z 1138, 1938
03 (rm te ros 3 = 103 (= ae
oj 2 C1 a2
Biometrika x 23
178 Miscellanea
Hence if we asswme the mean of array of offspring to be given by
VW y
aa (Z +2)
(i) the second portion of the expression must be zero, i.e. mean of whole population of
offspring must coincide with mean of array of offspring where parents have the mean values and
(ii) we must have 73=723=%. In other words the form of our assumption involves both the
equal influence of the parents and the value of the parental correlation.
From the standpoint of heredity no such assumption is legitimate. Neither in Mendelian
theory nor in biometric formula, nor again in actual observation is it permissible to suppose that
the mean of the array of offspring is determined solely by the parents. Still less is it possible
to suppose the actual character of the offspring to be the mean of that of the parents (i.e. put
u=0). Ifit were we should have z=4(a+y), whence flow
mr 1 ’ ”
(Doan (12 + #2")
m” | , ” 1
Bs = (3 + ps3 ) tie e/6iaiulole ae elnie)sieje eisieieis.eiais/s sie jslelelelsisiainre (xiv).
i 1 , , ” ”
ba 76 (ua + Bpy me’ +4")
But these equations assume that py, w3'Y and py” are all zero—an absurdity in itself and
contrary to all experience, whether biometric or Mendelian. For non-assortative mating and
equal potency of parents, they lead to parental correlations of the order °7 and to an impossibility
of stability in any population*.
In fact any such relations as (xiv) are inconceivable on the basis of both biometric as well as
Mendelian theory and observation. Parental correlations have never been observed anywhere
near such a value as 0°7. Equations (xiii) are, however, suggestive ; they show that if the
parental distribution be symmetrical and mesokurtic, the array of offspring will remain so after
selection; but if the parental distribution does not possess these characters, then any selection of
individual parents will emphasize the asymmetry and the kurtosis in the resulting array of
offspring ; or continued selection of this type will lead to greater and greater divergence from the
normal or Gaussian frequency distribution.
* If we assume that the mean of the array of offspring of parents of characters x and y is given by
lx +my, it is only another way of asserting that the regression is linear and that
_ 712 — 713723 3 _ 713 — 712723 63
i n=
1-137 oy’
1-193? 9°
If we make /=m, or give equal weight to the parents, it is only rational to suppose that o;=c2 and
T12=113, Which lead us to
N16 O°
=m=—2 3,
1+ T93 01
‘ : ‘12 93
Hence the mean of the array is —— —(#+y),
1+793 04 :
and whether we make x constant and y constant or x+y constant leads to precisely the same variability
in the array, i.e.
: 1 = 19? — 1137 — 17932 + 27121713723 Qryo2
s$=03 > ~=03 1-_—_,
1 - 723" 1+793
If assortative mating be zero, this equals
03 VI = Qo?
and, if to reach the results for u’” given above we put this zero, we must have
r= N50=0°7 nearly.
Miscellanea 179
IV. The Hlimination of Spurious Correlation due to position in Time
or Space.
By “STUDENT.”
In the Journal of the Royal Statistical Society for 1905*, p. 696, appeared a paper by
R. H. Hooker giving a method of determining the correlation of variations from the “in-
stantaneous mean” by correlating corresponding differences between successive values. This
method was invented to deal with the many statistics which give the successive annual values
of vital or commercial variables; these values are generally subject to large secular variations,
sometimes periodic, sometimes uniform, sometimes accelerated, which would lead to altogether
misleading values were the correlation to be taken between the figures as they stand.
Since Mr Hooker published his paper, the method has been in constant use among those who
have to deal statistically with economic or social problems, and helps to show whether, for
example, there really 7s a close connection between the female cancer death rate and the quantity
of imported apples consumed per head!
Prof. Pearson, however, has pointed out to me that the method is only valid when the
connection between the variables and time is linear, and the following note is an effort to extend
Mr Hooker’s method so as to make it applicable in a rather more general way.
If 21, @, #3, ete., W715 Y2, ¥3, etc., be corresponding values of the variables 7 and y, then if
X, Ly, Xz, ete, Y1, Yo, Y3, etc. are randomly distributed in time and space, it is easy to show that
the correlation between the corresponding th differences is the same as that between « and y.
Let ,,D, be the nth difference.
For 1D, = 21 — Vo, reer 1D,? = ar? = 24 Wy +209,
Summing for all values and dividing by V and remembering that since 7, and «2 are mutually
random S (2, #2) =0, we gett
Again, Dy= I - Yoyo Dar Dy = "19, — L211 — M1 Yot L272.
Summing for all values and dividing by NV, and remembering that 7, and y. and a, and y, are
mutually random
FDigDy Dp {Dy 2ey Fx Fy
peie/ Tenis
DD, ey,
Proceeding successivel ? =? SNe SSAA oe Saisncuieeecnice Bata eters asia tons ID
2 y ne Dy, rest OF n-1Dy ad ( )
“rie
Now suppose 2), 22, #3, etc. are not random in space or time; the problems arising from
correlation due to successive positions in space are exactly similar to those due to successive
occurrence in time, but as they are to some extent complicated by the second dimension, it is
perhaps simpler to consider correlation due to time.
Suppose then v= X,4+bt,+ct?+dtitete., «,= Vo+bto+cty?+dt3+ete,
where X,, X,, etc. are independent of time and ¢,, ¢,, t3 are successive values of time, so that
t, —tp_4= 7, and suppose y,= Y,+ 04, +¢t,?+ ete. as before.
* The method had been used by Miss Cave in Proc. Roy. Soc. Vol. Lxxiv. pp. 407 et seq. that is in
1904, but being used incidentally in the course of a paper it attracted less attention than Hooker’s
paper which was devoted to describing the method. The papers were no doubt quite independent.
+ The assumption made is that n is sufficiently large to justify the relations
S4"71 (x) /(n — 1) = Sy” (x) (mn — 1) = Sy" (x)/n_ and S{"—! (a?)/(n — 1) = Sy” (x?) /(n — 1) = Sy" (x?) /n,
being taken to hold.
232
180 Miscellanea
Then 1D, = Dy -bT — eT (t, + te) — dT (t?7 + tte 4%”) — ete.
1D, =,Dx—{bT+cT? +dT3 + ete.} — & {2¢7'+3dT? + 4eT + ete.}
—t,° {3dT + 6eT? + etc.} — ete.
In this series the coefficients of ¢,, t2, etc. are all constants and the highest power of 4, is one
lower than before, so that by repeating the process again and again we can eliminate ¢ from the
variable on the right-hand side, provided of course that the series ends at some power of ¢.
When this has been done, we get
nDx=nDy +a constant,
nDy=Dy +a constant,
BO Hei apy = Ds »D % => lxy)
‘ 20) wd . =r Pn oo » (0) 5 rayle Ss 7
and of course PD ety =e UD DY for ,D, and ,D, are now random variables independent
zn
of time.
Hence if we wish to eliminate variability due to position in time or space and to determine
whether there is any correlation between the residual variations, all that has to be done is to
correlate the Ist, 2nd, 3rd...2th differences between successive values of our variable with the
Ist, 2nd, 3rd...xth differences between successive values of the other variable. When the cor-
relation between the two rth differences is equal to that between the two (n+1)th differences,
this value gives the correlation required.
This process is tedious in the extreme, but that it may sometimes be necessary is illustrated
by the following examples: the figures from which the first two are taken were very kindly
supplied to me by Mr E. G. Peake, who had been using them in preparing his paper “The
Application of the Statistical Method to the Bankers’ Problem” in The Bankers’ Magazine (July—
August, 1912). The material for the next is taken from a paper in The Journal of Agricultural
Science by Hall and Mercer, on the error of field trials, and are the yields of wheat and straw on
500 345 acre plots into which an acre of wheat was divided at harvest. The remainder are from
the three Registrar-Generals’ returns.
I Il Ill IV V VI
Correlation between ... Sauerbeck’s | Marriage} Yield of Tuberculosis Death Rate. |
Index numbers. Rate Grain
Infantile Mortality
and wits ... | Bankers’ Clear- | Wages | Yield of 5
ing House Straw | —
ULI ees Ireland England | Scotland
head
Raw figures ss — °33 — 52 +°753 3 "35 +02
First difference... +°51 + ‘67 +590 +°75 + °69 +°51
Second difference ... + °30 +°58 + °539 74 ‘74 +°65
Third difference ... + :07 + °52 +°530 — — —
Fourth difference ... +11 + 55 + °524 — — —_
Fifth difference ... + 05 + °58 — — — —
Sixth difference... — +°55 = = — —
Number of cases | 41 years 57 years Ly 42 years
ae eed years! plots Sa) ae
The difference between I and II is very marked, and would seem to indicate that the causal
connection between index numbers and Bankers’ clearing house rates is not altogether of the
same kind as that between marriage rate and wages, though all four variables are commonly
taken as indications of the short period trade wave. 1 had hoped to investigate this subject
more thoroughly before publishing this note, but lack of time has made this impossible.
Miscellanea 181
V. On certain Errors with regard to Multiple Correlation occasionally
made by those who have not adequately studied this Subject.
By KARL PEARSON, F.R.S.
(1) Iv is well-known* that if we endeavour to predict the value of a variate xv) from a
correlated variates 21, 2, ... Y,, by determining a linear function of #7, #2, ... v, which has
the maximum correlation #, with wz, then the value of #,,? is given by
:
‘ h,?=1—A/Ago,
where A is the determinant
— 1 5) TOL) M025 eee Ton
Pity 0 5) Ry oon Pate |
| Troy Tnty nds oe 1 |
and A,, is the minor corresponding to the constituent of the pth column and gth row.
The system I propose to consider is that in which all correlations like 7, are equal, whatever
p be, toa constant p, and all correlations 7,,,, where p and g may take any values from 1 to n,
are the same and equal to «. We now have for the value of A the expression
Wl Pye (Ps) ees Ps |
|
To evaluate this determinant add all the rows but the first together, giving
nap, l+(n—l)e, 1+(n—l)e, ... 14+(m—-l)6,
multiply the result by p/(1+(2—-1) €) and subtract from the first row. We have
np* ; | np?
= —— OFF Oe OF — aa 0.
l+(n—le’ ’ , | {1 11s x Aon
Ps l, ¢, ||
Ps €, 1, € |
BNajoidie sraseuiscia gee aneicstesejmee eelicow camer \
Ps €, €, 1 |
ane 2
Hence fe a (he = i a i
1c n ( T+(n—l)e AGEING Too eer (a),
/ n -
or R,t=p Tes (eat hPa wetiecees (ii).
* Biometrika, Vol. vit. p. 439.
+ The sign of R, must be determined from other considerations.
182 Miscellanea
Thus if 7 variates are equally correlated (e) among themselves, and equally correlated (p) with
another variable, we shall not indefinitely increase the accuracy with which the last variable will
be predicted from the others by increasing indefinitely the number of the variates 2.
illustration. The coefficient of multiple correlation is required as we increase the number of
brothers from whom a prediction of a character in a given brother is made. The fraternal
correlation =°5,
Number of Brothers R,
1 “5000
2 “5774
3 “6124
4 “6325
5 6455
6 6547
10 ‘6742
a “7071
Compare against these results ¢wo parents only in a population where there is no assortative
mating and the parental correlation="5. Here e=0, p="5 and n=2, .-. R=4$,/2=°7071, or two
parents will give more information than 10 brothers and sisters, and as much in fact as an
indefinite number. Suppose the parents tend to select their like, i.e. suppose there is assor-
tative mating in the population, say, e=+15, then with the same intensity of parental correlation
2 = 6594,
or, two parents will give us more information than six brothers and sisters.
Now this illustration brings out the real nature of the effect of increasing the number of
variables from which we predict. Such increase has very little value, if those variables are
fairly highly correlated with each other. To be effective they must be highly correlated with
the variate we wish to predict and correlated very slightly with each other.
Even in this case there is a limit to the degree of correlation reached when the number of
variates is indefinitely increased, namely p/,/e, and it is clear that if p be small and e fairly large,
no very great increase of correlation is obtained if we use an indefinitely great number of variates.
For example if p='05 and «=°5, we find &, =-0707 only. Even if p were ‘10, we should only
raise R to ‘1414, could we predict from an indefinitely large number of such correlated variates*.
Indeed as long as ¢ is not less than p we gain singularly little by combining large numbers of
variates, For example if p were ‘4, and e=°4 ten such variates would only raise the correlation to
5898, and an indefinitely large number to ‘6325, which is less than double the single correlation.
Yet there are apparently many persons who believe that by taking a number of low correlations,
a high relationship can be reached !
Actually there is a limit.to what relations can possibly exist between a variate xv and a series
of equally correlated variables x, ... 7,. Since & must be less than unity, we have
py aes
PON) SV Gielen ats
2
np* —1
or e> f °
n—-1
Thus if 72=10 and p=‘5, « must be >'1667. Or, it would be impossible for 10 variates to
have a correlation ‘5 with another variable, and a zero correlation with each other.
* Even if p were ‘10 and ¢ as low as ‘10 we should not raise R for endless variates of this order of
correlation above -3163, while from compounding ten such variates we should only obtain a correlation
about double that of a single variate, i.e. R=+2294,
Miscellanea 183
If we suppose a number of variates » to be uncorrelated with each other, but correlated
115 102) «++ “on With another variable w), then we have from the determinant as given below
2 cee + 2
==i(l » Tot» To2> e%* Von =(1 — 749 — lo Sesicsaee UI ) Dou.
iy bes 0) 5 oon 0)
eevee eee eee eee)
, ‘ 5 9
eae: he= More+ Poy? + seeita Tony
or R=Jn ih Myo tit FT on"
Therefore, if 2 variables, uncorrelated among themselves, be correlated with an additional
variable, it is necessary that the root mean square of their correlations should be less than
55 We see therefore that it must either be impossible to find a large number of variables
n
uncorrelated among themselves, which are correlated with an additional variable, or else their
correlations with this variable must be extremely low. The last result shows us the fallacy
of supposing that correlations are simply added together for a combined effect ; clearly when
the variates are uncorrelated among themselves, we add by the sum of the squares. For
example, if 7) =792=...-=70n='03 one hundred such variables would only raise R to 30, On
the other hand if the variates are highly correlated together, say e=°81, an indefinitely great
number of such variables would only raise the multiple correlation to ‘0333, if the individual
correlation were ‘0300.
We are now in a position to apply our results to the problem of the relative intensity of
heredity and environment. This problem has been singularly misunderstood especially by the
popular exponents of Eugenics. Some illustrations of this may be given here. Major Leonard
Darwin writes as follows in the Journal of the Eugenics Education Society: “It is impossible
to compare heredity as a whole with environment as a whole as far as their effects are
concerned ; for no living being can exist for a moment without either of them*. Moreover,
in order to compare two things so as to be able to use the words more or less in connection
with such a comparison, we must have a common unit of measurement applicable to them
both. But what is the unit by which both heredity and environment may be measured ?
I myself have no idea. May we not be discussing questions as illogical as enquiring what
portion of the area of a rectangle is due to its width and what to its length? Js 7t ever wise
to use words in scientific literature without endeavouring to attach a definite meaning to themt ?
It is hard to conceive a paragraph of the same length more full of evidence of complete
ignorance of the methods used in modern science for comparing correlated variates! Yet it
goes out as the opinion of the President of a Society which is endeavouring to spread the
scientific doctrines of Eugenics among the people! Major Darwin begins by stating that it is
needful to have a common unit of measurement in order to compare two variates. To begin
with we are not comparing two things, but we are comparing the influence of two things on
* There would in our sense be no heredity if the average child born to noteworthy parents was equal
to the average child of the whole community. Yet it is perfectly easy to understand how living beings
could exist under such a law of reproduction. Major Darwin seems to be confusing two things, the fact
that a man is born true to his species, and the fact that he resembles his immediate ancestry. It is
the latter fact only which concerns us when we compare heredity and environment, i.e. how variation of
immediate ancestry affects the individual’s physical or mental characters. But without such heredity
individuals might quite well exist.
+ The Eugenics Review, Vol. v. p. 152. The italics are mine.
184 Miscellanea
a third, ie. the intensity of a certain environmental influence and the intensity of a certain
somatic character in the parent, say, on the intensity of the somatic character in the off-
spring. Yet Major Darwin tells us we cannot do this because we cannot measure these
things in the same unit !—How suavely yet forcibly Sir Francis Galton himself would have
ridiculed such ignorance in high places as is passed by the Editor of the Eugenics Journal !—
We can hear him now telling us how the intensity of each character could be measured by
its grade, and how the problem turned on whether the same change in grade in the environ-
ment and in the parental somatic character produced greater or less change in the grade of
the filial somatic character. When we inquire whether inter-racially stature is more closely
related to cephalic index or to eye colour, are we to be met by the statement that these
characters cannot be compared because they cannot be measured in a ‘common unit,’ and
then be told that it is not “wise to use words in scientific literature without endeavouring to
attach a definite meaning to them?” Every trained statistician knows that each character
is measured in the unit of its own variability—in what he terms its standard deviation*,
and that this standard deviation provides him with a measure of the frequency of each value
of the variate in question. It seems to me that the only correct sentence in this paragraph,
is the author’s statement that he himself has no idea what unit is ‘common’ to heredity and
environment.
But our author continues :
“Take any quality, and we find that the human beings composing any community differ
more or less considerably as regards that quality. Now we can measure the correlation
between the differences shown in this quality and the differences of environment to which
the members of the community in question had previously been exposedt. This is one
correlation. Then we can also measure the correlation coefficient between, say, father and
son, as regards the quality in question. Here is a second correlation; and if we are told
that the relative influence of environment and heredity is measured by the ratio between
these two correlation coefficients, we certainly do thus get a clear conception of what is
+
meant }.
But has the writer really obtained a clear conception of what such coefficients of correla-
tion mean, when in the next paragraph he continues :
“Tmagine an ideal republic, in some respects similar to that designed by Plato, where not
only were all the children removed from their parents, but where they were all treated exactly
alike. In these circumstances none of the differences between the adults could have anything
to do with the differences of environments, and all must be due to some differences in inherent
factors. In fact the environment correlation coefficient would be nil, whilst the hereditary
correlation coefficient might be high §.”
Could any better evidence be adduced that the President of the Eugenics Education Society
did not know what a coefficient of correlation meant at that date? The coefficient of correlation
for the environment might be anything from —1 to + 1; the only obvious fact would be that you
could not find its value, except in the form 0/0, from an environment which precluded any
measure of variation. How again Sir Francis would have smiled at the notion that the
coefficient of correlation for a constant environment must be nil. Why should we follow such
* Of course he may or does need other constants to help in the description of the frequency.
+ loc. cit. p. 153.
+ This seems to contradict the writer’s previous assertion that two things are incomparable, if they
have not a ‘common unit’!
§ I wrote at once to Major Darwin pointing out the error of such a statement and he withdrew it in
the next number. But the harm done by an article of this kind cannot be reversed by correcting a
single misstatement.
Miscellanea 185
advice as that given by the President of the Society to avoid as far as possible “such phrases as
the relative influence of heredity and environment,” when on his own showing he does not in
the least appreciate the methods by which this relative influence is measured ?
Then Major Darwin continues : “Surely what we want to know is how we can do most good—
whether by attending to reforms intended to affect human surroundings, or to reforms intended
to influence mankind through the agency of heredity. But does this ratio [that of the environ-
mental and hereditary correlation] give us any sure indication of the relative amount of attention
which should be paid to these two methods of procedure?” Our only reply can be that these
correlations certainly do, and that as long as the President of the Eugenics Education Society
fails to grasp their meaning, he is doing grave harm to the science of eugenics.
We measure the change in the character of an individual which would be produced by a
change of a like or an allied character in a parent, such change being one of which we have
experience ; we measure the change which would be produced in the character of the individual
by changes in the environment such as we have experience of, i.e. when we move the individual
from a badly ventilated to a well ventilated house, from a back to back to a through house, from
a low wage to a high wage, and so forth, and we find the resulting changes are of a wholly
different order in these cases to what happens when we change the physical characters, the
health or habits which define the parents. It is on the basis of this that we assert that the relative
strength of heredity is far greater than the strength of environment. To this reasoning, apart from
such arguments as the above or those to be immediately dealt with, reply is only made by talk as to
the impossibility of an individual surviving if you deprived him of his normal environment! It
would be just as reasonable to assert that everything must be due to heredity, because a race of
supermen would breed supermen ! What the scientific eugenist has endeavoured to measure are
the influences of such range of differences in environment as occur in everyday experience and
are therefore producible from the political, economic and social standpoints, not the absence of
all environment at all. But while this is recognised by some of the popular eugenic writers, they
have approached the problem from another standpoint which indicates equally how little they
grasp modern statistical theory. We admit, they say, that the environmental correlations may
be of the order ‘03 or :05 and the inheritance correlations of the order 50. But this is the
correlation of one character in environment. You ought to take ten or twenty, and then you
will have multiplied up environment to be more effective than heredity, for 03 x 20=-60. In the
first place we may suggest that it would be just as reasonable, if the argument were a valid one
to multiply up the favourable hereditary characters, to take weight, height, muscular activity,
health, intelligence, caution, and many other desirable factors, and these not only in one parent
but in brothers, sisters, aunts, uncles and grandparents and treat the cross-correlation of these
with the character under discussion. But although every improvement in stock would reflect
itself in improvement in offspring, correlations cannot be added together—any more than forces
by simple arithmetical addition. You do not combine two hereditary correlations any more than
two environmental correlations by mere addition. You must proceed by the combinatory process
indicated at the commencement of this paper, which is one of course familiar to every trained
statistician. :
Yet here is a statement which the Editor of the Hugenics Review admits to its pages without
contradiction * ;
The point that we wish to make is this, In the face of so much ignorance concerning, not only
heredity itself, but also its complement, the influence of environment, how can any one be justified in
making sweeping generalisations with reference to these subjects?
Such generalisations, however, are made. It is said that we have a definite proof that inheritance is
of far greater strength than environment. This argument takes the following shape. The correlations
between parent and offspring for a number of features have been calculated, and the mean is found to
* Vol. v. p. 219, in an article by A. M. Carr-Saunders.
Biometrika x 24
186 Miscelianea
be somewhere about °5. Correlations between individuals and various aspects of their environment have
also been worked out—as, for instance, mental ability and conditions of clothing, or between myopia and
the age of learning to read*—and the mean value is found to be about -03. It is then said that the
mean ‘‘nature value” is at least five to ten times as great as the mean ‘‘nurture value,” and upon this
is founded the generalisation that ‘‘nature” is of far greater importance than ‘‘nurture’+. It may be
questioned, however, whether such a comparison does not involve a serious misiake. For if we consider
the two mean values that are compared, we find that, whereas the ‘‘mean nature value” is the mean
value of a number of observations, all of which provide a full measure of the strength of heredity, the
‘mean nurture value” is the mean value of a number of observations, each of which measures only the
strength of some one isolated aspect of environment. It would appear then that the full strength of
inheritance has been compared, not with the full strength of environment, but with the average of a
number of small isolated aspects of the latter. Asa matter of fact it is quite beyond our power at
present to sum up the full effect of environment upon the individual and compare it with the full effect
of heredity. We are, therefore, justified in saying that we neither know in particular cases how far the
environment can produce any effect, nor can we make any definite statement as to the comparative
strength of ‘‘nature” and “nurture.”
Now this is the doctrine passed by the Editors of the Hugenics Review, the journal of a
society, which has assumed the mantle of Francis Galton{, and it is passed, because the
editorial committee of that society does not grasp the meaning of multiple correlation! The
passages in italics have been so printed to draw our readers’ attention to them. In the first
place, of course, a single correlation coefficient does not provide a full measure of the strength
of heredity. In the table cited the coefficients are those for one parent or for one brother or
sister. Each relative—and those for independent stocks are either non-correlated or inter-
correlated very slightly—provides such a coefficient, and further each character in such relatives
may be correlated with the character under discussion in the subject in question. In the; next
place the environment factors do not consist of “some one isolated aspect of environment.”
All these factors or aspects are closely interlinked, and this was a fact well-known to the
workers in the Galton Laboratory. The real interpretation of such a difference as 560 and °03
in the average values of single coefficients can only be appreciated by those who are conversant
with the theory of multiple correlation, and it is quite clear that those who profess to guide the
public in this very difficult problem—which is essentially a scientific problem—lack any adequate
knowledge of the sole instrument by which any conclusion can be drawn. :
The writer appears to be wholly ignorant of the nature of multiple correlation in the first
place, and in the second entirely to overlook the very high correlations which exist between
environmental factors. Bad wages, bad habits, bad housing, uncleanliness, insanitary sur-
roundings, crowded rooms, danger of infection, etc., etc. are all closely associated together,
and while the order of correlation between environmental and physical characters is low, that
between individual environmental factors is in our experience very high. Thus the problem of
multiple correlation illustrates closely the theory developed in the first part of this note; we
have to deal with a low p and a high «.
For example, if we take the environmental factors to have an average inter-correlation of °70,
then an infinity of such factors for a mean environmental and individual correlation of ‘03 would
* As the writer phrases this correlation, it is very liable to be misinterpreted. What the Galton
Laboratory did was to show that myopia was very markedly inherited, and that the theory that it was
largely due to school environment was incorrect, because children who began to read late, i.e. went late
to school, were not less myopic than those who went early.
+ Karl Pearson, Nature and Nurture, Eugenics Laboratory, Lectures vi. p. 25,
+ If there was one point on which Francis Galton felt strongly and wrote it was on this point of the
relatively great intensity of ‘‘nature” as compared with “nurture.” I do not stand alone in recognising
it as an essential part of his teaching: ‘I am inclined to agree with Francis Galton,” writes Charles
Darwin, ‘‘in believing that education and environment produce only a small effect on the mind of
anyone, and that most of our qualities are innate.”
Miscellanea 187
only raise the correlation to 0359 against a s¢ngle parental correlation of *5000; if the correlation
was ‘05 instead of ‘03, we should have the total possible environmental multiple correlation ‘0598
as against ‘5000. Even if we raise the average environmental correlation to ‘1 and the inter-
environmental factor correlation be reduced to °5, the multiple correlation of an infinity of factors
is only :1414 as against the sengle factor of heredity °5000. Even if we could pick out one
hundred environmental factors which had no inter-correlations—which experience shows is
wholly impossible—and each of these independent factors was correlated to the extent of -05
with the mental or physical characters of an individual they would only just reach the hereditary
influence of a séngle character in a séngle parent.
Now let us suppose an absolutely idle case, namely that the environmental factors had the
same correlation as a parent, i.e. 5, with the character of the individual, and only a correlation
of °6 with each other, then if we could use an indefinitely great number of such factors the
multiple correlation would only be ‘5//°6=°6455, while the correlation with two parents, with
no assortative mating, would be -7071. Even with assortative mating, it suffices to take only
the four grandparents into account to show that heredity acts in excess of an environmental
scheme even so preposterous as is suggested above. If we take the parental correlations *50, the
grandparental ‘25, and those of assortative mating -15, we have for the determinant :
A=| 1, °50, 50, ‘25, -25, -25, -25
(60,1; 15). "50; “50, “0; 0
| 50, ‘15, 1, -0, . 0, ‘50, ‘50
5, 50, 0, « 2
i
25, 0, ‘50, 0, 0, ‘15, 1
Add together the second and third rows multiplied by 3951, and the fourth, fifth, sixth and
seventh multiplied by ‘0456 and subtract the result from the first. The first row then becomes
| 5593, 0, 0, 0, 0, 0, O|
the others of course remaining the same.
Hence NOD 9B Anne
and R?2=1— A/Apy =1—°5590 = 4407.
Therefore R:=='6639.
Or together grandparents and parents would influence a man’s character more than an
5S
infinity of environmental factors of the same grade of correlation, because the latter factors
are far more highly correlated together than several of our relatives.
Actually of course we are dealing with average values; the average value of environmental
correlation with individual character being in our experience of the order ‘03 to ‘05 and the
Lod
inter-environmental factor correlations of the order 5 to ‘7. But these averages enable us to
appreciate the total effect.
The doctrine taught by the writers in the Hugenics Review, that we know nothing of the
relative intensity of environment and heredity and that it is unwise “to use words in scientific
literature without endeavouring to attach a definite meaning to them” only demonstrate how far
the Editors of that Journal are removed from any appreciation themselves of modern statistical
methods. How far the doctrine is removed from the very strong views held on this point by
Francis Galton, only those who have studied his writings and know how strongly he felt person-
ally on the subject are in the least competent to appreciate,
24—2
188 Miscellanea
VI. Formulae for the Determination of the Capacity of the Negro
Skull from External Measurements.
By L. ISSERLIS, B.A.
§ 1. Formulae for the determination of the capacity of the human skull from external
measurements, were obtained by Lee and Pearson*. The material they employed consisted
of various series of measurements of Bavarian, Aino and Naqada skulls. Measurements of
Ancient and modern Egyptian and other non-European skulls were employed, chiefly for
purposes of comparison. The formulae, some of which will be quoted later, were intended
primarily for the prediction of the capacity of European skulls, from external measurements.
Doubt has been thrown on several occasions on the applicability of these formulae to the Negro
skull, one of the reasons alleged being the supposed difference in thickness of the bone of
European and Negro crania.
The publication t of the late Dr R. Crewdson Benington’s researches on the negro skull has
made it possible to obtain similar formulae for negro skulls, and to test how far these can
be applied to the prediction of the capacity of European skulls and conversely to test the
applicability of Lee and Pearson’s Equations to the negro skull.
§ 2. The material is fully described in Dr Benington’s Study. The crania dealt with in
the present paper are Benington’s series A, B, C.
A. Congo Crania in the Royal College of Surgeons. These crania provide 46 males and
and 21 females, as owing to various defects no capacity is available for numbers 25, 38, 48, 54
among the males and numbers 69, 72, 75, 79, 82, 85 among the females.
B. Crania from the Gaboon, Group I, brought by Du Chaillu from Fernand Vaz in 1864.
Of the 50 male and 44 female crania in the series, 2 males (numbers 3 and ?) and 1 female
(number 2) are defective, leaving 48 male and 43 female crania available.
C. Crania from the Gaboon, Group LI, brought by Du Chaillu from Fernand Vaz in 1880.
Two of the 18 males (numbers 12a and 20) and two of the 19 females (numbers 8 and 18)
are defective.
Altogether 110 male and 81 female crania have been dealt with. The correlation has been
calculated of the capacity (C) and the product of the breadth, length and total height (B, Z
and #), for each group and for the aggregates of 110 male, and of 81 female crania.
Correlation coefficients have also been calculated for the capacity and breadth, capacity and
length, and capacity and total height, but for the aggregates of the three groups only. Re-
gression formulae are given in all cases. It is to be observed that Dr Crewdson Benington’s
measurements of capacity were taken with mustard seed, packing and measuring glass and
that the error of measurement or rather his average difference as compared with other workers
in the Biometric Laboratory was under 10 cm*.
In comparing the regression formulae obtained here, with those given by Lee and Pearson for
European and other skulls it must be remembered that in all their formulae except (12) and (18)
of p. 247 they employed the auricular height and not the total height. In the present paper as
in Dr Benington’s study H denotes the total height. Lee and Pearson denote this by #’ and
use # for the auricular height,
It was not possible here to use the auricular height as it was not available for the whole of
the Gaboon series B and C.
* Phil. Trans. Vol. 196, Series A, pp. 225—264.
+ Biometrika, Vol. vit. Nos. 3 and 4, Dec. 1911.
Miscellanea 189
Taking first the male skulls, the mean value of the capacity and the product BLH, their
standard deviations and the correlations are given in the following table.
TABLE I.
Mean capacit | Mean value of : ; in
in as z BLH incom} | %e 2 em. nae "C, BLU
46 Congo Skulls... 1344 3303 126°22 282°99 872
48 Gaboon (1864) ... 1379 3295 108°30 | 230°30 *822
16 Gaboon (1880) ... 1447 3463 10960 266-42 308
110 Negro skulls... 1375 3323 120°74 265 °20 "842
The corresponding regression lines are
for the 46 Congo ae3 C=‘00038889BLH+ 59 +o SCRE EECCaTE (1),
n
48 Gaboon (1864) | C=-0003865BLH+ 105% 4 eee (2),
n
16 Gaboon (1880) | 0=-0003323BLH4297 + occ (3),
vn
+A
110 male negro skulls C='0003849BLH+ 96 ao web eneee caseneade (4).
n
Lee and Pearson’s corresponding equation for males is
C= 000266 L BH’ + 594-6* vo. ccscececsscesceccescescesscecenes (P).
This is not a regression line, but is obtained by method of least squares from the results for
various races in their table 20.
The formulae 1—4 can be used to predict the capacity of an individual skull from external
measurements. The probable errors of the mean were calculated by the formula 0°674490, —
N
where # is the number of skulls in the group to which the formula is applied. If we substitute
in (1)—(4) the mean values of B, LZ, H for the Bavarian male skulls used by Lee and Pearson,
Viz. :
B =150°5,
L =180°,
H=133'8,
5 41
we obtain, from (1), C=14744+—=
Vn
np Op C=1471 #82
Vn
» (3), c=1506+ ©
Vn
» 4) v=14964 >,
Vn
* Loe, cit, Equation (12). H’=total height.
190 Miscellanea
The measured capacities of these German skulls have a mean value of 1503 c.c. a result
which is in very close agreement with (4) the formula based on 110 skulls. 1508 is the mean
capacity of 100 skulls so that === 655. Thus the difference between the actual mean capacity
N10
of German skulls and the mean capacity estimated by the negro formula is less than 10 cm’.
although the mean capacity of German male skulls exceeds that of negro males by
1503 —1375=128 cm’.
If the above values of B, Z, H are substituted in Lee and Pearson’s formula (P) on p. 4
we obtain C=1492.
On the other hand if we substitute the mean values of the dimensions of the 110 male negro
skulls, B=137, 2=178, H=135 in formula P we obtain C=1400 as compared with the measured
mean of 1375.
This is not as good a reconstruction as our formula (4) or as the formulae of Lee and Pearson
employing auricular height, and is probably due to the fact that P is obtained by the method of
least squares from 11 means only.
§ 4. An approximation to the influence of the thickness of the bone of the skull on pre-
dictions of capacity from external measurements can be obtained by differentiating the equation
C=hkBLH+const.
and putting dB=dL=dH=t.
We obtain dC=k(BL+LH+ HB)t,
or if we observe that in the equations the constant is comparatively small
dC 1 1 1
C= G ry 38 7H) t
with B=150°5, L=180°6, H=133'8
de
1500 —
t (02) approximately.
Thus a difference of 10 cm*. in capacity corresponds to a difference of 4mm. in thickness
which is about 5°/, of the thickness (say 6 mm.) of the human skull.
We may fairly conclude then, that there is no appreciable difference in the thickness of the
negro skull as compared with the European.
§ 4. The female crania yield very similar results. The following is the table for the female
skulls.
TABLE II.
| Mean capacit Mean BLH : : Giana
in tnt s in cm.3 oD OS om. "cy BLH
21 Congo Skulls... 1206 2858 LOZ e Se 90
43 Gaboon (1864) ... 1232 2924 126°7 270°95 8814
17 Gaboon (1880) .., 1240 2964 97°31 265°8 8560
81 Negro skulls... 1227 2956 117 255°72 "7668 |
Miscellanea 191
The corresponding regression lines are
21 Congo skulls C= 0003645 BLH+164+ E> Maseeameneace these (5),
Jn
43 Gaboon (1864) | C=-0004122BLH+ 27+ aw Teen, (6),
Vin
17 Gaboon (1880) C= 0003134 BLH4+311+ = sea ea Sessatiente teenies (7),
vn
81 Negro skulls C= 0003508 BLH 42044 ieee (8).
vn
The corresponding Lee and Pearson formula obtained by the method of least: squares is
C2 OO0USC LBA 7812 cciccceccsscscovccessezsssesesccescenss (Q).
The mean values of B, L, H’ for the Bavarian female skulls discussed by Lee and Pearson are
B=144-11,
L =173'59,
Hf’ =128°07.
With these values, we deduce from 5—8 the following values for C.
45
(5) C=1331+
ww
(6) C=1347+ =
vn
(7) C=13154—
The mean of the measured values of the capacities of these skulls is 1337 and formula (8)
based on 81 negro skulls gives a result in very close agreement.
If the above values of B, L, H’ are substituted in Lee and Pearson’s formula @ we obtain
C'=1284 a result which differs from the true value much more seriously than the prediction by
the negro regression formula.
Again, if we insert the mean values
B =130°75,
£ =171°38,
H=129°'81,
of the 81 female negro crania in the formula Q we get C=1266 as against the mean of the
measured values which is C=1227, demonstrating again the fact that the formulae P, @ based
on 11 means are not as good as the regression formulae.
§ 5. We add tables of the correlation between capacity and breadth, capacity and length,
and capacity and total height for the 110 male and the 81 female skulls, and for comparison
reprint the corresponding value for German (Bavarian) skulls.
Miscellanea
TABLE III.
Correlation Males.
Negro German
Capacity and Breadth 4977 6720
Capacity and Height 6080 (total height) 2431 (auricular height)
Capacity and Length 7433 5152
TABLE IV.
Females.
Negro German
Capacity and Breadth 7578 | “7068
Capacity and Height 5450 (total height) | 4512 (auricular height)
Capacity and Length 6699 | ‘6873
The corresponding regression lines are given in the tables below :
TABLE V.
Males.
Negro German
(9) C=12°6356B-—3561+202 | C=13-432B—517°34
Vn
(10) C@=12-8301Z— 1087 + = C=9'892L — 289-55
7
(11) C@=15°3265H’— 694 + > C=5:264H + 86805 (auricular height)
nN
(H'=total height)
TABLE VI.
Females.
Negro German
(12) @=17°872B-1114 +7 C=15-716B — 927-66
nN
87
(13) C=12°46 L- Oe ae C=12°055L —755°53
nN
98
(14) C=10°871H’— 184+ J, | C=10993H+ 8213 (auricular height)
(H’ =total height)
Se =r
5 ‘
4
7
wu
at
Pes.
a Ps fed
; :
j
i
; :
- ay ra
: :
: 2
7 ‘
1 a
ie
7 '
: :
2 t -
1
=. ae
1h
A ¥
i
_
a
: _
1
=
;
:
=
7
om
- *
:
:
*
got
Biometrika, Vol. X, Part | Plate X
Dr Maynard’s Piebald Negro
Miscellanea 193
No great degree of accuracy can be expected in reconstructing the capacity of a skull from a
single measurement, but the remarkable difference of formula (11) for negro skulls from the
corresponding German formula is of course due to their referring to different measurements
of the height. If we insert H=133°8 in (11), which is the mean total height of the Bavarian
96.
skulls we get C=1356'7 + —= instead of the measured mean C=1503 cm’.
Ne
sos : ; 105. : F :
Similarly equation (9) gives C=1555°6 + a instead of 1503 when we insert the German
n
mean B=150°5.
Thus 9—14 are of little use for our purpose.
VII. Note on a Negro Piebald. (C. D. MAYNARD.)
THE remarkably interesting photograph of a negro piebald on Plate X has been forwarded
to the Editor by Dr C. D. Maynard. The native comes from the district round Chai Chai.
Dr Maynard writes from Ressano Garcia, and states that the hospital attendant took the
photograph. The extraordinary interest of the case arises from the fact that the thighs and
feet are of normal negro pigmentation, but in the other patches we have varying degrees of
pigmentation of the skin down to albinotic white. Unfortunately there is no dorsal view, but
the back is stated to be also affected with albinotic areas. The boy reported that he was
in the same condition when born, and that the nature and areas of the pigmentation had not
altered.
VIII. Note on Infantile Mortality and Employment of Women, from
the Report on Condition of Woman and Child Wage-earners in the United
States, Volume XIII. Infant Mortality and its Relation to the Employment
of Mothers.
By ETHEL M. ELDERTON.
THe author of this Report emphasizes the difficulty of determining the effect of women’s
employment and points out that
‘It would be possible to draw positive conclusions as to the relative importance of this particular
factor only by point-to-point comparison of the infant mortality for a period of years in two large
communities, or two classes of large communities, in which all the material conditions were sub-
stantially common, with the single important exception that in one a considerable proportion of the
married female population of child-bearing age were at work outside of their homes and in the other
community with which the comparison was made none of the women were so employed.
To admit of entirely sound conclusions, it would be necessary that the populations-—and especially
the women—of both communities should be of like ages, races, and physical health, that their living
conditions should be practically identical, and that, in a general way, the child-bearing women should
be of about the same grade of intelligence....... In default of some such comparison on a broad scale of
the mortality of the infants of working and non-working women of similar ages, races, intelligence, and
living conditions, no one can determine accurately how many of the deaths of working women’s infants
are due to the mother’s work and how many to the other conditions of their lives and environment.”’
(p. 18).
The author illustrates the point by taking the six New England States and giving the infant
deathrate, percentage of women of 16 years and over who are breadwinners, percentage of foreign-
born to the population and percentage of population living in towns of 4000 and more inhabitants,
and showing that, though the states with the highest infant mortality have also the largest
Biometrika x 25
194 Miscellanea
number of women employed, they have also the largest percentage of foreign-born and of those
living in urban surroundings, and that it is therefore impossible without further investigation to
assign the infant deathrate to any of these three factors.
A further investigation has been undertaken into the 32 Massachusetts cities and the death-
rate under a year is given, the percentage of foreign-born, the births per 1000 of the population*,
the percentage of women gainfully employed and the percentage illiterate, and a comparison is made
between the ten cities with the highest and the ten cities with the lowest infant deathrate and
percentage of women employed and the other factors enumerated. The conclusion is reached that
“These comparisons indicate, superficially at least, that a more direct relation exists between
infant mortality and the birthrate, the percentage of foreign-born, and the percentage of female
illiteracy than between infant mortality and the employment of women.” (p. 38).
There can be no doubt that a direct study of the infant mortality in relation to women’s
employment can only properly be made, when we confine our attention to women, employed and
unemployed, who are actually mothers and live in the same town, and when we correct for aget,
and if possible home conditions. Still if we take a series of different towns the right method
must be to correct by the method of partial correlation for such divergent factors as we are
able to ascertain and allow for in the series of towns investigated. I have endeavoured to apply
modern statistical methods to the data of this Report, taking as measures of the environmental
conditions in the towns: D the general deathrate, 7= percentage of illiteracy, f= percentage of
foreign-born population, e=percentage of females employed 10 years of age and upwards (note,
not percentuge of employed mothers, so we may be largely measuring effect of child labour on
future motherhood), and @=deaths under one year per 1000 births. Then we have for cor-
relations :
Tc = 68, Ta= “70, ldf= 74.
Hence numbers of foreign-born and of illiterate appear to be slightly more influential on infantile
mortality than employment of women. These values are certainly high and the first is the sort
of crude value which is used as an argument against the employment of women. Proceeding to
partial correlations we have
Pac= 36, a= “43, oS 48,
Ct ed "42, eas 57, Fide “Bl.
We next corrected for two factors and found :
iflde= “34, ef a= 12, i af= 43,
Thus we see that illiteracy has least influence on the infantile deathrate and the presence of
foreign-born most.
But even the presence of foreign-born and of illiterates is not a very complete measure of
environmental effects liable to influence the infantile mortality in different towns as apart from
employment of women. Many women employed means industrial conditions and possibly
generally bad environment. I have taken as a measure of this the general deathrate D and find
TDa= 71, 'De= ‘47, TDi= ‘60, Di “49,
Whence I find :
prac='97, pra="62, pras="75,
D'fe= “49, D'i= 61, Dit 68,
showing very substantial relations after correction for a general measure of poor environment.
* The author is not very confident of the full accuracy of the complete registration of births.
+ Young women are often employed up to the birth of their first one or two children, but the death-
rate of these elder-born is heavier than the deathrate of those who immediately follow.
Miscellanea 195
Next proceeding to allow for two factors we find
pDra="23, pora="44, ppl ae = "35,
the latter result shows that general deathrate and illiteracy are about equally influential on the
relation of employment of women to infantile mortality. Finally I corrected for all three factors
and found :
isDl de = "28
or 60 °/, of the crude correlation 72,='68 is due to women being most employed in towns where
the general deathrate is high, where illiterates are frequent and the population is largely foreign-
born. How much further the relationship would be reduced, could we equalise other features of
these Massachusetts cities, it is not possible to predict. The examination of the individuals im
one city appears to me to be the only satisfactory method of disentangling the numerous factors
which influence infant mortality. We commend, however, the study of the first part of this
Report, as it deals very clearly with the difficulties which arise, and will counteract the tendency,
which is prevalent, to assert causation whenever association is observed. The author lays stress
on avoiding such logical confusions.
Part Il of the Report deals with infant mortality and its relation to the employment of
mothers in Fall River, Massachusetts. In 1908 the attempt was made to visit the homes of
each of the mothers of the 859 infants who died during the year and to ascertain details con-
cerning her occupation, ete. In 279 cases the family could not be traced. In 266 cases prior to
the birth of the child the mother was at work outside the home while in 314 cases the mother’s
work was limited to household duties or other work carried on entirely at home. Thus only the
cases of deaths are dealt with and the causes of death are compared in the two groups of cases
(1) when the mother was at work outside the home prior to the birth of the child and (2) when
the mother’s work was carried on entirely in the home.
I hold that this method will never prove as satisfactory as that employed in districts in
England ; in England certain districts are chosen and every baby within that area is visited
and the deathrate per number born in one group can be compared with another and the
circumstances surrounding those babies who survive and those who die in the first year of life
in a given district can be analysed.
I do not think that the fact that a rather higher percentage of all deaths from gastritis etc. in
Fall River occur when the mother works away from home and a rather higher percentage from
congenital debility at birth when the mother does not work away from home will help us much
in discovering the influence of the employment of the mother on infant mortality, nor do I think
it will throw much light on the question of stillbirths with which the Report also deals. It is
found that there are no more stillbirths proportional to all deaths when the mother is industrially
employed, but it seems to me that this tells us nothing about the number of stillbirths pro-
portional to all births. The real question is whether mothers employed away from home in
factory or workshop, wuose other circumstances are the same, lose more children in the first year
of life or have more children stillborn than the mothers who are only employed in their homes
and I do not think a comparison of causes of death will lead us much further, and I think it may
lead to difficulties.
When dealing with the mother’s work after childbirth in relation to the causes of infant
mortality it is pointed out that the smaller percentage of deaths from congenital disease among
the children of mothers who returned to work after childbirth was owing to the fact that most
of the children dying from this group of causes died in the early weeks of life before the mother
returned to work. For this same reason the number of deaths from gastritis ete. of children
whose mothers returned to work is exaggerated, for we are missing out a whole series of illnesses
196 Miscellanea
which have ceased to add to the child deathrate by the time the mother returns to work and we
must increase in this way the percentage of deaths of any disease of the later months of a child’s
first year of life.
It seems to me that a comparison of deaths in this way will really give very little information ;
an excess of deaths from one disease means a defect in some other disease; it is shown that
when the baby is nursed exclusively by the mother 26-0 per cent. of the deaths were from
diarrhoea, gastritis, etc.; when partly nursed the percentage was 52°3 and when artificial food
was exclusively employed the percentage of deaths from diarrhoea etc. was 42°9; the baby
certainly dies less from gastritis when it is breast fed but it dies in greater numbers from other
causes. Here again there is a difficulty; deaths from congenital diseases fall on the first weeks
of life when breast feeding is the rule, while deaths from gastritis etc. fall on the later months of
child life when “partial breast feeding” has become more common and I do not think it is
possible to draw any conclusions from a comparison of deaths from one disease to deaths from
all diseases as to the importance of artificial feeding in relation to deaths from gastritis.
Interesting information is given as to the reasons for artificial feeding ; the numbers are not
large enough to justify any definite conclusions, but thisis such an important part of any inquiry
into the influence of artifical feeding on the infant deathrate that one welcomes its inclusion in a
report of this kind.
WE have been requested by Professor F, M. Urban to insert the accompanying announcement.
ANNOUNCEMENT.
A prize of One Hundred Dollars ($100.00) is offered for the best paper on the Availability of
Pearson’s Formulae for Psychophysics.
The rules for the solution of this problem have been formulated in general terms by William
Brown. It is now required (1) to make their formulation specific, and (2) to show how they
work out in actual practice. This means that the writer must show the steps to be taken,
in the treatment of a complete set of data (Vollreihe), for the attainment in every case of a
definite result. The calculations should be arranged with a view to practical application, i.e. so
that the amount of computation is reduced to a minimum. If the labour of computation can be
reduced by new tables, this fact should be pointed out.
The paper must contain samples of numerical calculation, but it is not necessary that the
writer have experimental data of his own. In default of new data, those of F. M. Urban’s
experiments on lifted weights (all seven observers) or those of H. Keller’s acoumetrical experi-
ments (all results of one observer in both time-orders) are to be used.
Papers in competition for this Prize will be received, not later than December 31st, 1914, by
Professor E. B. Titchener, Cornell Heights, Ithaca, N.Y., U.S.A. Such papers are to be marked
only with a motto, and are to be accompanied by a sealed envelope, marked with the same motto,
and containing the name and address of the writer. The Prize will be awarded by a committee
consisting of Professors William Brown, E. B. Titchener and F. M. Urban.
The committee will make known the name of the successful competitor on July 1, 1915.
The unsuccessful papers, with the corresponding envelopes, will be destroyed (unless called for
by their authors) six months after the publication of the award.
Corrigendum. Dr Derry has most kindly pointed out a slip on p. 307, Vol. VIII; the value
of 100 (B—#H)/Z for Congo female crania is +1:9 and not —1°9, which brings these crania nearer
to their proper place, and the remarks on this point p. 308 should accordingly be cancelled.
outs
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VOLUME X NOVEMBER, 1914 Nos. 2 AND 8
A PIEBALD FAMILY.
By E. A. COCKAYNE, M.D., MRCP.
In spite of the great interest, which they have always excited, well
authenticated examples of piebalds in the dark races have been found to be rare.
In the white races they are much less conspicuous, in part owing to the presence
of clothing and in part owing to the lack of contrast between the pigmented
and unpigmented skin, but the likelihood of their coming under the notice of
a skilled observer is much greater. The scarcity of records shows that piebalds
in the white races also must be very uncommon. Last year I met with a case
in a baby, and found that the child belonged to a family, many of whose members
showed a similar defect of pigmentation. The family, belonging to a farming
stock, originally came from the neighbourhood of Bury St Edmunds in Suffolk, and
the anomaly is known to have descended directly through six generations. The
oldest member, with whom I have talked, is fairly certain that it was present in
one generation at least before this.
Of the first two generations in the pedigree (see Plate XI), I could obtain no
definite information except the statement as to the existence of the piebaldism in
I. 1 and II. 1, but of the third, III. 2 is said to have had a frontal blaze of white
hair and white skin on the neck and forearms, which was very conspicuous owing
to its marked contrast with the neighbouring weather-stained normal skin.
IIL. 4 appears to have been the only member of the family who showed a
marked dislike to the condition and always wore a wig to hide the frontal blaze.
III, 2, whose family name was C—-—%*, had fifteen children. The first, IV. 2,
a male, with dark hair, married twice, and had eight normal children, five by the
first wife and three by the second. The second child, IV. 5, was a piebald, with a
large frontal blaze, white skin on the front of the neck and arms, and blue eyes.
He transmitted the condition to all his three children. V. 3, the eldest boy,
aged 22 and unmarried, possesses dark hair, with a V-shaped frontal blaze of
white or cream coloured hair, the apex of the V commencing near the coronal
suture and spreading out to a width of 34 inches, as it reaches the forehead. The
eyebrows and some of the eyelashes are white. The next boy, V. 4, is aged 18.
He has light hair and a very large blaze of unpigmented hair, which covers the
whole of the top of the head. His eyebrows and eyelashes are white, and the eyes
are blue. Both boys have white patches on the front of the neck and on the arms
(see Plate XII).
* Names preserved in the confidential register of the Galton Laboratory,
Biometrika x 26
198 A Piebald Family
Next in the fourth generation were twins, IV. 6 and 7, both piebalds. They
were evidently not uniovular, because one had dark hair, and one light, and the
white blazes were dissimilar in extent, but it is uncertain which had the larger.
Both died at an early age.
The next, IV. 8, a girl, was normal with dark hair and eyes and remained
unmarried. Next came a woman, IV. 10, who was a piebald with a large frontal
blaze, white eyebrows and eyelashes, and white skin on the front of the neck and
forearms. The right eye was blue, and the left brown (see Plate XIII (B)). Her
child, aged 13, is quite normal with light hair and dark eyes.
The next child, IV. 12, Mrs W——, has a large frontal blaze and dark brown
irides. There is a large irregular patch of white skin extending from just below
the chin to the heads of the clavicles, and round it the skin appears to be more
deeply pigmented than the rest of the skin of the neck. There are a few small
islands of pigmented skin near the edge of the unpigmented area. The skin of
the anterior aspect of the forearms is unpigmented from the elbows to the wrists,
and here also, there are some small islands of pigmented skin in marked contrast
to the unpigmented area, in which they lie (see Plate XIV).
The first two children of this individual were daughters, V. 8 and V. 9, both
piebald, the third a normal son, V. 10, and then three more piebald daughters,
V. 11—138. The first of the daughters, Mrs G——, V. 8 (see Plate XIV), is very
fair with a very large frontal blaze covering the whole of the top of the head,
and her eyebrows and eyelashes are white. Her normal hair has pale creamy
diffused pigment and, according to the individual hair, some to a decided number
of granules*. The hair of the blaze has no diffused pigment and no granules.
The irides are light brown, but the outer segments on both sides are paler and
greenish in colour. The skin of the forehead and base of the nose is very pale in
colour. She has a large white patch on the skin of the front of the neck, beginning
just below the chin and widening out so as to embrace that over the inner ends
of both clavicles. As in her mother there appears to be some concentration of
pigment round this white area, and there are small isolated areas of pigmented skin
near its edge. She has unpigmented skin on the anterior aspect of both forearms.
Of her two children the first, VI. 1, a boy aged 8, is normal, the second, VI. 2,
a boy aged 14, is a piebald (see Plate XIV). This child, VI. 2, was nine months
old when first seen. He had a very large frontal blaze, resembling that of his
mother and covering all the top of the head, the eyebrows and eyelashes were
white with the exception of some of the outer hairs. Hair, pale cream in colour,
said to be from the light area, has pale creamy diffused pigment and some granules
(8), the granules being very small. It was obvious, even at this age, that hetero-
chromia iridis was present. The right iris was pale except for a sector of dark
grey occupying the upper and outer quadrant, the left iris was entirely dark grey.
No difference in colour of the skin of the neck or forearm could be made out.
* B to y on the Galton Laboratory scale of granular pigmentation.
KE. A. CockayNneE 199
When the baby was seen after the summer of 1913, the grey portions of the irides
were becoming brown, the pale portion was still light blue. The face and arms were
sunburnt, and it was noticed that the forehead was paler than the rest of the face.
There was a pale area on the front of the neck, and the whole anterior surfaces of
the forearms were white, the edges being very irregular in contour. There was
also a white streak running obliquely right across the posterior or extensor aspect
of the left forearm, and this offered a marked contrast with the rest of the surface,
which was very brown. When the sunburn had died away the difference between
the pigmented and unpigmented skin could no longer be made out.
IV. 12’s second daughter, V. 9, aged 23, has only a small cream coloured
frontal blaze, and the rest of her hair is light brown (see Plates XV and XVI).
The eyebrows are composed of an even mixture of brown and white hairs, and
the eyelashes are similar, with brown and white hairs alternating. The irides are
grey and uniformly pigmented. There is a large irregular area of white skin at
the base of the neck.
The whole of the anterior aspect of the right forearm is unpigmented, and
there are similar small areas scattered over the posterior aspect (see Plate X VII).
The left forearm is white only on the anterior aspect.
The next girl, V. 11, is aged 9. She has a very small frontal blaze, but the
skin of the forehead is pale (see Plate XV). The eyebrows show a division into
two parts, on the inner halves grow white hairs only, and on the outer brown hairs.
The eyelashes on the contrary consist of alternate brown and white hairs. The
irides are grey and uniformly coloured (see Plate XVIII). There is only a small
white area in the middle of the front of the neck, but there are well differentiated
white areas on the anterior aspects of both forearms (see Plate XIII (A)), and on
the inner aspects of both upper arms. Her hair was examined and the first sample
showed very pale diffused pigment and some granules (8). Two more samples
were then examined, one from the blaze and one from the neighbouring part of
the scalp. The first showed no diffused pigment and no granules, the second
showed the majority of hairs with yellow-brown diffused pigment and a decided
number to plenty of small granules (y—6), but a few had no diffused pigment
and no granules.
The next piebald child, V. 12, died young. She had a frontal blaze and blue
irides. Some of her hair showed very pale diffused pigment, and some granules (8).
The next child, V. 13, also died young. She was a piebald nearer to the
classical type than any of the others. She had a large frontal blaze, white skin
on the forehead, and large areas of white skin on the front of the neck and
chest, and in addition a very extensive area on the abdomen.
Of the fourth generation the next child, IV. 13, was a male with dark hair and
eyes, who had 5 normal children; the next, IV. 15, had fair hair and died young.
Twins, IV. 16 and 17, came next and died in infancy*. They were heterogeneous,
* The tendency to twin in this family is worth noting.
26—2
200 A Piebald Family
a dark-haired boy and a light-haired girl. A girl, IV. 19, was born next and she
had twin sons, V. 15 and 16, who were also normal. The last three children,
IV. 20—22, a girl, a boy, and one whose sex I am unable to ascertain, were all
normally pigmented and all died at a very early age.
The pedigree confirms the strongly hereditary nature of piebaldism, and in this
as in other published cases the character can affect either sex, but has only been
transmitted by those affected. Unless we are to assume that in the case of such
a rare anomaly as piebaldism, I. 2, II. 2 or III. 1, were really unnoticed piebalds,
then III. 2 could only be heterozygous, or since piebaldism is dominant a (DR).
We must take IV. 4, IV. 9, IV. 11 and V.7 for pure recessives (RR). Thus the
number of piebalds in the five sibships of generations IV, V and VI should be one
quarter, Le. $(15+34+1+6+2)=7 nearly. We have actually 14 out of 27,
thus piebaldism does not seem to act numerically as a pure dominant.
The areas of unpigmented skin are less than in the classical piebalds, but it is
probable that in some, at least, they are larger and more numerous than I have
stated. On the covered parts of the body and legs, which I was unable to
examine except in the baby, they would not be very noticeable. It was not until
I had noticed the white skin on the neck and arms of one of them that I was told
anything about the existence of similar patches on the others. If true, it is
remarkable that none have had white patches on the legs.
With regard to the local distribution of the pigment, there appears to be an
excess at the edges of some of the unpigmented areas, as has been noted in other
cases. In the case of other pale areas, the demarcation between them and the
normal skin is very slight, and is probably due to the fact that they are not wholly
unpigmented. ‘This remark applies especially to the forehead, which in some of
them looks paler than natural, but not wholly devoid of pigment.
In some the eyelashes are alternately white and brown, and in others the eye-
brows are similar, and in one at least hairs growing on the scalp near the blaze
are in some instances entirely without either diffused or granular pigment. This
suggests that the skin beneath may show a deficient and irregular distribution of
pigment.
The most interesting feature is the occurrence in three members of the family
of well-marked heterochromia iridis, a character which has been met with in
members of a piebald family, but always independently of their piebaldism, never,
as in this case, in true association with it. It proves conclusively that these cases
are not congenital leucoderma.
There seems to be no association of piebaldism and general lack of pigmenta-
tion of hair and irides. Affected and unaffected members have been both fair and
dark, but the fairest piebalds seem to have the most extensive frontal blazes.
In the cases photographed the individuals were blonds and there has been great
difficulty in getting a good photographic contrast of differences of pigmentation
very noticeable in the living subject.
Plate X|
Biometrika, Vol. X, Parts Il and Ill
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Biometrika, Vol. X, Parts Il and III Plate XIl
V. 3 and V. 4 as children showing their marked V-shaped frontal blazes.
Plate XIII
Biometrika, Vol. X, Parts Il and III
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Biometrika, Vol. X, Parts II and Ill
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Biometrika, Vol. X, Parts Il and Ill
Plate XV
‘wo sisters with white frontal blazes.
dances
V.
Biometrika, Vol. X, Parts Il and Ill Plate XVI
V. 9. Showing white forelock or blaze,
Biometrika, Vol. X, Parts Il and Ill Plate XVII
Right forearm of V. 9 showing white patches on posterior aspect. The photograph is untouched and it is difficult
to bring out by photography the grades of pigmentation when the arm is untanned by the sun, although they
are quite clear on actual inspection.
Biometrika, Vol. X, Parts Il and Il Plate XVIII
Large photograph of Y, 11 to show paleness of forehead and white hairs on inner half of eyebrows.
CLYPEAL MARKINGS OF QUEENS, DRONES AND
WORKERS OF VESPA VULGARIS.
By OSWALD H. LATTER, M.A.
Upon the front of the head of Vespa vulgaris certain yellow markings stand
out conspicuously upon the otherwise black surface. Below the three ocelli
and between the upper portions of the two compound eyes there is a median
four-sided yellow patch, the “corona”; to the right and left of this, separated
from it by a fairly wide interval, and occupying the bay of each of the compound
eyes is a pair of elongated yellow blotches; while straight below the corona and
between the lower portions of the compound eyes is a very conspicuous yellow
area which “extends over the clypeus and down to the labrum or upper lip which
hes between the two mandibles. This clypeal patch of yellow bears upon it a
black mark which is subject to considerable variation. I distinguish in the queens
and workers five chief types of this black mark: see diagrams on p. 202. In
Type I a broad vertical black band extends right through the yellow patch from
the top to the bottom; a little below its middle the band bears to right and left
a pair of bluntly pointed and slightly upturned arms: the portion of the median
band below these arms is somewhat narrower than that above. Type II is derived
from I by suppression of the black portion below the transverse arms. In Type III
the extent of the black colouring is yet further reduced by the absence of the
upper half (or thereabouts) of the vertical band. In Type IV the lower part of
the vertical band re-appears, but the width of all the components is very much
less than in any of the preceding types. In Type V the component parts of the
black marking cease to be in contact; the upper portion of the vertical band
is interrupted by a broad belt of yellow; the two “arms” are separated from the
lower part of what remains and from one another; while there is no black at
all below these remnants of the “arms”—a feature recalling Types II and III.
Types IV and V are however represented only by single individuals in the series
examined.
Between these main types certain intermediates occur. Thus some individuals
have the black piece below the “arms” very narrow, approximating therefore
202 Clypeal Markings of Vespa vulgaris
to II, but conforming to I if we take extension of the black right through the
yellow as the criterion of I; such individuals are distinguished as 1+ II. Others
again conform to II but possess a slightly darker stain on the yellow in the line
where the distinctive lower black portion of I might occur; these are called I+ 1.
Similarly, intermediates between II and III are recognisable: in II + III the top
of the upper portion of the vertical black band is very narrow; while in III + II
there is a mere stain on the yellow of this region. A single instance occurs of an
intermediate between I and III (1+ III), where the vertical black band extends
right through the yellow, but is much narrowed at its upper extremity.
Front of head of V. vulgaris ¢.
SD Ocelli
Eye
Yellow
Patch tn Girsen
bay of Eye
(All parts left white are actually yellow.)
Clypeus alone
HF) (*) @)
Types I Il III IV W
4 VII) VII VIII (3 VII) VIII
Drawn by K. W. Merrylies.
My first examination consisted of about 200 tubes containing queens of
Vespa vulgaris from different nests. In the case of some queens the heads
were missing, and in the course of transit the contents of some of the tubes
had got loose in the jar. I have numbered these 199 to 208. The results are
given in Table I (p. 204) and the summary below:
O. H. Larter 203
Type I Pure Poh
‘. I+II 11 62
, 1+ Il] I
Type II II+I] 29,
5 Pure 94} 120
55 II + Il 4
Type III III + I a :
4 Pure 3, C4
III +1V 0|
Type IV IV +I1ll 0
a Pure | il
2! IV+V 0
Type V Vea Hi 1
r Pure 1
Total: 185
It will be seen that transitional cases undoubtedly occur. The bulk of the
queens, however, fall into Types I and II, or queens are very little variable.
To test: (1) whether this variability was still further lessened by taking only
the queens from a single nest, and (11) the relative variability of queens, drones
and workers, I now examined all the queens, workers and drones of a single nest
of V. vulgaris.
In this case all the 127 queens were of Type II*.
The classes of the workers are given in Table II (p. 205) and may be sum-
marised as follows:
Type I Pure 5) 10
§ v.s or (I+II)? 5|
Type II II+I 6
“ Pure at Ae
Total: 172
It will be seen that they are somewhat more variable than the queens of the
same nest, but not so variable as queens from different nests.
T now turn to the drones of this same nest. JI had 150 at my disposal.
The drones exhibit a very wide range of facial markings. In the material
examined comparatively few fall into the scheme of classification adopted for
the queens and workers, and it thus becomes necessary to resort to six types
of face which appear to be peculiar to the male sex. These are numbered VI,
* There were 129 queens in this nest, but No. 34 was missing and No. 98 had its head damaged too
badly for classing.
204 Clypeal Markings of Vespa vulgaris
VIG VID, VII, VII (4 VIID, VIII and IX, see diagrams, p. 202. In Type VI
there are two somewhat elongated black dots upon the yellow clypeus, one being
sub-central, the other on the ventral margin; in VI (4 VII) the ventral dot is
longer dorso-ventrally and a third dot appears upon the left side (right side, in
figure seen from in front) opposite the gap between the two previous dots;
TABLE I.
Types of Clypeal Markings in V. vulgaris Queens.
No. | No. | No No. No. |
| |
| |
| oe = 44 III 87 | 1+II | 180 = 173 Ill
hee II Dials 88 II 131 II 174, | Tees
| 3 | Il4I 46 II 89 ae a 132 II 175 II
| 4 1 47 = 90 alge 133 Il WG ho WIESE
ae ig alle AS. Wise Oe L 134 — 177 Tore
6 at AQ) 3) > Sel 92 et 135 I Tey) ~ UE
re |) IE BORE in ale 93) LEE in| 36 II 179 esi
3 ail 51 if 949 i 137 II 180 II+I
9 | II o2 II 955) ILE 138 I 181 II
10 || I 5a el 96.09) 139 II 182 =
11 Il+I 54a eee 97 I 140 I 183 | I1+II1
12 II+1 ii) 10 98 dee 141 I 184 =
Lea 9 al 56 md 99 |II+IIL} 142 Teen 185 I
14 oe 57 IL 100 I 143 I 186 Il
15 II 58 pa 101 ats 144 Il 187 =|) St
16 II 59 I 102 Il 145 II 188 | I
7 A eaeale 60) meet 103 I 146 I 189 I+II
18 Il 61 IL 104 IU 147 II 190 II
19 | II+I 62 II 105 Il 148 =e 191 I
20) ae 63 II 106 a 149 i 192 I
HS ee Al 64 | II 107 I 150 Il 193 III
22a mel (ij) = TU 108 Il 151 II 193.4 II
937 | 21 66 II 109 I 152 = 194 =
OAgea eles 67 = TOKO} 7 |) S50 153 II 195 | I+III
25 IL+I Con ay tll 111 | IE+T 154 1H 196 II
26 I 69 eet 112 Il sys | AU 19 jee
27 Il 70 113 I 156 Il 198 IIl+I
28 | I a Il 114 II 157 Til
29 TST 72 ] 115 |II+I0T} 158 IIl+I = -
30 I a eal 116 II 159 Il
31 II Mee letsel Has IU 160 I Loose |
32 II Aven AT 118 I 161 I
33 V 75 | II ie) |). HE 162 Il 199 I
34 Tal M6 | SUL 120 |II+III} 163 I 200 I
Si alipageeee i, ae 191. || 7 ale 164 II 201 I
36 iat AGH), eae 129° | 165 II HO i) ULE
37 I oe ee 122 -- NGS | IM 2O3ke ee lal
38 II 80 ee 124. I Gy | 7 204 Il
39 Il+I] Se eee 125 II 16S 205 II
40 ete 82 - 126 elie es 169 |) »— 206 Il
Alton lee 83 I NO Wel 70a) eae 207 Ll
AN Oyl ad 845 i 128 | II 171 I 208s
| 42 Il 85 128b Il 172 = 209) | sri |
| 43 I 86 | TSU 129 SHI | |
|
O. H. Larrer 205
TABLE ILI.
Types of Clypeal Marking of Workers of a single Nest of V. vulgaris.
No. No. No. No. No.
1 — 37 II 74 II 112 II 150 II -
2 — 38 IT 75 II 113 II 151 II+I1
3 — 39 II 76 II 114 II 152 II
4 _- 40 II 77 II 115 II 153 IT
5 — 41 II 78 — 116 II 154 IT
6 — 42 II 79 I 117 II 155 II
7 — 43 II 80 II 118 II 156 II |
8 —- 44 II 81 II 119 II 157 Il |
9 II 45 |Iv.v.s. 82 II 120 II 158 II
10 II [1+11] 83 II 121 II 159 | II
11 II 46.) I 84 II 122 II 160 II
12 Iv. s. 47 II 85 it 123 II 161 II
[I+11] ] 48 I 86.) 1d 124 I 162 | II
13 II 49 —— 87 II 125 II 163 II
14 II 50 IT 88 II 126 II 164 I+!
15 I 51 II 89 II 127 II+1 165 II
16 II 52 II 90 II 128 II 166 II
17 II 53 II 91 II 129 Il 167 Il
18 IT 54 II 92 II 130 II 168 IT
19 II 55 II 93 II 131 II 169 II
20 II 56 II 94 Il 132 II 170 II
21 II 57 IT 95 IT 133 II 171 II
22 IT 58 Iv. s. 96 II 134 II 172 II
93. | Tf Aeim o7 fo = | 135 | mm} ia | 1
24 II 59 II 98 II 136 II+I1 174 II
25 I 60 II 99 II 137 II 175s I
26 II 61 II 100 II 138 II 176¢" If
27 II 62 lil 101 II 139 1H ied, II
28 II 63 II 102 II 140 II 178 II
29 II 64 II 103 IT 141 II 179 II
30 Iv.s. 65 I 104 II 142 Il 180 Il
f[+it)| 66 -| IL 105 | II ea iii [er
31 II 67 II 106 II 144 II 182 II
32 II+I 68 II 107 II 145 II 183 II
33 II 69 I 108 II 146 II 184 II
34 — 70 II 109 II+I 147 II 185 —
35 IAs al II 110 II 148 II 186 —
(I+1I] } 72 II 111 II 149 IT 187 II
36 II 73 II
in VII the two median dots are united by a slender black line, and there is a pair
of lateral dots, right and left; in VII (4 VIII) the median line is of uniform
width, extending from about the centre to the lower margin, and to its left side
there is a single dot; in VIII the median line alone is visible, both lateral dots
having disappeared; while in IX there is no continuous median line, but merely
two black spots, one at the extreme dorsal and the other at the extreme ventral
side of the clypeus. It will be noticed that Types VII—VIII approximate to
Type IV in so far as the black stripe begins at about the middle of the clypeus
and extends right down to the ventral margin.
Biometrika x 27
206 Clypeal Markings of Vespa vulgaris
The data are given in Table III (p. 207) and are summarised below:
very narrow 6
ape a A VII) if f
Type II no horns 1, very narrow 1 2
Type III : : : : : 0
Type IV : : : : : 0
Type V ; : : ; 0
Type VI Pure oi 60
3 Vale vali 1
Type VII 5 : : . nee,
Type VIII (VIII near VI) 8
F (VIII + $ VIT) 27 58
, Pure 48
Type IX : : : : ‘ 2
Total: 151
It will be realised at once how far more variable the drones, of even one nest,
are than the workers or queens for this character. But their variability is rather
of a negative than a positive character, appearing to consist in more or less extensive
absence of the fuller markings of queen and worker.
The results here deduced for variability of non-measured characters do not
wholly agree with those found by Wright, Lee and Pearson on the wing measure-
ments of the same nest of V. vulgaris. They found that for absolute measurements
the variability as determined by the coefficient of variation was in every case such
that the worker was more variable than the drone and the drone than the queen.
On the other hand they found when they dealt with zndzces that the drone for
wing measurements was slightly more variable than the worker and the queen
less variable than either*. Possibly the divergence apparent here may be explicable
in tbe sense of the drone’s variability lying in the present case in an absence
of marking rather than in any positive variation. The drone’s variation is about
a centre of much dimimished marking. If we could measure the variation in the
total area of marking in queen and worker we might find it as great as the varia-
tions in the smaller markings of the drone.
It would be of much interest to investigate a series of drones from different
nests. It is clear that the clypeal markings form a secondary sexual character
and they would probably provide classifications for hereditary purposes.
* Biometrika, Vol. v. pp. 414 and 421.
O. H. Latrrer
TABLE III.
bo
Types of Clypeal Markings in Drones of a single Nest of V. vulgaris.
No. No. No. No. | No.
|
1 VIII 33 VII 63 VII 93 VI 123 VIII
near VI} 34 VI 64 Wi 94 VI 124 I
2 VI 35 VIII 65 VII 95 VI narrow |
3 VIII 36 VI 66 VIII 96 VII 125 VI
4 VIII 37 VI 67 I 97 il 126 VIII
5 VI 38 VII very very 127 VII |
6 VI 39 VI narrow | narrow | 128 VII |
7 VIII 40 VII 68 VI 98 | VIII 129 VIII
8 VIII 4] VI 69 VIII 99 VIII 130 VIII
9 VIII 42 VI 70 VI 100 VI 131 VI |
10 IX 43 VIII Hil II 101 | VI gy VIII |
11 VI 44 VI but no 102 AVIA 133} VI
12 VIII 45 VIII horns 103 VIII 134 VI
13 VI 46 VIII 72 VIII near VI} 135 VI
14 Vall 47 VI Wp} VI 104 VI 136 VIII
15 VIII 48 VIII 74 VIII 105 VIII 137 VI
near VI} 49 VII 75 VIII 106 VII 138 I
16 VIII 50 VIII 76 VIII 107 VIII dots of
17 Vi 51 VII 77 VI 108 VIII VII
18 VI 52 VII 78 VI 109 VI 139 VII
19 VIII 53 VIII 79 I 110 VI 140 VI
20 VII 54 VI narrow Wadi WAL 141 VI
21 VII 55 I 80 VI 12 VIII 142 VIII
22 VIII very 81 VI 13 VI 143 VIII
23 VIII narrow | 82 VIII 114 VIII near VL
near VI} 56 IX near VI} 115 VI 144 VI |
24 VII 57 I 83 VIII 116 VI 145 VIII |
25 VII very 84 VII 117 II near VI
26 VI narrow | 85 VI very 146 VIII
De VI 58 VIII 86 VI narrow 147 VI
28 VI near VI} 87 VIII 118 == 148 VIII
4 VII 59 VIII 88 VII 119 VII 149 Vali
29 VIII 60 VIII 89 VI 120s ee Vall 150 VI
30 VIII 61 VIII 90 VIII 121 VI 151 VI
31 VI 62 VIII 91 VII 122 VI 152 VI
32 VIII iVII 92 VII
4 VII
27—2
“ar
TABLE OF THE GAUSSIAN “TAIL” FUNCTIONS;
WHEN THE “TAIL” IS LARGER THAN THE BODY.
By ALICE LEE, DSc.
In a paper published in Biometrika, Vol. v1. pp. 59—68, tables for the’ in-
complete normal moment functions were printed, and they have since been
reproduced in Tables for Statisticians and Biometricians recently issued from the
Cambridge University Press. From these tables values of the Gaussian “Tail”
functions were deduced and a short table of yy, and yy, appeared in Biometrika,
Vol. vi. p. 68. The value of these functions being demonstrated in practice
during the last few years, a more complete table of yh, W., Wy; has appeared in the
Tables for Statisticians and Biometricians.
In the introduction to those tables, however, Professor Pearson indicated that
it was important to have a similar table when the “tail” forms more than half the
entire curve, and gave the fundamental formulae for obtaining the numerical
values of the functions. The present table has been calculated to supply the
want thus indicated.
10
ff
y
Li
E B oO H Cc
Let the figure represent a Gaussian curve of total population WV and standard
deviation o. Let AB be the ordinate at which it is truncated and let
OB = haxaor
Let GH be the ordinate through the mean G of the truncated portion and BH =d,
the distance of the mean from the line of truncation, let } be the standard
deviation of the truncated portion about GH, and n=the area of the truncated
portion, or of the population observed. Then if any material be supposed to
ALICE LEE 209
form a truncated portion of a normal curve, d, n and = can be found (see Tables,
pp. xxvii and 25).
We have Apt ORY Sccaids oe str gh ssa oj Sow ee din «ies son (i),
Arn 10) pte sc rmas enee nice Se ncnen tageaaGs Antes (11),
Aira OV Ute cee cranes Gacis cuties aihsielstas vas b,c (iil)
These are tabled for each value of h’, at first proceeding by (01 and then by ‘10 as
unit. Now y, being known we find h’ from the table, and hence deduce yy, and
3. 2 gives us the value of o from known d. Hence h=h’ xo can be found,
lastly (111) gives us the total population from which n is drawn. Thus the constants
NV, o and h which fix the total Gaussian are determined.
It will be sufficient to illustrate the method of using the tables on certain data
as to the English thigh-bone, recently published by Parsons*.
Dwightt+ has adopted a method of sexing human femora on the basis of a
markedly bimodal distribution obtained by him for American bones. He terms
female any femur with diameter of head less than 45 mm., and male any femur
with diameter of head over 47 mm. Parsons follows this rule and sexes by other
points femora with heads from 45 to 47. As unsettled remainder he has 20 femora
of 45 mm. and he gives 12 to ? and 8 to #3; of 46 mm. and 47 mm. he has 41
femora and he gives 4 to ? and 37 to f. As a result of this process he obtains
a female frequency curve which rises very abruptly at high values of the diameter,
and a male frequency curve which rises very abruptly for low values of the femora.
But, if there really be any marked skewness in frequency of the parts of the
human skeleton, which is very unusual, we should anticipate that it would be of
the same sense. Parsons’ distributions are as follows (loc. cit. p. 256):
50 | 51 | 52
|
36 | 87 | 38 | 39 | 40 | 41 | 42| 48 | 44.| 45 | 46 | 47 | 48 | 49
1 1/—| 3 |] 8 |] 14; 12) 18 | 12] 12} 3
8 | 8
a 1 can
99) | 1741 Si || 19 i 10
bo
The ¢ 48 mm. femur according to.the rule should have been treated as a
male but presumably it had marked female characters. Were there no marked
male characters in any bone below 45 mm.? It will be seen that there is a
remarkable dip in the total material at 46 mm. which corresponds to Dwight’s
division. In material measured six years ago in the Biometric Laboratory, where
every bone in a relatively large series was measured, no such dip occurs and there
is in those data no justification for Dwight’s method of sexing. The group of
29 § bones at 47 mm. and the sudden cut off at 45 mm. seems to condemn this
method of sexing, at “ay rate from the statistical standpoint.
* Journal of Anatomy and Physiology, Vol. xiv. pp. 238—267.
t+ American Journal of Anatomy, Vol. tv. p. 19.
+ This material has been statistically reduced and will shortly be published.
210 Table of the Gaussian “ Tail” Functions
Without arguing this point out here, we may illustrate the use of the Table
(p. 214) of w’s by taking two of Parsons’ frequency distributions for females; we
will cut them off at the points suggested, and then investigate the total popu-
lations of females which result. Our author pools for these distributions right
and left bones.
Taking the diameter of “head of femur” for the females, we have
| |
| Diameter in mm. ... | 36 | 37'| 38 | 39 | 40 | 41 | 42) 43) gL
| |
Frequency ... sohe |Past 1/—]} 3] 8 | 14 12 | 18 | 12
These are exactly the bones the Dwight process gives as female. We find
>? = 28851,
d =2°6159 (measured from 44°5).
Hence wy = S2/d? = 4216.
Whence by interpolation from the table
h’ ='782, W,='864, W;= 1-278,
leading to o = 2:260, he edGn,
and Mean = 42°73 mm., NV =88-2.
Parsons gives for R. femur, Mean = 43,
L. femur, Mean = 42,
and the total number of bones dealt with 55 + 48 = 1083 (Tables, loc. cit. pp. 249—
251). In his frequency distribution (p. 256) he only records 85 female bones,
which give a mean of 42°54 and a standard deviation of 2078 mm. These values
are clearly not widely divergent from those we have found above by supposing
all bones under 45 to be female.
To test the matter further the 105* female bones of which the head was
measured by Parsons were taken out. They provide the distribution :
how |
| Diameter in mm. ... | 36 | 37 | 388 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 |
Sil
Frequency... ... | 1 | 1 | 4 | 109)18' | eulPotalarsiie4 3) == leer
These give Mean = 42°47 .
3 Dis 1-996; = 105,
* It is not possible to say whether he has omitted two queried measurements. He has not omitted
bones he queries in breadth of lower articulation.
|
|
| Frequency
ALICE LEE 211
Cutting off all bones over 45°5 we find
>? = 26392, d=2-6264,
leading to vr, = 3826.
Hence bh’ ='984, WW. ="783 and y,= 1195.
These provide for the non-truncated population,
Mean 42°48, s.D.=2:056, N=104,
which are in still better agreement with Parsons’ constants for the 105 bones than
the constants for the 85 bones were for their series. It would appear therefore that,
if we suppose all bones under 45 female and use our Tables, we get results in
reasonable accordance with Parsons’, and possibly by a theoretically more justifiable
method than endeavouring to sex the bones above 44 and below 48 from other
characters.
We have considered from the same aspect the character breadth of lower
articular end of femur. Parsons’ distribution of 89 female femora is as follows
(p. 257):
i | |
q . ; Ay SPL yee ya ae llie a wh | yw Prpiliay re
| Breadth in mm. ... | 60 | 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 69 | 70 | 71 | 72 | 73 | 74
| | | |
|
a : SeEeat
In this Table he has only one bone in excess of the numbers on which he
bases his means on pp. 250—1. If we truncate at 69'5, Le. reject all bones over
69 mm., we find
2?=3'9803, d=2:9058,
and pr, = "4714.
Hence we deduce
h’ =5295, We= 977, W,=1426.
These lead to
h=1:503, o=2:839, N=984, Mean=68-00 mm.
The actual values given by Parsons’ distribution above are
o=2571, N=89, Mean=67'54 mm.
Thus the agreement is not nearly so good as for the diameter of the head ot
the femur, being about 10°/, wrong ino and NV. It should give as good a result
if the method were quite satisfactory, for the bones have been sexed by the
diameter of the head, and the limit 44 mm. for diameter of the head corresponds
fairly closely to 69 mm. for the breadth of lower articulation.
As this paper is not intended as a discussion of Parsons’ data, to which we
hope again to return, we will only deal with one more illustration of the use of
212 Table of the Gaussian “ Tail” Functions
the Table. We take out from his Tables, pp. 244—248, the diameter of head of
femur for 174 male bones.
ee
Diameter of Head in mm. ... | 45 | 46 | 47 | 48 | 49 50 | 51 | 52 | 58 | 54) 55
=: Teas (sae
Frequency ... pb pee ||) c 33 | 17 38 | 20 18; 12; 6 | 8] 3
|
The constants of this distribution are
Mean=49'14, o=2377, N=174.
Truncating at 47°5 we have
>? = 3'4341, d=2°7869,
whence yw, = 4422, and from the Table
h'='679, W.=°908, wW,=1°331.
These lead to h=NT132
and Mean = 49°22, o=2°530, N=162,
Le. to a “tail” of 40 not one of 52 below 47°5. Actually this tail distributes itself
as follows:
Onder 45 45 46 Av
Gaussian tail 5 6 12 17
Against Parsons’ 0) 9 10 33
This confirms our previously expressed view that probably a considerable
number of the bones classed as 47 mm. are really female femora, and that the
male distribution runs considerably beyond 45 mm. into the range treated as
purely female.
Finally let us try the result of pooling male and female bones and breaking up
the composite frequency by the method of Phil. Trans. Vol. 185 A, p. 84.
We have now 279 bones distributed as follows:
ane ; | | ie |
HER UES ser 36| 37 | 38| 39 | 40 | 41 ca 43 | 44| 45 ale 47 48| 49 |50| 51| 52| 53| 54| 65) Total
ea ke
oe =
Frequency {1} 1]—/ 4 [10 Bee) 13/33/18| 3820/18/12! 6 | g|3| 279
|
The constants are Be = 46°63, s.D.=3°93,
by = 15°4040, bs =— 67791,
= 5415162, ws = — 5380°5339.
The nonic is
gq.’ — 596189," + '0689¢.5 + 9°83579q. — 3:°4275q.' — 8: 2041¢,? — 30209,"
+ :0144g, - 0097 = 0,
giving the root q,=— 934 and p,=— 9°34,
ALICE LEE 213
and ultimately the two components :
Male Female
Mean ‘ 2 : : f 49°83 43°72
Population : ; ’ 133°25 145°75
Standard Deviation . d : 2B 2°662
Max. Ordinate . . : : 23°83 21°84
While the means agree roughly with those obtained by Parsons’ sexing (49 and
43), we see that this analysis much more nearly equalises the number of male and
female bones, and indeed makes the female population rather larger than the male,
while Parsons has 79 °/, more males. The “truncated tail” method would probably
give results in better accordance with the present had we not truncated at the
quite arbitrary Dwight-Parsons’ divisions.
These examples may suffice to illustrate the application of the Tables to
anthropometric measurements on man, where we can feel fairly confident that
the material, if sufficient in quantity, would be adequately described by a Gaussian
or normal distribution. Such cases may arise when material for the two sexes,
or for two races, is commingled and we can be fairly certain that one or other or
both “ tails” of the material present homogeneous parts of the mixture,
Another illustration drawn from Galton’s data for American trotters will be
found in the Tables for Statisticians, p. xxvi. The chief weakness of the method,
besides the assumption of the Gaussian, often quite legitimate, is the absence as
yet of the values of the probable errors, which values must be very considerable
for slender material such as that used above.
See following page for Table of Gaussian “Tail” Functions.
Biometrika x 28
214 Table of the Gaussian “ Tail” Functions
Table of Gaussian “ Tail” Functions, “ Tail” larger than “ Body.”
W aT (—) Ay Yo (—) Ape Ps (-) Ay h’
F OF aDe “f)
00 | BS |. 008 le Bee coogi mh 200) (| oer samme
OL |) 860" 5 Oa aly 248 algae oan imn capes 01
03 | °567 2 | 1-949 1-969 02
002 006 : O15
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0, |) -5e4".| 22 | eBay Ia te len Osea one 04
oo | bea" (|) 7Pe ly eebi emcee ell wipe ual sae 05
06 || 2560, La lonm |) 1210 ae eal mtn weeny 06
O7 i558 | oa || aie coer ee ay OV
08: | 55m eee || q-bosmiaw 1880 | | 08
09 | °BBB sae 1203)741), oan el E86 a 09
“1 Cie te ate SIS? «lie 1-352 | “1
2 535 oe 1143 As 1°726 Was 2
3 DIG W/aore (C1000) | pee, eee eee 3
4 497 | Cra |, 1040 |, poh aki eae 4
of Avs d . 0
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cioaadhe can re ae eves ee 6
7 BE ae 899) || cocce || SteBLOnel uae 7
9 oy, 4 “QR “Oo é
2) | | S| Se | Ge | |
1-0 380 | ‘O19 17 eee rises Le kg
11 Br | elk FAO) | Ooh a alta a eee alate
12 949 | (OTR S Srod yl 088) Siciso We en are
E ‘018 ss 71 033 1-107 023 ry
: i a | ae He Ph | acy PO | ‘4s
15 88 | Ol? “10” || 080 S| jo7a 4) ClO,
ze ‘O17 : 028 014
ane eu 016 Be cin eae I nin |) Le
v7 Doe ee Ce eerie) ep aces. ||. 27
fe 4 4 “—° 6
| 32] oe | S| Bs | tee | ae |
2-0 =O alee 487 | 022 | 1-993 | 006 | 96
2] ay ee 466 | 020 | j.o18 | 095 4 947
a2 |--1g4 | Ols aap | LO i014 |e eles
2-3 We alee A009) || 02085 alone adores
ee eae ‘O11 as ‘017 1-008 :003 a
oe 151 | 010 397 | 18 | ioe | 02 | a4
Pye ae 010 ae ‘O15 ee 002 i
26 Laie ieee ref tds | ee | oe
"7 Teg | so8 369. | ota. | HROOBA IE ope alnne dy
2°8 124 | - cos A eis | ean (Ree
2-9 SECs | eae eee || leony | ear fe
3-0 109 333 1-001 3-0
See Zables for Statisticians and Biometricians, Introduction, p. xxvii.
CONTRIBUTION TO A STATISTICAL STUDY
OF THE CRUCIFERA.
VARIATION IN THE FLOWERS OF ZLEPIDIUM DRABA LINNAUS.
By JAMES J. SIMPSON, M.A., D.Sc.
CONTENTS.
Pact
I. Introductory : ; : : d ‘ : : : . 215
II. Botanical . , : : : : : : : ; . 217
1. Specific Gharatters. : ; : : : : ; 5 aly
2. Morphology of the Flower. ; . : : : 2 218
3. Conception of Chorisis . ; : : : : : . 219
4. Orientation of the Flower. : : ; : : . 220
III. Examination of the Data . : F : ‘ : ; ; . 222
1. Classification . : : : F : : : : . 222
2. Analysis . 6 : : ‘ F , ‘ : ; . 223
IV. Statistical . ; : : . 242
1. Study of the cane and Semidand Deviations : ; . 242
2. Study of the Correlation Coefficients . : : . . 245
V. Morphological Significance of the Statistical Results : ; . 251
VI. Variation in the Gynecium : : F : , . 255
VIL Suggestions for Future Studies in ine Plane 257
I. Inrropuctory*.
In the summer of 1905 Professor J. W. H. Trail drew my attention to an
extraordinary example of variation which occurred in the several organs of the
flowers of Lepidium Draba Linneus. At that time I examined in detail 1832
individual flowers taken from a single plant growing in a piece of uncultivated
* Tam pleased to have this opportunity of expressing my great indebtedness to Dr J. F. Tocher for
invaluable assistance in the biometric part of this paper. The correlation and other constants were
calculated in his laboratory, and without his assistance the publication of this paper would have been
greatly delayed. I must also thank Professor Karl Pearson, in whose department in University College,
London, the statistical study was originally undertaken, for reviewing this paper for publication and also
for much kindly criticism and advice. To Professor Trail my thanks are also due for many botanical
hints.
28—2
216 = =©Contribution to a Statistical Study of the Crucifere
ground in his garden, and the results of these observations form the basis of the
present contribution to the study of the variation in the Crucifere.
Botanical problems, which have been hitherto attacked from the biometric
standpoint, have been comparatively easily handled, because the material has
been more or less homogeneous in character. For example, variations in the
number of sepals of Anemone nemorosa* or in the number of ray-florets of
Chrysanthemum leucanthemuwm+, and the consequent distribution of these are
capable of direct treatment by Pearson’s well-known method of fitting frequency
curves,
The only work comparable to the one in hand occurs in Biometrika, Vol. It.
p. 145 (Variation and Correlation in the Lesser Celandine), but in this case the
numbers of members in the calyx, corolla and androecium have been examined
as a basis for a study of homotypic correlation and in this flower each of these
organs consists of a single constituent with numerous members.
The problems studied in this paper, however, are more complex inasmuch as
they deal not with one organ of the flower but with all the organs, their con-
stituents and members both separately and collectively.
It is also, I believe, the first biometric work of its kind on a cruciferous flower
and embodies a study of chorisis, that is, “the splitting up or division of one or
more components of a flower into two or more equal or unequal parts ”—a factor
which is supposed to have been of the utmost importance in the evolution of the
natural order—Cruciferee. A complete discussion of this phenomenon is reserved
until the flower is studied in detail.
It would be well here to emphasise the fact that the flowers examined for this
study were not taken from different plants but, on the contrary, were obtained
from several inflorescences growing on stems which had arisen from buds on the
roots of a single parent plant. This mode of reproduction is rather unusual, but,
in the present instance, is of particular interest imasmuch as it gives greater
homogeneity to the material.
The parts of the flower which have been considered are (a) the perianth, which
consists of (1) the calyx and (2) the corolla, (b) the andrcecium and (c) the
gynecium.
The functional differentiation of these organs is of great importance in the
interpretation of results so that it might be well to recall the particular réles
which these play in plant economics.
The gynecium and the andreecium are respectively the female and male organs
of reproduction and consist of carpels and stamens, while the perianth forms a
protective covering for these delicate structures. The calyx or outer organ of
* Yule, Biometrika, Vol. 1. p. 307.
+ Biometrika, Vol. 11. p. 309 et seq.
J. J. SIMPSON PANE
the perianth is concerned solely in the protection of the flower in the bud, but
the corolla, in the open flower, also serves, along with the honey-secreting sacs at
the base of the stamens, as an attraction for insects.
The characters which have been taken as a basis for this study are numerical,
eg. the number of petals in the corolla, but no measurable characters, e.g. the
length and breadth of the petals, have been considered, although, as will be pointed
out later in connection with possible future studies in this flower, these characters
might also with advantage be taken.
The Crucifere, as an order, are usually regarded by botanists as being very
definite in type and no observations have been recorded to show to what extent,
if any, deviation from the recognised botanical floral formula exists, so that the
main object of this paper was to determine the frequency of the variability of
the parts of the various organs and constituents, and also the degrees of correlation
existing between the organs themselves.
The mode of observation is worthy of remark, however, as it might well be
argued that if the flowers used for examination were fully “blown” deficiency in
the number of parts might be due to post-developmental fracture, but in all the
cases here recorded the observations were made on flowers in bud or only half open
so that the influence of wind or other external agency is altogether discounted.
The material was also examined microscopically in all cases so that there should
be no possible doubt as to the exact origin of any member. The importance of
this will be seen in the details of the analysis.
II. Bovranrcat.
1. Specific characters.
The generic and specific characters of Lepidiwm Draba may be obtained in any
complete systematic botanical work so that it is unnecessary to repeat them here,
but a few notes bearing especially on the study in hand may be of value.
It is a perennial about a foot in height and is covered by a minute down from
which its popular name, the hoary cress, is derived. The inflorescence is a raceme
not much lengthened and so forms a broad, almost flat, corymb-like termination.
The individual flowers are small, white and numerous. The constituents of calyx,
namely the sepals, are green; they are short, nearly equal and bear no pouch at
the base. The petals are small and white; they are equal in size, obovate,
undivided and generally stalked. The stamens are six in number; the filament
is simple, 1.e. it bears no appendages, and is shorter than the petals; the anther
consists of two roundish lobes. The pods are “ broader than long”; they are com-
pressed laterally at rght angles to the narrow partition. The thick valves are
boat-shaped and sharply keeled but not winged; each valve contains a single seed.
218 Contribution to a Statistical Study of the Cructferce
2. Morphology of the flower.
The typical flower consists of six whorls, made up in the following manner:
(a) Calyx 2.
(b) Corolla 1.
(c) Andreecium 2.
(d) Gynecium 1. (Plate I, fig. 1.)
(a) Calyx. This organ is composed of two whorls each consisting of two
sepals. The outer pair arise at one level on opposite sides of the flower and are
inserted on a slightly lower plane than the inner pair; they are parallel to the
plane of compression of the gynecium. The inner pair are also situated opposite
one another but in a plane perpendicular to that of the outer pair; they are thus
at right angles to the plane of compression of the gynecium. These whorls are
denoted on Plate I, fig. 1, by the Roman numerals I and IT respectively.
(b) Corolla. This organ consists of four petals all inserted at one level and
alternating with the position of the sepals; they thus constitute a single whorl.
(See III, Plate I, fig. 1.)
(c) Andrecium. Six stamens form the andreecium ; they arise at two different
levels and thus constitute two separate whorls. The outer whorl, which is lower
down, consists of two stamens which are shorter than the others and corre-
spond in position to the inner sepals. The inner whorl consists of four stamens,
arranged in pairs which correspond in position to the outer sepals. (A reference
to the figure (Plate I, fig. 1) in which the two whorls are marked IV and V
respectively will make this clear.)
(d) Gyneciwm. This organ consists of two carpels forming the sixth or
innermost whorl. (See VI, Plate I, fig. 1.)
It will be seen from the foregoing description that the order of the six whorls
here detailed is that in which they would be found were we to strip the flower of
its components at the different levels consecutively from below upwards. It is
also the order in which we would find them, passing from the outside to the
centre, were we to cut a transverse section through the flower.
Another point, however, which is not so obvious but one which has special
interest in our study, is the fact that this is also the order in time of development.
The actual sequence in which these constituents of the flower appear in the
bud is therefore :
I. Outer whorl of Calyx (Sepals).
II. Inner whorl of Calyx (Sepals).
III. Corolla (Petals).
IV. Outer whorl of Andreecium (Stamens).
V. Inner whorl of Andrcecium (Stamens).
VI. Gynzcium (Carpels).
J. J. SIMPSON 219
3. Conception of Chorisis.
Chorisis or reduplication is generally looked upon by botanists as a means of
multiplication of the parts of a flower. It consists in the division or splitting
of an organ in the course of its development by which two or more organs are
produced in place of one. Chorisis may take place in two ways:
(1) transversely—when the increased parts are placed one before the other,
that is, the resulting components are on the same radius ; this is known as vertical,
parallel or transverse chorisis ;
(2) collaterally—when the increased parts stand side by side, that is, on the
same circumference.
Transverse chorisis is supposed to be of frequent occurrence ; thus the pistils
of Lychnis and many other caryophyllaceous plants exhibit a small scale on the
inner surface at the point where the limb of the petal is united to the claw. The
formation of these scales is supposed by many to be due to the chorisis or unlining
of an inner portion of the petal from the outer.
Collateral chorisis is seen in different natural orders. In Strephanthus, in place
of two stamens there is sometimes a single filament forked at the top and each
division bears an anther. This is usually supposed to be due to collateral chorisis
arrested in its progress.
The flowers of the Fumitory are also generally considered to afford another
example of this type of chorisis. In these we have two sepals, four petals in two
rows and six stamens, two of which are perfect and four more or less imperfect.
The latter are said to arise by collateral chorisis, one stamen being divided into
three parts.
Collateral chorisis may be compared, according to Bentley, to a compound leaf
which is composed of two or more distinct and similar parts.
Let us now consider chorisis in its bearing to the flower under consideration.
In the description of the morphology of the flower we noted that in the inner
whorl of the andrcecium there were four stamens arranged in pairs while in the
outer whorl there were only two stamens situated singly. Various opinions have
from time to time been advanced to explain this anomalous structure so that it
might be well to briefly review these. Of the andrcecium of the Cruciferze Oliver
says: “The two pairs of long stamens are generally thought to be due to chorisis
or the division in the course of development of single antero-posterior stamens.
Others have thought that the six glands represent abortive stamens and that these
with the six stamens make up a normal series of twelve in three whorls.”
De Candolle held the view that the stamens formed a single, originally
tetramerous whorl alternating with the petals in which the median members,
Le. the anterior and posterior, were cleft (chorised) in two. Since however the
lateral stamens are inserted lower down than the median stamens and are also,
220 Contribution to a Statistical Study of the Crucifere
as already pointed out, formed earlier in the bud, this view is clearly untenable.
Two whorls must be taken into consideration owing to the difference in the levels
of insertion, the single stamens being lower down. Kunth, Wydler, Chatin and
others regard these two whorls as typically four-membered (tetramerous), those of
the outer whorl corresponding in position to the sepals, those of the inner whorl
corresponding in position to the petals. To arrive at a typical cruciferous flower
from this, two stamens in the outer whorl abort, while the individuals of the two
pairs of the inner whorl come together. (Plate I, fig. 3.)
Others (Krause, Wretschko and Duchartre) regard the outer whorl as typically
dimerous (i.e. with two constituents) and the inner whorl as typically tetramerous
(i.e. four-membered).
The more modern view, however, regards both whorls as dimerous but the
inner one chorised collaterally thus giving the typical cruciferous flower.
The reasons put forward to support this theory are as follows:
(1) The upper long stamens are usually paired in the median line, also
sometimes coherent. Further, in place of one or both of the pairs, there occurs
sometimes a single stamen—a hint at reversion, or one or both pairs may be
replaced by three or more—a suggestion of further chorisis.
(2) In the earliest visible stage of development in the bud it may be seen
that each pair of stamens arises from a single wart-like projection and that division
is therefore a secondary result. This is not very easily demonstrable in the
Cruciferee but is more evident in a closely allied family, the Capparidacee.
Since the present study includes numerical variation in the different constituents
and positions of the andrcecium it will be interesting to note to what extent any
one of these theories is borne out by the variations in this flower.
4. Orientation of the flower.
Having defined the positions of the various stamens relative to one another,
in what is usually regarded as a normal cruciferous flower, let us now consider the
different possibilities when the flower is abaormal.
Suppose that one of the pairs of stamens of the inner whorl is represented by
a single stamen, that is, suppose that chorisis had not taken place. Now with
regard to the peduncle of the inflorescence this stamen might be placed in two
diametrically opposite positions, namely (1) it might be adjacent to the peduncle
(Plate I, fig. 4) or (2) it might be on the distal half of the flower with reference
to the peduncle (Plate I, fig. 5).
Two questions now arise, (1) do non-chorised stamens occur as frequently as
chorised stamens on the side of the flower next to the peduncle? or (2) do either
of these occur with greater frequency in this adjacent position ?
According to which of these questions is answered in the affirmative must we
conclude whether there is any connection or correlation between the proximity of
J. J. SIMPSON 221
the chorised stamens to the peduncle and chorisis. The former would suggest no
correlation, whereas the degree of correlation hinted at by the latter would depend
on the frequency of the occurrence.
We have so far considered only two possible positions, viz. a non-chorised
stamen adjacent to the peduncle, i.e. in the proximal half of the flower, and a
non-chorised stamen opposite to the peduncle (i.e. in the distal half of the flower
with reference to the peduncle), but the question naturally arises “ Are these the
only two possible relative positions which might occur?” Might the petiole not
twist so as to bring the hypothetie non-chorised stamen into any position varying
from 0° to 180° with reference to the original plane ?
Let us illustrate this by means of the Figure 6, Plate I.
Taking the position of the peduncle as our fixed point the non-chorised stamen
might occupy the “adjacent” position a or the “opposite” position al. A rotation
of the petiole, however, might cause this stamen to occupy any of the positions
marked a2, a3 or a4 or even any intermediate position between a and al on either
side of the vertical plane A—B, in the horizontal plane a, a4, a2, a3, al.
In a study of the variations in this flower, this is precisely what was found to
occur, i.e. the distribution was equal round a fixed point so that we are unable to
say whether there is any connection between the proximity of the non-chorised
stamen to the peduncle and chorisis or not.
But the full bearing of this consideration does not end here. The orientation
of the flower is of practical importance in fixing a basis on which to establish
a grouping of the different variations. Any analysis of the data is impossible
unless some definite part of the flower be agreed upon as a starting point.
Now we have seen that the position of the peduncle with respect to any
definite stamen does not require to be taken into consideration. Consequently
we may take either of the two stamens of the outer whorl, which correspond in
position to the outer sepals and which are “normally” non-chorised, as our fixed
point and call it 1; the stamen opposite, i.e. in the same whorl, we shall call 2;
the chorised pair of the inner whorl to the left (or in the floral diagram above)
may be termed 3 and 4; while the corresponding pair to the right (or in the floral
diagram below) would thus be 5 and 6 (Plate I, fig. 7).
Where variations occur in any of these stamens we shall hereafter refer to
those as occurring in “position” 1, 2, 3, 4 and 5, 6 respectively.
On this basis of symmetry, it will simplify matters considerably if we regard
as 1, in flowers in which either of the two outer stamens is modified, that one
which still maintains its original character while, on the other hand, if both are
modified, that one which retains the greatest approximation to normality, e.g. if
one be chorised while the other is not, the latter would be in position 1; or if one
Biometrika x 29
222
Contribution to a Statistical Study of the Crucifere
were chorised while the other was only partially chorised* the latter would again
be in position 1.
Following on this it is at once seen that where both are normal or where both
are equally abnormal it makes absolutely no difference which position we choose
as.
TEI:
1. Classification.
EXAMINATION OF THE DATA.
Considerable difficulty was experienced in classifying the variations owing to
these occurring in so many different forms yet with so few characteristics in
common as to warrant their inclusion in definite classes.
The total number of flowers examined was 1832, of which 1062 had the
accepted normal structure (see page 218). The remaining 770 showed variation
in different degrees of advance or regression, i.e. there was an excess or deficiency
in the number and structure of the members of the various organs. Thus we see
that there was a deviation from the accepted normal structure in over 42 per cent.
of the individuals examined.
The perianth has been selected as a basis for classification and Table A shows
the sub-divisions which have been adopted. Amongst those flowers in which the
TABLE A.
Number Number Number Number | Variations
of in in in in the
Variations Group Sub-class Class Class
Class I. Perianth normal hele — — — 1687 =
Sub-Class A. Gyneecium normal -— _ 1680 — a
Group (a). Andrecium normal 1 1062 — — —
Group (6). Andrecium abnormal 57 618 — — —
Sub-Class B. Gynzecium abnormal — -- 7 — =
Group (a). Gyneecium one carpel 2 4 — — =
Group (6). Gyneecium reduplicated 2 3 —_ -- 62
Class II. Perianth abnormal at ee be — — os 115 —
Sub-Class A. Calyx normal, corolla abnormal — — 55 — ~
Group (a). Gynzecium normal iil 54 - _ =
Group (0). Gyneecium a single carpel : 1 — =
Sub-Class B. Both calyx and corolla abnormal — — 60 — =
Group (a). Gynzecium normal : il 46 — — =
Group (b). Gyneecium a single carpel 6 14 — — 29
Totals 91 1802 1802 1802 91
* For the present we use the terms ‘‘chorised” and ‘‘chorisis” in the sense of the definition already
given.
J. J. SIMPSON 223
perianth was normal there were no fewer than 62 different types of variation, and
amongst those in which the perianth showed a departure from the accepted
normal structure there were 29 types of variation. Thus of 1802 flowers examined,
1062 had the typical cruciferous structure, 625 had the perianth normal but the
andreecium and gynecium modified in 62 different ways and 115 had all three
organs modified in 29 different types of variation.
The remaining 30 individuals are not capable of classification under the fore-
going scheme but have been grouped into three classes as shown in Table B.
TABLE B.
Number Number
of of individuals
| Variations | in the Class
Class III. Reduplication of parts but flowers not separate ... 10 11
Class IV. Reduplication of parts with flowers separate... 6 17
Class V. Part of a flower replaced by a flower see oe 2 2
Totals... se ae a ae aes oh 18 30
Altogether, therefore, there are five separate classes which give a total of 109
different modes of variation.
2. Analysis.
In the further reduction of the data it is essential that we consider the
variations in the stamens, and for this purpose we must naturally commence
with Class I, Sub-class A.
To avoid describing each of these in detail, it is necessary to have recourse to
a graphic method of representation. Several such methods suggested themselves
and although none are ideal we have chosen one which may help to give a true
impression of the various modifications assumed by the andrecium. We shall
also give a few examples by another method which might have been adopted but
which seems to us to be even more complicated.
Let us, in the first place, consider in what directions abnormalities have
occurred. A typical stamen consists of two parts, (1) the filament and (2) the
anther.
(1) Filament. This may be of its normal length or less than its normal
length or altogether absent.
(2) Anther. This may be present or absent.
But other complications arise. As already explained, in the accepted typical
cruciferous flower, chorisis has taken place in positions 3.4 and 5.6 so as to give
29—2
224 Contribution to a Statistical Study of the Crucifere
rise to two stamens in each of these positions. Now, we find that, in certain
flowers chorisis has only partially taken place and in others it has not occurred
at all so that we have thus another three possibilities to consider.
In describing the androecium, therefore, we must (1) define the position of
each stamen to which we refer, (2) state the nature of the filament, (3) note the
presence or absence of the anther and (4) emphasise the nature of the chorisis.
Let us use the following symbols 1, 4 and 0.
Filament, indicates that it is present and complete.
1 with reference to 4} Chorisis, indicates that it is total or complete.
Anther, indicates that it is present.
jeulaetent, indicates that it is only half-length.
4 with ref oe Suleesee se :
g Wim retenence to (Chorisis, indicates that it is ouly partial.
( Filament, indicates that it is absent.
Chorisis, indicates that it has not taken place.
Anther, indicates that 1t is absent.
0 with reference to
We have already fixed upon our nomenclature for the various positions; these
are 1; 2; 3.4; and 5.6. To avoid descriptions and at the same time give a
graphic representation of the floral formula of the andreecium the following
system might be adopted:
(1) Place the whole floral formula within square brackets thus [ _ ].
(2) Place positions 1; 2; 3.4; and 5.6 within curled brackets thus { }; and
(3) Place individuals, i.e. 1, 2, 3, 4,5 and 6, within rounded brackets thus (_ ).
Expanding this with reference to a normal flower we would have for the
andreecium only
[12} {2} {2)],
or still further in the order of Filament, Chorisis, Anther, Stamen
(1.071) G.0. 1) (1.1), Ge ea Gea i
Or, taking an actual example from our data :
Stamen number 1 is normal and complete, and there is no chorisis; stamen
number 2 has a filament only half-length but the anther is present and complete,
and there is no chorisis; stamen number 3 is normal and complete; stamen
number 4 is only half-length but with a complete anther—chorisis between
3 and 4 is complete; stamens 5 and 6 are only half the normal length but have
complete anthers—chorisis between 5 and 6 is complete.
This would be represented thus:
(ia.0.1) 0% DGD) iG eae a) Geel (a sea)
J. J. SIMPSON 225
Another graphic method and the one which we have adopted is as follows.
Each flower is represented in a table similar to the following:
Number of
Diagram Stamen Filament Chorisis Anther
Frequency
The first vertical column gives the frequency of the variation or the number
of individuals examined with this structure. The second vertical column gives
the number of the corresponding diagram in the plates. The third vertical
column gives the individual stamens in the positions already defined while the
other three columns denote the various factors to be considered. The different
possibilities of variation in these may be shown by the symbols 1, 4 and 0 as
already defined. It should be noted, however, that in positions 1 and 2 a dash
(—) will be placed in the chorisis column to indicate that these are typically
non-chorised stamens and that absence of chorisis does not therefore indicate
abnormality. Representing the same example as before, by this method, we
would have:
Frequency eee Stamen Filament | Chorisis Anther
1 1 — 1
2 s = 1
3 1 1 il
4 4 1 1
5 i 1 1
6 4 1 1
The following table shows graphically the types of variations illustrated in
Figs. I—LVIII, ie. Class 1, Sub-class A, flowers in which the perianth and
gynecium are both normal.
It will be seen that in the flowers illustrated in Figs. XLVIII—LVIII another
complication has crept in. Stamens 8, 4, 5 and 6 have themselves sometimes
undergone partial or total secondary chorisis. In the tables, therefore, by sub-
dividing the squares containing the details we can thus adhere to our initial
nomenclature. Let us take the three most difficult examples to illustrate this.
(1) Fig. XLVUI. The division corresponding to stamen 8 is sub-divided. This
would indicate that in this position there were actually two stamens, The nature
of each of these individual stamens is, as before, given in the sub-divisions. In
Contribution to a Statistical Study of the Cruciferae
226
TABLE C.
ToYUY
Free | fmm at Ct |
ad et ae reed Fl me
Ls te ie en
Saenn es
aoe
ant Ono
meOnsa AS
SISTIOYS) | |=so0 | |aaae | |aHos | |azunoo | |oooo | loess lelisisters | |aaae
qUoULe ll Lo ee ee SS indin Lo ee ee eon >) SS itn iiaen lien linia! Seas HO Ss OHNO aie lee |
Lecleeueatel
mA Hd mA OH 1d mn Hud 6 mG O Huw 6 mM Hid 6c ra Go SH 1d 6O mA Hid © ram OD Hud 6
jo qoquinyy
Q, om lal >
—
Weisel iad ES LA I > HH bd ba
jo soquany = 4 x be > ba Ss
al a
)
Aouonboay oO oo t 00 o + a mo
+
JoyJay on aes HO aaa aeaee eS aoa tn oe oe ce ee aor OnO aaa
SISILOYD) | | Aa didn | poaeae | | HHOO | paaee | CS) j |oooo | ES solo Jaane
JUOTAC TL [eee ee oe ee IN See HHO IHN SS A HOHNO ts en I oe a ono mame OrO Sd a oS
uatey}9 |
mM & aH 1d 6 mao Hid mao Hud © mA +H 12d mA Hid © mA) 19 mG oO stm 6 HAN OD Hd
JO TOq UIT NT
Q, - a lam am 4
weASeIC, es = - = i > =
jo qoquinyy | > 4 bd oA a rs
, MN bd
I~
Aouenbea a = a =i oO = = =)
nN N Lal
ToyJay fo al es ft mee oe ee ee niles edie ieee! lito oon oe aoe HO Ses Me ee oe el eer Onn Cai en ie ie ie!
eco meee eo oe ioe) feos ihe) iesees| Gisceen pees
eee pon ae a SHIN aan a i ai Saeed CO SS Hin See Odea me AAAS
UsTLeYS z ie a -
jo roquinyy
mG oD st im 6
mG. OD StH 1d
maa oO Hid ©
TBIsRIC
jo raquinyy
I
VII
XVI
XIX |
XXII
fouonbed,y
1062
38
2
2
J. J. SIMPSON
TABLE C—(continued).
Sees HO
SeOnsa eS
Ses SSeS
woyqUy ee See aS a See ee
|
SISIOyD | |e Hoo | | aaa | |paunoo | | aaa | |auHoo | | aaae | | aaa | |aaeeaee
QUOTAVT TY SANA RS mS eS in re HN Hae © Rae SS SHINAI RI O qeHOn saan HNO A SaaS KH eae eR Re
uoUIeIS “ é é ,
rma GN oD Hud € ma oD Hun moIAN oO Hid mA OO Hd rma OO Hud © mA OD Him 6o mW O Hid oN MO HTH OO ©
jo OQ TUN AT
tH
en b om FS
ae 5 s IIIXXX IAXXX XIXXX . 4 ILIATX
Jo qoquiny “A 2 aah AA PN oS? tA tA ’
iN r d
A : _ 7 as :
Aouonbaty ar) 4 ry cor) a 4 a 09
wayyy
foe ee oes ee Biel oe
ae Aaa
Se HOHO
ae
sone sO
SISLIOYO
| Jeaae
| |auoo
Jama
1S...
| [aaa
| joa
| Jaane
| |nnoe
quowUe [Ly
mS ina
aR a AIN©
mt eS AAA
ry AIAAING HINO
INGA Re HINAaI|oa
Ss On eee
me OO se AAR
AIO aS eS RINO
UdTUR]S
JO TOQTUAN NY
mG) SH id 6O
NN 1) Hid ©
NOD SH 10 0
my GI) SH 1d ©
mG OD Hud ©
mA OO Hid
aN Hd ©
ITXXX
JIIAXXX
XLIV
XLVII
a ee bd
WIeISVICT a iS
jo aaquam Ss ~
“ “
Aouenbaa yy a on
|
BHOnaae
3}
es Onr HO
TIQIUY Sonia ieee tiene ie! see HO Sr Sena HO ee | mOn AS
SISLIOUY) |oaae | [| aHooO | HH0O | | HHOoO laa | jaa | Hn oO | | HHoO ad
f
JUOUe TL IAS Ss in sda HO ta ic oS a AladiaO RIN SICIR a in re OnrnaH HOnm RANA IO eA AHO
WEUIey}G 9 69 +H 1m 6 Ho Hid ©
jo zaqmny mao 10 co mA Hid © mA Oo HD mA OD Hud mM” oO} Hid 6c MN oD Hid ce Seno 1 on
|
tH = 4
urease iS 4 Sy = >
jo aaqman yy 4 TITAXX “4 INGIPXOXEXG TIAXXX tA nS =
ms 4 ~ ey
Aouonbary ~ rt 0 ar) na 9 a) 4
ferce
t
Contribution to a Statistical Study of the Crue
228
TABLE C—(continued).
loyyuy
talib) hl!
tt tl ia tl tela
SISIIOYO | |aroor | |SSseoe
JVUOUTAe TT Cs es ee Be oe Cn is oes ee ee
ueureyg
aa st 10 aN tH 1D ©
jo qaquinn
|
WRIseIg = =
jo qaquiny | A =|
|
Aouenbary. 10 4
Toyquy Loon lites Mien len en een ee itn Bien ee eee
SISTIOY | |aaenoO | | OOnaAs
QUOULe TTY I See ae
TOUIeyG
JO Toq Un Ny
UIB.LORICT 3 =
jo zaqunyy 5
Aouonboty 9 a
TIGA Lelie en endl oe ee eel a
|
|
sismoyg | | | | Aa cmaieictmicn
|
JVUOULe LY See ea SS ea
uaTIeyS
JO JOQ wan py
UUIBLoRIT
jo roquinn
Aouonbar gy
TABLE C—(continued).
yay Sen One ot
SISTIOYL) | |oOORn AA | |pnoorn
JUOULC TL Sen Ona ROnnR ASS
Goutes aN ost 1D CO Pano st 10
N au
jo zoquinn
3 om o
UIRIOvICL s =
jo qoquiny ‘S) =
4
Aouonbea gy st a
Loyqay (nt oe en en ee en Le en A oe en ee
SISLIOY() | \Iesio.or | [pase
JFUOULVIL Lo on oon ts oe Co (eS oe en |
eats) mao st 19° 6 NOH 19 €
3 = ac al 0
jo coquinn im
UWIRISVI(T = =
o caquin am
| POR LOC LUNG 5
| Souonbory n 4
J. J. SIMPSON 229
this example what really occurs is: There are two individual stamens each equal
to the original length and bearing a complete anther and separated from one
another by secondary chorisis.
(2) Fig. L. Divisions 5 and 6 are sub-divided to show that there has been
secondary chorisis in both of these stamens. In the former, chorisis has been
complete but has resulted in one being full-length and with a functioning anther
while the other is only half-length with a functioning anther. In the latter,
chorisis has not been complete inasmuch as only the anther has been chorised.
(8) Fig. LIT. Divisions 5 and 6 are both sub-divided, consequently we may
infer that both of these stamens have undergone some stage of chorisis. In the
first column we see that all are full-length, in the third that all have functioning
anthers, but the second tells us that chorisis has been only partial in each case.
The symbol } between 5 and 6 indicates that between these two chorisis has also
been partial. Thus we conclude a state of affairs as follows: In position 5.6
(1) there arises a single filament which divides into two at some distance from
the base and (2) that each of these again sub-divides and (3) that on the end of
each of these four sub-filaments there arises a functioning anther. The others
may be worked out in a similar manner but a reference to the diagrams will at
once obviate any misrepresentation.
From the foregoing table and illustrations it is evident that further classi-
fication is possible but it would be well to point out here certain difficulties
which arise. As an example let us consider such a case as (using our original
terminology) that in which, in any of the positions (1; 2; 3.4; or 5.6), the stamens
are represented thus (1.1.0) (0.0.0), thus ($.1.1)($.1.1) or thus (4.0.1) (4.0.1).
Which shall have precedence? If we are to consider these variations as deviations
from the usually accepted normal cruciferous flower, then we may safely assume
that that flower which has the greatest number of functioning parts in a certain
position is less aberrant than one in which any or all of the parts are altogether
wanting; while, on the other hand, if in a position in which chorisis normally takes
place, we have defective groups like those in cases 2 and 3 cited above, in one of
which chorisis has taken place but not in the other, we must consider that group
in which chorisis has occurred as being the one less removed from normal. On
this basis then the above examples would be placed in the following order with
regard to normality :
GaGa yd. bl) (2). G.0.1)G.0. 1); (3) (1. 1.0)(0..0..0).
Similarly for any of the others.
Consequently we are now in a position to classify the actual cases under
observation.
So far we have considered only those flowers in which there was the typical
number of stamens, with their manifold variations in size and structure, but now
Biometrika x 30
230 Contribution to a Statistical Study of the Crucifere
we must classify those in which secondary chorisis has given rise to more than
the accepted number.
~
Let us take stamens 5 and 6 as our basis, ie. those individuals in which
position 5.6 is occupied by more than two stamens.
The relative frequencies of the different types of variation in the andreecium
in the 618 specimens so far considered (see Table C) are very interesting.
The number 1062 in Table F refers to 1062 flowers in which the andreecium was
TABLE D.
Number of
Variations in
Number of
Individuals in
: Sub- 9 Sub-
Section group Section group
Group a.
Whole of the andreecium normal. Fig. I — 1 — 1062
Group 6.
Andrecium variously modified ...
Sub-group a. 21 = 480
Outer whorl normal (5 and 6 variously modified) — — — =
Section 1.
Stamens 3 and 4 normal. Figs. [I—IX 8 — 419 —
Section 11.
Stamen 3 normal, 4 represented thus (4-1-1). Figs. X—XIV 5 _- 25 —
Section iii.
Stamens 3 and 4 thus {(1-0-1) (1-0-1)}. Figs. XV—XVIII ... 4 — 9 —
Section iv.
Stamens 3 and 4 thus {($-1-1) ($-1-1)}. Fig. XXII 1 — 2 —
Section v.
Stamens 3 and 4 replaced by one. Figs. XIX—XXI ... 3 == 25 —
| Sub-group £. 13 —_ 54
| Outer whorl represented thus {(1-—-1)(4-—-1)} —- — = =
| Section i.
Stamens 3 and 4 normal. Figs. XXJJJ—XXIX 7 _— 42 —
Section ii.
Stamens 3 and 4 thus {(1-1-1) (4-1-1)}. Figs. XX X—XXXIV 5 — 11 —
Section ili.
Stamens 3 and 4 thus {(§-0-1) (0-0-0)}. Fig. XXXV ... 1 — 1 =
Sub-group y.
Stamens 1 and 2 represented thus {($-—-1) ($-—-1)} ... — 4 = 16
Section 1.
Stamens 3 and 4 normal, Figs. XXXVI and XXXVII 2 — 11 —
Section ii.
Stamens 3 and 4 thus {(1-1-1) (4:1-1)}. Figs. XXX VIII and XX XIX 2 _ 5 =
Sub-group 6.
Stamen 1 normal, 2 absent ... — 5 = 29
Section i.
Stamens 3 and 4 normal. Figs. XL—XLIII 4 — 27 —
Section ii.
Stamen 3 normal, 4 thus (4-1-1). Fig. XLIV ... 1 — 2 —
Sub-group e.
Stamen 1 thus (4:—-1), 2 absent. Figs. XLV—XLVII — 3 — 6
J. J. SIMPSON
231
TABLE E.
Number of Number of
Variations in | Individuals in
Section eee Section oa
Sub-group 7.
Stamens 1 and 2 are normal — 9 — 30
Section i.
Stamens 3 and 4 are represented by three. Fig. XLVIII 1 = 3 —-
Section ii.
Stamens 3 and 4 are normal. Figs. XLIX—LI 3 —- 16 — |
Section iii. |
Stamens 3 and 4 represented thus {(1-1-1) ($-1-1)}. Fig. LIT 1 — 2 — |
Section iv. ; |
Stamens 3 and 4 thus {(1-0-1) (1-0-1)}. Figs. LITI—LV 3 = 5 —
Section v.
Stamens 3 and 4 represented by one. Fig. LVI 1 == d —
Sub-group 6. :
Stamens 1 and 2 thus {($-—-1) ($-—-]}}. Fig. LVII —- 1 — 1
Sub-group x.
Stamen 1 normal, 2 absent. Fig. LVIII —_ 1 — 2
TABLE F.
Frequencies more than 3 in order of magnitude.
(References have been made to the figures.)
Figure Frequency} Figure Frequency Figure Frequency
|
I 1062 XXVIII 11 XXV a
VIII 227 XXXVI 9 VII 6
III 130 VI 8 x 5
IV 38 XII 8 XLII 5
XX 21 XIII 8 LI 5
XL 18 XLIX 8 XVIII 4
XXII 15 Ix a LVI 4
normal.
Where variation occurs, the greatest frequency, namely 227, occurs in
flowers in which one of the pairs in the inner whorl is replaced by a single
stamen while the next highest frequency, namely 130, occurs in those flowers in
which partial chorisis has taken place in the inner whorl of the androecium.
Following this the magnitude of the frequencies diminishes rapidly. The next,
namely 38, occurs in flowers in which nearly all the parts of the androecium
are modified while, near this, is the frequency 21 which exists in flowers having
only one stamen in each position.
we find that stamens 1 and 2 are involved.
In the next two frequencies, namely 18 and 15,
30—2
232 = Contribution to a Statistical Study of the Cruciferce
From these raw data we can see that the inner whorl of the andrceecium is the
whorl most subject to variation and further that this variation is in the direction
of a decrease in number.
TABLE G.
Class I, Sub-class B: Perianth normal, Gyneciwm abnormal (see Table A).
saleseey Seales ae me 1S ob ede
et eS lie valle tes Willers eae Perse) Were eb |) ee tae il ae i) 3
5) os og 2 ne s a) Ons, lone 2 Z a
2) 2h |28| 8 | S| S72) 26 (48/8) 5] s
o| 82 (dela |S ) Sele ee ee Sake eee
| etal, a Bran sea |e
| 21) E
| 1 1 — 1 1 1 — 1
2 1 — 1 2 1 — i
‘ - 3 1 1 1 ‘a 3 1 1 1
¢ Pes 4 1 1 1 ; Ibs 4 1 1 1
| 5 1 0) 1 5 1 1 1
6 0) 0) 0 6 4 1 1
l 1 1 1 1 1 _— 1
ep tenis Ooa|ot )|s na
2 1 1 1 3 1 1 1
; 1 1 1 4 1 1 1
: SoM. Ona lpeil alee ee i. | lentil
1 SET 4 i 0 1 2 | LXITI 5 l 1 l
1 1 1 6 1 1 al
5 1 Lele 1 1 1
1 1 1
@ aa Oak a
Class II: Perianth abnormal.
Variations in the members of the perianth (calyx and corolla) have necessitated
the introduction of new symbols in the diagrams. These are shown in the com-
posite diagram Plate I, fig. 10, and are explained on p. 257.
It will be evident from Table H, p. 233, that the same type of variation in the
andreecium occurs with different types of variation in the perianth, e.g. in the
second and fifth figures no fewer than six different variations in the perianth
accompany a single type of variation in the andrcecium. Reference to the
diagrams in the plates will show what these variations are and will render a
detailed explanation unnecessary. The asterisk in Table H under LXXVIII
indicates that there has been adhesion between stamen 1 and one of the
stamens in position 8.4, in other words between one of the members of the
outer whorl and one of the members in the inner whorl.
Class ITI.
The members of this class are characterised by a reduplication of the various
organs but without separation into two distinct flowers. There are in all 11
individuals with 10 different types of variation. A word of explanation is
necessary with regard to the interpretation of the position of the various stamens
233
J. J. SIMPSON
TABLE H.
Class IT.
IaqQuy
SISTIOU/) | Jaane On saa Ornate | | HHOO | JaaaR
JUOUeT LA Bae i en ROn RA A qaIO eS SSS Se een eS me
U9TBYS
Jo Taqman Ny rae” O stud © rN tid © mA Hid © mG OD Hud 6 mM Oo tid 6
: 7 - 7 — 7 a ; at
iS ial =
4 | (ct i lool
El rbd | > i >
TWRASVL(T ie sis | rd S “4
O Joq Un Ny 5 od K | q q r
seats - re | bd < |
| 4 4 iS
Kouonbot gq 4 ao a 4 ar)
ee
TIQIAY BS SSS Sieh! HOnnHO CORR eA Ss SAHnROHO
SISLIOY() | Jara | Ce 4HH |onmoo | | aaa ae | |ooos
qFUOTAV TTY [on en es en ee HOnnHH BHOnHHO COOn nA HAH SseqeHOHO
UOUIeYS
mn Hid co mG oH 1d 6 ra Oo Hud aA SH WO HA OD 19 6)
Jo oquUINN
: = te
eaeten! > Sey = = 4
a lool lous Il > =!
TWRISeIT Matnlsl ote ea eal bo Fe be be
bd bad Atisiilets) bd sf (
joaoquny | KM OK SC I IOISIOn : Co bd iS
F etter | 4 3 PS bd PS bl LA rbd iS
et | 4 SoH ei 4 4
4 4 4
| | aa
Aouonbery pees eb ona re ~ ati ay
I3Yquy aA ODO tre emal et Tire) AON sae OO nan aaa HO
SISLLOY) | HarIND OOO | | HHOO One aA A | Jaa | | xHoO
Quogue TY re en) eee ee) oO AnAARaRA SCOn nse ile)
|
se tehesuc
BONES) aN MO H19 aA H 19 rN 0%) +H 10 6 mn St 19 0c mm OOD tH 19 ©
jo raquin yy
| = a
ey
=
acie . ig =
WeASVIy Ss Site wn BA >
jo qaquan yy 4 Hr Ss 2 le
4 S 4
S 4
Souenbeaq om OAH = 4 0
The stamens in the outer whorl are longer than those in the
Consequently if these were reduced in length they might easily be
mistaken for members of the outer whorl.
in these flowers.
inner whorl.
In all cases of difficulty, however, the
crucial test, the point of origin, was applied and the positions assigned to the
various members as shown in the diagrams were determined microscopically in
See Table I, p. 234.
this manner.
TABLE I.
Class IIT.
234
Contribution
to a Statistical Study of the Crucifere
ToyIUy Ls en A oe cn Be Be Le Ae oe Be Ln es oe ee ee ee
SISTLIOUL) aes Ae HnOO ANA Ar ae aes wae | ee! aa 4
JVUOULET TY E> ee De Le os Le | Sees As SS
UdTUeYS
jo qoquin jy La! nN fan) st awd we} La NO Ho Ne} aN SH 1d We)
tH HH
UIVISVICT > a 2
o Jaquin o ‘S)
JO 1oq TUN y ma
Aouonbo.ty = 4 -
| Toya y Ss ee oe Ln en es nn ce ne
SISTIOU/) Se Ane ARS AR OO Ae Ses en en en en ee oe ee
qVUOUMETTY Ca i A | tela feel Sool) ih Geli} abel
ue UTS N oD ~—s Ye) Ke) N ise) Q No}
rt 5 A) q | al
JO Taquin
WWIRISVICT ‘ws =
jo raqmmnyy o o
a
ouonbaty — 4
ToyyUaV Cn en Bc ne | oy P teil) tl atl hie! Cc ee ns ee |
STISILOUL() COO FRR AAR Ae COO mem OCC CO AF ae eqn ee
JFUOTAET LY Eo en On ne sear eo | ae sans me
UdUIvYS
0 roqmanyy La ae) st re) Ne} aN oD st Nes) Ne} Lan) GW OD sH 19 co
Ri h |
rn
areasely a a =
oO
jo taquinyy a a o
q
: i)
Aouonbaty 4 4 =
|
Toy ae A Be oe Be ne Do eo Leen en ec ee a
SISLIOU() SO ROR NN OR ee | ee De
qFUOULe LY Ee en ne ee ee ee ee Do eo Se Se ee RR ORR Ae
U9U’YS
jo Toquin yy Lal AN oO st New) Ne) = ie ine) SH ite) We}
= =
TIVISEL(T iS >
jo Jaqmnn S oO
oA a
ouenbaa yy a 4
TABLE J.
r
Class IV
J. J. SIMPSON
jo roqmunyy |
ma Hid
JIYJUV aad aes HO aan One
ssuoyg | | |arnan | | [ance | | loons
PUNTA HY AAR AARAAAy ae AHO Neo OSS
Lecslandental
mo GOD SH 1d © oA Hid © aA Hd ©
JO ToQuUInyy
rH
wIRLsRly > = =
; Ss ein Q
jo coquin yy oO ro) a
eS)
TIyqJUY SSS See (oomat (ree Leen Veer Leal poo)
UOTE TY oes As ee Se eee ANA aa
Ce ubS roa GI OD) SH 1 mA Hud QO stud ©
GY.) GN 6%) S ce oY)
jo Joquinyy
UIeISvIC, > = =
: ay > ease
jo zaquin yy Oo So >
ie)
Aouonbeay 4 Ne) =|
Toyyuy Sessa See eS ae ORO
SISLIOYY) | [aaa | |oorn | losers
qUuOULe TTY Len oon Ben Aloe el Cs in ie ee ae | aeHOnO
waIeyS
| mao Hud
wmeasvlqT a Eis ae
jo toquinyy | oe) oO is
JIYUV loealiaen ies aon iii Salil leet ae ee! ae ee
que] Lp
reStasoedqet i ed
Sasser
SS St
ume
48 mao Hid aN OD Hid © mM Oo Hud 6
jo rtoquinyy
ureaselcy = > =
jo raquin yy 5 > ON Es ‘
ouenbar yy Ney o ry
236 Contribution to a Statistical Study of the Orucifere
Class IV.
In this class there are 17 individuals giving six different modes of variation.
Reduplication has taken place to such an extent as to give rise to two separate
flowers on one pedicel. Each of the flowers was diminutive in size. Two tables
are thus necessary for each “ flower,’ A and B: see Table J, p. 235.
Class V.
This class has been formed to include two very aberrant flowers showing two
distinct variations. In both cases part of the flower has been replaced by another
flower, in one case normal in the other slightly divergent. CIX is one of these
in which the original flower is normal except that one of the carpels has been
replaced by a small flower (see Table K and diagram, Plate X). CX is the other.
In the original flower stamen 1 has been chorised and one of the chorised parts
has given origin to a separate flower (see Table K and diagram, Plate X).
TABLE K.
Class V.
o v o a Ric o os os a a
2 ££ | od q 5 ~S = 2 &p 2a g qe 3
S| eS es s s 5 ge | 8s| 3 | 5
A = Bn = c y TD a es Ss)
cap elena, Se ee hee Fe
1 1 me 1 1 1 aa 1
2 1 = 1 2 1 ae 1
eC 3 1 i 1 ; CIX 3 1 0 1
Z 4 1 1 1 B 4 0 0) 0
5 1 feta ear 5 1 0 i
6 1 Ik 2p Pak | 6 0) 0 0)
; 1 OA ahs al 1 1 25 1
1 (0) * 2 1 = 1
; en 3 1 1 1
a a ee 1 B 4 i 1 1
ay 1 1 1 5 1 1 1
1 | CX] 8 1 1 1 6 1 1 1
A
4 1 1 1
|
1 1 1
1 1 1
|
6 1 1 Weed
|
The asterisk denotes the position of the origin of the secondary flower.
This concludes our analysis of variations LIX to CX both as to perianth and
andreecium, but before proceeding to the statistical part it is desirable that certain
peculiarities should be observed and that an understanding be arrived at with
regard to the interpretation of these.
J. J. SIMPSON 237
In this procedure 44 variations, namely LIX to CII, must be dealt with.
When we study the number of parts which occur in the position of individual
members of a whorl and then try to draw conclusions as to normality or abnormality
of the whorl itself we find the following difficulties.
Let us take the outer whorl of the andrcecium as an example.
(1) If in the position normally occupied by stamen 1 there were two stamens
and in the position normally occupied by stamen 2, no stamen occurred, then with
regard to the whorl the total number of stamens would be two. Now this is the
accepted normal number of stamens in the outer whorl, so that if number alone
were considered the inference would legitimately be drawn from the table that the
whorl was normal. But this is not so!
Or (2) If in position number 1, one normal stamen occurred and in position
number 2, one functioning stamen, with the filament only half the normal length,
occurred, then the number of functioning stamens in the whorl would be two, Le.
the accepted normal number. But again, on the basis of number alone, we should
not be able to say whether the whorl as a whole was normal or abnormal.
Now as this state of affairs exists not only in the whorl under consideration
but in all the whorls of the flower, we have thought it not only advisable but
necessary to emphasise these abnormalities as a safeguard in the interest of
systematic statistical treatment.
For this purpose, therefore, small diagrammatic formule have been drawn up,
and these have been given in conjunction with the diagrams: see Plate I, figs.
11, 12.
We have already defined the positions of the various parts of the androecium
but have hitherto refrained from naming the different constituents of the perianth.
In the cases under consideration, however, it is necessary to do so, and Plate I,
fig. 11 illustrates how these are definitely determined.
The two outer sepals are named A and B (see Fig. 11). A corresponds in
position to stamens 3.4 and B to stamens 5.6. OC and D are the two inner
sepals ; C’ corresponds in position to stamen 1 and D to the stamen in position 2.
The petals are named A’, B’, C’ and D’ and lie respectively between sepals A and C,
Band D, A and D, and B and C.
The actual order of all the parts is summed up in Plate I, fig. 12 (1—14).
Biometrika x 31
Contribution to a Statistical Study of the Crucifere
TABLE L. Class V.
238
| ouenberq
WIeIsVlCy
jo qoqruny
LXI
LXX
[eulIouqy 410
[BULIO NY ‘PIO AK,
[OU M,
ul SA9Q (AO TAT
iS K
gx <=
ie] st
UL taquin yy
| uorqtsod yore
Sse KH SaaS So SN
Len en oe ee ll ee ee oe CS |
Eee | Seater oS a) a a he aL = mee on enna |
Se se Ree On eR NN
UL SaOq UOT
Aonenbet y oa
| f
7
UWIRLSVICT bd =
jo qaquim yy 4 a
4
|_Tetusoaqy 0 = SS = ~ x
| [BULION TIO A,
TLOCTEAN = an ©
worqisod yore
UL TaquanN
Aouenbaatyy
Se oe Be ee oe oe eS eS |
TURISVI]
jo qequiuy
N
“
loot
|
‘|
i S me }
le sR
a nN oD Nn wt
ee ce ee oe oe oe ee eC ey |
6
ee ee ee ee ee eS ey |
| |
[euLtouq Vy 10
[VULI0 NY ‘[LOT] A.
N
N
TEOMA
Ul STOq THOTT
n |
TACT |
2S ~ So |
< ase aes =
a nN + a =H |
uorqtsod yao
UL faq un jy
[ZOU
| jo raquioyy
Se ee ee |
Se ee AR eR NN ANS
[OU M
jo zaqumyy
II
IV
[i ce A oe oe Be ee eS
WQ OD HSA GA
Ese a a SES HO
FSO SROs Se
—| —! Ky
a FR & F&F &
239
J. J. Srmpson
TABLE L—(continued).
Souanbeat yy 4 | =) = 4
TURAISVICT = pes ee yw
jo gaqmany NOON TAX NEDEXAXGT EXSXOXGETI
[eultouqy 10 x = i ; . = pe es ae, 7 Py ‘ | : » > -
[VULIO N ‘TOT, = =i q | q | al Po) 1 NX | io | LT NX xq qq N x NX q Ss
Ou AN es] a fe a oD | a fe] a o a = nN oD a 10 | 4 a rex 4 ra
UL Seq WoT
uotyisod yore
UT TOquIn
a
Se ae Ss OO He HN | SS ese ROO HO NN HO Fa FOR HH HO MN sO Fe On OR FRO Ae
Aouonbaty | a = = 12
is CARNE es —__~ =e Ea = = Z
UIeLSPICT DA LA
jo aaqrany A A EN XEXGT IXXXT
J a |
. ab R iM -
|
[euLIouq Fy 10 , n _ ‘ » is K R N S
peumon jou | 4 Se ~ ras ~ x = x 41 x ~ OR
a ah ge
see — a = | |
| me rae |
TOU M. s AG er m
an ) > aA) N yD A ies io) © o) a] > G > >
UL S1oq Wd PT : r 2 = re ¢ = | ~ = F = Ei
i aaa at” = i es
uortsod yore
as tol GJ ie Se iil KO | rt rst St SO GUD OOO. | SO Fe FR Or Fee Ae SHO FR OCOORF BO NN
UT TOQUIN NT
Aouonboay = | 4 ™” a
5 : ~ ee -| = = : e _ c =
WRIsViI ~ ae
jo zaquiny ~“ | INTEXEXET TENCX SCAT XxXXT
= | —
[eulouq Vy 0 ‘ re pa. si = K fs r
remmon qaoyyy| 4 as a x i SO nee x ~ se, Sars ~ se aS
TOT AA, % an ee m=
UL SIOQ UII] 5 eS ve aa ~ 7a N = an Pu a a oD 4 + 4 er) al 4 Ney
uortsod yoro
Quts SSO OR RRO ORO AN lil Lot ee) hh (Re -—-O Sm FR On FO NN sO Fn RR On FO MN
UI JaquIn
[LOT AA. aS. See HO Se HO ee ee +H ae +
Jo zaqtueyy AIS SSP Silas ee WR OQ WRSQA WAN EO =~ SA FROR RIA Resse SR CIQ Sasa oN fis
TOU. = Se a
sas Sg Fa ba See NG OS A) eS A ee he ieee he on St oii age
31
v
fere
y of the Crue
Contribution to a Statistical Studi
°
TABLE L—(continued).
240
UL S1Aq WATT
Aouionbaay = 8 a o
eee AXXXT HIAXXX'1 S 5
jo toquin ai A
| eee a = x = 8 x x be x S} oS Pe = x hg
TOU AL ~ re an a n a Se = — ON ve) ds
|
UL Taq uIN N
uorytsod yore |
NO Ae Few COO NM
Se ae OOR Se SBR NN
OO8 Ss Ol Oi Oe eGliGl
|
AN Aw BANS ANN OW
UL SLoqure fT
Aouenbe.lyy a 4 o =
} ee | : —!
WRIST it paste S) =
Jo zaquan x. AIXXXT TIAXXX'T nA g
d
yeuLtouqy 10 es = seal ae ie we ee is ta;
came drem |) OS x = Ste. oa ‘ = { = q saad ~ & ‘ Sees
TLOmT AN ° ~t a a tH a - a Ney oD 4 a) a 12 oD 12
uorqtsod yore
ul Jaquinyy
IO) tls! eS Se INS NIN
Le re ee ee CS |
a0 Fe FONN FAN HO
Tc eee
AN FAR BRN NT NM
ouenbad yy 4 4 Ne) 4
UIBASRI(T owe wXY XIXXXT :
46 sequin TIIXXXT IAXXXT DOO S
s
ean 2 ee [ss ss oo ee
IOUA
TOU M, o <H a o nN ° = a a a nN || ca ~ o9 a +H
UI Sdoquey |
| uorztsod yore Sieiis: an HH OO HHO0O OO SNS) eS Sate het lie ep Seno Amb abr R Tey“ leNten
ur taquainn
[LOU AA ae oe co) a Oe Lae <HO eae +O
Jo toquiayy SS SSNS e316 MAS Set OS eee SS OR PRSQ WS coig WR OM WHSOQ HA 2S
TOU = is = S =
| HOU - Ee se eee ee | eee eee ase Se
JO Toquin Ny
241
J. J. SIMPSON
TABLE L—(continued).
Aouon boty - o
=
TRIS] >
jo gaquiny 2) S
~
= —— ——— — |
- |
eutou JO
: eS ~ 5 x A = x ~ =z |
[CULIO N “PLOT AA, >
Toy A
eu tH a Ve) st oe) fe] st Ne) a We)
UL Staq Ute JY
wortsod yore
url faqmunN
CE EE ee
AN FA ANAK ANNAN WH
Aouonba.ty oq 4
[aml h
UTRISVICT > “4
aa © S
oO qaquInN
J aN A a
RULIOUG W LO ‘S
U Lv Ssh Ee x Yo oS Sl Ss a eS se|
[BULION “PLOT AK
TOUAUN Uh ee = as |
UL SIOQUOyY z é |
9
5)
4
6
N ioe)
uorisod yora |
Ur JaquuN
eee SS...
BN AN ANRwe Fe OM
Aouonbeady
ureLseiq,
jo raquin yy
Aouonbedy = a
WIBLSRL(T is was
jo raquunuy A OX
[eULLoUq YW «0 . = ke
‘eumoy qroym | “ S 7 hem aS se x <a
[OGM : =
UL SIOquayy ae i = wa So Bch SS 2 Sh = Sh
| uortsod yowo | eerie a ee eh Rie cok asi ane era
|_ ur equim
| touAA eee 1 Saints <0
| Jo zaqmeyy SEES HONS NSIS ISVS cag SS SS re oars
HOUMA 4 = > 4 =
jo coquan yy Bi a = mF She 5 eae
[euadouqy 10
[BUOY “ETOU AA.
[PROG LANN
UL SIOq WOT
uortsod yore
UL TaquinN
Aouonbaay
N
A
A
A
A
i s (=) lan) pro)
See ANN CDOO0O FAN MAN
UIRISVL(T
jo zoquin
CI
[euLtouq Fy 10
[BUI N AOU A
ROTA
UL SOQ UIOTT
uortsod ToRe
UL JoquInN
ae NAM ANNA NN TS
Ou. ae Sones Ke)
jo daquieyy BS SEO Seeks
TZOUIM [Psat ts = eo iss
= =
JO TOQuan Ny
|
242 Contribution to a Statistical Study of the Crucifere
IV. STATISTICAL.
The analysis which we have given of 1813 flowers is sufficient to show that the
idea of a definite fixed number of sepals in the calyx, of petals in the corolla, of
stamens in the androecium or of carpels in the gyneecium of cruciferous plants is
not upheld by an examination of a large number of flowers of this species. In
less than 1 per cent. of the flowers examined there was an increase* or decrease
in the number of sepals in the calyx; in less than 1 per cent. there was also an
increase or decrease in the number of petals in the corolla, but in 2 per cent. there
was an increase in the number of stamens in the andreecium, while in 22 per cent.
there was a decrease in the number.
Since then the number of sepals, petals and stamens is not absolutely fixed for
any of the organs it becomes necessary now to consider whether the number of
members in one organ is related to the number in the others.
As has already been pointed out we have not only to consider organs as a
whole, but, in the case of the calyx and the andrcecium, the constituents of these
organs, owing to the fact that these organs are each divided into two separate
whorls which are inserted at different levels and are placed in directions at right
angles to one another.
Further, a special study has been made of the various positions in androecium
to ascertain to what extent bilateral symmetry may be regarded as an inherent
character of the flower under consideration.
By this means also it seems that some defimite information might be obtained
with regard to the perplexing and, at present, hypothetical theory of chorisis, the
reasons for the existence of which have been summarised on p. 219.
The statistical part has been divided into two sections:
(1) astudy of the Means and Standard Deviations, and
(2) a study of the Correlation Coefficients.
1. Study of the Means and Standard Deviations. —
Although it is obvious from the analysis of the data under consideration that
the numbers given for the botanical floral formula, namely, Calyx—4, Corolla—4,
Stamens—6 and Gynzcium—2, are the nearest integers, it is not at all certain
from a mere inspection of the tables whether the actual means deviate from this
number in the direction of excess or deficiency.
* Where chorisis of a sepal or petal has resulted in two or more distinct individuals we have regarded
each of these as a distinct sepal or petal in recording the numbers. This method is natural however
inasmuch as it is the only means by which we may possibly trace reduplication of parts.
J. J. SIMPSON 243
Consequently the mean and standard deviation for each of the organs and its
constituents have been calculated and these are given in the following table:
TABLE M.
Means and Standard Deviations of the Number of the Organs and their
Jonstituents.
es : ee es — ed oo
; : | Coefficient
Organ Constituent Member ae Sh cn of |
| ae Variation
Calyx ne. fe — -- 3°9796 vaya 67415
i Outer whorl — 1°9757 "1979 10°020
5 Inner whorl — 2°0039 1304 6°507
Corolla... igs — _ 3°9520 *3523 8914
Andrecium ... — -- | 5°8092 "7567 13025
Ss Outer whorl — | 199570 | :2704 13°817
“ = “Stamenl | 9840-1602 «=| «16-280
Fe —_ —Stamen 2 ‘9713 | ‘1915 19°715 |
Fs Inner whorl | — | Sane 76588 | 17077 |
: _ | Stamens 3.4 | 1:9950 ‘2728 | 13674
me == | Stamens 5.6 | 1°8627 | ‘4863 | 26°107
(1) The most obvious result which is revealed by these constants is the fact
that in all cases (except the inner whorl of calyx) the actual mean of the organs
is less than the recognised typical number, thus :
The mean number for the calyx is 3979 instead of 4.
The mean number for the corolla is 3°952 instead of 4,
The mean number for the andrecium is 5°809 instead of 6.
(2) The inner whorl of the calyx shows the smallest departure from the
accepted typical number, namely, 2:004 instead of 2.
Let us, however, test how far the differences in the character of the analogous
parts are significant by ascertaining the Probable Error of the difference of the
means of the characters.
TABLE N (1).
I. Constituents of the One
- 7~—
| Maan. stars
Constituent |} Number — Deviation
Outer whorl | 19757 | +1979
Inner whorl | 2:0039 "1304
|
The difference here is D='02813 and the a eratis error of the difference
ie = 0037; thus the value p= 76. The difference is therefore clearly
He D
m
significant.
244 Contribution to a Statistical Study of the Crucifere
It is worthy of notice that the outer whorl of the calyx is more variable than
the inner whorl and that it possesses on an average fewer sepals.
TABLE N (2).
Il. Members of the Outer Whorl of the
Andrecium.
Mean Standard
Number Deviation
Member
‘9718 “1915
|
|
|
Position 1
| Position 2
"9840 "1602 |
The difference here is D= ‘01268 and Ep =:00391; the value ——
is Di
Thus when the two positions of the outer whorl of the andrcecium are taken
into consideration, a probably significant difference is found between the means of
the distribution of the parts of this whorl. Now in position number 1 there is
a greater approach to the accepted type owing to the fact that when the com-
ponent of one of the positions of the outer whorl was found to depart from the
accepted type, the other position was selected as the starting point for the
orientation of the flower and was called position number 1. It is all the more
noteworthy that the deviation for position 2 is not in the direction of greater but
of lesser frequency and the variability of position 2 is greater. We have thus
again a reduction in the value of the type with greater variability.
TABLE N (3).
III. Members of the Inner Whort of the
Andrecium.
Mem Mean Standard
Cen Number Deviation
Position 3.4 ... | 1°9950 2728 |
| Position 5.6 ... | 1°8627 -4863
Here the ratio eee is nearly 15 and therefore there is quite a significant
= (m,—™M2)
difference between the means of the distributions of the two members of the inner
whorl of the andreecium. In both cases the tendency is towards a suppression of
functioning stamens rather than an increase, together with greater variability
in the case where the reduction from the accepted type is more marked.
This difference in variability is, in the main, real and is not due to the arbitrary
selection of the 3.4 position. This will be evident from a study of Table XIX.
It will there be seen that there were 1754 cases where two stamens occurred in
J. J. Srmpson 245
one of the positions of the inner whorl of the andreecium. This is the number
in the accepted type, and thus there is no variability. What is the nature of the
distribution of the stamens in the other position (Table XVIII)? It is as follows:
1 | 2 | 3 4 |
|
287 | 1436 | 26 | 5 | 1754
The mean for this array is 18569 stamens, with a variability of ‘4116. Thus
when there is no variability in one position of the inner whorl of the androecium
there is a large variability in the other position.
Similarly we find the following distribution for position 2 in the outer whorl
of the andreecium when position 1 is of the accepted type, 1.e. shows no variability.
57 1708 1 1766
The mean for this array is ‘9683 with a variability of 1784. Again therefore,
when there is no variability in position 1, there is a reduction of type in position 2
with great variability.
2. Study of the Correlation Coefficients.
For the purposes of this study a number of correlation tables have been
prepared and as the results of these will have to be considered under different
groupings it seems advisable to tabulate them, and insert them consecutively.
The system which has been adopted to facilitate reference is to commence with
the outer whorl of the calyx and consider all cts relations with the other whorls of
the flower passing from the outside inwards; following this comes the inner whorl
of the calyx and its relations with the other constituents of the flower from the
outside inwards and so on.
The following table shows the characters studied and the correlation coefficients
found.
In order to make the comparison of the various correlations as complete as
possible it will be necessary to consider each constituent or organ with all the
other constituents or organs and to avoid overlapping as far as possible. The
most natural method would be to commence either with the outermost constituent,
namely, the outer whorl of the calyx, or with the innermost constituent, namely,
the inner whorl of the andreecium. For reasons of a morphological character, which
will be seen later, the inner whorl of the andrceecium has been chosen as the
starting point. :
Biometrika x 32
246 Contribution to a Statistical Study of the Crucifere
TABLE O.
Correlation Coefficients between the Number of Various Organs and their Constituents.
Table Yr.
The outer whorl of the calyx and the inner whorl of the calyx... is xa I 1957
The outer whorl of the calyx and the corolla ae w II "7275
The outer whorl of the calyx and the outer whorl of the andrcecium ae a Ill “5886
The outer whorl of the calyx and the inner whorl of the andreecium is ato IV 2613
The outer whorl of the calyx and the andrecium ... She Fae ee ids Vv “4371
The inner whorl of the calyx and the corolla Ae via a VI 2476
The inner whorl of the calyx and the outer whorl of the andrecium ase io VII 3229
The inner whorl] of the calyx and the inner whorl of the androecium wats moa). WELL +3905
The inner whorl of the calyx and the andreecium ... oh ee wee Ben IX “4592
The calyx and the corolla 2 aes ae see noe eA x 6926
The calyx and the outer whorl of the andreecium ne re es Bsc XI "6245
The calyx and the inner whorl of the andreecium ... a ute Me, ae XII -4014
The calyx and the andrecium ... nes ore ae a Boe || 2-SILILIL 5721
The corolla and the outer whorl of the ‘andrescium ante ame ES ae a XIV 4762
The corolla and the inner whorl of the andrcecium ... ats Ape are hee XV 1773
The corolla and the andreecium ae XVI 3174
The outer whorl of the andreecium and the i inner whorl of the androecium: XVII "1984
The inner whorl of the andreecium, Posie 3.4 and the inner whorl of the XVIIT| 4305
andreecium, position 5.6... a
The outer whorl of the calyx and the inner whorl of the andrecium, position 32 oxen 4646
The outer whorl of the calyx and the inner whorl of the andreecium, position 5.6 XX 2134
The inner whorl of the calyx and the inner whorl of the andreecium, position 3.4 | XXI 4634
The inner whorl of the calyx and the inner whorl of the andreecium, position 5.6 | XXII | 2519
The corolla and the inner whorl of the andrecium 3.4... ses is ... | XXIIT | +2558
The corolla and the inner whorl of the androecium 5.6 XXIV | 0903
The outer whorl of the andrecium and the inner whorl of the andreecium, XXV | -2661
position 3.4
| The outer whorl of the ‘andreecium and the inner » whorl of the andreecium, tXXVI +1539
position 5.6 soc
(a) The inner whorl of the andrecium.
From the standpoint of the systematic botanist the most anomalous constituent
of the cruciferous flower is the inner whorl of the andrcecium, inasmuch as in each
of the positions where one stamen would naturally be expected, the presence of
two is regarded as typical. It has been explained in a previous section that
botanists now usually regard this anomaly as having arisen by collateral chorisis
from what was originally a single stamen in ancestral forms. For the sake of
conciseness and in order to avoid unnecessary repetition the following abbreviations
have been used in Tables P—X.
O. W. Ca. = Outer whorl of the calyx.
I. W. Ca. = Inner whorl of the calyx.
Ca. = Calyx.
Co. = Corolla.
O.W. A. = Outer whorl of the andrcecium.
IW. A. = Inner whorl of the andreecium.
A. = Andreecium.
=
J. J. Simpson 247
The following Table, P, gives the correlation coefficients between the I.W. A.
and the other constituents or organs of the flower in order of position.
TABLE P.
I. W. A. and the other Constituents.
Constituent or Organ Table Correlation
OMWaiCan... As IV 2613
Hee Wit Cari ne Wang | "3905
Ca. site ae XII 4014
Co. Se oe XV | ‘17738 |
O. W. A. ... ves XVII 1984 |
The highest correlation between the inner whorl of the andrcecium and the
other constituents or organs is that with the calyx; next in order come the inner
whorl of the calyx, the outer whorl of the calyx, the outer whorl of the andreecium,
and lastly the corolla. In other words, we should be better able to predict the
number of stamens in the inner whorl of the andreecium from the number of
members in the calyx than from the number of members in any other constituent
or organ.
(b) Relations between the organs themselves.
Having thus discussed the inner whorl of the androecium with the other organs
and constituents it might lead to some useful result if we proceed to determine
the “organic correlation” existing between the various organs themselves. In
this connection we have to consider the calyx, the corolla and the andreecium, and
for this purpose the correlation Tables X, XIII and XVI have been prepared.
The character which has been selected for this study is the number of members
in each organ.
The following Table (Q) shows the results obtained :
TABLE Q.
Correlation Coefficients between
Ca, and Co. ... | "6926
Ca.and\ A.” ... | “5721
|
| Covand As... || “3174
| ee |
(1) The calyx and corolla are much more highly correlated to one another thau
is either of these with the andreecium. In other words, the two protective organs
of the perianth are more highly correlated to one another than is either protective
organ with the male reproductive organ. It is further evident that (2) the calyx
is much more highly correlated to both the corolla and the andreecium than are the
two last named to one another. From (1)it may be concluded that, on an average,
32—2
248 = Contribution to a Statistical Study of the Crucifere
an increase or decrease from the accepted typical number, namely four, of petals in
the corolla is accompanied by an increase or decrease in the number of sepals
in the calyx; while from (2) an increase or decrease in the number of stamens in
the andreecium will be accompanied, on an average, by a greater increase or decrease
in the number of sepals than in the number of petals.
(c) Relations between the constituents of organs.
The constituents of (1) the calyx and (2) the andrcecium will now be con-
sidered. ;
(1) Calyx. The outer and inner whorls of this organ are inserted at different
levels and have a decussate arrangement, so that, although the organ as a whole
is protective in function, the two whorls actually help to enclose the flower at
right angles to one another. The correlation between these two whorls is an
extremely low one, namely, 1957 (Table I), in other words, an increase or decrease
in the number of sepals in either of the whorls of the calyx is associated only in
a very small degree with an increase or decrease in the number of sepals in the
other whorl. Or again it may be expressed thus, the two whorls of the calyx
vary to a great extent independently of one another. This statement should be
taken in conjunction with that made on p. 244 with regard to their Means and
Variabilities and should also be borne in mind when the correlation between
these two constituents and the other parts of the flower are discussed below
(see Tables R and 8).
(2) Andreciwm. This organ is also composed of two whorls, an outer and
an inner inserted at different levels. Its function is of course reproductive.
The correlation between the two constituents is very low, namely, ‘1984 (see
Table XVII), and is almost the same as that between the two whorls of the calyx.
The inner whorl of the andrcecium shows greater variability than the outer whorl
and tends to vary independently of this latter constituent, just as in the case of
the two whorls of the calyx.
Having thus considered the organs per se, let us now compare the correlations
between each individual constituent or organ and all the other constituents or
TABLE R.
(d) Correlation Coefficients between the Outer Whorl
of the Calyx and
2nd Component Table 1,
UI Ne OP IL "1957
Co ner 11 7275
O. W. A Ill 5886
Te Wee A: IV 2613
A Vv 4371
J. J. SIMPSON 249
organs. For this purpose it will be necessary to tabulate the results in series and
consequently it might be well to start with the outermost constituent of the flower,
namely, the outer whorl of the calyx, and tabulate the correlation coefficients
passing inwards to the andreecium. The inner whorl of the calyx will next be
taken in relation to the other constituents and so on.
From the above table it will be seen that the outer whorl of the calyx is most
highly correlated with the corolla; it is also highly correlated with the outer whorl
of the andrcecium but much less so with the inner whorl of the andrcecium.
TABLE §&.
(e) Correlation Coefficients between the Inner Whorl
of the Calyx and
2nd Component | Table rs
COs. VI 2476
O. W. A. VII "3229
Wie Ae VItl “3905
TNS IX 4592
The low correlation between the inner whorl of the calyx and the corolla is due
to the close adherence of the former to type, that is, there is very small variability.
TABLE T.
Correlation Coefficients between the Calyx and
(f)
2nd Component Table rs
Come. x "6926
O. W. A XI 6245
IW. A XII 4014
AS sts XIII “5721
There is a higher degree of correlation between the two organs of the perianth
than between the calyx and the andreecium. The high correlation between the
calyx and the outer whorl of the andreecium is mainly due to the high value
obtained for the correlation between the outer whorl of the calyx and the outer
whorl of the andreecium.
TABLE U.
(g) Correlation Coefficients between the Corolla and
2nd Component Table is
OR Wards XIV "4762
I. W. A. XV 17738
Aes XVI 3174
250 Contribution to a Statistical Study of the Crucifere
The corolla is much more highly correlated with the outer whorl than with
the inner whorl of the andrcoecium, and the correlation between the corolla and the
andreecium as a whole is not very great.
A comparison of Tables T and U shows that there is a much greater correlation
between the calyx and the andreecium and its two whorls, than between the corolla
and the same constituents.
So far we have considered the relationships between the different parts of the
flower from the outside inwards, but when we examine these relationships, taking
the inner whorls as our starting point, some new aspects of* the problem become
manifest and, as these have been of great value in the interpretation of the results,
it has been considered advisable to tabulate them thus:
TABLE V.
(h) Correlation Coefficients between the Inner Whorl
of the Andrecium and
2nd Component Table is
Can ic ase XII “4014
TW C aremene VIII “3905
OMWis Cannes IV 2613
Oo AW tcAln tee XVII 1984
Co. =: mate XV ‘1773
TABLE W.
(2) Correlation Coefficients between the Outer Whorl
of the Andrecium and
| 2nd Component | Table i:
|
Carex: 586 XI 6245
[OW Cosme, Ala 5886
Cones. bee XIV “4762
[iti Rg Cay ee VII 3229
A comparison of Tables V and W shows that the correlations between the
outer whorl of the andrcecium and the other components are higher than for the
inner whorl of the andrcecium, except in the case of the inner whorl of the
calyx.
J. J. SIMPSON 251
TABLE X.
()) Correlation Coefficients between the
Andrecium and
2nd Component Table Ts
Ob aor oe XIII 5721
Te WieCare ere IX "4592
OW Can ce- V 4371
Co. ... ie XVI 3174
This table shows that when the androecium is considered as a whole it 1s mast
highly correlated with the calyx and least correlated with the corolla.
V. MORPHOLOGICAL SIGNIFICANCE OF THE STATISTICAL RESULTS.
It is quite clear from the tabulated results that there is a definite departure
from the usually accepted cruciferous structure in a very large number of the
flowers of Lepidium Draba which have been examined for this study. This does
not obtain merely in any one organ or constituent but in all the organs and
constituents, although not to the same degree in each.
The statistical results will now be examined from the standpoint of the
botanist in order (a) to note their morphological or genetic significance and
(b) in order to see whether these figures throw any light on the evolution of this
cruciferous plant.
It is almost axiomatic to state that the “purpose” of a flower is a purely
reproductive one and that therefore its existence is justified only in so far as it
serves to reproduce its kind. But not all the parts of a flower are solely repro-
ductive in function. Each individual consists of two parts, (1) Reproductive,
(2) Protective. (1) The reproductive organs are the gynecium (2?) and the
andreecium (¥), while (2) The protective organs (perianth) are the corolla and
the calyx.
One of the organs of the perianth, namely the corolla, is still further specialised.
The calyx consists of four sepals, green in colour, whose sole function is to protect
the flower when in the bud, and in many cases these are reflexed immediately after
the flower has opened up, and are of no further importance to it. On the other
hand the petals though essentially sepal-like in structure, in this as in the great
majority of flowers, are not green but of some other colour. In the species under
consideration they are white. Now although the petals are of great importance
in protecting the reproductive organs while in the bud their utility does not cease
252 Contribution to a Statistical Study of the Cruciferae
when the flower opens but, along with small nectaries at the base of the stamens,
serve as an attraction for insects whose visits are essential for cross-fertilisation.
The reproductive organs of what is regarded as the typical cruciferous flower
consist of (1) the gynaecium which is composed of two carpels and (2) the andreecium
which is composed of six stamens. The stamens are delicate structures and do not
hold an isolated position in the flower. When in the bud and immature they are
subject to external influences, for example, (1) they might be shrivelled up by the
heat of the sun, (2) they might be blasted by rain or wind or (3) they might be
attacked by herbivorous insects, so that the protective perianth plays an important
part in flower economics. Now what does an increase in the number of stamens
imply? It is obvious that if the number of stamens is increased the total volume
occupied by the reproductive organs is increased and consequently a tax is put
upon the protective organs if they are to fulfil their function adequately. If the
perianth does not respond to this tax from space considerations, the reproductive
organs stand a small chance of ever fulfilling their function, so that one would
naturally expect that variation of some kind in the perianth would follow variation
in the reproductive organs.
Another important point which must never be lost sight of when interpreting
the statistical results is the symmetry of the cruciferous flower. The calyx consists
of two whorls each with two sepals; the corolla of one whorl of four petals and the
andrceecium of two whorls of stamens, the outer having two members and the inner
four members (see Plate I, fig. 7). Consequently a cruciferous flower is bilaterally
symmetrical only on that vertical plane which passes through the division wall of
the carpels, between each of the pairs of stamens in the inner whorl, between two
petals on either side and through the middle of the outer pair of sepals, This
plane may be referred to as the “plane of symmetrical division.” Owing to the
fact that the corolla consists of only one whorl, the outer whorl of the calyx
corresponds in position to the inner whorl of the andrcecium, and the inner whorl
of the calyx to the outer whorl of the andreecium.
From a study of the Means and Standard Deviations of the various organs and
constituents we arrive at the following conclusions :
Calyx. (1) The greatest approach to constancy in number in the whole
flower is in the inner whorl of the calyx.
(2) There is a significant difference between the means of the two whorls.
(3) There is much greater variability in the outer than in the inner whorl of
the calyx and on an average it possesses fewer sepals.
(4) There is a tendency towards a reduction from type in the number of
sepals in the calyx.
Corolla. (5) There is a tendency towards a reduction from the accepted
typical number in the number of petals in the corolla.
J. J. Srpson 253
Andrecium. (6) There is a significant difference between the means of the
distributions of
(a) the members of the two whorls of the androecium,
(b) the members of the two positions in the inner whorl,
and (c) the members of the two positions in the outer whorl.
(7) From whatever axis we view the andreecium as an organ it is distinctly
asymmetrical in the distribution of its functioning stamens.
(8) There is much greater variability in the inner whorl than in the outer
whorl of the andrceecium.
(9) In both positions in the inner whorl of the andrcecium there is a tendency
towards a reduction from the accepted typical number of stamens and in the
position where this is most marked there is the greatest variability.
The interpretation of these results is not at first sight very evident.
Why should there be a tendency towards a reduction in the number of
members in the different organs of the flower and why should this tendency
be most marked in the inner whorl of the andreecium? As has already been
pointed out all the flowers examined were taken from a single plant which gave
rise to new stems by means of buds on the roots. May this tendency to reduction
in the parts of the flower whose function is sexual reproduction not be an expression
of a tendency towards an elimination of sexual in favour of vegetative reproduction ?
Another phenomenon which lends support to this hypothesis is the fact that in
this plant the percentage of “ pods” which attain maturity is extremely small.
Whether there is or is not a tendency towards vegetative reproduction, may
we not also have here a harking back towards an ancestral form in which the
number was less than the at present accepted typical number? In fact one would
expect that if the present constitution of the inner whorl of the andreecium had
been most recent in development, reversion would first take place in it, and
conversely one might reasonably conclude that since this whorl shows greatest
variability, and most marked tendency to reduction in the number of members, it
is more than probable that its present constitution was arrived at by an increase in
number from a more primitive type.
Let us now examine the deductions made from a study of the correlation
coefficients and see if they have any morphological interpretation.
(1) The calyx and corolla are more highly correlated with one another than is
either of these with the andrcecium.
(2) The calyx is more highly correlated with the andreecium than is the corolla.
In other words, the two protective parts are more intimately associated in increase
or decrease with one another than is either of these with the male reproductive
organs, and further the calyx which is solely protective in function is more
Biometrika x 33
254 Contribution to a Statistical Study of the Cruciferae
intimately correlated with the male reproductive organs than is the corolla which
serves as an attraction for insects as well as a protective covering of the bud.
(3) The two whorls of the calyx are not highly correlated, i.e. they vary inde-
pendently of one another.
(4) The two whorls of the andrcecium also are not highly correlated. Morpho-
logically this means that when there are two constituents in one organ, each having
the same function, they may vary independently of one another, so that although
an increase or decrease in the number in either may be correlated with an increase
or decrease in the number in any other constituent of the flower, the same does
not hold true with regard to the two constituents.
(5) The outer whorl of the calyx is most highly correlated with the corolla,
next with the outer whorl of the andreecium and lastly with the inner whorl of
the andrecium. The reason why the outer whorl of the calyx is more highly
correlated with the outer whorl than with the inner whorl of the androecium is
not at first sight very evident, but may be explained on the basis of its protective
power. The members of the outer whorl of the andrcecium lie in a plane parallel
to that of the outer whorl of the calyx, and are much more widely separated in
this plane than are the members of the inner whorl of the andreecium. Con-
sequently any increase in the number of stamens in the outer whorl would involve
a much greater increase in volume within the flower than a corresponding increase
in the number of stamens in the inner whorl. Thus we are not surprised to find
that such an increase in the outer whorl of the androecium is more intimately
associated with an increase in the outer whorl of the calyx than a corresponding
increase in the inner whorl of the andrcecium would be.
(6) There is very low variability in the inner whorl of the calyx and it is
almost equally correlated to the two whorls of the andreecium. The morphological
explanation of these facts follows as a corollary to that given above.
(7) The calyx is much more highly correlated with the andrcoecium as a whole
and with its two whorls than is the corolla.
As we have already said the calyx is the predominantly protective organ and
consequently this higher correlation has a physical basis. The corolla being partly
attractive does not enter so closely into space economics.
(8) The outer whorl of the andrcecium is more highly correlated with the
other components of the flower than is the inner whorl of the andreecium. This
again follows on the basis of space considerations. Any increase in the number
of members in the inner whorl of the androecium does not involve so radical a
change in the volume of the flower as does a corresponding increase in the outer
whorl of the andrceecium.
J. J. SIMPSON 255
VI. VARIATION IN THE GYNACIUM.
So far we have not considered the gynzcium on account of the small number
of variations which occur in that organ and from the fact that these do not lend
themselves to statistical treatment.
The gynecium consists typically of two carpels which are flattened in a vertical
plane parallel to those containing the pairs of stamens in the inner whorl of the
andreecium. The thin partition wall separating the two carpels therefore stands
at right angles to this plane.
Now when we examine the different types of variations in the structure and
number of the carpels we find the following: (1) a single carpel, (2) two carpels
(typical), (3) three carpels, (4) four carpels, (5) two sets of two carpels within
a single perianth, (6) two sets of two carpels within separate perianths but on one
pedicel.
Let us now proceed to examine each of these in some detail.
(1) The gyneecium consists of a single carpel (see Figs. LXXXVIIT—XCII).
In all these cases, except LXX XVII, as will be at once seen by reference to the
figures, the suppression of a carpel is accompanied by the suppression of some of
the members of nearly all the other organs thus:
In LXXXVIII two petals are absent and one stamen is aborted.
In LXXXIX one sepal, two petals and two stamens are absent.
In XC, XCI and XCII all the organs are deficient in members.
A noteworthy phenomenon in this respect also is that the suppression of
members which accompanies the suppression of a carpel is usually in the vertical
plane which passes through the plane of separation of the carpels.
(2) The gynecium consists of two carpels.
This is the accepted typical structure and the statistical study deals with these
in detail.
(3) The gynecium consists of three carpels (see Figs. XCIII and CII).
When three carpels occur in the gynecium they are never found co-laterally,
Le. the additional carpel is never found with its origin at the side of a carpel, but
always arising from the plane of separation, which is in the plane of greatest
variability.
(4) The gynecium consists of fowr carpels (see Fig. Cl).
Just as in the previous case the increase in the number of carpels takes place
in the plane of separation of the carpels—one on either side, so that a cruciate
structure is found. A reference to Fig. CI will make this clear. In both of these
groups it will be evident that an increase in the female reproductive organs is
33—2
256 Contribution to a Statistical Study of the Cruciferae
associated not only with an increase in the male reproductive organs but also in
an increase in the protective organs or perianth.
(5) The gynecium consists of two sets of carpels within a single perianth.
This is rather an anomalous group but is extremely interesting inasmuch as it
contains a series of annectant forms linking group 2 to group 6. What we actually
have here is a complete reduplication of the reproductive organs encased within
a single series of protective organs. In some of the flowers examined with this
structure it was rather difficult to determine the orientation owing to a torsion
of the thalamus, but in the types figured on Plates IX and X (Fig. XCIV and
Figs. XCV et seq.) the mode of origin of these is quite evident. Several
important observations on these forms may be stated.
(a) There are really two complete sets of reproductive organs and in one case
see Fig. each of these is of the typical cruciferous structure.
Fig. XCVII h of th f the typical f truct
(b) Increase in the number of the reproductive organs is accompanied by an
increase in the number of members in the protective organs.
(c) The increase in the number of members of the reproductive organs is for
the most part in the plane of division of the carpels, in other words, in the outer
whorl of the calyx and its associated petals.
(d) This is also the plane along which the separation of the reproductive
organs has taken place.
(e) This plane is the one which we have already shown in the statistical part
to be the plane of greatest variability.
(6) The gyncecium consists of two sets of two carpels within separate perianths
but on one pedicel.
In this group we reach the limit of variability in the material examined. In
place of a single flower consisting of calyx, corolla, andrcecium and gynecium we
actually find two complete sets of all these organs, on one pedicel (see Figs. CITI—
CVIIT), while in one case (Fig. CIII) each of the two flowers has the typical
cruciferous structure, so that were each of these separately examined it would
undoubtedly be regarded as a normal flower. Yet we must bear in mind that,
botanically considered, one flower and one flower only arises from a pedicel. Were
this, therefore, an isolated example, and if no annectant forms existed, the departure
might well be regarded as a “mutation,” but a consideration of the numerous
variations which we have already considered, taken in conjunction with group 5,
only serves to emphasise the fact that “the vertical plane which passes through
the partition wall of the two carpels and consequently separates the individuals
of the pairs of stamens in the inner whorl and passes through the centres of the
sepals of the outer whorl of the calyx is a plane along which this flower is in a
state of flux and is the plane in which it is probable that the flower has changed,
and is still changing, from some quite different ancestral form.”
J. J. SIMPSON 251
VII. SUGGESTIONS FOR FUTURE STUDIES IN THIS PLANT.
It must be very obvious to anyone who has perused this paper that the results
which might be obtained from a study of this plant are by no means exhausted.
An attempt, however, has been made to interpret the variability in its flowers,
both from a morphological and an evolutionary standpoint. Studies of a different
nature might be undertaken in order to test the results obtained, e.g. :
(1) What is the degree of fertility in the flowers of this plant? For this
purpose it would be necessary to find the percentage of flowers which produce
fertile seed.
(2) What are the variants, if any, which are associated with infertility ?
(3) What are the characters of the flowers which are produced from the seeds
of the different variants? If seeds selected from the different variants were grown
separately and self-fertilised, one could trace the variations in the flowers of the
next generation and see to what extent the different variations were transmitted.
This study is capable of much elaboration and is one which would be fraught with
great possibilities. It seems to involve a satisfactory method of determining how
far these variations are concerned in plant economics, and also to what extent
they have been instrumental in the evolution of the Order Crucifere.
EXPLANATION OF FIGURES 8, 9 AND 10. PLATE I.
FIGURE 8.
(a) Typical stamen (outer whorl).
(b) Stamen with half-length filament and complete anther (outer whorl).
(c) Typical stamen (inner whorl).
(d) Non-chorised stamen with two complete anthers (inner whorl).
(e) Stamen of inner whorl with two complete anthers but only chorised in the upper half.
FIGURE 9.
) Aborted stamen of outer whorl, i.e. filament with no anther.
(b) Absence of stamen in outer whorl.
(c) Full-length filament in inner whorl with no anther.
(d) Half-length filament in inner whorl with complete anther.
(e) Half-length filament in inner whorl with no anther,
(f) Non-chorised stamen in inner whorl with half-length filament but with two complete anthers.
FIGURE 10.
) Normal sepal.
(b) Sepal divided almost to the very base.
(c) Sepal completely divided into two distinct sepals.
(d) Sepal absent.
(e) Normal petal.
(f) Petal divided almost to the very base.
(g) Aborted petal.
(h) Petal absent.
Outer Whorl of
Andrcecium.
Inner Whorl of Calyx.
Number of Sepals
Contribution to a Statistical
bo
On
o.8)
TABLE I.
Outer Whorl of Calyx.
Number of Sepals.
0 1
1 3
y 1803 |
8 O |
J 6
Totals
TABLE IIT.
Outer Whorl of Calyx.
Number of Sepals.
ab Totals
3 5 0 5
5 Hh 1 85
Bs y) 1713 |
ES Mg 3
a 9g } 7a
<qiol est ‘
7, |Totals 1813
TABLE V.
Outer Whorl of Calyx.
Number of Sepals.
7 = s =a = |
Totals
ory 38
L 5 367
S| 6 1367
3 | c 19
Se 9 1
Se at0 2 |
= | 12 Om
af | 13 1
|. 4 1
18 2 |
Totals 1813
Study of the Crucifere
Corolla.
Inner Whorl of Andrcecium.
Number of Petals.
Corolla.
Number of Petals.
Number of Stamens.
TABLE II.
Outer Whorl of Calyx.
Number of Sepals.
QD
0
1
2
y
5
6
Totals
TABLE IV.
Outer Whorl of Calyx.
Number of Sepals.
0 dl 2 3 | 4 | Totals.
Cel a Tel
2 |= | 6 |.) 24s eae
3 | 2] 992 | — | —} .994
4 | 2 | 39:| 1399 | — |=") ago
5 2 25]; 1 | — 28 |
6 —|— 12) — | — 12
Sees Tye | a 4 |
9 1 1
10 —|— os 1 | — 1 |
1 f—|—| — 4a} 2 |
Totals] 2 | 50 | 1753) 6 | 2 | 1813 |
TABLE VI.
Inner Whorl of Calyx.
Number of Sepals.
DAW Se *M*HS
Totals
S
Nw
w
eS)
4 {Totals
Outer Whorl of
Outer Whorl of
Andrceecium.
Number of Stamens.
[=|
oO
ofS
a x |
ess
BO
a |
@3
aq 2
I
=
Z,
DB
g
ga
ee
eas
23
oH
oO
<q 2
q
fos}
Z,
TABLE
VII.
J. J. SIMPSON
Inner Whorl of Calyx.
Number of Sepals.
cS)
Q
ov
TABLE IX.
Inner Whorl of Calyx.
WOBANADAW
of 4
3 | 1803
Number of Sepals.
4 Totals’
Corolla.
TABLE
Calyx.
Number of Sepals.
| 7
7 |Totals
luner Whorl of Andreecium.
Number of Petals.
y
L
Inner Whorl of Andrcecium.
ber of Stamens.
Num
Number of Stamens.
> MA SS WH
~
Totals
259
TABLE VIII.
Inner Whorl of Calyx.
Number of Sepals.
g 3 4, | Totals
o| 1
|
|
|
}
Number of Sepals.
a
|
aay ties
=
|
1 3 | 1803 | 0 | 6
TABLE X.
Calyx.
|
'Totals} £
TABLE XII.
Calyx.
Number of Sepals.
Inner Whorl of Andreecium.
Andreecium.
Number of Stamens.
Number of Stamens.
Contribution to
TABLE XIII.
Calyx.
Number of Sepals.
TABLE XV.
Corolla.
Number of Petals.
Totals
1
Q
oS
J ie
‘5 2)
6 — |
8 Ik
9 Lem
10 1
VU 2
Totals
TABLE XVII.
Outer Whorl of Andrcecium.
Number of Stamens.
: —.
Totals
eh ]
5 wo
S81 2 30
es a 3 294
icp 4 1440
a a oO 28
LSC BG 12)
Sa| 8 A
eS 9 1
= ©
Boel ALO 1 |
SAl ty 2
= =
Totals 1813
Outer Whorl of
Inner Whorl of Androecium
Andrcoecium.
Number of Stamens.
aye
Number of Stamens.
Andrcecium.
Number of Stamens.
a Statistical Study of the Crucifere
TABLE XIV.
Corolla.
Number of Petals.
TABLE XVI.
Corolla.
Number of Petals.
5
6 |Totals
_\Totalsy 4 | 3 | 18 34| 8 | 4 | 1813
TABLE XVIII.
Inner Whorl of Andreecium, position 3. 4.
Number of Stamens.
4 Totals
ng 1
i SuL7/
2 1446
3 37
4 0
5 3
6 0)
ff 2
Totals 1813
J. J. SIMpson 261
TABLE XX.
Outer Whorl of Calyx.
TABLE XIX.
Outer Whorl of Calyx.
Number of Sepals.
Number of Sepals.
Inner Whorl of Andrcecium,
position 3. 4,
Number of Stamens.
position 3. 4.
Number of Stamens.
Inner Whorl of Andreecium,
a 3 Totals
los FS
1 =. @ 0 1
2 eS 8 i 317
3 tion] 2 1446
4 oe oie 37
5 BS y| 4
6 eS es 5
7 BE oe 7
nO
1813 = Totals
— be
TABLE XXI.
Inner Whorl of Calyx.
Number of Sepals.
4, \Totals
TABLE XXII.
Inner Whorl of Calyx.
Number of Sepals.
Gaieeatiy cor ies q 0 | 1 2 | 31] 4 |Totals
=|
Saar _ 25} a
it || oe iis ea ee | 8 8] 0
2 — 3 1750 | -— 1 1754 oR) a y
Ba fe |e ha One et 32
ep re pu |e Us ese WS
i (ae ale | ot 2 Oe: |e
ee ey ol Be hs
7 2 2 hore) tab
H cS
Totals} 1 | 3 | 1803] 0 | 6 | 1813 3 4
: A Totals
TABLE XXIII.
Corolla.
Number of Petals.
5
a
oO al SS,
PE eecead
aw & 2
<a: 3
Sa we] 4
eins} A 5
BS 5 :
ah 2 6
Fag) 7
mn OA
5 Totals
S
— bi —
Biometrika x 34
Inner Whorl of Andrcecium,
position 3. 4.
Number of Stamens.
2 Contribution to a Statistical Study of the Crucifere
TABLE XXV.
Outer Whorl of Androecium.
Number of Stamens.
g
3
D ;
=
=| co)
Bo g
. @
“ls ae
M
Soa
OB w
ae)
mer
ee
on v
E a'g
bolts
3
8 OG
=}
q
ml
TABLE XXIV.
Corolla.
Number of Petals.
NWA AK O89 WHO
6 | Totals
TABLE XXVI.
Outer Whorl of Androecium.
Number of Stamens.
s
SP RAAB WON
a
a2]
QD
4
3 | 4 |Totals }
ee
42 Some aa0
1754 go g| 1
9 as#| ?
Fleeccts cul
ae Seeds
0) Sa ® 5
2 cae | 6
Fe seal. a7,
Prt 4 Totals
J. J. SIMPson
PLATE I.
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A
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t
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CIV B
CVI B
OVIIB
CIII A
Clu B
CVI A
CVII A
XCVI
NOCHMALS UBER “THE ELIMINATION OF SPURIOUS
CORRELATION DUE TO POSITION IN TIME OR SPACE.”
Von O. ANDERSON, St. Petersburg, RuBland.
1. Im Aprilheft der Biometrika, hat “Student” gezeigt*, daB das von Cave und
Hooker vorgeschlagene Verfahren, den Korrelationskoeffizienten zweier oscillieren-
der Variablen durch Berechnung erster Differenzen (also durch Ersetzung der
Reihe a, 2,... %,, durch die Reihe: A’a, =x, — x, A’x,= a, — &3,... An. = Bn —4n)
vom evolutorischen Element zu befreien, eine Verallgemeinerung zuliBt. Das
Verfahren ist nimlich, streng genommen, nur dann richtig, wenn die evolu-
torische Komponente durch eine lineare Gleichung darstellbar ist. Findet
letzteres nicht statt, kann also, z. B., jener nur eine parabolische Gleichung
hoherer Ordnung geniigen, so mu8 man zweite, dritte u.s.w. Differenzen nehmen
(also statt A’s,, A’x,... nehme man A”z, = A’a,— A’x,, A” x, = A’x, — A’as, ... etc.)
und danach Korrelationskoeffizienten berechnen. Letztere kénnen bald einen
konstanten Grenzwert erreichen, der das gewiinschte Resultat darstellt.
Unterzeichneter ist schon vor etwa 2 Jahren zu ahnlichen Schliissen gekommen.
Durch von ihm unabhangige Griinde wurde er aber bis jetzt vom Drucke seiner
diesbeziiglichen Schrift abgehalten. Da er bei seiner Untersuchung Wege ein-
schlagt, die von denen des “Students” sehr verschieden sind, und auch zu manchen
Schliissen kommt, welche letzterem unbekannt geblieben zu sein scheinen, so
kénnte vielleicht eine kurzgefaBte Darstellung der wichtigsten Resultate seiner
Untersuchung fiir die Leser der Biometrika von einigem Interesse sein.
2. Methode. Die englische statistische Schule vernachlassigt in ihren Unter-
suchungen ein Verfahren, das von russischen und deutschen Gelehrten oft ange-
wandt wird (Tchebycheff, Markoff, v. Bortkiewicz, u.s.w.) und neben groBer Strenge
und Exaktheit noch den Vorzug hat recht elementar zu sein—die Methode der
mathematischen Erwartung némlich, Mathematische Erwartung einer Grofe (4)
heiBt bekanntlich soviel als das Produkt aus dieser GroBe und ihrer Wahrschein-
lichkeit (w), also Aw. Wenn eine Variable eine Reihe einander ausschlieBender
* Biometrika, Vol. x. Part 1, 8. 179, ‘‘ The Elimination of Spurious Correlation due to Position in
Time or Space.” By ‘‘Student.”
Biometrika x 35
270 «=©Die Verallgemeinerung der Cave-Hookersche Methode
GréBen annehmen kann, so ist deren math. Erwartung als die Summe der Erwar-
tungen aller dieser GréBen definiert. Wir werden hier die mathem. Erwartung
iiberall durch das Symbol #( ) bezeichnen. (A) ist also, z. B., gleich Aw.
Die hauptsichlichsten Satze tiber mathematische Erwartungen diirften als
bekannt angenommen werden. Um aber die Nachpriifung der Formeln dieser
Schrift zu erleichtern, werden wir hier die fiir uns wichtigsten Satze noch kurz
andeuten :
(Ql) H(@t+y-2ztu-t...)=H(a)+HYy)-H(a)+H(u)-H)....
(2) Wenn uw, y, z,... von einander unabhingig sind, so ist
H(e.y. 2... =f (a) Ee EZ) sea.
(3) EH (k«)=kK (x), wo k const. ist; und daher auch :
iE (hy) = he:
(4) Wenn eine Variable X die Werte 2, 2,...%, annehmen kann, so ist
die Wahrscheinlichkeit W, daB die Differenz «;—H(«) zwischen den Grenzen
—aV HE (a?)—[E(2)P und +a V# (a*?)—[E(a)f enthalten sei, gréBer als 1— =
ae
wo a gréBer als 1 sein mu (ein Theorem von Tchebycheff).
In unserer Untersuchung werden wir iiberall statt des wahrscheinlichsten
Wertes einer Gréfe deren mathematische Erwartung berechnen.
Bestimmen wir zuerst, wie sich der Korrelationskoeffizient zweier oscillierender
Reihen verhalt, wenn man deren GréBen durch Differenzen Ai. (Naa Nace
A’y, A”y, A’’y, ... ersetzt, und darauf untersuchen wir die Frage von den Grenzen
der Anwendbarkeit der verallgemeinerten Cave-Hookerschen Methode. Um Raum
zu sparen, werden wir nur die endgiltigen Resultate der Berechnungen anfiihren,
ausgenommen die 3 ersten Formeln, deren Bestimmung als Beispiel der Rech-
nungsmethode dienen mige.
3. Definition. Unter einer oscillatorischen Reihe werden wir eine solche
Reihe
XH, Ua, Uy, eee, Uj, ove, Ln
verstehen, bei der
HE (a) = H'@) =... = Ea) =. =f (a) =) const.
und alle einzelnen Glieder von einander vollig unabhingig sind, so dab
Ef (#;4;) = E («;). E (a), wobei 7+ 7.
Solchen Bedingungen wiirde zwm Beispiel eine Reihe geniigen, deren Glieder
die Resultate emer Versuchsreihe mit konstanter Wahrscheinlichkeit darstellen,
etwa Resultate von Ziehungen aus einer Urne mit m weiBen und n schwarzen
Kugeln.
O. ANDERSON 2M
4. Mittleres Fehlerquadrat.
Bezeichnen wir a;—H(#) durch &;, so ist
KE, = E (a; -— E (a)] = # (a) — EH (2) = 0.
Da die einzelnen & von einander vollig unabhéingig sind, so ist
E (&&) = # (&). H(&) = 0.
Das mittlere Fehlerquadrat der Reihe & ist gleich
Seine mathematische Erwartung wollen wir (nicht ganz in Ubereinstimmung
mit der iiblichen Bezeichnung) o,? nennen.
1 n nu
eran PO 1 [Se] 28 [Se]. sereatoo
= E (a*) -(H (@)P,
ein Ausdruck, der oben im Satze 4 (§ 2) unter dem Zeichen der Quadratwurzel
steht.
[St-2@r| {ee|
Andererseits ist aber 1 (Gone gleich A oe | und daher
o2= B(E).
5 nN
Untersuchen wir den Ausdruck abs (a; —- My], wo M, das arithmetische
ak
iL
Sa
SAD
Mittel der Reihe 2, also = bedeutet. Da
Barr t+ (a) + et... +H (a) + En _
nN
o— M,= HE («)+&- E;— M:
n
> &;
(wenn man JM; fiir +— einsetzt), so haben wir:
n
E E (ee My» | ae 1, Be st my | _E E aoe oe]
acl al 1
See—2366)|
=n B(E)— 0B (Me) =n (&) — nF | fant @)- B®)
ie
b[ Sq -any |= =H Be)
if
Um £(&) zu bekommen, muf man diesen Ausdruck durch (nm — 1) dividieren.
35—2
272 Die Verallgemeinerung der Cave-Hookersche Methode
Ks ist also auch
Um das Fehlerquadrat o*y,z fiir die erste Differenz A’xv zu erhalten, bertick-
sichtigen wir, dab
A’x; = (#i — Bir) = (Ei — Eis)
und E (A'a;) = E (&:) — & (Ein) = 9.
Daher haben wir:
n= ; } n—-1
[‘Stwn- Bay] ['S&- si
aes H | im a eee n—1
Sa | |S ee _2E Be fin | +E | 2e°]}
n—l 1 1 2
= +> (m1) B(E*)— 0+ (n-1) B(E)} = 28 (B).
Ks ist also Can — on
Nach demselben Rechenschema ergiebt sich fiir das mittlere Fehlerquadrat
der zweiten Differenz A’x; der Ausdruck 60,2
, dritten ms DG aes i 200,7
» vierten a PN ie = 700,2
2k!
» k-ten . Aas, »» AA On:
Wir konnen daher folgende Gleichung aufstellen
re One = o'a'a ahs aN Se: CNL ra 7,0, 2
Oy, = 2 = 6 = 20 = 70 pe are a sre eeeee (1)
i k )
welche exakt ist, und folgende
n n-1 n—2 n-3 TN n-~k
3 (0;—M,) “SA'a? “LA's? “SA” ae "Sama SNC
1 a 1 = u —_— —— u — 1 —— pe pe
n—1 2(n—1) 6(n—2) 20(m—3) 7O(n—4) “Qk! k ay *
TEE a
welche nur anniihernd richtig’ ist.
n-k n-k
D> Atk) x2 D2) (A®e -M (h Ve
5 5 1 J 1 A‘ ‘a,
* Ks ist vorteilhafter yeaa und nicht —— zu berechnen.
i :) raga ee
pea
—a
O. ANDERSON 273
LHYs
Ss . 1
5. Das mittlere Produkt mo
Wenn zwei Reihen
X, Xo, X3, shaiie, Xn»
und Yuva Gas Uae Uns
beide im Sinne des § 3 oscillatorisch sind, und eine Korrelation nur zwischen
GréBen mit gleichen Indexen, also zwischen 2, und y;, « und y, 2; und ¥;, u.s.w.
bestehen kann, so ist es leicht ersichtlich, daB
E ((x— E@)\[yj— Ey}, =0, wenn i4j,
Bezeichnen wir y;— H'(y) durch y;, #;— E(w) wieder durch &;, so finden wir
leicht folgende Ausdriicke :
Ei
Pay =H | — =H (Ei),
n
| = E (Eni) = pay,
n—l
> A’a, A’ Yi
1
Pareny= 8 |—Gaa— | = BP
[E@%— Me) yi- My)
E | ,
ree a ee ee rs
("S'a 8) op, A) yy 2k!
Paw, Agee Nae hee
Wir konnen daher wieder zwei Gleichungssysteme :
_Pyraty Parcary Pavrra’y Pavve atrry Pawa ay
k! k!
und
nN
n n—-1 n—-2 -3
> [w—E(«)|[y-H(y)]) DA'aA’y, TAwA"y, & AGA’;
1 1 1
n—1 "7S © M2)" = 20 = 3)
"S Awa, Ativ)y, AM x Ay,
= t= ———— =p (2a)
TO(n—4) 7" 2k! eS
Et kl yi i” —k)
aufstellen, von denen das erste exakt und das zweite angenihert ist, und welche
den Gleichungen fiir o,? (also auch fiir 0,2) des § 4 genau analog sind.
274 = Die Verallgemeinerung der Cave-Hookersche Methode
6. Das Fehlerquadrat der Fehlerquadrate.
Betrachten wir jetzt den Bereich der Schwankungen der GréSen der Systeme
(1a) und (2a) um deren mathematisch zu erwartenden GréBen in (1) und(2). Mit
anderen Worten, gehen wir (mit Riicksicht auf Satz 4 § 2) zur Darstellung der
math. Erwartung der Fehlerquadrate der genannten Groé8en iiber.
S [v; — £ (x)P 4 2) ]2
Fiir 2 7 ergiebt sich das Fehlerquadrat BSD) a (é )] P
3 [2:— Ma}
Fiir * : ergiebt sich das Fehlerquadrat Ee ACO ap aes] :
al n n(n—1)
S [A’ajp
Fir DED ergiebt sich das Feblerquadrat
(2n — 8) (EB (&) - [A (E)P}+2(a— DBE)?
2(n—1) ,
"S [Av aiP
Fiir aCe 2) ergiebt sich das Fehlerquadrat
{P=
(9n — 23) (H (E+) —[# (&)P} + '7n — 42) LE (&)P
9(n —2)? ,
n—-k
S(A® x;
: apeal
Und endlich fiir Ohi
aan (n—k)
Tere ae Oe 24) HE) — (BEN) + 4 EYP LAS (2+ 1
+ A? (n—2kh4+2)+4+ A)? (n—2k4+3)4+...+ 472 (n—2k4+h)]
+ 2B¢ (2 (8) —[H(E)K +8 [BOR Bat Bet + Bea)
ergiebt sich der recht komplizierte Ausdruck :
Wenn Jy, b,, bs, ... by die Koeffizienten der Zerlegung des Binoms (1 +1)*
k(k—1)
darstellen, also 6,.=1, b,=hk, b= 1g? USW., 80 ist hier
Qh! 2
Ag=(b2+b2+b2+... +02" = | eral
ht 7
Aj? = (doby + bb, + babs + 0. + Dp adn? = Fsivese ;
Qk! 2
A? = (by bz + bib; +... + dy ob)? = Feeeerd
Smet w ee ee mere e eer e ee eaee eee essere eeeereeeerereseseeeeeeeeeseerese
' 2k! 2
A? = (bb; ar bibj45 ap ooo ar by—jbx)? = Feel >
Cr ee i ee
O. ANDERSON 275
(Oy) (Oe Or) + (by Oy + 6") Fe. + (Oy + be + bP +... + Be 1)%,
B? = (by bi)? + (bob: + br be)? + (do bi + 1 02 + 6565)? +...
ar (by b, ar b,b, + bibs <font Op-20R-a) 5
Be = (b,b:)* + (bob; + Bibi gs)? + (Bob; + D1 Di4a + Dabiye)? +...
hy (by b; a b, bein + babi+2 tee + Dee Ueen)e
ee cy
By, a (bo by).
Wenn die Verteilung der « (und dies ist der fiir uns interessanteste Fall)
“normal” ist, so kénnen obige Formelnu betrachtlich vereinfacht werden. Da man
in diesem Fall # (&) gleich 3 [H (&) oder 30,* setzen kann, so haben wir:
3 (a — E («)P
1
4
Fir << das Fehlerquadrat 2 (vergl. Biometrika, U1. p. 276).
nv ;
> («; — M,) Qa!
” ial ee 1 ) > he 1 .
n-
var Cle Dae oder angenéahert eee
»” “2(n—1). ” ” (n = 1) ge ’ahner rae 1 ‘e
SN
ae): (85n — 88) 04 do!
> 6(n—1) P i 9 (n— 2) ‘ “4 n—-2°
n—-3
=f wy, .\2
PS eo) (231n — 843) oy! Sox!
e201 = 2) a 7 50 (n — 3)? 4 i, n—3°
n—k
> (A® x;
Fiir aa endlich kann man das Fehlerquadrat in solcher Form dar-
ma —*)
stellen :
@ oo Nin ~ k) + 2(n—b—1) (Epi) +2@-k-2) C Ses »)
+2(n—k—3)( k. (ke —1).(k— 2) ee
(k +1). (E+ 2). (E +3)
alle OSCE
(h+1). he +2). (6 +3)... 4g))
k.(b—1).(k—2)...2.1 yt
(k +1). +2). (6 +3)... (+h) 5°
+2(n-—k—jJ) (
+2(n—b—b) (
276 Die Verallgemeinerung der Cave-Hookersche Methode
Ks ist also klar, daB zusammen mit dem endlichen Differenzieren die Unsicher-
heit der Bestimmung von a,” stetig wichst, anfangs etwa im Verhaltnis
Nee NTR De ISY Ae
7. Das Fehlerquadrat des mittleren Produktes.
s Ei 2 7 2
urs == orpicbe sich das imonleriudine eerie ete
S (= Ma) (ys — My)
Fiir + ear ergiebt sich das Fehlerquadrat
B (Epi) — (BE P| Eis) P+ on2o,/
n n(n —1)
n—-1
> A'a;A’y;
Ace aa . Pee Pee f
Fir SON ea ergiebt sich das Fehlerquadrat
(2n — 8) {EB (E2r2) — LE (Ea +(e — 1) (LE EW) P+ 02207}
2(n —1) j
n-k
> AMa, Ay;
Im allgemeinen Fall Soi = ae erhalten wir fiir das Feblerquadrat folgenden
Sh)
kik!
Ausdruck :
Timp {48 — 28) UE EWE) — LE EP
+ 2 {LE (Eh) P + 0,70,7} [A (mn — 2h +1) 4+ A? (n-— 2k 4 2)4+ ... + Ae (n— 2k +k)]
+ 2B? (E (E2yr2) — LE (Ei) P} + 4 (LE (Esti) P + oe2o,?} (BP + Be +... + Bi} ,
wo die Koeffizienten A,’, A,’,... B,?, By, ... dieselbe Bedeutung haben, wie in § 6.
Wenn 2; und y; einander vollkommen gleich sind, so ist
[E (Emi) P={H(P)Paorts HEM A)=H EA); Foy = a7",
und obiger Ausdruck fallt mit dem in § 6 zusammen.
Fiir den Fall der “normalen” Verteilung kénnen wir auch alle diese Ausdriicke
betrachtlich vereinfachen, besonders wenn wir # (&;;) und #(&?.?) als Funktionen
von rz, darstellen. Olbne aber hier darauf einzugehen, wollen wir jetzt tiber den
Korrelationskoeffizienten ins Klare kommen.
8. Definition des Korrelationskoeffizienten.
Der Korrelationskoeffizient R wird gewoéhnlich nach der Formel
2 (a — Ms) (yi— My)
R= Z berechnet.
/S@- May 3 i,y
1 1
-O. ANDERSON 277
Zu welchem Ausdruck ist er nun als empirische Anndherung aufzufassen,
S (a; — Mz) (yi — My) H E (a; — Mz) (yi - m,)|
1 e 1
2
oder zu
zu = : :
r/ See —M,) S (y; - M,) Neo tas M,)"] E(S (y:— My]
1 1 1 1
Beide Formeln sind durchaus nicht mit einander zu identifizieren und fallen
nur in erster Anndherung zusammen. Da die zweite aber bedeutend leichter zu
handhaben ist und dies auch mehr den tiblichen Rechnungsmethoden der englischen
Schule entspricht, so definteren wir ry, als
EK E (a; re M;) (Yi -_ |
1
ve E E ee Mz) | E E (y;— My} | |
E : p :
Anders ausgedriickt ist yy = Pay , WO Dzy, Gz, Fy die Bedeutungen haben,
Oxy
welche wir ihnen oben in §§ 4 und 5 beigemessen haben.
9. Das Verhalten des Korrelationskoeffizienten zweier oscillierender Rethen
x und y, wenn man deren GroBen durch Differenzen ersetzt.
Fiir die k-te endliche Differenz von z und y haben wir
an ee : . Qe!
y a A! ao, A 4 } — i)
- ( 1 fe _ Paha, AMy kt. ey Pe
Afiz, AMy eg n—k nk = NE 2 2 = 9 ye
SR ae id A TAlkin + T Ah) 2k! 2k!
E\ > Ae2|.H| & A®y2 Ax ly 2: ee
paar ae lel lo” EL Ri’
So Gee
Tar, Ay = Guan oy Pay:
Wir haben also ganz allgemein das genaue Resultat :
Try = Ya'e, A’y = TA" x, A’y — TA" x, A”'y a T atk), Ally:
Da aber diese r unbekannt bleiben und wir fiir ein beliebiges Taide, A@y DUT
n—t .
AMZ, AMY,
dessen Anniherungsformel a, a —— kennen, so miissen wir
wiederum feststellen, inwiefern man sich in der Praxis auf die Ubereinstimmung der
empirischen Koeffizienten mit deren mathematischen Erwartungen verlassen kann,
wie gro$ also die Unsicherheit ihrer Bestimmung zu schatzen ist.
Biometrika x 36
278 Die Verallgemeinerung der Cave-Hookersche Methode
10. Mitileres Fehlerquadrat des Korrelationskoeffizienten der endl. Differenzen
zweier Rechen.
Aus der Formel &;=—-~ : , kann man folgenden Ausdruck
a
vi 5, Aa > A by
1 1
ableiten :
n—k n-k n—-k
> AMa A® y; > A a2 > Aye
- Parc k : or Alk : o
ae = ‘bt i
IR Ty ay Ac, Ay ni Aa m Aly
- —— ee, a) eT) aT aa ate p)
ee Pawea®y 20° Aix 20° Amy
der nur in erster Annaherung und, wenn alle 4 Briiche des Ausdrucks echte sind,
richtig ist; dies ist bei groBem n der Fall. Und ferner ergiebt sich daraus die
Formel :
+6 (1 =? xy)? ; Pe ee iden: ene k.(k—1) 2
Gas sisi i Ol eel =e " 2) ey cea
ig (Pan (er
+2(n— 22) agen (k +2). Gaal ox
k.(k—-1).(k—-2).. ;
+2(n—k— ®) (ary. (K+2).(k+38).. 21
"3 (Ama)
7
ii a = k)
(vergl. dazu die Formel fiir in § 6).
Die Formel fiir on ist immer nur dann giltig, wenn man
ea) ee (Eepe) —[H (Ea? , HE) — [LA (e)P ie E (p*) — [EB Gp?)P
n LE (Evi)? 4[H (é)] 4 [EB (p*)P
_ Ege) — Hkh) EE) BG) = BE) BOY)
_Beye)- E©). st
2H (&*). Bp’)
gleich Sal evar setzen darf (vergl. Biometrika, Vol. 1x. p. 4).
Aus der Formel fiir oR erhalten wir:
byes (1 = xy)?
Ry n ;
jar 1 Lata) 3n a 4
Ry n-1 °2(n—1)’
"i (l=) 7 80% 88
Rs a ee 718 (m=2)-
_Gd=—ry)? 231n — 848
Ryn —3 * 100 (n—8)’
u.S.W.
2
O. ANDERSON 279
Die Fehlerquadrate der Korrelationskoeffizienten aufeinanderfolgender Diffe-
renzenordnungen verhalten sich folglich zueinander ungefahr wie 2:3:4:5....
Die Unsicherheit wachst also mit zunehmender Differenzenordnung etwa im
Verhaltnis
N DEEN! O uA) AES Nf Dyions
ll. Korrelationskoeffizient zweier zusammengesetzter Rethen, die aus oscillato-
rischen und evolutorischen Elementen bestehen.
Da “Student” diese Frage treffend dargelegt hat, konnen wir uns kurz fassen.
Wenn wir in Betracht ziehen, daB fiir uns die evolutorische Komponente einer
Reihe schon dann in der Praxis verschwunden ist, wenn sie im Verhialtnis zur
oscillatorischen Komponente so klein geworden ist, daB sie nur die 3, 4", u.s.w.
Zahlenstellen des Ausdruckes fiir R beeinflussen kann,so kommen wir zum SchluB,
da8 nicht nur Komponenten, die durch eine Parabel héherer Ordnung darstellbar
sind, sondern auch solche, denen nur transzendentale Gleichungen (z. B. Sinus-
reihen) gentigen, beim endlichen Differenzieren eliminiert werden. Ja mehr noch,
man kann beweisen, daf iiberhaupt alle mehr oder minder “ glatten Reihen,” alle
bei denen eine geniigende positive Korrelation zwischen den Nachbargliedern
bemerkbar ist, fiir die Praxis beim endlichen Differenzieren verschwinden. Das
verallgemeinerte Cave-Hookersche Verfahren ist daher augenscheinlich ein sehr
universales Mittel, die Korrelation oscillatorischer Elemente aus zusammengesetzten
Reihen herauszuschiilen. Es hat aber einen Haken, auf den hier noch hingewiesen
werden muf.
12. Kann man aus dem Verhalten der Rethe R,, R,, R,, ... Ry bestimmen, ob wir
den Korrelationskoeffizienten rein oscillatorischer Reihen vor uns haben? “Student”
scheint zu glauben, daB8 wenn irgendein Rf; seinem Vorgiinger R;, gleich ist, wir
es sicher mit dem Korrelationskoeffizienten oscillierender Elemente zu tun haben.
Vor einem solchen Schlu8 ist nachdriicklich zu warnen. Wie es meine (fiir diesen
Artikel etwas zu langwierigen) Berechnungen zeigen, kénnen zwei Nachbarkoefti-
zienten h;, R;, auch bei stark evolutorischen Reihen einander ungefahr gleich
sein, und die Wahrscheinlichkeit eines solchen Zusammentreffens ist gar nicht sehr
gering einzuschatzen. Nur wenn wir, von irgendeinem #; angefangen, immer
dieselbe GroBe fiir R erhalten, also Aj = Rji.= Rj4.= Rj4;=, wird ein solcher
SchluB berechtigt sein, und je linger die Reihe gleicher R, desto wahrscheinlicher
wird dieser Schlu8.
36—2
STATISTICAL NOTES ON THE INFLUENCE
OF EDUCATION IN EGYPT.
By M. HOSNY, M.A., B.Sc.
The statistical returns for Egypt are—as compared with European data—still
in a somewhat elementary stage. Age-distributions are of very little value, and
in the case of infantile mortality we have only information for certain towns.
Further, in the larger towns there is a considerable cosmopolitan element, which
gives them a widely different character from the often sparsely populated rural and
desert districts. Education is not compulsory, and schools and literacy are largely
confined to Cairo, Alexandria and the Canal Government, even when we exclude
all foreign scholars. In the same way criminality* preponderates, in an inverse
order it is true, in these three districts, but it is not absolutely certain whether this
is due to their more efficient policing, to the presence of more foreigners, or to a
real absence of crime in the rural populations. Crime does not appear to arise in
Egypt from poverty or drunkenness, two of the main factors of its origin in
Western Europe. The criminal, indeed, is rarely habitual; he is an amateur,
rather than a professional, and criminals are more often well-to-do, their crimes
arising from motives of revenge or passion.
The fact that criminality in Egypt is highly correlated with literacy and
scholarship would be noteworthy and might possibly be used as an argument
against education, did not the association of crime and education arise from the
prevalency of both in the more populated districts, where again we find the
greatest abundance of foreigners. Naturally such questions arise as:
(i) Are the foreigners—and if so, which section of them—to any extent
responsible for the prevalence of crime in the districts frequented by them ?
(ii) If we allow for urban conditions, will there still be found a high asso-
ciation of crime and education ?
It is perfectly easy to obtain from the Egyptian Census-. we used that of
1907—the number of foreigners of each denomination in the various Egyptian
* We understand by “criminality” in this paper, not commission of but conviction for crime.
M. Hosny 281
governments. The only difficulty here was the presence of British troops in
Cairo and Alexandria, which placed that nationality in an anomalous position.
These were estimated approximately and subtracted. The following groups of
foreigners were then dealt with: (a) Ottomans, (b) British subjects, French,
Austrians, Germans and Russians*, (c) Greeks, (d) Italians. The Greeks and
Italians were separated from the general European group (b), because they
are largely differentiated, the Greeks being frequently small traders and the
Italians often manual workers. Their large numbers also justified a separate
classification.
Table I gives the foreigners per 10,000 in the 17 Egyptian districts we
were able to deal with. It will be noted that the Greeks far outnumber other
TABLE I.
Foreigners per 10,000 and Population per sq. kilometre.
Europeans other | Population
Governments Ottomans | than Greeks and Greeks Italians | per sq.
Italians | | kilometre
Cairo .... 0 a 453, 312 | 298 204. 6060
Alexandria oe god 661 514 745 482 6780
Canalt ... Bas aon 416 583 846 445 7666
Beherat ... Si hee 43 23 31 11 178
Charkieh ae ae 29 5 24 | 1 257
Dakahlieh and Damietta 18 5 | 18 3 346
Gharbieh 2 0) g 0) 226 |
Kalliuhieh 8 3 13 2 | 469
Menufieh 2 1 7 0 | 618
Assiutt ... + gz 3 il | 454
Assuan ... he 8 5 13 | 4 | 533
Beni Suef Aeneas 15 6 9 1 | 351
Fayoum ... 10 3 4 0 | 255
Gerga 2 0 2 0 532
Guizeh .. 6 6 4 33 447 |
Kenat ... 5 4 4 2 339
Miniat 10 5 7 1 458 |
foreigners, but that all foreigners are concentrated in the Cairo, Alexandria and
Canal governments.
It was far more difficult to obtain a measure of urban conditions. We had
to take very rough measures of the density of the population, because the limits
of certain areas are too vaguely defined to be of any service. El] Arish has been
excluded from the Canal district, Suez and Sinai have also been excluded as there
is no enumeration of them with respect to criminality, literacy and scholarship.
These densities, w.th such value as they have, are given in the last column of
Table I.
* The contributions from other smaller nationalities were omitted.
+ Various approximations and omissions occur in these cases in obtaining density.
282 Statistical Notes on the Influence of Education in Egypt
Table II provides the number of male criminals per 1000 of the male popu-
lation, the literacy or number of male persons able to read and write per 100 of
TABLE II. Hducational and Criminal Indices.
| Male | Titer Male Scholars
Governments Criminals per : er acy. 5—19, per 100
| 1000 males | P* 100 males boys of those ages
Cairo ar er “ae 12°90 28°03 30°20
| Alexandria ae aA 14°15 | 30°09 19°99
Canal and E] Arish ... 22°30 23°39 8°54
| Behera ... ae Sac 5°30 | 9°29 1-01
| Charkieh aie ae 4°20 9°09 1°66
Dakahheh and Damietta 4°35 5 {salts} 1°76
| Gharbieh ; a 5°65 8322, 3°04
| Kalliuhieh a fs? 6°65 8:13 1°39
| Menutieh bee ae 3°85 8°45 1:06
Assiut sate 5°45 701 4:02
Assuan ... Are well 4°20 7°68 0°82
Beni Suef Dial 8:42 2°03
Fayoum ... 6°85 6°54 1°85
Gerga 3°95 5°84 2°22
Guizeh A 5:10 6°38 1:15
|} Kena... Re ie 3°45 5-34 1°66
| Minia 5:20 (es Sule
the male population*, aad the number of male scholars aged 5 to 19 per 100 of
the native boys of those agest.
We shall use the following symbols to denote the factors which occur in
Tables I and IT:
O = Ottomans, G=Greeks, J = Italians,
= Europeans other than Greeks and Italians.
C= Criminality, 2 = Literacy, S=Scholarship, D = Density of Population.
Each government was treated as of equal weight, although the populations
vary from 233,000 in Assuan to 1,485,000 in Gharbieh. The standard-deviations
and product-moments were found without grouping. The following results were
obtained :
Means Standard Deviations Correlations
No = TO15, el 4-791, Noh + 8450) aE ‘0468,
1 — WOO, op = CoAl: Tos = + 6242 + 0999,
mg = 5031, og = 7'735, rrs = +9028 + 0308,
Mp = 1528, op = 2475°5,
Correlations :
rpc = +9614 + 0124, rps= + °8097 +:0566, = rp, = +9563 + 0138.
Now at first sight these results would seem to indicate a very bad influence of
education on crime. Where literacy and scholarship are greatest, there criminals
* Kgyptian Census, 1907, p. 99.
+ Foreign male scholars are excluded in the case of Cairo, Alexandria and the Canal. They have
no sensible numerical existence elsewhere. Criminals and scholars are taken from the Annuaire Statis-
tique de VEgypte, 1912, pp. 95 and 135.
M. Hosny 283
are most numerous! And a superficial argument might be used to condemn the
character of education in Egypt, or education in general. But it will be clear
on examination of the isolated values that the observed high correlations arise
solely from the urban character in Egypt of both criminality and education. We
have endeavoured therefore to correct this by finding the partial correlations for
constant density of population.
There now result
prog = — 9554 + 0143,
pcr = — 9231 + 0242,
blips = +7480 +0721.
Thus, while there still remains a quite considerable relation between the prevalence
of literacy and scholars for constant density, we find that for a constant degree of
urban conditions, the greater the literacy and the greater the amount of education
the less will be the criminality. The negative correlations are now even higher
than the uncorrected positive ones and of course are markedly significant. While
admitting the slender nature of the Egyptian data, we think that this swinging
over of the relation of crime and education when we correct for density is sug-
gestive, and it would be of interest to work out similar correlations for states
in which the statistics are of a more ample character. It does, however, appear
reasonable to assert that there is no evidence to indicate that education leads
to criminality—rather the reverse—in Egypt.
We will next consider the influence of the presence of foreigners in Egypt.
We find:
Means Standard Deviations Correlations
Mo= 99°53, oo — 195-60; Too = +°8425 +0478,
Mp= 86°88, Bea lleeyidy rao = +°9546 £0145,
Mg = 119-41, og = 25661, roo= +9429 +:0181,
m= 68°24, oa, = 152-04, rio= +9192 + 0254.
Correlations :
po = +°9575 +0136, rpm = +:9844 + 0050,
Ppa = +9491 + 0162, rpr= + ‘9617 + 0123.
Here, if we judged by the raw correlations only, we must assert that the corre-
lations of crime with the presence of foreigners are so high, that the foreigners
must be corrupting the Egyptian population. But again the association only
arises because the criminals and foreigners are both prevalent in the big towns.
If we correct for density of population, we find the results are very different. Thus
we have:
broad >= — ‘9811 +0061, preg=t+ "1692 ap ‘1591,
p’ac = +°3524 + °1483, pro = — 0718 + 11628.
It is now obvious that the correlation of Europeans other than Greeks and
Italians with criminality has become insignificant having regard to its probable
error; the correlation of the presence of Italians and criminality is now negative,
but less than its probable error. Thus of Christians only the presence of the
284 Statistical Notes on the Influence of Education in Egypt
Greeks may possibly, but not certainly, be detrimental. The Ottomans have
now a large negative correlation of a quite significant character, or we might
assert that the presence of Ottomans tends to diminish criminality. The Greeks
are frequently moneylenders and alcohol dealers, and the Ottomans, especially
the Arabs, have among them a good many religious teachers.
We have, however, to note that criminality is greatest in the Canal Govern- .
ment, where Europeans and Greeks are most frequent, while the Ottomans are
most numerous in Alexandria, where crime is almost 40% less than in the Canal
Government. To test the influence of the three densely populated governments,
we put the Canal proportion of the Ottomans at Cairo, that of Cairo at Alexandria
and that of Alexandria at the Canal. There resulted:
Toc = +°9707, instead of +°8425,
Tpo = +'9870, instead of + °9575,
leading to pYoc= + °4918,
or we may safely say, that if the proportions of Ottomans at Alexandria and along
the Canal were interchanged, then no relation between the presence of Ottomans
and the absence of criminality would exist, indeed the relation would probably be
reversed. The prevalence of the Ottomans in Alexandria has been attributed to its
more temperate climate. There is certainly a large Ottoman element in Alexandria,
there being 21,827 Ottomans out of a population of 332,246, and it is larger
than any other foreign element except the Greeks. In Cairo, with 29,516 out
of 654,476 inhabitants, the Ottomans exceed any other single foreign element.
It is conceivable, therefore, that they may be able to influence the moral tone of
those towns. It must be borne in mind, however, that crime is far more frequent
in the Cairo and Alexandria governments than in the more purely rural districts,
and we can scarcely suppose that Cairo and Alexandria would reach the still higher
criminality level of the Canal, were it not for the presence of the Ottomans. In
the Canal Government there are exceptional conditions, and we can hardly assume
that a transfer of the Ottomans from Alexandria to the Canal would interchange
their proportions of criminality. Greeks no doubt flock to the Canal for business
purposes, the other Europeans largely for control purposes; the Ottomans,
relatively speaking, avoid it. Without further analysis it would not be possible
to assert definitely that the presence of Ottomans reduces crime. It may be
doubted whether the presence of foreigners, with the possible exception of the
Greeks, is really associated with the extent of criminality in Egypt.
A further investigation was undertaken in regard to the possible influence
of education on infantile mortality. The birthrate and deathrate in Egypt are
both remarkably high. Thus for the years 1899-1909 inclusive the average
rates were:
Births per 1000* Deaths per 1000 Excess of Births over Deaths
Cairo 40°7 35°7 +50
Alexandria 38:0 31°8 + 62
* Still births not included.
M. Hosny 285
Many European towns with half the above birthrates have considerably greater
excesses of births over deaths.
Unfortunately the infantile mortality is only recorded in Egyptian towns and
not in the governments at large or in the rural districts. We are obliged, there-
fore, to deal with these only when considering the relation of education to infantile
mortality. From p. 49 of the Annuaire Statistique de V Egypte, 1912, we obtain
the infantile mortality for 1911, and note at once how extraordinarily high it
stands. From p. 286 of the Census of Egypt, 1907, we take the percentage of
male literates in the total male population, and from the Statistique Scolaire,
1912-1913, p. 74, the percentage of scholars in the total population*. As it
was possible that the density of town population might influence the results, we
took the number of persons per house, which was about the only social factor
available. . This will probably represent fairly closely the average size of family.
This was taken from the Census, 1907, p. 286. The mean, 5°64 persons per
house, suggests that the average number of living children can hardly exceed
three. The marked relationship that occurs in European towns between gross size
of family and infantile mortality cannot be satisfactorily tested on the Egyptian
data, because we cannot ascertain the infantile mortality in each size of family.
The number of persons to the house is indeed rather a measure of net than
gross family, and we only know this as an average value for each town. It does
not follow that a town with a low number of persons per house is one with
small gross families; the low number may be due to the heavy infantile mortality
itself. Accordingly the correlation between persons per house and infantile
mortality is not necessarily even a measure of the influence of overcrowding on
infantile mortality (although this is often supposed to be the case); it is con-
ceivable that a high infantile mortality might be the source of a low number of
persons per house, and the unravelling of cause and effect is only possible
where we know not only the number of persons per house, but its relation to
both the gross and net family of that house.
Let J = Infantile mortality, Z = Literacy, S = Scholars, P = Persons per house.
Then we have the following results :
Means Standard Deviations Correlations
M, = 29°30, o, = 7608, ry, = —'1040 + 1500,
M,, = 21:96, 7 ouligue Trg = +5809 + 1278,
M,= 5°569, og = 2951, ris = +0093 + 1508,
Mp= 5°643, op = 1428, oe
Correlations :
rprp= + 1675 +°1487, rp, =— "842141487, rpg = +0296 4 °1508.
* The scholars were taken for 1910-1911, the year of infantile mortality, but this involved
the assumption that the foreign scholars were the same in numbers in 1910-11 and 1912-13, probably
not a very inaccurate assumption, which in any case affects little more than Cairo, Alexandria, Suez
and Ismailia practically. It is the number of Egyptian scholars that is rapidly changing and the
scholars dealt with in our ratio are Egyptian only.
Biometrika x 37
286 Statistical Notes on the Influence of Education in Egypt
TABLE IIL.
Infantile Mortality. Persons per House and Education.
Infantile Male Literacy ‘ cholars Persons
Town Mortality per 100 of per 100 of per
per 100 births population population House
Cairo ee. 32°9 28°03 6°12 4°62
Alexandria Sec 26°9 30°09 3°41 8°43
Damietta 18°1 7°06 1°89 6°96
Port Said ae 21:0 24°13 2°64 4°14
Ismailia ... 26 16:0 28°05 1:22 4:10
Suez oe See 26°9 25°74 0°64 3°85
Benha ... sre 29°6 20°54 4°87 5:47
Zagazig ... we 27°9 25°67 6°47 6:07
Tantah ... ae 29°6 25°62 9°45 5°18
| Mansorah sae 21°4 26°77 6°20 4°60
| Chibine El] Kom 16°1 18°71 5:10 4°92
Damanhur ee Oieo 19°54 3°55 7:27
| Guizeh ... a 35°6 19°23 eS 6°12
| Fayoum ... fee 40°1 15°55 4°78 9°34
Beni Suef Sur i7/ 2il 21°15 7:19 511
| Minia... ae 38°2 21°96 8°89 4:96
Assiut ... ist 33°6 20:92 11°48 6:00
Sohag... ee 29°0 DOS | 5°58 6°15
Kena see oe oe 16°76 4°80 5:14
Assuan ... ae 41°1 21°31 5°78 4:09
hi 1
It will be clear from these results that there is no significant relation between the
literacy of the male population and infantile mortality. There is also no significant
relation between the number of persons to a house and the number of scholars,
Le. it does not appear to be the more crowded towns which have the largest
percentage of scholars to the population ; Alexandria and Damietta, for example,
have considerably more than the mean number of persons to the house and
relatively few scholars. On the other hand a larger number of literates marks
less crowding. Crowding and infantile mortality are slightly related, but con-
sidering the probable error, not with definite significance *.
While literacy has no relation to the infantile deathrate, it is noteworthy
that there is a significant correlation (+°53809 +°1278) between the number of
scholars and the infantile deathrate, which is greater where there 1s more education.
Now this either suggests that many scholars mean large families and large
families correspond to increased infantile mortality, which is usual, or that the
towns in which there are the classes who educate their children have a higher
infantile deathrate. The only means, and those inadequate, of testing the first
* This agrees with the result for overcrowding and infantile mortality in English manufacturing
towns, where the correlation is very small and sometimes has one sign and sometimes the other.
M. Hosny 287
assumption are to take the partial coefficient between scholars and deathrate for
constant number of persons per house.
We find*: pris => + 5336 + 1107 5
similarly pry, = — 0504 + 1543.
There is thus a slight increase in the relation of scholars to deathrate when
we take a constant number of persons per house, and it is hard to believe that
the relation is indirectly due to size of family. The second result shows that
literacy has no relation to the infantile deathrate. Towns like Alexandria,
Damietta, Port Said, Ismailia, and to a less extent Suez, with a low infantile
deathrate have a low education rate, and towns like Cairo, Guizeh, Beni Suef,
Minia, and Assiut, with high infantile deathrates have high education rates. The
first towns are on the sea or the canal, the second in the Nile Valley; it is con-
ceivable that the latter are the more unhealthy for the infant; it would need
special local knowledge to explain why education has been most accepted above
Cairot. There does not, however, seem any relation between ignorance, as
measured by literacy, and a heavy infantile mortality, nor on the other hand can we
assert that education and European influence have certainly increased criminality.
* The value of p7,5 is + °0207 +1508, and is therefore not significant.
+ It is noteworthy that there is no relation between literacy and number of scholars, i.e. education
of children does not appear to follow the power to read and write in their parents, that is to say
if we judge by the averages in towns and not by individuals.
37—2
HEIGHT AND WEIGHT OF SCHOOL CHILDREN
IN GLASGOW
By ETHEL M. ELDERTON, Galton Fellow, University of London.
In 1905-6 an enquiry was made in the Public Schools of the School Board for
Glasgow as to the height and weight of all the scholars, the occupation of the
parents, the number of rooms occupied etc. By permission of Sir John Struthers,
of the Scottish Education Office, these schedules were most kindly placed at the
disposal of the Galton Laboratory.
The number of children concerning whom the enquiry was made is over
seventy thousand of ages 5 to 18 years. The schools from which these children
came were divided into four groups according to the district in which the schools
were situated.
Group A comprised schools in the poorest districts of the city.
ee a » in poor districts of the city.
ee. ‘i , in districts of a better class.
eed 3 » in districts of a still higher class with which are
included four out of five Higher Grade Schools.
The data were originally used by the Galton Laboratory with the object of
discovering how far the physique of school children, judged by their height and
weight, is affected by the occupation of the father and the employment of the
mother. With this end in view the necessary data were entered on cards.
Children over 14 were excluded and all children who had not both parents alive
were also excluded; this left us with 30,965 girls and 32,811 boys.
The object of the present paper is to ascertain what is the average weight of
a child of a given age and a given height.
The first step in this enquiry was to sort the cards and form tables giving the
distribution of weight for each height at each age in each school group, and this
laborious work was carried out very largely by Miss Augusta Jones; this step
necessitated making 72 tables, and she is responsible for 58 of them while the
EK. M. EvprErtTon 289
remaining 14 are due to Miss H. Gertrude Jones; I have to thank most heartily
these colleagues for their very efficient help in this matter.
The three factors with which we are concerned are the age, height, and weight
of school children. The instructions issued to the teachers in the schools for
recording these three facts were that ages were to be given to the nearest year,
weights to the nearest pound, and heights to the nearest quarter of an inch*. The
method of recording ages is very important. Ages being recorded to the nearest
year, this means that children classed as 6 years were from 5°5 years to 6°5 years ;
and the average age of this group was 6 years; this is not the method most
frequently employed for recording ages; “age last birthday” is generally used
and if “age last birthday” be given as 6 years then the children of that age are
from 6 to 7 and the average age of children in this group is approximately 6°5 years.
It will be seen at once that a comparison of weights and heights of two groups of
children of 6 years cannot be undertaken until we know which method of recording
ages has been adopted. The height and weight of these Glasgow children have been
compared by Dr Leslie Mackenzie and Captain Foster+ with the height and weight
of children as given by the Anthropometric Committee of the British Association
and it is pointed out that at each age the average weight of the children is
uniformly below the “standard of the Anthropometric Committee,” and that
generally speaking the same thing applies to height. As a matter of fact this
point as to age has not been noticed by these writers and children whose average
age is 6 years in Glasgow are compared with children whose average age is
65 years, naturally the younger children are shorter and lighter. There is further
an important question to be asked: Which standard of the B.A. Anthropometric
Committee ought to be selected? To this point I return below.
As I have said the Glasgow children’s ages were recorded to the nearest year},
but the Anthropometric Committee recorded age last birthday, and before these
children can be compared the six months extra growth must be allowed for. This
is quite easily done by finding the regression of height and weight on age and
adding half the regression coefficient to the height and weight of the Glasgow
children. We have found the regression for children of 5 to 14 inclusive to be as
follows:
Boys Girls
Regression of Weight on Age ... 4564 4916
5 Height on Age foe 1:807 1937
* Tt is not known what record was made when an exact half year, an exact half pound or an exact
quarter inch occurred.
+ Report on the Physical Condition of Children attending the Public Schools of the School Board for
Glasgow, by Dr W. Leslie Mackenzie and Captain A. Foster. Wyman and Sons, 1907.
+ The actual wording of the Glasgow direction to school teachers runs: “In recording age, disregard
months and record to nearest year; thus 6 years 7 months record as 7 years, 8 years 3 months
record as 8 years.” It is not clear how 6 years 6 months would be recorded; we have assumed as no half
years are entered in the schedules that an exact calculation was made in the case of each child of
doubtful age to ascertain whether it was or was not past the half year.
290 Height and Weight of School Children in Glasgow
This means that we must add 2°28 lbs. to the weight of the Glasgow boys and
‘90 inches to their height and 2°5 lbs. to the weight of Glasgow girls and ‘97 inches
to their height before we can compare them with the Anthropometric Committee’s
standard. The Glasgow children still fall below the “ Anthropometric Committee’s
Average” but not to the appalling extent shown in the diagram at the end of
Dr Mackenzie’s and Captain Foster’s Report. Personally I should hesitate to
compare actual height and weight of Glasgow school children with the so-called
Anthropometric Committee’s standard. The so-called standard is taken from the
Final Report of the Anthropometric Committee of the British Association, 1883.
In Tables XVI—XIX, the average heights and weights at different ages of males
and females of different classes of the population of Great Britain are given. For
example in the case of stature we have four classes: Class I, Professional Classes,
Town and Country, 10,739 individuals, ages 9 to 60; Class II, Commercial Classes,
Towns, 5472 individuals, ages 8 to 60 (5 below 8 are of no service for means);
Class III, Labouring Classes, Country, 8727 individuals, ages 3 to 70 (8 below 8
are of no service); Class 1V, Artizans, Towns, 126,236 individuals, ages 3 to 60, and
451 babies at birth. All these data are pooled and the column headed “ General
Population, All Classes, Town and Country,” and it is this “General Population ”
which is so frequently cited by various medical authorities, including Dr Leslie
Mackenzie and Captain Foster, as the Anthropometric Committee's “standard.”
What they understand by such a “standard” it 1s impossible to say. It does not
represent the “General Population” of Great Britain, but the total population
measured by the Committee. In this all the babies are artizan babies, there
are only 8 children from 0 to 2 and these belong to the labouring rural classes, and
there is no professional class contribution until after 9 years of age. Then the
various age groups are made up from various social classes in proportions which
bear no relation whatever to their actual proportions in the kingdom at large. For
example, the average height of lads of 18 is determined from 1724 of the pro-
fessional, 62 of the commercial, 148 of the rural labourer, and 371 of the town
artizan classes! It will be quite clear that a “standard” reached in this way
means absolutely nothing at all, and yet this is the “standard” which, attached to
numerous weighing machines is posted in innumerable public places up and down
this country. It does not in the least represent any “General Population” of
Great Britain. To be a standard of the general population each class should have
been properly weighted, and this cannot be done as in certain classes certain ages
are quite inadequately represented, or not represented at all. There is in fact no
such thing as an “ Anthropometric Committee’s standard” for either height or
weight. The only thing that is possible is to compare the corresponding social
class in that Committee’s measurements with the measurements under considera-
tion. In the case of Dr Leslie Mackenzie’s and Captain Foster’s data, this is
undoubtedly the Class IV, “ Artizans, Towns.” Such a comparison is made in the
accompanying diagrams. It will be seen that the Glasgow children as far as
height is concerned are the equals if not the superiors of the Anthropometric
EK. M. ELpERTON 291
Committee’s artizan class. In weight they appear to be somewhat less, but here
Dr Leslie Mackenzie and Captain Foster have overlooked the fact that the Glasgow
children were weighed without boots, but the British Association Committee weighed
in ordinary indoor clothing, i.e. with boots or shoes on. Now girls’ boots weigh
as much as 11 to 21 lbs. and boys’ boots 1? to 34 lbs.* Hence in comparing
children in Glasgow with those six months older, Dr Leslie Mackenzie and Captain
Foster have dropped 21 lbs. in weight, while in comparing children without boots with
those with boots they have dropped another 14 lbs. to possibly 3$ lbs. We should
anticipate therefore that their readings would be 3} to nearly 6 lbs. too small f.
There is in our mind very little doubt that the weight of the Glasgow children is
at every age equal or superior to the weight of the artizan children measured
by the British Association Anthropometric Committee and the statement of
Dr Leslie Mackenzie and Captain Foster that “at each age from 5 to 18 the average
weight of the [Glasgow] children is uniformly below the standard of the Anthro-
pometrical Committee{” arises from their having entirely overlooked the con-
ditions as to class, age and manner of weighing which were adopted by that
Committee, a knowledge of which was essential to any comparison with the Com-
mittee’s data. In the diagrams on pp. 292-3 we have given the Glasgow measure-
ments set against those of the artizan class of the Anthropometric Committee,
and the reader will see clearly how all the arguments based on differences between
the Glasgow and the “ Anthropometric standard” fall at once to the ground.
There is nothing exceptional in the Glasgow data, they differ of course from data
for the children of the professional classes, but this difference is not confined to
Glasgow. Apart from this point it is essential that the ages of the two groups of
children should be the same and not differ by six months.
In the data used for this paper, children of 5 were omitted; they are few in
number and are not therefore likely to give such reliable results when each age
group is used separately. The mean weight for each height in inches was then
found and the regression equation calculated. These equations are given in
Table I. It will be observed from these equations that, though some irregularities
occur, generally speaking weight increases more rapidly for a given height in the
better school groups, at the later ages, and for girls more than boys except at
ages 6 and 7. ;
We can see from these equations that the multiple regression surface for
weight on height and age is not absolutely planar. It can be shown that it is
* New ‘‘tacket ” boots for girls of five in Glasgow weight 1 lb. 5 oz. falling to about 1 lb. 3 oz.
when the tackets are worn down ; for girls of fourteen 2 lbs. 6 oz. falling to about 2 lbs. 2 oz. For boys
of five years new tacket boots weigh 1 lb. 14 oz. falling to about 1 1b. 11 0z. when worn down; for boys
of fourteen the former weigh 3 lbs. 9 oz. and the latter about 3 lbs. 30z. We have to thank Dr Chalmers,
M.O.H. for.Glasgow, for this information.
+ Many public elementary school children have great masses of metal on their boots. Undoubtedly
the older children have heavier boots, and we can see from the diagrams that the divergence of the
Glasgow children from the Anthropometric Committee’s artizan children increases with age.
£ Report, Scottish Education Department, 1907, p. iv.
292 Height and Weight of School Children in Glasgow
| WRIGHT AND HEIGHT OF GIRLS
90 Glasgow Data, All Schools.
B.A. Anthropometric Committee,
85 Artizan Class.
65
Weight in lds,
>
Ss)
Ot
Or
Sera
Ge Ve
S)
Height in inches.
a eat nant tte
E. M. ELDERTON
9} WEIGHT AND HEIGHT OF BOYS.
90 Glasgow Data, All Schools.
B.A. Anthropometric Committee,
Artizan Class.
293
oS!
Or
Or
Weight in lbs.
or
oO
Orv oO
oO Fe
HHIGHT.
abe) 6 a 8 9 OMe = Ae Ss) U4, tS
Biometrika x 38
Height in inches.
294
TABLE I.
Height and Weight of School Children in Glasgow
Glasgow.
| Age Group 4, Boys Group B, Boys Group C, Boys | Group D, Boys
6 | W=—-21:16441:503H | W=—22°57641532H | W=— 24:4894+1591H | W=— 25°6784+ 1-603
7 | W=—-21-0654+1519H | W= —26:308+1°635H | W=— 27°39041667H | W=— 34:819+1°818H
8) W=-19°38141:-495H | W=-28:127+1695H |W=—- 6°64041:227H |W=—- 3277741 790H
| 9| W=—24°65241:635H | W=—32°8264+1-818H4 | W=— 20°67141562H W=-— 36:030+1:883H
10 | W= —29°589+41-768H | W=—35°696+1:899H | W=— 42°94242°055H | W=—- 51:63642:218H
| 11 | W=—54:91042302H | W=—39°72542-:005H | W=— 43-628+2-:088H | W=— 67°66442°546H
| 12) W=—49°5134+2:217H | W= —62:8904+2-476H | W=— 55:°386+42°337H | W=— 62:052+2-450H
| 13 | W=—65:467+2°547H | W=—-63°51642511H | W=— 81:342+2°854H | W=—- 99:908+43°160H
14 | W= —83°7844+2°888H | W=—76°749+2775H | W= —103°661+3°251H | W=— 126°446 + 3°633H
|
Age Group 4, Girls | Group B, Girls Group C, Girls Group D, Girls
6 | W=- 14561413298 | W=—-15:98541:345H | W=— 23°72841:551H | W=—- 27°5634+1624H
7 | W=-19°762+1465H | W= —24:13041:556H | W=— 29°312+1693H | W=— 30°22641694H
8 | W=—20-7214150383H | W= —30°6224+1-718H | W=— 29°113+1:691H | W=— 40°156 +1:927 H
9| W=-30°1334+1:730H | W= —29-0814+1°709H | W=— 38°62441917H | W=— 45:0664+2-045H
10 | W=—36-4784+1:878H | W=—38:8664+1:925H | W=— 46-263+2-088H | W=- 62:015+2397H
11 | W=—-48-70742:153H | W=- 43:005+2:034H | W=- 51:146+2209H | W=— 57°754+2'330H
12 |, W= —58°2774+2°360H | W= —63:908+2°465H | W=— -77°316+2°735H | W=— 84:298+42°8597
13 | W=—%74:15642694H | W=-—88-043+2:939H | W=— 83:960+2:892H | W=—103°594+3:229H
14 | W=-95:4644+3:084H W=-84:49642:906H | W= —106:1334+3°317H | W= —134:197+3'804H
; |
W is weight in lbs., H is height in inches.
The ages are central ages, and to obtain the weight corresponding
the child should be taken to the nearest whole year.
weight on age and not height on age which is non-linear.
to a given height
The departure from
linearity is not great, but Mr H. E. Soper, in order to smooth the material, fitted a
parabolic surface to the regression surface of weight on height and age.
Let W be as before the weight in lbs., H =height in inches, and y equal the
age of the child measured from 10*.
linear.
Then
W=—¢.(y) + ¢o(y) H
is the form of the surface when the relation of W to H for a given age is sensibly
Mr Soper now assumed:
hi(Y= At nyt+t ay, do(y) =b + hy + boy?
and determined a, a, and as, b), b; and b, so that:
> {n (bi (y) — a — Hy + aoy”)} = minimum,
> {n (db. (y) — b, — bry — b.y?)} = minimum,
where » is the number of individuals in any age group.
* Thus y takes every value from — 4 to +4 and we have nine equations to deal with.
Biometrika, Vol. X, Parts I and III Plate XIX
EPORT . 1906)
(CLASS B, CLASCOW R
. Model of Regression Surface, giving mean weight of Girls of Class B of Glasgow Schools for a given
Height and Age. The mean weight is the vertical coordinate and each section parallel to the front
of the model gives the mean weight for the several Heights of Girls of a given Age. See p. 295.
E. M. ELpERTON 295
Now ¢,(y) and ¢,(y) for given ’s are the values determined in Table I for
the constants at each age of the regression lines
W=-—-A+8H,
and n is the number of children dealt with at each age. Our type equations are
then of the form:
Up + 4a, % (ny) + 4a.> (ny?) = 42 (A),
hay’ (ny) + 4c, (ny?) + ae (ny!) = 43 (Ay)
4a) & (ny?) + 44, (ny?) + ta. (ny) =1> (Ay),
and similar equations for b,, b,, b., with B for A.
When these constants had been determined we put Y=10+y and obtain the
equations given in Table II.
TABLE II.
Glasgow. School Children.
W= Weight in lbs. Hf =WHeight in inches. Y=True age.
Boys
Group A W= {02181 242-214 —-2554Y} x H- (11327524 67°542-14-7417 F},
B W= {01533 Y2+1:900—-"1493V' x H— { 83314Y7+ 53:101- 9°8662Y},
C W= {03990 Y243-614— 5796 Y} x H - {2:08397 Y2+4 139-456 — 31-6570 F},
D W= {02983 Y?+2°799 — 3636} x H— {176624 Y24111-407 —24-0174Y?.
”
Girls
Group A W= {01880 Y?4+1°657—1644V} x H— { :96454¥?+438:119— 9°7305 7},
B W= {02081 Y?+1:907 —-2043 Y} x H— {1:16330 V2 + 60-230 — 13°6925 7,
4 C W= {02315 V242°222 —-2457YV} x H— {121642 VY? + 67626 — 14°3416)",
D W= {02701 Y?+2:385 — -2832V} x H— {1:56165 VY? + 87-624 — 18°9239 Y}.
A model of the surface for Glasgow Girls of Class B has been made by
Mr Soper. Allowing for the points based on few observations at the end of each
regression line of weight on height for constant age, the scroll represented by black
threads is quite a good fit to the observations represented by card sections. The
model is, however, difficult to photograph in a manner which shows effectively the
approximation of the thread scroll to the cut card sections. The reader should
note that an additional thread is placed between each of the threads which
graduate the regression lines for the different ages.
Further eight tables (Tables a—0) have been constructed in order that the
average weight of any boy or girl of a given height and age can be read off at once.
See pp. 8300—303. It has been stated before that the age groups in Glasgow are
from 5°5 to 6°5 years etc.and that 6,7, 8 etc. are the centres of each age group,
38—2
296 Height and Weight of School Children in Glasgow
but since frequently the centres are at 6°5, 7°5 etc. we have constructed Table III
which enables anyone to find mean height and weight at any age between 5°5 and
14°5 years. The regression lines are calculated from the original tables in which
children of 45 to 5°5 years were included. The regression lines omitting the
TABLE III.
Glasgow.
| Mean Height : Mean Weight
y Ge es sl er D Ae a8 G D
Boys: 5:°5— 6°5 41°3 42°1 42°] 43°0 40°9 42°0 42°'5 43°3
6Ob— 7h 43°0 44°0 44:0 44°8 44°2 45°6 45-9 46°6
T'5— 8:5 45°1 45:9 46°2 46°9 48-0 49°6 50°1 51°2
8'h— 95 47°0 47°7 48°1 49°0 52°3 53°9 54°4 56°3
9:5—10°5 48°8 49°5 49°9 50°9 56°7 58°4 59°5 61:2
10°5—11°5 50°6 51°1 51°5 52°6 61°6 62°7 63°9 66°3
11°5—12°5 52°3 52°8 53°5 54°2 66°4 67°8 69°1 70°8
12°5—13°5 53°8 54°3 55°0 55°9 lier 72°9 75°6 76°9
IZ*5—14'5 55°2 55'5 572 57°7 75°6 77°3 82°2 83°2
Regression on Age | 1800 ins. | 1°728 ins. | 1°847 ins. | 1°846 ins. | 4°305 lbs. | 4°395 Ibs. | 4°772 Ibs. | 4°914 lbs.
Girls :
5'b— C5 41°0 42°0 41°9 42°7 39°9 40°6 41°3
65— 75 42°9 43°7 43°7 44°8 43°0 43°9 44°7
7'5— 8:5 44°6 45°6 45°6 46°4 46°4 47°7 48°1
S:5— 95 46°6 47°4 47°6 | 48:6 50°5 51°8 52°7
9'°5—-10°5 48°5 49°2 49°4 50°4 54:7 55'8 56°9
10°5—11°5 50°3 51°1 51:2 52°2 59°5 60°8 61°9
11°5—12°5 52°4 53°0 53°3 54°1 65°3 66°8 68°4
12°5—13°5 54°4 55°2 55°4 56°5 72°4 74:3 761
IS*5—14'5 55°8 57°71 57°0 58°7 76°8 81°3 83°0
Regression on Age | 1°914 ins. | 1°859 ins. | 1°903 ins. | 1°943 ins. | 4°551 lbs. | 5-083 Ibs. | 4-944 Ibs. | 5°489 lbs.
41°8
45°6
49°3
54°3
58°8
64°4
70°5
78°8
89-0
children of 4°5 to 5°5 years were worked out for height on age and weight on age
for boys in Group A, and were found to be 1°81 instead of 1°80 for height and
439 instead of 4°31 for weight, but such differences are not great enough to
matter and the remaining regression coefficients were not calculated with children
of five years excluded.
In connection with the tables (1 to 72, pp. 304—3839) it should be noted that in
transferring the data for boys from the original sheets to cards, ‘75 of an inch was
included in the inch above; for example, 30°75 inches was entered as 81 inches
and the centre of the group of 30 inches is 30°125 inches. The data for the girls
KE. M. ELpERTON 297
were transferred to cards much later and the simpler method was employed, and
30:75 was included in the 30 inch group and the centre of this group is 80°375.
Through the kindness of Dr Priestley, School Medical Officer for Staffordshire,
we have been able to obtain the regression of Weight on Height for certain
age groups of boys and girls in that county. Staffordshire is a county of very
various occupations and contains an agricultural as well as a mining and factory
population.
The children measured are “entrants” and “leavers” and a further group of
children, namely those from 8 to 9 were measured. The “leavers” include
children of 12 to 14 years, “since in general the only ‘leavers’ at age 12 to 15
are rural, and the only ‘leavers’ at age 13 to 14 are urban*.”
The children were of the age stated, 5 and not yet 6, 8 and not yet 9, on
January 1, 1911, but the actual day of weighing may have been any school day from
January to December, so that a child entered as 8 may have been only a few days
short of 10 when it was actually measured, and therefore the mean age of the
group of children of 8 to 9 will be 9 years. “In the case of the group of leavers, 18
to 14, no child can have been more than 14, because on attaining that age the
children are entitled to leave school, and generally do leave. With these the mean
height and weight in our tables refer to the true mean of the years of the group,
viz. 13 and a halft.” We shall table to the middle of the group, namely at ages 6,
9, 138 and 134.
The children were weighed and measured without shoes, but in ordinary indoor
clothes. The figures were read to the nearest quarter of an inch and to the nearest
quarter of a pound.
Staffordshire Children.
GIRLS Boys
Ages | | |
Regression of | Regression of
Mean Mean Taio | Mean | Mean Poe
Height | Weight | eee Height | Weight ane
|
| |
6 419 | 398 | 1-705 49-1 | 41:0 1-741
9 47°7 51:1 | 2024 48°1 | 53°0 2°120
13 56°7 (ek | See, ays) | Thos} 2°811
13} 57°1 81-0 3°360 56°3 | teat, 3°166
|
It will be as well to compare these means with those for all Glasgow ; so far we
have not given them in this paper for all the schools taken together but only for
each school group.
* Staffordshire County Council, Annual Report of the School Medical Officer for the Year 1911.
J. and C. Mort, Ltd., 39, Greengate Street, Stafford, 1912.
+ Ibid. p. 25.
298 Height and Weight of School Children in Glasgow
Tlasgow Children.
GIRLS Boys
Ages
Mean Mean Mean Mean
Height Weight Height Weight
6 41°7 40°5 41°9 41°8
9 47°3 51°3 AlT(OT | 53°7
Le Yar 74°8 54°6 | 73°6
Girls in Staffordshire are taller at ages 6, 9, and 13 than girls in Glasgow, but
they are lighter at ages 6 and 9. We might argue from this a lack of physique in
Staffordshire girls who are absolutely ‘7 Ibs. lighter at age 6 than Glasgow children,
and relatively to their height even more than this amount. At age 9 the absolute
difference is less and at age 13 Staffordshire girls are heavier than Glasgow girls
but they are 14 inches taller, and since the regression of weight on height at
age 13 for girls is 3°272 lbs. we should expect Staffordshire girls to be 49 lbs.
heavier than Glasgow girls, but they are not so much. I should hesitate to say
that the physique of Staffordshire girls is inferior to that of Glasgow girls; the
difference probably is one of race, but such questions must remain unsolved till we
have a far wider range of anthropometric data than is available at present for all
the districts of Great Britain. Boys show the same characteristics to a lesser
extent; Staffordshire boys are taller at ages 6, 9, and 13, but they are lighter in
weight; at age 6 they are ‘8 lbs. lighter than Glasgow boys; at age 9 they are ‘7 lbs.
lighter and at age 13 they are 1:7 lbs. heavier, Again relative to their height
Staffordshire boys are lighter than Glasgow boys at the three ages for which a
comparison can be made.
Comparing boys and girls in Staffordshire we find that girls of 6 and 9 are
shorter and lighter than boys of the same age, but at 13 and 13% girls are both
taller and heavier. At 6 and 9 years the regression of weight on height is
practically the same for both sexes, but at 13 and 13} the regression of weight on
height is greater for girls than for boys; girls are heavier proportionally to their
height than boys are. For girls of 13 an additional ich in height should mean
3°3 lbs. more weight while for boys the additional pounds expected are only 2°8,
while for girls of 134 we expect 3:4 lbs. increase for every inch of growth and for
boys 3:2 lbs. increase. A comparison of the regression coefficients with those given
for Glasgow in Table I will show that the coefficient is higher in Staffordshire for
children of 6 and boys of 9 than in any of the school groups in Glasgow. The
regression coefficient found for girls of 9 and 13 in Staffordshire is practically
identical with that found in Group D in Glasgow, and boys of 13 in Staffordshire
would seem to be most like boys of Group Cin Glasgow.
EK. M. ELpERtTon 299
In a Drapers’ Company Research Memoir * recently published tables are given
showing the height and weight of boys and girls of 12 to 13 years who were
members of the Worcestershire public elementary schools. These tables are
XLIV and LIX and will be found on pp. 100 and 107 of the work cited; we
have calculated the mean heights and weights and the regression coefficient of
weight on height as we have done for Glasgow and Staffordshire. The mean age
of the group of children of 12 to 13 years is 12°5 years, so allowance must be
made for the six months age difference in comparing with the Glasgow data. We
have already given the mean heights and weights of Glasgow and Staffordshire
boys and girls of age 13 so we will calculate what the height and weight of
Worcestershire children at age 13 would be. An additional year makes a difference
of roughly 1:9 inches and 49 lbs. in the height and weight of a girl and of
1°S inches and 4:6 lbs. in the height and weight of a boy.
GIRLS Boys
, Regression ; | Regression
een eee of Weight Ea ioa Gone of Weight
eigh eig on Height eigh eig On EeiHe
12°5) Eat Sen ee 55°2 72°9 2°829 54°6 eal 2°800
emcee estes Fea | 75.8 = BBS | 744 em
18 Staffordshire 56°7 780+ = 55°8 ToL3 =
13 Glasgow 55°2 74:8 — 54°6 73°6 —-
Worcestershire children are taller than Glasgow children but slightly shorter
than Staffordshire children. They are also rather heavier than Glasgow children
but not relatively to their height. The height of Worcestershire children of
12°5 years is the same as the height of Glasgow children of 13 years, but the
weight of girls is 2 lbs. less and of boys is 14 lbs. less. Worcestershire children
are lighter than Staffordshire children, but when allowance is made for the
difference in height the Worcestershire children are not much at a disadvantage ;
girls are a pound lighter and the weight of boys is practically the same.
The differences we have found between the Worcestershire, Staffordshire and
Glasgow children may well be due to differences of local race, and not be the
results of differential environment or nurture. We should have little hesitation in
applying the returns for Glasgow children as an approximate standard—say to the
lb. and inch—for all British children of the artizan classes.
* «A Statistical Study of Oral Temperatures in School Children with special reference to Parental
Environment and Class Differences,” by M. H. Williams, Julia Bell and Karl Pearson. Studies in
National Deterioration, IX. 1914. Dulau and Co., Ltd., 37, Soho Square, W.
+ This weight appears somewhat exaggerated. It may in part be due to local differences in the
average ages of ‘leavers.’
300
Height and Weight of School Children in Glasgow
TABLE a GLASGOW. BOYS. GROUP A*.
Weights for Height at each Age.
Actual Age.
6 7 8 9 10 11 12 13 14
Oo gees.5 = =| ea = — -- _ —
34 | 30°0 | 31-0 — | = — — — — =
86 Weslo i325 = == — = — — —
Ae | SERB oe esi ip) — — — —
oy, 34:4 35:4) |) = 35:8a — | = — —- | —
38 35:8 36°9 | 37:4 3iE3 ne — — |) =
39 37°3 38-4 39°0 | 39:0 38-4 | — —_ — —
jo | 388 | 39°99 | 40° | 406 | 40:2 | 39:3 = — =
Bt | 40:3) |) 14 498] AD 30 249 es eles = = =
y2 40°7 | 499. | 43°71 “4ahOs |ex43-08 | BaB:4ye) w49-a = —
48 | 43°2 | 44-4 | 45:2 | AB 7 4537 9) 45-4 |) 44-7 AO Gum
44 | 44:6 | 45°99 | 46°83 | 47-49) 47-6" || 47-5 |) 47-0) 46-2 eee
S| 258) 46s Ay a de Ae eo 494 | 49° | 49:3 | 48-7 | 47-9
"oo | 46 | 47:6 | 48-90) 49:9: | 50-7 =| 5IS3: | 51-5) || (51-6) || IcSenenoee
"©.| 4% | 491 | 50-4 |" 61-5 9) 5974 5351 11 653-6" | 538) | 53 :omen aa
| 48 | 50S | 51:9 | 53-1 | 54: 54-9.) 556 | 56-1) 756 Deamenoee
49 _ 53-4 | 54°6. | 55°83 |. 56:8 | 57-7 | 58:4 | 59-1 | 59°6
50 -- 54°9 56°2 57:5 || 58:6 59°7 60°7 61°6 62°5
51 = = 57°8 | 59-2 | 60:5 | 61:8 | 63:0 || 164-2, Senet
52 = = 59°38 _| 60-8 | 62:3 |-63°8 | 65°3° || 66:8. j\m68:3
53 dn a 60:9 | 62:5 | 64:2 | 65:8 | 67°6. | 69:40") sD
54 | 64:2 | 66:0 | 67:99 | 69°9 | 72:0 | 74-1
55 | 65:9 67°8 69°9 72°2 74°5 77°0
| 86 | 67°6 69°7 72-0 745 77:1 79°9
eae: 69:2 | 71:5: || 74:0 —| 716-7 | 70-7 aalesono
eta te 2 alee tales 734 | 761 | 79:0 | 82-3 | 85-8
59 fy aisle 81°3 84°9 88°7
60 ae a 83:6 | 87-4 | 916
61 : 90°0 | 94°5
put = | : ae ae = 97-4
TABLE 6. GLASGOW. BOYS. GROUP B.
Weights for Height at each Age.
Actual Age.
| asta 7 8 9 10 u 12 13 U
|
[ego Sle 7c5 — | — = = = — = —
34 | 29:0 — | — = = = — — —
Sh. Nis80-C@ie ola pa ae = = — — =
cies epi | Soni Wy: as — _ —
73307 ile sasGe | reso = — —
SBN) 8572) 13620 36.6 = — — — — —
39. | "36°8 Nav:8, noes a a = — = =
40 | 383 | 39°4-}-40:0 | 4071 | 39°8 — — — _
41 | 39-9 “| 40:0.) 47a S41) eae — — — —
42 | -A0-5: |) 42:6) 4) AB *Ses | A387 eo 8 = —_ —
8 | 43-0 | 44-9 4570 1 45-5 45-7 45-5 — — —
yb | +446) | 45:8 9 467) 47 -Sie4 76am 47-6 — — —
a | ae | 465 1 474 4849 40-159) 49-5) 40-7 eoes = =
| 46) 47:7. | 49:02) 5051) 50297, 51-5) eS ma eee sel MD IEG —
20) 47 | 49-2 | 50-6) | 51:8 | 52-7 | 253-40 653-0 erode | ede? cal serorad,
| 48 | 508 | 522 | 53:5 | 54:5 | 55-4 | 560 | 565 | 567 | 568
49 | 82:4 | 53:8 | 55-2 |) 56:3 | 5763 9) 87s | 68'S | 59:3 ailoore
50 | 53°9 | 55:4 |.-56°8 -| 581 | 59:29) 60:2 ss6l-1 | Ci-8 mao
51 — 57-0. | 58'5 | 59:9 || 61-20) (62745 63:4) |) 64-4en Ga
52 a 58°7 |) 6020 Glave ose 645 | 65:7 | 66:9 | 68-1
58 — = 61:9 | 63:5 | 651 | 66°6 | 68-1 | 69:5 | 70:9
54 65°3 67:0 68°7 70-4 72°0 73°7
55 6771) | G8r9= | 70:8. 7277 || e746 vers
56 — = — —- 70°9 72:9 75°0 dell 79°3
57 — = — -— 72°8 75°0 77°3 om 82:1
58 Tel) 79261), (82520 ee beO
59 a = | 79°3 | 81:9 | 84:8 | 87°8
60 — = — | — —- — 84°3 87°3 90°6
61 | = — 89°9 | 93-4
62 | — — -- — 92°4 96°2
63 | — —- 95°0 99°0
* Throughout weights are given in lbs.,
heights in inches.
E. M. Evprerton 301
TABLE y. GLASGOW. BOYS. GROUP 0.
Weights for Height at each Age.
Actual Age.
eee
6 Ge 8 9 10 ol 12 13 10}
35 30°5 as = =— = = — a —
36 SOE al ae = = = = = —
37 33°7 36-0 — = = — — — —
38 35:2 37°5 38-6 = = — — — a
39 36°8 39-0 40°1 = = = = —~ =
40 38-4 40°5 41°7 41°8 = = = = =
41 40-0 42-0 43-2 43°5 = == = = =
42 ALD 43°5 44°7 45‘1 44-7 = = = =
43 43:1 45-0 46°3 46°7 46°5 = = = =
th 44:7 46°6 47°8 484 48:3 47°5 ae = =
45 46:2 48'1 49°3 50:0 501 49°6 48°5 = =
4G 47'8 49-6 50°8 516 51:9 51°7 50:9 = —
Wee 49-4 51-1 52°4 53-2 53-7 53°7 53°3 = =
We 3 51:0 52°6 53-9 54:9 55-5 55°8 55:7 55-4 =
2) _— 54:1 55-4 56°5 57°3 579 58-2 58-2 =
"en | 50 _ 55‘6 57:0 58:1 | 59-1 59°9 60°6 51-0 as
®| 51 ere ae 58°5 59:8 60:9 62:0 63-0 63°8 64:6
a) 52 tes yes 60:0 | 61:4 | 62:7 | 641 | 65-4 | 66-7 | 67-9
53 as = 61-6 63:0 64:5 66°1 67'8 69°5 71:2
54 as = 63'1 64:7 66-4 68-2 70:2 72°3 74°6
55 = aah == 66°3 68-2 70°3 72°6 75°1 77-9
56 = os = ae 70-0 72°3 75-0 77-9 81:2
5Y = a 74:4 77°4 80°8 84°5
58 =e ae = aa a 76°5 79°8 83°6 87°8
59 ane ii ese z 82-2 86:4 91:2
60 a a8 846 89-2 94°5
61 el rane = at 92+1 97°8
62 as aes es diss 94:9 | 101°1
63 = Se a = 97°7 | 104°4
64 aos ae = = Zs Js = — | 10738
65 as = z = <e = as = 111-1
66 ate = A — | 114-4
67 —* aE yey
TABLE 6. GLASGOW. BOYS. GROUP D.
Weights for Height at each Age.
Actual Age.
6 oy, 8 9 10 aD 12 13 Lh
37 31°7 = = me: = ae = = 3
38 33-4 =a aa we = me = Z te
89 | 35'1 37°1 = _ = — — — — |
40 36°7 38°8 39°6 = re: = = — —
41 38-4 40°5 41:4 aa = 2 = — —
42 40°1 422 43-2 43°3 =e = = — _
43 41°8 43-9 45-0 A Ole _ = — —
dA 43°5 45°6 46°8 47°1 46°5 = =e — —
45 | 452 | 47:3 | 486 | 491 | 48-7 = _ — =
4b 46°9 49°1 50-4 51:0 | 50°8 49°8 = — —
Ay 48°6 50°8 52-2 53-0 53-0 52-3 50:8 — —
48 50°3 52-5 54:0 54:9 55'1 54:7 53-5 = a
19 ae 542 55'8 56:9 57:3 57'1 56:3 54-9 =
| 40 = 55:9 576 58'8 59°4 59°5 59-0 58-0 =
Bo | ol a ze 59-4 60°7 61°6 61:9 61°7 61°1 59:9
Ra D2. — =. 61-2 62°7 63°7 64:3 64:5 64:2 63°5
| 53 = ae = 64°6 65'8 66°7 67:2 67°3 67:0
54 66°6 68°0 69°] 69°9 70°4 70°6
OMe es — SMES OR Me pOg emi. | WG. || o78:5° |) 97422
56 va oe = a 72°3 73:9 75°4 76°6 Wiel
5Y s = —_ cy ae 76:3 78°1 79°8 81:3
58 a _ 78°7 80°8 82-9 84:8
59 as =e uae = =o 81:2 83°6 86-0 88-4
60 ets pla ms 86°3 89°1 91:9
61 os ae ie ans aa = 89-0 92-2 95°5
62 a = == Lee aes = — 95°3 99-0
63 we a es rs = = fs 98:4 | 102°6
64 an = = a ii Sy a == 106-1
65 aa ae oo as = ze = te 109°7
66 ba) oes — mY a = i _ 113°3
67 23 = ct as he an aa = 116°8
Biometrika x 89
302 Height and Weight of School Children in Glasgow
TABLE « GLASGOW. GIRLS. GROUP 4.
Weights for Height at
Actual Age.
each Age.
6 2 8 9 10 11 12 13 1h
33 | 30:0 — — — -- _ — — —
8) 81:3" | 3ic3 = — — — — — _
35 32°7 32-7 3251 = — — — — —
36 3420) |) adel 33°6 = — — = — —
Bre W354) 85:5 35°2 34:2 — ~ = == =
38 | 36°7 SycOresGen 35°9 — — — = —
89 "| 38:1 3874: Wl 38:2 37°6 36°6 = = — —
40 | 39°4 | 39:8 39°8 39°3 38°5 = st ae Es
Ad 40°38 | 41-2 41°3 41°0 | 40:3 39°3 — — —
By) “42-1 42°7 429 42°7 42°2 | 41:4 = _ —
Galle Oreo) ececil 44:4 | 44:4 | 44-1 43°5 42-6 = —
yh 44°38 | 45:5 46-0 46°] 46°0 | 45°6 45-0 = —
3 | go | 46:2 47-0 W475 47°8 | 47-9 478 | 47-4 46°7 —
| 46 47°5 48°4 49°1 49°5 49°8 499 49°8 49°4 —
OU 7 = 49°8 50°6 51-2 51:7 52:0 52°1 52-1 =
x 48 = 51:2 52:2 52:9 53°6 54:1 54°5 54:8 55:0
49 = = 53°7 546 | 55-5 56°3 56°9 575 58-0
50 == — 55-2 56°3 57-4 | 58-4 59:3 | 60-2 61-1
51 = — = 58-0 59°3 60°5 61°7 62°9 64:1
52 = — = 59°7 61-2 | 62:6 | 64:1 65°6 | 67:2
53 — — = 61:4 63°71 64:7 66°5 68°3 70°2
54 — = = = 65:0 | 66-9 68:9 71-0 73°2
55 = — = = 66°8 69-0 Tes 1331 76°3
56 ss = = = 68°7 fad 1377 7674 79°3
57 = a 73:2 76°1 79°71 82°4
58 Ss = — a — 75°4 78°5 81°8 85-4
59 = — — 80°83 | 84°5 | 88-4
60 — = — = = = SBE IE 91°5
61 = a = oe 2 a as 89:9 | 94:5
62 = = on = = a = 92°6 | 97-6
63 = = = = = = —_ = 100°6
TABLE ¢ GLASGOW. GIRLS. GROUP B.
Weights for Height at each Age.
Actual Age.
6 fg 8 9 10 il 12 13 Uy
33 | 272 — = — = = = — =
3h 28°7 — = -- — _ — — —
35 30°1 31-0 = — — — — — =
| 36 31°5 32°5 as — = — — = =:
3 33:0 | 34:0 34°2 — — — — — —
38 344 35°5 35°8 35-4 — — — — —
39 35°8 37-0 S74 eer oree 36-2 = — — —
40 1 372 38°5 39:0 | 38-9 38-2 = = — —
| 41 38°7 40:0 | 406 40°7 40°1 38-9 = -
| 42 40°1 41°5 42°2 APACHE A9: OMe All = =~ =
UB A Ate 430 | 43:8 44:2 44:0 | 43:3 | 42:0 = —
Ah 43-0 44:4 45-4 45-9 45°9 45-4 | 44:4 = —
ous 44-4 45°9 ATO: |, e477 47-9 47-6 | 46°9 — --
rots) 146 45°8 47-4 48°6 | 49:4 | 49°38 | 49:8 | 49:4 | 48-5 —
2 4 47°3 48°9 50°3 51-2 51°8 52-0. 4 \5.51°3 51:3 ==
mr 48 48°7 50°4 51°9 52°9 53°7 542 | 54:3 | 54:0 —
49 = 51-9 53°5 54°7 55°7 boo 56-7, 56°8 56:6
50 = = 55°1 56°5 57°6 58°5 59-2 59°6 59-7
51 = a 56°7 58-2 59°5 60°7 61°6 62°3 | 62-9
52 = = a 60:0 | 61:5 62°9 | 64:1 65°1 66-0
58 — = = 61°7 63:4 | 65:0 | 66:5 | 67:9) "edu
54 ote Ze, = a 65:4 | 967-2 69-0 70°6 72:2
55 = = = = 673 | 69:4 | -71-4 73°4 754
56 = os = -- 69°3 TUGeal es Foro 76°2 78°5
57 = = a — = foul 76°3 78°9 | 81-6
58 = ae 75°9 78°8 81°7 84-7
59 a — = = = 78:1 81:2 | 84:5 87-9
60 = = a 83:7 87-2 91-0
61 = = = — = — 86°1 90°0 | 94:1
62 a = = = ae = = 92°8 | 97-2
63 — = = — — — = = 100-4
lk
HK. M. ELDERTON 303
TABLE 7» GLASGOW. GIRLS. GROUP C.
Weight for Height at each Age.
Actual Age.
Height.
Height.
7 8 9 10 ib 12 18 14
30°0 — == oes = a — aad ad
315 = — — — = = = =
33°1 33°7 = = ra a er aa aps
34:7 35°3 35°3 — = = = = —
36°3 37°0 37°0 — — = = = =
37°9 38°6 38°8 38°3 —_ = = = ee
39°4 40°2 40°5 40°2 — —_— _ = =
Al‘O | 41-9 | 42:2 | 4271 415 — = — —
42°6 43°5 44-0 44:0 43°6 42°7 — — —_
44:2 | 45-1 45°7 | 45°9 | 45:6 | 45-0 = — —
45°8 46°8 47 °4 47°8 47°7 47°3 46°6 = =
47°3 48°4 49°2 49°6 49°8 49°7 49°2, — =
48°9 50°1 50°9 51°5 51°9 52°0 51°8 —_ —
Blew soa, at | |ebacO 9 big | ebdea | bas —
53°3 54°4 55:3 56°0 56°6 57°0 57°3 —_
— 5671 57°2 58°1 58°9 59°6 60°2 —_—
— 57°9 59°1 60:2 61:3 62°3 63°2 64:0
— — 61°0 62°3 63°6 64°9 66°1 67°3
= — 62°8 64°4 65°9 67°5 69°0 70°6
— —- _- 66°4 68°2 70°1 72°0 74:0
— — — 68°5 70°5 72°7 74:9 77°3
— — — 70°6 72'9 75°3 77:9 80°6
75:2 779 80°8 83°9
—- 775 80°5 83°7 87°2
- “= 79°8 83°1 86°7 90°6
= — — — — 85°7 89°6 93°9
— — = 92°6 97°2
— — 95°5 100°5
— — — -- — oe 98°4 103°8
— — = - — — -- 107°2
TABLE 6. GLASGOW. GIRLS. GROUP D.
Weight for Height at each Age.
Actual Age.
7 8 9 10 11 a2 13 14
29°4 — — — = —_— —_ — —
31-1 — — — — — — —_— —_—
32°7 34°0 —_— — = — —_— = —_
Sule Ml Sein — — -- — — — —
36'1 37°4 37°8 = —_ — —_ —
or. | a9 396. | — we = = = =
39:4 || 40:9 | 415 ,| 41-2 =
41:0 42°6 43°3 43°3 — — — — --
DOT at Seno A Abs leaaey = = m= —
44°4 46°0 47°0 | 47°3 46°9 — — — —-
46°0 47°8 48°9 | 49°3 49°2 48°4 — = —
47-7 | 49% | 50°7 | 514 | 514 | 509 | 49°88 me =
AG Sete pine |) “52°Gr tlk (bo4e sles 7 Bale leony — —
52-9 | 54:4 | 554 | 559 | 560 | 55:6 = =
54°7 56°3 | 57°4 58°2 58°5 58°4 57°9 —
— 581 | 59°5 60°4 611 61°3 61°2 60°7 |
— - 615 | 627 | 63:6 | 64:2 | 64:4 | 64:4 |
= = 63-5 | 65-0 | 66:1 Gil. 6 W681
— — 65°5 67°2 68°7 69°9 71:0 71°8
— —_— —_ 69°5 We2 72°8 74:3 75°5
ly a= = Seer iss ier is. |) 79%3
76°3 78°6 80°8 83°0
— 78°8 81°4 84:1 86-7
| 8l°4 84°3 87°3 90°4 |
= i= = — 87°2 90°6 94°1
— — —_ = = 90°1 93°9 97°8
= = = = — — 97-1 101°6
ee = = = _ — | 100-4 | 105-3
jo = = — — —- | =— 109°0
= = = —_ — — — 112°7
304 Height and Weight of School Children in Glasgow
TABLE 1. Glasgow. Height and Weight of Boys of Group A.
Weight of Boys of 5:5—6°5 years
eve !
OC en ror mey Ow ho kh abel | f [Totals
& 29 ~~ Dd my § Ww = on) ban! Cale is) fa “ey el Sai Mel Re ios
R®eaens HS HH HH SF SHSsBsoy 6 HH Ss
1 1 — Se ST 2
— 1 — 1
—1-— 3 2 1 SS 5 = U
—— 455 1—- — — —- 15
—— 2 25 8 3 I] —_— 21
— 1— 21714 10 6 1 1 - - 52
——— 1725 16 15 5 1. 1——~— 11—-—— — 7
——— 11020 36 34 15 16 2 — 134
———— 415 32 48 35 20 4 2 - — — — 155
———— 1 565 Ill 54 61 52 27 2 1— 1 —- 1 216
——— 1— 1 6 22 56 52 3116 7 1 1— —— — — 194
———— 11 2 10 21 54 42 26 7 1 2— — — — — 167
- = 5 7 22 2229515 7 31 1—— — 108
1 4 4 11 714 7 2—— —— — 50
1 1 2 4 810 5 24 —— — — 37
—- — -— T—- 142 —— — — 8
= — —_- —— = 1 — 1
— if 1 — 1 3
1 3 7 15 52 91 117 192 206 224 145 87 54 22 16 8 2 1 — 1] 1244
TABLE 2. Glasgow. Height and Weight of Boys of Group A.
Weight of Boys of 6°5—7°‘5 years
poheds side
J
1
>
OP a
a
Wi A ~T~119
e —_—
Pte
bo bo © ~71 Oo
153 220 225 152
Totals
E. M. ELprrton 305
TABLE 3. Glasgow. Height and Weight of Boys of Group A.
Weight of Boys of 7°5—8°5 years
Beit yy yt | | bol 2 eae eae ee
WOH HHH HH HW H G BH BW BW 6 15 GH GH YH H 8 BH HB 19
S Sas es os es + SRE Sees eEsassssseRr
P53 |e — — —— - = 1
aie 2 — -———— — — 3
oh | || a es = = 2
OF 0 || area ces ee SS 4
eee 1 10
liars ——— 265 43 2 2 = = — 18
Cee 12 6.13 3 2 : a == ==) “95
Cae 3 1010 8 4 — = = 2 32
gO, {——— 111119 1 7 6 38 = 64
Mipeete = 2 4° 5 18 23 92 11 7 2 5 pe et 95
eee 9 4 14 36 53 24 87 «(4 — a fe Se eee aie!
eee A 39 44 60 438 15-1 - : 222
Lie — on 1 1 22 46 52 53 34 18 611 a 244
S| _—————s Ti 69 3054-43 83°91 6 8 211
LE ,, ee rd) 14. 81 33.234-19 19.8 1 1 164
2? oo) oe eM ly Weise Sih 16 99 T8 7 T 124
Bo) -— ———= 1 ee o ee WG IAN 4: O19 Seno be 74
S| = Se ee leh SiMe 2 =e ee eee 31
Oe = jt ee et ee Omi OS i eg eH 15
| 7 oe oa a
Se ee le
Totals] 1 1 1 18 24 56 89 157 194 202 249 159 150 83 78 47 21 8 8 1 — 1 — 1 | 1535
TABLE 4 Glasgow. Height and Weight of Boys of Group A.
Weight of Boys of 8'5—9°5 years
Height} | | | | | Peeeiee Oe l wlied Rl beth ht SL | op Petals
DD WS eS ee Se De ee te
Ras es 5 O22 BR SR BS BESSSSESRRKREELNS
—~ | aaa zs = 1
le SS i 1 _ = 3
= 1 Sie == 1 2 = = = 5
1 1 a= iz 1 = = 3
= == I) aah a ee 1 = (14
= = 1 SS eee — SS SS = is
Pmt 469 9 6-4. 1. 8 = 1 = 39
a 21710 8 Ae AA = 49
eee 9-11.98 95 18° 15. 6 > 4 99
= = a 28.19 83 42 80 18 6. 2 160
ot 615 23 43 45 33 23 11 6 2 “ —— | 907
B —— — 2 — 1 5 10 27 44 49 37 2510 3 2 = 216
AT yy J— — — — — elon 5 23ne40Ne 360952) 20) ON 1 - 206
48 4, | — 1 Sao 16m 2) A026 Reso] = 152
49, J —— — — — — — — 69 20 1914 18 4 Or 100
50 ,, = 1 Ceo miomONN wtcci = 1a he
51 ,, — — eee OM geeteG al or aie ee se 30
52 ,, — — ee Nee re re eee Te od a 12
53 J—— — — — — — 1 TMs 92} Smeg es ee eek ee 7
54 ” — —= 1— — 1 1 — 1 1 — 5
55 ,, = — = 1 3 — 1 — —- 1 6
56 ,, 2
60 ,, 2
Gils 1 |
1e2s 6-3-9 18 67 93.118 162 176 145 165 98 56 51 5% |
Totals
306 Height and Weight of School Children in Glasgow
TABLE 5. Glasgow. Height and Weight of Boys of Group A.
Weight of Boys of 9°5—10°5 years
ee ee eee de Pe Pt te fe ry
REY, TELIA CLOSE aE LC ILC) wD i) > iY) WwW WwW ay eiay Wey ey eye ev Eley ale)
SMR AA HQKR &@ HH © K ® HH HK QAAH
He HH HnmeS Ss FS WD WD vey wD SOF RO GECOMICOU ECO] oe 9 e
35 ,,
36 ,,
or is
eae = iT = ‘ =
89, he I 1 .
Oe. nse ele aoe al Dek - a
ie ee gle oo eee eel Be boda
\42, [—-——2 3325 2 1 : :
43 ,, ae ee eee Eee i Bye I sl ile = ge:
i ee eS PA ee et
Les, | = — — == 2 1024 oS rei ele omenon | 2
Hekapeny || (ees ee L Be 142 oon Oo) 19 OG) 7 46 aT zs BS
fier 3) 2 8 : ]
10 > 1 318 6 I
50 1 929. % 19).19' 27 381616 6 4 — (==
51s fe al ea el aispalsy okie oy)
b2) 5. = - 4 al 2 4° 7-9) 19" 9) 7 13. 03)
53 ,, 8 44 51 3) eee
= eA Ss i Tl
Totals
2 1 3 9 23 19 45 95 114 126 142 205 142 119 121 77 59 38 15 9 10 3
TABLE 6. Glasgow. Height and Weight of Boys of Group A.
Weight of Boys of 10°5—11°5 years
Hoan 1 tl ba) Dea
1 1D WH 1D 19 WH 19 1 1D WH IO 16
mM MWA BH DWH KR OADM HOR
© Dow Oe DMOMOeRe eRe eRe RH TH HH O&O
348 1
Cue 1
38 ,,
39 5, 1 3
| 40 ,, Leal! 2
|41, ._—1—— 22——— 1 = = 6
| 42 5, an 3 = ee Se 4
eS ECT a Wee, BOA Bey — al 14
2A oe | a es iy Gr By 4 = = 24
Vaio =e eS Ol yl oO mae ma 2 o- ae ae 55
46 cee =. ie iy 7” 0 eSal4 er Geen ome ee? == oe = 72
[gee PCS ae Oe 8 e181 3028 eae 103
oe ——— By Din OHS be OAs hy ily a 8} a ss =) \ as
en (en ee nn ee Me epi atn Reap) GC) se) 191
| 50,5. [= —— — — F 2. 3 7 116 193) 727 48 oS 181
Pye apr |f eae Ces age, See at V— 1 5) eS, 18) (354 eC Oniieoneo 3 167
52 ,, = 9 = 188 A RCO RICE OO een 117
| 53 ,, = : CE Gh MES ees TE IG BRS) BY i 99
Be | = 1 Xk 3 3 H Sig BA = i= jl 44
55 4, ae SS ee es ee 25
56 ,, feces CE ee re Ie ee eee ame 1 oe ae 10
15 tena (pea OR eg ——— = = 1 Se | ee 4
| 68, J— — — — — — — = == = — —
P9ns, afte ee SS aah ea ee 2
_ a Ee Ln ee
Totals] 2 2 4 6 15 41 35 61 73 158 127 125 160 114 82 76 70 48 321817 1 3 5 2 1 | 1278
EK. M. EvprErton 307
TABLE 7. Glasgow. Height and Weight of Boys of Group A.
Weight of Boys of 11:5—12°5 years
Lees Cee ll bl Lat | pQotals
HS Be A 1) Go Wey Rey Be SE eS ap pete tae ears aie)
DBO RwHaeHHSA K SF H® BO DBRMOAHHFMFR OH ODA
QO SFS Soo © SF GF GF GCKHKHHDHAAAAAS
as ere ES sy =
—— — — IL — =
———— 242 1 — 1 — = =
——1%13844 2 —
—--— 1 212 6 5 — 1 2 ==
—— 1l— 4722 14 7 1— 1—
———/— JF slr20 16 ll & 3 1 1 -
——— 1— 717 27 40 16 14 2 1 l= —_—
— —— —— 110 33 41 33 22 12 3— 1 -
—- —- — — — 1138 14 30 41 36 20 7 6 1 -
Oe. == Il = Go 1G Sl 38 si lb 7 1 1 2
iS, 1 5 8 16 28 312718 4 2 2
a — — — pe ge 8 1 93 99coT 4 9
6 ——————————_ udder Some 1O.uS ie Oa b i= oh Ip ee ee Oe
56 = 1h, lee Une Gat |
Si | te 1 ie ee ey oti Oe eee |
58 ., Tine et er
59 ,, 1 1 |
60", = |
Gi, |
62 ,,
6355,
O4 ”
Totals 16 50 98 160 163 143 91 78 43 26 12
TABLE 8. Glasgow. Height and Weight of Boys of Group A.
Weight of Boys of 12°5—13°5 years |
|
Height} | | eee ee il oe Tt el ts otals i
ye ON Ia 29) SIO! AG a Ag; AG, | 1D 9 ©) ©) W WwW HW WwW |
2 ££ 4 8&8 ® ® FH © DB ® BH BH B® HF B
SS ey Oe PR Se BO" OG SG SS
rn ee ea |
”
4A ” aa 1 5 cas 6
LEB, og ft — 1 3 1 —_- — 6
4G ,, — 2 4 5 1 12
oe = 2 8 13 68 = = = 27
48 ,, 2 ay Sy dkey yi 1 1 —_- — 41
HOS ee 2s 18s 7 Ti <6" Al 66
Oem ce = 7 97 29 «17 «10, 8 : 93
BIS 1 — 1 8 26 34 40 22 6 1 1 — 140
02 5, — — — 38 21 46 51 33 22 6 4 2 188
oe oe 7 84 AB Be TI 8 i 1 160 |
pe Ad, 20) 45. 87-18. 2 1 140
ene) 8 8, 10 138° 88516. 10 FO. 8
56 ,, ae uF eo Mtoe eG I) Qo 89
es 1 ee 10. 10" <8) 4 8 = 49
5B || aS ee ee Oe eee eas eee ee a a 33
KD a a a oo Ge: 22
60 ,, a Wor 23 4
61 .,. a a oe eh ee at 7
62 y, = ee ees We ae 2
3 ”
64 ,, —_— — — —- 1— — 1
Totals 4 8 21 67 189 172 205 222 155 71 59 37 16 17 7 1 «1 1202
308
Height and Weight of School Children in Glasgow
TABLE 9. Glasgow. Height and Weight of Boys of Group A.
Weight of Boys of 13°5—14°5 years
wD
S
ne Se ye Bp eS ee J S 1 ad u
aS Wey) D> 99 ™~ wD Sy 99 — ia wD d Sa) = >
Sop sys tis Sy ee! re) tase tase | IS) or ES SSS aS 2
42: : -
43 [ane ae ee
44 1 1 =
| 45 —- — — 1] — - = =
| 46 — # 1 -- - — =
ee = a a = 2 Se
Sins — = ta al — —
49 ,, 1°05 ahD 0 dered a eo =
50 ,, cme eee eyercc oO = = =
il x, — — 1 6 5 10 8 8 == =
ID — — — 411 22 4 1 #1 -
teh — == 1 42 AGS oe 4 eis ee ae =
bh, Wee ee Se ee eel SS a
eee (eee es Mee rl le 2S |
56 ,, 2: M4 10) 10" 1a Ge Lj = Sea
alte, 1 B62) Oe h8u 5) 4s = I —
| Res os = — 565 6 4 2 2 4 1 — —~ ~— —
59 ,, = = 1M Sie 1s 2 eee
| 60 ,, —- —-— 2— 212 1—- => —
61 ,, = Phat 1 =
62 ,, =e = DS
(Ge —- -—- — > HF ~— ~ —S — 1TH ~~ ~ ~ ~ ~ — 1
| O4 ” = eT = a
Totals BD 98 Il 28 46 77 56 GI 45° 365 T1715) 10" 825) lt
TABLE 10. Glasgow. Height and Weight of Boys of Group B.
Weight of Boys of 5°5—6'5 years
Height] Pt ot LS
1 A A 1G, 8 1S tS So Hl) tS od a) as ome
298 1 = = 1
30 ,, — —_— - — —
31 ” =, = Tay ae, | aaa aa sats
32 ,, | — —- -—- -— ~~ — — 3
33 4, 1 — 1 2 — —- — — 4
Shs, eS eageiyen p> sO ee = 5
39 4, — 8 4 4—- ~ ~ ~ ~ ~ ~ ~— ~ ~ ~ ~ — 11
36 ,, — — 6 14 3 1 — 1 25
OT ss — 1 3 1441 6 1 3 — — 43
38 ,, = 2D ISS OR ol 2 —_- —- — 67
39 1 — 1 8 29 31 23 11 4 2 — ~ ~ ~—~ —- ~ — 110
40 ,, — — 1 4 23 39 61 49 9 4 1 — 191
AL; — — — 6 9 2 46 59 28 14 2 2 — —~ — — — 192
2 — — — — 2 14 81 71 44 22 22 206
1 ae 6
1
69 97 138 188 239 140 97 88
TABLE
thik
E. M. ELpErRtTon 309
Glasgow. Height and Weight of Boys of Group B.
Weight of Boys of 6°5—7‘5 years
Height Totals
298 ae
30 1
31 ”
32 ”
33, 5
ey 2
35 Sal 6
36, 3% 13
Gress bang 23
38, 6 8 38 |
39». aay) 47 |
HO Bone 2: 103
Gis Sy ily 44 96 16 6 — 1 1 146 |
42 ,, i Tl Hes 59, Ge CO. Me a Stes, aah 8S
43 ,, a 2 47 63 59 36 16 3) 1 1—- —- — — 246
AL ,, 1 Oy, AAD 00 27 5 5 2 1 — — — PANG
ie Demon eocee ssa 7 Gee = TAO
UO - a Y; lO eZ AO OD: ele 12 —-—- — 114
ie - 1 4 by AIA iN) 1B} (3) 4 — = 60
1S... 2 : ee ede Sree Ok pear, ee = 27
oo i ER a a ee 9
50 ,, Lf =
Totals 44 93 115 189 218 215 174 119
TABLE 12. Glasgow. Height and Weight of Boys of Group B.
Weight of Boys of 7°5—8°5 years
Height
a)
ad
>
2 3
5 5 l
2 3 ire
41, J~——-—-—— Av “ey. a ak
Vp ee 897 18) 2b) 68: (3- —= =—= == — e
ee = DS 972 198 43° 96 9 98 1 5
444. | —— — — — DW NG Bt cl (les Bs) ily ey
45 ,, a fy iN) Bi tsfsh hy? YO) ills} © 9) = |
46 ,, - 1 De loue4e3Si bse 268 2909) oe |
47 ,, i 4 &A 18° 22 Bh BY Gb Tie & aS =
48 ,, i) SBR IO) 1K ay ORY ye)
49 = 5 oy 6G Lt, GBS en |
50 ,, 1 1 1 eA gO a) eo |
Bil |p = _ 2 2 iy sy
~ = = 1
Totals} 1 1 4 3 10
30 51 118 126 185 222
174 158 107 108 76
~
26 21 14 4
Biometrika x
310 Height and Weight of School Children in Glasgow
TABLE 138. Glasgow. Height and Weight of Boys of Group B.
Weight of Boys of 85—9°5 years
ee
B4e
35 ,,
36 ,,
87, 2
38
39 ,, 1
40 ” 4
ies 5
4», 5
3 ” 4
ee 6
Ds, 3
Oz .
5
Ss
Se)
> TIA Co HW @
=~
MAMA AAaawewHePS>
2 N CoN
lee
Totals] 1 3 3 11 32 85 100 156 160 162 161 198 124 89 73 45 17 14 7 4 2-1 1 | 1449
TABLE 14 Glasgow. Height and Weight of Boys of Group B.
Weight of Boys of 9°5—10°5 years
~
L
| | Pe
wD I ID IQ 19
4M HR OH
© 6 SG DO GR
37 1
39 1
HO |e le es = ——-
Tie ee | ere a eS =
Wor ss ——-— 2 22 3 511 1 1 -——
48, [~———-——-—— by Ge 8} Gy =
|| = SSO EB Bl — —
45 ., ——=— — — 1] 4171213 10 9 2 2 1 — - -
40.5, ———=— 1] 2 2 6129°94 19 26> 21 8 fF 1 1 -—- — _ —
Li ty 2 91620 34 39 20 13 1 2 Q — - -- —
48 ,, = 1 31118 381 44 48 95 28 4 2 —-
49 ,, - 3 1 5 16 48 45 40 3414 6 3 1 -
BI) - 2 UW PA ily PE apy sy CHL ey alah 7 —_ =
ihe 1 a 4 49 8) 1S 24 26 enero mele 1
52). = 1 Le a 4 2 ISOS ae Om Cee Ome. 4
ie == 2. 2 — 8 2 49°97 58 Yoo 4 OF 1) =a
54 5, 1 1 1 1T— 22 8 2 J — — — — — — — —
55.) _ 2 1 2 1 —-
ob = = 3— 1 --
Dil sy - - = - 1 —_
8 ,, = 1 = 1 =
Totals] 2 — 1- 6 13 19 26 63°80 94 133 188.159 133 132°82°50 51 S18 124 5 2) le
EK. M. Eberron 311
TABLE 15. Glasgow. Height and Weight of Boys of Group B.
Weight of Boys of 10°5—11°5 years
|
Blea ce.
oO SP)
»~D q Ne)
= |
e
Bre KH NODUD We |
| Ww bo bo
Or OU SUNT CO
vo bo | Ww bo
1 6 8 24 31 55 71 124 134 128 171 128 105 88 88 57 31 32 10 %
Glasgow. Height and Weight of Boys of Group B.
Weight of Boys of 11°6—-12°5 years
Sere ete ile eel) je | petals
Lg te Fy: On Or tS iW oD OH WwW KD ike they, Site, Ks
1
1 — = 3
il 1 - = 3
1 1 = = 3
8 1 1 1 13
7 3 2 2 20
So" 9 14 6 2 1 1 - 43
OM 4a 5) S08> alli 4 — — - — 74
tp) lS) Ghoys YAS Nils 10), 2) 128
2 13 25.35 29 17 #9 4 1 1 137
2 4 25 36 438 34 24 0 2 1 1 1 180
— 4 § 20 41 55 36 12 12 2 1 188 |
1 2 5 19 34 61 37 19 5 1 184 |
— —: il 2 4 16 22 30 17 7 38 2 — — Ll — 107 |
1 5 38 9 12 26 138 9 2 80 |
4 (6 lb B36 6. 18) 8 56
i 28) of 360 40 OB 35 |
= i—a—s 4A 4 = 14
—- —- — — — — 1— 1 —~ — 38 1 i, al 1 1 10
pa a ae 1 1
— = 1 1
— 1 1
eS | ES
13 48 70 1380 155 172 175 169 118 102 49 26 22 138 18 2 4 1 1282
312 Height and Weight of School Children in Glasgow
TABLE 17. Glasgow. Height and Weight of Boys of Group B.
Weight of Boys of 12°5—135 years
Height
5 ih il
iS ta
love) S
wn
tnt Os
Ss
WAND OW WOND ABS Ss WHO O
bo
a
DOS SRE EY
—
c
S
De HR ow
bo
1
1
3
8
of
9
2
5
3
3
1
feet
a
mw oognok dr
=
Totals —— 1 6 17 65 113 150 219 223 171 114 78 4613 8 9 3 2 1 #1
TABLE 18. Glasgow. Height and Weight of Boys of Group B.
Weight of Boys of 13°5—14°5 years
Height Totals
u
>
=
428 2
43 5, ]
44 ” =
45 ” az ae: ae ia
46 ” a 1 = a = 1
47 ,, : 2 I 3
ie. a = 5
i ae = ee See 10
50 ,, Sa ee Sa, SP Brae? 18
Due, SS Se al 410 7 5 2— ~ ~ ~ ~ ~ ~ ~ ~— — 29
52 See ee a 28
53 = 61s <4 Ons 1 — 35
54, fo= = SS 4, eS 8) To ye ee een
56, Pi Se 6 AS 6 a
56 ,, 3°23 SG Sede = AT
57 ,, lo 6 ©9212 26) ae ee)
58 ,, ~ 4 2 4290550 2) -— — 27
59 ,, = ae 9 3 2 “1 3.93) -— 1. See
60 = 1 l= = 2
Gil = 1—- 2 — — 2 — — 5
PB d |eee eo = 1 1 ae
68 ,, ee er a 1
O4 ” : = ro 1 1
Gos Me 1 1
SSS SSS SSS
Totals 1 — 1 8 23 40 54 54 67 44 39 30 155 7 9 1 383 — 1 397
eno At RAS
EK. M. Euprrton 313
TABLE 19. Glasgow. Height and Weight of Boys of Group C.
Weight of Boys of 5°5—6:5 years
Eewamieieeriiert ae let f= ee ay fy | | | Totals
GMa oe PG BIG a se ye FAG Ay | uD
Peal Ay NS SSE ery ae Sel esis fey ss QV S)
8) ial Gabo Gayo Se eas sre ub i ears) ZWD
TABLE 20. Glasgow. Height and Weight of Boys
Weight of Boys of 6°5—7‘5 years
|
bo
He
—
or
ee
“IO bo
We
|
|
|
Ww
i
w
Or d> CO GC or
lad
| | | | pow mi ator
Totals
314 Height and Weight of School Children in Glasgow
TABLE 21. Glasgow. Height and Weight of Boys of Group C.
Weight of Boys of 7°5—8°5 years
}Height ty fo St eh ail, 7h el eels gat ae ee dh | jTotals
I HH DH HH 19 1H 1H 16 H HH 1H WH 1 19 Ib | 1S
Ts S9) AO) eS So Spiel imiteemena si a) sy tos Ne GSP Sa) ey GSS Sa)
9 9 H HH © SS YS SH S59 ©) DH ) H) © OS © =
308 1
ohh Ss
36 ,, 1
37, 1 1
CT an Pee ah ke CMO did ihe oi) 6
29 Le To A ee 11
Gor A ao oe 10
pt eer Ge owe 32
42°, Se Sa 43
5S a We =, Soe peed ally 66
eg (mes sear 5 TT 94
| 455, Pe 1 010. 22) 7a 14) 3. an 104
6 Se ES ape PRS 90
47 ” Tl ae Sat Ge Se PSS SC ee ee eee: a ot eee ag ena 2
4S ” ae q
49 ”
| AO op
| ol ss
| 52 ,,
NODE Gs
54 ”
95.5;
56 ,,
57 4,
Ooi,
Totals} 1 1 5 8 23 37 66 75 94 88 61 72 52 2617 7 2 4 1 1 — 1 +4 642
pei el [see = = = = = neon
TABLE 22. Glasgow. Height and Weight of Boys of Group C.
Weight of Boys of 8°5—9°5 years
Height Totals
an
wer
RHE HEH Oo Ss % OS SS
ZA NMW Ss WHS OAWAND
—
47
|
PTTL LET] al ee] erro more! =! |
wel |
i
WD re Ob
1
1
5
4
3
6
5
2
1
1
1
| Se
NORE wWoOORR
Bere
E. M. ELpErRtTon 315
TABLE 23. Glasgow. Height and Weight of Boys of Group C.
Weight of Boys of 9°5—10°5 years
Height | | | | | | | | | | | Totals
enh ty ess eye athe key Ney Uy Ne ea io)
m 8 C f= SO ) ~
Oo © Oo © 2 ~
392 -— 1
AD op || Le re are 2
Aly J — — — ee er ee =
Oe. : 33
43 | | Sapa aaa, 1 1 4
4A 4, 5 3 3 17
4S OS = ae eae 1 5 4 6 4 4 2 ——s —— comes —) 26
cme el Oo. 3 9) OMA 7" 7. 5 = = SSeheil 158
in ee ee 8 7 ay 10 4 Tod == 1 a 65
48, | — — — — —- — — 665) 2011926 18e1t 2° —= 1 2 = = 110
49 ,, — LB Gu Bis 1 ee wl wk By | Se Se
50, f— — — — — — — — — 5 Gi lOnls,o3 10. 8-6) 3 ==] 88
aL 4, = i i BO) By 13} MO) WO) ak 62
b2 ,, — 1 @ © yf BR BP ak a ee et
53 ,, ———— Oe Oe eStats 20)
54 J —— — — — — — — = eee aaa Be ss 11
55 ” nay) = as i rl = SS SS 1 1 — « —— 1 = 3
} == i l 2
2 i a ee i
6 19 32 33 66 69 91 68 81 37 616
TABLE 24. Glasgow. Height and Weight of Boys of Group C.
Weight of Boys of 10°5—11°5 years
Height Totals
Ss al | it ! | | I i u i d Ney Al id a wD il ry if it rl il L | J a
3 SHA SHS DR AHH HK DAH 4 S
=> S OB, i 68S 6 6 MM BR RM NR Go oo H is)
252 1
oe) oe ik
40 4, —_
41, J—— 1 —~ — — — — — — 1
i 2
we" 1)
4h ,, 2 4
45 ,, a) 9
46 ,, Oy BY) 9
AT ,, Be 9) 4.2 Selo Al = = 36
48 ,, 6G 16slhel2 == 8) 2 == 2) = = 65
49 ,, J — — — — — AWS AWeO iS eal 3a ae 61
50 ,, 55 LO O9al'7 to 1k Bad) = = 96
51 ,, LG) aly WO GT 2 80
52 ,, = TL AIL Wee GY IS ee) = 75
53 ,, — lS Le a OURAN Meron sees We 3 = S59
54 4, | — = 1 Oy ese 8) Gel al BP Oe a a a | BG)
55 ,, oe ul eo sot b) 08) Orr Fe
56 ,, a et eee 0 yi Se
Se SS SS 2 [=|
39 44 25 18
316 Height and Weight of School Children in Glasgow
TABLE 25. Glasgow. Height and Weight of Boys of Group C.
Weight of Boys of 11:5—12°5 years
certs ee ie ee EN ek of" Howls
Pe i eta Sie Dae Sagal Ue eto) bal ak Us) sy tS) | > pe SS
Le OQ SH SSeS Ss 897 8S 1G) RE RD CO SS Oe SO Sa) | an
Shr Ser Ui AOI Gi 6 8) SO! KO) UBS) BSE Ee Co Co ROOM EC) EOD EOS = oF
35% —— ee Dee 1
4l, Jl —— Se 1
oes ] = == Sa OS OS SS SS i
Jhy J- 1 — —-———— = 1
ee ee ee ee 5
46, |— 1 2 — = —_—— -— 3
Vy derma Va ee a sy ye lee) lg Se — —] 13
RS EE a SS a ee ee 26
10) NS a onlay — —{ 41
Oy —— 17 91812 611 2— 23—— —— 1] — — — — — 69
ol ., —— 2 2 9 14 2823 7 7— 1—— —— — — — — — — 93
52. ———— 315191819 461 1— = 86
co 6 715 18.17 96 2) 2 — 72
i Te aye ePBy als, (Ge Oho Ts ee 81
DD, 1445 8 6 4 32 ———— —— — 37
D0 ne —- 1— 4 3 56 810 3— 2 — | —— — — 37
Sr APO Se eS. tO meal = 17
AS = — 39 WS 7 ee 7
59, — -— l — lili1ii1i— 1 1 8
60 — ee oh eee zee
]
1
Totals] 5 4 13 29 36 74 86 83 88 54 42 30 3011 7 3 1 3 — 1 — 1 [601
TABLE 26. Glasgow. Height and Weight of Boys of Group C.
Weight of Boys of 12°5—13'5 years
66
| 67
ea a en ey EN é
cee ee 80
4 4—- — — 1] — — — = —-} 62
15s¢ [2 oe ee
16 9 9 = SSS SS eae
6 199 493 Pie
913 6), 222)
By Sou ieeercbe le Sle) ae
a 2 vO aS ois 62) al
— — 1°41 2) 2 = = =
— — — 1_m— 2@ ii —
— —- — — —~ ~ — 1 2 ~~ —°1
— — 1
—— 1
i
| Totals 9 5: 90 32° 77 94 “881 80 154. 58 95 19978) 10) Gun ae en
E. M. ELpERTON 517
TABLE 27. Glasgow. Height and Weight of Boys of Group C:
Weight of Boys of 13°5—14°5 years
Height| | i ee aS ee eee ae | | | Totals
EME rea et tay 8 AS i AS aR AG |
9 DS 89 > ™ itn) er) Sp) aa = » D> Sa) fag al wD = Be) |
ite} “WD Ney NS ~ ro ie“e) ive) Dp 5 je) S S i | ™ In| 2
SSS Slee ot |
i) = Se 1
49 ,, J ae Ser i 1
50 | a pe ame 4
ee 2 TY 8 6
SD os | ees ea ee Ha = 17
53, 31° CRS as ine spay eh E 26
55. ee oe G6 ae VG aA 2 lho
55 ee Fh 816 4 ee 35
Pe ee eT 6 10 18 a Se 1 Bt ee 41
Die a TU 8lUCUGCBCOCC CB TD 33
Se 08 at eed. Bk ye
Pe ee a Ge 15
60 ,, eee Vere s eo Roi SN Pare Se ON ag
61 ,, = ea a ee eee ee 4
ee le leo. eae 6
Oo. a Ae ee es ee 3
64 ” aie => 1 asi tees Caw * 1
65 ,, ae ee ee ee 3
(50 Ne ene a 1 Ee 1
69 ,, —_—- — 1 lL |
Totals Ae Open led) 49) S00 29 9? 1b. 8. 8, GB BD, 259 |
TABLE 28. Glasgow. Height and Weight of Boys of Group D.
Weight of Boys of 55—65 years
Height a ie eee lim Serre a
She ee yeh ee ey ee Slike 2 Sages £ : :
365— ea 2
a7, 1 2 9
355, 48 15
39 6 4 17
40 5, 6 16 i 54
ee 2 6 8 al
Das 1o8 if 92
43 ,, I 8 71
44 ,, 1 44
45 ,, 7 25
46 ,, 1 12
47 ,, 4
48 ,, 1
Totals 417
_-Bicmetrika x
~Z OD Ti Us ¢
a)
OH OH GUE BE REE
S
We DWN OD
318
Height and Weight of Schooi Children in Glasgow
TABLE 29. Glasgow. Height and Weight of Boys of Group D.
| Weight of Boys of 65—7‘5 years
ae mens Oe eee oe le ee eee Seo
Yee et Moe rye Sie Oey Ye Mah 9s Mays ky
Lee StSS Ne ope Gey Tey) Tee Fey Sy 9G sy) OS
sh} 6 © He SBS SE SF SF YF HO 6 G
1—-— 2 ~~ —~ ~—~ ~—~ Tre ~—- ~~ ~—- —
— 1 = ——— oe
— — 2 2 2 1 1 =
— 2 OD) 2 eee ee 1—- —- — — —
te: A) =e lpn aS aoa eee Maem ages
om | ep 2 Sao al re el Oe) 1 =
ey ee ee a MA wey OIG oh Si :
44 4, SS 1D ly BR OB se
Feats (emcee nate ay) ear eH OP MEL ACY §. Bh) Ge
46 , 3) 8 Sallis Osean Oom a8) ae Tee
awe 1 ieee Gy ise eee eee
WRX = ee ees
ve a ot ee ey eee Se
505, — 1 —
Totals 1 4 138 14 32 65 90 84 80 44 29 20 19 6
TABLE 30. Glasgow. Height and Weight of Boys of Group D.
Weight of Boys of 7°5—85 years
| J Is J i al ’ | \ | i Y | | il ; l J 0 | u | I! f
ey <3 5S U RS Fey iS > > S : Q
x 9 §9 WD ~
~~
Ss
a 1 a= fae 1
1 = 2 _ 3
ae gh a = = 1
Se ee OS ae 3
ee eet N Fil ai ee) al é eee 8
ss 2s 159 ROM ON hae — SS | 58
Occ 16G ble £6) ae opeel eG
Seg eRe) icteats oor) OYh RB. Be = 71
Hh) tee = Lae 8) Soe eee ed == || 8
= 1 1 98) 20729 28\rs oa Se aoe
: 9; 0 0F 16-220mo0 100
1 2 tA er Ommnly 64
= Sos see eee Oe a 38
16
4
1
2
—.2 h&. 4 83. 42. 55. 94°72 80 69 68 26 14 9 5° ‘Ge ==ameieminnen
EK. M. ELpErRtToNn 319
TABLE 31. Glasgow. Height and Weight of Boys of Group D.
Weight of Boys of 85—9°5 years
ig Totals
SEEN ak sta el ee eee | f
Gy ae Sa) ey SS a els Wea ee bent Sy res tee he Spy ey te 39
3 > S19 5 Bo SCS ES $
373 1
38 ,, os
39 ,, iia
40 ” a
4l ” 1 2
42 , 1 2 1 6
We & 4 a Ze
Dh « —- I 235 4 2—- —_——
fo Po 2 WO 1Oe Ge 1 Ba04 |
46, J——— 7131310 7.5 3 1 — |
ieee = 15 24 17 1b 8 8 2
eee = 421615 01-19 1619 7 1
49, J—— — — — i T2210 12 th 8 3 I
50 ,, | — — — — — Bees 2) el 20-0) 19) 13 — —
IP 2) 2) 2 ADT 33. 5 3
14 6 4 2 1
12121 1 2—-—1——
8 16 36 54 614 61 71 5! 9
52 41 25 15
TABLE 32. Glasgow. Height and Weight of Boys of Group D.
Weight of Boys of 9°5—10°5 years
Height [inca ah fea eee ease eae | Totals
: - ey ey > 9 \ yD
3 2
36 3 1
oe
3 ” 1
43 5, 2
44 55 8
45 ” 2 8
46 ,, 6 5 1 23
Ty 8 8 Ale 3 48
48 ,, 4 12 Bia eal 5D
49 ,, 5 6% 20 15 94
50 ,, 2 22 26 115
61, | —— — — — 1 917 84
52 ” AO OMA SeG 453) — 60
Dy 3 40
54 ” 1 12
55 ,, 2
56 ” 3
Siew. 1
58 ,,
59 ”
BO
Cn. 1
Totals} 4 8 10 24 27 34 68 50 6: 52 45 25 ¢ 558
320 Height and Weight of School Children in Glasgow
TABLE 33. Glasgow. Height and Weight of Boys of Group D.
Weight of Boys of 10°5—11°5 years
Te
DQ 9
J!
iG
i 1
io 13
49 ,, ‘ ray mies
Ol, Bediky py aye ty ey YE ak bal
Sil — /610920 14516) 9! 6553
52, J—-——- - 1 3 41014 380 17 1211) 4 7 1 2 —
53, | ———-—- — Ne es) oh kay ye se Gy OS} Te TI
Bee — 5.5 6 8 410°563 253 =]
55. 9 4°36 4 4 3 900 2
He) <p ] 1122 3 1—— 1} —— ~— —
57, 3 ah
1
bo
|
e
1 1 11 9 13 26 49 53 64 65 75 49 48 31 3118121210 7 2 3 1
TABLE 34. Glasgow. Height and Weight of Boys of Group D.
Weight of Boys of 11°5—12°5 years
Serine tal ye
SESTESSS SRCRKRSITESRSESS F
l =—— =
= ] ss _
1
1 2 —_—
— 8 Se
Ue Bie le Bl —
aa 2) 2 Oe Cale eee —-—— =
= e215 e ON Ge 7 a
—— 3791212 6 6 38,1 = — —
= sl 1 39 8 18020 ot 6. soe
—-— 12 7 527 3714 8 2 1 —
eS) Sle PEyP Py ail ce ET
1 Let 3) 15) 1 s10%) S; aeela sie
-- - 1 5 99 71051023 I
erie. Se 2
1 22 aoe ee
be Bo 2 l
2 l i ===
: : ae
1 — — =
Totals] 1 — 1 10 13 39 44 54 83 112 77 50 3292113 6 7 — 3 2 2— 1
es
K. M.
ELDERTON
TABLE 35. Glasgow. Height and Weight of Boys of Group D.
Weight of Boys of 12°5—13°5 years
Height Totals
443— 1
45 ”
40 ” a
4i ” 2
48 ,, 3
49 5, a 7
50 ,, 6 16
51 ,, 6 37
52 7 48
53 ,, 6 1 72
Cah 1 = 84
55 4, 1 4) 1 101
56, nie 2 =e 80
On ss 7 ae 68
58 ,, 4 See 42
59 ,, 4 3s 21
60 ,, 1 Tesi 1 14
Bi. 1 De ee 9
Go. — elt, i
cee ieee age 4
O4 5, 1
Totals 32) 31 59" 779 TO 119 65 62 22) 26 611
TABLE 36. Glasgow. Height and Weight of Boys of Group D.
Weight of Boys of 13°5—14°5 years
Ese] ol Ce a a ee
y VE) UD 2D 1 WD voy ey ite) Lays los oy
= a m 2 i
5 ee Ol
51 | os ara ee a
ae 5b 6 = —
eee — 1-6 4-10 2 -3
54 5, —— 1 45 815 4— — —-
DS io ——— 2 3 41310 4 2 1
56s || i ee toe) ee ee a pe
Di § ———— 1412 97TFT— 2— Q
XS || —— 2 16 OMI SL2s 245 Go we al 1
59. | — — — — — — Op By= 8). 1K0) 2 aie
60 ,, Teja oeeo eh Geet Tha .
ot 24a 1 SS 3 ee)
62 ,, peers Oh sult oy ae ee eT
63, le Oe re as
O4 ” Win’ = iat Sane eg ae eT Oa ee eae ee ee
65 ,, 1 lo il Se Se
66 ,, — — = ——— = = = |
10 5 | — — — - —— — ——— 1
a
Morals i we 1299-95 47 6415940 41 1396 14 7 6 5 2 1 2— 2 — 1
322 fleight and Weight of School Children in Glasgow
TABLE 37. Glasgow. Height and Weight of Girls of Group A.
Weight of Girls of 5°5—6°5 years
hel
a ss
29 4, =a) = ae
30 ,, ee ee = =
Cig. = — — 3
oe — 2 eT 1 6
he — — — — 38 1 i— il ili — = = 8
Bh yy —-—— 2 8-4 5 1—-— 1— —~ — — — — — Ff ,16
85... i ee 88 DOR aie Le aera 1 — 44
96 — — — — 8 122 122 1 5 1 — — — 49
ee Le 1 = 38> 9530 sd BOD ole 1 82
38 ,, —— OTA eke) a Sib a8 8} — 124
oo 4-13 43 59 23 19 6-=— (9° = j= 170
40 ,, 1 7 24 42 55 46 14 6 2 — — — — 197 |
Via hae 3 8 6-20) 45> 757.) 19) bn Sa oe 165
Lge 1 1) 49902351) 929) S17, Ik oe eee 146
4B ,, ieee gi Ahkeiloe uch Yi Oc — ie
As, Px SB) Gp 07> SO pee = 38
45 ,, = | 1 1 1a oe il
Vow - 1 -= “2552 6
hae = I Ae 3
Totals 1 1 4 8 25 50 110 146 183 186 205 100 47 41 24 8 2-2 1143 |
TABLE 38. Glasgow. Height and Weight of Girls of Group A.
ae
Weight of Girls of 65—7°5 years |
Height ik ih al i ‘I it 1 | 1 | L | L | al a Hl al t il | al ab Boels
yk aie nest Daas OE ; oe 5 ey
t aS > > Ss wb WwW Ns)
30 &— 1
COL” ex. 1
82... 4
Some 3 4
o4 ” 2 er 6
85 ,, Do 18
BO, abe ali 23
BT. 5 ee KG) = 38
BET 1153 1 69
39 ,, 4 16 1 119
40 25 3 = 164
tee = 8 4 a 218
HD — 2 2 2 1 — 215
43 ,, l Oe ay = Il == 211
44 ,, A kB 166
wa) Ley sy" 7 91
4G ,, SOs oo 34
LP ee ya 8 ll 17
1 iL & 6
=
1? 2)
|
Totals 3 BY tb 95 153 189 265 191 1438 143 : ‘ i g 1405
E. M. ELpErton 323
TABLE 39. Glasgow. Height and Weight of Girls of Group «A.
Weight of Girls of 7°5—8°5 years
j la al : 7
ae ee ee eo eh ek te ae
SY DB ms DS © & > ™ SA) 9D co SH Ba ay Rey Ge Sen tsa ey aoe UD
RNR VR RN 5 46H & Say = > = = os got a Nene es hn Feng ken Se Ss eee eee co
1
1
2
4
1
A
5
10
3
6
1
AaNOoORe wy
Totals] 2 — 1 7 25 39 71 108 183 199 189 197 182 80 71 46 29 5 5 1131—41 13938
TABLE 40. Glasgow. Height and Weight of Girls of Group A.
Weight of Girls of 85—9°5 years
Peon st | | | td | lard eet | [Totals
Eee oe eNO 2S Rotor IG) US is. Was MS SS |
yar es ID sy BA mw ID a~ ™ 89 cys 2S) 9 9 ~
Q wm s 2 8 3 :
328 1
3S 99 nn 0 nie! apenas ce ee ON a GR ae a ee ea See il |
OW yy il)
a5 3i|
36 ” 4
Q = ] 5
6 3 14
5 1 14
Lessa lees 2 — — — = 26 |
36.112 4-56. — 46 |
SoM PS eS: oy Gas” I - 72
A tae) 07 DOR 20m Gre 3 121
— — — — — AW 22e 73840 Aer ee D8 1 160
== — — 2 712 33 51 46 33 10 10 3— 2 — — —— — —J] 209
45 15, 42> 40°30 32.98.10 2 3 — — — — — 1 |, 219 |
— 12 7 li 26 32 23 3012 6 6 1————— 163 |
2. 2) VO Ge fo 2438.15 12 4° 3 1 —.—— — — || 133),
— 7 3811 91697 2 1———— 65
—— — Ih Sol Miler =—elleSslO S.Quil 2° Bo 4. 1) a a 37
— — eS pS ee ee ee Ss 17
— 141— 7
—_—— — = SS ee ee 2
a Mi ene 1 |
90 135 214 170 129 112 122 68 40 28 16 10 1314 |
TABLE 41. Glasgow.
Height and Weight of School Children in Glasgow
Height and Weight of Girls of Group A.
Weight of Girls of 9°5—10°5 years
n)
8
SS
Diy
Ce ¢
a)
ec
Ys
RR & aw
VS OS
es)
SS
nS
_
Nt~ 9
a
1
4
0
6
3
6
3
1
IL
3
womornw rs
it~ Ce 0
FNL OR KWH
| me Ore Phe
| |
epity
ax
AAAAA AGH
Cs
eo)
22 36 45 74 131 107 147 135 153 112 100
74 48 30 2!
TABLE 42. Glusgow.
Height und Weight of Girls
Weight of Girls of 10°5—11°5 years
ae ee |
Mote viet oe Woreierilap ape at Wel iene 1 ike)
STEAD aah SGN OS) Galo Nee RS GS
3 St SS + J Ye 3
U
eee
ns
x
OS
on
bo
Gg Cs Ce oe oe
& s
SS
exes
Wi Le CH
DE Db orp w
wo BATON TR ee
CON
roe | eo OUR
=
a)
HN eS HSH
S
Mts Og ]WWN
an
e
w
et
1
8
6
18
16
9
5
PpoaT@o
BSH CODDNWe
Oo ND
AA AAN
S
Xe)
| Totals
5134) 18022) 55) 3) 272 ol
EK. M. EvprErton
325
TABLE 43. Glasgow. Height and Weight of Girls of Group A.
Weight of Girls of 11°5—12°5 years
Height Totals
TL 1S Se GS tana ran es
lim twos OS ace l
35 ,, il
38 ,, ]
39 ,, =e
40 ,, 2
41 ” il
42 5, 2
43 ” a
44 yy 13
45 ,, 20
4O 4, 32
47 5, 65
48 5, 83
49 ,, ) 139
50 ,, 4 162
267-1 A l 161
20 ieee ol 153
— — — — — 112 7-1 1 — — 133
3a Orb — I le 92
4b Oh by cay kh 93) = eS SS 65
All) 83, PE ie 1) DE Oh. ee 41
TRA a a) see i eee ea ee 27
Sot elle lle ———t le a 1 9
as ES — i
| 2 ee 1 |
3
a 1
20 41 70 OD De Otsolelo elo Ome seem 63) doe 1s 7 1214
129 159
of Girls
of Group A.
TABLE 44 Glasgow. Height and Weight
Weight of Girls of 12°5—13°5 years
Height} | | | eae | | leeelee lie le al : Totals
Se] WW WH H WW 6 BH WH lH H H we ww IO WD |
Reese es DS 1 SS RS SS
Se See Se ay Ss) SOP OP at ROMO” Os BCI ho a
139 120
= It]
4 | _ —
2 3 —
8 4 1 — —
Ob 43 2, =
13.1310 1 1 =
712 4 12 — =
3) 6B) 1 il eS
12 6—11—2 ——— —
Ae Deedee Dt 2, 1 —
es
een ES EG Ye es ee ae ee
Biometrika x
326 Height and Weight of School Children in Glasgow
TABLE 45. Glasgow. Height and Weight of Girls of Group A.
Weight of Girls of 13°5—14°5 years
a 3
> ~
Ks) io)
464 — — 2 —-—--—- - -— 2
47 ” 1 a ah SSS 3
48 ,, Boe ee 2
vom GBM Se Sip ROMREee gs =o tamer 10
50, ==, A OC Ogee Se pasty vel at = = 13
bl | eee 8 68 ea be 2 es
Vee) Weer oe i sO) Gy A _ = 33
58, | — — — 2 2 3 2 6 4 4 3— 1~—~ —~ —~ — — — | 27
hy | — — — — 1 5 1 10 8 56 5 1 ~ 1 ~ ~ ~ — — Ff 37
55 ,, = 1-9 10. -% 80 8 ee en
56 ,, — 2-167 8.8 4 6 6-—9 — — Wee
Cygne (eee ee ee Po Se a eT OS
a (i ee ee ee Peay oy ae OO |! Be
BYP) 3 —- — — — — 1-— — — 1 4 3 «4 a — o al 2 22
Ce | = ee eee ee SS 3
61 ,, = = — Ss 1 Flt Sse a] = 5
2 ” Fay = ch Ges Osa eee 2—- —- —- —- — — — 2
| 20 30 22 28 11 291
TABLE 46. Glasgow. Height and Weight of Girls of Group B.
iF a |
Weight of Girls of 5°5—6°5 years
Height rama lave lem Totals
: eo) Re RD
isp) 9 2 jer)
39 > =>
Diet em gal ae ah 2 ee 1
30 ” 1 4 ae .- Lo ae 1
Bie ——— a= ae = = = SSS
32, 2 = 2
By 2) 4
Bh ” os 2 1 =a = a d
Soe Ne oe ade Ome eee a 14
86. 1 oO --6 10-9 = SS ee 28
B75 Se BOB 6) I ee ee 49
38, = OY 29. 86 24.10. 5 8 ee eee 78
ore 1’ 93s 18°238" 888033). 14 924) ee rn 152
40°, [oS TD 84 P81 536 80) 13) 29 se en eT
le. — — — 6 18 48 89 45 22 6 3 1— — — — — 188
42 ,, —_- — — 1 4 9 31 53 34 14 5 — — — —~— — — 151
43 ,, — — 1-— 2 6 12 37 36 22 21 4— 1 ~—~ ~— — 142
44 5, — — — — 1— 4 14 17 18 12 8 — — 1 — — 70
45 5, — — = — 1 1, 4 8) 5 8 7 Se 24
Wi eee Oe ees Te 8
Vat pee = SSS Se Sat 3
54 5, =a mM ee eS Oe SS SS SS 1
55 ” Peat, Se ee eb See a gee a ea = 1 — 1
Totals{ 4 17 37 93 148 171 198 201 133 61 55 18 3 2 8. 1 Teme
KE. M. ELprERtToNn 327
TABLE 47. Glasgow. Height und Weight of Girls of Group B.
Weight of Girls of 6°5—7°5 years
Height
327— Seeger
$3 ,, Dive VS
34 5, a 4 2 1
35 ., —= Sey
BS os ee = 5 =
ny ae as a= $2 2%5 A SF ah ee ae =
gee eee me 7a OPC IO) 4 Ae 93) lt Se
89 ,, > Do Sly bay al) ye 8 ae!
Oe i 3) 20. 36 298 280° 8 8 1. 1 1 ae aa
41 ,, = ToS A NO). Zu 8h) Dy i OS Se
42 ,, ee elo oh 160) 47 2 lo) 9) 9
4a, | — — — — 1 2 2 52 57 45 23 7 3B ~— 1 ——
ane 3G) 80) 56 465 36 20 8 38 2 ————
45 ,, = 1 9 14 30 31 23 15 4 @2 = =
Cn a 8 9) 8) 18. 13) 8) 5 =
ies — — aS > Re EG OD il
48 ,, —— SS SS SS 1 1 — 83 2 4
499, | — — — — —~ —~ ~ ~ ~ 2 ~ ~ — 1 1
50, J — —- — — —- — —- —- —- —- —- — — 12~2 1
157 224 215 54 Ql
TABLE 48. (Glasgow. Height and Weight of Girls of Group B.
Weight of Girls of 75—8-5 years
Z x PRS om, i a See a os 4 |
Height | ot aan Same a ae (aa eile | | [Totals
Us Be a ‘ DD DM kD w% |
Dd 89 E Sa) a o> |
a) = ID UG ie)
|
3 1
34 ,, oa
Oy ‘
36 ” 1
aT ” 8 |
RE ss 13 |
OY 5 34 |
40 ,, 45 |
4l ,, 87 |
42, J~—— 1 2 3 622 35 30 16 9 2 J—— — — — — — — 127
meee G88: 89. BI -36..80) 16 oh) 8 9 = 177
44 ,, t= 208
4d 4, fee al 229 |
AND sy. | a ral yes 166
47 yy J — — — — — — 10 2 4 132
48 ,, Ona 69
ID | | SS SSS SS Al cd 33 33
50 Sup 16
Bil S279) 1 4
52, J———— — — 1
53, | ——— —— — — Lo
Totals} 1 1 4 15 25 51 78 155 16: 159 115 96 30 19 12 1356
8 Height and Weight of School Children in Glasgow
TABLE 49. Glasgow. Height and Weight of Girls of Group B.
Weight of Girls of 8°5—95
years
DD
=>
I
fos)
4
Ke}
Coe Orb bo
| snaozaal |
Hanooquere
Totals} 2 1 1 38 15 42 74 92 123 163 161 142 156 132 98 58 44 23 8 8 5 4 1 2 2 | 1360
TABLE 50. Glasgow. Height and Weight of Girls of Group B.
| Weight of Girls of 9°5—10°5 years
Height) | Pd lb al a ee ee
1 19 ID 19 AS 2D WO AD Wg AHA ADAG WAG AS ts to Oe oe Romeo mS)
a eS depremab Say ats) =~ on) ™ SS) ID NS ory asl SSsaleeewatss] les) Gash Ga) US) oO
HH H H SS SS Suen HIN SHKCONE cE IRS Ns) To sy AS aS RS CS
367 ] = 1
a7 ae 1 : = 2
eoe. il 1 1 - - == 3
295 el 1 1 =_ 5
40 ,, 2 1 1 ae 4
ee Cee ee ee alee oh = 10
IO Neale De eae ieee _ eon
ie ae eM ye a. od 2 eee
i ate rem besa ealeg alls Oy. 1B) ee ih he es : = 62
DHSS ——— 1 2 317 30 23 22 to 2 wl — —_— 117
4O ,, ———— 2 217 22 28 22 20 13 8 4Y— 2 — 142
Wie Soyo} eis Sh) Zbl Bey Bey = 4 Hl —- —_— 194
ke 5 | SS SS L169 le 23830 AC Sl Ae eee ee 190
49 ,, 1. 8 811 1b 44 41.99 76.8 6 7 eee 185
00%, ] ay ES) lise Xo} iO) AT 0) GO ee 136
ons ] 2 1 1 By ey Ty aly IG) Wk il By HY sD — 1 110
52 ,, 3 2°98 9 9 9 7 8. 3— Teneo
53. Pipe lbes) Pe aNey in lee I] Lael 24
ae 1 1 1281 eee 8
| 55 ,, 1 |e ee ee 2 = peal 8
item (eas oe ae ai ee ee lo 1 = 6
Ce aoe 9 Sy ee 4
3 4 11 24 35 76 103 127 136 126 176 138 116 81
K. M. ELDERTON 329
TABLE 51. Glasgow. Height and Weight of Girls of Group B.
Weight of Girls of 10°5—11°5 years
[Eh PT a a ce
Totals
ee oy, eg on see Se ORO, WS te “tt a ie Ass G.
Soe Oats SO LOR re Oy ry) Sd) Ce. Od
6 % SD) RCOle eS Gece, 2S IS CO! 109) 109) 65. 09
7
9
41 5—
43-5 —
91-5—
3
10
caw | vo
w
aI
—
He bo
|
| ox
a
WOanNTRARR
See
— OO
eel
RP Ol fp bb
Sa ew oror-1 © © bo
| FNWWNHNYe
| ern ores =t | [oi Hel I
| mies arorosen |
eae COS) Ee)
a= or CO
1 1
years
Totals
|
fo)
»
I~T He GO) COT IO
Tee
S
Oro
|
{
edie
J OV
Se POH Oke
=
=
_
w
w
bore bb b
i
—
i
WMOWDQO of
u
or
ES)
Sw
i
b
| |
| = ro | ATR NI OD bo
| =i enon bo be
Totals] 3 4 14 29 54 103 140 162 171 129 131 86 74 59 361231212111 3 3 2 — 1 | 1252
OW PENH EH
330 Height and Weight of School Children in Glasgow
TABLE 53. Glasgow. Height and Weight of Girls of Group B.
Weight of Girls of 12°5—13°5 years
o
cia (Me ee eh ob has oe ak
Rel NSPS GSE ast WOU ONEMICRYS Tue) Moh (SN IGA tS Gal) Rey ea Ses Se) NS
“pb ost oF 1 iD Tope resets es) usc). sy Key SN S = = Tel gtel x X
SSS ==
Dh A iae |e eee na |e ee ee a a
43 ” = sy a oat a eal
ga Nias ee Dad agar ees ae =e a
Pitino et lye ane mes Se e es —
YGe A reseed ee -—— —
Gp oe Se eO! GAs Eo ameD ee = —
YSt ieee baie weed eee eas — —
$95, = =e OD: SISA SS Ope ee
BOs 0 Wo = 2 ALO", 1 Gomes Ge ee = —
ble, (oe LD S00 een = —
525, = 2D) 198 82 SP GOSme inert tee = —
Ce | eee 2: 5, (2333931 lo o6y ole =
54 ,, ee eI eS ale RO ayay ey I) sy zl | =
Ds ——— 12 5 14 30 48 941610 4 1 — 1 =
56 2 1S) 3) 10) Mibg 92407 68S Ay en ee
Ye | ee CREO Se ie iiy (9G Seo)
58 ,, — 3 20 10N 7 1314 on ie
BOs Wretntos a eae —= 5 6104s
60 ,, eet Lge Pgh Se ep
Gi 1 ey a ye she OY ire
62 ,, = 1 1 SS) Sea
Gia —— 1 ——
60 ,, 1 —
67 ,, = zal _ =
OS ” os 1— =
69 ,, = = Catal apa ahh Se
Totals] 1 4 18 34 87 144 133 180 154 125 93 63 63 231612 7 3 3 2 — 1
TABLE 54. Glasgow. Height and Weight of Girls of Group B.
Weight of Girls of 13°5—14°5 years
Sa kod Wk hs 4 hobs " Sg | A
$38 6 8s 8 SRkkSseRseSESS x
sf—| — — — 1 — — = a
| 46 ” 1 ae rae LS Ss as aa ee oa aN
4? ,, = = =) Se = a
48 ,, a i1—- — — — i — =
oe oS ge ee
GO. i ete nl ln ee — ees
sf, | = '— 2-9 6 9 2)6 7 ee
50. Ns ee 8 0 GaGa mee — — — — —
53 ,, Se ee sy Oe) I A Oh i ae Se SS
54 ,, SD oe LT SC oe one ele eel
DOI. = Te A a ida: SS
56 ,. ee ee ee cieihien) Oe eS De
Oe, —- 1— 1 TL a Ge Ge 2 lO ae Oe 2 ee eee | —
58 ,, 1 ae er i BY ay GT eee
59 ae 1 Th Fe Ae lla
| GO ,, = it oe 4 il A SI
61 ,, = 1 peeeen MO Si RB Aye Op ee ee —
ie 1 eo
680 = = SS ey a ee Ss
a LD |
| Totals 1 1 7 11 21 25 44 59 44 45 47 31 21 21 10 5 8 — 2
E. M. Exvprerron . 331
TABLE 55. Glasgow. Height and Weight of Girls of Group C.
Weight of Girls of 5°5—6°5 years
Height | | | | | | J Totals
. ae
= <s
i 1
32 ” oa
3S 4,
Bh 5,
§ a
22
43
56
18 19 13 3 69
9
PEL TT LL eno om te |
a
are D :
[oe]
Sil
He
|
|
|
|
|
| Hee 100 ox |
| ec
TABLE 56. Glasgow. Height and Weight of Girls of Group C.
Weight of Girls of 6°5—7‘5 years |
Bega te l-t tI leery ee lade Ste) dee he he ap Botals
Sh) Toy A Mae ipl aise Belo e alop tte
. . . . . . . . . . ~ a . . . . |
i
2
3
7
19
34
ae 50
4l ” — 105 |
42 4, Boao, Oil fee Ion
ipsa Lop Shoe es 03
4h y, 77 7 41 — — 79
Hie (SetGu 1 232 oY 61
ye heyy CG oee Gan) ein it
ia Messe te or Or et 14
38 ee ea A ee 9
49 ” Py a a a See =
50 ,, —- — — —~ — 1 — 1
|
Totals 617 |
332
Height and Weight of School Children in Glasgow
TABLE 57. Glasgow. Height and Weight of Girls of Group C.
Weight of Girls of 7°5—8°5 years
’ |
5 9 4 to) AS
a gyi FAs =e Ney J i! |
& LD ~N fon : ™ Say
= Sie ast © <6
V7 5 —
ec Yree aes iineee Lol Se 6 2 Se ee 3
Peer berenaree z = SSeS St
Gia Ye Se Ee Pe a ES a 4
|. 39 == wile ob gael eae 2 eee — 12
$0. 9 — BR = 7 98 4 3.99 9 2 ee een
Paces i. 1 8 eR = Bpemomme enon no 2 39
| 42 ,, 4 AG SS AS Se ees ee SEP 66
one — "= 1-8 8319931 1619 =3 3240 ee en
L444, gf) ete = 19) 96? 30° 98-1 7 a) ee ee
| Yo. Gp ee ee 2 4 1-1) 87 S18 ig) 8 a ee
Ihe) il Sy ile AV eb 118} 5 2 2 — — = —
lene 2 1 7858 I yoy 42
See. = 4st 405 6 S10) 23-8 3 oe
or 1 Li 8) 382
50 eal ergs SO Me gl ee
51, = 1
Doak. — =
Doms — —
o4 ” are es
Totals 65 87 129 66
TABLE 58. Glasgow. Height and Weight of Girls of Group C.
Weight of Girls of 85—9°5 years
ff = 8 e : L i) 1 i d i if J J uD 1 [ al ul i J a el is) wD J J Xe) J if To t als
32E 1
pis = 2
some ye 7
THO) i Ba il 6
Tie Sa ogee 9
2. 2 2483 ele a l4
sen 8 6.94 62. 2 4 22 37
Te 4 6 916 9 56
iar i 2 9 11 20 20 95
pte Rei eee I “6 6 21 31 2 111
ie Sy ei) oe Ge i eS 102
Ban BES te nG 89
yo 2 2 59
| 50 ,, a 27
51 ,, a 16
D2. 4
58, F—— ee er eee ee I ee ET ET 2
i —
De op | he es ee eS 2
| Totals 11 23 38 46 78 97 iO) TEE a OE 639
EK. M. ELpERTON 333
TABLE 59. Glasgow. Height and Weight of Girls of Group C.
Weight of Girls of 9°5—10°5 years
1
Height feel |
1
| | a NOUNS Eee BS).
Ro ONGy eS bor
Hr Tbe
mw
eR we We = bo
bo
bo
14 21
TABLE 60.
Glasgow. Height and Weight of Girls
36 45 62 64 58 70 57 60 25 18 15 6
8
of Group C.
er bd Ww Ww
me OCR SE bo
|
|
=
Il
2
2
6
7
8
2
bo
=
WDOONWN WwW
| =|
Oe
KH bw Boe
ee
KH POOWr OBE
crane | ree
—
Ot en)
Ros Ee or OUD
ail era
Hep Roe
Totals
3.8
10
83 75 54 44
34 21 18 12
Biometrika x
334 Height and Weight of School Children in Glasgow
TABLE 61. Glasgow. Height and Weight of Girls of Group C.
Weight of Girls of 11°5—12°5 years
| Totals
|
iy SS SS ES 1
43 ” re a i mae ae a ae coe a ee Sa Tee
| 44 ” are 1 1 ae i 3. oh ee 2
[ie || pee Se ee ee Se 4
[i Skil ee oD 1 — — — — — 7
Pyare (eee ray 26 9a a = <=seas
| YSin 2 == (2° 4" 8B baal Ss SS SS SS 32
pee Pe eA TO i. Sh 7 Beg |
VrO es jhe al Lh TNS IPS BY) ee 71
51 sella =, S20) One dome seo wal = = 71
62.;. (=== = “197 21 14,00 165, Bes 2 96
5S, a = 07 15 610 10M On 2s) ie 79
bh. Na 92 5 10,11 1696 6 48
Do tlh ae a aoa 13 510198 6 eS 44
56 a 2 6 664.3 1079.9 222 2 See ee
Bi Ki eee Se ee ae 1 a3 12 (or ee ee ee eee 1 9
[aoe 3.21 3 ————— — 9
59 a a Se ee ee 3
60 ,, l
ol ,,
G2 ” 2
| 63 ” a OT
64 99. Ee ens a ee ae Sage ag A ne ral Y 1
Totals] 3 9 14 39 66 83 74 68 62 46 41 28 23 12 10 595
TABLE 62. Glasgow. Height and Weight of Girls of Group C.
Weight of Girls of 12°5—13°5 years
Heo g iat | ole heh Sel gel call alan ein ae a ea (sheet eal eiberae
2 PW WW 1 |} 8 |G WW 1 1 1H 1B 1G 1H 18" 6 1D 19
SSSBERESSSRESSSSSESSSSS
38 | —
3d),
40 ,,
| 46 ”
47 5
48 5, —
49 5, 2 2
205; 4 =e
AL ,, 14 1 =
52 5, 7 1 1
o3 ” 17 4 3 2
Sh, 24 18 13 a
55 4, 919 11 6h al
56 ,, 2 5 9 4
57 ,, 2 10 10 12 6
58 ,, 4 7 8
59 5, — 1 5 -1
CL; 2
| 62 ”
63 ”
| Totals 1 2 5 10 29 58 57 81 84 55 48 462515 9 7 4 5 2 1 544
EK. M. ELpERTON 335
TABLE 63. Glasgow. Height and Weight of Girls of Group C.
Weight of Girls of 13°5—14°5 years
Totals,
| |
LD wD
a 35
3 S
|
Th eee Sh AE yet ee Ee ee 1
ae = Nl
ee a nee 3
eee ohh OeeaeeCe ee 2
ike deo ee 2 oe 6
5 I Ee Dt eee le eR 1a Lt
le 934 I 1 8
ee GO oe pao eae” ee ea |
ee Oe eG Oe oy aS) et) og]
— = 1 I= 6 5 4.2 —.2 2 — — — | 93)
es ee ae a on ee ee ee
eo oy Gl 28 ees es 8
ee eae Ore et. De ede ee 17
ase yet 1 | Oca pay eae gn 7. |
me a 1 OepeeaiG dae a 5 |
eae: prs
TABLE 64. Glasgow Height and Weight of Girls of Group D.
Weight of Girls of 5°5—6°5 years
Height
u al d a al wD il il al n ! 1 | il | il i
ee oe eS ees
eS ai Ey ae Reet
35a as) ST ee ee ee ae
36, Soe I coe Be ee a a
37, Sole yal mee Ay eens Uh eet eee ee lee
88.5 Dee ee ihn see Sea Se Se a ee tes
89 4, iD ee Ae ee Big Oa aE idoh a ne Seca tas im [oe area gee Oe a
40 ,, Se ge! oe OY he E Mase ea elig) -2 2 Geen ren ee ef Soe
WL, - - el (ye ke XS 9 4 1 —
Dak ee ele i OOM Ome Geclibmee Os fe oe. Me eee
ee ae ee ea) oe Ge ee a
4h 45 ee l Aye Ae Oh eg A LL —-— — —
45 ,, STS eal a ae or ee rea Seo 3 4 6 3 ——= 2 — =
46 ,, = eS ee
Ll, es [oe MS VE
336
TABLE
Glasgow. Height
Height and Weight of School Children in Glasgow
and Weight of Girls of Group D.
Weight of Girls of 6°5—7°5 years
Height | | | | | | | | | | | | [eee"| Lotals |’
wD Uw, UW wD iy LD LD WD LD wD id iD wD uw WD
HR Sk SSS SESE EE EB
37 EF 1 I — — 1 SS SS SS 3
38 ,, 1 2 = SS = 3
oo), —- — 2 4 2 3 2 —- ~ ~ ~—~ ~ ~—~ — — 13
40 ,, — 4A A Gp ed — = = 16
ilies eH 5 US er Sh 8} i 42
vi. i Ope 4 209 20 ei a3 eas Wo 74
Ide Vo = Se Da 298 0, So 5 ee
44 ,, —- — — 1 She Mil ily y aifshs Igy (S) 1 Bo = = 76
ote, — Oe San l4 Seli9) tel By A ee 66
UB a. —- —- —- — 1 1 2 inl Oa tT Go 2B. Ae ll 1 40
Ts = a - 1 1 A Oe Ogee Ong 2 1 26
AS ,, -— 1 a I 6
Os - — — SSS =
50 ,, co a ee ee a :
TABLE
Glasgow.
Height
and Weight of Girls
Weight of Girls
of 7°5—8:5 years
WwW “WD i
9S OW
3 9 8
i ea
Wy) SD ) D
tS 39 9 ion D>
= => = > >
1 | ’ | al it uw? il al il | JL
Sie AS eh ial 9 ™
iD KD D 9 2D © S
1
le Oe 4
} E29 ee a 6
Bet ID 2a des Cia nO, tee ge en pee ee eee 22
ee ee ee ee ee ee eee i eee eee |
nae ee. “OO eS eae ee te ee = = 34
a SD 24 1G; 9) 0 5 eee
ae ee ne ime Oey Ime Gs Oe) 1 — | 8
ee ee Pe NN sy Oph pe go live
ae ee SD A 75 162 1G ey ee
= 1 =) 8 7 59°99 38 64° enn
ey Be eS ee |
a ee eee Oa eee Ss i
en ee 2
iss 2
E. M. ELpErtToN 337
TABLE 67. Glasgow. Height and Weight of Girls of Group D.
Weight of Girls of 8°5—9°5 years
Benet ed fh. all do periisetly salerale ols alae | | | | Totals
| lp 18) 1p 1S 1S th 1s 1s 1 As tp 1B 1s 16 1 tH 1H 1H 1G
Pap OS EO ESO eer eOp eT: Or NS) 2S) OX st SD, AQ eS SS) Sf So) La eS Gy
Se SON west est estes stg CY, UG Ry GS LG ECS) “RO GOs KO AO! t= RS RS GAS, BS CD
1 1
= 1
1 De ee
= 2165 :
a 56 4
ee hes 17) 19013) 8-8 ae ae = | 59
ope 3 13 1819 Gar gaat Ee ee 70
VN et 1 POG HIG WT 1Oje7 —— 2) VA a Ti
Vee eT 8 1813. 17. 84 3 5 8 Se 73
HO) | | gig ie an aera WaOb en dome, Ord At ee 69
| Sea oMlOL bo ya O64 a Te ed Ag
S| (ti em esl el ee Dron a: Ae el ee re 15
Ome ee eave alle ele wi af ee ME SD
5 — ee (OSes ae = Se ee ee 1 6
ee a ee De Aiba [yee ee ice eee 4
Oe |e ner ee ee ee ee Sk oe 1
TABLE 68. Glasgow. Height and Weight of Girls of Group D.
Weight of Girls of 9°5—10°5 years
Height eee yd potas
WW UD My AQ AQ WO ID ID WD ID ID W ID iD
iS > 1
e) a) a)
41§—| — 1 — — SS SS 1
42, | 2 — i Sse a _ 3
eal he = = ee ener ae
Abby a. lle at UGens Hote a Lees — — — 9
45 5, — 5 2213 1—-—— 2Z— — — — —- — — — 16
HS 5 aes UALS ey OSE, Ssh ots}e a} ee — — SS 44
Ula eA AG Ged no, ook 3 SS 68
48 ,, —— 4 4 612 919 15 4 3 2—— ~— — — — — — — — — — 78
Loe. — — —— 4 713 21131714 6 4 2 — 102
50.,, | — — — — — Gree As opal SA 7, — il — — 92
él, J|————— — = IG Gi 18 2 64
52 45 fF — — — — — — — — é 2 8 62 21—-—1%1—— 42
2.3 213 —-—— — — — 1 18
— 2 By i 6 k= 12
| — al — 2 — 6
eS ee ee eee ee ee See) Se 1
24 30 49 61 57 52 560
338
Height and Weight of School Children in Glasgow
TABLE 69. Glasgow. Height and Weight of Girls of Group D.
Weight of Girls of 10°5—11°5 years
wel Ca
yor lon = ame ee Ss ee ee eee
eg |. eae a AS ee ae
aay Wes, Nor Me Mites 5 2 Wa re 2
La) —— 1 2 Lm 4
Je. | 3.19 2 1 oe SS SS SS eee 11
Ue = 0 96. ae gel ee oe =o Sees
4g, [ee Ll -4 510 4827053 1 ie eee 39
Or ———— 1 9141614 5 5 38 21 1’ — — — — — — — 72
20. ;, —— 1 —— 4 7 8141015 7 1 131 1TH— 1 ~— — — — — — 74
41, J — — — — — 1 6 6 412141011 4 5— 2 2 — — — — — — — — 77
Gy ie Nee See 1.413 8 8 810 °3 2°20 = oo = ee
Bore oo a a ee ae 8 3.77 3059" oe Gael eee 55
TS Raien| Pee, fon, ey Le 391-347. 5 29'9 4 3 oe eee
Lem (enema ee a ee, ee at Sa 3156 1. 3.49) 2 1 18
Cauca (eee seer ee Say, hat Q: 4. = SE ae 8 ie a Ss ee eee
Td Ws eae es Se eS ee i 2
a ee ihe 4
Se eS a 1
Totals} 1 2 9 10 15 24 44 43 44 54 47 43 34 35 30 1418138 4 5 3 2 1 2 — 1 | 498
TABLE 70. Glasgow. Height and Weight of Girls of Group D.
Weight of Girls of 11°5—12°5 years
Height Totals
a a a a a
eh Ss fee Sa ee ce ee
6i— a 4
eee 2 i 15
48 ” 4 = = ll 18
on 7-5 11 el 36
50 ,, a 12 1 ied 9.02. 55
areas 8 514 Ose 78
52 ,. = 8 By a) 9S) 1 63 :
58 == 9 ee) 78
ob 5; — 5 21-17-15 4 95
55 —— 1 45/6; 72 5 10 53
56. eee IGS Be Sib} 47
evans eG 4 27
aoa ee 4 11
59 ,, ee al i
nial) 5
6. 1
62 ” The
68: ,; 1
O4 ” Ti
Paes zs
| Totals} 4 15 30 42 61 77 76 65 62 47 26 38 594
E. M. ELpERTON 339
TABLE 71. Glasgow. Height and Weight of Girls of Group D.
Weight of Girls of 12°5—13°5 years
i Vege ee ee ee ee ee 6
- By a hy hs Se a a 16
., 3) cit heey Weare = —— 9
.; 8 9 FT 4— 2 2—- —- — — — ~— — — 33
» Creo e Say 68a 4 |, 37
5) 2 10 14 13 8 2 Loy’ ~~ ~~ ~—~ ~— — 51
» — 6 4 19 1 13 6 1 3— ~ ~ ~— — — 67
3 1 1 38 11 22 9 10 4 4 ~—~ —~ —~ 1 ~— — 66
> —-— 1 6 18 11 14 9 8 1 1%1—-— JI 71
” — — — 38 383 12 20 12 1446 6 2 1—— I 76
” — — — — 4 7 4711 1 2 6 1m i — 46
” — — — 1 2 38 4 7 56 56 22 1 — — 32
5 Se ee ee a en ee Ee 14
Mt ss —_- — — — 3 2 38 1.1— — 10
52, fe he ee ae i = 2
63" ,, oS ee a — aH 1 1 —_ 2
— SS
Totals 538
TABLE 72. Glasgow. Height and Weight of Girls of Group D.
Weight of Girls of 13°5—14°5 years
TEESE PE alba rlPs P aeoca
ID ID WH 1D 1D 1D ID ID 1H ID 1S ID 1H 1H HH 1H 1H WH | 19
AMA HH BDH ~ HH DH AK HTH DDB HF >
Sig bb © S Bee a Cope CON OO COMES OG! A a Se SE 9
i a rr SS ar a SSS SSS SS eS 1
op een me See 3
Pag me pace SUR aA SMe ye ey yt Ne ere ae ey See 2
— — — — — J ~— ~~ — — — 1
ee ae eS 10
io Nee as 10
OMe ees ee ete 32
ag eee et 35
Gare) fee = =e a ee 45
ee 15 51
= eee ees 10 a bees eal Ar
= a 5 52
pe re eo See 3 39
ee 1 eee as ee ee | 0)
es 2S a a 1 in
Se a ee a 7
2 oe ee Se ee, eg Se ee 1
aa pe eee 2
Totals} 1 1 3 6 7 19 43 33 41 47 50 24 30 20111310 5 2 2 — 1 369
NUMERICAL ILLUSTRATIONS OF THE VARIATE
DIFFERENCE CORRELATION METHOD.
By BEATRICE M. CAVE anp KARL PEARSON, F.RBS.
In 1904 Miss F. E. Cave in a memoir on the correlation of barometric heights,
published in the R. S. Proc. Vol. Lxxiv. pp. 407 et seq., endeavoured to get rid of
seasonal change by correlating first differences of daily readings at two stations.
A similar method was used by Mr R. H. Hooker in a paper published some time
later in the Journal of the Royal Statistical Society, Vol. LXVII. pp. 396 et seq.,
1905. This method was generalised by “Student” in the last number of
Biometrika (Vol. X. pp. 179, 180). He showed that if there were two variates
x and y, such that
x= (t)t+ X,
y=fO)+ ¥,
where X and Y are the parts of # and y independent of the time ¢, then the
spurious correlation arising from a and y being both functions of the time could
be got rid of by correlating the ditferences of # and y, and that ultimately, when
m is sufficiently large :
Tamg Amy = VamHyamty = etc. = 1yy,
so that the correlation of 2 and y, free from the spurious time (or it might be
position) correlation, Le. ryy, could be found by correlating the successive differ-
ences of « and y. When the correlations of the differences remain steady for
several successive values, then we may reasonably suppose that we have reached
the correlation 7yy*.
This method is still further developed by Dr Anderson of Petrograd, who in
a valuable memoir published in this Jowrnal has provided the probable errors of
the successive ditference correlations of a system of variables :
xX; AGS O05) XG
Vis Vay OO) NG.
* Having been in communication with ‘‘ Student,” while he was writing his paper, I know that the
interpretation put by Dr Anderson (Biometrika, Vol. x. p. 279) on ‘‘Student’s” words (Ibid. p. 180) is
incorrect. ‘‘Student”? had in mind, if he did not clearly express it, the ultimate steadiness of
. . c 7
T amy Amy for a succession of values of m. K. P.
Bratrick M. Cave and Karu PEARSON 341
where the correlations of random pairs of values of the variates, or the product sums,
S{(X,— X)(X,—-X)},
SHCis= OrOe = 9a)e
are both zero.
Dr Anderson has further provided us with the values of the standard deviations
of the successive differences, Le.
aud ¢
Tamx Amy?
which represent the ultimate values of o,,,, and o Amy? when we have carried m so
far that the time effect has been eliminated.
The new method appears to be one of very great importance, and like many
new methods it has been developed in a co-operative manner, which is a good
reason for not entitling it by the name of any single contributor. We prefer to
term it the Variate Difference Correlation Method.
With the exception of a few illustrations given by “Student,” no numerical
work on the correlation of the higher differences has yet been attempted. It is
clear that much numerical work will have to be undertaken before we can feel
complete confidence in our knowledge of the range and of the limitations of the
new method. We have yet to ascertain how far in different types of material
a real stability of difference correlations is ultimately reached, and how far
various assumptions made in the course of the fundamental demonstration apply
in dealing practically with actual statistical data. One of the most important
assumptions made if there be n values of the variates is that arising from the
reduction in the number of values as we take the means which occur in successive
differences, and a like assumption is made in the case of standard deviations.
Thus for example:
Ue ede © ee
{8(X) =X,
n-1
but = S(X 5 — X54) = (X1— Xn) /(n— 1), and will not be sensibly zero, although
Te ke cal
it is assumed to be, unless n be very large. Similar remarks apply to the sums used
‘ erat 2 : ey ee 1 n a =
in the standard deviations, i.e. we assume in the proof S i) = oar js (Xe a)
eal coe float
Ultimately with the mth differences we come in the proof to relations of the
type
1
NES iit
3 my g (X 5)
1 Ny
and
nM = 1 id Van)
re - § (X,).
il
nm—-m 4 ?
Biometrika x 44
342 Illustrations of the Variate Difference Correlation Method
Now such relations will undoubtedly be very approximately true, if the X’s
are random variates uncorrelated to each other, and provided m is small compared
with x. These conditions seem amply satisfied when we proceed to fourth or sixth
differences in barometric pressures, taken, say, over ten or twelve years; the
addition of four or five daily pressures will hardly affect sensibly either the mean
or the standard deviation. But such extensive data, while not only involving a great
deal of labour in the difference work* are not those which, perhaps, most frequently
demand the attention of the statistician, whether he be economist, sociologist
or a student of scientific agriculture. In such cases it not infrequently happens
that the available data only provide a range of 20 to, perhaps, at most 50 years ;
and we need to discover whether there is a true relationship between our variates,
apart from a continuous change in both due to the time factor. At present
accurate statistics of annual trade or revenue, or satisfactory annual demographic
data hardly extend at most beyond a period of 50 years. Very often—under
even approximately like methods of record—we shall hardly have more than
twenty years’ trustworthy returns. Not only has the method of record been
changed, but the conditions of transit and trade may have been immensely
modified and in a manner which we could not suppose to be even approximately
represented by a continuous function of the time.
The object of the present paper is to illustrate the theory of the variate
difference correlation method in its present stage of development on a short series
of economic data, in order to test what approximation there is in such short series
to stability, and further how nearly Dr Anderson’s values for the successive
standard deviations apply to such cases. We have selected as our data ten
economic indices of Italian prosperity for the years 1885 to 1912, together with a
“Synthetic Index,” formed by taking the arithmetic mean of the ten economic
indices referred to. These eleven indices are given by Professor Georgio Mortara
in an interesting memoir: “Sintomi statistic1 delle condizione economiche
qd’ Italia” which was published in the Giornale degli Economisti e Rivista di
Statistica, for February, 1914, and form Tabella I, of that memoir, which we
here reproduce in part as Table I. The indices in each case are obtained by
dividing the returns for any year by the means of the returns for the years
1901—05, inclusive, and multiplying as usual by 100.
The indices are for returns of (1) Gross Receipts of Railways, (i1) Shipping,
loaded and unloaded at the ports, (i11) Effective Revenue of the State, (iv) Inter-
national Commerce, Value of Imports and Exports, (v) Number of postal
Letters and private Telegrams, (v1) Amount of Stamp Duties, (vii) Savings Banks’
Returns, (viii) Impo: tation of Coal, (ix) Gross returns of consumption of Tobacco,
(x) Returns of Coffee imported. Professor Mortara has drawn attention to the
very high correlations of these individual indices with each other, and of each of
them with the “Synthetic Index.” The latter correlation is, however, to a certain
* A discussion of the correlations of the higher differences in barometric pressures will we hope be
shortly issued.
Ce a
Beatrick M. Cave AND Karu PEARSON 343
extent spurious. For if J,, Z;,... J be the individual indices and J, the synthetic
index, then J,=75(44+ 4+ ... + Ji) and any individual index J,, if there be no
correlation between such individual indices, would give
1 a 7 T i! T \2
7b Us — Ls) La — La) = z9,, 8 Fa — 40)
I
9.
a
2 I Ti 1 2 2 2
Ca poss SU; —I; = 100 (Cie Ose eed)
and accordingly
1 72
my
; Wie 2 :
Cae toe UG ea aac
eee)
co
Tho
TABLE I.
Professor Mortara’s Table of Index Values of Italy.
Numerical Index (Mean 1901—1905) = 100.
|
A | “a » vw | | | S.5
Paani eee Soul ste Vee eta | ia | 8 | eg | 2 eae
Year z a ® 3 & oa} as ax Ss ois 2 Opes ae
eee eo acess! Sees e | O ) Ss |S, | ee ee
a B a 20 aS ES) | ess a BA 4g |
} |
1885 61 |-- 63 78 72 | 38 82 47 53 82 98 67'4
1886 62 63 |. 79 | 74 40 87 53 52 86 94 69-0
1887 68 73 | 82 78 42 94 BY 64 88 91 73°5
1888 Omg ples” 83 =)" 162 43 98 57 69 86 81 72:0
1889 71 Me | 285 70 45 99 58—| 71 86 78 74:0
1890 (2 78 | 86 66 46 97 59 | 78 87 81 75°0
feomaneer2 72 | 85 \| 60 |: 48 | 96 | 60 | vo | 88 | 80 73°1
1892 lye 7 86 | 64 bl 96 64 69 89 80 745
1893 "On 970 8 | 64 | 55 95 65 66 90,5], 73 Toes
1894 72 a 72, 86 | 63 57 93 65 84 89 71 75'2
1895 73 76 | 89 | 66 60 92 68 a 88 70 759 |
1896 75 ves 90 | 67 64 94 70 rp So ee Tec
1897 78 80 90 | 68 | 68 96 73e |e 76 Site (ro 79:1
1898 81 84 STE aan 183 96 75 | 79 | 89 78 Sp |
1899 86 88 | 93 | 88 75 96 70. iN 87 \ar91 82 | 86°5
1900 89 89 | 94 | 91 78 96 83 | 88 92 82 | 88:2
NOON |) 90" <I, Ol" “96. 1h 92. | 852 || (96."| 87.4 86 | 95 | 92 91-0
1902 967 |) 99" | 198 95 | 93° | 97 92 | 96 | 97 | 94 95°7
1903)" | 10 | 103 | 99° | 99 | 103 | 99 99 99 | 99 | 102 | 100°3
1904 | 105 |-102 | 101 | 103 | 111 | 101 | 107 | 105 | 102 | 103 104°0
WoOpsaiel0s | 105" 105 | 10k | 0S" i07 | tts | 114 | 106 | 109 108°8
1906 | 119 | 123 | 108 | 132 | 107 | 115 | 127 | 137 | 109 | 118 119°5 |
1907 | 126 | 125 | 108 | 144 | 114 | 120 | 144 | 148 |,115 | 125 126°9
1908 | 134 | 129 | 113. | 139 | 122 | 120 | 155° | 150. | 124 | 132 131°8
NOOR 139) 40 121 } 149° 132 193, 168 | 165° | 131 | 140 140°8
1910 | 146 | 146 | 129 | 159 | 140 | 130 | 180 | 166 | 138 | 147 148°1
HOU \al56 156: | 135 | 167" 1149 \136 [18% | W71 | 144. | 153 155°3
1912 | 165 | 169 | 188 | 179 | 158 | 141 | 192 | 179 | 151 | 160 163-2
{ | | | |
Mean | 94°8 | 116°3| 97°6 | 96°4 | 82°3 | 103°3 | 95:9 | 99-0 /100°6 98°6 96°5
344 /llustrations of the Variate Difference Correlation Method
which would have a substantial, but spurious value amounting to ‘316 if
Cp Op Gy If there be high correlation between the individual indices, of
course, the correlation of each individual index with the arithmetic mean or
synthetic index will also be high. Thus our Table IV shows it to range from ‘952
in the case of Coffee to ‘998 in the case of Railways. Possibly a third of this
correlation may in some cases have a spurious origin. But the individual indices
are very highly correlated together; only two such correlations are below ‘9, and
the lower of these two is as high as ‘885. We are accordingly left with fifty
correlations ranging from ‘885 to ‘997 between the individual indices, and if we
accept these as true measures, then it is clear that any one of these ten indices
might be used as a reasonable index of Italian prosperity; it would for practical
purposes be idle to calculate them all or to table their arithmetic mean.
But the high correlations found lay themselves open from our present stand-
point to some suspicion of being solely due to the fact that during the 28 years
under consideration Italy has progressively increased in population and_ac-
cordingly the consumption of innumerable goods and the means of interchange
have all grown together with the time. In other words the correlations we give
under the heading of “quantities” in each separate section of Table IV are very
high solely because the individual indices are variates increasing one and all as
continuous functions of the time.
The material therefore seems especially suited to the application of the
variate difference correlation method. For example, the correlation between the
indices for tobacco and savings is ‘984; are we to interpret this to signify, that,
if there are large savings this means that much will be spent on tobacco? Or,
is this high correlation simply in whole or part spurious, merely indicating that
both savings and consumption of tobacco increased markedly with the time ?
Actually the correlation of first differences drops from ‘984 to ‘766, that of
second differences is negative if insensible, while from there onwards it steadily
increases negatively, till with the sixth differences we reach —‘431, which seems
to indicate that, when time has been eliminated, expenditure on tobacco in any
year means less money saved. Again the coffee and tobacco indices appear very
highly correlated, ‘955, but by the third ditference correlation we have reached
about a third of the relationship, ‘319, which is scarcely altered in the sixth
difference correlation, 326; we may assume therefore that there is probably a
moderate “organic” relationship between the expenditures on coffee and tobacco,
but the association is nothing like as close as would be suggested by the corre-
lation of the raw indices.
The work has been done in the following manner: The successive differences
of the indices up to the sixth were found. The means and standard deviations
of these differences were calculated, and the correlations were then worked out
in the product-moment manner. This involved the very laborious work of
determining 385 coefticients, and then to these coefficients were added the
Beatrice M. Cave AND Karu PEARSON 345
probable errors as found by Dr Anderson’s formulae. These probable errors are
of course those of the correlations of A™X and A”Y, and will not be the correct
values for the probable errors of the correlations of A™# and A™y, until A" = A”"X
and Amy =A”Y, ie. until m is sufficiently great for ¢ to have been eliminated.
’
Further their accuracy depends on the vanishing of the means of the differences
or on the equalities of the sums like
1
w= n
S Xe) ah | . (XG), ete.
1
n—1l
which, while true on the average, will only be approximately true in the actual
instance if n be large. We give in Table II, the Mean Values of Index
\
Differences.
TABLE II.
Mean Values of Indices and their Differences.
S| = oF ine val, ‘1 7
| 2 bo ES | i hes (3s $8
ees | 8 ok Ss 2. “ip 2 P ey 3p fet
ee ee eee ase 2 | 3s i.e |eee8
seles les Gales |ae | sa) °.| Ss | 8 |e 88
ee fs Et et | | @ <5
us zal : :
Quantity ... | 94°8 | 116°3| 97°6 | 96:4 | 82°3 | 103-3 | 95-9 | 99-0 100°6 | 98°6 96°5
Ist difference |— 3-85 |— 3-93 | — 2:22 | 3-96 |— 4-44 | — 2°19 | —5:37 | — 4°67 | — 2°56 | — 2°30 — 3°55
prides 5, — 35/— 50|— 08|- -38/— 27) -00/+ 04/— -35|- ‘12/- -42) - -24
3rd, + 16/+ -28)4+ -20}— 08] -00/+ 12/-— -08|+ -40)/— 12) ‘00; + ‘09
4th 3 — 33/-— 88/-— ‘13}-1:17 00)/— -28)/+ -04]/-— ‘79|- ‘16;-— 42} — ‘40
5th ” + 67/4226 + -26/+2°52/4 30 00 |— °52)/41°87)+ 22/4 87) + °83
6th 5 SCD Ne 2L00 F008) 735) — 05 |4+2°00;-— °95)-168) -—1°10
It will be seen from this table that the means of the differences are far from
zero even when we have reached a difference for which we may suppose the time
to have been eliminated. This arises from the smallness of the series dealt with
and shows us that we ought not to anticipate more than a rough accordance with
theory, or only an approximate steadiness, for sums like
may grow less and less steady as m increases,
Similar considerations apply to the standard deviations of the differences.
These will not at first obey Dr Anderson’s formulae because they are the values
for om, and T amy? and when we have taken m sufficiently high for
and o i to be theoretically equal to c,,,, and o,n5,
Tamy Am
346 Illustrations of the Variate Difference Correlation Method
and accordingly the correlation should have begun to be steady, then some failure
to obey Dr Anderson’s formulae will arise, because the means of the differences
are not truly zero and equalities of the type
el n
n—-1
Use
=i 1 1
8 Ce) — ar s (X2;.5)) ete
will not be satisfied when n is relatively small.
Dr Anderson gives the value of omy I terms of o°,, but we do not of course
know o?,, which will be very different from o?,, and can only practically be found
from the value of o? itself, after that value has become equal to Oo my» Le. after
Ar
steadiness has set in. In order therefore to test the formulae we have formed the
ratio of
O AmylF jm, from m=1 to 6.
: Dies : ; -
This equals seater in Dr Anderson’s formulae for Can Cention and therefore
we have a good measure of the approach of Aw to AX, or of the growth of
steadiness as apart from the correlations. The following Table III gives the
values of the ratios of the squares of the standard deviations, theoretical values,
actual values and the mean value for each of the differences of the ten
individual indices.
TABLE III.
; 2
Values of o?4my/O%4m-1y and their approach to mer ;
S234 | fa Mean of 10
§ se 3 , | oS! ¢ 8 2 ewe a 8 & Index
lm | 2-2 a8 e a S gq a £2 g 3 2 BSI Difference
3 8 aa on] ate | ag S = s Pe S 2 = Standard
rch ean a en) 25 a n a Deviation
= & Ratios
1 012) °012 031] -019) °038} -009]) *040) -O10} °035 | °022) -036 | 025
2 | 3 705 | °708 | 1°834| °763|1°720| -799 | °585 350 | 2°074| °352| °843 1:003
3 | 3°333 | 37107 | 2°816 | 3093 | 2°124 | 3°032 | 1°959 | 1°660 | 2-214 | 3°075 | 2-213 | 3°307 2°549
4 |3°5 3°167 | 3°128 | 3°174 | 2°747 | 3°213 | 2°597 | 2:008 | 3°106 | 3°379 | 3:025 | 3-619 3°000
5 | 3°6 3143 | 3°449 | 3°189 | 3:020 | 3°104 | 3°010 | 2°328 | 3°275 | 3°580 | 3-117 | 3°701 3°177
6 | 3°667 | 3°149 | 3°711 | 3°195 | 3:164 | 2°881 | 3:208 | 2°499 | 3°455 | 3°682 | 3°101 | 3°791 3°269
It will be clear that until we reach the ratio of the square standard deviations
of the third and second differences, there is no general approach to steadiness.
After m=8, however, for m= 4, 5 and 6, the ratio of the values for the mean of the
series of individual indices to the theoretical value is ‘86, ‘88 and ‘89, respectively.
Thus, there is increasing approach to agreement in the observed and theoretical
values, but the approach is slow, and we believe that there is greater steadiness than
is really indicated by this test. The source of this apparent unsteadiness lies we
BrEATRICE M. Cave AND Kart PRARSON 347
think in the relative largeness of m compared with n (Le. at a maximum 6 as
compared with 28), rather than in our not having taken sufficiently high
differences.
We now turn to the correlations. These are given in Table IV, the actual
values of the standard deviations of the quantities and their differences being
recorded along the diagonal cells, while the other cells contain the correlations of
each pair of variates and of their successive differences.
We will now consider these correlations in detail.
(a) Synthetic Index (Arithmetic Mean) with Individual Indices.
We see at once that the Synthetic Index is highly associated with Shipping
(>°85), with Importation of Coal (? >°75), and International Commerce (? < ‘68),
and fairly highly with Revenue (c. ‘55). On the other hand the sixth difference
correlations with Post (c. 15), with Stamp Duties (c. ‘24) and with Savings (> °23)
are all such (i) that they might well have arisen from the spurious element in the
Synthetic Index correlations, and are all less than their Andersonian (steady
value) probable errors. Almost the same may be said of the Railway Index ; it
is not beyond suspicion of being spurious, and is scarcely significant having regard
to its probable error. The Consumption of Coffee is also not very closely associated
with the Synthetic Index ; it is only about twice its probable error (427 +°205),
and a good deal of its value may be spurious. Further in the case of both
Coffee and Railways, the correlations are still falling between ‘04 and ‘05 for each
difference. The last individual index remaining is that for Consumption of
Tobacco and although the correlation of sixth differences is not really significant
it is negative (— ‘247 + 235), and is exhibiting a steady negative rise.
Stripped therefore of the common time factor the Synthetic Index will be seen
to be no very appropriate measure of trade, business activity, and spare money for
savings and luxuries. With Post, Stamp Duties and Savings, it has probably only
a spurious relationship, expenditure on railways has little influence, that on
luxuries is very slightly significant, or indeed in the case of tobacco negative. It
is, however, closely related to variations in external trade, i.e. imports (including
coal), to exports and shipping and to effective revenue. It appears to us that a
suitable general index of prosperity, which will distinguish between a continuous
growth in all factors with the time, and favourable and unfavourable fluctuations
from this growth, can only be obtained, when there has been far more ample
study of the associations of individual indices among themselves, and of these
indices after they have been freed from the time factor, i.e. of associations
between high difference correlations. From this standpoint the study of General
Index theories is at present in its infancy.
(6) Railway Index, This index is very noteworthy in the nature of its
associations after removal of the time factor. We have reached a steady
correlation (c. 62) with Shipping, but beyond this no values of first class im-
portance appear. The relation of Railways and Revenue after falling practically
ce Correlation Method
ereé)
te Diff
1a
ions of the Vari
Tllustrat
348
Weg toe
OO PS Ao eS
GVB-+ZPl- | OGG. FIO. | EPS. +TLT- | 16.49L3- | 961-FC9F. | FOL. FoLr. | 68.68 | 6ez.FPIZ. | LL1.FUrG- | 0¢z. ¥L00.—| cet. FOg9. . ¢
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349
BEATRICE M. CAveE AND Karu PEARSON
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45
Biometrika x
350 Illustrations of the Variate Difference Correlation Method
to zero, now stands at something greater than -42 and might rise higher, but
the relation to International Commerce as a whole is zero, which suggests that
the goods imported and exported are not in the bulk carried by rail. Further
althoagh the final value of the Railway and Post correlation is scarcely sensible
(— 214 + :259), it has been continuously negative from the second difference, and
thus suggests that increased expenditure on the post means lessened profit for
the railways. This might be interpreted in two ways: (i) that business con-
ducted by post or telegram lessens rail intereommunication by person, or (il) that
in the case of state-railways, there is not an increased profit to the railways from
carrying larger mails. But still more remarkable are the negative correlations of
Stamp Duties, Savings, Tobacco and Coffee with Railways; none of them are very
large, and all but savings, perhaps, of the order of their probable errors. But
taken as a whole they suggest that when the Italian spends little money in going
about, then he saves more, or spends more on such luxuries as tobacco and coffee.
Lastly we have the Coal Index. It might be supposed that a year with great
coal importation would signify great railway activity, and this-is the judgment
which would be made from the raw correlations of these variates. But the actual
facts are exhibited in a correlation still falling at the sixth difference and hardly
significant having regard to its probable error. The inferences formed must be:
(i) that imported coal is used largely at the ports of disembarcation or travels
inland by other than railway transit, (11) that the imported coal is largely used on
the railways themselves and that its cost is a heavy tax on their resources.
(c) Shipping Index. As we might anticipate this is highly correlated with
(i) Railways (c. 62), (ii) Revenue (c. °75) and less highly but very significantly with
(111) International Commerce (c. 54) and (iv) Coal (c. 58), but it appears to have
no relation whatever with Post, Stamp-Duties and Savings, and when we come to
luxuries, their importation is clearly not a factor of shipping prosperity. Neither
in the case of J’obacco nor of Coffee are the correlations really significant ; with
the former we have an increasing negative correlation and with the latter a
decreasing positive one already below its probable error. Thus we see that
neither directly by bulk of importation nor indirectly by immediate increase of
consumption, does a rise of shipping mark significant rises in the use of luxuries
such as Tobacco and Coffee. It would be of interest to ascertain whether in-
creased consumption of luxuries does not rather follow than accompany favourable
trade fluctuations.
(d) Revenue Index. This index as we might expect is fairly highly cor-
related with Shipping (c. °75). It has relatively small relation to Railways
(422 +206) at least at the sixth difference and a somewhat similar value
(c, 42 4°20) for Coffee. Thus the suggestions arise that revenue is but little
produced by the railways and that coffee is not a very large factor of the custom
dues. It is astonishing to find, however, that Post, Stamps and Savings have
negative correlations with Revenue of —°888 +°213, — 255 + 234 and —:154 4+ :244
respectively, which, if scarcely significant, have been in each case for several
Bearrick M. Cave and Karu PEARSON Bay
differences persistent in sign. Even the correlation with Tobacco is small, falling
and insignificant (< 115 +247), and that with Coal which might be supposed to
be high as marking good trade times is hardly significant although apparently
rising (? > -270 + 232). Lastly the correlation of Revenue with International
Commerce is again small, falling, and insignificant (< ‘214 +°259). Thus Revenue
or the “entrato effectivo dello stato” seems to provide an index which has little
valuable relation to any other characteristic of prosperity beyond shipping.
(e) International Commerce. Here we find no single final individual index
correlation greater than °54, which is that for Shipping. The next most important
correlations are with Post (>°47) and Stamp Duty (c. 46). With Railways the
correlation is zero, and with Revenue also falling and insignificant. With Savings,
Coal, Tobacco and Coffee the correlations are all insignificant; in fact in the last
three cases not only are the values less than their probable errors, but they are
still falling. It is thus clear that in Italy the total of Exports and Imports is
no measure of all-round prosperity, they do not immediately increase either
savings or the consumption of luxuries.
(f) Post and Telegrams. Here we have the lowest series of correlations we
have yet reached, Post values have no significant relation to fluctuations in
Railway (c. —:20 + °24), to Shipping (— 059 + 249), Stamp Duties (—027 + °250),
Coal (— 050 + 250), Tobacco (+ 108 + 247) or Coffee (— 183 + :246) Indices. It
is significantly correlated only with International Commerce (> + ‘47 +'19) and,
perhaps, significantly with Savings (+ 336 +°222) but negatively with Revenue
(c. —'38 +:21). In short the number of letters and telegrams in Italy is hardly a
mark of any other favourable fluctuation in prosperity, beyond International
Commerce.
(g) Stamp Duties. This Index is correlated positively and significantly with
International Commerce (c. +'46 +°20) and positively, and doubtfully with Savings
(c. + °85 +°22). It is correlated insignificantly and negatively with Raalways
(—'261 +233), Shepping (— 009 + -250), Revenue (—'255 +°234), Post (—'027 +250),
and Tobacco (—'129 + 246); it is correlated positively and insignificantly with
Coal (+ (052 + ‘250) and Coffee (+ °222 +238). Thus again freed from continuous
time changes, fluctuations in the Stamp Duty Indew are of small value as a
measure of contemporaneous general prosperity.
(h) Savings Bank Index, There are practically only two correlations of any
importance with Savings and these are both negative, namely those with Railways
(— "431 + 204) and with Yobacco (—°431+4°204). Hence it would appear that
when the Italian people is in a saving mood, it spares on transit by rail and on
the consumption of tobacco, and when it expends on these luxuries, then it does
not save. Savings have small and possibly not significant correlations with Post
(> +°33 4°22) and Stamp Duties (< +'353 + °219), and insignificant and positive
correlations with International Commerce (>+°27 + °23), Coal (> +19 + °24)
45—2
352 Illustrations of the Variate Difference Correlation Method
and Coffee (c.+°05 +°25); they have insignificant negative correlations with
Shipping (< — 03 4°25) and Revenue (< —‘15 + °24).
Savings are thus—apart from continual time change—no very satisfactory
measure of general prosperity, and a fluctuating increase is usually accompanied
by a reduction of luxuries.
(t) Coal Index. The importation of coal has little relation to any factor of
prosperity besides Shipping (c.+°58 4°17). With Railways the correlation is
not quite double the probable error and the value, even at the sixth difference,
appears still falling. The correlation with Revenue only just exceeds the probable
error (+ '270 +232). With International Commerce (+171 + °243), Stamp Duties
(+ 052 +°250), Savings (+ 196 +241) and Coffee (+ 152 + °245) the correlations
are less than their probable errors, small and in some cases still falling. With
the Postal Index, the correlation is negative, insignificant and falling. Alone in
the case of the Tobacco Index does the correlation appear to be nearly as
significant as in that of Shipping, but it is negative and increasing* (—°514 + 184),
while in the case of Shipping it was steady. It is singular to find that Coal, the
increased import of which should mark increased industrial activity, is, beyond
the naturally influenced Shipping, alone effectively associated with the con-
sumption of Tobacco.
(j) Tobacco Index. This is of considerable interest as marking the association
of indices of trade prosperity with the consumption of a luxury. With four
exceptions J'obacco is negatively correlated, although often insignificantly, with
the other indices. Revenue (+°115 + °247), International Commerce (+ °015 + ‘250),
and Post (+ °108 +:247) are all positive, insignificant, and in the first two cases
still falling. The correlation with Coffee is positive and might, perhaps, be
significant (+°326 +°224), but it appears to be still falling. With Coal and
Savings there are probably significant negative correlations (—°514+°184, and
— 431 +°204 respectively); with Railways (— ‘243 + ‘236), Shipping (— ‘271 + °229)
and Stamp Duties (— ‘129 + ‘246) there are insignificant negative correlations, but
they tend to confirm each other in sign. Thus we see that the consumption of
tobacco can hardly be considered as a measure of general prosperity; it appears
to be greatest when trade conditions are unfavourable, and in particular when
savings are least and manufacturing conditions as measured by the importation of
coal are slack. The result suggests the pipe of the unemployed at the street
corner, rather than the increased expenditure of the fully occupied artisan.
(k) Coffee Index. 'This is another luxury and the results are very similar.
There appears a significant correlation with Revenue (+ °400 + 210), which might
easily be explained, and there is a falling but possibly significant correlation
with Tobacco (+326 +°'224), With all other indices the relationships are
* It is, perhaps, hard to believe that so much smuggling could be carried on in colliers, that it would
seriously affect the profits of the tobacco monopoly !
—_—_ —
BeaTRicE M. Cavk anp Kart PEARSON 353
insignificant. Railways (—‘204 +240), Shipping (+ °2382 +°237), International
Commerce (+142 +245), Post (—'133 4246), Stamp Duties (+°222 + 238),
Savings (+°046 +250) and Coal (+°152+°245). Apart therefore from the
general increase of consumption with the time, during which time the general
prosperity of the nation has increased, it would not appear that the consumption
of a luxury has any organic relationship to prosperity. We do not find that a
favourable trade fluctuation is associated with increased consumption of luxuries.
In fact the suggestion arises that in the case of tobacco the consumption may be
greater in a period of depression.
Conclusions. While we lay no special stress on any of the results suggested
by the difference correlations above studied—far more intimate economic know-
ledge of Italian affairs and methods of measurement would be requisite—we yet
venture to insist on one or two general considerations.
The very superficial statements, so frequently met with, that such and such
variates, both changing rapidly with the time, are essentially causative will
doubtless cease to have any scientific currency, directly the method of variate
differences is fully appreciated. We shall no longer assert that the fall of
the phthisis death-rate can be off-hand causatively associated with the con-
temporaneous rise in the number of persons dying in institutions, or that the
increased expenditure on luxuries is necessarily a measure of increased national
prosperity.
If we turn as in the present paper to the actual correlations of the indices
themselves, we find in every case an arid and scarcely undulating waste of high
correlation. No one can obtain any nourishment whatever from the statement
that the Tobacco Indez is correlated with the Revenue Index to the amount of :983
and with the Suwings Bank Indeaw to the extent of ‘984! The organic relationship
between these variates is wholly obscured by the continuous increase of all three of
them with the time. But when we proceed to sixth differences and see that the
consumption of tobacco has little, if any, relation to revenue, and is associated
substantially but negatively with savings, we seem to touch realities, and realities
of some worth. Again what can we learn, if we are told that the Shipping Indes is
correlated to the extent of ‘99 with both the Revenue and the Savings Bank Indices?
We might imagine, that increase of shipping was not only the primary cause of
increase in Italian revenue, but also the essential origin of any increase in the Italian
peasant’s and artisan’s savings! An appeal to the variate ditference method shows
how fallacious such imaginings would be! An examination of the sixth difference
correlations shows that while prosperity of the revenue is closely associated with
trade as measured by shipping (77), the correlation is not nearly perfect; on
the other hand there appears to be no significant organic correlation at all
(— 154 + ‘244) between the prosperity of the revenue and the savings of the
Italian populace. As we have noted a knowledge of local conditions and methods
354 Illustrations of the Variate Difference Correlation Method
of reckoning quantities might enable us to put other and, perhaps, more luminous
interpretations on our results. But there can be small doubt that to proceed from
the actual correlation of such indices to the correlations of their higher differences
gives the feeling of clearing away the sand of the desert, and reaching all the
ordered arrangements of an excavated town below; the slight undulations of the
waste above are really fallacious, and enable us to appreciate nothing of the
actual topography of the city.
The method is at present in its infancy, but it gives hope of greater results
than almost any recent development of statistics, for there has been no source
more fruitful of fallacious statistical argument than the common influence of the
time factor. One sees at once how the method may be applied to growth problems
in man and in lower forms of life with a view to measuring common extraneous
influences, to a whole variety of economic and medical problems obscured by the
influences of the national growth factor, and to a great range of questions in
social affairs where contemporaneous change of the community in innumerable
factors has been interpreted as a causative nexus, or society assumed to be at
least an organic whole; the flowers in a meadow would undoubtedly exhibit
highly correlated development, but it is not a measure of mutual causation, and
the development of various social factors has to be freed from the time effect,
before we can really appreciate their organic relationships.
In the present paper we have dealt only with very sparse “ populations” (only
28 values of the variates), but this has enabled us to consider not only a very
large number of correlations, but to see the practical influence of terminal con-
ditions on our theory. This may we think be summed up in the statement that
the Andersonian formulae for the standard deviations will hardly in many practical
cases be more than very roughly approximated before the size of the population
becomes too small to make the deductions reliable. Further in most cases our
difference correlations have hardly even with the sixth differences reached a steady
state. Possibly they have done so in the cases of Rail and Shipping, Shipping
and Post, Shipping and Coal, Revenue and Post, International Commerce and
Stamp Duties, International Commerce and Savings, Savings and Coffee, and in
one or other additional pair. But in the great bulk of instances there is still a
more or less steady rising or falling appreciable in the difference correlations, and
all we can really say is that the final value, the true ryy, will be somewhat
greater or less than a given number. From an examination of the actual
numerical working of the correlations, it appears to us that the terminal values
are in the case of these short series of very great importance. It is further clear
that the theory as given by “Student” depends upon certain equalities which
are not fulfilled in practice in short series. We await with much interest the
complete publication of Dr Anderson’s work, and hope to find a fuller discussion of
the allowance to be made in short series for the influence of the terminal state of
Beatrick M. Cave anp Karu PEARSON 355
affairs* on the steadiness of the series and on the approach to the standard-
deviation formulae. But apart from these lesser points, our present numerical
investigation has convinced us of the very great value of the new method of
Variate Difference Correlations.
* For example if we measure X from its mean,
J] “1 = afi n n m
oa pang 8 (Xe Kou)? (AR P= 5 (8 (9) - Ky 8 (A) MP) - AN
1 1
~n—-1
n-1 : r, 2 , V"\2
since S (X,X,,;) is by hypothesis zero, Sg ee Ng teary, 1 {o2x,— 4 (X?+X,?)} —(4X)?. The first term
1 2 or :
: aoe Hi n—1 = = 1 r r
207y. gives Dr Anderson’s value of o?,y. Now AX equals nisi 8 (X, — X544) = (X,-X,). Thus the
v—1
ee. 2 ail
remainder is ——> [ ox,-3 oe? a ala
i (X,- x, | . Now the average value on many trials of
4 (X72+X,2) will be oy, and of (X;-X,)*, 20, so that the full value may be 2 77x and small
2
(nv —1)
for n large; but for n small as above such a relation as o®, y=20%, and the similar but more complex
relations of the standard deviation formulae for the higher differences need not hold for any individual
case, and thus the steadiness of the difference correlation series, and the approach to the Andersonian
formulae are very far from attained.
AN EXAMINATION OF SOME RECENT STUDIES OF THE
INHERITANCE FACTOR IN INSANITY
By DAVID HERON, D.Sc.*
In the last few years a number of studies of the inheritance factor in insanity
have been published in America, Germany and England. The value of investigation
of such a topic cannot be overestimated. We are quite certain that the prevalence
of insanity is not falling; many of us indeed believe that the statistics suffice to
demonstrate that it is substantially increasing, and that we can attribute this
increase not in the first place to the intenser strain of modern life, but to the
greater power of modern treatment to check or temporarily cure attack, and thus
allow wider possibility of reproduction to members of affected stocks. Indeed the
problem seems closely associated with an essential difficulty of modern civilisation,
the greater protection of physically and mentally degenerate stocks unaccompanied
by any adequate limitation of their thereby increased power of procreation ; the
inheritance factor thus tends to aid the relatively greater survival of the socially
unfit. The studies we have referred to would be of great importance from this
aspect of eugenics if (i) the data were collected without conscious or unconscious
bias, and (ii) the inferences drawn from them followed logically from the data thus
collected.
Unfortunately it is not only in the interpretation of statistics that adequate
training is required. It is equally important that in the actual collection of them
we should proceed, not only free from the bias which arises from the hurried
acceptance of dogmatic theories of heredity, but what is often still more needful,
free from the bias which is almost certain to waylay our progress, if we have not
initially considered with trained insight the fallacies which may result from our
method of recording or even tabulating our material. The day of the amateur in
svience is gone; no one now pays any attention to men who propound elaborate
atomic theories or stellar hypotheses, without having had preliminary training in
physical or astronomical science. There are still, however, some who appear
willing to accept the statement of statistical data or the inferences drawn from those
* This paper formed the second portion of a lecture given at the Galton Laboratory on March 3, 1914.
D. HERon 357
data by men who have clearly had no adequate training in statistical science. The
craniologist, the anthropologist, even the biological student of heredity and evolu-
tion are recognising that a statistical training is needful for the true interpretation
of many of the facts in their special fields of research. The physiologist still
appears to believe that he can deal with the average effects of diverse dietaries or
the pathologist with the “ mass-phenomena” of the hereditary factor in insanity
without any training in statistical method. A physicist might just as logically
assume that without mathematical training he could give an adequate mathe-
matical account of a physical phenomenon, or a cosmic theorist suppose that he
was effectively furnished for astronomical research by the perusal of a popular
primer on the stars! The statistical calculus cannot be mastered by any easier
road than the differential calculus, or, to put a more apt illustration, statistical
training is as needful a preliminary to the handling of statistics, as time spent
in a physiological laboratory to the effective handling of tissues. In twenty years
it will be unnecessary to insist on these points, they will be universally recognised
in the courts of science; but at present it is not only necessary to reiterate
unpleasant truths, but to emphasise their validity by illustrations which bring
home forcibly to scientist and layman alike the danger of amateur statistical
handling. To state that a man is in error is not sufficient, if he continues time
after time to repeat his assertions, apparently under the belief that incessant
repetition will convince the world of the value of his theories.
In the case of the inheritance factor in insanity we are not dealing with any
purely academic question of science. We are up against one of the most difficult
problems of modern life, where true advice is of urgent importance to the nation
as well as to the individual. It is not only the medical man but the layman who
seeks guidance in the question of the marriage of members of insane stocks, and a
laboratory like the Galton Laboratory knows how often advice on such points is
sought. It is disheartening when help is rendered to the seeker to be faced with
the criticism: “ But Professor says I may marry if I take a wife of sound
stock,” or “Dr recommends marriage, although my father was insane, because
I am over twenty-five and still sane myself.” When teaching of this kind, arising
solely from false interpretation of defective data, is spread widecast in a dozen
different papers or journals, it is not sufficient to issue a brief statement of its
futility. It is needful to give it the coup de grdce by a more lengthy criticism of
its fallacies and their illustration in a form more likely to impress the imagination.
The attempt is made in this paper to deal with only one of the authors, who have
contributed fallacious eugenic rules to those seeking knowledge on the influence of
the hereditary factor in insanity.
In a long series of papers Dr F. W. Mott, Pathologist to the London County
Asylums, has stated that when the children of insane parents become insane, they
do so at a much earlier age than did their parents, and on the basis of this assertion
he has drawn some very sweeping conclusions for practical conduct. Thus in the
Biometrika x 46
358 Recent Studies of the Inheritance Factor in Insanity
British Medical Journal of May 11, 1912 (p. 1060), he states that “this signal
tendency of insane offspring to suffer with a more intense form of the disease and
at an early age, as shown in the above figures and tables, is of great importance
for the following reasons: first, it is one of Nature’s methods of ending or mending
a degenerate stock; secondly, it is of importance to the physician, for he can say
that there is a diminishing risk of the child of an insane parent becoming insane
after he has passed 25, a matter of great importance in the question of marriage ;
thirdly, it is of importance in connection with the subject of social surgery of the
insane, for when the first attack of insanity occurs in the parent the children for
the most part have all been born....Sterilization would therefore be applicable to
relatively few parents admitted to asylums.”
Put briefly, Dr Mott’s views are that in “Antedating” or “ Anticipation,” in
this alleged tendency of the offspring to become insane at any earlier age than
their parents, we have Nature’s method of purifying degenerate stocks, that the
children of insane parents who are still normal at the age of 25 may safely marry*,
and that it is useless to take any special measures to limit the reproduction of the
insane since nearly all their children are born before the onset of insanity.
These conclusions, if proved to be correct, would be of the utmost importance
to the Eugenist. If the Law of Antedating or Anticipation really acts in the
way Dr Mott has suggested, then it would seem to be unnecessary to take any
special Kugenic action in the case of the insane and indeed the “ Law” has already
been used in support of this view. Thus in a leading article in the British Medical
Journal+, which deals with Dr Mott’s work, it is stated that “This intensification
of mental disease in the young—this ‘ anticipation’ as it is called, which is one of
Nature’s methods of ending or mending a degenerate stock, is specially important
in connection with sterilization, as the figures given by Dr Mott show that when
the first attack of insanity occurs in the parent the children have for the most part
all been born. Sterilization, therefore, would be applicable in relatively few
cases.
It is at least obvious that when views such as these are taken of the “Law of
Anticipation,” it merits the most careful examination. Let us consider, then, first
of all, Dr Mott’s presentation of the case for anticipation. For some years past
Dr Mott has been engaged in the collection of cases in which two or more members
of a family are or have been resident in London County Asylums, and has noted
wherever possible the age of onset of the insanity. Information was thus obtained
regarding 217 pairs of father and offspring, and 291 pairs of mother and offspring
and the results are summed up in the following table.
Thus in comparing the age at onset of insanity in father and offspring, we find
that among the fathers only 1:4°/, became insane before the age of 20, while among
the offspring the percentage was 26°2. These figures are also shown graphically in
* See for instance Problems in Eugenics, p. 426.
+ May 11, 1912, p. 1089.
D. Herron 359
Figs. 1 and 2*. Here the horizontal scale represents the age of onset in 5-year
groups—the vertical scale the percentages of cases occurring in each age group.
TABLE I.
Percentages of Cases whose First Attack of Insanity occurred
within Various Age-periods.
Age-periods Father | Offspring | Mother Offspring -
Percent.|} Per cent. | Per cent. Per cent,
Under 20 years ig 26°2 0°6 27°8
20—24 years... 0-4 18-0 374 15°7
pting 1-4 | 18-0 44 18-2 \ Adolescence
30—84 ,, 9°6 13:0 4s) 13°4
35—39 ,, 11°5 73 9-2 10°0
40O—-AL,, 9-2 6-4 10°3 58
4I—49,, 14:3 6:0 12°0 3'7 | Involutional
50—54,, 17°5 0-9 12°3 2°4 t period
55—59 13°8 3°7 14:0 Py)
60—64 ,, 10°1 — 11°6 1°3
65-—69 _,, 5:0 _- 8°8 —
70—74 ,, 46 0-4 31 —
75—79 ,, 0-4 _- 1:3 =
80 - 0:4 — 0°6 —
I have been obliged to follow Dr Mott in treating the “under 20” group as a
5-year group as otherwise my diagrams would bear no resemblance to his, but this
procedure is far from satisfactory when such a large proportion of the cases in this
group are congenital cases in which the age of onset should be taken at 0 years.
The tables and diagrams show that among the parents more than half the cases
occur after the age of 50, while among the offspring, more than half occur before
30, and this is taken to prove that there is Anticipation or Antedating in
Insanity.
This will perhaps be made more evident if the percentages of those who became
insane before the age of 25 are given in each case. Among the fathers, 2°/, and
among the mothers, 4°/, became insane before the age of 25. Among the off-
spring, on the other hand, the percentage is 44. Another way of looking at the
matter is to take the average age of onset of insanity in each case. Dr Mott
gives a Table showing these averages but unfortunately has omitted the congenital
cases so that the extent of anticipation is considerably under-estimated, and the
form in which the data are given does not permit of an accurate calculation of the
actual averages. From the information given it appears, however, that the average
age at onset of insanity among the parents is about 50 years, among the offspring
about 26 years, showing an anticipation or antedating of some 24 years.
* I am very grateful to Miss H. Gertrude Jones, the Hon. Secretary of the Galton Laboratory, for
the diagrams which illustrate this lecture.
46—2
360 Recent Studies of the Inheritance Factor in Insanity
et
at Ons
1 and 2. Diagrams to illustrate the Distribution of Age
+ and Offspring. (Mott.)
ty in Paren
of Ingani
Fathers [_]
Offspring FY
=
[i
D. HERON 361
Now these conclusions, if satisfactorily demonstrated, would obviously be of
the highest importance, but they were immediately challenged by Professor Karl
Pearson in a letter which appeared in Nature of November 21, 1912 (p. 334).
Professor Pearson’s letter is as follows:
On an Apparent Fallacy in the Statistical Treatment of “ Antedating” in
the Inheritance of Pathological Conditions.
The problem of the antedating of family diseases is one of very great interest, and is likely
to be more studied in the near future than ever it has been in the past. The idea of antedating,
i.e. the appearance of an hereditary disease at an earlier age in the offspring than in the parent,
has been referred to by Darwin and has no doubt been considered by others before him. Quite
recently, studying the subject on insanity, Dr F. W. Mott speaks of antedating or anticipation
as “Nature’s method of eliminating unsound elements in a stock” (“Problems in Eugenics,”
papers communicated to the First International Eugenics Congress, 1912, p. 426).
Iam unable to follow Dr Mott’s proof of the case for antedating in insanity. It appears to
me to depend upon a statistical fallacy, but this apparent fallacy may not be real, and I should
like more light on the matter. This is peculiarly desirable, because I understand further
evidence in favour of antedating is soon forthcoming for other diseases, and will follow much the
same lines of reasoning. Let us consider the whole of one generation of affected persons at any
time in the community, and let , represent the number who develop the disease at age s,
then the generation is represented by
Noy Nyy Ng_ vos Ngys0- N00, Say.
Possibly some of these groups will not appear at all, but that is of little importance for our
present purpose.
Let us make the assumptions (1) that there is no antedating at all; (2) that there is no
inheritance of age of onset; thus each individual reproduces the population of the affected
reduced in the ratio of p to 1. Then the family of any affected person, whatever the age at
which he developed the disease, would represent on the average the distribution
Po, PN, Pla, +++ PNs5 +++ PN100-
The sum of such families would give precisely the age distribution at onset of the preceding
generation.
Now let us suppose that for any reason certain of the groups of the first generation do not
produce offspring at all, or only in reduced numbers. Say that g, only of the n, are able
to reproduce their kind; then of the older generation, limited to parents, the distribution
will be
Jo ot M121 + GoN2t «e+ YgNgH «+» + Y100%1005
but the younger generation will be
D (GoM + G12 + YoNgt 0. + Ye Nat --- + G100%100) (pM +... HNg+... +2100);
i.e. the relative proportions will remain absolutely the same.
The average age at onset and the frequency distribution of the older generation, that of the
parents, will be entirely different from that of the offspring and will depend wholly on what values
we give to the g’s. If frequency curves be formed of the two generations they will differ
substantially from each other. This difference is not a result or a demonstration of any
physiological principle of antedating but is solely due to the fact that those who develop the
disease at different ages are not equally likely to marry and become parents.
362 Recent Studies of the Inheritance Factor in Insanity
A quite striking instance of the fallacy, if it be such, would be to consider the antedating of
“violent deaths.” Fully a quarter of such deaths in males, nearly a half in females, occur before
the age of twenty years. Consider now the parents and offspring who die from violent deaths ;
clearly there would be no representative of death from violence under twenty in the parent
generation, and we should have a most marked case of antedating, because the offspring
generation would contain all the infant deaths from violence.
In the case of insanity, is the man or woman who develops insanity at an early age as likely
to become a parent as one who develops it at a later age? I think there is no doubt as to the
answer to be given ; those who become insane before twenty-five, even if they recover, are far
less likely to become parents than those who become insane at late ages—many, indeed, of them
considering the high death-rate of the insane, will die before they could become parents of large
families. Now Dr Mott took 508 pairs of parents and offspring, “collected from the records of
464 insane parents whose 500 insane offspring had also been resident in the County Council
Asylums,” and ascertained the age of first attack. As at present advised, it seems to me that
his data must indicate a most marked antedating of disease in the offspring, but an antedating
which is wholly spurious. There is, I think, a further grievous fallacy involved in this method
of considering the problem, but before discussing that I should like to see if my criticism of this
method of approaching the problem of antedating can be met.
KARL PEARSON.
Biomerric LABORATORY,
University Cotieer, Lonpon,
November 11, 1912.
Dr Mott has referred to this letter in his Report for 1912*, but it will be more
convenient to deal with his reply after we have examined the method by which his
data have been collected and the use made of the data. Let us consider first of alli
how the data were obtained. Dr Mott in describing his material says that it
consists of a collection of cases in the London County Asylums where two or more
persons are related to one another. Thus Dr Mott has dealt—not with a series of
complete pedigrees in which every member is included, whether insane or normal,
but with a series of cases in which two or more members of a family are known to
have been in London County Asylums. No notice is taken of those who are
normal throughout their lives and no allowance is made for those who are normal
at the time the record is made but who may afterwards become insane.
Do cases selected in this way provide a complete or impartial view of the facts?
Some of Dr Mott’s own comments on his data throw a considerable amount of
light on this point. In his Report for 1909+ he says: “From all the Asylums
I have received valuable reports, but in the case of the older asylums it has been a
matter of the utmost difficulty to trace the records of so many years back,” and in
his Report for 1910+ he says, ‘Some of the asylum authorities have gone through
their case books for a number of years back, but the results have not been
satisfactory owing to the difficulty of obtaining particulars without a living repre-
sentative of the family being resident in the asylum—for instance, 110 old cases
* Annual Report of the London County Council for 1912, Vol. 11. p. 62.
+ Twentieth Annual Report of the Asylums Committee of the L.C.C., p. 90.
+ Twenty-first Report of the Asylums Committee of the L.C.C., p. 94.
D. HERon 363
reported from Bexley have been rejected as the relatives in the other London
County Asylums could not be traced, for no instance has been included unless full
particulars could be obtained.”
It is thus clear that not all the cases could be traced and that there was
special difficulty in tracing the older cases. What is the effect of a selection
of this kind? A study of the following hypothetical cases may serve to throw
some light on this point.
TABLE ILI.
Anticipation or Antedating in Insanity. Hypothetical Laamples to
show the Effect of Dr Mott's Selection of Cases.
|
First Example | Second Example
Mother: Born Sod ss we oi te. | 1873 1833
Married... a ey aay oe 1893 1853
Became Insane and admitted to Asylum 1913 1873
Age at First Attack seis ise Pot 40 40
Died fee oe if ave eee 1914 1874
Son: Born... ace 900 See Scie oc 1894 1854
Became Insane ... “ine ies nor 1894* 1914
Admitted to Asylum... apo oe 1914 1914
Age at First Attack ... a ne 0 60
The mothers in those two examples have exactly parallel careers. In each
case the mother became insane at the age of 40 and only lived one year in the
asylum. In the first case the son was a congenital idiot but was only admitted to
an asylum at the age of 20. The age of onset in this case is taken at 0 years and
the case shows marked “ anticipation.” In the second case the mother also became
insane at the age of 40, the son not till the age of 60, 40 years after his mother’s
death. The second example thus tells against the Law of Anticipation. Are
these two cases equally likely to appear in Dr Mott’s data ?
In the first case mother and son are in the asylum at the same time and were
admitted within a year of each other. It is very improbable that the relationship
would escape notice and such a case is almost certain to be recorded. In the
second case, however, the son is not admitted to an asylum till 40 years after his
mother’s death. Even if the family remained in the same area for 40 years after
the mother’s death, it would obviously be very difficult to connect the histories of
mother and son. This case, which tells against the Law of Anticipation, is
almost certain to escape notice. A spurious anticipation or antedating is thus
inevitable owing to the method of collecting the data.
It has also been pointed out that Dr Mott has made no allowance for those
who are mentally normal at the time the record is made but may subsequently
* Congenital Idiot.
364 Recent Studies of the Inheritance Factor in Insanity
become insane, and this introduces further spurious anticipation. Another hypo-
thetical example will perhaps make this clear. Let us take the case of a mother
with six children, five of whom have become insane as follows:
TABLE IIL.
Children
Mother eis bal 5 |
ipl | 2 eat biel wee) 5 6
Born 506 dot one 1830 1850 | 1852 1854 | 1856 1858 | 1860
Became Insane ... | 1860 — 1872 1896 1914 1888 1860+
| Age at Onset of Insanity | 30 —* 20 | 42 | 58 30 0
| a
The extent to which this family would show anticipation or antedating
would depend very largely on the time at which the record was made as is shown
in the following table.
TABLE IV.
Age of Onset of Insanity in
Date of Average for | Amount of
Record Children Anticipation
Mother Children
1860 30 0) 0) 30
1872 30 0, 20 10 20
1888 30 0, 20, 30 16°7 13°'3
1896 30 0, 20, 30, 42 23 0 7
1914 30 0, 20, 30, 42, 58 30 (0)
If the case were noted in 1860 then the age of onset of insanity in the mother
is 30 years—of the child 0 years—a clear case of anticipation, and nothing
would be known of the fact that four other children will afterwards become insane
and will bring the average age of onset in the children up to 30 years—exactly
the same as that of the mother. Nor is the record even now complete for if the
eldest child ever becomes insane, the age of onset in his case must be at least
64 years and this will further increase the average age of onset in the children.
It is thus clear that in dealing with incomplete families and ignoring the possi-
bility that those who are normal at the time of record may afterwards become
insane, Dr Mott has introduced a further spurious anticipation or antedating.
If we examine carefully the first pedigree given by Dr Mott at the Eugenics
Congress}, we see clearly how probably much of the anticipation recorded by
* Alive, 64 years of age and still normal.
+ Congenital Idiot.
+ Problems in Eugenics, p. 413.
D. Heron 365
Dr Mott has arisen. Unfortunately this is the only pedigree for which sufficient
details have been given to enable its completeness to be tested. The pedigree
and Dr Mott’s description of it are as follows :
“A.B., an alien Jew, aged 54, was admitted to an asylum for the first time suffering from
involutional melancholia ; he has a sister who has not been in an asylum, but, as events turned
out, bore the latent seeds of insanity. The man is married to a healthy woman who bore him a
large family ; the first five are quite healthy, then comes a congenital imbecile epileptic (cong.)*,
then two healthy children followed by a daughter who becomes insane at 23, then a son insane
at 22, and lastly two children who are up to the present free from any taint. The sister of A.B.
is married and has a family of ten, seven girls and three boys; one of the females was admitted
to the asylum at the age of 19, and since this pedigree was constructed a brother of hers has
been admitted aged 24. Half-black+ circles are insane. The pedigree is instructive ; it shows
direct and collateral heredity ; it also shows remarkably well the signal tendency to the
occurrence of insanity at an early age in the children of an insane and potentially insane
parent.”
3 Brothers
6 Sisters:
23 22 19
@ = Insane.
13 children: 9 Alive, 4 Sons, 5 Daughters. 4 Dead. 3 Insane.
1
Cong.
Fig. 8. Pedigree to illustrate the effect of Dr Mott’s selection of cases.
F. W. Mott: ‘‘Heredity and Eugenics in relation to Insanity.” Problems in Eugenics, p. 413.
This pedigree was given as above in July 1912, and in an address previously
delivered before the Manchester Medical Society on Oct. 4, 1911, Dr Mott gave
the same pedigree, but without any reference to the nephew of A.B. (brother of
the girl who became insane at 19) who became insane “since the pedigree was
constructed,” so that this man became insane between 1911 and 1912 and this
serves to “date” the pedigree.
Now it should be noted that at least five of the children of A.B. are over 23 years
of age and up to the present time healthy. But all these children are alive and if
any one of them afterwards becomes insane, the average age of onset of insanity in
the children will be raised—and it is clear that the more incomplete the pedigree
the greater the amount of spurious anticipation. Again Dr Mott states that in
* This does not agree with Dr Mott’s pedigree which gives the congenital case as the seventh
instead of the sixth child.
+ According to our usual custom, they are represented by full black circles in Fig. 3.
Biometrika x 47
366 Recent Studies of the Inheritance Factor in Insanity
nephews and nieces the age of onset is earlier than in uncles and aunts. In 1911
this pedigree gives a case in which an uncle became insane at 54, his niece at 19—
but one year later a nephew who became insane at 24 has to be added, thus
raising the average and there are eight more children some at least of whom
may become insane at later ages. As before the incompleteness of the pedigree
introduces an artificial and spurious anticipation or antedating. The remedy is
obvious ; we must only deal with completed families.
A further fallacy involved in Dr Mott’s method of work must now be noted. In
directly comparing the age of onset in parent and child, Dr Mott has ignored the
fact that in the parent the incidence of insanity is for all practical purposes
limited to the age of 20 and over since cases of congenital defect and of adolescent
insanity hardly ever marry. Among the general population of asylums, however,
12°/, become insane before the age of 20 and in Dr Mott’s selected data the
percentage rises to 27—or more than a quarter of the whole become insane before
20. This in itself causes a very marked spurious anticipation. As Professor
Pearson has shown (p. 361 above) if we were to investigate the age at death in
parent and child from accident or violence, we should find the same spurious
anticipation.
There are thus three fallacies involved in Dr Mott’s work. In the first place a
spurious anticipation or antedating arises from the inclusion in the record of
families whose history has not yet been completed, for those who become insane
at late ages in the younger generation do not appear. Secondly, even with families
whose history is completed, those cases in which the insanity of parent and child
is contemporaneous are far more likely to be recorded than those in which the
child becomes insane long after the parent*, and thus the cases which show
anticipation are more likely to appear in the record than those which tell against
Dr Mott’s views. Thirdly, by directly comparing parent and child, he has practi-
cally limited one of the two groups which are being compared to ages at onset of
over 20 years and has thus obtained further spurious anticipation.
Dr Mott also lays stress on the appearance of insanity in a more intense form
in the younger generation. “I have proved,” he sayst+, “that there is a signal
tendency in the insane offspring of insane parents for the insanity to occur at an
earlier age and in a more intense form in a large proportion of cases, for the form
of insanity is usually either congenital imbecility, insanity of adolescence, or the more
severe form of dementia praecox, the primary dementia of adolescence, which is
generally an incurable disease.” But we have already seen that Dr Mott’s method
of collecting his data is such that an enormous preponderance of early cases
of insanity in the younger generation is inevitable and of course such cases are
largely incurable. Type of disease is very closely related to the age of onset and
* Dr Mott states (Archives of Neurology, Vol. v1. p. 82) that ‘the main bulk of the cards (i.e. his
records), however, refer to parents and offspring admitted to the asylums within the last fifteen years.”
+ Archives of Neurology, Vol. v1. p. 82.
D. Herron 367
by selecting the latter we can alter the proportion of any particular type of
insanity. Dr Mott has obtained his material in such a way that, in the younger
generation, cases of insanity coming on late in life are much less likely to be
recorded than those which appear in early life, and hence the early cases are in a
majority, but the change in age of onset, and consequently of the type, is entirely
spurious and arises solely from the way in which the material has been obtained.
We can now deal with the reply Dr Mott has made to Professor Pearson’s
criticisms. In his Annual Report for 1912 (p. 62), Dr Mott says: “ Professor
Karl Pearson, writing to Nature, November 21, 1912, ‘On an apparent fallacy in
the statistical treatment of “ Antedating” in the inheritance of pathological con-
ditions,’ criticises on mathematical grounds the evidence of anticipation. I do not
feel myself competent to reply to the opinion of such an eminent authority on
mathematics applied to biometrics, but it does not militate against my conclusions,
nor explain away the fact that a large proportion of the insane offspring of insane
parents are affected with imbecility or adolescent insanity; for granting the
assumption that there is no antedating at all, we might rightly expect the ages
at onset of insane offspring of insane parents to be comparable with the ages at
onset of all the admissions to the asylums during the same period*. This is by no
means the case, for amongst the insane offspring there is a far greater proportion
atfected early in life, as is shown in the following figures and curves” (they appear
here as Fig. 4 and Table V).
According to these figures the onset of insanity among the recorded insane
offspring of insane parents is considerably earlier than among the general admis-
sions to asylums, but it has already been shown that this is due to the fact that
the data have been selected in such a way that the early cases in the younger
generation are the most likely to appear. Further, if Dr Mott’s argument be a
valid one, we might also expect the ages at onset of the insane parents of these
insane offspring to be comparable with the ages at onset of all the admissions to
asylums during the same period. This is by no means the case as is shown in
Fig. 5 below (see also Tables I and V). We see here that the insanity of the
parents comes on at a much later period than among the general admissions to
asylums and that there is a far less proportion affected early in life. If Dr Mott’s
method of argument be sound, he has not only to deal with an antedating of
insanity among the offspring but also a post-dating of insanity among the parents.
Both are of course spurious and arise from the peculiar selection of the data and
from the fact that, owing to differential death-rates, the ages at onset of “ admis-
sions” will never be the same as the ages at onset of the admitted—i.e. the asylum
population—at any time.
* ««We might rightly expect”’ these ages to be different, because ‘‘admissions”’ are not the same as
the population in the country who have at one time or another been insane. The percentages of total
cases of acute mania, of senile insanity, of congenital idiocy, and of melancholia, who reach the asylums,
are not the same. The reader has to distinguish between the population of admissions, the population
of admitted, and the insane population of the country. A sample of the latter may be reached from
completed family histories, but not from records on admission or from records of an asylum population.
47—2
368 Recent Studies af the Inheritance Factor in Insanity
45-7 45
Ko All Admissions [__] Ns All Admissions [_ ]
/ Insane Children Wy fe Dar
yyy YY Insane Parents of “Wy
354 Yy of Insane Parents 35 Insane Children Yd,
30 30
g $
> >
i)
D> Ss
= 20- =
5 §
Q AY
=
an
t
VaR" 25- 35- 45- 55- 65- 75-
Age at Onset of Insanity. Age at Onset of Insanity.
Fig. 4. Fig. 5.
Diagram to illustrate the Distribution of Age at Onset of Insanity among:
(1) The Insane Offspring of Insane Parents.
(2) The Insane Parents of Insane Offspring.
(3) All Admissions to L.C.C. Asylums. ;
TABLE V. Percentage Comparison of the Age at time of Onset of Insanity in
the Insane Offspring of Insane Parents and the General Admissions to
the London County Asylums.
MALE FEMALE Toran
Age at
Onset of | 4489 direct | 274 insane | 5097 direct | 389 insane | 9579 direct | 663 insane
| Insanity admissions | offspringof | admissions | offspring of | admissions | offspring of
during last insane during last insane during last insane
four years parents four years parents four years parents
Under 25 20:0 43°8 20°2 44-2 20°1 44:0
25—84 19°9 27°7 19°9 28-0 19°9 27°9
35 --4h 21°9 13°8 21-5 16°7 21°7 15'5
45—B4 Teel 7) 10°2 18°6 7°4 18:2 8°5
55—64 1353 3°6 12°4 2°8 12°7 3:2
65—7T4 SPH O-'7 59 0°8 5°8 O'7
75 45) — 1°6 _- 1135) —
41 male imbeciles out of 274 offspring
54 female 389 ”
”»
95 male and female , alae 663
D. Heron 369
It is possible to illustrate the various fallacies which vitiate Dr Mott’s conclu-
sions regarding anticipation by considering the age at death of parent and child.
I do not know whether it is generally recognised that it is exceedingly difficult
to get any considerable body of data in which the ages at death of a parent and all
his children are given, for of course the record is incomplete and biassed until the
death of the last surviving member, and in some cases to get a complete record we
must trace the history of a family for over 150 years. George the IIIrd, for instance,
was born in 1738 and all but one of his 15 children were still alive in 1810, 72 years
afterwards, and the last surviving son, Duke of Cumberland and King of Hanover,
did not die till 1851, 113 years after his father’s birth—and this is by no means
an extreme case. In the material I am about to describe I found one case where
the interval was 160 years.
Another difficulty which arises is the tendency in practically all family
histories to omit infant deaths, so that we do not get a complete record. It
seems probable that the deaths of minors are not represented in such records
in anything like their true proportion and that the differences are greater than
might be expected to arise from differences of physique and nurture due to
class. Thus records of the Landed Gentry give 31 deaths per 1000 males under
20 years* while actual experience shows 163 to 197 per 1000+. But in the
records of the reigning families of Europe we get a practically complete record of
all members and therefore from von Behr’s Genealogie der in Europa regierenden
Fiirstenhdusert{, I have extracted particulars of the age at death of over 2000
individuals—all belonging to the 18th century. There was here no selection—
every child was entered and every family had been traced from the birth of the
parents till the death of the last survivor.
Now in Dr Mott’s data we have already seen that cases in which the age at
onset of insanity in parent and child is contemporaneous are most likely to be
recorded. We can test the effect of a selection of this kind by investigating the
effect of selecting, from our data regarding the age at death among those royal
families, only those individuals who died within a certain number of years of their
father’s death, and the results are given below in Table VI, p. 370.
When we deal with the whole of the data, absolutely unselected, every family
being complete and traced to the death of the last surviving member, we find that
680 out of 1829 or 37:2°/, died under 20 years of age. Let us now apply a very
slight selection to the data and reject the 92 cases in which the interval between
the deaths of father and child was at least 60 years. We find now that 680 out of
the remaining 1737 died under 20 years of age—or 39:1 °/,. Thus the effect
of a selection of this kind is to cause a slight increase in the proportion of deaths
at the early ages. If we make the selection slightly more stringent, by taking only
those who died within 40 years of their father’s death, the percentage of individuals
dying under 20 years of age rises to 46°7 and if we go still further and consider
* See Pearson: Proc. R. S. Vol. 65, p. 291. + Statistics of Families, p. 73.
Pp ? p
+ Tauchnitz, Leipzig, 1870.
370 = Recent Studies of the Inheritance Factor in Insanity
TABLE VI.
Illustrating the Effect of Selection of Material on the Distribution
of Age at Death.
(Reigning Houses in Europe—18th Century.)
Children who died:
All Cases ie
Unselected within 60 years | within 40 years | within 20 years in their
Age at Data of their of their of their father’s
Death father’s death father’s death | father’s death lifetime
Numbers} °/, |Numbers| °/, |Numbers] °/, |Numbers| °/, |Numbers| °/,
Under 20 680 | 37°2 680 39°1 680 | 46°7 680 | 6274 648 82°7
20—389 277 15:1 277 15°9 277 19-0 254 | 23°3 121 15°4
40—59 336 18°4 336 | 19°3 274 | 18°8 127 Igoe 15 1:9
60—79 450 | 24°6 395 | 22:7 | 214 14:7 29 2°7 — —
80and over 86 4°7 49 2°8 | 10 7 — — = =
Totals 1829 | — 1737 | — 1455 — 1090 - 784 =
Average |
Age at 35°9 33°7 26°9 16:2 eu
Death*
only those who died in their father’s lifetime, then the percentage rises to 82°7 °/,.
Looking at the matter in another way we find that the average age at death has
fallen from 35°9 years to 7°7 years.
The same facts are given in Fig. 6, which shows that as the selection of cases
becomes more stringent, there is a regular increase in the proportion of deaths at
the younger ages. In exactly the same way, the fact that cases where the insanity
of parent and child is contemporaneous are the most likely to appear in Dr Mott’s
records causes a spurious exaggeration of the cases of insanity at early ages in the
younger generation and consequently a spurious exaggeration of the number of
cases of imbecility and adolescent insanity.
We can also investigate directly the question of anticipation or antedating
on this material. In order to avoid the heavy weighting of large families which
would arise if every child were entered, I have taken only one child from each
family. Let us consider first of all the distribution of age at death of Fathers
and their First-born Children. The facts are given in Table VII.
We have altogether 294 cases in which we know the age at death of a father
and his first-born child. None of the fathers died before 20 but of the children
* These averages were calculated, not from the five age groups given above, but from the same
material classified in 15 age groups.
D. Herron B yall
106 out of 294 or 36:1°/, died before 20. The average age at death among the
fathers is 61 years, but among the children it is only 36 years, so that there is an
anticipation of 25 years. To borrow Dr Mott’s words, the figures clearly show
the signal tendency among the offspring to die at a much earlier age than
their parents; that is to say, anticipation or antedating is the rule.
Age at Death.
O= 205 40- 60- 80-
A OO
== 40
as
20
Children who
2 group
died within
40 years of their
vn €AC
Fathers’ death
Percentages dying
h age
anu ee on
i ee aS
died within : * 60
20 years of their = 40 ae aa
2 ey
Fathers’ death &.= 20 oy
E
TTT TT
EET EEE
S$ 80 a ee
Fs
er ge 60
~o~
pba Ae SES
Fathers’ lifetime $§§ a Soe
LC a
QS . —
Age at Death.
Fig. 6. Diagram to illustrate the Effect of Selection of Material upon the Distribution of
Age at Death. (Reigning Houses in Europe, 18th Century.)
Now in this material there is no selection of families. Every family was taken
and the age at death of every first-born is known, so that we are only left with the
372 Recent Studies of the Inheritance Factor in Insanity
TABLE VII.
Showing Anticipation in Age at Death. A. Fathers and Children.
(Reigning Houses in Europe—18th Century.)
é __ | First-born First Sons who
Age at Death Fathers Children | Fathers Hadeckularen
0— 9 — 95 s oo.
10—19 — 11 os as
20—29 6 21 4 8
30—89 16 18 8 15
40—49 45 31 31 39
50—59 70 34 54 39
60—69 tid 37 58 44
70—79 62 33 46 54
S80—89 18 14 12 13
90 and over _ — — 1
Totals 294 294 213 213
Percentage dying under 20 0 36:1 0 0
Average Age at Death ... 61 36 60 59
| Anticipation de es 25 1
third of Dr Mott’s fallacies, in that no allowance has been made for the fact that
the parental group is limited to ages over 20 while more than a third of the off-
spring die under 20. The effect of this selection can be removed almost entirely
by taking instead of the first-born child, the first son who married and had at least
one child. There are in all 213 such cases and we see that there is now no
anticipation. The difference between the average ages at death is less than a year
and by removing the artificial selection we have got rid of all anticipation or
antedating.
These facts are also shown graphically in Figs. 7 and 8. The horizontal scale
gives the age at death in 10-year groups while the vertical scale gives the actual
numbers of parents and offspring dying in each age group. The diagram on the
left shows marked anticipation, and should be compared with Dr Mott’s diagram
(Fig. 1) in which the ages at onset of insanity of father and child are compared.
When, however, we get rid of the selection of cases by taking only sons who have
had children, then there is no anticipation.
If we compare the distributions of age at death in mothers and children we get
exactly the same results. The facts are shown in Table VIII.
We see that the first-born children died on an average 18 years before their
mothers, but when we compare the age at death of mothers and the first son in
REIGNING HOUSES IN EUROPE — I8™ CENTURY.
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AGE
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Fig. 7.
374 Recent Studies of the Inheritance Factor in Insanity
TABLE VIII.
Showing Anticipution in Age at Death. B. Mothers and Children.
(Reigning Houses in Europe—18th Century.)
First Sons to
| First-born
|
|
, . |
| Age at Death Mothers | “Ghijdren’ | Mothers | pave Children
0— 9 —- 122 — =
10—19 2 13 hers —
20—29 47 26 21 8
380—89 | 49 22 30 16
4O—49 | 43 32 A) 41
| 00—o9. |aeom 35 39 | 40
60—69 | 80 42 52 46
70—79 leer O2. an anes 4] 54
SO—89 Po eS ear Wa 10 14
90 and over | Die eal oH 1 1
| a |
| Totals | BE) Sar 220 220
Percentage dying under 20 6 39°1 5 0)
Average Age at Death ... | 53 35 55 59
Anticipation Ss aoe 18 —4
each case to have children, then the sons live four years longer than their mothers.
It would have been better in this case to have compared the mothers with the
first daughters to have children but unfortunately von Behr gives very little
information regarding the female lives, except in special cases. The figures show
a marked anticipation in age at death when we directly compare, as Dr Mott
has done, mother and child, but this vanishes when we remove the arbitrary
selection. The same facts are shown graphically in Figs. 9 and 10.
If we combine these figures we can compare the age at death of parent and
child and the results are shown graphically in Figs. 11 and 12.
Fig. 11 shows that Dr Mott’s limitation of one of the two generations he is
comparing to adults, without imposing a similar limitation on the other generation,
introduces an artificial and spurious anticipation. The average age at death of
the parents is 56 years and of their first-born children only 35 years—so that we
get an anticipation of 21 years. If, however, we make the two generations
almost directly comparable by dealing only with sons who have children—there is
no significant difference between the two averages (58 against 59 years).
In these cases we have dealt only with completed families and have taken
every family without selection. If, however, we consider only the cases in which
CCA
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376 Recent Studies of the Inheritance Factor in Insanity
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the eldest child died in his father’s lifetime the amount of anticipation is greatly
increased. The facts are shown in Table IX and in Fig. 13.
REIGNING HOUSES IN EUROPE — I8™ CENTURY.
ACE AT DEATH OF FATHERS & OF FIRST BORN CHILDREN.
WHO DIED IN THEIR FATHERS’ LIFETIME,
AVERACE ACE AT DEATH OF:- FATHERS
FATHERS 62
CHILDREN 10 CHILDREN
ANTICIPATION §2
100
FREQUENCY
We see here that among the fathers none died under 30 while 87°/, of their
children died under 30; the average age at death among the fathers was 62--
among the children only 10, showing an anticipation of 52 years.
378 Recent Studies of the Inheritance Factor in Insanity
TABLE IX.
Showing Anticipation in Age at Death. C. Fathers and First-born
Children who died in. their Fathers’ Lifetime.
(Reigning Houses of Europe: 18th Century.)
First-born Children
Age at Death | Fathers | dying in their
| Fathers’ lifetime
|
0-9 | et | 89
LO==19 | — 11
20—29 — 16
380—89 | 6 10
4O—49 | 20, 5
50—59 | 32 2
60—69 3D —
O79) 32 =
80—89 10 | a=
Totals 133 133 |
|
Percentage dying under 20 | O a)
Average Age at Death ... 62 10
Anticipation oe aae 52
It is now possible to illustrate the effect of the principal fallacies which vitiate
Dr Mott’s conclusions. In the first place he has dealt with families which are
largely incomplete and has collected his material in such a way that cases in
which the insanity of parent and child is contemporaneous are the most likely to
be recorded ; in the second place he has directly compared parent and child with-
out allowing for the fact that practically no parent can become insane before 20,
while there is no limitation of this kind among the offspring of these insane
parents.
In Table IX and Fig. 18 we see the effect of dealing with incomplete families
in which the children died in their fathers’ lifetime. There we get an anticipa-
tion of 52 years. If we get rid of the first and second fallacies involving a
selection of cases by dealing with every family, as shown in Table VII and Fig. 7,
the anticipation falls to 26 years. If we get rid of the third source of fallacy
also, by comparing the fathers with the first sons who have children, asin Table VII
and Fig. 8, then the anticipation falls to less than a year. The Law of Antici-
pation or Antedating has thus in Dr Mott’s case no foundation, in fact it is a
spurious result of the mode of collecting and interpreting data.
Now Dr Mott has not only asserted that this “ Law” applies to insanity but
has also drawn the conclusion that the offspring of insane parents if still normal
D, Heron 379
at the age of 25 may safely marry. In an address delivered before the First
International Eugenics Congress*, he said: “ You will observe that 47°83 °/, of the
500 offspring had their first attack (of insanity) at or before the age of 25 years
and as you see in the curves of parents and offspring, the lability of the child of
an insane parent becoming insane tends rapidly to fall. Now besides the fact
that this shows Nature’s method of eliminating unsound elements of a stock,
it has another important bearing, for it shows that after twenty-five there is a
greatly decreasing lability of the offspring of insane parents to become insane
and therefore in the question of advising marriage of the offspring of an insane
parent this is of great importance. Sir George Savage recently said that this
statistical proof [sic !] of mine entirely accorded with his own experiences, and that
if an individual who had such an hereditary history had passed twenty-five and
never previously shown any signs (of insanity) he would probably be free and he
would offer no objection to marriage.”
Now I entirely fail to understand how anyone could recommend marriage in
such cases, even on Dr Mott’s own figures; for if it be true that 48 °/, become
insane before 25, it must be equally true that 52°/, become insane after that age
and this very important point seems to have been forgotten. These figures,
however, are taken from Dr Mott’s selected data, selected.in such a way that the
early cases are enormously exaggerated. Until Dr Mott publishes a series of
complete pedigrees, it will be safer to assume that the age at onset of insanity
among the offspring of insane parents does not differ widely from that of all
admissions to Asylums and there we find that only 21°/, become insane before 25,
and 79°/, after 25.
But surely at a Eugenics Congress of all places some thought might have been
given to the mental condition of the children resulting from such matings, before
advising marriage. It would not have been difficult for Dr Mott to have extracted
all the available cases of this kind from his collection of pedigrees, i.e. all cases in
which an individual had an insane parent and was normal at the age of 25, and so
have discovered the probable fate of the offspring from such matings.
Unfortunately the details given by Dr Mott regarding his pedigrees are usually
so scanty that little use of them can be made, but two at least show the danger of
the matings Sir George Savage and he sanction; these two pedigrees were given
by Dr Mott in his lecture on Heredity in Relation to Insanity, delivered to the
members of the London County Council. The first is shown in Fig. 14. (It
appeared as Fig. 11, p. 18 of Dr Mott’s lecture.) In the first generation a man
who became insane at 70 had four children. The eldest, a girl, became insane at
68 and was therefore normal long after the age of 25. Dr Mott does not state
whether the marriage of this woman preceded or followed the onset of insanity in
her father, but even if her father had become insane before her marriage, Dr Mott
* Problems in Eugenics, p. 425. This is one of many illustrations of the evil done by that Congress ;
attention was directed and much weight given to hasty statements and ill-digested material.
380 =Recent Studies of the Inheritance Factor in Insanity
would have raised no objection to the marriage since the woman herself was not
insane. There were in all six children from this marriage of which Dr Mott would
have approved. Two became insane, three were blind and five are said to have
been paupers.
= Insane P= Pauper B= Blind
Fig. 14.
el
Til
IV
ie 3
®&) = Tuberculosis gy = Suicide
& = Insanity Con = Congenital
ZG) = Unknown
D. HErRon 381
The eldest child remained normal till the age of 34 and although both his
parents became insane Dr Mott apparently would not have objected to his marrying.
He did so and one of his children became insane and eight out of nine are said to
be paupers. These nine children are apparently still young so that their ultimate
fate is still uncertain.
The second pedigree I shall quote was given as Fig. 28, p. 33 of Dr Mott’s
lecture, and appears here as Fig. 15.
A man who had an insane father and an insane grandfather became insane at
the age of 55. He was therefore normal at the age of 25* and Dr Mott would
have sanctioned marriage in his case. He actually married twice. His first wife
was tuberculous but not insane; they had two children, both insane. His second
wife was normal and it is definitely stated that there was no insanity in her
family; they had five children and one of these became insane. Yet Dr Mott
would permit the children of insane parents to marry if only they are normal
at the age of 25!
Again, Dr Mott has stated that it is useless to attempt to limit the fertility of
the insane since most of their children are born before the onset of insanity,
and therefore before any action can be taken. From his statistics of relatives in
L.C.C. Asylums, Dr Mott has calculated the proportion of offspring who were born
after the first attack of insanity in the parent and found that “Forty-six offspring
out of 581 were born after the first attack of insanity in the parent, 1e., 7°9°/,.
That is to say in the case of 529 insane parents, the birth of only one-twelfth of
their 581 insane children would have been prevented by sterilisation or life segregation
of the parent after the first attack of insanity. These figures refer to the offspring
which become insane, but there are a large number of offspring which do not become
insane and these would be cut off if life segregation or sterilisation were adopted +.”
But here again Dr Mott is using the data obtained from his index of relatives
which shows a greatly exaggerated number of cases at the earlier ages among the
offspring, and he thus greatly exaggerates the number of cases in which the children
were born before the onset of insanity. No conclusion can be drawn from any but
complete records of families. But apart altogether from this, many of these parents
are themselves the children of the insane and much could be done to discourage
such marriages. Unfortunately as we have seen Dr Mott directly sanctions
marriage to those who remain normal till the age of 25.
In further support of his view Dr Mott has stated that out of 642 females
admitted to three London County Asylums in 1911, 148 were recurrent cases and
of these 32 (21°/,) had children between their respective dates of admission.
“The inference that can be drawn,” he says, “is that about one-fifth of the
recurrent cases, or approximately one-twentieth of the female admissions have
* If the term ‘‘age at onset”? has any real meaning.
+ The italics are Dr Mott’s.
Biometrika x 49
382 Recent Studies of the Inheritance Factor in Insanity
children after their first attack of insanity and of 31 such cases examined, 73 chil-
dren were born after the first attack of insanity in the parent.”
But have these 148 recurrent cases been followed up to the end of the repro-
ductive period? Not at all. No ages are given and the cases are merely those
which were admitted to Asylums in 1911, Dr Mott’s remarks being made in
June 1912, so that no attempt has been made to follow them up. There is no
justification for Dr Mott’s advice.
There are many other points in Dr Mott’s work which deserve detailed exami-
nation, but time will not permit more than a brief account of a few of them.
It should be noted, for instance, that Dr Mott has used his index of relatives
in London County Asylums as an argument in favour of the importance of the
inheritance factor in insanity. His argument is as follows:
“At the present time in the London County Asylums there are 725 individuals
so closely related as parents and offspring, brothers and sisters. A priori, this, to
my mind, is striking proof of the importance of heredity in relation to insanity,
for we cannot suppose that 20,000 of the 44 millions of people in London brought
together from some random cause would show such a large number closely related
eis i0 9) oc
But Dr Mott has not attempted to give, and I doubt if he ever will be able to
give, a satisfactory estimate of the number of relatives in even a random sample
of the population, and the population of asylums is far from being a random
sample of the general population—there is for instance an extraordinary divergence
inage. Yet without definite information on this point it would be impossible to
say whether insanity is inherited or not—that is if we had to depend solely on
Dr Mott’s data,
It should also be noted that in these cases Dr Mott has clubbed together
every form of insanity, from congenital idiocy to senile dementia, except of course
cases due to specific infections or trauma. I myself think that course is the only
possible one. To anyone who has studied even a few pedigrees of mental defect,
nothing is more striking than the extraordinary number of different forms of
mental defect that may appear in the same family.
Seven years ago, in a First Study of the Statistics of Insanity and of the
Inheritance of the Insane Diathesis*, I was confronted with the same problem,
and after a full consideration of all the available data and of the opinions of those
medical men who were best qualified to express an opinion came to the conclusion
that the only possible course was to group all forms of insanity together, with,
of course, the exceptions I have already indicated. The whole question was dis-
cussed very fully in my paper and it was there suggested that an even broader
classification might be of service. This point of view met with some criticism at
the time but nothing has occurred to alter it, and the study of the inheritance of
* Galton Memoirs, No. II. (Dulau and Co.)
D. Herron 383
insanity in general or of an even broader degeneracy must always remain the
first object of our studies.
Any investigation of the inheritance of special types of insanity or degeneracy
can only be carried out however on unselected material—on the records of com-
plete families. The type of insanity is so closely related to the age of onset that
any tendency to exaggerate the number of early cases, as in Dr Mott’s material,
will entirely vitiate the conclusions drawn. Thus Dr Schuster’s conclusions as to
the inheritance of special types of insanity based upon Dr Mott’s data* must also
be rejected on the above grounds.
Dr Mott’s index of relatives in London County Asylums is unfortunately of
very little value in the study of inheritance in insanity. Progress can only come
from the study of complete pedigrees in which every member of the family is
entered, whether insane or normal, and the ages of the normal at the time the
record was made are just as important as the age at onset of insanity in the insane
members, for a statement that a young man of 20 has not been insane is of a
very different degree of importance from the statement that a man of 70 has
not been insane.
In the papers I have cited the children of the insane if normal at 25 are
advised to marry, and it is asserted that it is useless to attempt to discourage the
reproduction of the insane since most of their children are born before the onset of
insanity, and that we should rely on the Law of Anticipation to end or mend
degenerate stocks.
I have shown, I think, that the Law of Anticipation as applied to the insane
has no foundation in the facts provided and that the advice given as to the marriage
of the insane and of their normal offspring is fundamentally unsound and directly
eacogenic. Much yet remains to be learnt regarding the inheritance of the insane
diathesis, but no one who has studied the family histories of the insane can doubt
that in ivheritance we have by far the most important element in the production
of insanity, and in view of all the facts it is the obvious duty of the Kugenist to
discourage, rather than to encourage, procreation by the insane and even by those
of their offspring who appear to be normal.
* Report on the Statistical Investigation of Relative Cards, 21st Annual Report of the London
County Council Asylums Committee (1910), p. 95.
49—2
ON THE PROBABLE ERROR OF THE BI-SERIAL
EXPRESSION FOR THE CORRELATION COEFFICIENT.
By H. E. SOPER, M.A. Biometric Laboratory, University of London.
In a recent paper* Professor Pearson shows that where one character is in
multiple graded grouping and the other in alternative categories, greater or less
than a given magnitude, the correlation coefficient admits of simple expression ;
the assumptions being that the unmeasured character, B, has a normal distribution
and that the measured character, A, has linear regression upon B. Under these
conditions the data required are the numerical ratio of the alternative groups,
the standard deviation of the measured character and the deviation from the
general mean of this character of the mean of one of the groups.
This expression is subject to greater fluctuations of value in samples of NV
of the population than is the product moment form, especially where one of the
groups is relatively small; and it is proposed to find formulae for the mean and
second moment of the errors from this mean to a first approximation, that is to
terms in 1/N. These will appear in terms of the correlation coefficient, r, of the
original population (which will be supposed normally correlated) and the fractional
frequency, f, in that population of the group possessing the Bree or positive
intensity of the character put into two classes.
Let y be the graded character and # the alternative character the intenser
value of which is possessed by the fraction f of the population. Let %, y be the
general means and o,, o, the standard deviations of « and y. Then it is shown
in the paper that if 2’, 7 are the means of the group /,
pelt ee (1),
cy La
on the assumption that the regression of y upon « is linear; and that if 2 be
normally distributed this is equivalent to
ona x Se ee HONORE AOR ON NEDSS SC > (2),
Cy Z
* Biometrika, Vol. vu. (1909), p. 96: ‘‘On a New Method of Determining Correlation between
a Measured Character 4, and a Character B, of which only the Percentage of Cases wherein B
exceeds (or falls short of) a given intensity is recorded for each grade of 4.”
H. E. Soper 385
where z is the ordinate of the normal curve cutting off the area
f=4(1—a) or 4(1 42)
as defined in Sheppard’s Tables of the Probability Integral.
Now if Do, Pi; Pa, ete.
be the moment coefficients of the whole population with respect to the character
y defined by
0S (Mats) Nett cess cat mener rience ersta es (3),
[see bi-serial table below which is here to be looked upon as representing the
general population] and
Dy; Dia! Pz ete.
are moment coefficients of the group n’(=/N) defined by
(i 2 1S) (COOP I BL geese en erm gona races anesoo00 (4),
we may write f= pp, fy’ =pi, J = Pr. Ty = V(po— pr) and
mye oe, Ay at Ms
aE scl AOE he) De 2 aR oe (5).
V(p2— pr") 2
—+ Grades of y in Bi-serial Table.
[tea ES 1 ea ae, ae
V ” ” ”
< Ny Nye 3
on |
°
2 ny Ns! Ns
as}
S
5 |
oO
Ny Ny Ns
n'/N=f
In samples of WV the frequencies n,, n,, ns’ and consequently the moment
coefficients p and p’ and the ordinate z are subject to fluctuation and the values
of the correlation coefficient calculated from this formula will have a distribution
of errors. Let 7 be the mean value in such samples, ér the deviation from this
mean value in any sample when 6dpp, dp,, dp), ete. dz are the deviations in moments
and ordinate. Then
= 3 Pi + Spy’ — (po + 8p) (pi + 8p)
7 +6) : oe geet te hess Cormeen (3):
/ {po + Sp. — (p, + 6p,)"} x (2 + 62) SP
To express 6z in terms of deviations’ of the moments we have
Ls,
Ufa a (Fee ae 8 bh
V Qar 8)
386 Probable Error of the Bi-serial Correlation Coefficient
Hence to second order terms in powers and products of deviations
1 1 2
240s = ——— en eee
N Qar
=z{l—aéa—}(1—a’) (Sa)}
at+éa 1 AD
Sie eee
a /Qar
sa] m
= —— Sa(@e ay
0 J 2a 5
Cy
=2["(1-ag-40 -@) B+...J de
=z |da—4a(da)}.
It follows that .
etb2e=2+a8f—5 (8f)
,_ i ine
=z+adp — oF (Ope )8 kei beeeeeee (10).
At the same time that this value is put in (7) we may simplify the expression
and subsequent algebra by supposing the graded character y to be measured
from its universal mean value as origin and in terms of its standard deviation as
unit of measurement in which case
n=0; pr=of=1, and by () py 2) ieee ees (11),
and since p, =f (7) becomes,
mM zr + op,’ — fdp, — Spy Sp,
Expanding to second order of deviations we find,
P+ OF HT 64 Oy disaw- ccnp ease oven eeeeeeeee (13),
where 6,, 6, are the first order and second order expressions
1 ; '
o= a {dp,' — fdp, — zr dp, — ar 8p,'},
fa
1 :
n= : {har8p, dpy — - Spy Spo + = 5p, dpy — $6p' Sp, + 4 fOp, Sp.
— Spy dp + $2r (Op,)? + S er (dp)? + (1 + 2a?) _ (5p. Seek (14).
Taking mean values
T= MEAN Og cooler eae eaten (GUS),
mean 6, being zero since by (3), (4)
mean dp, =S (mean dnsys”)/N =,
mean 6p,’ = S (mean 6n,’¥;”)/N = 0.
H. E. Soper 387
Mean 6, is to be evaluated by the formulae
mean 67,,5p) = (Putv — PuPo)/ N
mean Op, dpy = (puso — Pu Pu )/N |
mean 6p. 6p. =(p'utv — Pups )/N
of piich the first two are well known* and the third may be proved thus:
N8Spu = S (Sngye) = 8 (Eng ys”) + 8 (bn ys"),
N Sp, =S8 (6ns' ys”),
*, N?Spudpy =S8 {(Sns )? yst*} + S {Ong Sng’ ys yy} + S [Ons Sng Ys’ Ys},
where in the third sum s’ may or may not equal s.
But mean (6n;)? =n; (1 — n/N),
mean (6n; dry’) =— ns ns’ /N,
mean (dn; dny”) = — ns'ng/N,
the last whether s=s’ or not. Thus we findt
mean dpudpr = (puso — Pu Py — pu" Pv) | N = (p'usrv— pupo)/N.
Evaluating mean 6, es these formulae we find,
7 Saye 73 tt ar (p2’ — P2Po ‘) SES = (py — Pi Po)
A (pr — Pipo) — & (ps — py Po) + of (Ps — Pips)
; ; as ; ee 7
—(pr — Po Pr) + $27 (ps — pr’) + 3 2r (pa po!) +1 + 2a?) 5, (Pr — Po Y ena let);
in which the undashed moments, being those of a normal curve with unit standard
deviation, about its mean, have the values
Pi=9, Pr=l, ps=0, pr=3,
and the dashed moments beyond the first two,
Sy i eae
have values depending upon the nature of the frequency distribution of y and z.
Assuming «, y normally distributed t
? =|, depareat ay YT dandy,
ae 1 :
pi = i | tice TOA og diay
a -2 Tv Se
* See Biometrika, Vol. 11. (1903), p. 275: ‘‘On the Probable Errors of Frequency Constants.”
K. Pearson. The second follows in exactly the same manner as the first, since the constancy of the
total frequency dealt with is only involved, in deducing the relations (i) and (ii) of p. 274, of that
paper.
+ Dae =8 (n,/y.")/N.
+ Since the moments appear in the term containing 1/N any errors in their calculation due to
incorrect assumption of normality will not affect the present approximate formulae provided such errors
are of the order 1/N.
388 Probable Error of the Bi-serial Correlation Coefficient
Putting y=rex +7 and integrating with respect to 7 for constant 2,
ie l Ly?
pi=| (ray +(1 ae r)} ——— @ 2% da
V Qe
ps = {(rx)3 3 (rv) (1 Re r2)} oe Pat ce ia
Sat A eeett alta a) | ee ON ANET Oo ovctc caaiSookdde cane (19).
When these values are put in (17), and terms collected, the mean value of the
bi-serial correlation coefficient in samples of V is found to be
Mi : eee
a i ae fs cenaaee i: | Pees. (20),
where f =1—f= WN. ‘
In the work of obtaining this approximation all powers and products of
deviations above the second order have been neglected. The means of such
terms in samples of NV involve second and higher powers of 1/N* and the present
result is correct to the first approximation.
Again squaring (13) and taking mean values and subtracting the square of
(15) we find to the same approximation as before
G,7 = mean (or) —imeanyo,” seen ene ene eee (21).
The evaluation of mean 6, being carried out precisely mm the same way as
mean 6,, the result is the second moment of deviations of the bi-serial corre-
lation coefficient in samples of J,
ofa (e- i s (1 = (1 +f)! ror| ieee (22).
Writing the two results (20) and (22)
r=T {1 + (be 4P 1)
of =a Niyee Wir Erk ee (23),
the values of 1, Yat and y, for values of $(1—a) [=the smaller of n/N, n/N]
from ‘50 to ‘01 are to be found in table (24).
* See Biometrika, Vol. 1x. (1913), pp. 97—99.
+ xa for $(1+a) was tabled in Biometrika, Vol. 1x. p. 27, and the table is reproduced in Tables for
Statisticians and Biometricians, p. 35, Cambridge University Press.
H. E. Soper 389
$(1-a) fa Xa" va 3 (1—a) fa Xa” a
50 0354 | 1:5708 | 2:5000 20 — ‘0871 2:0414 | 2°8578
“49 ‘0353 15711 | 2°5003 ‘19 — -1001 | 92-0898 | 2°8951
“48 0350 | 1°5722 | 2°5011 ‘18 — 1146 | 2°1437 | 92-9364
47 0346 | 15741 | 92-5024 “17 — ‘1308 | 2:2035 | 2°9825
“46 0339 | 1:5766 | 2°5043 16 — 1490 | 2:2703 | 3-0341
“4S 0331 1°5799 | 2°5068 15 — 1696 | 2:3453 | 3-0923
“hh ‘0321 15839 | 25098 Ly = 1931 24303 | 3°1582
“43 0309 | 1°5886 | 2°5134 13 — +2201 25272 | 3°2337
“42 0295 | 1°5943 | 2°5177 ‘12 —~ -2513 | 2°6389 | 3°3208
“41 0279 | 16007 | 275225 ‘11 — 2880 | 2°7687 | 34224
“40 0260 | 1:6079 | 2:5279 10 — +3317 | 29221 | 3°5427
39 0240 | 16161 | 2°5341 095 | — -3568 | 3-:0095 | 3°6115
38 0217 | 1°6251 | 2°5409 090 | — +3844 | 3:1057 | 3°6873
3 ‘0191 1°6351 | 2°5484 085 | — 4153 | 32119 | 3°7713
36 0163 | 1°6461 2°5568 080 | — +4499 | 3°3299 | 3°8648
85 0132 | 1°6582 | 25659 075 | — -4887 | 34620 | 3-9696
‘Bh 0098 | 1°6714 | 25759 070 | — -5324 | 3°6110 | 4:0879
33 0062 | 1°6858 | 2:5868 065 | — -5822 | 3°7806 | 4:2294
32 ‘0021 1:7015 | 2°5986 060 | — -6403 | 3°9748 | 4:3777
‘81 |—-0023 | 1°7186 | 2°6115 055 | — °7083 | 4:2002 | 4°5584
30 |—-0070 | 1:7371 | 2°6256 050 | — *7897 | 4:4652 | 4°7723
‘29 + |—-0122 | 1°7573 | 2°6409 045 | — °8868 | 4:°7829 | 5-0283
emt —-Ol79.) || 17791 2°6575 040 | —1:0053 | 51715 | 5:3410
27 | —-0241 1°8028 | 26755 035 | —1:1577 | - 56568 | 57362
26 |—-0308 | 1:8286 | 2-6952 030 | —1:3556 | 62859 | 6-2485
25 |—-0382 | 1°8567 | 2°7166 025 | —1°6308 | 7°1347 | 69481
‘24 |—-0462 | 1:8874 | 2-7399 020 | —2:0272 | 83600 | 7:9572
23 =|—-0550 | 1°9208 | 2°7654 015 | ~2°5263 | 10:3024 | 9-4975
22 | — 0647 19574 | 2°7933 010 | —3:8889 | 13°9393 | 126086
21 |—-0753 | 1:9974 | 2-8240
|
ee (24)
The bi-serial value of the correlation coefficient has the standard deviation
Op = (Ke — Wart ry/ VN,
whilst that of the product moment value is
op =(L—)/VN
In table (25) a comparison of the values of the numerator is made for five
values of r, for divisions at 0, ‘5, ...2°5 times the standard deviation from the mean
of the ungraded character.
Values of ,/(xa2— par? +74) for $(1-a)=
Values
? of |
(a “500 “309 “159 067 023 -006
‘00 1:00 E25 1°31 1°51 1:93 2°76 4°5
225) “9375 1:19 1°25 1°45 1°86 2°68 4:3
“50 ‘750 1:00 1:06 1°26 1°65 Dea, 4°0
wy) "4375 “69 ‘75 94 1°30 1:95 3°2
1:00 ‘00 OF oo “49 ort 1:13 1°8
Seahie (25)
Biometrika x wy
390 Probable Error of the Bi-serial Correlation Coefficient
Thus the effect of grouping and applying the bi-serial value of the correlation
coefficient is to add 25°/, to the probable error in the most accordant case where
r is zero and the division equal, whilst if r is as large as ‘5 and one group as small
as 10°/, of the whole the probable error is nearly doubled. For higher values of
r the errors of sampling, in the case of the product moment formula, grow smaller
and ultimately vanish when 7=1,; but the bi-serial values are not invariable in
samples drawn from a perfectly correlated population but possess a variability as
high as *27/s/V in the most favourable case when the grouping is equal.
If the standard deviation be calculated from the approximate formula,
Graf [2 —7) IN wee (26),
which may be written * =
Or = (Na — 12) NO eee CAD
the error of computation will not be great for values of f and r commonly met
with as the following table compared with the last will show:
Values of ya—-7? for $(l-a)=
ip
“500 309 "159 “067 ‘023 -006
‘00 1°25 1°31 1°51 1:93 2°76 4:5
"25 1-19 1:25 1°45 1°86 2°70 4:4
“50 1:00 1:06 1°26 1°68 2°51 4°2
‘75 69 wi) 95 1°36 2°20 3°9
1:00 "25 31 51 93 1-76 3°5
soo0n (28).
The difference between the two expressions only reaches 5°/, when the
smaller group is less than 7 °/, of the whole.
It will not be necessary, excepting in small samples, to apply a correction
to the bi-serial formula for r in virtue of the mean of samples differing from the
population value. The correction is less than 1/Nth part of the value calculated
unless one of the alternative classes is as small as 4°/, of the whole.
I have to thank fellow members of the Staff for assistance in calculating the
tables.
* For a table of xa see Tables for Statisticians and Biometricians, p. 37.
ON THE PARTIAL CORRELATION RATIO.
PART I. THEORETICAL.
By L. ISSERLIS, B.A.
§ 1. The theory of non-linear regression in the case of two correlated variables
is due to Prof. Karl Pearson*. He shows that regression ceases to be linear
when the correlation ratio » differs sensibly from the correlation coefficient 7 and
establishes criteria for parabolic, cubic and higher forms of regression.
The present paper deals with the regression surface of three correlated variables
x, y, 2, Where, though the regression of z on a, y cannot be adequately represented
by an equation of type
Ze
= ETE ed (1),
om ox oy
the regression of z on w for a constant y and of z on y for a constant « is linear.
A large proportion of the non-linear cases that occur in practice fall into this
class. It will be remembered that Z,, in (1) denoting the mean of the array of z’s
for a given w and y the coefficients y;, y;’ are partial regression coefficients and it
will appear that just as it is necessary to introduce the correlation ratio 7 for an
adequate description of non-linear regression of two variables, there must be
introduced multiple or partial 7’s for the description of such regression in the
case of more than two variables.
We recall the definition and principal properties of .7,—the correlation ratio
of y on 2, oq, being the square root mean weighted square standard deviations of
the arrays of y:
i S(Nzo"n,) “ SS {roy (y - In)
(_—-7’)o,/°= Ory V Notes (2),
Om, S (nz Ya — Y)}
Cee Umea
7 ae Naeem yer an Tecrinas treats (3),
and NGG ST) Cy Sua ne) ys wacnaweeteace seals (4).
* Drapers’ Company Research Memoirs. Mathematical Contributions to the Theory of Evolution.
XIV. ‘‘On the General Theory of Skew Correlation and Non-linear Regression.” 1905,
50—2
392 On the Partial Correlation Ratio
' Here we are dealing with WV pairs of two characters A and B. ng of these have
the character w of A. Y, is the mean of this #-array of b's. o,,, is the standard
deviation of this array, om, is the weighted standard deviation of the means of the
arrays and Y is the value of y given by the regression straight line, Le.
This is the “best fitting” straight line (in the Gaussian sense) to the means
of the arrays, and is the regression line when the regression is linear.
§ 2. Consider now three correlated characters A, B,C. If N combinations
of A, B, Care taken, we may denote by n, the number of these which have the
character A =a and by nz, the number in which A has the value w and B has
the value y. Let %, 7, Z be the mean values of the total population, and let Zz,
be the mean of z for a given w and y. The frequency of A=a, B=y, C=z
1S alte
We define the correlation ratio of zg on « and y, which may be denoted by
vyll;, or if no confusion is likely to arise by H,, by the equation
esl] Seen une = 20) > (6).
The triple sum in the definition can be written
SSS {Nayz (2 — Z+ 2 — Zry)?}
= SS {Ngy (Z — Zry)?} + SS {(Z — Zay)} X S [Maye (2 — Z)} + SSS {Nays (2 — 2)}
= SS {nzy (Z — Zzy)?} — 2SS {gy (Z — Zey)?} + No?.
SYS Oe Zo
= eee De G2 GSE a oe oe (7)
This is a generalisation of the property of ,, given by (3).
Hence pilates
Further, the “best fitting” plane tou the means Z,, is given by
2-2 L—2 592)
== Wh Sf eyg OP ctahnsnetes See eee 8
Gy a On ue Cy (8),
Te Pyan
where fg ven vealid tn ctde debe ne (9),
ary
> __Vyz— Vaz xy
2 aa (a (10).
Tay
Let ,,R, denote as usual the maximum correlation of z with any linear function
of « and y, then
rayliter = 32a + Ys Tzy PIA MOOD CAS AHOC CI OO ORDO GODOOAt.00 (a):
y2 2 > > >
_ Pye Fen = WV yz x28 ay
Ley
u ™
L. ISSERLIS 393
Subtract (11) from (7) after replacing 7, and rz, by the appropriate sums, and
we obtain
Laifet aS Ayla | N oy
= SS ay (Z — Zn)? -—S | = (2—2Z)(a@—@) vs} == Jew ge (2-Z)(y-y) ve}
a y
Big ee ce Opie RNa 2s. Cs ye 2 PT Day
= Sis ay (2 — Za)” — Nay = (Zxy — 2) (@ — @) Y3 — Nay 7 (Zay — 2)(Y— 9) Ys | .
a y
Using (8) this can be written
[eyll? — my R?| No? = SS {nay (Zry — 2) (Zay — Z)}
= SS {Nyy (Zry — Z)?} + SS {xy (Z— 2) (Zn — Z)} ...(13).
But SS {ney (Z— Z) (Zxy — Z)}
=SS | (1
9 yp / * ane 12
= No, (Ys02" zx — Ys’ F2 — 33 Vay Fz + V3 T2V yz — V3V3 Vay Fz — V3 oz)
“LB 1¥—Y
Oz+ 42 J
Ox Cy
Co See wey
Oz i eta z= Cray 3 is Fa)
x y
PMG aval ine — Va is May) ict 8 ys — Ya — Yalay)) <seessoersccseeeenies (14).
Using the values of y; and y,’ given by (9) and (10) we see immediately that
Tea — V3 — Ys Tay =Tyz— Ys — ¥3V xy = 0.
Hence (13) becomes
(cyl Ey reg RENIN op ISI SI OPI C29) 1 ee (15).
This is the generalisation of (4). We deduce that ,,H,=,,R, if and only if the
regression is strictly linear, that otherwise ,,H,>,,R7 and by (6) that ,,H?< 1.
§ 3. These properties and definitions can be extended to the case of m
variables #,, 2, ... fm. We now use » mt, for the mean of a, when a, a3, ... &m
are given and denote by S a summation extending to the variables a, a, ... wp.
Le. 497
If we define the correlation ratio of x, on a, #3, ... 2m by the equation
UN Git mace eda) iS IC gel ali) Maan easiest lesesed (16).
1...m
We can deduce in the same way as in § 2, the relation
AM ote mev ds Wioc=3 112) (Gp coe tn i Oe (17).
2...
In order to generalise (15) we recall that the “best fitting” linear function of
the variables a, 3, ... 2m to the mean o5 mi, 18
Oe ye ie oeay meee ong ew me (18),
on O71 G2 Oo; om
394 On the Partial Correlation Ratio
where by, = a and Ryq 1s the minor with its proper sign of the element in the pth
row and qth elamn of the determinant
= ecg rime
Fayre ec eee cere Pee:
i ry
lm Vme OOo Ih
while the maximum correlation between any linear function of a, 73, ... ®, and
TIS ast le, Were
R-R
—o3,..miy? = pe = bisa Ost is os ie Olin Tun ate oe (20),
ll
S {ny (@, — DB) (2, — %)} S {ns (a — Z,) (#3 — X,)}
= by ce a Dis 6 7
Noo, Noo;
8 {7m (@ — &) (2m — Lm)}
+... + Dig —— Neto = aa ee (21).
Subtract (21) from (17), noting that n.= S {rom}, etc., we obtain*
3...m
(, 3, eek FS iOn3 30. plate) No?
= = oO a i
= 8 {ae (%— oan | =f 2 |r = Bis (a, — X;) (a, a x)
P 2
2...
oO 3 ae
apiece Ar S {ran --. Dine (ay ae 7.) (Gan ro Zn)
Im om
as SENG oO — ae =e
== {No mM (&, = away +8 {n, ee Dis (a. = Lz) ae a x)| Tee
9 C2
2...m
m ne
oO pee = =
+8 \" a bin (Ca ee Lm) ( m& fad a)
= = Np o = 2 =
= 8 |» (2, oe 2...) + Nem 4 Dy (2, aaa 2) (2. mA ae ;) 7 gsi |
2... 2
2... 2
Oj —t
= 8 tm (Gone 2) jaan — a+ Drs = (2 — ae) Tee
m
Ge ee
a Dim es (Ge zm) eae eee (22),
using (18) this equation becomes
(s, 3, Ravel motores ana) No? > (Hen Cena at 2) (2, me — X1)}
2...
= 7/8) gen (om = DG) + 8 {fea an CG a X,) Goan aan X,)}
2...
2...m
* By an extension of the notation described at the beginning of this section S3., denotes a
summation with regard to the variables 73,14, ... 2,3 11,2,...m iS the frequency of a particular com-
bination of the characters w,, x2, ... x,, While ny, is the frequency of the combination 2, 2.
L. IssERuis 395
But S {ts on (X, a ZL) Cann =e X,)}
2. IN
( = ee. geen
Ly — Ly Xz — Vs er Ly — Ve
=S ie (- Dig ——— 0, — dy O—---) (2. my —Mt+ Dis O,+... (24),
o on o
2...m ( 2
and -Noyon%is= S {2m (@, —%;) (o—- )
1...
=S (eee (a2 = Hy) Cant aa X,)
2...
with similar values for 7,3, 7.3, ete.
.. the right hand side of (24)
= = by? yp — Dy? oP — 043012047123 « --
oe Gy Osis bs0 or To3 — bis Oe Bis Dr Cie vaee
= GOS (- Tro — Oya — Bis 05 — Oya t'n4g — « — Gina)
+ o;7b,s (= 113 — Dis — Diy Ig — «- )
Goh Gam - ~.nens seen nee eIAeER SIR CIMA CE Ie eRe Eero (25).
Each line in (25) is identically zero from the definition of the b’s and the properties
of the determinant in (19).
S {Ns mM @ m% a aa)
2 2 2...m
Hence payrelae = 2 ealagi => —— = WN Fe aii RIM 8s oie sks (ouslekele| oxsiejsie’s (26),
Orie
so that the fundamental properties proved by Professor Pearson in connection
with the correlation ratio 7, hold for the generalised H defined in this section.
In particular equation (26) shows that a necessary and sufficient condition for
linear regression in multiple correlation of m variables is that
O35 Pilla == pyr means
For in this case the mean value of any array of 2, will lie on the “best fitting ”
m-dimensional plane.
§ 4. The regression surface of z on xy being assumed of any particular type
the constants in the equation may be determined (i) by the method of least
squares, i.c. by making the sum of the squares of the deviations 2,,—@(xy) a
minimum, z= ¢(w#,y) being the regression surface, or (11) by giving such values
to the constants that the correlation between z and (a, y) shall be a maximum.
When ¢ (a, y) is of the second degree the two methods lead to identical equations
for the determination of the coefficients.
The same equations are also obtained if the surface be “fitted” to the means
by the method of moments. There is, however,a distinction to be observed. The
equation z= (a, y) when the regression surface is of specified degree contains a
definite number of constants and the first two methods will give exactly as many
independent equations as there are constants to be determined. The method of
moments will give as many equations as we please if sufficiently high moments
396 On the Partial Correlation Ratio
are used, including of course the equations given by the “least squares” method
or maximum correlation method. Even without introducing high moments, when
there are three variable characters new equations may be obtained by the method
of moments, by combinations of characters which do not arise in the other methods.
The method of moments is most convenient for our purpose, but we shall only
employ those equations which can also be justified by (say) the method of least
squares.
For convenience let the origin be taken at the mean of the three characters so
that2—y—2— 0) Jet atytze denote
SSS {Mayet y'2"} _ Pasty (27).
a ee
Noiiay Ge Ox Oy Cz
With this notation 7,, and ga, are identical; when z does not appear in the
product, it is sufficient to write @,s 4
The most reasonable next approximation to make when a linear function
(a, y) does not adequately represent the statistics is (a, y)=a quadratic
function of a, y.
a 2 2
Let ELI rg OEE a (28).
om Ox Gy Gxoy Oz Oy’
Multiply (28) by mz, sum for all values of w, y, z and divide by V
OS + Cry HOA SF 200. -2. sess dea near eeeeee (29).
’ : ales x De Z
Multiply (28) in turn by “ae times —, 7, ae ae x and sum as before,
ap IO i) Oa
and we obtain
Toy = 0 + OT yb CQnry Ap CGasit Pf nyte s0+efecioes seine nee enna (30),
Pyz SO Olay + COmp + Oxy Hii Qys seceese osese nee Gnee eee eter (31),
Qayz = Uxy + Aqary + OY aye + CYary + CYasy tI Yays veeceeceveseees (32),
Gare = A+ Agast+ gary + CYary + CQat Hf Qary? vrecerececrscecscneas (33),
Pye = AA Adaye + bq ys + COays + CYnry +S Qys -cescersensercereoees (34).
Actual numerical fitting shows that in many cases e and f are small compared
with c*. This is the case when the regression of z on « for a constant y and of
z on y for a constant « is linear. We shall therefore confine ourselves in the
present preliminary paper to the case where we may write
2 gi BY ee (35).
Oz Ox Cy OyGy
Here for constant # or y the regression of z on y or of z on @ is linear.
* Cf. Census of Scotland, 1911, Vol. 111. p. xuvi1. where Mr G. Rae obtains by moments the
regression of fertility on age of husband and wife. Let W=age of wife, H=age of husband, C=number
of children in completed marriages. He finds
C yy = 20°149493 — 0°555812W — 0173804 — 0:002846 W2 — 0:003494.H?2 + 0:012675 WH.
See also the paper by E. M. Elderton in the current part of this Journal, pp. 291—295.
L. IssERLIS 397
Equations (29) to (32) become, when the regression surface is given by (35),
ORICA CTir lia oe ME sia a i sremiebe cin ccs ara ie aheteers sie yee scsgiee (36),
Me MAO OI <IaICQ Bayh SAAR acide cing Sammarn imum netics tee’ (37),
fig, OS OMe Cis, a apeons chochaed ciaden Mace se aeeog sare (38),
Ynys = ON ny + BGary + OQmya + CQ ary? — szerereveecsteseerens (39),
= — OF xy + Gary + bgaye + Care by (36).
Solving these equations we obtain
a oe b ee
Wey ae. day Paz | ie | ary 1 Voz
1 Yay? Vyz Yay? Tay Tye |
aye § Jary2— Pry — Jaye | | Gury? — Tay Ya2y — Yacye |
Cc . 1
MMe te ol. a Gy ceases (aw):
Pry 1 Vyz Poy 1 Tay?
Gary = Yay? aye Gary Yay? Yay? — Tray
we have already denoted the partial regression coefficients by y; and y;' so that
_ Paz Vy2V ay + Tye Vaz ay
(i acm ae and y; eure rie (9) and (10).
In addition let VEO al) A ae (41),
1l—Pryy
and BeNLaE ies DOI Df ter oh ik gihee ct es (42).
Ll Pry,
After some reductions the determinants in (40) yield
CYNIC Umm eee Nett ee Aah MN otto ena te 1 hil ve (43),
b = Ys + ch sLbiaisiisteteraip/siiaretehareta'atalstanets seve) a seksi) eaactve rege cera ia tis yay fins (44),
Or z2 ar br yz F Yxyz
oS : Far ASE ae ee ee 45),
Bary + Pay + Qary — Tay (
also NC pa Sete Cte ee Me aa cag Giasiis Gusta ot (36).
Note that Vs Gary shits Qe N= NO ag AID Tay) eeosscswsodiscest sree (46),
and Ys Vza a Ys. Vey = pln: (cf. eqn. 11 Js
oy: S LY =F Y i
§5. By definition (1—-yzH?)= ue ee : Zmy)?} ;
oy
nin using (35)
ql H, ) =5 ea (= === ee = ug. CLY
ayttz
N \o, Cia aye SCRAORY
=1-2812 (d+ 44 art.) Pash
CO; Oy Cy Oxy N
++ 04+ 04 Cqgaye + 2dergy, + 2abray + 2acqry + 2beqzy,
Biometrika x 51
398 On the Partial Correlation Ratio
= gy? = — Arey, — Wry, — WAayz+ C+ B+ + Cqary2'
+ 2cdryy + 2abrey + 2acqary + WCqayr-..-+-+- (47),
= W(— Pag t A+ Orgy + CQa2y)
+b (= 1yz +0 + zy + Cay?)
+ (= Gayz + Ar zy + gary + bqays + C422)
+d (d+ cryy)
— ON 24 — Diyz = COpye «2-2 ee ee a ..(48).
The first four terms vanish by eqns. (36)...(39),
"sagy ll? = Oz +: Ola) HCOnye ees ee ee (49).
If we now insert the values of a, b, ¢ from (43)...(45) in (49)
ay, = (3+ CO) Pon + (ys + ch) r zy + CY xyz
= Ys" en + Ys Tye + 6 (Ore + Oye + Gayz)
(Or ze + PV yz + Vaye)
ary? + Qayeh + Garye — Mary’
(Gaye (1 = Pry) = Gay (ToeTay — Tyz) — Yaty (TyeTay — Tae)\ (30).
2 ey? + Qa — 2x2 27 %
| = Pig) i ay? + q a == y Vary | (1 — r2,y)
ay
== aylees ete
or yl a
It follows from (50) and (15) that
2 a2 % = 2 2 Ta.
a (51)
Yox2y2 — Vary >
If we eliminate q,,z (which is a triple moment troublesome to caleuiae)
between equations (45) and (49) we have !
ay? = OP zy + Dr yz + CGayz
= OP ag + Dr yz + C (Gary — May) + ACQary + beqaye
= Tox (3+ CO) + Tey (ys + Ch) + C (Gay2 — Tay)
+ (Y36 + 09) Yary + (Ys'¢ + CD) Qanye
=O [Gaye — Tay + Oqary + Gaye] + ¢ [Ore + Pry + Ys Gary + 9 dey]
+ Ys" ex + Ys Vey 7
= yl? + C [Gare — Pry + Oqary + $Yxy2] by (46) and (11),
— Pay t Fay — 2qery? Qty =.
ay Ht? — ayl? at | dev — Ty
tea
He RZ :
Hence C= $$ et ee (52).
2 Paty + Paye — 24ary2QaruT ay
Gary? — Pay — = Seeage
me ay
This value of c? is positive by (51).
Equation (52) shows that ,,H7=,,R? is a necessary condition for linear
regression, which we have already proved in equation (26).
L. IsskEr.is 399
The regression surface of z on a, y is, with the values we have now obtained
for the constants
BZ Vg — Tyel ny © © Vyz—Vax2V ay Y
as = = te
om 1l- Prey Ox Us rey ae
ar we el 2 ale: : {= ay? = Yary & e + Vay Yary — Cay Y — y
Ee ee EY et ee a = pee
4 7 Payt + Paty — 2Qa2y Yay?? ay pales ae ae 7
sys 1 ay Ye
1 a Tey (
(w7—%)(y—-Y)
+ — pS “ey piavetanallslasons (53)
Tiley
The terms in the first line give the ordinary regression plane. In most cases
the regression does not differ widely from linearity so that ,,H?—,R? is small.
§ 6. We must now get some idea of the relative magnitudes of qa, Jay, Ge2
and moments of lower order.
First with regard to g.., which is equal to Leis
: On Oy
aye S (Nyt? y) = S (Nz ix") (54)
Vary No? oy No? Oy Si aheieje<e/e\ere:e. 6 cise: e-0c8 Blexeie reece .
If the regression of y on w be linear
(1 7y ory]
Ox : =
Vary = S = Vyy VB,
No,20,
so that q,2, is zero if the regression is linear and the frequency of a symmetrical.
In fact 22 — 1m VB, = 0 is the same as Pearson’s criterion for linear regression given
by €=0 (Skew Correlation and Non-linear Regression, p. 30, Eqn. (1xix)).
We may obtain a good approximation for qx, by considering the regression of
y on « to be parabolic. This is a natural assumption to make if the regression
surfaces of w on z, y and y on z, be also of the hyperboloid type we are discussing
for the regression of z on a, ¥.
For with origin at mean we may write
@, Zia,
Bip eg IS de”
Ox Cy Ox Oyoz
Hence keeping y constant and summing for the 2’s
Bh Ss, WEI hyey
(on Oy Oe Gy Ozs
Zz y V yne — 7 ye (Y == i
But rye Pye" Yee Nymd Pye 2 _ y: ie a V By’ Shee) | x
Ge © By — By —1 (ey Cy |
* See Pearson: l.c. Eqn. (lxv) (where Y,, is a misprint for X,), and py”, 8; refer to the distribution
of z.
Px
400 On the Partial Correlation Ratio
“" is a quadratic expression in y if we remember that h being of order
Ox
Ci SS Wan ane oa peas
NE ay is of the same order as Vyn2—7yz*.
But the relation $2 (x? — 12y) — € =9 is satisfied when the regression of y on
# is parabolic t.
Here d,=8.—Pi—1, €=€— Tay VB, €= qayt.
aes Qaty — Ty VB, = Vang — Try V Bo — By — 1 veteeee eee eee eens (55).
Similarly Quy — Tay V BY = Vn =a V By By = 1... were (56).
The use of these approximations will save the direct calculation of gq, and
zy provided we can determine the signs to be attached to Vz,?—7'sy and
Vine — 1x. This is often easily done by inspection of the regression curve whose
a y: we — Tay | ae — &
= a5 —* a- -———1;8
oy : ox B.— By —1 lox" VB, ox 5
We can approximate to qax,2 as follows||:
SS (tay ay?) _ S {nea (o*y, + (Va — yh
da? = = :
avy Nox2o,2 NoZo,7 )
equation is
where Y; is the value given by the regression straight line and oy, x nz the
second moment of the array of y’s for a given x, about the point Vz.
S {n, 2 (Y2—Y)? ;
But . - i 2 _ 8 (matrtve, Vo
a Oy
= Saag . 7
S (Ny, 2? a7 yz)
Thus ee ae mae
Gary? Nox? oye +P Be vy >
Bhd S(n yo? )
Similar] gf a Zs
: Me Yaty Nozo/ Be Pay;
or, so far without approximation
ai 9 _<¢ Y 5 ¢
us = S(nz,a?o7y,.) + S (ny yxy)
i oe cy)
Ae 2Naeo,7
ap $ (By ote B,’) Preys
* It is noteworthy that the hypothesis that regression of z on a, y although of 2nd degree is such
that regression of z on x for a constant y is linear leads to the result that the total regression of z on x
is parabolic.
+ Pearson, l.c. p. 28, Eqn. (lxiii).
{ Pearson, l.c. Eqns. (li), (xlv) and (xiii).
§ Pearson, l.c. Eqn. (lxv).
|| It can be found fairly directly by tabling to the squares of the variates, when we need a simple
product moment. In a later part of this paper some comparisons of actual and approximate values for
numerical cases will be found.
L. Isseruis 401
Now the mean values of oy, and o?y, are known to be
Cy liter) -and,-o,° (L—73,,,),
and the deviations from these are usually somewhat irregular. Rarely can we
do anything better than assume them to vary with a slight linear variation from
the mean. For example
Oy, = Oy (1 —7°gy) (1 + Ax),
where A is small. In such a case
Were (1 — rg) (144.8)
or to a fair degree of approximation*, we may put
: Vx2y2 = 1 Tey ar 4 (B, ar (SEO) Pevy,
and thus write
Quy — May = 1+ %(B2+ Bs — 4) Pay
= 1+ y+ $(8,— 3+ Bo — 8) roy.
The latter part of this expression vanishes if the frequency of the # and y variates
be mesokurtic. It can of course be retained if desired but its product with
V wy? — ey RZ will usually be of the second order. If we write
v=t(Bo- 3+ B, — 3),
we find the approximate regression surface
Lim aa PyzVay (@ — @) ns Pyz — VazV ay We Y)
Oz 1- Tey Ox t= Tey Oy
/ se — ah? (w— 2) (y-Yy) PG (57)
Tee Was | os = ae :
This equation (57) enables us to express approximately the multiple ,,H, in
terms of the simple 7, 292, xMys yNx-
+
§7. To obtain this connection between the multiple ,,H, and the simple 7’s
we may proceed as follows :
OF
Zay =d f= ae
, origin at mean,
oz ox Gg y Oy
Hence keeping y constant and summing for the «’s
“Ud 4 a — +2 uf ce ae ee ee (58).
z & y FnOy
» _ SSS {ney (2) — 2) les SSS (ayz 27) «¢
ue Luar aon No? No? ie
SSS {Nayz (Zey — 2)? — SSS {Nay2?
pee Yy. y fend aye” oi
ead ay He = No? BeUNag -~
* T.e. we are neglecting terms of the second order as AV Bi.
402 On the Partial Correlation Ratio
page ls — yn? = SSs ka (“ (wy) | cy (a — =») (2d+a @ tty | 4 by
ae ox oe oy oy Cy
| Gyan
= 5 (22) (a+ 2) [2 (a4 on) (24%)
= ss |" (FS Ox #) (a+) 2 d+) + (a2) Ox ) .
by By 0, — 2,2 CY\? Nay)
99 a ie ay y Cy Nay
s {(a+ ale jo Ses Ox ta + as |e On? (a+) N}
by CY\" | Nay LU — X,?
=0+8 \(a+ 2) (s Neorg )t.
Now S {Naya} = 8 {(w — dy + Ly)? Noy} = ny (on + 0+ 2,7),
x we
a : CY \7 Ox,? N,
ay H? — yn =8 (a+ “f) at :
On
ns fo+ 70-9
y
= ee + 2et 0),
Cy oO,
ay? — ne = (1 = yn2) (EO)... eee (59).
Similarly eye ee = (8 —27,))(0F aC) nee eee (60).
Remembering that a= y;+c0, b=+¥;' +ch we get from (59) and (60)
a ra) be ay? — Ne
woe See ee : SF le ¥5%3 (sh — ys 9)
1- Qa
; tiGe {ys o— 39 — Ob (ya = Ys, 0)} AERC (61).
From the values of ys, y;, 9, @ in (9), (10), (41), (42) we obtain easily
yzYary — Vaz Vay?
/ yz 4ary az Yay
Yah — ys = "een Teeter
Seay
/ Vaz Q x2 oan Tye Jay?
y3 6-730 =
1 — 1 xy
Ya — 29 — Of (ys — ys 9)
— PaeQary — TyeGiy? (Txy Yay? — Yary) (Cay Jary — Yay?) Cyz ary — Vaz Gay?)
LP ey (1 — rey)?
and c? is given by (52). Hence (61) may be written
(Qe —Vxz Try) (Tey Vay? — Yury) (7 az — Ty2? ay) @ wy Yay? — Ya? y)
x lai x R? = x = ~
ie es (1 = xy)? CL = ya") (1 — ray)? (Ll — amy’)
(TxzGary = Tyzay ) ne (Vey ary? = Qa? 2y) (? "ary Yay y — Yay? Cae, = "ae Gey))
6 tly (L= 722) f
2 24 ae ay? —s 2 wy2Y a2 Vg )
(Gu22 — Tay) — CZ yt GF a = wary Vay |
ay
L. ISSERLIS 403
mau yz — “a2: ey) (7 xy Vary ane dynes CG wz — Tye ‘ny) (Vay Vay? — = x2 a) ane
(1 ile oe (1 — yNax’) (_l— Trey) d ort y N22)
(122 — TyzTay) (yz — Vr2V ay) (Tyz x2y — 22 J ny?)
( es Tr)
pis (re = Vaz" ny) Pay ary — Ixy) xz —Py2"ny) Cry Yow? — dev} (62)
avis a ; aa eee i
(1 — rx)? (1 = yn2?) (1 — ray)? (1 — any’)
<In general the square of any correlation ratio (ordinary or generalised) differs
little from the square of the corresponding correlation coefficient; also we have
seen in (55) that an approximate value for q,2, 1s
Vey VB, — Vem; = Tey VB. om fen —l.
Now 8, is itself in general small, so that without making any assumption as to
the relative order of magnitude of 6, and ,»,?— 12, we may safely treat ¢2,, and
Yay: aS small quantities.
+
Thus it appears that (62) is an equation in which all the terms involved are
small, and a certain amount of care is required in deducing from it an approxi-
mation to the value of »,H— »,R,.
We have
1 ms 1 y Ne — T ny
fe 9 = i 2 7 ) ar sa
Lyn” LP gy nc —yn2)(1— Tey) 1 — Pgy Si Say (63)
1 1 1 “Ny? =e te wl. = g/ releist er >
1 ays L— 1 yy eed) May ;
yz ee. ( 1 ) 2 2 ( 1 i
ay ee Smee eat yaar =r ar 1 > 4 2 Ar
lie ya WMI May 1 (ume a) ie rey &)
oy at ha SE Aan re a (64),
Lips
where 2, = Tay (ya! oe ay) ye ~My and d= (yn? —T ‘ew) (y Ma" zs ny)
G2) Gary) | lr, (1 — ynz’) (1 =1%xy)
: ane oy / , Seay
while = Sire alae vay ats So Seat ease reine (64,
where ),’, A. may be Agere on Ai, A» by an interchange of # and y in the
suffixes.
1 1
Finally ~ ee ate antonnes (65),
Quy? Fay t Paty — 24ay2 2x2 ihe. ~ ee ye r xy
CAE xy 2
L— Prey
where
Ko = Pay + Prey = 24ay aryl xy
Fay? t+ Pay = ‘Qaey? Ta? y “ay) zis
a2
pal
The suffixes of &,, &/,-’,, Xo, Ay, Xo’ and «, denote the order of smallness of
these terms.
(1 77 5))(Qa242 — Tny) (dur eee
404 On the Partial Correlation Ratio
We make the corresponding substitutions in equation (62), including the value
Tee hale ee 27 eV yx Manz Se
ay
age R? on the right-hand side and obtain the following
et Tey
accurate formula for ,,H?—,,R2:
¢ ; (7 "ye — Vaz i) © “ay Vary Yay2) G 1 )
2 2 ve \" vy x en
(ay Jal z wy R, ) | qd Tay 1—r &
(xz yz s Gi ay a ary) (= ti)
(1 cay Pe De an
ie) {ste ~VyzFay? (Tyz Qa2y — Vaz Jay? 2) (7, ay Jay? — Ya? y) @ "ey Vary — Gy)|
The ray (l-r wan) )
1
————
Qaty? — ay
= (Tyz i? NazT ey) (Try Qaty ay?) ( re + Ny + 7)
(1 = rxy)? a7,
ne meine! yeVay) (Pry Jay? — x2 y) ( ers , ‘
(1-7) ee
nae yz Jat ira ne Voesj2) (7. ez Speier) (Giip = ieee)
(_-r a
— ye tae — Bsa ye ay | ce = ae ey) (Tay Ja2y ~ Gov) ( oe )
(= 7a) (Lr)? barges
(Taz — TyzT xy) (Tay Try? — uty) ( Se eae ‘| "
v6 (l—7,,)° las + EY lili ees (66).
The right-hand side of (66) apparently contains terms of the first order, while
on the left the lowest order occurring is the second. But the coefficient of qz2, in
the first order terms is
1
= yy ae aa Vee lacy) TayT zy ar (Cie — Pyzt 7) Tog + Lyz (Px oar Prehiey) (Gere a Toe Tay)
xy
1 :
ee (1— rzy)! {yz —Vxz ie) Vey + (Taz — Lyz Tny) Gz + Px — Papel Piha)
— Pry
and vanishes identically. Similarly the coefficient of qg,y: is zero and thus
equation (66) is a relation between terms of second and higher orders.
A first approximation then may be obtained by equating second order terms.
This gives
(ay H 2 ~~ ay
R2) (Tyz— Paz xy) (Vay Yary — Yay?) — (Vez — Vyz" ay) ("xy Qry? — Yay)
3 (1 = 7x)?
— Paz Qary — Vyz ay? |
(1 = ray) (Gaye — May)
_ (Vyz = Vaz xy) (Pay Very — Yay’) E yz + ae — = 2x27 yzV ry iB |
= al a Tay)? 1
oy (ae an VyzV ay) Gra Gav aa ary) [a fol r yet T 2% — 2Px2VyzT ny é| (67)
(1 = ,,? E A hens AA eat ore ;
Ta
L. ISSERLIS 405
The coefficient of ,,H,—,R,/ on the left reduces to
1 1-7,
( — ry) [Qu ylaz — Yay? 2Tyz| (1- cs i
dhe ya Pony
Pye tT a2 — 20 x2P yzP oy = Ne — Tay é (TyzV xy = Vee)”
ae Uae Ee
—
1-7, Lr ay l= Pp
a yNe — 3 ”, aii (ya — Tiny) (fer ie =) (68)
Bee (1 y 1 — yn Teter oes Gane ,
Similarly
a tape ye a De ce Nye see (Ue ae TNC
eee uz wel yz! wy er _ ale za wily z( pesca ) me
1 it os Tn) E, 1 —/ a il oof Tye 1 a fy (69).
We shall still be correct to second order terms if when using (68) and (69) in
(67) we replace 1 —,,y,? and 1— any occurring in denominators by 1 —7%,,; so
that
ae
(oy He — sy 2) oe
ey
w-¥*
Yr, ay V2 yo Yay? Lr Ye — Vyz21. xy 9 9 Vyz Vay — Vaz 9 9
= z ; = : 5D Gne == T*2y) a Ti = Fr Gar = Te)
ay E
Qa2yVaz — Yay?! a 7 ay
Vay Vay? — Ya2ry Ve, = VyeT xy 2 3 YoeVay ~ Tyz .
a (Car i Tt) a 1— 7, Ment = Ta)
Yay?’ yz ~ Yx2y Vaz 1 — Pay
§ 8. This result is of importance, as it shows that the heavy labour of the
direct calculation of the generalised correlation ratio can be replaced by the
calculation of four simple correlation ratios.
The coefficients involved are the ordinary coefficients of linear regression
denoted above by y;, y; and expressions involving product moments of orders
3 and 4, To these latter we may approximate by the methods of § 6
Qaryx- 1. Array
A good approximation for —“*—— 1s _ . Ifgreater accuracy be needed
Gary? — May + py
Jury — 1 aa (4(A.+ Bo - 6) + ZT ay
Gury — Pry 1 +1 ay +4 (Bs + Bs — 6) Pay
We saw in § 6 (equation 55) that
Vaty = Vay VB, + VN Gp= a ay VB.—Bi—1 oS )
Yay? = Vay vB an Nae Tey v By — 6-1,
approximately. 8, and @,’, which are zero in normal correlation, will in general
be very small compared with gn’? —72y and yn? — 7xy so that
Voy Vary ~ Yay? be) Vey V (Be zm 1) (cy? = Ty) a V (By —1) G Ne — ry)
QaryVaz — Yxy?"yz Tzz V (Be — 1) @Iy? = Pay) — Tyz V (Be —1) (yn? — xy)
we may use
and
Voy Vay? — Yary _ Vay V(By = OGie = fay) = V(Bs —1) (7,7 — Poy)
Gay? yz — Quty lz Vyz Vv (22 Be —1) Gis Toy) V(Bo —INiGny Tv ee
Biometrika x 52
406 On the Partial Correlation Ratio
In the important case 8, = 8,/=0 and ,.=2n,=Te, these approximations
fail, and so does the process by which (70) was obtained as @ and ¢ vanish and
(61) is indeterminate.
We must then fall back on equation (59)
xy H?—- ye = (a? + c*) (1 — yz”)
=(ys +c’) (1 —1xy) since 0@=0 and 9, = Tay
= 3 (1 — Mey) + = (1 + #3),
Gary? } ar y
1-7,
cy
2
Ya2ry? — Tay
or Cay He TF wy Te?) € <a ) = yNe 7 ay Jay =e ys" (1 >a iea))
neglecting 38rd order terms.
The right-hand side reduces to , — 7?,,, so that
9 2 ¢ BE i: 4 9 9
ae GS) (ne =1)) ine (71)
Yxty2 — 1
PA 9 2 bs
ae aaa (yn? — 1°,,) approximately ;
xy
of course in these circumstances (60) would lead to the value
29, ) a 5 5
ee = TF cet ean PeR PROBA ARIA on. Jas0005000%0 (72),
ay
showing that if 8, = 8, =0 and if (,.7,? — 7,7) — (yn2? — Mxy) is of higher order than
the first then (,7.2— 1.2) — (,n2 — 7°.,) is also of higher order than the first.
We shall now seek relations between the six correlation ratios of three
)
“hyperbolic” variates.
From (59) and (60) we get, on eliminating ,,, H/,
ye — af =P > COLO Une — 29) Heyes — 0 sn,
= 3? — ys + ys ye? — Ysa Ny” +2 (ys9 y nx = Ys Px’)
+0 (0 — 6 + ya? — any + Pyne — Fany?)
= (5? — ys") (L= ay) + Ys" (ya —Pny) — Ya 2 (ey? — Tay)
— 2¢ (79 — yb) (1 — xy)
+ {209738 (y nx? — xy) — 2erys'h (omy? — Tay)
+O (P — b+ ye = aty + Pyne — Px Ny’)
The terms in the second line are second order terms. Neglecting these and
noting that
(Ys? — ys?) (L = Pay) = yz — Mae and (39 — ys'b) (1 — ray) = Pye Gay? — Tez Qary,
we obtain the following equation for c:
(yne — Tey) — (ane? — Mex) — Ys? (ya? — Tray) + Ys? (ey? — Tay)
=D Vary — Tye Vay) eee (73).
L. Isseruis 407
24/2 P i
Now (eH? - mR 2) fee 2c?7*,, 1f we neglect second order terms. If
ary? a ey
we use the value of c given by (73) in (70), and adopt the notation ,U, for
ane — Vx, We have
Tvy ty Up: 2 U, rae Ys y OF aF pee Gi
= 2 (TuzQery — Py Gay?) (Pay Gary — Gey) Ys (ye — Ys? y Ur)
— (Pry Gey? — Yury) Ys (@Uz — Y2?2Uy)} 0... (74).
Let us write »x, for ,U,—;?,Uy and yx, for ,U, — y3°,Uz, then (74) becomes
(pe =e apeay Tey = 2 Gee Gy — yz Jay?) {rys. Cary oF Joy?) yXz— Ys Cosy Yay? — Yury) aXet
(75) is a relation between second order terms and it is sufficient to use
equation (55) for qaz, with 8, replaced by zero and B, by 3, so that
arty = "2,0 y,
ay? = V2,Uz,
(X2— y Xz)? Tay = 4 (Taz VUy eye VU) Ls y Xz (Tay VpUy i VU 2)
= Wane Vay Vp Um— Nay) Weoeeon (76).
This identity between »7-, yz, xn, and ,7, is symmetrical in w, y. Two more
such identities may be obtained by interchanging the letters «, y, z in cyclic order*,
There are therefore three identities between the six correlation ratios :
yNa>. «Myr, yz. zNyr az, za
I have not so far succeeded in reducing them to simpler forms, although possibly
such exist. In special cases simplifications result. These are illustrated in the
following section.
§9. We defined y;, y,;' the regression coefficients of z on w, y by the equations
Vaz — VyzVay 1 Tyz — VazVay
(eer Ch te LE (10).
a lay
ee diaellv 1 Vee ~ Vy yz
Dae Pe Se le a a ae
Pye SE (77)
_ Vyz — Vaz ay 1 Vay — ye" ee
Dee 76 2 nk amare 2
— 1" xg 1-7,
It will simplify the algebra if we use X”, p?, v°, XN, w, v? for yUy, -U,, 2U2, Ue,
Buy. je respectively and P,Q, RP”, QR’ for yes eXys aXe aXe» aXu» vXe
jeapectively so that
P= - ye”, Q= fe Ya? W?, R=r—- YH)
Dee Ne yy? oe, Q’ an Me es yo? y?, Rea nyt? J Siesietase seats
* We are supposing here that the regression surfaces of # on y, z and of y on z, x are also hyper-
boloids of type similar to (35).
52—2
408 On the Partial Correlation Ratio
The three identities connecting the six simple 7’s become in this notation
(R— RB’) ray = 4 (rz f= Pyzd) {ys BY (ray pw! — 2) = aR (Tyr — bY} --(79),
(P= PP rye = 4 (ye ¥! = Tex) (yy P" (Pye ¥ = Bb) — HP (Tyee — v’Y} ---(80),
(Q a Or =4 (12 — Vy V) {2 Q GaN —V) 9, Q Ca = ny} ...(81).
(i) We can deduce from (79) that if the correlations of both # and y on z be
linear and equal, then the correlation ratio of 2 on y and y on @, i.e. ny and yz
are equal. Thus in biparental correlation, if the regression of the child on each
parent be linear, then the correlation ratio of the father on the mother is equal to
that of the mother on the father.
For under the conditions stated
az = yNz = Vea = Vey
Hence y,= 3 =- Je and (79) becomes
1 ae Voy
st DP = pee P rey = Arye (Mo A) [= 8A? (Pay — A) + YH? (Tay r — BY},
oe 3% — BYP A+ BYP Pay = Ary (X= WY {May Apel — 2? — Dye! — 7h,
Us (A — bw’)? a (A+ WP — 4 (ray rp! — Ape’ — DV? — we?) | = 0,
ey
which reduces to
No RG +2?A+(2—717%,,) pi ti 4” (27 ay + 3) (Tay + aa
A-pwy ~ — : ; : = 0.
( i) | (es ap i) (Tay ap 2)?
But rzy is numerically <1. .*. the factor in curved brackets is positive. Hence
Niles Ope Cee eT
(ii) An interesting deduction from the identities (79)—(81) is the following:
“Tf any four of the six regression lines that occur in the mutual variation of three
variables are linear, so are the other two.”
We have to prove that if any four of the six quantities ), w, v, X, mw’, v’ vanish,
then the remaining two vanish as well.
Hirst let 4p = — Ae 10;
(79) gives Toy? (Ys fe? + VP = Are w [Ye VEN ny ft — Yas 1°},
(80) NV Ty? = — ray nn”,
(81) bg =
From (80) yy [Aes Pye + yr 27y27] = 0.
But Ary, Vay ar wees = (ee ee — at) >0.
Wy =),
and 0)
and these satisfy (79).
There are three cases of this type.
L. ISseriis 409
The case X’= p’ = vr’ =X = 0 is proved in the same way. There are three cases
of this type as well.
Next take the three cases of type
N=p=0,
N= 0.
Equations (79)—(81) become
(79) i — se se ep yr.
(80) Ny ry? = ey’ (yyy? (— vf,
(81) ys”? = 4(—Tryv) {— Yeo y2V? (— vd},
(80) leads to (ry By Array eya) yt = 0.
We have already shown that the first factor is positive,
pv =0,
and hence v= 0,
and these values satisfy (81).
The three cases of type X == pw’ = v'=0 lead to
(79) yt 7, = 0,
(80) Ny, = 0,
(81). (ye? v? = y2?X?) Pye? = A (Pye — Pryv) [= 2 Yo? (Tae =v) + 222A? (Trev — VY},
whence \’ =v=0.
There remain the three cases of type
N—i— 0)
(0.
Here (79) is satisfied identically.
(80) becomes OM? = YP YP Pye = — Aeryzps (ya (A? — 1?) (= wh,
(81) (WP = YP A?P Tae = Aryzd" [= Yo (Mw? = yo ®”) (= VI,
which reduce to
(A? = 2?) [Pyed? = (WT yz + Ae’ Taz) w} = 0,
(w? — 2? d’?) et — (279 2 + Ayo) "yz) 7} es
The only common solution of these equations is
0.
We have thus accounted for all the fifteen possible cases.
(iii) Three regression curves linear.
In six cases out of the possible 20 cases the linearity of three only of the
regression curves involves the linearity of the remaining three.
410 On the Partial Correlation Ratio
Let X= p= 0 and either v or v’ = 0.
It follows from (79) that R= RB’, ie. v= v’.
“. both v and v’ are zero. We have now four linear regression curves, .’.
six are linear.
Le p=v =0 and p =—0:
Since; = v= 0, 3) P= or N= eesontliat
P= — yee NeSore,
P22 = per — yo Ren
(79) becomes (v? + 932A?) ey = 40 yes? (Tay? — Ys"¥s V2),
(80) becomes
(ye? v? — rye? MP)? Pye = A (Tyzd — Pry V) Yayo ¥? [(Yov — Ya’) + Taz (q2'V — Y2X)]-
The first reduces to
{x Val Cee + 1 yP xy) Tey — : 2ryz
Ven == (0)
Se) a we
and the second to
(av? — yo! rN)? Ie Ayarye, (Mz — Tey vy) v? = 0,
all
Hence either 7}=v=0 or there must be a very special relation between
Vays Vyzs Vee
If instead of uw’ =0 we take v= 0 we get similar results, ie. in general the
vanishing yu, v’, w’ or w, v’, v involves that of X, v, ’ or A, pw’, V.
This accounts for six more cases.
There are eight left. Of these six are typified by
fp =7=0=y 50m ut
and lead to the same conclusions.
The remaining two are
N= — 7 — 0 Or Ay
The first supposition, \=~=v=0 gives
Pa=-yv?, Q=— 9°, R= yp,
P=nr2, One RR’ = p?,
leading to
(79) (vps My? Tye = bres Wye! (Pay? — Ys 7},
(80) 2+ yy?! 3 y, = Atay VO [Tyed? =v? |
(81) (2 + yg? A)? 729, = Ary Neyo" {Taz fl'? —. Yas NJ,
which give A’ =p’ =v'=0 or a very special condition to, he satisfied by the
correlation coefficients ry, Tyz, Tex:
L. ISSERLIS All
We may conclude then that in general the linearity of any three of the six
regression lines involves that of the remaining three.
(iv) If the regression surface of z on #, y reduces to a plane, the regression
curves of w# on y and y on & reduce to straight lines.
We have as in § 7
z, Goya) Gj
LU eg gph OEE Ne als, (58).
on: Ty, Cy OxFy
@, 4 ya Pay ie > Y
But Pe ear EEE EEA CHa AB YC ces if,
ox aca “ Bi Bat oye Ce
By d+a ie here ee Date, (av B, +¢) Vee Me — peal Ped
Oz oe B, =e oy é ne -8, =
Op = B (yt? — 9" UR a), ies Poy “}
cemeg WC aye C aye ae +a SET
oy (( i B2— 8-1 Pris
+ terms
Now the regression
coefficient of y? is zero.
and thus
Sunilarly
But c¢ vanishes when
Hence if
it follows that
of higher order.
of z on y for a constant « is linear, Therefore the
To first order terms we may put 8,=0 and £,=3,
aaa
CV ey ats a ee = (0,
ae — x
Clay £ b ne i 9 =f =0,
ga hee = lags by (52).
pal = uy Bi,
“Ny = yNa = Vay:
We thus see that if the three generalised correlation ratios »,H,, yzHx, xH,
are equal to »,fz, yx, zx, respectively, the six correlation ratios yn, Nz,
“Mz, 2x» ys yz reduce to the corresponding correlation coefficients xy, Tz, Tyz
and that the “linearity ”
the six regression lines.
of the three regression surfaces involves the linearity of
MISCELLANEA.
I. On Spurious Values of Intra-class Correlation Coefficients arising
from Disorderly Differentiation within the Classes.
By J. ARTHUR HARRIS, PH.D. Carnegie Institution of Washington, U.S.A.
WHEN the constants of the and y characters of the population in 7, are quite indistinguish-
able symmetrical tables* may be used, but not otherwise.
Primarily and for the most part, however, the use of symmetrical tables has been restricted
to cases in which the degree of interdependence between the measures of all possible pairst drawn
from a considerable series of associated individuals—in short to intra-class correlations {—is
sought.
The dangers of spurious correlation due to the artificial symmetry of the surface is then much
greater §. Pearson|| long ago pointed out that when intra-class differentiation exists, for example,
because of age in the case of characters determined upon the members of a fraternity, or of posi-
tion on the axis in the case of serial organs, the values of 7 may be to some extent spurious.
In the cases considered by Pearson differentiation is an orderly phenomenon, i.e. the magni-
tudes under consideration increase or decrease with age, position on the axis, or some other
extrinsic characteristic with such regularity that the relationship can be expressed by an
equation which may be used in correcting the raw values of 7.
In other cases, the problem is not so simple. Ditferentiation within the class may exist, but
it may be difficult or impossible to arrange the individual measurements by any character outside
of themselves to obtain the constants necessary for determining the true correlations from the
spurious values deduced from the tables.
Illustration [. The correlation between yields of wheat in variety, testing.
In variety testing, the experimenter seeks (or should seek), among other things, to determine
the correlation between yields of varieties in different years. If this correlation be 0 (and regres-
sion be linear) it is clear that the yield of a variety in one year furnishes no basis for prediction
* R. Pearl, Biometrika, Vol. v. pp. 249—297, 1907; H. S. Jennings, Journ. Exp. Zool. Vol. xt.
pp. 1—134, 1911 ; J. Arthur Harris, Biometrika, Vol. vit. pp. 325—328, 1910.
+ K. Pearson and others, Phil. Trans., A, Vol. cxcvu. pp. 285—379, 1901; K. Pearson and
A. Barrington, Eugenics Laboratory Memoirs, No. V, 1909.
+ Biometrika, Vol. 1x. pp. 446—472, 1913.
§ With only one pair of measures the probability of spurious correlation is, in cautious work, very
slight, for the possibility of differentiation can be easily tested by the critical comparison of the physical
constants,
|| Pearson, K., “On Homotyposis in Homologous but Differentiated Organs.” Roy. Soc. Proc.
Vol, LxxI. pp. 288—313, 1903.
Miscellanea 4138
concerning its yield in any subsequent year. If, on the other hand, the correlation be high,
prediction from a few years’ test may be made with great probability of certainty.
Given a measure of the “performance” of a series of varieties during a number of years it
would at first seem quite allowable to form symmetrical tables or to use the intra-class formulae
of a former paper* to determine the intra-varietal correlation, and to regard this as a satisfactory
measure of the differentiation of the varieties and of the average prediction value of a year’s test.
Such is, however, not the case, for while there may be no orderly change in yield throughout the
period under consideration, the individual years differ greatly in their average yield for all the
varieties. The influence of this “disorderly differentiation” upon r is admirably shown by
A. D, Hall’s+ table of the yield in bushels of wheat in the Rothamsted experiments.
Let b=yield in bushels per acre of any one of m varieties in any one of n years, 71, Y2 be the
“first” and the “second” years of a symmetrical intra-varietal correlation surface, v,, v, be the
“first” and ‘second” varieties of a symmetrical intra-annual correlation surface. Then Toy, oyy
will be a (spurious) measure of the (persistent) differentiation of varieties, Ty byg? 2 (spurious)
measure of the differentiation (in the yield of all the varieties) of years. Applying formulae
(v)—(ix) of Biometrika, Vol. 1x. p. 450, to these data, I find
S[n (n—1)]= 2128,
S[(n—1)3 (b')]=83122°5, S[(n—1) 5 (b2)] =3483626°4,
S [5 (b') P=3610204:57, S [5 (b)]=370820°13,
6=39:0618, 0)? =111°257328,
"by, Py = — 032.
The result is obviously spurious, for mere inspection of the entries in the table shows that
some varieties regularly give heavier yields than others. The source of the spurious value is to
be seen in the fact that an intra-class coefficient has been calculated from a symmetrical surface
formed from classes (varieties) represented by a series of yields differentiated by annual variations
in the growing conditions. By correcting for this source of differentiation by expressing each
yield as a deviation from the mean yield of all the varieties for the particular year, i.e. b”’=b—b,,
where the bar denotes a mean and the subscript y that it is for all the yields of a year, I have
found ¢
"by, byo= 266.
Measuring the differentiation of years in terms of intra-annual correlation (intra-class correla-
tion in which each class is defined by the year and its individuals are the yields of the different
varieties grown), I find from Hall’s table
S[m (m— 1)]=4440,
S[(m—1) 3 (b)]=174129-2, S$ [(m —1) 3 (b)] =7317531:92,
S[s (b) P=7586436'21, S[> (b)]=370820°13,
b=39-2183, 042=110-017719,
Ty, by = "791.
Since the varieties have been shown to be differentiated, this result must also be spurious.
Let b”=b—6, where the v indicates that the mean denoted by the bar is for the yield of the
* Biometrika, Vol. 1x. pp. 446—472, 1913.
+ Hall, A. D., The Book of the Rothamsted Experiments, p. 66, 1905.
+ Science, N. S. Vol. xxxvi. pp. 318—320, 1912. Probably a better method of dealing with such
cases will sometime be found. So far I have not succeeded.
Biometrika x 53
Miscellanea
414
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69-G089I | 6-SIh | IL | Ltr | 0.18 | 0.06 | 88h | G28 | 9-9F | L-9e | @&1¢ | O28 | Ber | ZTE | * °° (eqtG AA) eq ATOOM
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GG-OLILL | &-617 | IL | 62> | LbS | Sl | 8-Lb | 0-8b | Gch | 0-6€ | 1-29 | G-ze | L-2F 2 ay M sAasep
LE-TIEL. | &-b3h | IL | & Lh | 9-61 | 0.Fo | &-6P |6-0F | L-6€ | G-8e | I-I¢ |¢-8e | ¢-0F a (poy) 3ormorg
O€-8P80L | 0.-€46 | 2 — |01> | 8er | &re | 8-8F | &-Ge | 0-Le (OMA) BEYO poy
OZ-1618l | v-rEF | IL | 6-Gh | 3-83 | 0.36 | L-2¢ | 91h | or | GEE | LS¢ | L-Le | 8-eF "7 dapuoM poy
€v-OPZ8T | €-Geh | IL | Z-9F | 9-08 | 9-16 | 1-69 | 1-Fr | Sr | 9-1e | ree | G.6e | F-FF “pay JoIsig,
99-0€I8T | 8.Geh | Il | &8h | 9.86 | BGs | 8.09 |63r | Ger | 6Fe | Leo | Le | 8-eF ee wavysuey pay
O€-F8LLT | 9-007 | OL |. 8-Sh | #86 |.¢8 | 0-49 | F-9F | 0.0F | FLE | BES | &-9F a “7 *** MO0ISOY POY
00-L66LT | 9-€0F | OT | 9-Sh | %-B | ¢-16 | 0-79 | Lor | 0-FF | BEE | 9-6F | 0.ZF | ¢.9F — |" (071M) eseopreyy
€6-89r6l | o.pay | IL | $-9F | 97S | OTE | 8BG | BF | F1h | Beh | 18h | Soh | Bsr | ose | + (pay) oprorg sejog
86-88018 | 8FO7 | IL | 80¢ | 6-8T | O18 | 82d | o-6F | HBF | 186 | BIG | Br | B6r | G68 | SHeTTeH (pey) dorq uaploy
6E-S19GS | 6-9LF | IL | Feb | 7-91 | GES | 0-19 | 96h | 94h | 9-9h | 9-69 | Gly | 8.67 | O98 | °° ** (poy) yeoumA QnID
B6-E4881 | Z-86¢ | 6 G-bG | 1-86 | 82 | 0-69 | 8h | ¢-6F | OF | 1-9 | 9.07 = om * (pay) Beyo ey M
6E-E9Z1Z | SIF | 6 GSS | F-2S | O91 | 1-99 | 9.6F | Ger | o-8F | 0-29 | L-8F = cat ee sess (pay) qoary
a : | Feces ]
(A) z (4)< | story | Test | OssT | 64ST | SLST | LUST | QLET | Gost | FLeT | exeT | sLeT | TZST | Aqarre a
‘supa x quaiafuy ur qoayy, fo sayaruny fo pjargx
IT ATaviL
Miscellanea 415
variety for all the years it was grown. Correcting for the influence of the differentiation of
varieties in this way I have found*
= Ooi
Pett Ug
Thus season is a far more important factor than variety in determining an individual yield.
Ittustration II. Influence of Personal Equation upon the Correlation between the Grades
assigned to the Same Paper by a Series of Instructors.
Stripped of the verbiage in which it has been clothed in discussions among pedagogues, one
of the chief problems concerning the reliability of the grades assigned in examinations resolves
itself unto the statistical question: What is the correlation between the grades assigned to the
same paper by different instructors ?
Let g be the grade assigned to any one of m papers by any one of 2 instructors, let 7,, ¢ be
the “first” and “second” instructor (of a symmetrical intra-class table) passing judgment
upon a paper, ~1, ~2 the “first” and “second” paper graded by the same instructor. Then
from Table I of D. Starch+ I deduce, by the intra-class formulae (v)—(lx) of Biometrika, Vol. 1x.
p. 450,
‘O71.
= '659,
Vo. gy /E =
ij Vig 9p "no
By using the deviation method as illustrated above, I have found
P94 !ig = TB2 Pol'y Ol pg = B86:
TABLE II.
Grades of Papers Assigned by Various Instructors.
Instructors.
pe
8G aes eae | 80
|
|
Papers.
SDWNA AW Co MH
Both of these results, in which an attempt was made to correct for the personal equation of the
instructors in determining the correlation between the estimates of different instructors on the
same paper, or to correct for the differences in merit of the papers in testing the individuality of
the instructors, are higher than the raw values given above, which are clearly spurious. Similar
results { are obtained from Jacoby’s astronomical grades§.
* Science, loc. cit.
+ Science, N. S. Vol. xxxviu. p. 630, 1913.
+ Personally, I can attach little pedagogical significance to series as short as those of either Starch
or Jacoby. They serve here as illustrations of method merely because I know of no more extensive
series.
§ Science, N. S. Vol. xxxr. p. 819, 1910.
53—2
416 Miscellanea
The essentials of this note may be summarized as follows:
In using Intra-class coefficients care must be taken to guard against spurious values arising
through differentiation among the individuals of the class.
Besides the orderly differentiation (due to age of individuals, position of organs on axis, etc.)
for which Pearson has determined corrective formulae in terms of correlation coefficients, a
disorderly differentiation for which such corrective formulae have not as yet been found some-
times obtains. Illustrations of such cases are here given.
Probably the empirical methods used here in correcting for this disorderly differentiation
should be replaced by formulae with a sounder theoretical foundation. This I have not as yet
been able to do.
The purpose of this note will have been served if it directs attention to a source of danger
which may sometimes be encountered in the use of serviceable formulae, and indicates a method
by which in the absence of more perfect methods practical results may be secured.
CoLtp Spring Harpor, N.Y.
February 3, 1914.
II. On an Extension of the Method of Correlation by Grades
or Ranks.
By KARL PEARSON, F.R.S.
In a memoir published in 1907* I have shown how, on the hypothesis of normal distribution,
the true correlation of variates » may be ascertained from the correlation p of grades. If
g, and gz be the two grades, v, and v2 the corresponding ranks, # and y the corresponding variates
_ with means # and y, and standard-deviations o, and o2, while
2 Ore 2
NV 1 tialR- a)
2
= C102 02
270102 vies
is the normal frequency surface of the variates, then
Fe LG Ney 2-1
N=3N=%H, Fg) =F =7:N,
ei Laas
1 —-R=1=>F— | e° 91" da,
N20, / 0
oy, = 02x, = 7b (W2—1).
Further I showed in the memoir just cited that
y=2 sin (F ?)
* «On Further Methods of Determining Correlation,” Drapers’ Company Research Memoirs (Dulau
and Co.), pp. 11, 12.
Miscellanea 417
where a convenient method of finding p was by the formula
: ey OSI De
or again by =1- WN?)
The problem has recently occurred of dealing with data where:
(i) One variate is given quantitatively, the other variate is given by ranks.
For example, place in school-class has to be considered in relation to marks in examination,
or the rank in a teacher’s general appreciation has to be considered in relation to marks in
examination.
(ii) One variate is given by broad categories, the other by ranks.
For example, five or six categories of general intelligence are given as the basis of the
) 8 g
teacher’s classification of intelligence, and this has to be considered with regard to rank in,
say, class or examination, possibly with regard to a special subject.
We require in both cases to deduce from the data the true variate correlation.
Case (i). Let x be the character measured by its grade, y the character given quantitatively.
Then with the notation above, if p’ equal the correlation of grade and of variate, + the corre-
lation of the two variates :
1 Pay
ar Noyo,’
where
$2 [te 2 =
Pay= | a —_ 2(Y¥-Y)(W-H) aedy,
pz, , +0 [to 5 0
Pot is ie (y-¥)%q ap rey:
Integrating by parts after putting y¥=0 and writing
de _ de»
dr 1 dady?
qh aco pets di, dz
‘Baw. | ie 10x F a
Integrating again by parts:
dr Lol Sone
: ry)
+0 [+o =
sos] al e ~ozdady
aoe
2-r 2rx'y’ y?
2 [+o [+o 4 12,
__ o9iV? Ie | 1 = yes ee =) dx dy
NOR. —« 2m FS
o, N? 1 o V2
oot i Nr
re” ary
Hence dip! _ 7 (ae \= NV =
No204, Wr oy,
dr adr
* Phil. Trans. A., Vol. 195, p. 25.
418 Miscellanea
Thus since p’ vanishes with 7,
p’ = amie
Thus finally
r= us 5 P= 10233 p’,
-)
ne aoe 1-0233 p”.
It will be clear from this that the correlation p’ between rank and quantitative variate can
never be “ perfect,” for it cannot exceed the value ‘9772, otherwise the correlation 7 would exceed
unity. It will be seen that for practical purposes 7 is very close to p’, but still from the
theoretical standpoint, it is not without interest to discover that the correlation between
ranks and a quantitative variate can never be perfect. For example, it is impossible to have
perfect correlation between place in class and examination test, even if the boys were in the
same order in class and examination. The defect, however, will be very slight.
Case (ii). Let the subscript C refer to any “broad” class and let n be found from either
of the formulae
12 1 — =\2
ae S {te Gog)
ee 12 — =
or 7 WV (¥2 —1) Be a
the first applying to grades and the second to ranks; then
12 ges
r= 1:0233 wi Wa 8 (ne Go- 9)",
Ghee =
ee = 10233 rl n S {te Vo-v)};
according as grades or ranks are used. In actual practice the values of 7’ or 7” should be
correct for number of classes and for “broad” categories. See Biometrika, Vol. vit. p. 256 and
Vol. rx. p. 118.
Numerical illustrations will be provided later.
III. Correction of a Misstatement by Mr Major Greenwood, Junior.
In a recent paper by Mr Major Greenwood and Mrs Frances Wood “On changes in the
Recorded Mortality from Cancer and their Possible Interpretation*” occur the following words :
‘The case is evidently analogous to that studied by Professor Karl Pearson in his pamphlet,
The Fight against Tuberculosis and the Death-rate from Phthisis (Dulau and Co.). Professor
Pearson published three diagrams: (a) the general death-rate of England and Wales; (6) the
phthisis death-rate ; (c) the ratio of phthisis deaths to all deaths. The original figures seem to
have been the crude rate for males and females separately from 1835 onwards.” The “ evident
analogy” with what appears to me the wholly fallacious treatment of the authors in their paper
above cited I do not now stay to discuss, but I wish to draw attention to the words: “The
* Royal Society of Medicine, Proceedings, Vol. vir. Section of Epidemiology, pp. 79—170.
March 27, 1914.
Miscellanea 419
original figures seem to have been the crude rate for males and females separately from 1835
onwards.” Why the writer of these words should have assumed them without any inquiry
of me, or any examination of the values of the crude death-rates (which are accessible to every-
body) to be “crude death-rates,” I do not know, but they illustrate his readiness to form a
biased judgment when his feelings are stirred by unfavourable criticism. As a matter of fact
the rates were standardised rates reduced to the population of 1901, and most kindly
provided at my special request by the General Register Office. It is of interest to observe that
Dr Weinberg of Stuttgart—recently made precisely the same charge as Mr Major Greenwood
with the same over-hasty assumption that the reality must be the desired, if undemonstrated,
error*, With the German as with other foes, it is well to leave ample opportunity for their
assuming you to be foolish ; their assumption may lead them to run against hard reality.
Kees
IV. Note on Reproductive Selection.
By DAVID HERON, D.Sc.
The fact that in the case of mant fifty per cent. of one generation comes from twenty-five
per cent. of the preceding one was first noted by Professor Karl Pearson in the Chances of
Death (Vol. 1. p. 80) and in dealing more fully with this important generalisation in the Ground-
work of Eugenics, p. 27, he said: “It is very difficult from any English statistics to determine
how many adults never marry. No information on this point is asked in the death schedule for
males ; it is asked but imperfectly answered in the case of the schedule for females.” In
a footnote he adds: “The Registrar-General informs me that the record of civil condition
in the case of female deaths is worthless and that no useful return can be made from it.”
He found that in the Argentine and in Scotland 60 per cent. died unmarried, in the United
States 51 per cent., and from the last two English Censuses and the Annual Reports 48 per
cent., and added “ This indirect method of reaching the result is, however, not very satisfactory.
We may, I think, conclude in round numbers that 40 per cent. of the population dies before it
reaches the age of 21 and that probably another 20 per cent. are never married.” On this
assumption Professor Pearson proceeds to show that “about 12 per cent. of all the individuals
born in the last generation provide half the next generation.”
Some data published in Bulletin of Population and Vital Statistics No, 30 for the Common-
wealth of Australia (Tables 48 and 84 a and b) prove that the assumptions made lie very close to
the facts. The data are shown in the following table which gives the conjugal condition and
issue of the males and females who died in Australia in 1912. From this we find that half the
total number of children came from 3337 of the parents (all those who had at least 9 children
and part of those who had each 8 children). It thus appears that of the males 17,404 out
of 30,285 =57°5 °/, died unmarried while half the total offspring came from 25-9 °/, of those who
married and 11:0°/, of the whole number of males, so that approximately three-fifths of the
males born die unmarried and one-half of one generation comes from one-quarter of the married
population or from one-ninth of all the males born in the preceding generation. The diagram
gives a graphical illustration of the argument.
In exactly the same way we find that nearly one-half of the females born in Australia die
unmarried and that one-half of one generation comes from one-quarter of the married and from
one-seventh of all the females born in the preceding generation.
* Archiv fiir Rassen- und Gesellschafts-Biologie, 1x. Jahrgang, 8. 87. Leipzig, 1912.
+ It has also been dealt with in various mammals. See the Groundwork of Eugenics, Eugenics
Lecture Series 11 (Dulau and Co.), p. 29.
420 Miscellanea
Conjugal Children in || Deaths of Total Deaths of Total
| Condition | Each Family Males Children Females Children
Single 0 17404 —_ 10011 —
Married 0 1422 — 1317 —
5 1 1036 1036 1083 1083
45 2 1098 2196 992 1984
x 8 1127 3381 1050 3150
S 4 1147 4588 1001 4004
- 5 1070 5350 976 4880
s 6 1058 6348 1013 6078
5 tf 1040 7280 974 6818
ry) 8 992 7936 881 7048
Ar 9 819 7371 799 7191
A 10 | 801 8010 622 6220
5 11 473 5203. 469 5159
" 12 394 4728 314 3768
ne 13 196 2548 193 2509
- 14 109 1526 101 1414
Bs 15 50 750 57 855
16 | 27 432 22 352
a 7 119 8 136
H 18 5 90 3 54
x 19 6 114 2 38
” 20 3 60 2 40
» 21 = — 1 21
a 22 -- —_— 1 22
iY 23 J 23 — —
Totals 30285 69089 21892 62824
Diagram to illustrate the fact that three-fifths of those born die unmarried and that one-ninth of
one generation produce one-half of the next. (Deduced from records of males.)
First Generation.
Percentages.
60 70 80 90 100
O 10. 20 30 40 50
: More |
a
Families of 8 and
upwards
Families of 1—8
6 10° 20" 3040 0 150 COM OMG OMNES OMIOO
Percentages.
Second Generation.
Journal of Anatomy and Physiology.
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Carnegie Institution of Washington, U.S.A. Artie Ta * cee
es On an Extension of the Method of Correlation ie Grades or Hanke by |
Kart Prarson, F.R.S. sales Genk rat wal AS 416.
(iti) Correction of a Misstatement made by Mr tion OL es Junior. K- P. 418 ©
(iv) Note on Reproductive Selection. By Davip Huron, D.Sc. ', _ 419
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VoLUME X MAY, 1915 No. 4
Tne
by
« ;
{* wie 915)
ASSOCIATION OF FINGER-PRINTS. é
3 Nations bul
Z
By H. WAITE, M:A., B.Sc.
1. Introduction. Certain papers have been published in recent years giving
the results of research on the variability and correlation of the hand, notably
(1) “A First Study of the Variability and Correlation of the Hand,” by
Miss M. A. Whiteley, B.Sc., and Karl Pearson, F.R.S., Proceedings of the Royal
Society, Vol. 65, pp. 126—151, and (2) “A Second Study of the Variability
‘and Correlation of the Hand,” by M. A. Lewenz, B.A., and M. A. Whiteley, B.Sc.,
Biometrika, Vol. 1, pp. 345—860. In the former the writers urge “the import-
ance of putting on record all the quantitative measures we can possibly ascertain
of variability and correlation ” of characters of the human body. Although Finger-
Prints, the characters dealt with in the present paper, cannot strictly claim to
_ be quantitative it is hoped by the writer that the results may prove of some
interest and use in the solution of the great Problem of Evolution in Man,
especially when compared with the results obtained from the study of other
measurements of the hand.
The principal motive underlying most of the work which has been done in the
past on the subject of Finger-Prints has arisen from the development of means of
identification and it was based on the fact that the general pattern and character-
istics of the finger-prints of any individual are persistent throughout life. As far
as I am aware, however, no paper has yet been published attempting to measure
the association between the various types of finger-prints in an individual or com-
paring these with the relations which have been found to exist between other
measurements of the hand. These are the objects of the present paper.
2. Primary Classification of Finger-Prints. As primary classification
Purkenje proposed nine types, Galton* three—each being divided into twenty-
four sub-classes,—and Henry+ four, these also being sub-divided into a number of
classes. For the purposes of this paper I have adopted the method of dividing all
the prints into four primary classes; I have also adopted Henry’s definitions and
* Fingerprint Directories, by Francis Galton, F.R.S. Macmillan, 1895.
+ Classification and Uses of Finger Prints, by Sir EK. R. Henry, C.V.O., C.S.I. Wyman and Sons,
Third edition, 1905.
Biometrika x 54
422 Association of Finger- Prints
nomenclature as far as they are required, and these follow in general those of
Galton. Secondary classification with its minute details is not used in this paper.
The four classes referred to above are Arches, Loops, Whorls and Composites.
In Arches the ridges run from side to side, consecutive ridges being roughly
parallel and the curvature increasing in general from the base to the tip.
(Plate XX. Fig. 1.)
In Loops some of the ridges are doubled back upon themselves making a half
turn or a little more, the two parts of the doubled ridge diverging from each other
at the centre of the pattern. (Fig. 11.) Consequently this pattern has an open
mouth directed downwards either towards the right or towards the left of the
finger. The direction of this opening supplies a means of subdividing Loops into
Radial and Ulnar Loops according as the direction is towards the radius or towards
the ulna, that is, towards or away from the thumb. As will be seen later (p. 422B)
the proportion of Radial Loops is very small except in the forefinger, so that this
method of subdivision has been used only in dealing with that finger.
In Whorls some of the ridges make a complete circuit, either as closed con-
centric ovals or as a more or less continuous ridge forming a spiral. (Fig. iii.)
Composites consist of combinations of two or more of the other patterns.
(Fig.iv.) In this class are also included those finger-prints which are too irregular
in general outline to be placed in any one of the other main groups.
This class also includes the bulk of those patterns about which Sir Francis Galton,
in his book on Finger Prints*, p. 79, states—* They are as much Loops as Whorls,
and properly ought to be relegated to a fourth class.” It is possible, however,
that some of Galton’s “ambiguous cases” may have been classed in this paper
with Loops.
For further details of these principal classes with their modifications and sub-
divisions reference may be made to the works mentioned in the footnotes on
p. 421.
3. Material. The material on which this investigation is based consists of
two thousand complete sets of finger-prints of adult males, part of a much longer
series in the Biometric Laboratory of University College, London. They belong
to the lower type of artisan and labouring classes. No selection whatever has been
made, except that a few sets, which were incomplete or which contained prints so
damaged as to be indecipherable, have been rejected.
4. Symbols, The following symbols are used :—A = Arch, SZ =Small Loop ;
LL = Large Loop (see p. 423); W= Whorl; C= Composite; L,= Radial Loop;
L,,= Ulnar Loop; R=Right Hand; Z=Left Hand. &,, R,, R;, R,, R; designate
the thumb, forefinger, middle, ring and little finger respectively of the right hand,
and L,, L,, L;, L,, L; represent the corresponding fingers of the left hand.
* Finger Prints, by Francis Galton, F.R.S., Macmillan, 1892.
Plate XX
Biometrika, Vol. X, Part IV
Loop.
Fig. (ii).
Arch.
(i).
Fig.
Composite.
Fig. (iv).
Whorl.
Fig. (iii).
Illustrations of the four fundamental types of Finger-Print.
H. Waite 422A
5. Distribution of Classes of Finger-Prints. A preliminary survey of the prints
brings to light a considerable clustering together of prints of the same kind.
Thus, each of 241 sets contains prints of one class only ; each of 329 sets has nine
prints of one class, and each of 194 sets contains eight out of the ten prints of one
class; that is, each of 764 sets, or over 38°/,, has at least eight prints of one class,
the large majority of these being loops. Again, each of 892 sets contains prints of
two classes only, so that each of 1133 sets—or nearly 57 °/, of thre whole—has
representatives of not more than two of the four classes. On the other hand all
four classes appear in only 95 sets, while the number of single hands, each of
which contains at least one of every class, is only 23.
For the calculations which follow it has been found advisable to subdivide the
loops into two classes, Small Loops and Large Loops (p. 423). Considering these
as separate classes, giving five types in all, the distribution of numbers of types
for the two hands is shown in the following Table:
TABLE 1.
Distribution of Types in Right and Left Hands.
Number of Types in Right Hand.
[=| 7
= 1 2 3 4 5 Totals
a
Ewe 1 37 84 47 6 = 174
As 2 65 | 465 | 360 61 4 955
sM| 3 15 | 256 | 347 | 96 2 716
2 4 1 36 83 30 1 151
2 & 5 = 1 2 1 =
= Totals} 118 | 842 | 839 | 194 5
q eee patel =
In this Table, taking as origin the cell (3, 2) containing 360 types, we have the
following results :
Mean of Left Hand Types, 428
oy, "7628
Mean of Right Hand Types, — 435
a ‘7608.
We thus find the correlation coefficient (7) to be ‘281 + 014.
The contingency coefficient (c), corrected for the number of cells, is °289. Hence
we conclude that there is a distinct, though not very great tendency towards
equality in the number of types in the two hands of an individual. It appears,
however, that the divergence is rather greater in the right than in the left hand.
The question now arises whether the difference in divergence in the two hands
for the samples taken is significant. I have tested this by the method proposed by
Professor Karl Pearson*.
* «©Qn the Probability that Two Independent Distributions of Frequency are really Samples from the
same Population,” by Karl Pearson, F.R.S., Biometrika, Vol. vu11, pp. 250—254, July, 1911.
54—2
4228 Association of Finger-Prints
TABLE 2.
Divergence of Types in Right and Left Hands.
Number of Types.
i 2 3 4 5 Totals
Right Hand _... 118 842 839 194 vi 2000
Left Hand wo 174 955 716 151 4 2000
For this Table
x? = 33°72,
whence FP is less than ‘000,005.
That is, the odds are more than 200,000 to 1 against the occurrence of two such
divergent samples if they were random samples of the same population. In other
words the right hand generally tends to have a greater divergence of types than
the left.
The following Table gives the distribution of classes of prints for the various
fingers of both hands:
TABLE 3.
Distribution of Classes of Prints.
A is rie W c
| Ry = 46 1104. | 1 649 200
1 lS) 537 456 481 174
Re a 212 1399 38 274 ai
Vee se 63 1015 17 729 176
Re a 31 1631 3 228 107
‘aoe pe ers - is
| Totals Mes 704 5686 515 | 2361 734
L, on 91 1311 3 341 254
Ly ae sl3e || 732 | 383 437 135
Ls oe: PAS} 1408 35 240 102
In aeS 66 1283 | 12 491 148
Ls Ses 35 1727 — 150 88
Totals us 720 6461 433 1659 727
Totals for both hands} 1424 12147 948 4020 1461
The most striking feature of this Table is the uneven distribution of the
various classes, especially the large proportion of ulnar loops and the very small
H. Warts 423
number of radial loops except in the forefingers. A comparison of the distribution
in the two hands shows considerable differences; e.g., in the left thumb the num-
ber of arches is about double the number in the right; again, the whorls in each
finger of the right hand are greatly in excess of those on the left, while the left
hand has, in every case, an excess of ulnar loops.
If we arrange the numbers of each class in order of magnitude, we see that the
order for the arches is identical for the two hands and also for the ulnar loops.
In each of the other classes there is one exception to the “identical” order.
I have tested these distributions for each type by the method referred to in the
footnote of p. 4224, with the following results :—In the arches the odds are more
than 500 to 1 against the occurrence of two such divergent samples which are
random samples taken from the same population; in the ulnar loops the odds are
more than 200,000 to 1; in the radial loops about 5 to 2; in the whorls more than
1,000,000 to 1, and in the composites more than 1300 to 1.
We may thus fairly conclude that with the exception of the radial loops the
frequency distribution of the classes between the fingers is different in the two
hands and the radial loops are so few, except in the forefinger, as to be almost
negligible.
6. Subdivision of Loops. The great preponderance in the number of loops
and the insignificance of the number of radial loops, except in the forefinger, make
another subdivision of this class necessary. The method adopted is as follows :—
All loops, in common with whorls and composites, contain certain well-defined
points; these are (1) the “ delta,” or “outer terminus,” and (2) the “ point of the
core,” or “inner terminus.” [See Henry, pp. 22—24.] The number of ridges
mtervening between the delta of a loop and the point of the core may be anything
from one up to about thirty; in only 38 cases out of the 13,095 loops does the
number of ridges exceed 25; two of these are over 30, one being 32 and the other
35. The complete distribution of ridges is given in Table 4 a.
In dividing the loops into two sub-classes according to the number of ridges
the nearest approach to equality is obtained by taking (a) those containing from
1 to 12 ridges, and (6) those containing 13 or more ridges. For brevity I have
called these classes (a) Small Loops, and (b) Large Loops; the terms “Small” and
“Large” have no reference to the relative sizes of the patterns. The numbers in
the two groups, thus arranged, are 7033 and 6062 respectively.
Table 46 gives (1) the number of loops for each finger, (2) the means, (3) the
standard deviations, and (4) the coefficients of variation in the numbers of ridges.
Examining the Table below consider first the means. The order which is
identical in the two hands runs:
(1) Thumb, (2) Ring Finger, (8) Little Finger, (4) Middle Finger, (5) Index.
It will be noticed that this order of the means is quite different from that
of the relative areas of the patterns.
424 Association of Finger-Prints
TABLE 4a.
Distribution of Ridges in Loops.
Right Hand. Left Hand.
Ridges | Ft) | (Be -| Bel seeaieleee, NE | len |) es
1 3 19 18 8 a 20 21 5 10
2 7 68 47 27 24 79 65 25 32
3 7 76 50 26 49 Tey 54 40 42
4 8 71 66 35 68 ga 60 33 56
7) 24 67 57 58 70 70 52 45 69
6 14 47 80 36 92 54 47 32 69
tf 24 45 76 44 77 64 | 75 47 70
8 29 50 85 45 73 67 70 34 91
9 25 40 97 36 83 65 77 53 85
10 43 47 109 55 94 73 117 61 122
al 53 55 116 48 100 76 124 89 156
12 53 69 120 71 133 77 125 119 157
3 65 62 116 66 111 65 141 94 134
14 89 60 128 76 143 67 139 111 150
15 68 60 84 79 103 49 99 94 131
16 91 43 75 73 106 35 66 117 136
1 86 38 62 60 97 26 47 86 88
18 84 33 28 65 73 12 37 60 64
19 100 16 11 34 46 11 15 58 33
20 61 12 5 34 48 a 6 33 17
21 49 a 4 18 19 4 22 6
22 34 2 2 19 3 6 1 12 4
23 33 4 — 8 4 a — 9 2
24 22 1 1 4 6 1 1 8 1
25 ll -- = 2 2 = —~ 2 2
26 7 _- —_ 1 2 = == 1 =
27 7 — — 1 1 -— — 1 —
28 7 — — — -- = = 1 —
29 1 = = 1 =
30 ae rig | Seas 1a Sas eee ones
32 techs | Meee ee | a
35 as ee ae
Totals | 1105 | 998 1487 | 1032 | 1634 | 1314 | 1115 | 1443 | 1295 | 1727
TABLE 40.
Nee of Means Standard Deviations Coefficients of Variation
oops
Ree R iB R is R ij
Thumb 1105 | 1314 | 15°52++10 | 13-27+-09 | 5°174°07 | 4°634°06 | 34°34+ °53| 34:85+ °51
Index .. {| 993 | 1115 | 9°69+:12) 8834-10 | 5-41+°:08 | 4°88+ °07 | 55°82+1-08 | 55°24+1-00
Middle Finger | 1437 | 1443 | 10-41+-08 10°55+-08 | 446+ ‘06 | 4°53+ :06 | 42°80+ ‘63 | 42°914 °63
Ring Finger... | 1032 | 1295 | 12'374°12 12°774°10 | 548+ ‘08 | 5°09+°07 | 44°31+ °78 | 39°85+ °61
Little Finger . | 1634 | 1727 | 11°75+-:08 | 11°53 + -07 | 4°97 + ‘06 | 4°46 + -05 42°30+ °58| 38°71+ ‘50
le
H. WaAITE 425
Comparing the two hands we see that the differences in the middle, ring and
little fingers are insignificant ; in the thumb and index, however, there is a marked
difference in favour of the right hand.
The order of the standard deviations in the right hand is:
(1) Ring Finger, (2) Index, (3) Thumb, (4) Little Finger, (5) Middle Finger.
In the left hand the order of the last two is reversed, but the difference is
small.
With the exception of the middle finger, where the difference between the two
hands is only about equal to the probable error and is therefore insignificant, the
standard deviation is in every case greater for the right hand than for the left ;
the differences are all of the same order of magnitude and range from about
39 to 54,
Coming now to the coefficients of variation—the order in the right hand is:
(1) Index, (2) Ring Finger, (3) Middle Finger, (4) Little Finger, (5) Thumb.
In the left hand the order of the ring and middle fingers is interchanged.
Comparing the two hands we see that in three cases—the thumb, index, and
middle finger—the differences are each less than the probable errors; in the other
two cases the variability is considerably greater in the right hand than in the left.
I have carefully revised the calculations involved but have been unable to
detect any error; neither can I suggest a reason for the large differences.
In “ A First Study of the Variability and Correlation of the Hand” (see p. 421),
the writers find that the variability of bone lengths is closely related to the
relative utility of the fingers, the least variability being that of the most useful
finger. There appears, however, to be no such simple relationship between the
ridges of the loops and the relative utility of the fingers.
I have compared the distribution of ridges in the loops of the thumbs by
Professor Pearson’s method (p. 4224, footnote), which gives y? = 166°64; hence the
odds are much greater than 1,000,000 to 1 against the occurrence of two such
divergent samples if they were random samples taken from the same population.
The distribution—absolute and percentage—of the five groups is now as
follows (Table 5).
In comparing the large and small loops it will be seen that in both hands there
is an excess of large loops in the thumb, ring and little fingers, and an excess
of small loops in the index and middle fingers. The order of these classes agrees
in the two hands with one exception in each case.
An approximate measure of the relationship existing between the various
combinations of digits is given by the number of cases in which two particular
digits on the same or on opposite hands have the same pattern. Table 6a gives
the percentages for the same hand and for digits of the same name on opposite
426 Association of Finger-Prints
TABLE 5.
Arches Small Loops | Large Loops Whorls Composites
No. Oe No. oie No. Sh No. Os No. oe
Ry me 46 2°30 290 14°50 | 815 40°75 | 649 32°45 200 10°00
Ry aise 352 17°60 | 654 32°70 339 16°95 481 24°05 174 8°70
Rs ... | 212 10°60 | 921 46°05 516 25°80 | 274 13°70 77 = 3°85
Ry eet 638 315 489 24°45 | 543 27-15 729 36°45 176 ~=—880
Rs nit 31 1°55 870 43°50 | 764 38:20 | 228 11°40 107-5585
Totals aio 704 3224 2977 | 2361 734
Es 91 4:55! 547 27°35 | 767 38°35 | 341 17:05 | 254 12°70
bs 313 15°65 | 833 41°65 | 282 14:10] 437 21°85 | 135 6°75
: 215 10°75 | 887 44°35 | 556 27:80] 240 12:00] 102 5:10
fe 66 3°30 | 583 29:15 | 712 35°60 | 491 24:55 | 148 7-40
Ve 35 1°75 | 959 47:95 | 768 38:40] 150 7-50 88 4:40
‘ i = -
Totals = 720 3809 3085 | 1659 727
Totals for both hands | 1424 7033 6062 4020 1461
hands; the readings for other combinations of digits on opposite hands are given
in Table 6d, p. 431, where all the patterns are grouped in three classes for the
sake of comparison with Galton’s results.
Remarks on Table 6a. (a) The percentages vary greatly with different
combinations and with different patterns.
(b) The means and totals for digits of the same name on opposite hands are
all much greater than the corresponding readings for the right or for the left
hand; the means, with one exception, and also the totals for particular combina-
tions on the left hand are all greater than the corresponding readings for the
right.
(c) The order of magnitude of the totals is nearly the same for the two hands,
those of the combinations including the thumb being, with one exception in each
hand, the lowest. Hence, judging the relationship by the totals, it appears that
(1) digits of the same name on opposite hands are the most closely related, the
magnitude falling in order from the little fingers to the thumbs; (2) omitting
the thumbs, two consecutive digits are generally more closely related than others
more widely separated ; (3) the digits of the left hand are more closely related
than those of the right.
(d) The relationship between the thumb and any other digit seems to be less
close than that between any pair of digits not including the thumb; also, in both
H. WaAItE
427
hands, the thumb appears to be most closely related to the ring finger, then to
the little finger, next to the middle and least to the fore-finger.
Another method of investigating the approximate relationship between the
various digits is by means of a “centesimal” scale, as in Galton’s Finger Prints,
Ch. Vit.
Table 66 gives such scale readings for small loops, large loops and
whorls, for pairs of digits on the same hand and for digits of the same name on
opposite hands.
Percentage of Cases in which various pairs of Digits possess the
TABLE 6a.
same Class of Pattern.
I have not considered it necessary to include other couplets
Right Hand Left Hand
Couplet Totals Totals
A SL | LL W Cc SL | LL W Cc
Thuinb and fore-finger 15] 63) 775) 13°0} 1:4) 29°7 J 2°4/15°7|) 7:2} 8:3) 1:3) 34:9
* middle finger [1:3] 8°7|11:2| 80| -7| 29°9 }1°6/ 16-2) 13-4] 4-9] 1-4] 37-5
+ ring 5 71) 6:5 | 12:4) 17-8) 1:1) 38°5 71132) 17-1) 7:9} 1-5] 40-4
is little . ‘4| 99/150] 6-9] -9| 33:1 } -7/| 19:0) 15-7] 2:7] -9| 39-0
Fore-finger and middle finger | 6°4 | 22°7| 7:2) 8:°9|1:2| 46:4 [5°8/26°3| 7:°9| 86/1:2) 49°8
5 ring ,, 2°3/13°7| 6°4/16:3| :9| 39°6 }2°5)17°8| 7:1) 12°7|)1°5| 41°6
‘ neler 1:2/ 20:1] 9:1] 6-8] -9| 38:1 [1:3] 26-3! 8-2] 4:3] -3] 40-4
Middle and ring finger 2°5| 16°8| 91)12:0) 7 | 41°1 §2°6 | 21:1] 15:4) 8:8) -8) 48°7
- little ,, 1:0 | 26°8| 14:0} 4:6) :4] 46°8 [1-2 | 28°9) 15°3| 2:9] -6| 48-9
Ring and little os *8 | 20:0} 15-7} 10°0 | 1:0 | 47°5 ‘9 | 23°7|19°5 | 6:0} °8]| 50°9
Means 1°8 | 15-2 | 10°8 | 10-4 9} 39:1 2-0} 20°8| 12°7| 6°7/1°0 | 43°2
| |
Couplet A Sia |) oda) He C | Totals
Two thumbs 3 1°56 | 10°2 | 23°4 | 13°5 | 2°7] 51°3
» fore-fingers ... 9°3 | 22°7 | 5°6.| 14:4] 1°74) 53:4 |
, middle fingers 5°8|31°8|14°8| 7:0] 9} 60°3
» ring - 1°9 | 18°2 | 18°4 | 21:2 | 1:5} 61°2
Seilittle, =": 9 | 36:1 | 27-2) 5:0| 1:4) 70°6
Means... 3°9 | 23°8 | 17°9 | 12°2 | 1°6 | 59°3
from opposite hands, because, as is shown later, the relationship between any
pair of digits from opposite hands is practically the same as between the corre-
sponding pair on the same hand. I have also omitted arches and composites from
this part of the inquiry as the numbers belonging to these classes are, as a rule,
comparatively small.
Biometrika x
55
428 Association of Finger-Prints
The scale reading for any pair of digits is calculated as follows :—
Take, for example, the whorls on the right thumb and right fore-finger; the
former has 32°5 and the latter 24 per cent. of whorls, while 13 per cent. of right
hands have whorls on both thumb and fore-finger. Now from independent pro-
bability we shall expect BEES x 100, or 7°8 per cent. of “double whorls” in
this combination of digits and we therefore conclude that the remaining 5°2 per
cent. of double whorls are due to a relationship between the digits. If we set
aside the 7°8 per cent. out of the 32°5 and 24, we see that from the remaining
24-7 and 16:2 per cent., the greatest possible percentage of double whorls would
be 16:2; but as the actual percentage in addition to the 7°8 is 5:2, the centesimal
5-2 x 100
measure of the relationship is Sr
, or 32, to the nearest unit.
TABLE 60.
Approwimate Measures of Relationship between various pairs of Digits
on a Centestmal Scale.
| Right Hand Left Hand
Couplet ]
SL | LL | W |Means} SL | LL | W_ | Means
Thumb and fore-finger ... 16 6 | 32 18 27 21 34 27
5 middle finger... 26 0 38 21 26 16 28 23
Re ring 5 27 8 29 21 27 16 29 24
little i 44 Oo aioe Ooms mn 4 | Jo30n sos
Fore and middle fingers... 44 22 54 40 34 39 64 46
* ring os Hh 30 15 49 33 33 23 44 33
‘ little a Bae 32 25 47 35 29 32 46 36
Middle and ring se, os 42 11 80 44 50 31 64 48
y, little ,, a 29 26 31 29 33 27 30 30
Ring and little . ae 67 32 81 60 64 26 74 55
Couplet SL LL W | Means|
Two thumbs me 59) | 34 | 69) || 54
» fore-fingers ... 48 27 55 43
» middle fingers... 47 41 52 AT
» ring ms 64 50 78 64
», little 53 67 53 62 61
Most of the remarks on Table 6a will be found applicable to Table 6 6, with,
at most, but slight modification; the chief differences are that the relationship
between the middle and little fingers is not so high in Table 6 0 as in Table 6a,
and the order for pairs of like digits is not the same in the two Tables.
H. WaitE 429
Comparison of results with those of Galton. In order to compare with Galton’s
results it is necessary to put large and small loops into one class and to include
composites with whorls. Making some allowance for the difference of classification,
and for any slight variation which may be due to the fact that the material is
drawn from very different classes of the population, it will be found that there
is almost perfect agreement between our data on all essential points.
The relative frequency found in the two investigations was :—
Galton Waite
Arches 65 per cent. 71 per cent.
Loops CGO GBA A
Whorls PABA QA
The differences are small in comparison with some found by Galton when
examining the finger-prints of different races. For example, 1332 Hebrew
children had arches on the right fore-finger in 13°6 per cent. of the cases, while
only 7°9 per cent. of 250 English children had arches on that finger.
TABLE 6c.
Percentage Frequency of Arches, Loops and Whorls on the different Digits.
Gatton* | WAItE
From observations of the 5000 From observations of 20000 digits of
digits of 500 persons 2000 persons
Arches Loops Whorls Arches Loops Whorls
Digit FARR Eile reali | | ote Wad | Oh RON
|
Fore-finger... | 17 | 17 53 53 30 pa i ad Ue 49°7| 55°7| 32°7| 28°6
Middle finger | 7| 8] 78] 76] 15] 16] 106 | 10°7 | 71°9) 72:2] 17°5| 17-1
Little se 1 2 86 90 13 8 1°5 a7 81°7| 86°4] 16°38) 11°9
Thumb 3 5 53 65 44 30 2°3 4°5 55°2 | 65°7) 42°5 | 29°8
Ring finger... 2 3 53 66 45 31 3°2 3°3 51°6 | 64°7| 45°2} 32-0
Totals SOs EsoMo2eooON 4 yell lise Sbs2el 8o:9) | SLO) 344275) 15479) 119°
ie = |
Galton arranged the digits as in Table 6c, in order to bring out certain
peculiarities. He says :—
“The digits are seen to fall into two well-marked groups ; the one including the fore, middle,
and little fingers, the other including the thumb and ring finger. As regards the first group, the
frequency with which any pattern occurs in any named digit is statistically the same, whether
* From Finger Prints, p. 116, Table II.
430 Association of Finger-Prints
that digit be on the right or on the left hand; as regards the second group, the frequency differs
greatly in the two hands. But though in the first group the two fore-fingers, the two middle,
and the two little fingers of the right hand are severally circumstanced alike in the frequency
with which their various patterns occur, the difference between the frequency of the patterns
on a fore, a middle, and a little finger, respectively, is very great.
“Tn the second group, though the thumbs on opposite hands do not resemble each other in
the statistical frequency of the A. L. W. patterns, nor do the ring fingers, there is a great
resemblance between the respective frequencies in the thumbs and ring fingers ; for instance,
the whorls on either of these fingers on the left hand are only two-thirds as common as those
on the right. The figures in each line and in each column are consistent throughout in
expressing these curious differences, which must therefore be accepted as facts, and not as
statistical accidents, whatever may be their explanation.” (Galton, Finger Prints, p. 116.)
These remarks apply with equal force to my figures although the actual
percentages differ somewhat in certain cases, the most marked being in the
middle finger arches and the little finger whorls.
The following points of agreement in the distribution of the patterns are also
noticed by reference to Table 6 c.
The frequency of arches on the fore-fingers is much greater than on any other
of the four digits. “It amounts to 17 per cent. on the fore-fingers, while on the
thumbs and on the remaining fingers the frequency diminishes in a ratio that
roughly accords with the distance of each digit from the fore-finger.
“The frequency of Loops has two maxima; the principal one is on the little
finger, the secondary on the middle finger.
“ Whorls are most common on the thumb and the ring-finger, most rare on the
middle and little fingers.” (Finger Prints, p. 117.)
In discussing radial and ulnar loops, which Galton describes as loops having
“inner” and “outer” slopes, respectively, he says :—
“Tn all digits except the fore-fingers, the inner slope is much the more rare of
the two; but in the fore-fingers the inner slope appears two-thirds as frequently
as the outer slope. Out of the percentage of 53 loops of the one or other kind on
the right fore-finger, 21 of them have an inner and 32 an outer slope; out of the
percentage of 55 loops on the left fore-finger, 21 have inner and 34 have outer
slopes. These subdivisions 21-21 and 32-34 corroborate the strong statistical
similarity that was observed to exist between the frequency of the several patterns
on the right and left fore-fingers; a condition which was also found to characterise
the middle and little fingers.” (Finger Prints, p. 118.)
These statements are true, in general, of my Table 8, but my percentages on
the right fore-finger are 22°8 radial and 26-9 ulnar; on the left they are 19°2 and
36°6 respectively.
Close agreement is also observed in Table 6d which shows the tendency of
digits to resemble one another in their various combinations. Galton omits
combinations into which the little finger enters “because the overwhelming
H. Waite 431
frequency of loops in the little fingers would make the results of comparatively
little interest, while their insertion would greatly increase the size of the table.”
(Finger Prints, p. 119.) I have included them, however, for the sake of com-
parison and completeness.
My percentages are readily obtained from Tables LVI to C in the Appendix.
TABLE 6d.
Percentage of Cases in which the same Class of Pattern occurs in various Couplets of Digits.
GaLTON* WaItE
Arches in Loops in Whorls in Arches in Loops in Whorls in
Couplet
Same | Opposite | Same | Opposite | Same | Opposite] Same| Opposite Same Opposite Same Opposite
hand | hand hand | hand | hand | hand fhand| hand |hand| hand |hand| hand
Two thumbs — 2 — 48 — 24 — 1°6 = 47 °4 = 24°5
,, fore-fingers = 9 — 38 — 20 — 9°3 — 36°2 = 20°4
» middle fingers — 3 — 65 -- 9 — 5°8 — 60°6 — 10°5
» ring + re 2 — 46 = | ae _ 1g, — 46°3 — 27°9
» little Sac || — ~- — — — —_— ‘9 = 63°2 — 63 |
Thumb and fore-finger | 2 2 35 33 16 15 1SOi) WeSh 36:8 Shey i828 1745
Ps mid-finger 1 1 48 47 9 8 1-4 15 47°0,; 46°7 |10°9} 10°5
x ring finger 1 1 40 38 20 18 af 6 41:0} 39°4 | 20°8| 19°0
Fore and mid-finger . 5 5 48 46 12 a 6:1 5°5 ANID} || alos | ileats) || 12383
H a ring finger ... 2 2 35 35 ily 17 2°4 2°3 36°5 | 35°7 | 20°8} 20°2
Middle and ring finger} 2 2 50 50 13 12 2°5 2°4 48°3) 47:1 | 14:7} 13:7
Thumb and little finger | — -- — —- | = oo "52 "45 |54:2|) 53°6 8°8 81
Fore and little finger... | — — eel eae —_— 1-20} 1:15 |47°7) 47:2 9°3 8:1
Middle and little finger el | = il 1-0 63:9 | 63-5 6°5 6'1
Ring and little finger... | — — — — | — — 8 7 56") 54:8 | 12°9')) “11-8
|e
In commenting on his results in Table 6d, Galton says:—*“ The agreement
in the above entries is so curiously close as to have excited grave suspicion that
it was due to some absurd blunder, by which the same figures were made in-
advertently to do duty twice over, but subsequent checking disclosed no error.
Though the unanimity of the results is wonderful, they are fairly arrived at, and
leave no doubt that the relationship of any one particular digit, whether thumb,
fore, middle, ring or little finger, to any other particular digit, is the same, whether
the two digits are on the same or on opposite hands.”
It will be noticed, however, that while exactly half of Galton’s eighteen pairs
of percentages, which are worked to the nearest unit only, are in strict agreement,
in all the other cases the result is one or two units less for two digits on opposite
hands than for the corresponding digits on the same hand. In my figures the
percentage for two digits on opposite hands is in every case the lower, and
* Finger Prints, p. 120, Tables VIa and YI Db.
432 Association of Finger-Prints
although the differences are small, ranging only up to 1:8 while four-fifths of
them are less than 1, the consistency of the results suggests a slightly closer
relationship between a pair of digits on the same hand than between the corre-
sponding pair on opposite hands. This view is further supported at a later stage
of this paper. (See Remark (d) on Tables 14-16, p. 450.)
One further comparison is of interest, namely, the measure of relationship
between the various digits on a centesimal scale. It should be noted, however,
that while Galton’s means are based on loops and whorls only, omitting arches
from his three groups, mine are based on small loops, large loops and whorls,
omitting arches and composites from my five groups; also Galton gives no results
for those combinations which include the little finger.
TABLE 6e
Approximate Measures of Relationship between the various Digits,
on a Centesimal Scale.
Gauton* WatrtE
| Couplet
Means Right Left
Thumb and fore-finger ... 24. 18 27
i middle finger... | 27 21. 6 \e Tosa
a ring finger... 39 21 24 |
Fore and middle finger pac 60 40 46
» ring finger... ne 23 33 33
Middle and ring finger aa 52 44 48
Right and left thumbs Soh 61 54
se fore-fingers ... 48 43
i middle fingers 43 47
aH ring fingers ... 65 64
For the reasons given above we could hardly expect that these readings would
be even approximately equal, but for all that, the same general relations are seen
to hold good in the two sets of results.
It is convenient at this stage to summarize a few of the most important
points which have been brought to light in the foregoing pages. These are:
(a) <A greater divergence of types in the right hand than in the left.
(b) A clustering of the same type in the hands of an individual.
(c) The uneven distribution of the various types in the different fingers,
especially the almost entire absence of ulnar loops except in the index.
* Finger Prints, p. 129, Table VIII.
H. WaItE 433
(d) The ditferentiation of types in the two hands, in particular the large excess
of whorls in the right hand and of arches in the left thumb.
(e) Where there is any significant difference in the means, standard deviations
and coefficients of variation in the numbers of ridges in the loops of the two hands
those quantities are always greater for the right hand than for the left.
(f) The relationship between digits of the same name on opposite hands is
closer than that between any others; the digits of the left hand are more closely
related than those of the right; and two consecutive digits, whether on the same
or on opposite hands, are generally more closely related than others which are more
widely separated. The relationship between the thumb and any other digit is less
close than that of any pair not including the thumb.
We may thus conclude that the left hand in its distribution of patterns is
differentiated from the right and that individual fingers are associated in a differ-
ential way with special types. We know that the right hand is differentiated from
the left in use, and it would seem reasonable to suppose, even if we cannot account
for the adaptation to use, that the finger-prints have been differentiated in accord-
ance with this use differentiation.
It may be suggested that the finger-prints if differentiated in accordance with
diversity of use of the several fingers and of each hand follow a law of differ-
entiated utility and not as the bones a law of maximum general utility of the
finger.
7. Correlation between the Classes of Finger-Prints. The object of this section
is to obtain the associations between the various classes of prints and on the basis
of these associations to enquire whether any Natural Order exists in which a
certain degree of continuity may be assumed. For a complete investigation of this
problem fifty-five Tables are necessary. They are:
(a) 10 Tables of Classes for the Right Hand.
(b) 10 ” ” 3 Left Hand.
(8) PA, “ » Right against the Left Hand.
GP 10; 55 both Hands together.
These Tables are given in the Appendix, pp. 453 et seq.
The correlation coefficients and the contingencies have been calculated for the
whole of these Tables. For all the restricted Tables, I to XX, and XLVI to LV,
and in certain of the remaining Tables where the results of the other two methods
are widely divergent, the correlation ratio has also been found. In these cases
I have obtained 7 in both directions, the values of 7 given in Tables 8 and 9 being
the square root of the product of the two 7’s for each Table *.
* The arithmetic instead of the geometric mean might have been taken, and there would not have
been very marked differences. But the geometric mean has the advantage of a symmetrical value, i.e.
Re Oe see) Ie
i Ox Fy
which has certain analogies with a coefficient of correlation.
434 Association of Finger-Prints
8a. Method of Calculating the Coefficients of Contingency in Restricted Tables.
It will be noticed that the Tables I to XX and XLVI to LV, given in the
Appendix, differ in general character from most correlation tables since the whole
of the cells in the lower right-hand portion are necessarily empty. Consequently
the usual method of finding the independent probability numbers for the purpose
of calculating contingencies is not applicable to those Tables. The method which
has been employed was suggested to me by Professor Pearson. It is as follows:
Consider Table VI, Appendix, p. 454, which gives the distribution of small
loops and whorls for the right hand. Commencing with the 45 hands which con-
contain 5 small loops each, it will be seen that the independent probability
number is the same as the observed, since a hand which has five prints of one
class can have no other. In the next column the distribution of the 211 prints by
independent probability is not in the ratio of 861 to 497 since 45 of the 861 have
already been disposed of, but in the ratio of 816 to 497, that is, the numbers in
the two rows are 1311 and 79:9. Again in the third column from the right con-
taining 3 small loops, 45 and 131-1 of the first row are accounted for and 79°9 of
the second row; hence the independent distribution of the 306 in the third
column is in the proportion of 6849 : 417-1 : 292; that is, the numbers are
150°3, 91:6, and 641; and so on.
It should be noted that the same independent probability numbers are obtained
if we commence at the bottom of the first column with the 50 hands each con-
taining 5 whorls and work horizontally instead of vertically.
The differences between these independent probability numbers and the
observed numbers are then used to find the contingency in the same way as in
the ordinary contingency Table.
No correction for the number of cells has been applied to the contingency
coefficients in this type of Table as we have, at present, no appreciation of what it
should be.
The complete contingency Table, worked as described, is given below.
Note on Calculation of Contingency Coefficients. It should be borne in mind
that in finding the independent probability numbers in all contingency tables as
well as in calculating the standard deviations, it is assumed that the distribution
of the marginal totals is in the same ratio as would be the case if the whole
population were taken ; in other words, that if n, is the total of an array when a
sample NV is taken and m, the total of the corresponding array when the whole
population M is taken, then it is assumed that
Ne sie
Evidently the correct value of the independent probability number in the (s, s’)
cell of an ordinary contingency table would be
N
/
MO eS OED. a
‘ <M /N
WV or MUN gt WE
W horls, R.
H. WartE 435
and the contributory contingency
7 kee
Nyy — Ms M's He
rey, ,
MsM gy We
TABLE 7.
Contingency Table.
Small Loops, R.
0 i 2 3 4 5 Totals
= eee
Gee 78 144 204 211 179 45 861 |
B 200°6 167°4 166°6 150°3 131°] 45
— 122°6 —93°4 37°4 60°7 47°9 —
y?/3 74:93 3°27 8°40 24°51 17°50 —-
1 106 }lkiay33 126 80 32 — 497
122-2 101°9 101°4 91°6 79°9 —
—-16°2 | 51-1 24°6 —11°6 -—47°9 —
2°15 | 25°63 5°97 1°47 28°72 —
2 130 92 55 15 -— — 292
85°5 lca: 71:0 64:1 — —
44°5 20°6 -—16°0 —49°1 — —
23°16 5°94 3°61 37°61 — _:
3 125 38 7 — — — 170
63°8 53°2 53°0 — — —
61:2 —15°2 — 46:0 — — —
58°70 | 4°34 39°92 =. _: —
y 104 3, 26 = = 2s — 130
70°9 59:1 — — _ —
Bio yall —33°1 — — aoe —
15°45 18°54 = — mee =
5 50 — — — — _- 50
50 | — — — _ —
Totals 593 453 392 306 211 45 2000
“19991
x2=S (y2/8)=399'82, ¢=72/n='19991, = mE = 4082.
one oe N
Similarly the quantities m, MW’
i AN
eS ye
7
119991
etc. would be the correct marginal totals
to use in finding the independent probability numbers for the restricted contingency
Biometrika x
56
436 Association of Finger-Prints
tables and in obtaining the standard deviations, instead of the observed totals
Ns, Ny, ete.
However, as we do not generally know M, m,, m’y, etc., we are obliged to use
the observed marginal totals as the nearest approximation we can get to the
J : IY ae :
correct values, although n, is not, in general, equal to m,; Vk A similar assumption
is of course always made in the formulae for the probable errors of samples, where
the sample value is put ultimately for the population value.
8b. Correlation Ratio of Restricted Tables. It is obvious that the ordinary
method of calculating the correlation ratio also requires modification with Tables
of this type; for this method is based on the differences between the means of
the marginal totals and the means of the arrays. Now, in restricted Tables it
would be impossible for the means of about half the arrays to approximate to the
means of the marginal totals and it would be fallacious to base any conclusion on
the deviations of the observed means from impossible values.
A nearer approximation would be to take the pseudo » from the formula
y= S {ne Ya a¥i)}
Nao,? :
where qj; is the mean of an array of the independent probability numbers; but
the denominator of this formula must be modified in such a way that in a case
of perfect association, 7 = unity. The desired result is obtained if we put >?
instead of o,?, where
>? = SS (y = ali)
N
We may write
xs SS(Y= Tat a ~ a
- N
= SS (y ge Ya) ak S (Re (Yau me ai)’ a 28S (y ae Ya) Ya x a¥i)
* N N N
S Cs Gia) S {Ny (Ya = ai)?
Fu. ain ie toes oe Nie eae
since the third term vanishes ; hence
2 SiMe (Go ai) YN
Jas (Nz Fq7)/N +8 {nz Ya- ail} /N
Stn Oa)
But DAAC) | eee
u No,? 1-7,
and S [Ne (Ya = adi)”
Novae eam
where , is the crude 7 found by the ordinary method.
H. Watt 437
We have, therefore, the value of the correlation ratio of restricted Tables
given by
7? Sey ee Ny a
1— 9." + 9)" ;
or ae
7 a
V1 — ei + Hp"
The correlation ratio has been found by the method described above for all the
restricted Tables; it has also been determined by the ordinary method for a few
of the other Tables, but no correction for number of arrays has been applied.
The results, together with the coefficients of correlation and of contingency,
are given in Tables 8 and 9.
Regression curves for all the restricted Tables are given on Plates (a—e). The
continuous line is the independent probability curve and the broken line the
curve of the observed means. It follows that the area between the curves,
weighted, of course, with the marginal totals, gives a measure of the correlation
ratio between the two characters.
Each set of three figures for two particular characters, namely, those for the
right hand, left hand, and both hands respectively, will generally be found to
resemble each other closely. Irregularities occur chiefly with composites but this
is not surprising if we consider the nature of this class.
Sc. Coefficients of Correlation of Restricted Tables. A glance at the diagrams
of means of the restricted Tables, Plates (a—e), shows that the regression is
generally non-linear; it is also evident that a sensible value of r is introduced by
the restriction*. Hence the value of 7 as found by the ordinary product-moment
method is (1) too small because of the skewness of regression and (ii) too Jarge on
account of the restriction. These two contrary causes render the coefficient of
correlation of restricted Tables unreliable and therefore quite valueless ; for even
if it sometimes agrees fairly closely with the correlation ratio and the contingency
coefficient, this agreement is probably due to the fact that the two sources of
error counterbalance each other.
In the remaining Tables, for which the results are given in Table 9, the
regression is frequently skew; for this reason and for those given above, I have
rejected the values of the coefficient of correlation in the sequence and have based
my conclusions on the contingency coefficients, confirmed in general by the corre-
lation ratio.
* For example, in small loops and large loops, left, the case in which the difference between r and ¢
is the greatest, the independent probability numbers have the correlation coefficient — 512 (instead of
the theoretical value zero), as compared with —-+507 of the observed numbers. In the case of arches
and small loops, both hands, 7 for the independent probability numbers is — ‘148, as against + °147 of
Table 8.
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‘saAmng [enjoy = seury usyorg ‘saaang AypIqeqoig yuepusdepuy = seurry snonutyu0D
yfaT ‘saysodumoy
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“ybry ‘sdooT abav'yT
‘SJULIG-doSUly IO} soaang uowsseisoy
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qybuz ‘sajrsodwop
H. WailtTE 443
TABLE 8.
Table in
y yee | yalp | yx) | ayTe | xyIp | xy? 0 Cc Appendix
Arches and Small Loops 062 | (192 | :242 | -227 | °228 | -266 | -264 | -251 | -305 I
5 Large Loops 273 | °273 | °194 | °198 | °298 | °215 | *220 | -209 | -246 IT
Z % Whorls 317 | °359 | *283 | °290 | °353 | *286 | °292 | 291 | 335 III
3 Composites : 146 | °154 | °139 | 139 | °157 | °140 | -140 | +140 | 154 IV
| Small "Loops and Large Loops 422 | 438] ‘074 | -082 | *430 | °073 | 080 | 081 | -166 Vv
a=) $6 Whorls 585 | 622 | °313] °371 | 574 | 315 | °385 | °378 | °408 VI
a0 Composites 270 | °273 | *207 | *210 | *273 | *197 | 201 | -205 | *228 VII
| Large Loops and Whorls -234 | -239 | 079 | 081 | °305 | 112 | 116 | -097} -162| VIII
Composites 093 | °138 | :093 | °093 | °104 | 045 | 045 | 065 | °128 IX
Whorls and Composites 020 | °162 | °117 | °126 | (032 | ‘061 | °061 | -088 | 137 xX
Arches and Small Loops 038 | 197 | °271 | +267 | *249 | *301 | -292 | +281 | °335 XI
a Large Loops *333 | 344 | -253 | -261 | °375 | -293 | -302 | -280 | °309 XII
Whorls "255 | -286| °232 | -235 | *281 | *232 | -235 | +235 | -274 XIII
'e ‘ Composites 154 | +159] +144 | 7145 | 164 | -147 | -148| -146 | 162) XIV
s Small Loops and Large Loops |— *507 | 534 | ‘095 | °112 | °510 | °028 | 025 | -055 | *120 XV
os ¥ Whorls *565 | 623 | °364 | *422 | °570 | °310 | °353 | °386 | -419 XVI
o Ne 365 | +392] -298 | -309 | 401 | 281 | 293] 301) 311}; XVII
4 | Large Loops and Whorls 160 | °193) °129 | -131 | *260 | °157 | -160 | °145 | -236| XVIIT
Composites 080 | -131 | -063 | -063 | ‘088 | ‘007 | 007 | 021 | -108 XIX
Whorls and Composites 115 | +208 | °217 | °217 | 173 | *202 | -201 | -209 | °244 XX
Arches and Small Loops "147 | -340 | -387 | -381 | °279 | °358 | 350 | °365 | -440 | XLVI
B 6) Large Loops 364 | 375 | -298 | 306 | 482 | °364 | °384 | 343 | 383} XLVIT
f= ‘ Whorls 319 | -409| +355 | °363 | 397 | °341 | 348 | 355 | 402 | XLVIIT
3 Composites 203 | -227 | 220 | 220 | 226 | -212 | -213 | -216 | 239} XLIX
& | Small "Loops and Large Loops A471 | °527| 162 | *187 | 476 | -059 | -O71 | *115 | °234 L
ee ,, Whorls ‘638 | *707 | +421 | 511 | °670 | -412 | -478 | -495 | -503 LI
a Composites 382 | 393 | °331 | °339 | -402 | 333 | °341 | *340 | °365 LII
Large Loops and Whorls 147 | "194 | °178 | °178 | °323 | °228 | -234 | -204 | 333 LIII
Composites 020 | 109} -085 | 085 | :087 | ‘066 | ‘066 | °075 | *181 LIV
Whorls and Composites 150 | 280} -295 | *294 | °195 | -285 | +233 | *260 | 320 LV
Remarks on Table 8. A comparison of the Correlation Ratio with the Con-
tingency Coefficient of the Restricted Tables.
(a) The values of 7 and C are generally in very close agreement.
(6) The value obtained for 7 is, however, always less than that for C.
(c) In only three cases does the difference between y and C exceed 011.
probable error of 7 ranges from ‘015 for the smallest values to 011 for the largest ;
it will also be remembered that no corrections have been applied to 7 nor to C,
since we do not yet know what these corrections should be for restricted Tables.
We may assume, however, that, as with ordinary Tables, correction would modify
m less than it would diminish C, and the corrected values of 7 and C would thus,
in all probability, agree somewhat more closely than at present.
Biometrika x
The
57
444 Association of Finger-Prints
TABLE 9.
Right and Left Hands.
Table in
Oe Ct 1 Appendix
Arches 2 and Arches Z ... ... | +°686 + 008 664 688 = XxI
ie Small Loops Z ... | +°160+°015 "285 302 | -2344+-014| XXII
a Large Loops Z... | —"297+ 014 "322 337 — XXIII
a Whorls ZZ... .... | —°257+°014 ‘290 | -307 -- XXIV
Composites Z ... | —*140+°015 “118 ‘161 -- XXV
Small Loops R and Arches L ... | +°185+-015 *309 325 | -2834+:014| XXVI
A Small Loops Z| +:711+ 007 631 635 2 XXVII
ss Large Loops Z| — °378+°013 382 “393 = XXVIII
by Whorls Z ... | —'494+ ‘011 “499 506 = XXIX
Composites Z | — -290+-014 292 | -309 aoe XXX
Large Loops Rand Arches L ...| —'275+:014 297 314 = XXXI
i Small Loops L | —-217+-014 | -262 | +282 ee 0.8 Gk
" Large Loops Z| + °550+:011 519 525 — XXXIII
2 Whorls Z . | —*123+°015 ‘210 235 | 159+ 015 | XXXIV
Composites ZL | —-017+°015 ‘000 ‘089 — XXXV
Whorls # and Arches L ... ... | — 3808+ 014 BB B51 = XXXVI
5 Small Loops Z — 555 + °010 534 “540 = XXXVII
5) Large Loops Z +°021+°015 283. ‘301 | 170+ °015 |XX XVIII
x Whorls Z . +°741+:007 ‘670 672 = XXXIX
Composites ib +°280+ °014 296 313 = XL
Composites Rand Arches L ... | —°146+ °015 15 |, 59 = XLI
Small Loops L| —:188 +°014 72 203 = XLII
x Large Loops Z|} + 131 +:°015 25) ‘166 a XLII
Whorls £ . | +°059+°015 PLA 168 | -105+:015| XLIV
- Composites Z... | +°250+°014 367 379 =— XLV
Further Remarks on Tables 8 and 9. The results given in these Tables show:—
(a) A general agreement between the correlations for the same pair of classes
of prints whether obtained by different methods from the same Table (omitting
values of r in Table 8), or from different Tables, the principal exceptions being
those for which the correlation ratio has been calculated in Table 9.
(b) A wide range in the magnitude of the results for different pairs of prints.
(c) ‘The association between any class of print in one hand and the same class
in the other is, in general, as might be expected, much higher than any other
association of these Tables. Omitting the composites the remaining four con-
tingency coefficients between the same class in different hands are, with one
exception, each greater than any others; the same may be said of the correlation
coefficients, the exception in each case being the correlation between whorls in
the right and small loops in the left hand, which is slightly greater than the
correlation between the large loops in the right and left. Even with the com-
posites the contingency for the two hands is greater than that for composites with
* Values of contingency coefficients corrected for number of cells.
+ Values of contingency coefficients not corrected for number of cells, given for the sake of comparison
with other Tables.
+ The value of 7 is in all cases Nyx Ney
H. WattE 445
any other class found from any of the Tables, while the correlation coefficients have
five exceptions to this general rule.
(d) The contingency coefficients given in Table 8, where the two hands are
taken together, are, with two exceptions, greater than the corresponding coefficients
in other parts of Tables 8 and 9. The exceptions are (1) the contingency co-
efficient ‘234 for small loops with large loops of Table 8 is slightly less than those
in Table 9; and (2) the coefficient 503 for small loops with whorls in Table 8 is
rather less than that for whorls (right) with small loops (left) of Table 9.
A further study of the above Tables shows that :—
Large loops are closest to arches.
Arches * r whorls.
Whorls FS FA small loops.
Small loops ": a whorls and then to arches.
Composites is » small loops and then to arches.
The suggestion thus arises that arches and whorls have the closest natural
resemblance to intermediate sized loops, and also that the “natural order” of the
classes of finger-prints is :—
(1) Large Loops, (2) Arches, (3) Whorls, (4) Small Loops, (5) Composites.
This is more clearly seen from the following arrangement of the contingency
coefficients.
TABLE 10.
Contingency Coefficients of Right Hand.
Large | arches | Whorls | S™2ll Composites
Loops Loops
Large Loops _... 1 246 162 166* 128
Arches ... aes "246 1 °335 “305 154*
Warorlsine.. as 162 "335 1 408 137
Small Loops _... 166 "305 408 1 228
Composites ie 128 154 137 228 1
TABLE 11.
Contingency Coefficients of Left Hand.
ieee Arches | Whorls ae Composites
Large Loops... 1 309 | +236 120 103
Arches ... ee “309 1 | °274 tooo "162
Whorls ... Bee "236 214 et “419 244
Small Loops... *120 335 =| 419 1 “311
Composites... 103 162 | 244 ‘B11 1
L te
* Coefficients which do not agree with the proposed ‘‘ natural order.
57—2
446 Association of Finger-Prints
TABLE 12a.
Contingency Coefficients of Right Hand with Left.
Right Hand.
; Te Arches | Whorls tae Composites
vz
a
tr | Large Loops _... 519 "322 283 *382* *125*
eS) SArchesayen: Ai °297 664 337 309 “115
3 Whorls ... og 210 “290 670 “499 127
1) Small Loops... "262* "285 534 631 172
Composites wa “000 ‘118 *296* | :292 367
(Corrected for number of Cells.)
TABLE 120.
Contingency Coefficients of Right Hand with Left.
Right Hand.
os rae Arches | Whorls Tae Composites
ae
eB
hi | Large Loops _... 525 337 301 *393* 166*
is Arches ... aoe 314 688 “351 325 159
EN Wo WRI eae ac "235 307 “672 ‘506 "168
I | Small Loops... "282% | -302 540 "635 2038
Composites ao 089 161 "313* | 309 — 379
L$ $$ —$__ J}
(Not .corrected for number of Cells.)
TABLE 13.
Contingency Coefficients of both Hands taken together.
rae Arches | Whorls nee Composites
Large Loops _... 1 “383 333 234 181
_ Arches ... hee 383 1 "402 *440* 239
Whorls ... ie 333 "402 1 503 *320
Small Loops... "234 “440 503 1 . 361
Composites aa 181 | 289 *320 361 1
The contingency coefficients of the right hand with the left have been given
both corrected and uncorrected for the number of cells and both sets of results
point to the same conclusion.
* Coefficients which do not agree with the proposed ‘‘ natural order.”
H. WalItTe 447
The proposed “ natural order” of the types is supported by the above Tables,
only eight coefficients out of the fifty-five not being in complete agreement. In
four of these cases the difference is very small, most likely well within the probable
errors, and they may therefore be regarded as insignificant.
A similar arrangement of the correlation coefficients still further supports the
proposed order, though not quite so conclusively, probably on account of spurious
correlations.
9. Association between the various Fingers. In this section I have calculated
the contingency coefficients only, the classes being arranged in the order found in
Section 8, p. 445.
It would, of course, be possible to obtain Tables with much finer grouping
either by further subdivision of the loops or by making use of the “secondary
classification” described by Galton or Henry (see footnote, p. 421). All such finer
grouping would raise the contingency; the extra labour involved by the addition
of some three or four rows and columns to each Table would, however, be so con-
siderable that the question arises whether some allowance can be made for the
coarser grouping employed. ‘This can only be done if we may suppose a “natural
order” of some kind with a frequency roughly approaching the normal. This
gives a rough upper limit to the contingency and is the purport of the work in
the earlier sections on “natural order” and corrections.
As an example of the effect which finer grouping has on contingency I have
found the contingency between the index fingers of the two hands by means of
a “seven by seven” Table, the radial and ulnar loops being separated, and also by
means of a “five by five” Table in which no distinction is drawn between the
radial and ulnar loops. The results in this case, not corrected for grouping, are
653 and ‘626; when corrected for grouping these results become ‘704 and °698,
respectively. ‘They are so nearly identical as to suggest that no very material
advantage would be gained by a further subdivision of classes.
On the assumption that there is a certain degree of continuity in the distri-
bution I have corrected all the results for grouping as well as for the number of
cells. The method employed for the former correction is fully described by
Professor Pearson in Biometrika*.
The following Tables give the contingency coefficients for each finger with
each other finger. The two sets of coefficients are included, viz. those which are
not corrected for grouping, that is, which are obtained without any assumption of
a “natural order” and those which are so corrected, in order that the conclusions
based on the latter may be compared with those based on the former.
* «On the Measurement of the Influence of ‘Broad Categories’ on Correlation,’ by Karl Pearson,
F.R.S., Biometrika, Vol. 1x. pp. 116—139.
448 Association of Finger-Prints
TABLE 14a.
Contingency Coefficients of Right Hand.
R, Ry Rs Ry R;
Rk, 1 429+ 011 *455 469 A73
R, 429 1 “645 576 519
hs *455 645 1 665 565
Ry “469 576 “665 1 *690
Rs 473 519 565 “690 1
(Corrected for Grouping.)
TABLE 140.
Contingency Coefficients of Right Hand.
R, 1 373 +012 “379 *400 385
Ry 373 1 561 ‘S11 441
Rs 379 561 1 568 *460
R, “400 ‘511 568 1 576
Rs; 385 441 ‘460 “576 1
(Not corrected for Grouping.)
TABLE 15a.
Contingency Coefficients of Left Hand.
L, Ly Ls; Ly L;
I 1 503 “465 “474 508 + ‘012
Ly 503 1 675 “609 539
Ls 465 675 1 “724 585
‘ZL; 474 -609 "724 1 “711
Le “508 539 585 ‘711 1
(Corrected for Grouping.)
TABLE 150.
Contingency Coefficients of Left Hand.
L, Ly Ly Ly Ls
Ly 1 435 390 “401 ‘410+ 014
LT, 435 1 582 529 447
ip -390 582 1 ‘611 ‘471
; -401 529 ‘611 1 ‘577
i “410 “447 ‘471 ‘577 1
(Not corrected for Grouping.)
—
eA RENE SEER NI OEE NG A ac al bl mr tals SS aa pie:
H. WAITE 449
TABLE 16a.
Contingency Coefficients of Right Hand with Left.
R Ry Rs R, R,
Ds; 177 441 -440 446 | -424
Ss 698 [5x5 Table . se |i @ose
L, “479 an iE oe Table 640 559 521
Ls "427 ‘608 ‘786 | ‘669 -| 561
D4 "446 ‘587 663 | °814 | ‘675
Ls “501 ‘515 OM OLS | “899
(Corrected for Grouping.)
TABLE 16).
Contingency Coefficients of Right Hund with Left.
R, Ry R; R, Rk;
I “649 *B85 368 383 "347
arGtG 626 [5 x5 Table] KS hick ie
ie 412 653 [7x7 Table] | 2°) 493 439
Lz *356 “530 656 572 “459
4 “375 514 558 “702 "556
Ts “402 "432 431 534 ‘707
(Not corrected for Grouping.
Remarks on Tables 14a, 15a, and 16a. (a) It will be seen from these
Tables that the association of types between corresponding fingers of the two
hands is, with one exception, always closer than that between any other pair of
fingers. The order of magnitude of these associations is :—
(1) Little Finger, (2) Ring Finger, (3) Middle Finger, (4) Thumb, (5) Index
Finger.
(b) If we omit the thumb for the present, leaving it for separate comment,
and consider the association between corresponding fingers as of the “first order,”
that between fingers of consecutive rank, such as R, and R;, or R, and L, as of the
“second order,” and so on, we notice a significant relation between any particular
association and its “order.” Thus: j
First order associations range from *899 to “704 or ‘698,
Second o i - = ‘724 to 608,
Third 2 : ra K 609 to °537,
Fourth Fe A ; Pr 539 to 515.
The amount of overlapping in these ranges appears to be quite insignificant.
450 Association of Finger-Prints
(c) It follows from (6) that if in any of these Tables we start from a first order
association and pass in any direction through those of other orders we find a
continuous and rapid fall; that is, a finger is always more closely related to a
consecutive finger than to one more remote (but see (a)); and the greater the
difference in rank between two fingers, whether on the same or on different hands,
the less close is the association between them.
(d) The association between any pair of fingers in one hand is, in general,
closer than either of the corresponding associations between a finger of the right
and one of the left hand. There is one exception to this rule in associations of the
second order, one in the third and one in the fourth.
(e) The associations of the left hand are in every case closer than the corre-
sponding associations of the right.
(f) The associations of either thumb with any finger all fall below those of
the fourth order of (b), and the range of the sixteen coefficients is only from *424
to ‘508. As it is difficult to base any conclusions on these figures as to the
relations between the thumb and the various fingers, I have carefully checked
them by reworking the whole of the calculations involved, but have in every case
arrived at the same result. I have also found the probable error* for the largest
and for one of the smallest coefficients of the set. As the contingency coefficients
are all of the same order of magnitude and the number of individuals the same in
all cases, the probable errors of all will be of about the same magnitude and it is
unnecessary to calculate more. The probable errors in the two cases being of the
order ‘011 the differences in the contingency coefficients may be regarded as
insignificant. Although in three cases out of the four the contingencies of the
thumb with the middle, ring and little finger respectively are in ascending order
of magnitude, the differences are so small in comparison with the probable errors
that no conclusion can be drawn as to the relations between the thumb and the
various fingers. We may notice, however, that the rule (d) holds good for the
thumbs with but two exceptions.
The contingency coefficients given in Tables 14 b, 156, and 16, are all smaller
than the corresponding results of the other series, but a careful study will show
that the remarks (a) to (g) almost invariably apply to these Tables also.
Note. In some preliminary work on this paper I classified the types as
follows :—(1) Arches and loops with 1—3 ridges, (2) Loops with 4—10 ridges,
(3) Loops with 11—14 ridges, (4) Loops. with 15 or more ridges, (5) Whorls,
(6) Composites. With this classification the following contingency coefficients
were found for corresponding fingers of the two hands:—Thumb ‘686, Fore-
finger 642, Middle finger ‘686, Ring finger °730, Little finger ‘738. These results,
which were not corrected for grouping, are seen to agree very closely with those
* The method employed is that given in Biometrika, Vol. v. Parts 1. and u., ‘‘On the Probable
Error of Mean-Square Contingency,” by John Blakeman and Kar! Pearson.
H. WaAITE 451
of Table 166, the values being rather larger probably on account of the slightly
finer grouping.
10. Comparison with Results of Previous Work. It would be well to compare
briefly some of my results with those of the two works mentioned on p. 421.
Whiteley and Pearson arrived at the following conclusions :—
(i) The hand is a very highly correlated organ, far more highly correlated
than the skull and even somewhat more so than the long bones.
(11) The parts of the left hand are distinctly more closely correlated than
those of the right.
(ii) The order of correlation of the first finger joints is identical for both
hands. This order is as follows :—
(a) The external fingers have the least correlation and the little finger always
less than the index.
(6) <A finger has always more correlation with a second than with any other
finger from which it is separated by the second.
(iv) With corresponding members on both sides the extreme pairs show least
correlation, and the pair of middle fingers higher correlation than the pair of ring
fingers.
In the paper of Miss Lewenz and Miss Whiteley the chief results which are
comparable with those for the finger-prints are the following :—
(v) There is a slight, but we cannot say definitely significant, preponderance
in the correlations of the right hand bones over those of the left.
(vi) Dividing the hand into marginal members, i.e. thumb, index and little
fingers, and central members, i.e. middle and ring fingers, and the bones into
“lower bones,” i.e. distal and middle phalanges, and “upper bones,” i.e. metacarpal
bones and proximal phalanges, the correlations roughly speaking are highest for
the upper bones of the central members and become less as we move out from this
upper centre towards the lower and marginal parts of the hand. This is true
whether we take pairs in lateral or in longitudinal series.
(vii) The highest correlations occur between corresponding bones of the right
and left hands.
(viii) Generally there is a “rule of neighbourhood,” i.e. any bone is more
closely correlated with a second of the same series than with any other from which
it is separated by that second.
The above conclusions are to a certain extent mutually corroborative: e.g. (v1)
and (iv) are in agreement, and (vill) agrees in substance with (111)). Again (vil)
agrees with Table IV, p. 130, of the “ First Study,” while (aii a) is in general sup-
ported by Table XXII of the “Second Study.” On the other hand (ii) and (v) do
not agree. It should be noted, however, that the “ First Study ” was based on the
measurements of the first finger joint only of both hands of 551 women, while for
the “Second Study,” in which all the finger bones were measured, only 37 to 44
Biometrika x 58
452 Association of Finger-Prints
skeleton hands were available. The writers of the latter paper state that in con-
sequence of the comparatively small number of bones measured they look upon
that study “as one of suggestion rather than of definite statistical proof,’ and it
is possible that with more adequate data their results might have been somewhat
modified and exceptions less numerous.
There appears to be no direct connection between finger bones and the patterns
of finger-prints, but it is distinctly interesting to find that some of the most
striking relations discovered amongst the former also exist in the latter. In
particular, my conclusion (a) agrees with (vii), (c) with (1116) and with (viii), and
(e) with (i1) but not with (v).
11. Concluding Remarks. The most important conclusions reached in this
paper have been summarized on p. 432, and in the Remarks on Tables 8 and 9,
pp. 443—445, and on Tables 14—16, p. 449; it scarcely seems necessary to re-
capitulate them, but a comparison will show an almost perfect agreement although
the sets of results have been obtained by entirely different methods.
The essential results of the present paper are that finger-prints are not scat-
tered at random over the fingers; certain types are more or less peculiar to certain
fingers, and further the appearance of one type is correlated with the appearance
of asecond. In this respect certain fingers are more closely related to each other
than to any third finger, and the distribution of this relationship is in general
similar to what is known of the like distribution of the correlations of the bones of
the same fingers.
It has been already stated that the material used is taken entirely from adult
males of the lower type of the artisan and labouring classes; it would be of
interest to compare the results obtained with those found from the finger-prints
of females of the same grade of society, and also when the material is drawn
from the professional classes.
Tables I to XX, and XLVI to LV, are of a type which I have not previously
met with; novel methods have accordingly been employed in calculating coefficients
of contingency and correlation ratios from those Tables. The general investigation
of Tables of this type offers an interesting problem, demanding further study.
I am deeply grateful to Professor Pearson for placing at my disposal the
necessary material together with a number of books and memoirs bearing on
the subject, and for much valuable assistance given during the course of the
investigation.
It can scarcely be expected, with such a mass of numerical calculation involved,
that the work should be entirely free from inaccuracies, but I trust that no serious
errors have escaped detection. The laborious arithmetic has been much lightened
by the use of a calculator, for the loan of which my thanks are due to the Govern-
ment Grant Committee of the Royal Society of London.
The Tables on which the preceding calculations are based are given in the
Appendix, pp. 453—478.
Large Loops, BR.
Whorls, R.
Small Loops, 2.
Mise Co WHOS
H. WaItTE
APPENDIX.
TABLE I.
Arches and Small Loops, Right.
Arches, R.
0 1 2 if 3 4 5 Totals |
453
Composites, R.
Totals} 1541 | 294 111 33 16 5) 2000
TABLE II.
Arches and Large Loops, Right.
Arches, R.
: 0 1 2 3 4 5 Totals
0 | Qe 16 457
1 489 | 114 46 10 O — 659
2 400 66 12 O — — 478
3 245 31 4 a — _ 280
5 TORS |) Sp ae eS eR eee Tats
5 18 18 |
1541 | 294 111 33 16 5 2000 |
TABLE III.
Arches and Whorls, Right.
Arches, R.
0 il 2 3 4 5 Totals
o | 512/215 | 86 | 27 | 16 | 5 | 8e1
1 406 61 24 6 0) — 497
2 275 16 Le — — 292
3 168 2 0) = — — 170
J One On | a0 130
5 50 | — — | — = = 50
Totals | 1541 | 294 ial 33 16 5 2000
TABLE IV.
Arches and Composites, Right.
Arches, R.
Totals |
0 1423
1 442
Q 118
3 14
4 3
5 0)
Totals | 1541 | 294 111 33 16 5 2000
58—2
454
Association of Finger-Prints
TABLE V.
Small Loops and Lurge Loops, Right.
Small Loops, R.
@) 1 2 3 4 ii) Totals
RG | Lad
ze 0 104 | 69 67 81 91 45 457
rales od 144 | 99 | 142 | 154 | 120 == 659
2 2 143 | 141 | 123 71 — — | ‘478
ea 8 119 | 101 60 280
© 4 65 | 43 = = == = 108
OA eas 18 | 18
fan] | Mees s
es Totals} 593 | 453 | 392 | 306 | 211 45 | 2000
TABLE VI.
Small Loops and Whorls, Right.
Small Loops, R.
—__——
0 1 2 8 4 5 | Totals
as | ? 78 | 144 | 204 | 211 | 179 | 45
Zi 1066 153 126 60! ao ee
oe |\ 2 TBONO2 G55 <2 5 a5) ee ee
B87 A125 438 (| eae
| 4 | 104 |-.26
= 5 50 | — = - — --
Totals } 593 | 453 | 392 306 211 45
TABLE VII.
Small Loops and Composites, Right.
Small Loops, R.
0 1 2 3 4 5 Totals
= 0 353 | 294 279 257 195 45° | 1423
a 1 165 121 | 93 47 16 — 442
2 2 62 34 20 2 — — 118
g 3 10 4 0) = = = 14
euliely 3 Oni | 2 Se eee Sale alee 3
5 0 )
i)
iS)
Totals | 593 453 392 306 211 45 2000
TABLE VIII.
Large Loops and Whorls, Right.
Large Loops, R.
0 1 | -2 3 4 5 Totals
es | 197 | 289 | 162 | 133 62 18 861
7 86 | 130 | 148 87 46 Ss 497
& AT 84 | 101 60 = = 292
2 27 76 67 = = == 170
a=} 50 80 130
S 50, | sae = ene 50
457 | 659 | 478 280 108 18 | 2000
H. WaItEe 455
TABLE IX,
Large Loops and Composites, Right.
Large Loops, R.
0 | 1 2 3 4 5 Totals
RS} 66 )6f 37 | 471 | 318 | 205 | 94 | 18 | 1498
ae) oi Some SSe TSO ltsy | eid, keke |) dag
Ee alae? le if com ouraids. jer len | ais
Bl 3 9 2 Be ah tev lte Seo, bs 14
Sih 2 2 gh | eee petty ty a2 = 3
Bl 5 RIN cease | hcp Ae ca 0
S
Totals | 457 | 659 | 478 | 280 | 108 | 18 | 2000
TABLE X.
Whorls and Composites, Right.
Whorls, R.
= 0
ig os
a= DQ
ay 3
4
el 5
ie)
Totals
TABLE XI.
Arches and Small Loops, Left.
Arches, LZ.
By Totals
nj :
o 0 496
Qy 1 425 |
Sale? 366 |
4 2 315
= 4 283
a| 6 115 |
NM
| Totals 2000 |
TABLE XII.
Arches and Large Loops, Left.
Arches, L.
1
Large Loops, L.
Me Cs W@RS
re)
°
co
©
=
DR
456
Association of Finger-Prints
TABLE XIII
Arches and Whorls, Left.
Arches, L.
Totals
Whorls, L.
Tit Ss WHR DS
Totals | 1542 102
TABLE XIV.
Arches and Composites, Left.
Arches, L.
| areal ee | 4 | 5 | Totals
20 4 1436
lo
Oo
1 Oe ely OA
2 eas |} nes
3 22
5 =) = 4
5 =| = i
Composites, L.
Totals
1542 | 290 102 42 | 20 0) 2000
TABLE XV.
Small Loops and Large Loops, Left.
Small Loops, L.
[hg 0 1 Bie ie 4 | 5 Totals
Sj |
- 0 82 48 | 63 83 | 124 | 115 515
2, 1 121 73 94 | 130 | 159 = 577
S 2 97 | 116 | 108 | 102 =e = 423
4 3 100 | 110 | 101 = =e = 311
ee 4 63578 141
0 5 335 | == = = a = 33
Si
Totals | 496 | 425 | 366 | 315 | 283 | 115 | 2000
TABLE XVI.
Small Loops and Whorls, Left.
Small Loops, L.
MA Ss WHOS
Whorls, L. Composites, L.
Composites, Z.
TABLE XVII.
H. WaAITE
Small Loops and Composites, Left.
Small Loops, Z.
TABLE XIX.
Large Loops and Composites, Left.
Large Loops, L.
0 1 2 3 4 i) Totals
0 234 292 264 264 | 267 115 | 1486
i 176 110 84 48 16 — 434
2 64 19 17 3 = — 103
3 17 4 1 — — -—— 22
J 4 one 4
5 ig |p c= ahs ee ee 1
Totals | 496 | 425 366 315 283 115 } 2000
TABLE XVIII.
Large Loops and Whorls, Left.
Large Loops, L.
| 0 1 De ae ep Py ese notals
0 338 315 195 165 106 33 1152
1 63 94 114 102 35 — 408
2 22 58 79 44 —_ — 203
3 26 63 35 — — — 124
y 45 | 47 92
5 21 21
Composites, L.
0 1 | 2 3 4 5 | Totals
0 399 | 276 1436
1 102 124 | 119 66 23 — 434
2 23 41 24 15 — — 103
3 7 Mil 4 — — — 22
4 2 2 — 4
5 ie ee atl 1
Totals | 515 577 | 423 311 141 33 2000
TABLE XX.
Whorls and Composites, Left.
Whorls, L.
0 1 2 3 4 5 Totals
ON, Fra aa ae es as | Nin!)
0 924 | 249 120 66 56 21 14386 |
1 173 | 120 59 46 36 — 434
2 44 26 21 12 — — 103
3 8 al 3 — — = 22
4 2 2
5 1 — — = — =
Totals} 1152 | 408 | 203 | 124 | 92 | 21
457
458 Association of Finger-Prints
TABLE XXI.
Arches, Right, and Arches, Left.
Arches, R.
0 1 2| es 4 | 5 | Totals
0 1369 | 145 24 3 1 0 1542
See 146] 97 | 40 a 0 | 0 | 290
wn 2 24 45 26 5 2 0) 102
2 3 2 6 16 12 6 ) 42
Sala es Oylcanl 5 6 ae 20
<q bi 0) 0 (0) 0 2 2 4
Totals J 1541 | 294 111 33 16 5 2000
TABLE XXII.
Arches, Right, and Small Loops, Left.
Arches, R.
0 1 2 3 p 5 | Totals”
Qj — aaa
Pen 473 | 14 5 0) 2 2 496
al 342 | 47 17 10 6 3 425
Sere 249 | 68 29 15 5 0) 366
Hl 3 209 | 74 28 2 2 0 315
= L 187 | 66 26 4 0 ) 283
= 5 81 | 25 6 2 1 0) 115
SI
n |
| Totals} 1541 | 294 | 111 33 16 5 | 2000
TABLE XXIII.
Arches, Right, and Large Loops, Left.
Arches, R.
Totals
nD 515
or 577
8 423
4 311
o 141
2 33
eS
2000
TABLE XXIV.
Arches, Right, and Whorls, Left.
Arches, R.
g Totals
. 1152
> 408
r) 203
B 124
a 92
= 21
Totals
H. WaltTE 4
TABLE XXV.
Arches, Right, and Composites, Left.
Arches, R.
2 3 4 5 Totals
= Obie esr 16. |) 5. fide
2B 13 4 0) 0) 434
2 2 1 0 0 103
2 1 0 0 4) 0 22
Q4 0) 0 0 0 4
g 0 0 ) 0) 1
[e) = aD
Totals J 1541 | 294 111 33 16 7) 2000
|
TABLE XXVI.
Small Loops, Right, and Arches, Left.
Small Loops, R.
0 1 2 3 4 5 Totals
558 369 QO liter 129 30 1542
ot Dae aon |) va | me | by | tL |b 290
gf Mieilbesisfe nis) ul) aoe |) 01 4 | 102
3 Tech aley ihe ie 3 0 49
e 3 5 7 4 1 0 20
“a 2 2 0 0 0 0 4
593 453 | 392 306 211 45 2000
TABLE XXVII.
Small Loops, Right, and Small Loops, Left.
Totals
496
425
366
315
283
115
Small Loops, L.
Aes WeRS
Totals 2000
TABLE XXVIII.
Small Loops, Right, and Large Loops, Left.
Small Loops, &.
Totals
515
577
423
311
141
33
Large Loops, L.
2000
Biometrika x 59
9
460
Whorls, L.
Arches, L.
Small Loops, L.
Composites, L.
Association of Finger-Prints
TABLE XXIX.
Small Loops, Right, and Whorls, Left.
Small Loops, BR.
0 1 2 3 J 5
0 159 217 286 | 249 198 43
i 130 143 79 | 48 12 1
Q 112 58 19 13 I 0
} 92 25 6 | 1 10) 10)
y 79 10 eo 0 1
5 21 0) O 10) 0 0)
Totals | 5938 453 392 306 211 45
TABLE XXX,
Small Loops, Right, and Composites, Left.
Small Loops, AR.
0 1 | 2 8 | 4 5 Totals
0 | 323 | 311 | 299 | 267 | 191 | 45 | 1436
1 194 LOD IF 82 32 | 17 0) 434
2 60 22) | ll | vi | 3 0 103
3 12 10 | 0) O 0 0 22 .
J 4 0 | Oo 0 0 0 4
5 Oneal 0 0 OueaO 1
Totals | 593 | 453 392 306 | 211 45 2000
TABLE XXXI.
Large Loops, Right, and Arches, Left.
Large Loops, &.
| 0 | 1 | 2 3 | } | 5 Totals
0 290 483 394 | 255 1030 ell 1542
I 22 ;
3
J
ii)
Totals} 457. | 659 | 478 | 280 | 108 | 18 | 2000
TABLE XXXII.
Lurge Loops, Right, and Small Loops, Left.
Large Loops, R.
Totals
496
425
366
B15
283
115
Totals 3 2000
— oe
H. WaAItTE
TABLE XXXIII.
Large Loops, Right, and Large Loops, Left.
Large Loops, R. —
a baer eral | | |
lO A OR is | isa) 3
ome 7 121 | 956 | 135 | 59 5
Sa 2 53 | 135 | 149 | 67 | 18
4 3 17 | 66 | 101 77 =| “Al
) 4 2 20 36 AT 32
SS 5, 0 4 ¢
4
Totals | 457 | 659
. TABLE XXXIV.
Large Loops, Right, and Whorls, Left.
Large Loops, R.
Whorls, L.
Meer Cs DMS
Totals
TABLE XXXV.
Large Loops, Right, and Composites, Left.
| | 0 | ae 3 J | 5 | Totals |
——— —
. 0 Boel ATeeasoSn elo 1 Sle | 13 1436
cs 1 90 | 140 | 107 70 93--) 4 434
| 2 Hh eels ees, alg) 4" 103
Pima 4 St Rates Ce eal Olle 20 22
S 4 2 2 Olea OF Oe |e 4
5 5 Ss ieee Oita |e): OM | 102 Inn 20 1
O | | aot
Totals} 457 | 659 | 478 280 | 10s | 18 | 2000
TABLE XXXVI.
Whorls, Right, and Arches, Left.
. Whorls, R.
S 0 | 526 | 405 | 262 | 169 | 130 | 50 | 1542
al 193 t6On hoya sie O:| 0 4 290
Ms 2 80 | 19 3. || 20 One 20) AP 102
S| 3 38 4 0 0) One) (0) 42 |
J 20; nO OF). 20 OF 0 20
nl ode deh OG |e 20 O20) 70%), 10 4
| Totals | 861 | 497 292 |170 | 130 | 50 | 2000
462 Association of Finger-Prints
TABLE XXXVII.
Whorls, Right, and Small Loops, Left.
Whorls, R.
S—
& O
nD |
jor
2 |
alii | 80 5
= 202 | 65 15 1
S 99 | 16 O 0
g
NM
Totals 497 292 170 130 50
TABLE XXXVIII.
Whorls, Right, and Large Loops, Left.
Whorls, R.
3
Large Loops, L.
| Totals | 497
TABLE XXXIX.
Whorls, Right, and Whorls, Left.
Whorls, R.
Whorls, L.
Mey to OWNS
| Totals | 861 497 292 170 130 50 | 2000
TABLE XL.
Whorls, Right, and Composites, Left.
Whorls, R.
0 i 2 3 4 5
NS ‘y Lard 9°
2 0 744 | 331 | 185 93 55 28 | 1436
ey 1 98 | 132 78 58 52 16 434
| 2 16 28 19 16 19 5 103
oa 3 el es 7 2 4 1 22
Es 4 Ouran a1 2 1 0 0 4
5 5 0 0 1 0 0 0) 1
S | ES EET SEES
Totals | 861 | 497 | 292 | 170 | 130 50 | 2000
H. Walt
: TABLE XLI.
Composites, Right, and Arches, Left.
Composites, R.
0 il 2 8 4 5 Totals
|
- 0 1042 | 378 | 107 12 3 O | 1542
ial ae 933 | 46 9 2 0 0 | 290
a 2 84| 16 2 ) 0 0 102
a 3 40 2 0 0 ) ) 42
S 4 20 0 0 0 0 0 20
<q 5 4 ) 0 0) 0 6) 4
2000
TABLE XLII.
Composites, Right, and Small Loops, Left.
Composites, R.
0 i 2 BS 4 5 Totals
Sy See
s 0 496
Qy 1 425
=) 2 366
4 3 315
= 4 283
s 5 115
5
asta 2000
TABLE XLIII.
Composites, Right, and Large Loops, Left.
Composites, R.
; 0 1 i, 3 4 5 Totals
| pent =e
Spano 416 | 80 | 17 2 0 0 515
= 1 416 | 121 36 3 1 (0) 577
io) Zz, Arey || alial 27 6 1 (0) 423
4 3 210 | 79 20 1 1 0 311
0) 4 86 Bi 16 2 0) (0) 141
orl > pals ae 2 Ole 0 0 33
5 |
Totals | 1423 | 442 118 TA tS (0) 2000
TABLE XLIV.
Composites, Right, and Whorls, Left.
Composites, R.
Whorls, Z.
ABW Ss WHO
Totals J 1423 | 442 118 14 3 0) 2000
463
A464
Small Loops.
Large Loops.
Association of Finger- Prints
TABLE XLV.
Composites, Right, and Composites, Left.
Composites, R.
0 eee 3 J
= 0 1100 12765 | 53) 4 6 1
a 1 958°| 130°) 432) 43 0)
ay 9g by 28 14 | 3 1
S 3 8} 8 4 2 )
= 4 0 0 3 lal 30 1
3 5 ) ) eid 0 0)
0
Totals | 1423 | 442 118 14 3
ar TABLE XLVI.
Arches and Small Loops, Both Hands.
Arches.
i) 1 2 B 4 5 6 if 8 9 10 {Totals
0 361 7 1 1 1 |-0 0 0 0 0 2 373
1 205 | 19 7 6 3. k= 30 ) 0 0 5 — 245
g 168 33 12 7 ‘| (0) 5 2 5 — — 223
} 139 36 13 9 S| 6 4 10 — — — 225
105 40 25 10 2. | 6 10 — — —_ — 198
102 | 33 25 18 10 10 — — — — — 198
88 | 37 21 18 15 — — — — — — 179
74 47 23 21 — —- — — — —_— 165
638] 298. |. TS ha) ee eT
45] QI = =< pe = = _ = _ = 66
19-|° = = — se = = =5 — a _ 19
1369 | 291 | 145 90 40 22 19 12 5 5 2 | 2000
TABLE XLVII.
|
Arches and Large Loops, Both Hands.
Arches.
§
0 109 | 32 24 32 19 13 12 11 5 5 | 2 264
if 164 | 55 34 23 ll 6 5 1 0) 0 — 299
2 230 70. _|__33 18 4 3 2 (0) 0) — — 360
3 225 41 27 9 4 (0) 0) 0) — — — 306
4 212 | 45 16 5 1 0 0 = = = — 279
5 160 | 22 9 2 OS XO = _ a= = = 193
6 119 13 2; 1 1 | — — — — — — 136
7 86 9 0 0) —_— — — — — — —_— 95
8 49 3 (0) — — — — == =e
9 1 — — — —_ — pes
10 sy Pees ee | ey ee =
oe Sle
ee
=.
eS ae Ven eee
H. WaAITE 465
TABLE XLVIILI.
Arches and Whorls, Both Hands.
Arches.
| 0 | 2 1 pee eS ee ae ae ee
0 351 | 160 100 66 30 21 16 12 ii) 5 2 768
1 240 69 25 16 7 1 2 0) O O — 360
2 191 38 ll 5 2 (0) 1 0 O — — 248
BS 8 146 14 8 Dy 1 10) 0) O — — — 171
Fly 124] 6 1 1 0 0 63 = 132
2 5 86 3 (0) (0) (6) O — — — — — 89
= 6 ah 1 (6) (0) O — — — 78
t 71 O O 10) — — — —_ — — — 71
8 47| 0 0 ae = 47
9 23 (0) — = — — | 23
10 13 Sly _— — 13
| Totals | 1369 | 291 145 90 40 22, 19 12 5 5 2 2000
|
TABLE XLIX.
Arches and Composites, Both Hands.
Arches.
eee ee ote.) 7 |. 2 |
650 |193 | 101 | 62 | 33 1
40g | 66 | 34 | 19 6
‘ 202 | 92 ale ty i
8 3 1 g
2 0 1
oy 0 0 0
g 0) (0) (0)
} 0 0 ot
e) (0) — a
40
TABLE
Small Loops and Large Loops, Both Hands.
L.
Small Loops.
0 1 2 3 4 5 6 ” 8 9 10
0 44 13 10 18 17 20 16 | cot 31 30 19
1 44 19 13 19 14 30 37 45 42 BG alse
a 2 60 94 | 24 S10) | Bin 42 54 55 SG ip hee
a| 3 5) 31 25 39 43 47 32 Onn ees || ee eee
5 y 49 34 | 41 44 | 46 40 25 ea mene iL se
om 5 35 39 38 33 29 19 = = = = aa
AG 34 | 32 34. 22 TOES | st TR ee al a ec a (ae
= 7 24 | 99 | 29 Oe | reise fl, (aed RS ae ne
eS 8 16 20 13 ye ee oh | eee ee
9 9 Oe RS cee Se tL" a a (bee
10 By a a a ee ee
= j |
Totals] 373 | 245 | 223 | 225 | 198 | 198 | 179 | 165 | 109 66 19
Totals
264
299
360
306
279
193.
136
95
52
13
466 Association of Finger-Prints
TABLE LI.
Small Loops and Whorls, Both Hands.
Small Loops.
Whorls.
Totals 24! 2° Hi 179
TABLE LILI.
Small Loops and Composites, Both Hands.
Small Loops.
| Beales | 7 | 8 | 9 | 10 |Totals
0 1098
1 536
E; a 240
mt Mer es
nD 4 6
S. i)
= ‘
j=)
iS)
Totals | 373
TABLE LIII.
Large Loops and Whorls, Both Hands.
‘Large Loops.
an si 2 8 J a | @ | y | 8 9 | 10 [Totals
0 156 138 133 91 73 51 43 42 29 9 3
1 24 45 54 63 56 39 30 29 16 4 —
2 12 24 36 35 50 3) 30 19 7 — —
a 3 12 16 30 21 35 35 17 5 — a= —
cele 8 7 | 20° | 32 | 96 | 23 | 16
fe 5 3 8 20 20mg 928 10 — -- — — |—
= 6 4 9 24 30 11 — —_—
if 10 18 29 14 — — — aoe
8 11 22 14 — —
9 11 12 — a = -- — —
10 13 (etna ie | ra eee Ba ||
—
Totals J 264 | 299 360 306 279 193 136 95 52 13 3
Composites.
Composites.
H. WaAItTE
TABLE LIV.
Large Loops and Composites, Both Hands.
Large Loops.
Whorls and Composites, Both Hands.
TABLE LV.
Whorls.
0 tr | 2 3 5 5 6 : By ano 10 |Totals
0 26 1098
1 40 536
2 12 240
3 8 85
y 3 26
5 ) Tl
6 ai 6
7 as l
8 ai 1
9 ae 0
10 be 0
Totals 248 | 171 | 132 | 89 2000
TABLE LVI.
Right Thumb and Indea.
Right Thumb.
A SL LL W (6! Totals
Pe
= A 29 97 148 50 28 352
a Sh 12 125 320 139 58 654
x LE 2 27 149 125 36 339
a W 1 26 144 | 260 50 481]
20 C0 2 15 54 75 28 174
aa aD |
Totals | 46 290 | 815 | 649 200 | 2000
Biometrika x 60
468
Right Middle Finger.
Right Little Finger.
Right Middle Finger.
Right Ring Finger.
Association of Finger-Prints
TABLE LVII.
Right Thumb and Middle Finger.
Right Thumb.
A f
SL 18
LL 1
W 2
C 0)
Totals 46
290
SL
LL
428
223
69
20
815
W
31
215
208
160
35
649
200
Totals
TABLE LVIII.
fight Thumb and Ring Finger.
Right Thumb.
|
A SL bh We C Totals
BL 5 60 | 248 166 64 543
W 8 55 | 929 | 355 82 729
eee 2 22 73 57 22 176
| Totals | 46 290 | 815 | 649 | 200 2000
TABLE LIX.
Right Thumb and Little Finger.
Right Thumb.
A SL LL Ww G Totals
A 8 ‘14 5 3 i 31
Se 30 198 | 414 | 162 66 870
BD, 5 61 300 | 304 94 764
W i 10 58 137 22 228
Oo 2 4 38 43 17 107
Totals | 46 290 | 815 | 649 | 200 2000
TABLE LX.
Right Index and Middle Finger.
Right Index.
A SL THs W C Totals
A 127 69 9 5 2 212
SE 187 | 453 119 104 58 921
Tes 30 | 108 144 | 1792 62 516
W 4 16 48 | 178 28 274
C 4 8 19 22 24 thy
Totals | 352 | 654 | 339 | 481 | 174 | 2000
—_— sl a
H. Waitt 469
TABLE LXI.-
Right Indea and Ring Finger.
Right Index.
Totals
Right Ring Finger.
TABLE LXII.
Right Index and Little Finger.
Right Index.
ie isn en ewe) eo leretats
Right Little Finger.
TABLE LXIII.
Right Middle and Ring Fingers.
Right Middle Finger.
Right Ring Finger.
TABLE LXIV.
Right Middle and Inttle Fingers.
Right Middle Finger.
| Ww
Right Little Finger.
60—2
470
Association of Finger-Prints
TABLE LXV.
Right Ring and Little Fingers.
Right Ring Finger.
S SL Totals
&
ey 31
o | 870
= | 764
tS | 228
7m | 107
a aad
i] 543 2000
= |
TABLE LXVL.
Left Thumb and Index.
Left Thumb.
A SL LIL W C Totals
eal ne A733 78 25 | 30 313
SS SL 32 313 355 68 65 833
iG LL 5 40 143 AT AT 282
2 W 3 46 136 166 86 437
ro) C 4 15 55 35 26 135
=|
Totals | 91 547 767 341 254 2000
TABLE LXVII.
Left Thumb and Middle Finger.
Left Thumb.
i SL Totals
to} 0)
A=] y
ew
© SL
es | ie
o| W
= C
3 | Total
| otals
Left Ring Finger.
| Totals
TABLE LXVIIL
Left Thumb and Ring Finger.
Left Thumb.
7
264 | 198
152 | 342
60 | 172
33 48 |
547 767
5
38
110
158
30
341
Totals
66
583
712
491
H. WaItE
TABLE LXIX.
Left Thumb and Little Finger.
Left Thumb.
a Totals
oO
bp
42) 35
Falla aCye 959
ae LL 768
BS W 150
I C 88
ney
3 Totals
TABLE LXX.
Left Index and Middle Finger.
Left Index.
3 A SE ae W © | Totals
=
ty A 117 84 9 Z 3 215
© SL 166 525 80 79 37 887
oS IDE, 19 187 157 139 54 556
ac) Ww 6 Q5 21 171 17 240
SHieo 5 12 15 46 | 24 102
~
8 Totals } | 282 2000
TABLE LXXI.
Left Index and Ring Finger.
Left Index.
Left Ring Finger.
SL LL
833
TABLE LXXII.
Index and Little Finger.
Left Index.
Left Little Finger.
SL LL Ww
Totals
471
472
Left Little Finger. Left Ring Finger.
Left Little Finger.
Left Thumb.
Association of Finger-Prints
TABLE LXXIII.
Left Middle and Ring Fingers.
Left Middle Finger.
Totals
| A S L
13
421
311
88
54
215 887
LL W
556 240
102
Totals
66
583
712
491
148
2000
TABLE LXXIV.
Left Middle and Inttle Fingers.
Left Middle Finger.
SLI
W
Totals
Totals | 215 887 556 240 102 2000
———
TABLE LXXV.
Left Ring and Inttle Fingers.
Left Ring Finger.
| PO Si Pe | W | Cc | Totals
c (0) 1 35
99 44 959
212 73 768
120 15 150
60 15 88
Totals | 66 | 583 | 712 | 491 | 148 | 2000
TABLE LXXVI.
Right Thumb and Left Thumb.
Right Thumb.
A Gn i Wee | motte):
A 91
SL 547
LL 767
Ww 341
C 254
Totals 2000
aa de
H. WaItE
TABLE LXXVII.
Right Thumb and Left Index.
Right Thumb.
A SL LL WwW C Totals
3 A 313
s SL 833
& aE 282
ie W 437
Gay
D 135
om e [SS
Totals 290 | 815 | 649 2000
TABLE LXXVIIL.
Right Thumb and Left Middle Finger.
Right Thumb.
S A Speen | Ww Cc | Totals |
E Fes
fy 23 60 86 32 14 215
ie 17 167. | 414 | 210 79 887
= 0 40 | 232 | 214 70 556
oS 5 18 57 141 19 240
S 1 5 26 | 52 18 102
2
3 46 290 815 | 649 | 200 2000
TABLE LXXIX.
Right Thumb and Left Ring Finger.
Right Thumb
.
Left Ring Finger.
TABLE LXXX.
Right Thumb and Left Little Finger.
Right Thumb.
ra
o
Ey
o_ A
Ee isu
foe a,
~~ W
4 C
ae
os Totals
A SL LL W C
10 17 6 2 (0)
29 207 468 181 74
2 54 275 341 96
3 8 40 80 19
2 4 26 45 11
46 290 815 649 200
473
474
Left Index.
Association of Finger-Prints
TABLE LXXXI.
Right Index and Left Thumb.
Right Index.
Totals
A SL LL W C
a
lalla 91
= SL 547
ale Ae 767
iS W 341
o C 254
SSS SST
Totals 352 654 339 481 174 2000
TABLE LXXXIL.
Right Index and Left Index.
Right Index.
| aol st. sz, \ cee) re wr omlinee
A 313
SL, 295
SLy 538
1H 88
LL, 194
hee 437 |
C 135
Totals 352 282 372 174 165 481 174 2000
TABLE LXXXIII.
Right Index and Left Middle Finger.
Right Index.
Be
on
S
i A
© SL
= LL
Ss W
S| @
—
3 Totals
TABLE LXXXIV.
Right Index and Left Ring Finger.
Right Index.
Left Ring Finger.
Totals |
66
583
712
491
148
H. Wattr
TABLE LXXXV.
Right Index and Left Little Finger.
Right Index.
TABLE LXXXVI.
Right Middle Finger and Left Thumb.
Right Middle Finger.
x | ee Se ye ech Totals. |
0 | |
= A ¢ 35 |
Ba SE) i 236.425 12 nae) 192 50 959 |
=| LL 81 | 183 | 176 | 224 | 104 768 |
Ss W 5 21 30 86 8 150 |
HH 6 6 5 17 49 11 88 |
= | |
—j | Totals}: 352 | 654 | 339 | 481
Totals
fe)
S 91
= 547
a 767
a 341
Bs 254
4
2000
TABLE LXXXVII.
Right Middle Finger and Left Index.
Right Middle Finger.
Hea aed W. | .0% | -Totals: |
i I |
i | A 10 | 176 | 21 1 5 313. |
TSM) VG 86 | 529 | 177 2a 7 833
eS | LL 7 95 125 42 13 282
2 | Ww 5 80 144 180 28 437° |
Ss C 4 41 49 27 14 135 |
TABLE LXXXVIILI.
Right Middle and Left Middle Fingers.
Right Middle Finger.
5 A SIAL, kW. C | Totals
oe e
eee A 115 94 4 0 2 215
ay UE 84 635 129 25 14 887
wr | LL 8 152 295 70 31 556
ae) W 3 20 65 140 12 240
Ss C 2: 20 23 39 18 102
&
SI Totals 212 921 516. 274 77 2000
Biometrika x
61
476
Left Thumb. Left Little Finger. Left Ring Finger,
Left Index.
Association of Finger-Prints
TABLE LXXXIX.
Right Middle and Left Ring Fingers.
Right Middle Finger.
A SL | LL W C Totals
A 49 16 1 0 0 66
SL 110 398 | 59 8 8 583
LIL 39 346 | 241 63 23 712
W 9 102 165 184 31 491
C 5 59 |. 50 19 15 148
Totals | 212 921 516 274 77 2000
TABLE XC.
Right Middle and Left Inttle Fingers.
Right Middle Finger.
SL Totals
13 35
959
289 260 137 47 768
30 57 56 3 150
18 3l 28 8 88
2000
921 516 274 77
TABLE XCI.
Right Ring Finger and Left Thumb.
Right Ring Finger.
Totals
17
SL 121 49 547
LL 262 Tes 767
W 3 28 88 199 23 341
C g 130 254
Totals 729 2000
TABLE XCII.
Right Ring Finger and Left Index.
Right Ring Finger.
4 ae aa w | @ | Totals
313
833
282
437
135
2000
Left Thumb.
Left Ring Finger. Left Middle Finger.
Left Little Finger.
Right Ring and Left Middle Fingers.
Right. Ring Finger.
H. Waire
TABLE XCIII.
| Ay ese eet ye Ce “1 Totals
| 47 122 SS | Peg) race et 215
15 886 meas |e ais 83 887
1 24 | 204 | 266 61 556
0 5 96 |), 1949/15 240
0 2 1 77 12 102
63 | 489 | 543 | 729 | 176 } 2000
TABLE XCIV.
Right Ring and Left Ring Fingers.
Right Ring Finger.
A | SZ | LL | w | Cc | Totals
| 66
| 583
| 712
491
Totals
148
TABLE XCV.
Right Ring and Left Little Fingers.
Right Ring Finger.
| A SL LL W Cc Totals
A 17 15 3 0 0) 35
SL 42 399 250 194 | 74 959
LL 4 68 276 339 81 768
W 0 5 9 125 11 150
C 0) 2 5 71 10 88
Totals 63 489 543 729 176 [ 2000
TABLE XCVI.
Right Little Finger and Left Thumb.
Right Little Finger.
in soi
W
Totals
91
547
767
341
254
61—2
477
4
(
8
Association of Finger-Prints
TABLE XCVII.
Right Little Finger and Left Index.
Right Little Finger.
pA uate ST | LL Wo} C€ | Totals |
a 22 | 204 76 6 5 S130 .|
3 8 477 | 269 44 35 833 |
aa 0 76 | 155 34 17 282.
2 1 73) |e 198e) a128 37 437 ||
o 0 40 66 16 13 seul
4 eer | ‘ = |
| Totals} 31 | 870 | 764 | 228 | 107 | 2000 |
TABLE XCVIII.
Right Inttle and Left Middle Fingers.
Right Little Finger.
5 ASM asia eee W | @ | Totals |
ee 3 ——
a A 17 5) OSes 2 3 215
ie Ys 12 507-291 42 35 887
sally ea) 0 152-301 71 32 556
Sy) 2 al eo a4 26 240
sl 0 14 48 29 11 102
= | | |
3 Totals | 31 870 | 764 | 228 | 107 | 2000
TABLE XCIX.
Right Inttle and Left Ring Fingers. |
Right Little Finger.
i SEE W C | Totals
a | |
= 50 0 1 66
os 440 10 9 13 583
op P 255 | 398 30 27 712
= G0. | ST 78e leas 50 491
pa 1 36 79 16 16 148
2
o | |
=a) 31 870 | 764 | 928.0!) 107
TABLE OC.
Right Little and Left Little Fingers.
Right Little Finger.
2 | | 4 | sz | zz | w | @ | Totals
2 3 :
= A 18 17 Oy | 0 0 35
3 SL 11 ol a6 21 30 959
= LL 1 115 | 543 74 35 768
= W 1 13 22 99 15 150
4 C 0 4 23 34 27 88
cs Ss | ee
| Totals | 31 870 | 764 | 928 | 107 | 2000
i
ON THE PROBLEM OF SEXING OSTEOMETRIC
-MATERIAL
By KARL PEARSON, FE.RS.
It is well known that anthropometric, particularly craniometric measurements
give frequency series, which for moderate sized populations follow closely the normal
or Laplace-Gaussian distribution. Measurements of stature, cubit head-length,
cephalic index, etc., etc., obey with sufficient accuracy for most purposes of science
the normal law. This statement may with a high degree of certainty be extended
to practically almost all measurements on the adult skeleton. But a new difficulty
arises in dealing with the parts of the skeleton: the sexing of the several bones of
the human body is by no means certain, and this is especially the case when we
come to deal—not with the cranium or the pelvis but with the long bones. In
order to get over this difficulty, and to find the constants for each sex, it occurred
to me some years back when the sexing of the long bones had presented this
problem very forcibly to workers in my laboratory, that the method of my first
contribution to the mathematical theory of evolution* might be applied. Namely,
we might take the unsexed material and assume it to consist of a compound of
male and female data, the frequency curve for each of these being normal ; the
two components might then be found in the manner of the paper just referred to.
The method was especially likely to be successful, when. the series was otherwise
homogeneous, the numbers large and the character dealt with substantially diffe-
rentiated sexually. Of course the method does not give the sex of each individual
bone, but I have shown in another memoir+, how four to six characters thus
resolved form a basis for determining the probable sex of each bone, and this with
an accuracy which is very probably as great as, or even greater than, anatomical
appreciation unbased on a system of numerical measurement.
One of the few objections to the method is the labour involved in the process.
While the analysis required in the application of the method is not so severe that
it has not been applied in a large number of cases by workers in the Biometric
* Phil. Trans. Vol. 185, A, pp. 71—110.
+ To appear in the next number of this Journal.
480 On the Problem of Sexing Osteometric Material
Laboratory, it is still considerably beyond the powers of most of the present
workers in anthropometry, and probably no anatomist of the present day has the
mathematical knowledge requisite for the solution of the reducing nonic, or the
arithmetical patience required for the calculation of its coefficients. It has occurred
to me, however, that the work might be considerably shortened by the following
considerations. The bones usually dealt with are those found in ancient cemeteries,
in plague pits, clearance pits or crypts. It is probable, though by no means
certain, that adult female bones in such cases would be rather more numerous
than male. On the other hand being somewhat smaller they are asserted by some
writers as likely to be more frequently broken, and they certainly may more readily
escape preservation or measurement. If we take these two causes as counter-
acting each other, we may assume as a first approximation that the numbers of
male and female bones will be equal. In the next place it is a result of much
anthropometric experience that male and female variations, i.e. their standard de-
viations, are closely alike. These again we can take equal to a first approximation.
Accordingly, to this first approximation, our osteometric series may be considered
to consist of two equal normal components with different means. Let the mean
of the unsexed material be J/, and let the actual means of the sexed components be
M,, Mz, their standard deviations be o,, o., and their total frequencies 7, and n.,
where the subscript 1 refers, say, to the males, and 2 to the females. Then m,,
Mz, 01, Fz, NM, and n, are the quantities we desire to discover. Let the moment-
coefficients of the total material be, in the usual notation, p., Ms, Ms, Ms and let
N(=7,+%:) be the total unsexed population. We shall write as customary
Bi = bs?/s?, Bo = Mal oo’, Bs = Msfls/uo. Then, if our hypothesis be correct and the
material consist very nearly of two equal normal distributions, 8, and ®; ought
to be very small, while @, will be large in relation to them.
It is convenient also to write:
f, =4(38-8,), C= LOB — Bye aene: eae oe eee (1),
m=M+m, DAU PPS Saerinang oper capo Wedoanoe: (ii),
Jo = Yr'Po/ f2, Gy = (Gare a) fae eee eee eee (111),
Ce AU RRR AR ALB GRHAB Me clin OnISHE GrO4090 5» (iv).
Then the fundamental nonic may be written:
: Keg 3 o.°
= OR ahr 2 Bi ge® —3 (6, oar 5G?) qo ae (878, & Ste i) qe"
+3 (482 — 30, 6-36") q+ 3 (Bi & — $ Bib") qe + 8B fq — Be =0...(v).
VB. {8,— 66.0. 3& oe — sash
Further: a= ana ee ae QA),
where the sign of /8, is determined by that of ps.
Again i EE ip i DON Se a (vii).
Kart PEARSON 481
Lastly : Cie — fa + Go) — $ poly —$ Vion |
os = py (1+ G2) — Malt — Vary |
Equations (ii), (iii), (iv), (v), (vi), (vii) and (viii) form the complete solution of
the problem when we make no approximations whatever*.
If, however, 8, = 8; =0, then, the two components being equal, we have+:
ese,
VGA UU 0 Veit J tan sg Sais abs (ix).
o= o=NV py {1 — Vo,}2
It will be seen that it is needful in order that the solution may be real that
should be positive or 8, < 3, 1.e. the total frequency should be platykurtic. Now
let us suppose that the values given by (ix) are a first approximation and that we
need a second approximation in which the two normal curves will be unequal in
frequency, mean and standard deviation. Write:
n=4N, y=VmG3, c=Vu(1—-VG)P ee (Gh: Ohiat
and suppose :
m=n+ 6n,, N= N+ONy,
Voie eels) Oa)
o,=0+ 60;, O,= +560,
where the differentials represent small quantities of which the squares and
products may be neglected to a second approximation.
Our equations aret:
m+n =N,
MY + Neo = O,
ny (y? ar o;”) + Nz (Yo? + O27) = Nps,
M (or? + By, 077) + M2 (2 + 3y,0.?) = Nuss,
1 (yi' + Gyo? + Bay!) + Ne (24 + Gy? oe? + 80,!) = Nyy,
2 (91? + 10y2 oy? + L5y, 0,4) + ne (y2? + 10y.20.7 + L5y.o0!) = Vu.
We now differentiate these and after differentiation put
ie wW=-yY= ono.
Hence we find:
OMe ONEE Hla ciple ed Ae oe cae aii (x),
@ (Oyi + OFs) 4 2yony= 0 cc vageseeecee stones pee (xi),
Zny (Sy, — Oya) + Zo (Sa, + Soy) =O... eeeveceeeeceseee (xii),
3n (Oy, + Sy.) (y? + a?) + 2dr (9? + 30°) + Brox (dc, — da.) = Nuys... ..(xili),
y (y? + 80”) (Sy, — Syz) + 80 (9? + a”) (60, + 60.) =0 oe ee eee. (xiv),
n (Oy; + Oy) (Sy! + 80 yo? + L504) + 28ny¥y (yt + 104?o? + 150%)
+n (60, — 02) 20yo (y? + 8a?) = Nps .....c eee. (xv),
* They are, in a somewhat better form, those originally given by me in Phil. Trans. Vol. 185, A,
1894, pp. 71—110 ; see Nquations (14), (15), (18), (19), (27) and (29) of that memoir.
+ Loc, cit. footnote, p. 91.
iwioch cit. p: 182.
482 On the Problem of Sexing Osteometric Material
where it must be remembered that the differential terms are introduced solely to
account for the asymmetry as represented by p; and y;, assumed to be zero to
a first approximation.
But (xii) and (xiv) show us that we must have :
by, = Oy, 60, = —So,.
Hence from (xi): On S = NOG | Y cecu eee a eee (xvi).
(xil1) now becomes : 2ry? dy, + Garda, = ps,
and (xv): Ary? dy, (9? + 507) + 20a (9 + 3807) da, = p;.
Whence solving we find:
ss ue) ao Ms 3 Ms i
by; by 5 (1 +3 =) oT Sy oe (xv),
~o\ Ms 1 ps
60, = — ba. =-2(1452)% Sante ee (xvill)
6n, = — 0m = — "Oy EAS MEMO ee EME ERR RIE rio 90 (xix)
sy,
These form together with (ix)’* the complete solution of the problem.
The following example illustrates the procedure: 541 measurements were made
of the bicondylar width of English femora, right and left, male and female being
mixed. The frequency below resulted.
Frequency Distribution of 541 Femora for Bicondylar Width.
mm. Frequency mm, Frequency mm. Frequency
61 | 1 71 23
62 1 72 33°5
63 1:5 73 25
64 5 74 22
65 13°5 75 36
| 66 14 76 25°5
67 15°5 Wh © 29°5
68 22 78 32°5
| 69 31 79 19°5
| 70 19 80 33
The constants of this distribution were :
M=75'8152,
jy = 37°692,112, jiy=— 2'587,693,
ju, = 3020°898,695, u, = — 83-260,992.
Hence we deduce:
B,= 000,125,047, 8, = "000,106,750,
Bo = 2°126,349, €, = °436,8255,
¢, = 001,143,72.
Kart PEARSON 483
Clearly 8, and ®; are so small that the distribution fulfils our condition of
being very closely symmetrical. The nonic, equation (v) above, is:
qs) — 3:057,789q.’ + 000,18757¢9.5 + 2°858,8174q,°
— -009,8679,! — °754,678q.' — 002,50114,"
+ :000,000,0546q. — ‘000,000,000,005 = 0,
the last two terms being written down to many figures to show their inappreciable-
ness. The root required is:
qo = — 65679,
which by (vi) leads to:
y? + 558,0507 — 24°755,802 = 0,
and provides the solution :
Females. Males.
Mean: 70°547 mm. 80°526 mm.
Total Frequency : 255°4 285°6 ee nen cron (A).
Standard Deviation: 3°4842 mm. 3°6944 mm. |
Modal Ordinate*: 29°24 30°84
We have now to inquire how far the same result would be reached, if we had
supposed as a first approximation equal Gaussian components and then proceeded
to determine a second approximation by aid of (xvii) to (xix).
Equations (ix) give us:
ies — 1 — Oro:
W=—y=y = 49912,
Gy= o2=3'5750.
Thus to a first approximation :
Females. Males.
Mean : 70°824 mm. 80°806 mm. |
Total Frequency : 270°5 270°5 esca acre (B).
Standard Deviation: 3°5750mm. 35750 mm.
Modal Ordinate : 30°19 30°19
(B), statistically speaking, is so close to (A) that it gives every confidence of
a second approximation practically reproducing (A).
We find: |
Ms _ _ 020.8112 Ms _ _ .026,0386
Cy oe ry ) a5 Ve ’
of ry
2 Dj
a= 519.0874) Y = 1:948,228.
Ine o
* nv
Yo=——- of the normal curve.
Hy v/ 210
Biometrika x 62
484 On the Problem of Sexing Osteometric Material
Hence by (xvii) to (xix):
SL es Sir MaRS SOLO),
60,=—60,= oo X‘029,809= 1066,
én, = — Ong = +n X 056,308 = 15°231.
It will be seen from these results that:
bo, _
o
O% _ _ 0563, 0298, % 0568
oY,
may be considered fairly small quantities, and that they justify our assumption.
We have accordingly:
Females. Males.
Mean: 70°543 mm. 80°525 mm. |
Total Frequency : 255°27 285:73 > ¢.| WAR eee (C).
Standard Deviation: 3°4684 mm. 3°6816 mm.
Modal Ordinate : 29:36 30°96
It is clear that the solutions (C) and (A) are for all practical purposes identical.
Thus the short method is justified in the problem of sexing osteometric material.
An improper extension of the method to material in which the sexes occur in very
unequal groups may be guarded against by simply observing whether 8, and £,
are very small quantities.
In conclusion it may be desirable to compare the values of these sex-constants
as found mathematically with sexing by anatomical appreciation. I owe an
anatomical sexing of the same bones to my colleague, Dr Derry.
The following values of the constants resulted :
Females. Males.
Mean: 70:098 mm. 79°764 mm,
Total Frequency : 221 320 GRAB Kiso (D)
Standard Deviation: 3°5148 mm. 4°1254 mm.
Modal Ordinate : 24°55 30°95 |
It will be seen that the mathematically deduced constants are not widely
divergent from those obtained anatomically, but the accordance if fair is not ideal.
The accompanying diagram exhibits the differences in the frequency distributions
found by the two methods of sexing. The chief difference lies in the transfer by
the anatomist of the larger female bones of the mathematical sexing to the male
group. I do not propose to discuss here the relative advantages of the two
methods, but would draw attention to a few points of interest :
(i) The svlution (D) makes no appeal to measurement in the sexing, it is
based purely on an anatomical appreciation. It would therefore be subject to
485
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L Hl 1 1 L ! n L L Nl L 1 | i I |
KARL PEARSON
i ail [eens zi 1 falas as 1 = L Tt Sheet
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486 On the Problem of Sexing Osteometric Material
personal equation, depending on the features upon which the experience of the
individual anatomist leads him to lay most stress. The solution (C) is unique,
that is to say, given the same data, all statisticians would reach the same values, of
Frequency Distributions of Bicondylar Width in Male and Female Femora
sexed by Anatomical Appreciation.
mm. ie) 3 mm. f°) 3 mm. f°) 3
61 1 = 71 20°5 2°5 81 = 28
62 1 = 72 29°5 4 82 — 23
63 15 = 73 16 9 83 = 19
64 5 — 74 115 10°5 84 1 16°5
65 13°5 = 75 10 26 85 — 19°5
66 14 = 76 7 18:5 86 _ 16°
67 15°5 77 2°5 27 87 = 75
68 22 — 78 I 31°5 88 —_— 3
69 31 — 79 1 18°5 89 — 3°5
70 15°5 3°5 80 1 32 90 — 0°5
course apart from errors in arithmetic or from the number of decimal places
retained in the working. It eliminates the factor of personal equation.
(ai) (C) would, however, be influenced by the fact that our material is not
perfectly homogeneous except for sex ; because (a) there is a mixture of right and
left bones, and, to judge by the anatomical sexing, this may involve a difference of
‘7 to ‘9 mm. in the means and ‘08 to ‘24mm. in the standard deviations; this
would add to the heterogeneity, (b) our bones may be due to somewhat mixed
classes and possibly mixed periods, (c) the bicondylar width is liable to be injured
by rough treatment of the bone, and this injury will most affect the weaker, and
therefore probably the younger, bones. These bones might then be treated as female,
a classification which most anatomical sexing also favours. While the total number
of these London femora is nearly 800, the bicondylar width could only be measured
in 541 cases. This selection will not necessarily be random as to size or sex,
and may modify our constants found mathematically from the distribution. On
the other hand it would affect also the anatomical appreciation of sex, but only
in as far as it was based on the size of the condyles.
(iii) We know from very considerable sexed data that the variation of man
and woman is very nearly the same. The coefficients of variation measured in the
usual way, i.e. by 100 standard deviation divided by mean, gave:
Mathematical Sexing. Anatomical Sexing.
9 492 ff 457 SOL my
I= G5) A=- ‘16
Karu PEARSON 487
There was thus closer sexual accord from the anatomical method. But when
the same anatomical sexing was applied to the character of the head of the femur
in the vertical plane, I found for right bones:
Q 5:05 J 637 A =— 1°32,
and for left bones:
9 4:91 PAY) A=-1°19,
differences far greater than occur in the mathematical sexing from the bicondylar
widths. Accordingly no great stress can be laid on inequalities in the coefficients
of variation deduced from either process of sexing.
It. would appear to me that we have reached on the whole a reasonable
biometric method of sexing. To what extent it can replace the sexing by
anatomical appreciation must be left to the future. But it is clear that when
anatomists themselves prefer to that appreciation an appeal to a single character,
e.g. to the measurement of the femoral head, and only settle by anatomical appre-
ciation the sex of femora with diameters between 45 and 47 mm., then they do
not show much confidence in their own method of sexing. An interesting experi-
ment could be made if some 400 to 500 sexed bones were available, and then,
without knowledge of the real sex, two or three anatomists and a statistician were
to be asked independently to determine the mean and variability of two or three
characters of the bones of each sex in this material.
I have cordially to acknowledge the help of my colleague Mr E. Soper in the
determination of equations (xvii)—(xix) and in their solution (C) in the numerical
case for which I had reached the solution (A); also the labour of my colleague
Miss H. Gertrude Jones in the preparation of the diagram which contrasts
graphically the mathematical and anatomical solutions of the problem.
FURTHER EVIDENCE OF NATURAL SELECTION
IN MAN.
By ETHEL M. ELDERTON, Galton Research Fellow,
AND KARL PEARSON, F.R:S.
(1) ‘The second author of the present paper writing in 1894 a commentary on
the statement that “no man, as far as we know, has ever seen natural selection at
work,” remarked : “ Every man who has lived through a hard winter, every man
who has examined a mortality table, every man who has studied the history of
nations has probably seen natural selection at work*.” The emphasis is here to
be laid on the word “ probably,” because the seeing depends on the power and
validity of the scientific means adopted to analyse the observed facts. In a paper
communicated by the same author to the Royal Society in June 1912+, it was
shown from the Registrar-General’s series of ten yearly life-tables that when
allowance was made for change of environment in the course of the fifty years a
very high association existed between the deaths in the first year of life and
the deaths in childhood (1 to 5 years). This association was such that if the
infantile deathrate increased by 10°/, the child deathrate decreased by 5°3°/, in
males, while in females the fall in the child deathrate was almost 1°/, for every
rise of 1°/, in the infantile deathrate. The method of investigating by life-tables
could not be extended beyond 1900, because the life-tables for the next ten
years (1901-1910) were not then out, and indeed have only just appeared
(December 1914). While the infantile deathrate as shown from the life-tables
had risen from 1871-1900, the child deathrate had fallen for the same period.
During the next decade 1900-1910 both deathrates have fallen together; such
a secular change does not in any way modify the argument of the paper, which
lies in the statement that whether two deathrates rise together or rise and fall
simultaneously we can draw no inferences at all, wntil they have been corrected for
secular change. Most economic, demographic and physical variates are changing
continuously with time, and no comparison of time graphs or calculation of .
correlations will demonstrate of necessity anything but spurious association, until
* The Chances of Death and other Studies in Evolution, Vol. 1. p. 166.
+ ‘“The Intensity of Natural Selection in Man.” R. S. Proc. B. Vol. 85, pp. 469—476.
Erue, M. ELpERTON AND KARL PEARSON 489
the time factor has been eliminated. It is the deviations from the continuous
curves of secular change which may turn out on careful analysis to be truly
indicative of causal relationship between the variates under consideration.
The first attempt to get rid of secular change by a method of differences was
made by Miss F. E. Cave in 1904 in a paper on barometric correlations*, and
shortly afterwards Mr R. H. Hooker published a paper dealing with the same
pointt. Both these authors used only first differences and gave no general theory
of the method. Quite recently “Student” has published a paper{ giving the
fundamental formulae, and indicating how by taking successive differences of two
variates and correlating them, we free ourselves from the time or locality influence,
and approach the true and probably causal relationship between them. When the
correlation of the differences becomes steady, then we have reached the actual
correlation of the variates corrected for the time factor, provided an assumption is
made which we shall discuss at greater length below: see footnote, p. 495. Mean-
while Dr Anderson of Petrograd has been working on the subject, and in a
most valuable memoir§ he has added to “Student’s” results a number of new
theorems; for example, the probable errors of the successive difference corre-
lations when they become steady, and the relations which should be fulfilled
between the squares of the standard deviations of successive differences, when
the series has become steady. We have thus a double means of ascertaining
whether the desired object—the elimination of the time-factor—has been approxi-
mately achieved. A third additional test will be indicated in this paper.
This new statistical process has been termed the Variate Difference Correlation
Method||, and there is small doubt that it is the most important contribution to
the apparatus of statistical research which has been made for a number of years
past. Its field of application to physical problems alone seems inexhaustible. We
are no longer limited to the method of partial correlation, nor compelled to seek
for factors which rendered constant will remove the changing influence of environ-
ment. In the present case, that of the influence of imfantile mortality on child
mortality, Pearson endeavoured to eliminate the influence of continual environ-
mental improvement by making the expectation of life at six years constant.
Snow achieved the same object by correlating the deathrates of one sex for a
constant deathrate of the other**. In both these cases substantial evidence of
Natural Selection was obtained from the mortality tables. The object of the
present paper is to demonstrate by the still more complete elimination of the
* R. S. Proc. Vol. uxxiv. pp. 407 et seq.
+ Royal Statistical Society Journal, Vol. uxvitt. pp. 396 et seq. 1905.
+ Biometrika, Vol. x. pp. 179, 180.
§ Ibid. pp. 269—279.
|| Pearson and Cave: ‘‘ Numerical Illustrations of the Variate Difference Correlation Method.”
Biometrika, Vol. x. pp. 340—355.
| R. S. Proc. B. Vol. 85, p. 472.
** «The Intensity of Natural Selection in Man.” Drapers’ Company Research Memoirs, Dulau & Co.,
1911.
490 Further Evidence of Natural Selection in Man
time factor involved in the variate difference correlation method that a selective
deathrate plays even in highly civilised states a marked part in the natural history
of man.
(2) The material dealt with in this investigation consists of the Registrar-
General’s returns for births in England and Wales and of deaths in the first five
years of life from 1859 to 1908 with the addition of as many years before 1859
as were requisite to make our highest differences fifty in number, and with the
addition of as many years after 1908 as were requisite for following up the births
of that year to the fifth year of life. Thus actually our data extended from 1850
to 1912. The reason for this procedure lies in the desirability of using a constant
population, and not reducing by one a relatively small number like 50 on each
differencing. Asa result of this process we had to modify Dr Anderson’s values
for the probable errors for the steady values of the difference correlations because
‘In our case the size of the population does not change as we proceed to higher
differences*. The second cause which requires extension of the data is a very
important one, and must be illustrated numerically. Consider the table:
Deaths of those born in a given year.
7 Female ; 2
Year Births 0—1 1—2 2—88 3—4 4—5
1908 | 478,410 | 63,594 = = =
1909 — -_ 14,146 es eas
1910 = os = 5,020 =
1911 = = = = 3,449
1912 a =i = == =
Now the deaths of infants 0O—1 in 1908 are not necessarily of infants all born
in 1908, but the total deaths 63,594 must represent closely the deaths in the
478,410 infants born in that year. Disregarding immigration and emigration, this
gives a deathrate per 1000 of 107-495 and leaves 414,816 children alive. Of this
group 14,146 may be taken to die in the second year of life, giving a deathrate of
31-990 per mille. There remain 400,670 children who reach the third year of life
in 1910, of whom 5,020 die, giving a deathrate of 11:939, and 395,650 survivors.
These survivors are followed into 1911 and 1912 in the same manner, and thus
we obtain approximately the deathrate up to the fifth year of the male children
born in 1908. We thus in bulk follow the same group of children through the
first five years of life. Tables I and II give the deathrates for males and females
respectively under the heading of the birth year of each group. These death-
rates have been taken to three decimals places for the purpose of determining the
higher differences correctly to one decimal place. The successive differences of
* All the probable errors of the difference correlations given in this memoir are these modified
Andersonian values, i.e. they are the probable errors on the assumption that the difference correlations
have reached steady values.
Erur, M. EupERTON AND KaArL PEARSON
TABLE I.
Deathrates in each Year of Life for groups born
in the Year of 1st column.
_ Males.
|
Ga 1—2 2—3 oy I—5
1850 159°781 63°935 34°092 22°138 19°021
1 168°706 64:°977 33°882 26°657 18-209
2 173°324 63°533 40°936 24-316 16°022
3 174°808 74°802 35°464 22°298 16°720
4 170°890 60°598 30°740 21°688 21°329
5 169°151 59°696 33004 29°546 19°780
6 156°756 64°317 39°551 25°594 13°751
tr 168°486 67°928 36°777 19°471 13°923
8 172°591 68°712 30°566 19°857 16°268
9 167°106 58°887 31°650 22°325 21°962
1860 162°642 70°401 35 °297 30°083 21°325
1 167°634 65°785 41°390 27°654 | 16°236
4 156°684 73°848 37°326 22°0138 | 14°982
3 163°183 67°537 32°774 21-088 12°552
4 166°309 66°462 34°408 17°660 15°991
5 174°356 68°369 28°278 21°473 17°015
6 173°659 60°603 32°159 22-751 | 18°105
¢f 166°905 63°790 32°731 23°220- | 16°085
8 168-064 62-988 32°357 20°185 | 12°821
9 169°022 65°716 30°186 17-450 | 11°624
1870 174°287 62°401 26°760 16°344 | 16°052
1 171°840 59°853 25503 21°635 | 14°727
2 162°321 99 °267 30°754 18°731 | 12°655
3 163676 61°415 28°007 17°275 12°502
4 164:976 60°316 ZO Be, 16°028 13°499
5 173°145 58°422 25°400 18°848 13°187
6 160°415 56°088 27°992 177044 13°2638
tf 149 °627 63°344 26°104 17171 11°738
8 166°266 58°104 27°853 15°280 13°049
og) 149°754 66°188 22-245 16°741 12°115
1880 167°313 48°181 25°909 16°036 11°920
1 142°532 59-164 23°333 15°261 10°430
2 153°154 54°365 24°026 14°362 9°416
3 151°184 58°292 23°015 13°823 11°042
4 160°381 53°952 22°077 14°792 9°533
5 151°175 57°960 23°052 13°615 10°073
6 163°081 55°611 21:009 13:°974 10°569
th 158°243 50°713 22°169 14°394 9°894
8 150°177 56°882 22958 13°536 10°150
9 157°476 56°279 22°338 13°750 10°647
1890 164°757 59:098 22°604 14°551 9°623
1 163°761 54°255 20°821 12°615 8°757
2 162°112 52°776 19°442 12°497 10°604
3 173°333 47-035 20°343 14°121 8°546
4 149°633 55°787 21°271 11°750 8:439
5 176°280 51°404 19°535 11-912 8:973
6 160°989 50°293 18°742 11°802 9°366
ih 170°291 50°986 19°124 12°396 8°608
8 175°183 48°227 19°380 11°496 8°657
9g 176°606 49°837 17-094 11°565 7°165
1900 168°685 44°728 17-400 9°759 7°553
il 165°617 42°597 15°570 10°241 7°135
2 146°791 40°581 16°572 9°587 6°853
3 144°567 45°517 15°368 9-260 6949
4 158 °684 38921 15°234 10°194 6°458
5 141°1938 39°326 15°702 8°893 6'886
6 144°819 38 °037 14°222 8°858 5°655
tf 130°259 36°615 15°159 7643 6°230
1908 132°928 34°102 12°529 Salis 5°962
Biometrika x
63
491
492 Further Evidence of Natural Selection in Man
TABLE II. Deathrates in each Year of Life for groups born
in the Year of 1st column.
Females.
Oxsiely apes) 2-8 gay paws
1850 130°477 | 62°235 34°147 22625 19°278
i 138°306 62°106 33°992 27:°087 16°762
2 142°185 61°812 39°787 23°805 16°148
3. 144°270 | 71:°939 35°186 22686 16°948
4 Pa 52 59024 30°863 | 22°226 21°907
Oo} 3218 se eo elA|: 34°057 29°375 20°501
i
129°877 61-902 39°758 | 26°146 14°305
142°203 65°853 36°712 19-990 14°391
é 143°595 64°513 29°716 20-796 17-089
9 138°477 55°5385 32°024 24°485 21°454
1860 131°914 66°902 36060 29-941 20°765
“et
1 137 °339 61°860 41°243 27°727 16°287
2 127°203 70°313 36°543 21-905 15°398
3 133°321 63°438 32°637 21°623 12°702
4 138°232 63°395 34°846 18°217 15°547
5 145°388 66°401 28°484 21°437 17°145
6 144°879 58°180 32°392 23°086 17°536
7 137°712 61°641 33°243 23°081 15°653
8 141°756 58624 31°892 20-060 12°524
9 141°450 60°720 31004 18-108 11°732
1LS70 144°596 59°845 26°855 16°235 15°196
1 143°353 56°379 25°062 21°493 13°903
| 2 1367453 52°652 30°2138 18°454 12°571
3 134°079 57°739 27°549 16°950 11°421
4 135°939 o7°795 25°498 16-031 13°253
5 142°730 54°032 24°563 18°848 12°833
6 131°718 51°057 27976 16°685 12°651
i 121°936 59°5385 25°135 17441 11°214
& 137°958 52°463 27 °646 15°186 12°491
9 120°6438 62°324 21°792 16°911 11°880
1850 137°691 45620 25°319 15°556 11°335
1 117°221 55413 22°443 15°359 10°396
2 127°627 50°057 23°539 14°178 9°586
3 122°766 54°048 22°948 13°643 10°606
4 132°701 49-967 21°259 14°913 9°40]
i) 123°667 53°415 22°740 13°251 10°030
6 134°814 51°108 19°985 13°800 10°295
i 130°689 46-402 21°538 14°594 10°012
8 122°312 53°369 22°456 13°926 10°146
) 1297148 53°307 21°783 14°170 10°569
1890 135°839 54°850 21°607 14°731 9°478
1 132°763 51°567 20°841 12°646 8°900
2 132°414 49 °387 19°174 12°708 10°370
3 143°346 44°511 19°842 14°293 8°643
4 123°522 52°267 21°499 12°183 8°300
5 144°326 | 49°339 18°643 11°817 8°499
6 133°535 | 46°691 18°439 12°184 9°282
tf 140°755 47°998 18°109 12°739 8642
& 145-001 44-832 18°436 11-490 8°865
@) 148-000 45928 16°954 11°724 7164
1900 139°148 | 41-901 16°902 9°712 7611
e 1 136°346 39°527 15°156 10°606 77184
2 118°479 37 :064 16°168 9°891 6°792
3 118-004 42-270 14°774 9°315 7615
4 131°477 -36°598 14°136 9°793 6°453
5 114°641 37-084 15-066 8-710 6-902
6 119-668 36°006 13°552 9°168 5°513
@ 104°487 33904 13°789 7493 6°071
1908 107°495 31°990 11:939 8°546 5°939
Ernest M. Evperton AND Kart PEARSON 493
these deathrates up to the sixth and, in a few cases, to the tenth were then
formed. In our notation m, is the deathrate in the rth year of life, 1e. from r— 1
to r years of age, and 6,m, is the sth difference of this deathrate. As we have five
deathrates for each sex this involves 10 means, 10 standard deviations and 20 corre-
lation coefficients, but as we have used six successive differences these numbers
must be multiplied by seven. The calculation of these differences and of upwards
of 150 correlation coefficients has meant very strenuous labour. It must, indeed, be
admitted that the application of the variate difference correlation method is not,
even with small populations, a light task, but the change from the high positive
to low negative and then to high negative values of the correlation is of
extraordinary interest, and indicates the stages by which the associations are
freed from the spurious influence of the time-factor.
(3) All our correlations are given in Table III (p. 497), but it is desirable to
discuss in detail certain groups of them. We take first the correlations of the
deathrates in successive years. They are:
Male. iMemates
Mn Ms + 398 + 080 + 390 +081
Tg my +859 + 025 + 864 + 024
Pe +924 + 014 4-928 + 013
i + 911 + 016 +917 + 015
my Ms
All these are positive, all are significant and, the first excepted, are very high
correlations. There is no significant difference between male and female. The
least important is the relation between deaths in infancy and deaths in the first
year of childhood. We have in these correlation coefficients the numerical
expression of what is obvious in Tables I and II, 1. as the deathrate in any year
of age falls so does the deathrate of the same group in the following year. It is
this fact which has led to the erroneous idea that natural selection plays no part
inman. The fact, however, simply expresses the continuous change of environ-
ment which has been in progress since 1860. During the half-century improved
economic conditions, bettered sanitation, and developed medical care have lowered
the deathrate at each age*. It is therefore impossible to deduce any argument
as to natural selection in man from these correlations until we have removed this
continuous influence of the time-factor. This is achieved by the variate difference
correlation method. In every case a preliminary examination of Tables I and II
shows that the correlation of the first differences of the deathrates of successive
years is negative, and as we take higher and higher differences the intensity of this
negative correlation increases, until with the sixth differences it reaches to the
* As we have already remarked the infantile deathrate showed little of this improvement till 1905,
It was about this same year that the absolute number of births in England and Wales began to decline,
so that while the population has increased by something like 34 millions, that population produces
about 76,000 fewer babies annually.
. 63—2
494 Further Evidence of Natural Selection in Man
very substantial value of about —°7. In other words a rise in the deathrate of
one year of life means a fall in the deathrate of the following year of a most
marked kind. While with the sixth differences we are approaching fairly closely
steady values it may be doubted whether we have reached them in any case but
that of 154m. dgm;:_ Lhe following are the sixth difference correlations in the case
of the deathrates of successive years:
Male. Female.
Tagmy . Some — 688 + 090 —-719 + 081
"55 my « 5gm3 — 673 + 092 — 660 + 095
User tasene: — "703 + 085 — ‘731 +:078
T Samy « 5g Ms — 695 + ‘087 — 736 +077
Again the male and female results are in excellent agreement, and we grasp
the startling manner in which the new method reverses a judgment based on-
relations which have been deduced without any regard to secular change.
(4) The question naturally arises: How far are these the “steady” values of
the difference correlations measuring the organic relation apart from the time-
factor of the deathrates in different years of infancy and childhood ?
There are three fundamental tests: (i) The correlation coefficients of suc-
cessive differences should have ceased to be markedly rising or falling. Table II
(p. 497) shows that this is approximately but not absolutely the case, but we have
reached a stage in which any further changes are certainly of the order of the —
probable errors and thus of little significance. The unsteadiness as will be in-
dicated later in better tests is greatest in the differences of the deathrates in the
first and second years of life. Here the correlations were taken to the seventh
and eighth differences and gave:
Male. Female.
Lae ee — 696 + 090 — 729 + -082
Tsai . dams — 692 + 094 is ‘731 4 °084
which appear to have reached practical steadiness. Actually the final correlations
must be somewhat greater than those obtained from the sixth differences. To
push the process further, however, would be of small advantage because higher
differences involve introducing earlier data, and the birthrate data before 1855
become more and more unreliable. Again in the extremely high differences, the
additional year required for an additional difference if not appertaining to rela-
tively smooth data may in itself, when we have only a small total frequency of 50,
produce a certain amount of unsteadiness.
(ii) We may consider the mean values of the differences.
Erne, M. EvperTon AND Karu PEARSON 495
If our first variable be taken* as a= ¢,(t) + X, where X is the intrinsic value
of w as apart from the time change, then mean 6,4,” after steadiness has set in is
* One of the bases of the variate difference correlation method lies in the assumption that the
intrinsic variation is superposed on a secular change of a continuous character; the causes which
determined the intrinsic variation X are supposed to be sensibly independent of the time for the
period under consideration. We conceive the secular change as given by a parabola, say, of the
sth order, but the deviations from this curve are supposed in magnitude and sense to be independent
of the time, i.e. due to chance causes which are the same in 1850 as in 1900. This assumption
is an important one and must lead to our seeking relatively short periods consistent with a numerical
frequency sufficient for significance. It can be roughly tested, of course, by considering ox as found
from, say, the first and second halves of our observations. In our own case we found:
Values of ox deduced from Sixth Differences for 1st 25, for 2nd 25,
and for all 50 years.
(m1) | (mz) (ms) (mg) (m5)
a
= |
SoMa el ieee. lacs: igual ae: GO malh aes a) 1
é | seat Ds
Ist 25 years ... | 7°32 | 6-94 | 5-51 | 5-61 | 2-09 | 2-30 | 1°52 | 1:67 | 1:05 | 0-91 |
All 50 years ... | 8-61 | 7-83 | 4:71 | 4:63 | 1:59 | 1:77 | 1:17 | 1:28 | 0°86 | 0°78
2nd 25 years ... | 9°70 | 8°61 | 3°73 | 3:37 | 0°83 | 0-98 | 0°66 | 0-68 | 0°63 | 0°62
I |
These values are less steady than we had originally hoped for. Clearly the variability of the X
portion of the infantile deathrate has grown greater, and that of the four child deathrates has grown
sensibly smaller with the time. The fundamental hypothesis of the variate difference method is there-
fore only approximately true for this material, We have made some investigations on the assumption
that x= ¢,(t) + (a+ bt) X, but the values of a and b obtained were by no means satisfactory. We have
in hand a further investigation of the problem by the method, originally suggested by one of us, before
the difference method was started; namely to subtract from « the value obtained by the best fitting
parabola of the sth order in the time and so to reach the actual values of X. The relation of these
to the time can then be found with some degree of accuracy. To the male deathrates of the second and
fourth years of life we applied parabolae of the third order in the time, and obtained excellent fits ; we
then subtracted the ordinates of these parabolae from the deathrates and correlated the remainders,
dy and dy say. We found Wiehe +°312+-088, a value corresponding more nearly with "551mg d5neg than
T Sm dgmy? and indicating that we might more rapidly approach final values by this method than by
that of variate differences. But the fitting of high order parabolae is very laborious; at the same time
the graphs give excellent tests of the accuracy of the work, and we obtain the actual values of what
we have termed X and Y, as represented by dy and dy. We then correlated the numerical value of ds
with the time and found "dt = —°284+-089. It is clear that with correlations of this order with the
time, "dod, would not be modified by the extent of its probable error if we found the partial corre-
lation tdydy? or corrected the correlation of dy and d, for the time. There is another point, however,
which justifies us in disregarding this variation of X and Y with the time as of secondary importance.
The correlation of X with the time is positive in the first year’s mortality and negative in the following
four years; thus while it would certainly tend to give a negative value to 7x, for the 1st and 2nd
-years of life, it would tend to give a positive value to the correlation for all successive pairs of years
beyond the 1st and 2nd. Now all such successive pairs of years have high negative values, which are
therefore minimum values, but these values are all in excellent agreement—roughly equal to —-7—with
that found for the 1st and 2nd years of life. We therefore concluded that the influence of the time on
the deviations from the secular curve of change, although very sensible, is of no substantial importance
for he correlations.
496 Further Evidence of Natural Selection in Man
equal to mean 6,,,X, and this (taking, as we have done, ‘backward’ differences)
is given (the C’s being the usual binomial coefficients) by
= (X= Oa Cs Kg =) a Ae ee Oe Xp s09))
MN
Now if we remember that the X’s have chance values uncorrelated with each
other then we shall have for the squared standard deviation of the mean 6,4,X,
9
mean 6,.,,;X —
2c,? ( + CO? ae 1 Ce? + a0 + +C,?)
o
n>
Or, the probable error of the mean (7 + 1)th difference after the steady values
have been reached
2 =
= 67449 / 2122 ox
r[r
At first sight this appears of no value, because oy is unknown, but Dr Anderson
has ‘given os __,, in terms of ox when steady values have been reached *, 1.e.
Rat Le
when we assume steadiness reached.
The values of the means of the differences with their probable errors on the
assumption of steadiness are given in Table IV, and the ratio of the means to
their differences in Table V.
It will be seen that the positive and negative signs are not scattered quite as
much at random as we might have hoped and that this is especially the case in
the infantile mortality differences}. If we take all the ratios of the means to
their probable errors except the first difference, we find their average value 1°16 ;
it should be of course 1:18. Of these ratios 33 are positive and 25 negative. If we
omit the ratios for the first year of life, we find 24 negative and 20 positive, while
the mean value ="98 as against 1:18, the theoretical ratio of the mean to its
probable error. It is obvious that the infantile mortality differences are those
which are anomalous. Otherwise the mean differences vary fairly satisfactorily
* Biometrika, Vol. x. p. 272.
+ It may be noted that at the beginning of the period we have the disturbing influence of war and
at the end of the period wholly changed conditions due to a great limitation of births. The means
depend on differences of mortality under these conditions.
497
Erue, M. ELpERTON AND KARL PEARSON
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TH WiTdvi
Further Evidence of Natural Selection in Man
498
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Erak, M. EupErtTon AND KARL PEARSON 499
round zero in the required manner. The interest of this test is that we see that
the bulk of the time effect has been removed even when we reach the second
difference, a result confirmed by the fact that the correlation of the deathrates’
second differences is in every case already substantially negative.
(iui) A third set of tests are those which are based on the standard deviations
of the differences. In the first place if we assume steadiness to have set in, we
can calculate oy, the intrinsic standard deviation from the known value of 5 x» bY
means of Dr Anderson’s formula cited above (p. 496). Table VI gives the intrinsic
values of ox, i.e. oy as deduced from the variability of the differences. It will be
at once observed that for the third difference the mortality ratios of the third,
fourth and fifth years of life reach steady standard deviations. In the case of the
first year of life it is not till the eighth difference that this result is reached, while
in the case of the second year, it can hardly be said to have been obtained with
the ninth difference. A distinction should be noted here of which the exact
physical significance is not obvious to us. In the second, third and fourth years
the intrinsic standard deviations fall to steady values, but in the first and second
years they rise towards those values and these are just the cases where steady
values are not absolutely reached.
TABLE VI.
Intrinsic Standard Deviations (ox).
Year: O—1 (m;) | Year: 1—2 (mg) | Year: 2—3 (m3) | Year: 3—4 (my) | Year: 4—5 (ms)
Order of
Difference |
3 ¥. 3 ? CRAs se 3 4 3 a
2 ae = =
Ist 7°62 6°96 3°90 3°86 1°75 1°83 1°53 1°52 1°23 1:09
2nd 7°89 7:22 4°13 4:09 1°67 1°81 1:29 1°34 1:00 83
3rd 8:14 7°45 4°32 4°28 1°62 e728 heal 1:29 93 80
4th 8°34 7°63 4°47 4°42 1°59 1°76 1°18 1:29 87 77
5th 8°50 7°76 4°60 4:54 1°58 1°76 1:17 1:28 86 77
6th 8°61 7°83 4°71 4°63 1°59 Ea) erly 1:28 86 78
7th 8°66 7°84 4°80 4°72 _- —- ) — — — =
8th 8°68 7°85 4°88 4°78 — =
9th — — 4:97 4°82 — | — — — — =
(iv) There is another test for the standard deviations of the differences
deduced by Cave and Pearson from the Andersonian results and used by them in
their memoir on Italian Index Values*, namely as steadiness is approached the
ratio of the squares of standard deviation of successive differences should approach
closer and closer to 4, the exact value being
o"5.m 2
4-2,
s
2
o's ym
s
* Biometrika, Vol, x. p. 346.
Biometrika x 64
500 Further Evidence of Natural Selection in Man
Table VIL shows how rapidly the system approximates to the theoretical values
in the case of the higher differences.
On the basis of all the tests we have applied we may, we think, conclude that
by the sixth difference we have reached values for the correlation of deathrates
in successive years which are in all probability close to the organic or intrinsic
values. Only in the first and second years of life is steadiness not absolutely
reached, but for practical purposes but little change can be anticipated in the
correlation coefficients.
TABLE VIL.
Ratio of Squared Standard Deviations.
my my ms ma Ms Mean | Mean | Theory
3 : 3 f 3 ¢ ) f 3 f 6 |S
i
|
vl 949) 956] °354] °369] +142] °144|] +194) +192] -211] -181 | °370| ‘368) 2
2 | 3:199 | 3:221 | 3°374 | 3°384 | 2°731 | 2°934 | 2°127 | 2°317 | 1-996 | 1°728 | 2°685 | 2°717 3
8 | 3°547 | 3-552 | 3638 | 3°633 | 3-133 | 3-240 | 2-944 | 3-086 | 2°896 | 3:109 | 3°232 | 3-324] 3°333
4 | 3676 | 3673 | 3°754 | 3°741 | 3°363 | 3°428 | 3°341 | 3°504 | 3:054 | 3°227 | 3°438 | 3°515 | 3°500
5 | 3°738 | 3°723 | 3°811 | 3°793 | 3°591 | 3°604 | 3-509 | 3-562 | 3:504 | 3°641 | 3°631 | 3°665 | 3°600
6 | 3°756 3°848 | 3°828 | 3°690 | 3°683 | 3°708 | 3-648 | 3-691 | 3-758 | 3-739 | 3°730| 3°667
3734
| |
(5) We can look at the association of deathrates in successive years from
another standpoint. We can ask if there be an increase of 10 points in the
deathrate for a given year, what increase or decrease will there be of deathrate
in the same group in the following year ?
In Table VIII below the second column gives the spurious change which is
apparent in the crude data, the third column gives the real organic change which
is discovered when the time-factor is removed.
TABLE, VIII.
Association of Deathrates without and with Annulment of Time-factor.
Result of an increase of- 10 deaths per mille in one year of life on the deaths per
mille in the next year.
| Disregarding Time-factor Annulling Time-factor
Increase of 10 in Deathrate of | —
) ? 3
|
di)
Ist Year on 2nd Year
2nd Year on 3rd Year
3rd Year on 4th Year
4th Year on 5th Year
| Increase 3°3
Increase 671
Increase 69
Increase 7:0
Increase 3'8
Increase 6°6
Increase 6°7
Increase 6°8
Decrease 3°7
Decrease 2°2
Decrease 5‘2
Decrease 5:1
Decrease 4:3
Decrease 2°5 ,
Decrease 5:3 |
Decrease 4°5 |
Erase, M. Exvperton and Karu Parson 501
It is easy to see how those who contented themselves with crude deathrates,
making no allowance for the betterment of deathrates with the time, interpreted
a higher deathrate in one year to mean a higher deathrate in the next year of life,
and so questioned whether natural selection applied to civilised man. As a
matter of fact we see that the true organic relationship of deathrates is much
more probably summed up in the statement that a decrease or an increase of
deathrate in one year of infancy or childhood is in each case followed by an
increase or a decrease in the deathrate of the survivors of the same group in the
following year. Disregarding the time-factor we have a result quite incompatible
with natural selection; annulling the time-factor, we have a result not only
compatible with natural selection, but very difficult of any other interpretation
than that of a selective deathrate, 1.e. a heavy mortality means a selection of the
weaker members, and the exposure to risk in the following year of a selected
or stronger population, which has accordingly a lesser deathrate.
(6) We now turn to the problem of how far this influence extends, or
probably it would be better to phrase it: how far this influence can be traced.
It is not only that the age group we follow does not absolutely consist of the
same individuals but even with those members that are the same there is very
often change of environment due not to time but to a change of locality or
of economic condition affecting individuals. Added to this there is a continuous
immigration and emigration. But beyond these causes weakening the association,
there is another difficulty of great importance arising from what has happened
in the intervening years. We wish to find out how an increase of deathrate
in the sth year of life affects the deathrate in the (s+2)th year of life, but
the events in the (s+1)th year will largely dominate and, perhaps, screen the
results we are seeking. Such problems are always arising in statistical research.
For example, a child may resemble its grandfather simply because both grand-
father and child are like the child’s father. We know that the problem is
answered statistically by inquiring what is the relation between a character in
the child and the grandparent for a constant value of the character in the parent.
In precisely the same manner we must in the present problem inquire: What
is the correlation between the deathrates in the sth and (s+ 2)th year of life
for constant deathrate in the (s+1)th year of life ?
TABLE IX.
Influence of Natural Selection at Interval of Two Years.
Partial Correlation of For constant 3 Q
dom, and dgm3 sis... gig | — °4307 | — 5242
Ooms, and Sgmq dg73 —'2555 — 2058
Somz and dgm, —... Ogig — ‘1798 | — 3129
502 Further Evidence of Natural Selection in Man
We shall of course work with the sixth difference correlations in order to free
ourselves substantially from the time-factor.
Here again the judgment based on the partial correlation of the crude
deathrates is in all six cases reversed. For every one of the partial coefficients
of crude deathrates shows that for intervening year with a constant deathrate,
an increase of deathrate in the earlier year means an increase, not a decrease in
the later year. Actually an increase in the one year is shown in Table X in all
cases to be followed by a decrease at two years’ interval.
TABLE X.
Influence of Natural Selection at Interval of Two Years.
Result of an increase of 10 deaths per mille in the second following year.
For constant death-
Increase of 10 in Deathrate of SR é ce)
Ist Year on that of 3rd Year 2nd Year Decrease ‘81 Decrease 1:28
2nd Year on that of 4th Year 3rd Year Decrease ‘61 Decrease ‘52
3rd Year on that of 5th Year 4th Year Decrease ‘99 Decrease 1°4
It will be seen that these values are appreciable although far less important
than the decreases produced in a following year by an increase in the immediately
preceding year.
Thus we judge that a selection of the weakly children in one
year is largely influential on the deathrate of the immediately following year, and
diminishes, as we might anticipate, with increase of time.
Some objection might, however, be taken to the sixth difference correlations,
when we consider deathrates of the same group two years apart.
Ui dgmy . dgmg
i dg my . dgmg
V5qmg. Sgms
Male.
+227 +°159
+ 339 +149
+°397 +:142
They are
Female.
+°200+°161
+377 +:°144
+°393 +142
It will be seen that while they are all of the same sign and fairly accordant for
both sexes the probable errors are becoming very substantial relative to the
coefficients. We have indeed too limited a range of years.
(7) If now we take out the correlation coefficients of the sixth differences for
three years’ interval, and again for four years’ interval we find great irregularities.
Male. Female.
Togmy . dgmg +°205 +:'161 +035 +168
P3gme . Og™Ms a ‘030 + ‘168 ae ‘072 + ‘] 67
"gm « gms — 181 +163 — +251 +158
The correlations now do not agree in sign, they are insignificant having regard
to their probable errors, and there is no close correspondence for the two sexes.
Erase, M. Evperton anp Kart PARSON 503
We should need a far longer period than 50 years to determine certainly even the
signs of these correlations, and their real magnitudes would require still ampler
data. It would appear impossible to assert on the basis of the above values of
the correlations at three and four years’ intervals more than the insignificance
of the associations between deathrates of the same groups at intervals of more
than two years*. In other words the effect of intense selection appears to be
exhausted after an interval of two years. The word “appears” is used purposely
because there must be some spurious weakening of the effect due to our not being
able to follow absolutely the same individuals.
(8) We have further studied to some extent the relationship between the
male and female deathrates. There is almost perfect correlation between male
and female deathrates in any given year of life after we annul the time-factor.
Thus, if we represent female deathrates by m’, we have as illustrations:
= +9905,
Peqms. Some’ — ‘9880,
T5gmg .5gmg" + ‘9687,
TS mg. dgmy’ >= ak ‘9800.
Psgm, . dgmy’
Of course the sole significance of these values lies in the fact that years of
stress, whether due to climatic or epidemic causes, affect equally infants or
children of both sexes of the same age. But these very high values in our
opinion cast considerable doubt on the partial correlations derived from them.
We have in fact
— T1313 _N
= De
and if we suppose 7, and 7; nearly equal, then if 7; be of the above high value
N will be extremely small, but D is also, owing to the presence of the factor
V1—732, very small. Thus 37. although it may be very considerable is the ratio
3°12 =
* Actually the partial correlations of the sixth differences at three years’ interval based on the above
values are :
dgm, and dgnr4
dgmy and dgms
dgntg and dgmz +526 +°181
dgms, and dgmg | +°251 + °485
Correlation of For constant 3 @
|
These are certainly all positive, but they are irregular as between the sexes and probably quite
unreliable for the reasons already given. Should a more extended experience show that there is a
real if slight positive correlation between deathrates at three years’ interval, while there is con-
siderable negative correlation at one and two years’ intervals, we should be compelled to discuss
whether there may not be something periodic in the nature of the heavy and light deathrates of
infancy and childhood. We have been unable to trace any sign of such periodicity either in the
deathrates or in the graphs drawn, but we do not believe that a very short periodicity would be elimi-
nated by the variate difference method using any moderate uumber of differences. We cannot on
this point accept Dr Anderson’s view. See Biometrika, Vol. x. p. 279.
504 Further Evidence of Natural Selection in Man
of two small quantities and any disturbing cause which but slightly modifies the
value of either 7, or 7.; may even change the sign of NV and so swing sry. over from
a considerable positive to a considerable negative value*.
We can consider the correlations between the female deathrate in one-year and
the male deathrate in a second year, supposing of course time influence annulled.
We have
Tscmcoglia, aa ae ‘667 4 (Gaon Ogio sa 6879),
Tégmy’.dgm. — — 7337 Cae, Sgm! — 7188),
"Sgms dma’ — 7313 (Ts5ms 50g 745 ee 7032),
5 gms’. dg my — — 7278 (1"5, ms’. dgmg’ — "7313).
Thus we see that the same remarkably high negative correlations exist between
the male and female deathrates of successive years of groups born in the same year
as exist between male and male or female and female deathrates within the same
group in successive years. In fact in two out of the four correlations the cross
relationships are higher than the direct, although the differences are scarcely
significant. Here again there is nothing noteworthy, considering the very high
correlations just noted to exist between the male and female deathrates of groups
born in the same year. We can, however, endeavour to correct such values by
finding the relationship between the deathrate in females in the first year of life
and males born in the same year in their second year of life for a constant death-
rate of males in the first year of life. Or still more stringently between the
deathrates of females in the first year of life with males in the second year of life
for constant male deathrate in the first year of life and constant female deathrate
in the second year of life. We should anticipate that such values would come
out small or insignificant, if our interpretation of the high negative correlations
between deathrates of the same group in successive years of life be a correct one,
Le. that the high deathrate leaves a stronger population. For a heavy deathrate
in the females of one year should not leave a stronger population of males for the
following year after correction by partial correlation.
We obtained the following correlations :
5g my! Sgmy’ . Og mg tas 5240 + 0692,
=+°4665 + 0746.
dem! 65m . 56 My!
* The reader must note that we say a ‘‘disturbing cause”; it is not the mere result of random
sampling affecting N. The probable error of N=1rj2—133793 for a sample of size n is given by j
4 e
Jn
and is thus quite easy to calculate. We have tested it on a number of cases of partial correlations
worked out for this paper and find that if -67449cy is of the same order as N, then -67449c,,,, is
of much the same order as 379. In other words, if N is so small relative to its probable error that
it might easily have a reversed sign, then 37). is insignificant as compared to its probable error also.
For example, N=:0446 and D=-0956 leads to 3rj2=°4665 with a probable error of -0746. 3ri2 is
accordingly considerable and significant, but the probable error of N is only :0105, and we can hardly
suppose the sign of 3r1 due to a random sampling variation in the sign of N.
“674490 y= 67449 — {D2 — N2[2 (1 — 1452) + 2 (1 — 1992) +1 — ry? - 3]}?,
Erue, M. Evprerton anp Kari PEARSON 505
These values were so startling and so contradictory, that we proceeded to
eighth differences with the results:
gm" dg my’. dgmq — ‘60185 + 0609
gmz/""Sgmy . Sgmy’ — + 5481 + 0667,
which emphasised as well as confirmed the previous results.
Now it seems absurd to suppose that the deaths of female infants in one year
can organically influence the deaths of males of the same group in the next year,
or male infants the deaths of females in the successive year. But the extraordinary
feature of these results is that while a high deathrate of female infants lessens the
deathrate of males in the second year of life of the same group, a high deathrate
of male infants increases the deathrate of females in the second year of life of the
same group.
In order to throw further light on the matter we investigated male and female
deathrate correlations in the third and fourth years of life. We found
5gM3! Sgmg’. Sgmy — — ‘2640 + 0887,
— 0082 + ‘0954,
The second is practically zero, the first of no importance having regard to the
high values of the correlation of deathrates of groups of the same sex in the third
and fourth years of life (/:—°703 +085; 2:—°731 4 :078). Had we come to
these values at first we should have been content, but the cross relation between
the infant deaths of one sex and the deaths in the second year of life of the
opposite sex was undoubtedly puzzling.
5g mg! gms. Sm! —
We then proceeded to still further limit our conditions by determining the
partial correlation between female infants in one year and males in the second
year of life of the same birth-year when the deathrates of the males‘in the first
year of life and of the females in the second were both constant. We obtained
= +1632 + 0928,
= +2997 + 0868.
Og my - OG my! Sgmy! . Og my
5g my’. dg my! Ggmy . 5g ms!
Having regard to their probable errors these are of a quite different and
negligible significance when compared with the values of
an d 56 my
gm!) dg my’. dG my 'TSgmy. domo’
given above.
It is worth while noting that
Sg! 7 Sg my’ . 8gm, = — °2188 + 0908,
Sgma! demi «Sama = + 1088 + 0943
also give values of no practical importance. Or, to annul the spurious influence
of infantile deaths of one sex, A, on deaths in the second year of sex, B, of the
same group, it is more effective to render constant the deaths of A in the second
year of life than of B in the first year of life.
506 Further Evidence of Natural Selection in Man
In the light of this result we have found the correlations between deathrates
of sex A in the third and sex B in the fourth year of life, for constant deathrate
of sex A in the fourth year of life.
We have
= — 0818 + 0948,
Sgmy! Syms’. Sg my”
Some idemsstsen = ‘1477 + (0933.
Both of these may be taken as zero, having regard to their probable errors.
Thus on the whole, while the relation between the deathrate of a group of one
sex in one year and the deathrate of the remainder in the following year of life
appears after the annulment of the time-factor to be very considerable and
negative, there does not appear to be any organic relation between the deathrate
of sex A in one year and sex B in the following year, if we proceed by the method
of partial correlation. But at the same time we believe that this method must
be used with very considerable caution, and that to avoid erroneous conclusions the
whole problem must be investigated from a variety of standpoints in cases like the
present where one of the three total correlations is extremely high. The numerator
NV ranges in the cases we have been discussing from about ‘01 to ‘05 and with a
small total frequency like 50, any disturbing cause—apart from random variation—
may have marked influence*.
(9) The conclusion which we have formed is that in the present problem of
natural selection it is probably better to annul the environmental factor by
the variate difference method rather than to proceed by the method of partial
correlation as we have hitherto done.
By the former method we have shown that for both sexes a heavy deathrate
in one year of life means a markedly lower deathrate in the same group in the
following year of life, and that this extends in a lessened degree to the year
following that, but is not by the present method easy to trace further. It is
difficult to believe that this important fact can be due to any other source than
the influence of natural selection, i.e. a heavy mortality leaves behind it a stronger
population. Nature is not concerned with the moral or the immoral, which are
standards of human conduct, and the duty of the naturalist is to point out what
goes on in Nature. There can now scarcely be a doubt that even in highly
organised human communities the deathrate is selective, and physical fitness is
the criterion for survival. To assert the existence of this selection and measure
its intensity must be distinguished from advocacy of a high infant mortality as
a factor of racial efficiency. This reminder is the more needful as there are not
wanting those who assert that demonstrating the existence of natural selection in
man is identical with decrying all efforts to reduce the infantile deathrate.
We have to acknowledge the great assistance we have received from our
colleague Miss Beatrice M. Cave in the laborious arithmetical work of this paper.
* If F=N/D, where N and D are both small, but F finite, then 6f/F=6N/N-6D/D and small
disturbances produce great results in I’.
FREQUENCY DISTRIBUTION OF THE VALUES OF THE
CORRELATION COEFFICIENT IN SAMPLES FROM
AN INDEFINITELY LARGE POPULATION.
By R, A. FISHER.
1. My attention was drawn to the problem of the frequency distribution of the
correlation coefficient by an article published by Mr H. E. Soper* in 1913. Seeing
that the problem might be attacked by means of geometrical ideas, which I had
previously found helpful in the consideration of samples, I have examined the two
articles by “Student,” upon which Mr Soper’s more elaborate work was based,
with a view to checking and verifying the conclusions there attained.
“Student,” if I do not mistake his intention, desiring primarily to obtain
a just estimate of the accuracy to be ascribed to the mean of a small sample,
found it necessary to allow for the fact that the mean square error of such a
sample is not generally equal to the standard deviation of the normal population
from which it is drawn. He was led, in fact, to study the frequency distribution
of the mean square error. He calculated algebraically the first four moments of
this frequency curve, both about the zero point, and about its mean, observed
a simple law to connect the successive moments, and discovered a frequency curve,
which fitted his moments, and gave the required law.
Thus if a, #, ... 2, are the members of a sample,
NL = 0, + %,+...+ Ln,
and ny? = (@, — @)? + (&% —Z)P +... + (Lp — @)’,
the frequency with which the mean square error lies in the range du is propor-
tional to
: ms”
jee 20? du.
This result, although arrived at by empirical methods, was established almost
beyond reasonable doubt in the first of “Student’s” papers. It is, however, of
interest to notice that the form establishes itself instantly, when the distribution
of the sample is viewed geometrically.
* Biometrika, Vol. 1x. p. 91. + Ibid. Vol. vt. pp. 1 and 302.
Biometrika x 65
508 Distribution of the Correlation Coefficients of Samples
In the second of these two papers the more difficult problem of the frequency
distribution of the correlation coefficient is attempted. For samples of 2 the
frequency distribution between the only two possible values —1 and +1 was
7
2}
where p is the correlation of the population. Besides this theoretical result, |
“Student” appeals only to experimental data. From these he derives an
empirical form for the distribution when p=0, and makes several valuable
suggestions. It has been the greatest pleasure and interest to myself to observe
with what accuracy “Student’s” insight has led him to the right conclusions.
The form when p=0 is absolutely correct, and as a further instance I may quote
the remark* “TI have dealt with the cases of samples of 2 at some length, because
it is possible that this limiting value of the distribution, with its mean of
determined by Sheppard’s theorem to be in the ratio +sin'p : 5 — sin“,
AS : ‘ Dine 2 :
~ Sin~'p and its second moment coefficient of 1 — e sinp) , may furnish a clue
to the distribution when n is greater than 2.” As a matter of fact it is just these
quantities with which we shall be concerned.
To Mr Soper’s laborious and intricate paper I cannot hope to do justice.
I have been able to establish the substantial accuracy and value of his approxima-
tions. It is one of the advantages of approaching a problem from opposite
standpoints that Mr Soper’s forms are most accurate for those larger values of n,
where the exact formulae become most complicated.
2. The problem of the frequency distribution of the correlation coefficient 7,
derived from a sample of n pairs, taken at random from an infinite population,
may be solved, when that population can be represented by a normal surface,
with the aid of certain very general conceptions derived from the geometry of
n dimensional space. In this paper the general form will first be demonstrated,
and for a few important cases some of the successive moments will be derived.
Incidentally it will be of interest to compare the exact form with Mr Soper’s
approximation, and with reference to the experimental data supplied by “Student.”
If the frequency distribution of the popuiation be specified by the form
ot 5 my)” _ 2p (a —m,) (y — ms) (y - ot
df = 1 =e 1—p?2( 20,2 20102 2o>2 dady
Qra,0,V1 — joe :
where df is the chance that any observation should fall into the range dwdy, then
the chance that » pairs should fall within their specified elements is
if e =m)? _ 2p (w— my) (y — mo) a (y — me)?
1 a res p 2
(Qqro, om V1— pin 5 as cate gee He | dx, dy, ... dn AYn...(1),
and this we interpret as a simple density distribution in 2n dimensions.
* Biometrika, Vol, vi. p. 304.
R. A. FISHER 509
For the variables # and y it is now necessary to substitute the statistical
derivatives determined by the equations
Ne =
= Mea
(2), ng = &(y),
nus = (eB), mpd = xy ~ 9,
nN
MT Pa fy = = (a— %)(y—Y),
and it is evident that the only difficulty lies in the expression of an element of
volume in 2n dimensional space in terms of these derivatives,
The five quantities above defined have, in fact, an exceedingly beautiful
interpretation in generalised space, which we may now examine.
3. Considering first the space of n dimensions in which the variations of «
are represented, the mean and mean square error of n observations are determined
by the relations of P, the point representing the n observations, to the line
Ly = = Hy =... = Ly,
for the perpendicular PM drawn from P upon this line will lie in the region
B+ Het... tUy = NX,
and will meet it at the point M, where
N= eee Oe eC 5
further, since, PM? = (a, —%) + (a —@P +... + (@n— 2),
the length of PM is p/n.
An element of volume in this n dimensional space may now without difficulty
be specified in terms of % and y,; for, given % and w,, P must lie on a sphere in
n—1 dimensions, lying at right angles to the line OM, and the element of
volume is
Cu dp,da,-’
where C is some constant, which need not be determined.
65—2
510 Distribution of the Correlation Coefficients of Samples
The point in 2n dimensional space which is represented by the n pairs of
observations must be such that its projection on the n dimensional space, in
which « is represented, lies upon a certain sphere of radius ,/n, and on the space
in which y is represented, upon another sphere of radius p/n, and now, when we
come to the interpretation of r, we must observe that to each point on the first
sphere there corresponds a certain point on the second sphere, to which it bears
the relation
aa yaa Gna
WY yea Yo wee
In general this relation does not hold for the n pairs of observations, and the
two projections will not fall at corresponding points on the two spheres. If now
one of the spheres be turned round so as to occupy the same space as the other,
and so that the lines upon which a, and y,, and the other pairs of coordinates, are
measured, coincide, then corresponding points will lie on the same radii, and the
correlation coefficient 7 measures the cosine of the angle between the radii to the
two points specified by the observations.
Taking one of the projections as fixed at any point on the sphere of radius po,
the region for which r lies in the range dr, is a zone, on the other sphere in n — 1
dimensions, of radius “,VnV1—7, and of width gw, Vndr/V1—7r?, and therefore
n-4
having a volume proportional to pw,” (1—7?) 2) dr.
4, We may now turn to the direct simplification of the expression (I), at each
stage discarding any factors which do not involve r.
eee oe _ 2p (ema) y =me) won
eas 20102 202" S da, dy,da,dyy ... diary dyn
may be reduced to
n he =m)? + oy? — 2p {rpyme +(G~ my) (J — ms)} 4 (¥- m2)? aot
» 1—p?? 2072 20102 2o02 ee
AD AY py”? A pry fig"? A fly (1 — 7)
__n a _ 2pruame + i n-4
or to Ta pi (203° 20,93 203? fey eae ae (leo?) 2 du, du,dr.
In order to integrate this expression from 0 to 0 , with respect to mw, and pe, let
Habe oe Hao
O10," fed,”
and we have
0 09 = =a (cosh z— pr) ¢ a
OZ Wi iG One (1 -7’)
axes) 0
d n—-4
- is
z
or Se (a) 2
0 (cosh z— pr)” *
R. A. FISHER 511
which, on substituting cos @ for — pr, may be expressed in terms of a Legendre
function in the form
n—4
(i cosec 0)" Qn_o(t.cotO).(L—9®) 2 dr ceececccecesesseeceee ily,
Nak I a dz teed.
EES 9 cosh z+cos @~ sin @’
- dz 1 Git Eee Oe)
Eo bee I, (cosh z + cos 0)" D —2 (aa 78) sin 0’
and since this is a function of pr only, we may express the frequency distribution
by the convenient expression
Fart / 6
Ne ig Ra
aD or" fe a) ue
Professor Pearson has shown that this last result can be obtained directly
from Sheppard’s theorem* that
ao 1 Ma 2 be
1 [ [e ~ 3 ) (5 aes a)
Id, 2.V1— R? 0 Jo
making the substitutions
pol po
pnd pa = ose)
1 ” n
(=Rh)z2° d= p°)ay”
it s n
(— B)S)— poe’
R mrp
(—R)>,5, (1—p2)a,0,’
which give R= pr
and cos (— Rh) = 6,
we obtain
n My? 2prey fy | Me”
2 po — oe 2 2
i ia) sana 7102 F62
- ~ e 2
o1,02(1—p)Jo Jo
and hence differentiating (nm — 2) times with respect to 7, the required expression
is obtained.
6
OU se
5. The form which we have now obtained may be applied without difficulty
to all small even values of n, and in such cases is peculiarly suitable for the
calculation of moments.
When n= 2 the ordinate of the curve, with abscissa 7, is
SOE
(1— 71°) sin 8’
which becomes hyperbolic in the neighbourhoods of —1 and +1. The value
* Phil. Trans, Vol. 192, A, p. 141.
512 Distribution of the Correlation Coefficients of Samples
of r is, therefore, as we know, either —1 or +1, and the proportion, in which
these occur, depends upon p. The ratio of the infinite areas included with the
asymptotes of the above curve is
cos" p
eos (= p)’
so that the mean value of a number of observations is SU
2
When n=4 there is still no approach to normality, the curve takes the form
— (0 — 3 cot 6 + 36 cot? @),
which, when r is positive, increases regularly from its value of ;4 when 6=0, to
infinity, to which it approaches as @ approaches 7. Unless p is actually equal
to 1, in which case r is also 1 of necessity, the curve has finite ordinates at both
extremes. For calculating the number of values which should fall within any
given range, the integral, earl — @cot@), may be directly tabulated, as has
been done in forming the accompanying table of “ Student’s” observations, and
the corresponding expectations. The values given by Mr Soper’s formula are
apposed for comparison.
Table for comparison with p. 114, Biometrika, Vol. IX.
H.E.Sopevr’s
Calculated ae 3 i
r frequency | Observed ri Sees oe approxi- Diasec g
m : m mation g m
‘905—1 | 202-1 175-5 230°3 ge
*805—905| 1249 | 136-5 } eRe 69 | “98-9 } pie 20
705—805| 88-7 84 } 72°1 |
— 38 09 +203 | 318
-605— 65:1 66 57°6
Peoer =e a t +123. { 173 |. Zee \ +118 | 158
305— 30°6 245 34°3 x
205— 24°8 24:5 } = Oe Te | 99-7 i Boe 2
105— 20°5 19 25°6 2
aoe aa 7 \ =11-6 9). 2:55" sllemsets } 21:6 | 9-80
1-905— 145 22 18°8 2
1-705 — 10:7 ee oo NE at ee *
ef a
‘o re 5 . . i . .
oe he iG } 412-7 | 10-54 oe $121 | 9°21
Pape oe ca } +51] 219 He + 86 | 8-80
1105— 5-1 ES NS : 19 . ;
ie ee ae ot ee lets 2 } +105 | 44-10
— 745 — 23°61 — = 84°17
R. A. FIsHER 513
6. The direct process of integration by parts applied to such expressions as
-4
ae 1 (eB > : on 02 '
| ae ey, and es (1 —r*) T ani 9 Os
: : is: oP &
when n is even, merely introduces the sums and differences of the terms AP
at the extremes, where r is —1 or +1, with coefficients which are, in any
particular case, easily calculable.
Thus, » being 6,
OLE bat Cd AE CGE a. Oe yn
ie =) ga= ja 5 |. a - (25 oI,
= 2 x the sum of the extreme values of 6 7p 9 — 3 cot 0 + 30 cot? A)
(1 — @cot 8).
— 2x the difference of the extreme values of 3 r
If p=sin a, so that the extreme values of 6 are a= a and 5% a, the sums and
differences may readily be expressed in terms of a, and the first few may here be
tabulated: the table has been carried back as far as is necessary for the calculation
of the fourth moment.
sum difference
sin?6 (7+26? 7-66? m cot? 6
po Bee OROs= 2 = ————— 3 te an?
ap? a 36 cot 6 7 cot of m (a+3 tan a+3 a tan?a)
; 62 2
= {6 +#(1 -5) cot a cota (1+a tan a) cota {a-2ton a+ (F +0’) tan a}
6? 1 :
3 Zi +a? Ta
snd 5 mw tana 2a tan a
= 7 ae 6 cot 6) 2 tan? a (1+a tan a) m tan? a
— Gs 3 cot 6 +36 cot? 6) a tan? a (143 tan? a) 2 tan? a(a+3 tan a+3a tan? a)
a (4-96 cot 6415 cot? 6-156 cot? @) | 2tanta(4+9atana+15tan2a+ 15a tana) | r tanta (9 tan a+15 tan’ a)
There are here two natural series, which appear alternately as sums and
differences; the simpler, which may be expressed in the form
7 sin? a (. ae
2 cos ada ‘
514 Distribution of the Correlation Coefficients of Samples
is essentially a series of Legendre functions of the first kind; and may be
expressed as
Be
i
= . tan? a ea Py (1 tan a) ;
and it is these only which occur in the evaluation of the even moments.
7. It is, however, desirable to obtain general expressions for these integrals
in terms of 1 and p, and to evaluate them when n is odd.
For this purpose let us introduce a quantity ¢, such that
cos ¢ = cos 0 —k,
then, when & is sufficiently small, we may expand ¢? by Taylor’s theorem, so that
Pa Cee | aC eat a: oP NOE
gat * 59903 t/3 (sna) 3 to
Now let k= phv1—7,
eesti re Cn =") ( 0 y o
hen Bene Se anim s 2 sin 000) 21°”
and differentiating twice with respect to h
F ; 4) 2 ¢ pee 2 a) 2 @2 i 3 6) 3 @2
pa Teer) (e ae =e -)( sag) 5 thea) Gam! ge
whence, dividing by (1 — r)2, we obtain
p Gein) SOR arenes ( Was (sana) 5
as & a) 2 (1 — 72) \sin 7) Dal Ue sin na 2
pale 4 ( oe
Ee wate (sn 000) 3
a n—4 4 Q
: “3 3 gr e
so that {Ee r? (1-7?) aaa g ar
may be obtained by multiplying by |x —3 the coefficient of h”-* in
ef r? dr il —¢ cot d
=1 A) Pay sind’
when cos ¢ = cos 0 — phV1 —r?=—p(r+hv1—7°).
Our object might equally be achieved by the evaluation of the integral
ie ner ( (oy )
fe -1 1—7°\sing sin @/"
The quantity ¢ is determined by the equation
cos ¢ = cos 0 — ph V1 — 7°,
that is cosh =—p(r+hv1—7").
R. A. FISHER 515
If now r=sin B,
h=tane,
then cos = — psin Bf,
cos $=—pV1+h?sin(@+e)=—pV1+/?sin f’,
and as_ sr passes from —1 to +1,
8 passes from -5 tomer
2 y
6 from 5 —a to xt a,
Bp from —v 46 to = and thence to = Das
2 2 2
7 7
and co) from g—%tos +a and thence back to — st a,
where sina =p V1 +42, ¢ oscillates in the same manner as 6, with a somewhat
greater amplitude, and slightly in advance in respect of phase.
1 dr
P si? VI- 9
The expression
may now be reduced to
pe ae : pa ’
e| - aag (. 1 & g sin a sue .) ae’
= +e
sin? d zi —sin?a’ sin? 8’ (1 —sin?a’ sin? B’)?
us
2
ane] ats sina’ sin 8’ dp’
= (0 ie 2 28 : ; $
zs » 1 —sin?a’ sin’ — sin’ a’ sin (1 — sin? a’ sin? 8’)?
+p fi (f) sin a’ sin PB’ dp’
a an — sin? a’ sin? 6’)?
2 2 a7 4 7 2
_puw | 7p ae (=) + ™p (1 — cosa’)
cos a” cos?a’ \cosa cos? a’
pr sin a tan é€
~ cos? a’ cos a
but cos? a’ = 1 — p?(1 + h?) = cos?a— sin? a tan’e,
,ftil-¢cotd dr om tan? a
so that jes ——_ as
1 sitd vVl—,~ 1l-Atane
From this evaluation we deduce the general form
ie (1 ae Foe e Cie, CAN YO cy dae anlaw sie (III).
Biometrika x
516 Distribution of the Correlation Coefficients of Samples
The absolute frequency df, with which + falls in the range dr, is therefore -
n-1 net
—p?) 2 ) n—2 @
SES Co) ae ie aie
8. Ido not see how to integrate the other expressions of the type
en rv dr
es sin?d VJ — 7?’
although a form could probably be obtained when p is even. The general
expression for the second moment may, however, be deduced by means of a
reduction formula.
By a process of integration by parts it appears that, if we write
n-4 ne 23 Q
n—1 2
ie (ies 2)? ares Nl 2 dr =Ln.p,
then Lnse.2 = Into.0+ WLn,9 —u(n— Dies
: tan®
and since i — an (= “—tana+a),
we may obtain successively
tan?® tan?
I 6.9 = 247 ee eesee “+ tana—a),
4 3
an7 7? 3
io 20e (= green CPrges “—tana-+a),
6 5 3
and so on, yielding, when n is even, the expression
a
In.g = In. —T |n _ 2{ tan" xd,
era oe)
a form which may well hold when 1 is odd.
The above expressions are useful in tabulating the numerical values of the
second moment, 7+ 07, of the unit curve, which may easily be calculated in
succession for different values of n when tan?a is taken to have some simple
value.
9. Before leaving this aspect of the subject it is worth while to give a more
detailed examination of the mean of the frequency curves of r when n= 4.
Two formulae are arrived at by Mr Soper, which are equivalent approximations
of the second degree
a fat 3 PA elie 1—p? 3
L r= p|[1- ay {1+ 7 +8p)} [=p [1-95 {r+ pga +3e9} |,
1. F=p [1 HOF {tga M9 J=0[1-*Gf1-70-o}
R. A. FISHER 517
and these we shall compare with the form
ET: 7= 2 (a + cot a— acot?a),
p | 1000 | -2000 | -3000 | -4000 | 5000 | 6000 | 7000 | 8000 | 9000 | -9500
I | 0853 | 1710 | -2578 | 3463 | 4377 | ‘5333 | ‘6347 | ‘7443 | 8649 | ‘9304
II | 0847 | 1697 | 2555 | 3419 | 4310 5241 | 6236 | °7330 | 8566 | -9254
TIT | 0850 | 1704 | 2570 | 3451 | 4360 | 5301 | 6290 ‘7357 | 8540 | 9209
It will be observed that the approximations le on either side of the exact
value over the greater part of the range, and that the error of the first
approximation increases up to the value when p="9. The second formula
gives the correct value somewhere between ‘8 and ‘9, and is thereafter too
large.
For the particular case p = 6608,
I find (formula IIT) 7 ='5897, nearly the maximum difference from p,
Mr Soper gives (p. 109) the value 5933
and the experimental data ‘5609.
The two theoretical values are much nearer to each other than either is to
the experimental value. On the whole, it is obvious that even in this unfavour-
able case Mr Soper’s formulae possess remarkable accuracy.
10. The use of the correlation coefficient r as independent variable of these
frequency curves is in some respects highly unsatisfactory. For high values of r
the curve becomes extremely distorted and cramped, and although this very
cramping forces the mean 7 to approach p, the difference compared with 1 —p
becomes inordinately great. Even for high values of n, the distortion in this
region becomes extreme, and since at the same time the curve rapidly changes
its shape, the values of the mean and standard deviation cease to have any very
useful meaning. It would appear essential in order to draw just conclusions from
an observed high value of the correlation coefficient, say 99, that the frequency
curves should be reasonably constant in form.
The previous paragraphs suggest that more natural variables for the treatment
of our formulae are afforded by the transformations
ip
PEA oo irceaer(
l-r
p
7 = tan a= —___.,
“/
The expression for the frequency curve (II)
aaa 6
; moe n—-1 ff2 :
G8)” laa) ae
66—2
518 Distribution of the Correlation Coefficients of Samples
now becomes
( 0 ie 62 dt
" n-1
sin 000 2 (+e) =
and the range of the curve is extended from — to +o.
It is interesting that in the important case, r=0, the frequency reduces to
dt
~—~n=1 and the curves are identical with those found by “Student” for z,
d+?)
the probability integral of which he has tabulated in his first paper.
11. The moments of these curves are obtained by the evaluation of the
expressions
i ( 0 ie @ dt fe ( 0 Va @ tt
in@ ; a » \sin 000 5 haneeeaie
—« \sin@0@ 2 dQ+e)2 J —» \sin@00 2 +e2
and so on; of these the first is known already (III) to have the value
vg In —3
preareaenii=l
Chea yo)) 2
and the others may be obtained in succession, for
Greys i << Cre Cedi, oh Cm | Sut Ney G2 aeaateone
CP eee SiO OO) cara (14 pyr op”
aoe | ee eS
TO ptt as i tae ano p annem
so that the first moment
i ( ri) Nae tdt fy) 7 |r — 4 ain
. a 5 n=l ~ 9, ° n= n=4
sin 806 (1+ #)? 2 (Ch) (ae)
»n—1 n—1
eae 2 (1+#)2
4(n—2)p_
—@
geal gar ail ss
n-3V/1—— n-3
hence t= T.
The mean, therefore, is greater than the true value 7 by a constant fraction
of its value. And this fraction decreases in the simplest possible manner as n
increases.
In the same way, we may evaluate the second moment,
COs = array Hea) ag)
preatie| IH ise 3) all
and a= flts + Ooo ot
the third moment
ae (n—2)r Re au i ))))
VB.e ~ (n—3)(n—4)(n—5) |sa+e)+ (n—3p J’
and the fourth moment
R. A. FIsHEr 519
3 6 (n — 2) 7? ah
a Se eS 2\2
ae = (n—4)(n—6) {a a) GA — 3)(n — 5) (n— 3)!(n—5)
For high values of n, all but the first terms tend to vanish; §, tends to vary
as p’?, and @, tends to become independent of p. In effect for high values of 7,
where p? is nearly equal to unity, the form of the curve is nearly constant, but the
skewness measured by , decreases to zero at the origin, and changes its sense,
when 7 and p change their sign.
(l+7°)+
Tables are appended for inspection rather than for reference which show the
nature and extent of these changes in the form of the curves.
Table of o°.
P= ‘O01 03 ‘10 *30 1:00 3°00 10°00 30°00 100°00
8 °2531 2593 *2810 3430 “5600 1°140 3°350 9°550 | 31°250
13 "1123 °1148 1234 1481 2344 ‘4811 | 1344 3811 | 12°444
18 ‘07219 | °07372 | -07908 | :09438 | °1479 *3010 8365 2°367 7722
23 705319 | °05429 | -05817 | °06925 | -1080 2188 6066 1714 5°592
3S 03484 03555 | -03805 | 04518 | "7015 1415 3912 17105 3°602
43 02590 | -02643 | -02827 | °03353 -05194 1045 “2886 *8146 | 2°655
53 02062 02103 | °02249 -02666 | -04123 08288 | °2287 *6451 | 2°103
Table of By.
eS 01 O03 ‘10 30 1:00 3°00 10°00 | 30°00 | 100-00 oe
r=
& | 05685 | °1662 ‘5076 =| 1°230 2°450 3°788 3°965 | 4°153 | 4°184 | 4°252
13 | 01517 | 04776 1376 *3400 *7058 | 1:018 1°205 | 1271 | 1296 | 1°3065
18 | 008399 | :02463 ‘07645 | °*1914 “4016 5857 *6990 | °7395| °7546| °7619
23 | ‘005757 | ‘01691 05247 | °1317 “3016 4093 "4910 | °5208| °5314| °5361
38 | °003518 | 010385 038214 | -08100) ‘1731 "2559 *3031 | °3260} 3334 | °3366
43 | °002530 | -007435 | *02315 05841 | +1251 "1858 ‘2237 | °2376| °2429 | °2452
53 | :001973 | 005798 | *01807 04562 | -09800] +1458 1757 | *1868} +1910] °1928
Table of Bs.
7=| -00 ‘01 03 “10 30 1:00 3°00 10°00 30°00 | 100°00 or)
r=
8 | 60000 | 6°1137 | 6°3179 | 70179 | 8:4767 | 10°9668 | 12-9652 | 14°1116 | 14°5024 | 14°6508 | 14°7159
13 | 3°8571 | 3°8802 | 3-9248 | 4:0663 | 4°3770 | 4:9397 | 5°4240| 5°7147| 5°8186| 5°8578| 5:8750
18 | 3°5000 | 3°5121 | 3°5356 | 3°6104 | 3°7937 | 4°0828| 4°3532) 4:5186| 4°5783 | 4°6009| 4-6109
23 | 3°3529 | 3°3612 | 3°3768 | 3:4271 | 3°5556 | 3:7486 | 3°9356) 4:0511 | 4:0930] 4:1089]| 4:1159
33 | 3°2222 | 32271 | 3:2365 | 3°2667 | 3°3343.| 3:4619| 3°5773| 3°6493| 3°6756| 3°6856| 3-6899
43 | 3°1622 | 31656 | 3°1723 | 3°1938 | 3°2422 | 3:3261 | 3°4172|] 3:4692| 3°4886| 3:4958] 3:°4991
53 | 31277 | 3°1303 | 3°1356 | 3°1522 | 3°1898 | 3:2640] 3°3281| 3°3676| 3°3826] 3:3883| 3-3909
520 Distribution of the Correlation Coefficients of Samples
12. The fact that the mean value 7 of the observed correlation coefficient is
numerically less than p might have been interpreted as meaning that given
a single observed value r, the true value of the correlation coefficient of the
population from which the sample is drawn is likely to be greater than r. This
reasoning is altogether fallacious. The mean 7 is not an intrinsic feature of the
frequency distribution. It depends upon the choice of the particular variable r
in terms of which the frequency distribution is represented. When we use ¢ as
variable, the situation is reversed. Whereas in using 7 we cramp all the high
values of the correlation into the small space in the neighbourhood of r=1,
producing a frequency curve which trails out in the negative direction and so
tending to reduce the value of the mean, by using ¢, we spread out the region ot
high values, producing asymmetry in the opposite sense, and obtain a value ¢
which is greater than tr. The mean might, in fact, be brought to any chosen
point, by stretching and compressing different parts of the scale in the required
manner. For the interpretation of a single observation the relation between
¢ and 7 is in no way superior to that between 7 and p. The variable ¢ has been
chosen primarily in order to give stability of form to the frequency curves in
ditferent parts of the scale. It is in addition a variable to which the analysis
naturally leads us, and which enables the mean and moments to be readily
calculated, and so a comparison to be made with the standard Pearson curves, but
it is not, with these advantages, in a unique position. In some respects the
function, log tan 4 (a +5) , 1s its superior as independent variable.
I have given elsewhere* a criterion, independent of scaling, suitable for
obtaining the relation between an observed correlation of a sample and the most
probable value of the correlation of the whole population. Since the chance of
any observation falling in the range dr is proportional to
4
ee PW ie peat Be Oiage
rey ea) (a oe my
for variations of p, we must find that value of p for which this quantity is a
maximum, and thereby obtain the equation
n—-1
4) x mow : 4) n—l 62 =
dp ia moe fe 730) 3 a
Since ie dat aes 1 ( 0 es @?
o (coshaw+cos 6)” |n—1\sin@00 2
a = d
ye ee 2:
ge ke I, op \a p*) (cosh & + cos no ae
* R. A. Fisher, ‘On an absolute criterion for fitting frequency curves,” Messenger of Mathematics,
February, 1912. |
R. A. Fisner 521
which leads by a process of simplification to the equation
du
i (uabiz= are —p cosh 2) = 0.
Since cosh w is always greater than pr, the factor in the numerator, r—p cosh a,
must change sign in the range of integration. We therefore see that r is greater
than p. Further an approximate solution may be obtained for large values of n.
The integrand is negligible save when # is very small, and we may write
9
1+ 5 for cosh x
nar
and (1 — pr)” e*(1—P") for (cosh a — pr)”.
meena? ene
Then rf e Te | (1+ 5)e aC PO de,
0 0
and in consequence, as a first approximation,
1 = r
=p(l 2 ) :
r=p ( ats oF
The corresponding relation between ¢ and 7 is evidently
1
—is (1 ar x) Fi
It is now apparent that the most likely value of the correlation will in general
be less than that observed, but the difference will be only half of that suggested
by the mean, @.
It might plausibly be urged that in the choice of an independent variable we
should aim at making the relation between the mean and the true value approach
the above equation, or rather that to which the above is an approximation, or
that we should aim at reducing the asymmetry of the curves, or at approximate
constancy of the standard deviation. In these respects the function
log tan } (a + 5) that is, tanh p
is not a little attractive, but so far as I have examined it, it does not tend to
simplify the analysis, and approaches relative constancy at the expense of the
constancy proportionate to the variable, which the expressions in 7 exhibit*.
* [It may be worth noting that Mr Fisher’s ¢ is the ¢-square root mean square contingency—of the
more usual notation, and is the expression used in determining the probability that correlated material
has been obtained by random sampling from uncorrelated material. Eb.]
ON THE DISTRIBUTION OF THE STANDARD DEVIATIONS
OF SMALL SAMPLES: APPENDIX I. TO PAPERS BY
“STUDENT” AND ROA FISHER:
(EDITORIAL.)
CONSIDER the population distributed according to the law
—m)2
hy (x —m)
and let a sample of n represented by the variate values a, x... &, be taken from
it. Then the probability 6P that this sample will lie between
a, and 2, +6a,, x, and a+ da, ... % and a+ bap,
Nn _1 S(@,-— my
yet Boat
is = Ge Note Or;OTs 2Oce
= const. xX e
where %= - Sas) ae a— - S(a,;—Z)? we may write:
n=? n (Z&—m)?
a (3
2 D} 2
SP =const. x e g
) OD OL g's ss OL gl eee nee (iii).
Changing as Mr Fisher does (see p. 510 above) to & and & as coordinates
we have:
n=? n(xZ—m)?
= (bese
SP =const. xe ( os o ) >"-2 87 8>.
We see at once from this* that the law of distribution of samples of means is
the normal curve
YH We. 2 OE wee ee eee (iv)
* Of course the form reached above shows that for normal distributions there is no correlation
between deviations in the mean and in the standard deviation of samples, a familiar fact.
EDITORIAL 523
with mean =m, the mean of the population, and with standard deviation
=a//n, a well-known result.
On the other hand the distribution of samples of standard deviations is
This curve was first reached by “Student” as a highly probable result
following from the relations he had obtained from the moments of ?*.
Mr Fisher's work thus enables us to justify “Student’s” assumption.
“Student” has discussed at some length the distribution curve for >. He
has obtained the values of the moment coefficients p., ws; and py, and the
general expressions for the means when n is even and odd. The whole problem
is of such importance that it seems worth reconsidering, and providing tables
showing the approach of the distribution curve to normality as n rises from
4 to 100.
The following investigation largely repeats work given by “Student,” but it
expresses the values for ws, w,, and @, and 8, in a different formt. We shall not
use approximate expressions for the constants, for the order of terms in 1/n
depends so largely on the relative magnitude of their coefficients, that such
expressions become unreliable for values of n under 100.
Clearly (v) is a skew curve with range limited at one end, }=0, and not at
the other, = 00. See Figure p. 524.
We shall write the standard deviation of %, os, and the moments of the
frequency about the end of the range O as M,’, M,’, etc., while the moment-
coefficients about Q will be as usual w,(=0), 2, etc. Obviously w.=cs% It is
desirable to ascertain S, =, os and the skewness as well as §, and ®, for the
distribution. We do this to show the rapidity of change to a normal distribution.
It is well, however, to notice a priori that for n large the distribution does become
normal.
* «Student’s” approximate values for 6, and fy (loc. cit. p. 10) are, we fear, erroneous. He gives
iManeee aly but it is needful to have a further term in we
a) ene 72 2 order to obtain 6; and fy, correctly to
the second approximation in =. If this further term be p/n?, then:
1 64p —3 : A sere 3
oi on (a + i). as against ‘‘Student’s On (a - i) ;
16p 1
Bo oF lgerers ” ” ” 3 (1- 7a).
An examination of our table (p. 529) shows that ‘‘Student’s” corrections are not of the right sign to
agree with the facts, and that further no constant value of p would give good results even for fairly
high values of n, i.e. it is probable that the term in 5 in D? is of equal importance with that in
t+ “The Probable Error of a Mean,” Biometrika, Vol. v1. pp. 1—25, more especially pp. 4, 6, and
8 to 10. ;
Biometrika x 67
524 Standard Deviations of Small Samples
yb
|
i}
}
i}
'
|
i}
1
!
1
)
U
ie) P a
OP =mode=, O0Q=mean=2.
To obtain this approximation to (v) let us assume & =>+ e, and suppose e
small. Expanding log y we find:
log y = log y, +(n — 2) log 3S —4n(S/o) +
eo. -, n Ee
n-2o¢?
=
ne n—2 6? :
= @ + 5 )+ terms in e%
2c? n >
Hence since > is at our choice we will take it so that
and thus:
n>? e
ww _— 4 Ss
Y=Yur2e ~ 7 @ * o*/(2n) + ete.
Or, if ¢ be small compared with o, the distribution is the normal curve:
2
y=yle FR
=|
=
om
<—
<.
=:
=
=x) ce Ee
with mean at aay rma: and standard deviation o/V2n. If n as usual be
considerable, this agrees with the ordinary result, ie. Z=o and cy =o/V2n, the
distribution being treated as normal.
We will now deal with the full result (v). We have:
é nz?
M, = ySrds=y{ PAs al Pee CEP Rane pec n s ccc (viii),
0
and clearly M,’ depends on a knowledge of
Ly= | ule du detasen indeocer aaa eee (ix),
0
for we have:
P fon n+p—-1
Mea cal PAs
EDITORIAL
525
Integrating by parts we find:
Ly = (q—1) Ly-2
...d.1 L,, if g be even
(Ghee a q
Mea ep pe sad:
Now
I= [e-¥du=,/5,
0 2
L, =| ue *” du =1,
0
thus M,’ is determined, and will depend on whether 7+ p be even or odd.
poh cn n+1 ie oe n+1 J
But My = Yo a Ln = Yo C (n-1) Ens,
ims go \?-1
M, =H Yo (=) Una
—l1
pe = M,'/M)' = o° ; n
and
Hence
To find the modal value } we must differentiate (v), and we have
which gives
a result in agreement with the mean > of the approximate solution (vi), as we
should anticipate.
It now remains to find WM’ and WM,’ absolutely
on ea
M, =Y% \J/n HRS
n—1
My = % Ga Lino
Suppose n even, then
Lya=(n—2)(n—4)...2x1
TEN PSN ase 1 vA PT ee TL ee (x11).
2;
Hence for n even
3 p= MM = oe n—-2n—4 VE
eG eG eal we ttteesaeeees (x1V A).
Again for n odd
Ln = (n — 2)(n— 4)... 1 x Ve
Ins =(n — 8) (n—5)...2 x1,
and .hence
_ o (n—2)(n—4)...1 /o
ReaD Bel fe
14)
526 Standard Deviations of Small Samples
These accurate values of =, the mean standard deviation of samples, were first
given by “Student” (loc. cit. p. 8). Now by Wallis’ Theorem
Ny Gea £ product-of even numbers up to 2n
gv" ~ product of odd numbers up to 2n —1°
Thus (xivA) for n large tends to become
SS ae eeu
Vn He 5 me
Vaio N°
These values, however, really only suffice to show the approach of > to o, as
M
II
and (xivB)
they depend on the neglect of terms of the order * as compared to 1, and we should
get absurd results for os? by subtracting the square of the above values of > from
#2 in (xi). All they really tell us is that for n large & =o, but they give no true,
approximation in a
If we use Stirling’s Theorem up to the third termt, Le.
a 1
wl= Bmore (1+ 55+ Me
12% 288.2?
é = 3} (foes
we obtain L=a (1 Teamie 39,8) esi eee (xv),
2 1
oc? = om (1 — a) | deed telatne oe eee (xvi),
ape ae
M3 = ane? ie ee
Bar nomhouldet lled’to antreducouNeut _ 189" Sito, Sunnie
ut we snou e compe ed to introduce e term — 5184025 g
expression to reach the second terms in p; and «4. As we have indicated (p. 523,
ftn.), such a term, even if used, will not lead to profitable results. It is better
to work with the full formulae. It is desirable to find the full third and fourth
moment coefficients in order to determine @, and 8, and so measure as 7 increases
the rapidity of approach to the normal curve.
* « Student” has used an extension of Wallis’ Theorem, which will suffice for certain constants only.
+ We can write (xiv a)
< o gin-1 ln —1)?
>= ( [dese J: Fis aeteeseet cuneate en er ee eeeee (xvii),
Jn |n-2 T
and (xivB)
PO: (n-2)[n-38
= el ™ ee
Jn (g-3) | 3 (m ios 3))2 ie ig ieielsleieieleltis oloieseleieleteeieivieleleieieintelersteteletete (xviii),
and then apply Stirling’s Theorem.
EDITORIAL 527
We have: f é
N+3 N+3
May a Lins = Yo (5) (n+ 1) Ly
= = (n + 1) M,’.
2 ’ 4(n2—1 :
Hence ps = Mi /M, = = (n+1) pe = ae ee eros taste (xix),
n+2 N+2
M; = Yo ca Lnia = Yo a ) MEn-
= Ss nM’.
n
Hence |g EG DLE ope) RRR a eT reer eR OED (xx).
Transferring to mean:
b= bs. a 2a fer. a fe [ny
SS GnnY (1 — 2¢3°/o? — jae *)
S a (1 z =e eae an Ga
Thus yp; will grow small, not only owing to the factor z but because oc? tends
to equal o?/2n as n increases.
i os? \?
eae o>: ( iz OF
Now Bi mr bs? i. n2 os! i
Or B= 8n 2 (1 - Set aoa) Shoes ay Semana (xxil).
Here =/o is of the form 1— 2S and
pee (ay aXe
2259} ( 2) ‘
and thus @, tends as n increases to take the form 8y,?/n, but as y. may be a
considerable numerical coefficient 8y,” may be commensurable with n till n is very
considerable.
We next turn to «4, and shall endeavour to express it in terms of u.=<a3?.
Since fa! = fy + 3? = 0° (1 : 7)
by (xi), we have
= 1
S=o(1 --) — Mo.
N n
528 Standard Deviations of Small Samples
Further by (xx)
Hepa = 07)? = a4 ¢ 2 5) — ofa.
Thus
b= on x Apes’ by’ + 6 plo’ by” = 3 p74
aati =~ 4(1- 7) +4846 (1-7) ( --- 4)
nN nN oO 11) n Oo
2 2
~3(1--) +B (1-7) -3 3h
nN o~ nN oC
(5 1 3 os? 3 cs? \2
=o eee oon vee fae Neue ie i
2 [4r2 2n (4 =| (1 aa) An? (2 Tan BdoSodo00
Hence
By =
eaieiar ? sae (4 i ") € i oa) 8 (1 = | é.<( XX111));
Our results for ws and 4, although expressed in other notation, are in
accordance with “Student’s” (loc. cit. p. 9), so also are our results (xv) and (xvi)
although reached by a different method of approximation. We do not agree with
his approximate values for us, ws, or 8, and f,.
The calculations to find S/o, os/(c/s/2n), 8, and B, presented some trouble.
In order to be correct to the four figures of decimals in the tabled results, tables
of ten-figure logarithms had to be used in the logarithmic part of the work.
Formulae (xvii) and (xviii) of the ftn. p. 526 were adopted, using Degen’s Tables of
the Logarithms of Factorials. /,' was calculated to nine figures, and even then,
as n became large, the determination of the antilogarithms presented consider-
able difficulty. Further the powers of 1 — os?/(o?/2n) gave rise to trouble. The
numerical work was undertaken by Ethel M. Elderton and Beatrice M. Cave,
to whom very hearty thanks are due. We think the results may be depended
on to the figures tabulated.
It will be seen that by the time n =50 the mode is as close to the mean as we
should expect to find in any random sample of normal material; the average
mean > is only 1°5°/, from the usually adopted value o, and the average standard
deviation cy only 0°3°/, from its customary value o|V2n. Further 8, and f, are
0105 and 3:0003 respectively, or for all practical purposes have reached their
normal values. We think it must be concluded that for samples of 50 the usual
theory of the probable error of the standard deviation holds satisfactorily, and
that to apply it for the case of n= 25 would not lead to any error which would be
of importance in the majority of statistical problems.
On the other hand, if a small sample, n < 20 say, of a population be taken, the
value of the standard deviation found from it will be usually less than the standard
n~
deviation of the true population. If we take the most probable value, >, as that
EDITORIAL 529
which has most likely been observed, then the result should be divided by the
number in the column entitled mode &/o to obtain the most reasonable value for a.
For example, if } be observed, and n = 20, then the most reasonable value to give o
is /°9487.
The paper by Mr Fisher and the accompanying table more or less complete
the work on the distribution of standard-deviations outlined by “Student” in
1908.
Table of Values of the Constants of the Frequency Distribution of the Standard
Deviations of Samples drawn at random from a Normal Population.
caer Measures of Deviation from
eon Standard Deviation Normality
Sample Mode Mean =
n S/o Z/o
oy/o o3(o/V2n) Skewness By Bo
4 ‘7071 ‘7979 ‘3367 9524 2696 | -2359 3°1082
5 "7746 *8407 *B052 ‘9651 "2168 ‘1646 3°0593
6 *8165 “8686 *2808 9725 °1857 "1255 3°0370
7 "8452 "8882 "2612 ‘9774 "1648 | “1011 3°0251
8 “8660 "9027 "2452 “9808 "1495 "0845 3°0181
9 *8819 “9139 ‘2318 "9834. W188 je" -O725 3°0136
10 "8944 9227 2203 "9853 *1285 0634 3°0106
11 "9045 ‘9300 "2104 ‘9868 *1209 0564 3°0085
WD "9129 "9359 ‘2017 “9881 "1144 ‘0507 3°0070
13 "9199 *9410 “1940 “9891 "1088 0461 3°0059
14 "9258 "9453 1871 “9900 1041 0422 3°0049
15 “9309 "9490 “1809 “9907 “0998 ‘0390 3°0042
16 "9354 9523 1752 "9914 “0961 ‘0362 3°0036
17 *9393 “9551 ‘1701 ‘9919 0927 0337 3°0032
18 9428 ‘9576 "1654 9924 ‘0897 ‘0316 3°0028
19 "9459 “9599 ‘1611 “9928 ‘0869 ‘0297 3°0025 |
20 "9487 ‘9619 *1570 "9932 0844 ‘0281 3:0022 |
25 "9592 ‘9696 "1407 "9948 *0745 “0219 30014 |
30 ‘9661 ‘9748 "1285 *9956 ‘0674 ‘0180 3°0009 |
385 *9710 "9784 1191 "9963 0620 0153 3°0007 |
40 ‘9747 ‘9811 1114 *9967 0577 0132 3°0005
45 ‘9775 "9832 Odi 9977 0541 ‘O117 3°0004
50 9798 "9849 ‘0997 ‘9974 0512 “O105, 1) 3:0003) |
58 “9816 9863 ‘0951 ‘9977 "0488 ‘0095 3°0003 |
60 "9832 ‘9874 0911 | -9979 ‘0467 =| +0087 3°0002 |
65 "9845 “9884. ‘O875 “9980 ‘0447 “0080 3°0002
70 "9856 "9892 0844 | 9982 0430 0074 | 3:0002
75 ‘9866 ‘9900 0815 | -9983 0415 ‘0069 | 30001
80 ‘9874 "9906 0789 | -9984 | -0402 0064 == 30001
85 “9882 9911 ‘0766 "9985 "0389 70060 = 30001
90 ‘9888 ‘9916 0744 | -9986 | 0378 ‘0057 3°0001
95 "9894 “9921 0725 | ‘9987 03867 =| «0054 | 3:0001
100 "9899 "9925 ‘0706 "9987 0358 | ‘0051 3°0000
TUBERCULOSIS AND SEGREGATION.
By ALICE LEE, DSc.
(1) In his book The Prevention of Tuberculosis (London: Methuen, no date
on the issue we have used) Dr A. Newsholme has examined the influence of
segregation on Tuberculosis. This is the topic of Chapter xxxv. In the opening
of this chapter, he writes:
The exact measure of institutional segregation of phthisis is the ratio stating how many of
the total days’ of sickness (number of patients and number of days of sickness) are passed in
institutions. This ratio and the equivalents for it which have to be used in practice may
for convenience be called the segregation ratio. The need for equivalents for the ratio as stated
above arises from the fact that we are dealing with actual recorded experience, and the material
has to be taken from the records as they happen to exist. (p. 266.)
After noting the incompleteness of the records, Dr Newsholme continues :
It becomes necessary therefore to select other figures which vary approximately with the total
days of tuberculous sickness and the total days of tuberculous sickness passed in institutions.
(p. 266.)
We shall discuss below what “indirect measures of segregation” Dr Newsholme
selects, but he gives the following most proper caution with regard to them:
In using these indirect measures of institutional treatment of tuberculosis and of its pre-
valence it must be remembered that they are indirect and approximate. Thus, for instance,
figures for institutional treatment usually give the number of cases and not days of treatment,
and while they tell how many people were segregated in institutions do not’ show the average
duration, still less the quality of the treatment. Any of these indirect forms of segregation ratio
has therefore to be verified wherever possible by the application to the same community and
period of one or more other forms of the ratio, and checked wherever practicable by a special
examination of sample constituent communities whose figures are included in the total. (p. 268.)
Dr Newsholme in the course of his chapter gives a number of very high
correlations between the phthisis deathrate and the indirect forms of the segre-
gation ratio he has selected, and he interprets these as well as a long series
of graphs as demonstrating that institutional segregation has been a most
important factor in the diminution of the phthisis deathrate. Now any two
variates which are changing continuously with the time—say, the consumption
AticE LEE 531
of bananas per head of the population and the fall in the birthrate—will exhibit
high correlation and will show graphically very high association, if plotted to
appropriate scales and on a common time basis. Until the time factor has
been removed, either by partial correlation or otherwise, it would be most un-
wise to interpret such cases as providing any causal relationship.
It seemed accordingly worth while to reinvestigate Dr Newsholme’s problems
with the aid of a rather more adequate statistical apparatus.
(2) We must frankly confess at the outset that we have had great difficulty
in following Dr Newsholme’s description of the methods he has adopted to
measure the amount of segregation. His charts do not seem always in ac-
cordance with his tables, and both are occasionally out of agreement with his
definitions. As he does not give the raw data on which his correlations are
based, but only condensed versions of them in his tables and graphs, it is
impossible to test his conclusions without returning to the original sources,
which are not always stated, and when we have found them and our results
differ, we are unable to say whether the difference is due to failure in his or
in our arithmetic, or to divergences between his and our records.
Dr Newsholme uses in all some six measures of the segregation ratio, four
intentionally and two apparently by inadvertence.
Let P= total population of a given area, @ = the total number of annual deaths
from phthisis. Then ¢/P multiplied by 10,000 or 100,000, as the case may be, gives
the crude deathrate from phthisis. Let D; be the deaths from all causes which
occur in institutions and D the total deaths in the same area, then 100D,/D is
Dr Newsholme’s first approximation to the segregation ratio*. On p. 270 he
gives two tables which show in (a) England and Wales as a whole, (b) in London,
that, while in the course of forty years 1000¢/P has practically halved, 100D;/D
has practically doubled. The data, Dr Newsholme tells us, show “not only a
very close correspondence between the increase of total institutional segregation
measured by the ratio in question and the decrease of phthisis, but an even more
striking similarity in the ratio at which these changes have occurred” (p. 271).
This is illustrated by a graph on p. 271, in which the logarithms of the phthisis
deathrate are plotted to time against the logarithms of the indices of institutional
deaths to all deathst. We do not know why Dr Newsholme has chosen this
method of representation; it certainly, with his choice of scales, makes the two
curves roughly parallel, but this does not demonstrate the “similarity in the ratios
at which these changes have occurred.” For, if the actual values be plotted to
the time, the curve of phthisis deathrate is conver and the institutional death-
rate concave to the time axis, in other words while the rate of one is increasing,
* The assumption made appears to be that for the period in question D; is proportional to the
institutional deaths from phthisis,—a very big assumption.
+ The logarithms of the ratios of institutional deaths to all deaths appear to be either wrongly
plotted or wrongly calculated.
Biometrika x 68
532 Tuberculosis and Segregation
the rate of the other is decreasing during the period in question,—always on the
supposition that we plot the results as Dr Newsholme has done with reversed
directions of increasing scales for the two indices. He states that “the experience
is summarised in the high correlation coefficients of ‘91 for England and Wales
(1878—1903) and ‘90 for London (1866—1904)” (p. 271). The correlations
found from his actual tables do not appear to agree with these, being, for example,
— ‘93 for England and Wales with the negative sign as we should anticipate; but
as Dr Newsholme does not give the same years for his correlation coefficients as
in his tables, he may have worked out his coefficients for individual years. It is
impossible to test the matter, as neither the figures nor their source are provided.
If, however, we take his Tables LXII and LXIII, and apply the variate
difference method* to Dr Newsholme’s data as they stand in his book, which
are all the data available, we find
Correlation of Phthisis Deathrate and Ratio of Deaths in
Institutions to Total Deaths.
England and Wales: Third Differences — 174+ 293,
London : Second Differences —:094+:'252.
In other words the data show no significant relationship between this measure
of segregation and the phthisis deathrate, when the time-factor is annulled, even
with the early differences. It is impossible to press the matter further because
the data are far too sparse for difference treatment, but the results, such as they
are, are sufficient to indicate that Dr Newsholme’s high correlations are solely due
to the fact that both variates are continuously changing with the timef.
(3) As a second measure of segregation Dr Newsholme takes 100¢;/6 and
1000¢/P is then correlated with this, ¢; being the deaths from phthisis in
institutions. On p. 275 Dr Newsholme gives very meagre data for Brighton,
Sheffield and Salford in groups of years, six pairs of values for Sheffield, five for
Brighton and four for Salford. It is thus impossible to test these for annulment
of the time-factor, and no references are given to the sources of the original data.
On p. 276 we read:
Coefficients of correlation summarising this correspondence for long series of single years
work out at 67 for Salford from 1884 to 1904 and ‘80 for Sheffield from 1876 to 1905 f.
If the arithmetical values be correct, they should certainly have negative signs,
but even then they would not demonstrate anything but the increasing use of
institutions and the decreasing prevalence of phthisis during the years in question.
* Biometrika, Vol. x. pp. 179, 341.
+ These values might be modified if we could go to higher differences, but this is impossible on the
very limited data which Dr Newsholme provides. On these data all we can state is that no evidence
of organic relationship between the variates, such as is asserted by Dr Newsholme to exist, can be
demonstrated.
} There is no statement as to why Brighton has been omitted.
Autck LEE 5338
There is, however, a much graver criticism to be made of Dr Newsholme’s
method in this measure of segregation. He proposes to correlate
1006/6 and 10004/P,
and interprets the high correlations as a sign of the value of segregation in
reducing the phthisis deathrate. We have not his data to test his conclusions by,
but we can compare them against certain results for 88 years in (i) England and
Wales, (ii) Scotland, and (111) Ireland. Here they are:
Correlation of Phthisis Deathrate and Ratio of Institutional
Phthisical to all Phthisical Deaths.
Years District Correlation
1866—1903 Scotland —°9815 + 0040
1866—1903 England and Wales — ‘9750 + 0054
1866—1903 Treland — 8720 + 0262
1876—1905 Sheffield — ‘80+°0443
1884—1904. Salford — 67+°0811
The reader may imagine in this table a confirmation of Dr Newsholme’s
results, for the larger material gives higher values of the correlations. On the
contrary, these correlations have been obtained by taking as the measure of
segregation the ratio
Mean Institutional deaths per annum from phthisis 1866— 1903
eee Annual Total deaths from phthisis
Now it is clear that this index never varies with the increasing percentage
of institutional deaths from phthisis. Yet all the correlations are greater than
Dr Newsholme’s! We have little doubt that he would get higher values than he
has done, if he replaced the actual institutional deaths per annum by the constant
mean value. In other words the results reached by him are of no significance,
for we get higher correlations by putting a single fictitious value for the annual
institutional deathrate.
The real source of his result is not the strong influence of segregation on
phthisis, but the spurious correlation introduced by using the phthisis deaths, ¢,
in the numerator of one variate, 1000¢/P, and in the denominator of the other,
100¢;/¢. Thus no scientific results of value can be found from Dr Newsholme’s
second measure of segregation.
In discussing this second measure of segregation, Dr Newsholme lays great stress
on the part played by asylums for the insane in segregating the tuberculous. He
notes that the percentage of lunatics treated privately with relatives and others was
18:4 in 1859 and fell to 5°5 in 1902, thus marking increasing segregation during the
period of fall in the phthisis deathrate. He states (p. 274) that: “the deathrate
from tuberculosis in borough and county asylums in 1901 was 15°8 per cent. of
the inmates, and over ten times as great as in the general population.” Now
68—2
534 Tuberculosis and Segregation
Dr Newsholme’s figure appears to be quoted from the 56th Annual Report of the
Commissioners in Lunacy, and in this case it should read 15°8 per 1000 and not
per 100, and although Dr Newsholme appears to have made a similar slip in
dealing with the deathrate in the general population, he seems to be comparing
deaths from all forms of tuberculosis among the insane—some of which have
possibly a direct relation to their insanity—with deaths from phthisis alone in the
general population. Further he has made no allowance for the very marked
difference between the age distributions of the two groups he is comparing.
The difference is so great that a phthisis deathrate of 1:46 per 1000 in the
general male population is equivalent to one of 2°41 per 1000 among the insane
population of males. Even if the corrected deathrate among the insane for
phthisis were ten times its magnitude among the sane, we fail to understand
what Dr Newsholme means when he asserts that: “the segregation of each
tuberculous lunatic has been equivalent to the withdrawal of ten ordinary tuber-
culous persons” (p. 274). Because tuberculosis among lunatics is ten times as
frequent—judging by deaths, and accepting for the purpose of argument Dr News-
holme’s figures—why should the isolation of one tuberculous lunatic be equivalent
to the withdrawal of ten sane tuberculous persons? That must suppose a tuber-
culous lunatic capable of spreading ten times the infection of a tuberculous but
sane individual. All Dr Newsholme could say would be that from the standpoint
of segregation it is ten times more desirable to segregate any lunatic, than
any sane person, for the former is ten times as likely to die of tuberculosis.
Dr Newsholme brings no evidence to show that the individual tuberculous
lunatic is ten times as dangerous as the individual tuberculous sane person.
As a matter of fact we still need very careful investigation of the relation of
lunacy to tuberculosis, not only having regard to some forms of tuberculosis as
possible sources of feeble-mindedness, if not of insanity, but also having regard
to whether the old idea of asylum segregation as a possible cause of the spread
of tuberculosis among lunatics is wholly erroneous, and we might further examine
whether the new idea that the majority of tuberculous lunatics were tuberculous
on admission is in its turn wholly sound*. In the present state of our know-
ledge we think the assertion that the increased segregation of lunatics has
substantial relation to the decrease in the phthisis deathrate is quite unproven.
(4) Dr Newsholme’s third approximation to the segregation ratio is the
index 100p,/p, where p; is the number of paupers in institutions and p, is
the total number of paupers, indoor and outdoor. Unfortunately Dr Newsholme’s
usage does not agree with his definition. The index he appears to use is generally
100p,/p;, and the values of this are given in the last column of Table LXV
(p. 277) and Table LXVII (p. 279). In Table LXVI (on p. 277), however, the
100 factor is dropped and p,/p, again used in the heading to the central column,
* Many lunatics enter and re-enter asylums, it does not follow because they died of tuberculosis
and were tuberculous on last admission that their tuberculosis was there on first admission.
AuicE LEE 535
although the figures in that column appear to refer to 100p,/p;. Below this
table occur the words:
This experience for the entire series of individual years is expressed by a coefficient of
correlation of — ‘94 between segregation measured :by the fraction of pauper population treated
in institutions and the phthisis deathrate. (p. 277.)
The correlation to support Dr Newsholme’s views should be negative if
100p;/p,
has been used, and positive if 100p,/p; has been used. But as many of his
other correlations are given with the wrong sign, it is difficult to discover what
measure of segregation he actually has used. To add to the confusion the index
actually plotted is log p,/p;, and not 100p;/p,, which is what Dr Newsholme
defines as his index. We have accordingly in our analysis of the figures, to be
given later, used both indices 100p;/p, and 100p,/p;.
It is very difficult to appreciate how the ratio 100p,/p, can effectively measure
the segregation ratio—it is indeed impossible to agree with Dr Newsholme’s view
that any of his indices “ measure with approximate accuracy the ratio which states
how many of total days of tuberculous sickness are passed in institutions.”
The policy of compelling as many paupers as possible to go into the workhouse
was directly adopted with a view to diminishing the total pauperism as well as
abuses connected with outdoor relief, and that policy is the source of increase in
the index 100p;/p,._ Had Dr Newsholme examined his own Tables LXV, LXVII
and LXIX carefully, he would have seen that the percentage of indoor paupers
on the general population has remained almost constant for the period in question,
while the total paupers per cent. of the general population in England with Wales
and in Scotland have decreased. If the same relative number of paupers are segre-
gated now as formerly, how can this segregation have diminished the chances of
infection in the community? We can hardly assume that all paupers are tuber-
culous, or markedly so relatively to other men, so that the reduction of the number
of outside paupers by indoor segregation is equivalent practically to a reduction
pro tanto (note the extraordinarily high correlations !) of the number of tuberculous
in the community. If so, then the reduction of the tuberculous deathrate would
be due not to the segregation, but to the large decrease in the total pauperism
relative to the population of this country. The correlation, as we shall demon-
strate, is not between the segregation of paupers and the phthisis deathrate, but
between the diminution of total pauperism and the phthisis deathrate. We shall
investigate how far this relationship between total pauperism and the phthisis
deathrate is “organic,” i.e. continues after the annulment of the time-factor, or is
purely due to the fact that both pauperism and phthisis have diminished during
the forty-year period under consideration.
It was this third definition of a segregation ratio in conjunction with the
fourth segregation ratio to be considered later that led us to realise that the whole
536 Tuberculosis and Segregation
problem must be dealt with afresh, and the modern methods of partial correlation
and variate difference correlation applied to its various aspects. We have taken
the period used by Dr Newsholme, 1866-1908 inclusive, and have used the figures
for each individual year thus obtaining 38 entries, which are few indeed, but the
best we can probably do with data of this kind, and therefore directly comparable
with Dr Newsholme’s results, for he seems to have used individual years for his
correlations although he does not always say so (cf. pp. 271 and 280), and notwith-
standing that his tables are all given for five-year periods.
The population numbers for England and Wales (Table A) were taken from
the Registrar-General’s Annual Report for 1909, and the phthisis deaths from the
Reports for 1866-1903; the average of each five years’ period agrees with Dr
Newsholme’s values for phthisis, but the values for indoor and for total paupers
do not quite agree with his. Dr Newsholme was therefore written to and asked
whence he obtained his numbers. He was kind enough to reply, but said that
he was unable to refer at the moment to the original tables, but that undoubtedly
the data were the statistics given in the Annual Reports of the Registrars-General
for England, Scotland and Ireland. We then examined the Local Government
Board returns and found that Dr Newsholme apparently had used the pauper
returns for the January quarter of each year. We kept therefore to the Registrar-
General’s Report, as the numbers there given are based on the Local Government
Board’s returns for the whole year, which are a fairer measure of pauperism than
those for the January quarter alone.
For Scotland, our numbers (Table A) agree with Dr Newsholme’s for both
phthisis and indoor paupers, except when we take the first five-year period
(1866-70), where they differ slightly. In the case of total paupers for the periods
1866-70, 1881-85, and 1896-1900 our figures do not agree*. We cannot find
any reason for these divergences except a slip in his or our arithmetic, or the
possibility that a wrong number of outside paupers has been taken by one or other
of us. We do not think the differences in the values are such as to invalidate
a comparison of results.
In Ireland the only serious discrepancy in our values is in the total number
of paupers for the period 1876-80.
These discrepancies, however, emphasise the very necessary rules for statistical
treatment: (1) that the ultimate raw data should be published with every inquiry,
and (11) it should be stated exactly where they are taken from, and how they have
been treated.
Table A gives our raw data, Table B our deathrates and indices based thereon.
We have correlated the phthisis deathrate taken as 10°¢/P with 100p;/p, and
* We are unable to compare his and our data for individual years, because Dr Newsholme has only
published his data for five-year periods.
AuicE LEE 537
100p,/p;. Taking first England and Wales, and calling these three indices
respectively Jy, J; and J,, we find:
Correlation of J, and J; = — 9664 + 0072,
I, and J, = + 9298 + 0148.
Dr Newsholme gives — ‘94 as the coefficient of correlation “between segregation
measured by the fraction of pauper population treated in institutions and the
phthisis deathrate” (p. 277). Having regard to his confusion of J; and J, and
his frequent interchange of the signs of correlation coefficients, we can only say
our results confirm his high numerical value, but not his actual figure.
But does this actual figure mean that there is any real relationship between
segregation and the phthisis deathrate? To test this, we replaced the index J; by
I,, where
Mean number of indoor paupers per 10° for the population, 1866-1903
I, = 100 —
10° x p,/P
-100 (5)/().
In this index the relative number of indoor paupers is assumed to remain
absolutely constant. We found:
Correlation for England and Wales of J/g and J; = —‘9459 + ‘0115,
that is to say we get substantially the same value, a value higher than
Dr Newsholme’s, by putting the number of indoor paupers relative to the
general population constant throughout the period. It is very difficult, in the face
of such a result, to suppose that segregation of paupers has anything whatever
to do with the diminution of the phthisis deathrate. It is clearly due to a
; - F F 1 ae
negative correlation of a high magnitude between —, and ¢/P, or to a positive
(Me
correlation between Br and p i.e. to a correlation between a high total pauper
g paup
Ie le
rate anda high phthisis deathrate. Dr Newsholme’s result merely reduces to
the statement that total pauperism in England and Wales has diminished con-
temporaneously with phthisis. If the result has nothing to do with segregation,
can we assert that the reduction of phthisis is causally related to the reduction in
total pauperism ?
Overlooking for a moment a new objection to be raised later, let us apply the
variate difference method to the correlation of ¢/P with 100p,/p, and 100p,/p;
in the cases of England with Wales, of Scotland, and of Ireland; also to the
correlation of ¢/P with the index 100 (p;/P)/(p,/P) in the case of England with
Wales. The following are the results:
538 Tuberculosis and Segregation
TABLE I.
England with Wales Scotland Ireland
Correlation of 10°¢/P 10°$/P 10°¢/P 10°¢/P 10°¢/P 10°¢/P 10°¢/P
with elas ateie i I; Iie I; I, I; I,
Crude Indices | — 946 + 012 | — 966+ ‘007 |+°930+ 015 | — 952+ -010 |+ ‘920+ :017 | — °881 + 024 | + °893 + :022
Ay + 090 + °134 | — 258+ °126 | + 340+ °120 | — 265 + °126 |-+ °250+ °127 | — 280+ °125 | + -235 + 128
A, — 2014 °149 | — 4614123 |4+ °542+4°110 | — -2404°147 | 4+ °1824°151 | — 2644 °145 | 4+ °180+°151
A3 — 335 + °153 | —-508+4°127 |+°567+°116 | —:205+ :164 | + 086 + 170 | — 226+ °163 |+°162 + °167
Ay — 407+ °155 | — 518+ °136 |+ °547+°130 | —°186+°179 |+ 024+ °185 | — 182+ 179 |+°133 4-182
As — °475 + 7153 | — 528+ °143 | + 529+ °142 | -— 1824-191 | -—-003+°'198 |— 145+ °194 |+ 108+ °195
Ag — 538+ °149 |— 5434 °147 |+ °5834°151 — — "112+ °206 |+°081 + °208 |
Ay — 584+ °145 |— 5624 °150 |+ 539+°156 —_— —_— — +°044+°219
As — 614+ °143 | —°587+4°151 |4+°557+°159 —_— — — — 004 + :230
It will be seen from this table that whether we use the index J; or its
inverse I,, we get practically the same results—naturally with changed sign.
But the results themselves are of extraordinary interest. For both Scotland
and Ireland, when we proceed to annul the time-factor by correlating successive
differences, we find that the high correlations interpreted by Dr Newsholme as
marking a relation between pauper segregation and phthisis deathrate entirely
disappear or become less than their probable errors. There is thus no organic
relation between these variates as measured by the above indices. In the case
of England and Wales, however, while there is a reduction on annulment of the
time-factor to roughly two-thirds of the high value noted by Dr Newsholme,
this value does not tend to disappear with increasing differences. Thus in
England with Wales, as apart from the remainder of Great Britain, there would
at first sight appear to be an organic relation between segregation of paupers
and the phthisis deathrate. But our first column under the England with Wales
section shows that if we fix the percentage in the general population of these
indoor paupers and then annul the time-factor, we reach a slightly higher value
of this apparent organic relation. It has therefore nothing to do with segregation.
Thus Dr Newsholme’s interpretation of his original high correlations appears in
every case fallacious.
There are two methods of testing this result, ie. the absence of organic
relationship between indoor pauperism and phthisis. Suppose we correlate the
crude numbers of phthisis deaths per annum and of indoor paupers per annum,
the resulting coefficient will have very small logical value because both these
variates are continuously changing with the time*. But now suppose we annul
* It is noteworthy that the England with Wales and the Scotland correlation coefficients for these
crude variates are high and negative, but for Ireland the coefficient is moderate and positive. Thus
the factors at work must be totally different in the two Islands. Since indoor paupers relative to
the population have remained singularly constant the increase of phthisis deaths must have been much
slower than the population increase in Great Britain, but somewhat faster in Ireland.
ALICE LEE 539
the time-factor by correlating the differences of these variates, then we shall free
ourselves from the influence of the time-variate, and in doing this we shall also
free ourselves practically from the influence of change of population, which is a
time change.
The following table resulted from this investigation.
TABLE II.
Correlation of Crude Phthisis Deaths (b) and Indoor Paupers (pi).
TL
Variates England with Wales Scotland Treland
Crude — 9384+ 014 —°718 +053 +°457 + ‘086
Ay — 376+°116 — 206 +°130 — 092 +°134
A, — 302 +°'141 — 2194148 — 1038 +:154
As — ‘2134164 — 180 +°166 — 143+°168
Ay — '100+°183 —'157+°181 —'147+°181
A; — 016+°198 —°158+°'193 — 140+°194
It will be seen that for all three countries, whether we start with the positive
correlation of the Irish or the negative correlation of the English and Scottish
returns, there is no remaining significant correlation after annulment of the time-
factor between indoor pauperism and phthisis.
A second method of verifying our conclusions is to find the partial correlation
between indoor pauperism and phthisis deaths for a constant value of the total
population and a constant value of total pauperism. We thus ask the question
whether with a constant population and a constant amount of total pauperism,
an increase of indoor pauperism would organically affect the number of deaths
from phthisis. By making the population and the total pauperisin constant we
are largely producing an annulment of the time-factor and ascertaining whether
a change in the number of indoor paupers due to causes other than temporal
influences the number of deaths from phthisis.
The system of correlation coefficients given in Table ILI, p. 540, was determined :
Here the values of py 7», for England with Wales and for Scotland confirm
the conclusions we have reached by other methods, ie. there is no significant
relationship at all between phthisis and indoor pauperism. The value for Ireland
is, perhaps, significant, but having regard to its smallness (—‘3 +'1) and the size
of its probable error, no one can lay real stress on it, in opposition to the results
of the other two countries. In general the coefficients for the Irish data appear
very anomalous, and certainly divergent from those for Great Britain.
Thus our investigation of the relation between indoor pauperism and phthisis
appears to be entirely opposed to Dr Newsholme’s conclusions. We find the
segregation of paupers to have no substantial influence on deaths from phthisis.
The one outstanding point at present, the relation between p,/P and ¢/P after
Biometrika x 69
540 Tuberculosis and Segregation
annulment of the time-factor (see our p. 538), has no bearing on the segregation
problem of Dr Newsholme.
TABLE III.
Total and Partial Correlation Coefficients of Crude Numbers of Indoor and
Total Paupers (p; and p,), Total Population (P), and Phthisis Deaths (¢).
Coefficients | England with Wales Scotland Treland
"pio — 9325 014 — 718 + 053 +°457 +087
(7
"yp +°955 +010 +°831 4-034 +°763 + 046
uv
We Ok g ‘ > 1 AS : q
Total "op 950+ 011 — 896 +022 +:479 + 084
Coefticients ie —°544+:077 — +528 +079 — 251+:°103
(ai
"bp +577 + 073 +°780 +043 +:070+:109
7.
"py P — 674 + 060 — 805 + 038 — 684+ -058
Pints — 287 +°100 +111 +°108 +:162+°107
u
Partial p,! po 75+:073 4+:492+
Coefficients eho 5a A ie i
Py! pio alee as CS —-017 +109 — "305 + 099
Pr pi
To approach nearer to the meaning of the relation between total pauperism
and phthisis we determined the correlation between p, and ¢@ for constant P, and
found
Pp 6= — 277 £101,
which is barely significant having regard to its probable error.
Now after elimination of the time-factor, we found for the correlation of ¢/P
and J, at the eighth difference —‘614 +143, but this is the same as the corre-
lation of oe
significant, positive and of the order ‘6. Now if p, and ¢ after the removal of the
time-factor were practically independent of each other, there would be a high
positive correlation between p,/P and ¢$/P, due to the fact that P when it
takes—after annulment of the time-factor—any random deviation appears in
both variates’ denominators. In other words, we are inclined to believe that
the high negative correlation between ¢/P and J, is solely due to spurious
correlation arising from the nature of the indices used.
and @/P. Hence the correlation of p,/P and ¢/P must be very
To throw still more light on the matter we have investigated the correlation
between the total number of paupers and the total number of deaths from phthisis
when the time-factor approaches annulment. It will be seen from the table
below that for both Scotland and Ireland there is finally no relationship at all
ALicE LEE 541
between phthisis deaths and total pauperism. On the other hand, England with
Wales is tending to a value at least approaching to the crude correlation. We
have therefore this noteworthy result: England with Wales starts with a con-
siderable value and concludes with an equally great value, Scotland starts witb
a high value and ends with a zero value, Ireland starts with an insignificant value
TABLE IV.
Relation between Total Pauperism (p,) and Deaths from Phthisis (@).
England with Wales Scotland Treland
Crude values +:°5774:073 +°780 + 043 | +°070+°'109
Ai — 095 +134 +025 + 135 | +164 +°132
Ay +°1744°151 +°025+°156 +144 +4152
A3 +286 +°158 +:012+°172 | +131 £°169
Ai 4.347 4-163 +033 £185 | +:110 £183
As +413 +164 +027 +°198 | +090 +196
and ends with an insignificant value. If pauperism were causative of phthisis, it
is hard to believe that this would not manifest itself in the Scottish and Irish
returns; these negative any such hypothesis. It would appear that there are
essential differences in the treatment of pauperism in the three countries. I
suggest, but I cannot demonstrate the view, that phthisis itself leads to pauperism
in England, ie. that the relatives of the phthisical breadwinner more often are
allowed to become paupers in England than in the sister countries. In other
words, that the only organic relationship between pauperism and phthisis we have
been able to discover may be due to a relatively harsh treatment in England of
the dependents of the phthisical.
To show how effectively the variate difference correlation method removes
time influence, we may note that we correlated total population (P) with total
pauperism (p,) and total phthisis deaths (¢) with total population by this method,
with a view to ascertaining whether the relation between p, and ¢ would be
modified, if we determined it for constant population.
The following results were reached :
TABLE IV. England with Wales.
Total Population Total Population
and | an
Total Pauperism | Phthisis Deaths
(P and p,) (P and ¢)
Crude values — 674+ :060 — 950+ :011
Ay +°457 + °107 — 039 + °135
Ay — ‘016 +°156 — °205 + *149
As — °022+°171 — 089 +°170
4 — 031 4°185 + 002 + °185
69—2
542 Tuberculosis and Segregation
Thus we see that apart from the time-factor there is no relation whatever
between either pauperism or phthisis and population. In the relation between
total pauperism and phthisis deaths, no further correction for population is
needful than that obtained by the annulment of the time-factor as in Table IV.
Table IV bis shows us that neither pauperism nor phthisis is organically related
to population, although we might well have anticipated that greater density of
population would influence pauperism and provide greater chances of infection,
and so of deaths, in the case of phthisis.
(5) We now come to Dr Newsholme’s fourth and last measure of segregation.
It is “the ratio in which the number of paupers treated in workhouses and work-
house infirmaries stand to the total number of deaths in the community ” (p. 276).
In our notation this is p;/f, or as an index 100p,/¢. But in the figures actually
given in Table LXV (p. 277), and headed Segregation Ratio, Dr Newsholme
appears to be using 100¢/p;.. The same remark applies to Tables LX VIII and
LXIX (pp. 280—281). Thus it is difficult to be certain of what Dr Newsholme
intends to be taken as his fourth measure of segregation. In our discussion below
we have used both 100¢/p; and 100p;/¢ to provide for both contingencies and
to check our results.
Unfortunately Dr Newsholme makes little attempt to justify either his third
or fourth ratio as an approximate measure of segregation. It will be remembered
that he has defined the true method of measuring segregation to consist in
forming the ratio “stating how many of the total days of sickness (number of
patients and number of days of sickness) are passed in institutions” (p. 267). In
this fourth index of segregation he replaces phthisical patients in institutions by
indoor paupers, and total of phthisical patients by total deaths from phthisis,
dropping any question of the number of days of sickness. At the very least this
seems to involve two assumptions, (a) either that all indoor paupers are phthisical
or that for the period in question the proportion of indoor paupers who are
phthisical has remained constant, (b) that for the period in question the number
of deaths from phthisis has remained a constant fraction of the total number of
cases of phthisis. It is difficult to see how, without such assumptions, such figures
can “measure with approximate accuracy the ratio which states how many of the
total days of tuberculous sickness are passed in institutions” (p. 267).
Yet in another paragraph Dr Newsholme quotes with apparent approval the
statement of Mr Fleming, who speaks of the “great change in the character of
workhouse inmates during recent years....The able-bodied inmates are gone and
the sick inmates have come” (p. 273). Such a statement is absolutely inconsistent
with the assumption (a) above.
To justify (b) we must assert that for the last fifty years of the nineteenth
century there has been no change in efficiency of treatment in the case of tuber-
culosis, for without this we cannot assume that deaths from phthisis are even an
approximate measure of the number of cases (p. 267). The fact that the reduction
AuicE LEE 543
in the phthisis deathrate has been substantially different for the different age
groups, and is especially marked in the case of children, seems to indicate that
recovery, at least from puerile phthisis, is more frequent now than formerly.
However, not to spend more time on these assumptions—which, it appears to us
that Dr Newsholme has by no means justified—let us examine whither this fourth
method of approximately measuring segregation leads us. Table V gives the
necessary coefficients.
TABLE V.
Correlation and Difference Correlations of 10°6/P and 100p;/h or 100¢/p;.
; England with Wales Scotland Treland
Variate
10°¢/P as
ee 100046 1009/p; 100p;/9 100¢/p; 100p,/¢ 100¢/p;
Crude |—°760+ :046 |+°976+ ‘005 | — ‘861 + °028 |+ °944+ °012 |—°712+4 054 |+ 666+ ‘061
Ay — 868 + 033 |+°848 + °038 | — °755+°058 | + °772 + 055 |— °819 + 045 | + °707 + (068
A, — 879+ °035 |+ °875 + ‘037 | — 824+ °050 |+ °834+ °047 |— 922+ :023 |+°755 + ‘067
A3 — '895 + 034 |}4+ °874+°041 | —:809 + :059 | + °824+ :055 |— 954+ :015 |4+ °791 + :064
Ay — ‘895 + ‘037 | + °860 + -048 | —°811+ ‘064 |+ °805 + :065 |— 964+ °013 | + °805 + 065
As — °898 + 038 |+ °847 + ‘056 | —°786 + ‘076 |+ °788 + °075 |— 970+ ‘012 |+°831+4 ‘061
Ag — ‘907 + ‘037 |+ °850 + °058 | —°788+ 079 |+ "794+ 077 |—°973+°011 | + °848 + :059
Ay —'917+:035 | + °835 + ‘067 | — "792+ 082 |+ °791 + ‘082 — + °857 + (056*
Now this table at any rate demonstrates a very high correlation between $/P
and p;/, while the previous table for Dr Newsholme’s third approximate segre-
gation ratio led in the case of England with Wales to the value —‘587, and in
the case of Scotland and Ireland to negligibly small values! Dr Newsholme
himself writes: “ Any of these indirect forms of segregation ratio has therefore
to be verified wherever possible by the application to the same community and
period of one or more other forms of the ratio, and checked where practicable
by a special examination of sample constituent communities whose figures are
included in the total. This has been done so far as the information obtainable
allowed. It will be seen that the results obtained by applying different ratios to
the experience of the same country and period are usually, though not invariably,
in good agreement ” (p. 268).
What is quite clear from the above results is that, while in the case of
Dr Newsholme’s two chief measures of segregation, there is very sensible difference
in the case of England with Wales, there is an absolute discordance in the cases of
both Scotland and Ireland. Accordingly on the basis of his own axiom, that we
must check our results by application of one or more other forms of the ratio,
* This correlation continues to rise until it reaches 929 with the thirteenth difference, but with such
high differences the “population ” is so reduced that the method ceases really to be reliable.
544 Tuberculosis and Segregation
we are bound to reject these ratios as even approximate measures of segre-
gation*,
But it would not be satisfactory to leave the matter here and not provide some
explanation of why this fourth segregation ratio, both before and after the annul-
ment of the time-factor, leads to such high correlations. Luckily the matter is
capable of a perfectly straightforward and obvious explanation, which would have
been anticipated had Dr Newsholme had in mind the danger of “spurious cor-
relation.”
What he is correlating are essentially ¢/P and p;,/¢. The latter may be
written (p;/P)/(¢/P). Now pj/P is practically constant during the period in
question. Hence Dr Newsholme is correlating ¢/P with 1/(¢/P), or a variate
with its reciprocal. In other words we may anticipate something very closely
approaching perfect correlation. The deviation from such correlation arises from
the fact that p;/P is not absolutely steady, although its variations are very probably
nearly random. The assertion therefore that this fourth measure of segregation
assists in demonstrating the close relation between the fall in the phthisis death-
rate and institutional segregation is based on a fallacy which entirely overlooks
“ spurious correlation.” ;
It will be seen therefore that not one of Dr Newsholme’s methods of reaching
an approximate measure of the segregation is satisfactory, and they lead to con-
tradictory and inconclusive results. Whether there is any really substantial
relation between the prevalence of phthisis and institutional segregation we
do not yet know. All we can say is that Dr Newsholme has entirely failed to
demonstrate it, if it actually exists.
(6) Before concluding this paper it may be of interest to judge how far it
justifies the application of the method of variate difference correlation to such
problems as are here dealt with.
In the first place, the correlations of successive differences should approach
steady values. This is generally—as the reader can judge by examining Tables I,
II, IV and V—but not invariably, the case. The test cannot, however, be com-
pleted, as the method ought not to be pressed to such high differences that the
order of the difference is a large percentage of the original “ population.”
We doubt whether it is advisable to carry differences beyond the 8th in a
population of 38. 20°/, to 25°/, reduction in the population is as much surely as
it is safe to allow where the original population is so small in number. It is true
that a population of 38 itself is capable of exciting the derision of trained
* Under the circumstances it is, perhaps, unnecessary to draw attention to Dr Newsholme’s state-
ment that ‘‘the specific result of pauper segregation must have been lower in Ireland than in England or
Scotland” (p. 282). Free of the time-factor the correlations of phthisis deathrate and Dr Newsholme’s
fourth segregation ratio are higher in Ireland than in England or Scotland. This criticism as well as
Dr Newsholme’s original remark are of no importance, because the fourth segregation ratio correlation
is entirely spurious.
ALIcE LEE 545
statisticians, and ought never to be used where hard work can produce larger
numbers. But in annual returns, as has been indicated by others, a period of
30 to 50 years is often the maximum attainable, and we must take what we can
get. In the present case the probable errors of the difference correlations—based
on the Andersonian formulae for steady conditions—show us that we can form
fairly legitimate conclusions from the results reached.
A second test that we have applied is the approach to the theoretical values
in the function 0”; ,/o°5 .. where 6,0 is the mth difference of the variate «.
The following table shows that there is a reasonable approach to these
theoretical values in the calculated standard deviations of the differences, and
suffices to justify the application of the variate difference method within the
‘limits of practical statistics. We have continued the differences beyond the
values used in some of the correlation results to indicate the sort of irregularities
which may be expected to occur when using high differences in small populations.
Terminal irregularities then begin to affect the uniform rise of 05 ,/o°5
m1 *
Tuberculosis and Segregation
546
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Biometrika x
Tuberculosis and Segregation
548
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THE INFLUENCE OF ISOLATION ON THE
DIPHTHERIA ATTACK- AND DEATH-RATES.
By ETHEL M. ELDERTON, Galton Research Fellow
AND KARL PEARSON, F.R.S.
(1) Introductory. The problem of the advantages of isolation, not only in the
case of diphtheria but of other diseases of an infectious character, is hkely, owing
to modern views as to “carriers” and other sources of transmission, to be much
discussed in the near future. It is therefore well to consider what may be learnt
from the statistics available. The questions which naturally arise are of the
following kind:
(i) In districts with a maximum of isolation is there a minimum of incidence?
Gi) In districts with a maximum of isolation is there a minimum deathrate
from the disease isolated ?
There cannot be the slightest doubt that, if these two questions were answered in
the affirmative and we could show that the incidence was markedly less and the
deathrate significantly smaller in districts where isolation was most stringently
carried out, then these results would be advanced as a strong argument in favour
of isolation.
To the trained statistician, however, no conclusion based upon such results
without much further analysis would have any validity. To illustrate this point,
let us consider the hypothetical case that medical or popular opinion in a given
town has been persistently in favour of increasing the isolation-rate, and further
suppose that in this district improved economic conditions have increased the
immunity, or bettered sanitation lowered the incidence, while at the same time
new methods of treatment have lowered the deathrate of the disease; it will be
clear that in considering the statistical results over a course of years we should
find a high isolation-rate negatively correlated with both the incidence- and the
death-rates. Thus if we considered this correlational as a causal nexus, we
should be raising an apparently strong argument in favour of a maximum of
isolation, which would be based on the statistical fallacy, that when two quantities
are both changing continuously with the time, this must of itself denote a causal
relation.
70—2
550 A Study of the Effects of Diphtheria Tsolation
In precisely the same way a positive correlation between the isolation rate and
the attack- or death-rates by no means justifies us in asserting that isolation is
worse than non-effective. It is conceivable that in the period or the district
under consideration with an increasing isolation-rate there might be decreased
immunity in the population, greater virulence of the disease, or even a limit
to the available isolation accommodation, so that in the case of attacks of an
epidemic nature the isolation rate would not increase proportionately to the cases,
or indeed might even diminish*. Further, if apart from the changes in a single
district, we consider a great variety of districts, it may chance that the greatest
isolation-rate occurs in those districts where the disease has been found most
prevalent, because it appeared the most obvious remedy, and thus a greater attack-
or death-rate would be no real measure of the futility of high isolation.
If, however, it should turn out that on the whole the higher isolation rate is
associated with the bigher attack-rate or the higher death-rate then it will be clear
(i) that there is ground for demanding a closer investigation as to the advantages
of isolation, and (11) that we may be overlooking the real method, or at least one
or more important factors, of the transmission of the disease. It is conceivable
that isolation of all cases during attack may be of far less importance than isola-
tion of certain special cases for a shorter or longer period well subsequent to the
attack, and after they would normally have resumed their ordinary avocations fF.
The main problems which arise are accordingly these :
(i) Have isolation-, attack- and death-rates changed continuously with the
time, and are the apparent correlations really suggestive of causal relationships ?
(ii) Are associations between isolation-rate and attack- and death-rates really
spurious arising from the fact that where the attack- and death-rates have been
severe there the remedy which appeared nearest to hand was more isolation ?
* For example, if there were only 100 hospital beds available, and out of 100 cases 50 were sent to
hospital, the isolation-rate would be 50 °/,; but if in the next year there were 300 cases and all the beds
were used, the isolation-rate would be only 33 °/,. Thus limited accommodation may tend to produce
a negative correlation between isolation-rate and attack-rate, so that a positive correlation between these
two rates may be of more importance than its apparent significance. It is extremely probable that
some of the falls in isolation-rates are really due to an increase of incidence, so that the same
percentage of cases cannot be met by the available hospital accommodation.
+ It is, on the hypothesis of natural selection, a plausible view that the parasites—including under
this term all disease organisms—which ultimately survive must tend to become innocuous to their
hosts, and thus the decreasing virulence of certain diseases may be accounted for. The organism is
destroyed owing to the death of the host or its own death at his recovery, or it has been modified by
selection so as to become innocuous to its host relative to his immunity. But immunity is a matter of
personal equation, and thus the function of the ‘‘ carrier” in preserving and spreading a conceivably less
nocuous form of the organisrn becomes clearer. We are not unaware of the view that the organism
remains the same, but that the immunity is increased owing to ‘‘ practise” of the leucocytes, but such a
view requires the assumption of inheritance of acquired characters to explain reduced disease virulence,
and further compels us to assume two types of immunity, the one which destroys the organism, and
the other which without modifying it, establishes so to speak a mutual modus vivendi.
EruEe, M. ELpERToN AND Kari PEARSON 551
(ii) Are the districts which have adopted most isolation really urban
districts where isolation was easiest to adopt and where possibly economic or
social conditions favoured the spread of the disease or, in the case of the death-
rate, the disease encountered a less resistant population ?
(iv) What evidence is there to show that the districts which have rapidly
increased their isolation-rates have subsequently lower attack- or death-rates ?
If no one of these problems can be fully answered,—even in the case of a single
disease—with the data at present available, at least light can be thrown on the lines
which their solution in the future must take; and further something can be done
to prevent hasty generalisation and excessive dogmatism as to the advantages or
disadvantages of the isolation system. It can never be too strongly insisted upon,
because it is so often forgotten, that preventive medicine is essentially an
experimental science, and that in nine cases out of ten the efficiency of any line
of action can only be adequately tested by statistics and by statistics collected after
the expenditure of many thousand pounds, possibly spread over a long period of
years, in carrying out this line of action *.
(2) Material. In endeavouring to throw some light on the above problems we
have fortunately received data of very considerable value from Dr E. H. Snell, the
Medical Officer of Health for the City of Coventry. He obtained for a period of
nine years, 1904-1912 inclusive, for about eighty towns or districts of large popula-
tion but of very varying local conditions, (i) the annual number of diphtheria cases,
(11) the number removed to hospital, (iii) the number of deaths. We have added
to this material the estimated population of the town or district, and further
certain data as to the economic and social conditions. Unfortunately there is no
existing adequate measure of the general sanitary condition of individual towns,
although the construction of a general sanitary “index number” would be of
remarkable value in many forms of inquiry. We took as our measures of social
condition :
(a) Death-rate of infants under a year.
(6) Amount of overcrowding, that is to say the percentage of the population
in private families living more than two in a room.
(c) Density of population, i.e. the number of persons to the acre.
* Assert that it is most desirable to test the effect of sanatoria and of tuberculin in cases of
tuberculosis, but do not dogmatically proclaim them as “cures” for phthisis, until statistics have been
collected in sufficient amount and have been adequately and dispassionately examined to prove or
disprove your statements. Insist on compulsory inoculation for enteric in the case of all recruits, but
do not make it optional and then publish letters in the newspapers giving perfectly idle statistics, or
go round to the camps giving popular lantern lectures to the recruits showing the gravestones of
uninoculated persons, the portraits of persons dying of enteric, or much enlarged pictures of bacilli!
If you think it experimentally worth doing, inoculate ; but don’t bring inoculation about by emphasising
the dread of pain or the fear of death, both of which it is the first essential for a soldier wholly to
disregard.
552 A Study of the Effects of Diphtheria Tsolation
(d) Economic prosperity as measured by the number of indoor and outdoor
servants of both sexes per 100 private families.
Our data are baged on the census of 1911 as providing more ample information
on these points. It will we think be admitted that the list of towns dealt with
provides a very fair sample of the urban populations of this country. It ranges
from manufacturing towns* like Preston, Rochdale and Bolton, mining and iron
towns like Rhondda, Wigan and Middlesbrough, sea-ports like Hull, Liverpool
and Southampton, to county towns like York and Reading, watering places like
Brighton and Blackpool, suburban districts like Acton and Hornscy, and residential
towns like Oxford or Bath. We ought from such a list to be able to throw some
light on the relation of isolation to incidence under a variety of social conditions,
if indeed these latter are factors in the problem at all.
(3) What are the crude correlations between Isolation-Rate, Attack-Rate and
Mortality-Rate? The isolation-rate (7) has been measured as the average per
cent. of cases removed to hospital during a five or four year period. We have two
such periods, the earlier period 1904-1908 and the later period 1909-1912. The
attack-rate has been measured per 1000 of the population, uncorrected for age
distribution. Since diphtheria is largely a disease of infancy and childhood this
neglect of the age correction—the reduction to a standard population—may seem
serious. But in the first place we had not the age incidence in the individual
districts, and in the second place we satisfied ourselves that such correction, if it
could have been made, would not substantially modify any argument we have
based on our data. For we calculated the attack-rate (A’) on the population
under 15 years of age, as well as the attack-rate (A) on the total population of the
districts. We found the correlation between the two methods of measuring the
attack-rate was +°972, which indicates how close is the relation between the two
methods of measuring the attack-rate and how little influence small variations
in the proportion of less immune persons in the population due to age differences
could have on the resultst.
The attack-rate (A) has been measured as the number of cases per 1000 of the
population. The mortality-rate has been measured in two different ways; first as
the population mortality, the death-rate in the ordinary sense (J/) or the deaths
per 1000 of the population ; and secondly the case death-rate or the mortality (m)
per 100 attacked. We now give the crude correlations between J and A.
They are:
First Period: 1904-1908, ry = +°427 + 063,
Second Period: 1909-1912, Tra — +2290). 069:
* See table, p. 567, for 76 of the 80 towns, the four others with full data only for the second period
being Reading, Stoke, Dewsbury and Edmonton,
+ The formula giving the juvenile attack-rate A’ in terms of the crude attack-rate A is: q
A’=1°3094 A+ -0164
with a probable error of -:1369.
Thirty-three towns were selected at random out of the 80, and gave the following results for d’
calculated from 4 and A’ as observed. The theoretical mean error=*162; the mean error of the defects
Erne, M. ELDERTON AND Kart PEARSON 553
Thus both periods show significant if not very large correlation*. The difference
(1387 + 093) between the coefficients for the two periods is, however, probably
not significant. Thus in towns with greater isolation-rate there is certainly a
higher attack-rate, and equally certainly no argument can be based on the crude
figures to prove that the more the isolation the less prevalent is diphtheria. We
will now turn to the death-rate M, and we find:
First Period: 1904-1908, Tim = +°153 + 075,
Second Period: 1909-1912, Ny = ID Oho:
In the first period isolation was associated with a higher diphtheria death-
rate, in the second period with a lower diphtheria death-rate, but neither are
of any real significance. Thus all we can conclude from the crude figures is
that they show no evidence that isolation has reduced the general death-rate from
diphtheria.
We next take the case death-rate (m) and we have for the two periods:
an
a
First Period: 1904-1908, Tim = — ‘509 + 057,
Tam = — ‘d2T + 056,
Second Period: 1909-1912, Tim = — 084 + 054,
Pan = = 7405 4 -O5T.
is—°153 and of the excesses+-134; this shows very fair accordance, 17 deviations being positive and
16 negative. The greatest deviations occur in Hornsey, Bath and Brighton, where residential neighbour-
hoods show fewer children, and in Edmonton, Walthamstow and Rhondda where there are probably
Observed sree A | Observed rine ts | A |
| | |
Derby seal) Gee 4:25. | —:17 ||Edmonton ... | _ 1°36 1:80 | +4:44
Southampton ... 2°67 2710) ate OSe Wl Bath. soenl| aIsilfs} 87 = 83]
Hornsey Acni|| eects! 1:86 | —°82 || Newport cesta wht 1°25 +11
Bristol SS Alezioe 2°23 — 10 || Rhondda eae | ee lealte) 1°34 4°24
Reading Lele one 1:95 | —'18 || Bury e108 ‘Ole |). =-17
Nottingham ... 2°09 1:99 = 10 Rotherham... 1:06 1:18 +°12 |
Salford ee 2°04 2°16 +12 || Dewsbury ae 1-01 1:01 =-*00
Ilford see 1:98 2°09 +1] Blackburn julie 299 91 ==-)Si |
Brighton 1:94 1°68 —:26 || Manchester... 94 ‘97 JECaB}. |
Stockton 1:90 Is, +°22 | Oxford Fei||Ue eeaehy) "79 | =-10 |
Ipswich 1:87 1°89 +:02 |} Bolton a 86 85 — ‘01
Grimsby ee) 1:85 1:90 | +:05 || Rochdale oa "82 "75 —O7 |
Walthamstow... 1°85 2°10 +°25 || Northampton ... ‘78 “76 — 02
Coventry 1°84 1:93 | +:09 || Barnsley a To | -88" | ees
Plymouth 1°61 1°56 -'05 | Wigan ose 585 | <64 +°06 |
Wakefield 1°40 1°39 —‘O1 || W. Bromwich...| °45 53 +:08 |
Smethwick IS Se seed Se aCe ||
| |
excess of children. On the whole the general order is very well maintained, and the general attack-rate
closely fixes the juvenile attack-rate. In any further collection of material, it would of course be well
to have the age-distribution of cases.
* We endeavoured to see whether the correlation of isolation- and attack-rates would be modified if
we took the attack-rate on children under 15 years. This made little difference, 7 being raised only
from +'290+ 069 to +°315 + 068.
554 A Study of the Effects of Diphtheria Isolation
According to these correlations, when or where the isolation-rate is high, the
case mortality is low. Further when or where the attack-rate is high the case
mortality is low. Now we know that:
I =100 x isolated cases ~ all cases,
A =1000 x all cases + population,
m=100 x deaths + all cases.
Hence if we selected the number attacked at random and chose the deaths to
be simply some number less than this, we should expect to find a considerable
negative correlation between A and m; and as we actually do find such a corre-
lation, we cannot be certain that the actually observed values of r,,, are not due
to “spurious correlation.” If they were “organic” we should interpret them to
mean that a widespread epidemic (A large) was a less virulent epidemic (m small).
On the other hand the spurious correlations of J and m would be positive in value,
while the actual correlations are negative. Thus it would seem that while a high
isolation-rate is associated with high attack-rate, it must be “organically” asso-
ciated with a lessened case mortality. In other words while isolation does not, on
the crude figures, appear to lessen the frequency of disease, it does appear to
lessen the mortality among the attacked. This result appeared to be of such
very great importance, if thoroughly established, that we determined to inquire
into it further. It seemed reasonable to believe that the bulk of persons attacked
might have better care in a hospital than in their own homes and thus isolation
indirectly lessen the ill effects of the disease.
We accordingly endeavoured to approach the problem from a somewhat
different standpoint: Given two districts with the same total number of persons
attacked (a), will that district with more isolated (7), have fewer or more deaths (d)?
The answer to this question depends on whether the partial correlation coefficient
of total isolated cases with total deaths for constant number attacked is negative
or positive. We found:
First Period Second Period
Correlations 1904-1908 1909-1912
rq = Isolated Cases and Deaths ate + °860 + :020 + 867 +:019
Tiq =Asolated Cases and Attacked ize +°937+:010 +968 + 005
Tq= Attacked and Deaths at Se + 907+ :014 +°918 + °012
rectal hos end Datel} segsy-ort aor
Thus in the first period for a given number of attacked more isolation was
associated with more deaths, and in the second period for a given number of
attacked, with fewer deaths; but in both periods, having regard to the probable
errors, we cannot assert any real significance, or be reasonably certain that where
there is more isolation, there recovery is more likely to occur.
We shall see later that the correlation between J and m for constant total
number of attacks is not the same thing as the correlation of the total isolated and
Erne, M. Evperton anp Kari PEARSON 555
18
total deaths for constant total number of attacks. And this divergence, often in a
marked degree, of partial correlations for rates and for absolute numbers is not
unfamiliar to those who have had to deal with disease statistics. In the present
case it renders still more obscure any argument drawn in favour of isolation from
apparently lesser case mortality.
?
(4) On the degree to which “spurious correlation” may be influencing the
attack- and death-rates. It seemed desirable if possible to throw further hight on
this point and accordingly we correlated attack- and death-rates with the total
population. It will be remembered that:
A = 1000 x cases + population,
M = 1000 x deaths + population,
and accordingly if A and M be correlated with the population P, we might
anticipate that if cases and deaths had no relation to population, there would be
a high negative correlation arising from A and M both varying inversely as P.
We were comforted by finding practically insignificant positive correlations. Thus:
First Period Second Period
Correlations 1904-1908 1909-1912
’p4=Population and Attack-rate +:°137 +°075 + 054+ ‘075
7py = Population and Death-rate ris lsO70 + ‘116+ 074
7», =Population and Isolation-rate 52+ 075 +102 + ‘075
The last correlation cofficients show us that there is very little relation between
the size of a population and the amount of isolation practised. Further these
isolation correlations in which there is no obvious source of spurious correlation
are as significant as those of population with attack- and death-rates where the
possibility of “spurious correlation” is manifest. We conclude accordingly that
risk is more uniformly distributed over population than we had anticipated, and
that the correlations between the three rates J, A and M are really open to
?
“organic” interpretation.
The next point which arises for discussion is whether the presence of the total
number attacked (a) in the rates J and m can produce spurious correlation. If so
we should anticipate that the absolute number a would be negatively correlated
with both isolation and case mortality rates. We found:
First Period Second Period
Correlations 1904-1908 / 1~ 99-1912
Yaz =Total attacked and Isolation-rate 4+ :264+°072 -+°226 +072
vam= Total attacked and Case-Mortality = Ost Oe — 903 + :072
The first set of these coefficients are not even negative and therefore cannot
be due to “spurious correlation,” although such correlation may have reduced
their organic values. They admit, however, of an easy interpretation, namely
that: where the number attacked has been large the isolation has been more
practised. The second set of coefficients might be due to spurious correlation, but
they again admit of a simple interpretation as apart from “
namely that: when the attacks are numerous the deaths are relatively few,
spurious correlation,”
Biometrika x 71
556 A Study of the Effects of Diphtheria Isolation
because a wide-spread epidemic means a mild epidemic. All four coefficients are
significant, and pair and pair they are quite consistent but in no case are they
of any marked importance. They enable us, however, to correlate the isolation-
rate and the case-mortality for a constant total attacked, ie. to find the partial
correlation rm. We have the following results:
First Period Second Period
1904-1908 1909-1912
( Correlation of Isolation-rate )
ima a, a and Case mortality for — 474+ ‘056 —°512+°057
aa | constant number attacked j
while we have already found :
Correlation of number isolated
aa with number of deaths for +066 + :077 — 220+ :072
constant number attacked
Correlation of Total Numbers Isolated and Total Registered Deaths.
Total Numbers Isolated.
|
0
50 |
000
5
no)
~
T50
1250
Oo
4
QD
Totals
7
750-1000
5)
500
1250—1500
2750—3000
3000—3250
1500—1
1750—Z£
2000—2.
1000.
Deaths Registered.
0— 75 64 2 -= iL — | — | = | — 94
75—150 | 16 2 36
150—225 2 9
225—300 2 4
300—375 1 6
3875—450 2 5
450—525 = ile
§25—600 — 1
600—675 10)
675—750 (0)
750—825 1
Totals
; | sl | espcreldae
Means 54:2 | 59°8 | 112°5 | 133°9 | 932°5 | 312°5 382°5 at 2125 isolated 103°4
It will therefore be clear that removing the variation in number attacked has
made only shght reductions in the values of the correlation coefficients between
isolation-rate and case-mortality. The discrepancy between the absolute numbers’
and the rates’ correlations is not to be accounted for by “spurious correlation ”
involved in the use of total numbers attacked in both rates. It must therefore be
due to: (i) lack of linearity in certain of the regressions, (11) high values in the
coefficients of variation in certain of the quantities under discussion, or to a com-
bination of these causes. With the small size of the populations under discussion
it is by no means easy to test the true linearity of the regressions, even if we do
what appears legitimate in this case, namely pool our data for the earlier and
Eruet M. ELpERTON AND KARL PEARSON 557
later periods. Our actual correlations have all been found without grouping by
the direct product-moment formula, but we give on pp. 556 and 558 two grouped
tables to illustrate the difficulties which arise in analysis. Our first table is for
the total numbers isolated and the total deaths registered. It will be seen at once
that the marginal distributions are intensely skew, crowding up into the corner of
few deaths and few cases isolated, so that they appear to asymptote to the zero
Further, Diagram I shows that the regression curve
Total isolated.
O 250 500 750° 1000) 71250 1500 1750-2000: 2950 2500
values of the coordinates.
Total deaths.
Dracram I.
of deaths on total number isolated is, if just sensibly, still not markedly skew.
Turning to the actual numbers given by this table we have the following series
of constants :
Numbers Isolated (i) Registered Deaths (d)
Mean 460 406 ae S0 % =475°33 d =103°42
Standard Deviation ae, Be o; =571°25 og=118'61
Coefficient of Variation ... ies OF = | 140) = 1°15
(=8.D./Mean)
Correlation Coefficient and Ratio Tiq= '8348* + 0163 nai= 8564 +
* Agrees reasonably well with the non-grouped values for the two separate periods.
+ Found by taking means of all 13 column-arrays.
71—2
558 A Study of the Effects of Diphtheria Isolation
Clearly these results are of much interest; they show that the difference of 7
for deaths on isolation over 7 is not as great numerically as, perhaps, the graph
suggests, but they indicate the markedly high values for the coefficients of
variation. Now it is quite straightforward algebra to prove that
aVita, dla = aVi, do
provided we may neglect terms of the square and product order in v; and vg com-
pared with unity, and this is perfectly legitimate when these coefficients of
variation are, as is usual in anthropometric measurements, quite small quantities.
But in the present case these quantities are greater than unity and their squares
are not negligible as compared with unity, thus we need not be surprised at the
marked inequality of q?ija,aa* and 47;,af found above. The values of the former
show a marked relation between the case mortality and the isolation-rate, and the
values of the latter indicate no appreciable betterment in the deaths due to
increased isolation. Before we consider which of these coefficients gives us in the
present case the better result as a guide to practical conduct, let us examine the
correlation table for isolation-rate and case-mortality for the same 157 observations.
Correlation of Isolation-Rate I and Case-Mortality m.
Isolation- Rate.
0—10) 10—20| 20-30 | 30—f0 40—50 | 50—60 | 60—70 70—80 | 80—90 | Totals
| | .
wl 38 ee 1 ii 6 eae 3 28
ay cea 2 = 3 3 4 8 11 OV eee 45
“| 12-16] 5 | 4 3 8 3 8 6 GS Saal 44
Sel 7e=e20Nl) oa 2 1 2 3 3 i |) qo alee
S | 20-24] 9 = 2 i 2 = 2 1 a 17
= | 24—28 33 | 1 2 6
Seieeeeeoa. ene ee a hae 1
Ceres :
| Totals | 23 4 11 is |e} 26 28 22 | 16 | 187
Means | 19-04 | 14:00 | 16-18 | 15°60 | 14:00 | 11:08 | 11°71 | 11-09 | 9°25 13-26 |
The following constants were found for this table:
Isolation-Rate Case Mortality
Mean Sen es OW ee age m =13:26
Standard Deviation ae aes Gpo= 25:52 Om = 558
Coefficient of Variation... ; Oe = OP} Um = 0°42
Correlation Coefficient and Ratio TIm= — ‘5291 +:038 Nmrt= ‘5546
The graph of the regression of case mortality on isolation-rate shows small
evidence of skew regression (see Diagram II), and this is again confirmed by the
difference between 1;,, and 7,,; being fairly small. The marginal frequency dis-
tributions show, however, considerable skewness, and that for the isolation-rate is
lumped up at the end where there is no isolation: more than half the numerator
of 7,,; being contributed by the towns with little or no isolation. It is desirable
to consider these towns further. They have an attack-rate of "76, which is sensibly
* This is the a tm Of Our P. 556. ? + The values are given on our p. 554.
Erne, M. Evperton AND Karu PEARSON 559
less than the mean attack-rate (1°30), but they have a case-mortality of 19°04 as
against the average case-mortality of 13:26; the 17 towns* with no isolation at
all give a case-mortality of 19-4, It would thus appear that the towns with little
or no isolation are those with a lower average attack-rate, but with rare exceptions
their case-mortality is high.
Isolation-rate.
0 10 20 30 40 50 60 70 80 90 100
Case-mortality.
Dracram II.
To test the influence of these towns with little or no isolation, we have
removed the column 0—10 isolation-rate group and recalculated r;,, and 7,,;; we
find
Tim = — "4120 + 0484, 9,7 = °4810.
Thus while the correlations are somewhat reduced by excluding the towns with
little or no isolation there is still in the towns which do isolate a very sensible
relation between the degree of isolation and the case-mortality, and this relation
exhibits rather more skewness. :
We may sum up as follows: The relation between greater isolation and a
lessened case-mortality appears to be a real one. We have shown that it is hardly
due to spurious correlation, as this would have produced a positive correlation and
further no great changes are made when we correct for inequality in the numbers
* South Shields (1st and 2nd Periods), Sunderland (1st and 2nd Periods), Barrow (1st Period),
Preston (lst Period), Wigan (1st Period), Smethwick (1st and 2nd Periods), Walsall (1st and 2nd
Periods), West Bromwich (1st and 2nd Periods), Coventry (1st and 2nd Periods), Barnsley (1st and 2nd
Periods). Of these towns West Bromwich in the 1st period had the highest case-mortality recordeu of
any of our 80 towns, while Smethwick in both periods, and Coventry and Barnsley in the 2nd period
with no isolation had case-mortalities below the general average.
560 A Study of the Effects of Diphtheria Isolation
attacked. The regression is roughly linear and only very partially due to the high
case-mortality in towns with no isolation. It is probable that where there is
a large amount of isolation, the care of patients falls largely into the hands of
a few men with a more extensive experience of the disease, and that this reduces
the case-mortality.
Against this may be set the fact that the correlations between the absolute
numbers of deaths and of cases isolated for constant numbers attacked are in-
significant. The divergence between the two methods of approaching the problem
is, however, explicable because the coefficients of variation of the absolute numbers
are greater than unity, and the identity of the correlations reached by the two
methods depends on the neglect of the squares and products of the coefficients
of variation compared to unity. It may be asked: Why in this case we prefer
the partial correlation found from the rates to that found from the absolute
numbers? We reply: Because the partial correlation coefficient for the absolute
numbers depends on very high total correlations, and if these correlations be, as
we have shown, non-linear, then the partial correlation coefficient not only loses
its full meaning, but may, as experience has shown us, easily change its sign as
well as its magnitude. We would suggest that in a minor sense total mortalities
and total isolations are bound to give “restricted tables,” for deaths and isolated
cases are perforce less than the numbers attacked, and that in such “restricted”
tables, there is a general tendency to skew correlation and to a spurious factor*.
On the other hand it is true that case-mortalities and isolation-rates cannot
exceed 100°/, or fall short of 0°/,, but these limits are the same for every array
and do not vary from array to array as in the previous case. On the whole we
think it safe to say that isolation is associated with greater prevalence of the
disease and with a lessened case-mortality.
(5) Is there any significant Relation between Isolation-Rate and General
Diphtheria Death-Rate? We have seen (p. 553) that insignificant correlations exist
between J and M, and it is difficult to understand how a spurious factor could
have modified this result. In the first place the small values of rp, and rpy on
p. 555 show us that the value of p7,,, is sensibly the same as 7,,,; thus, for a con-
stant population there is no sensible association between diphtheria mortality and
isolation. But now let us ask whether for a constant attack-rate, isolation does
not lessen general diphtheria mortality. We have:
Correlation First Period Second Period
7,4 =AIsolation-rate and Death-rate ... +1532 — ‘0119
7,, =Isolation-rate and Attack-rate ... + 4268 +°2905
?y4= Death-rate and Attack-rate a +°6772 +6879
Hence
iim =Isolation-rate and Death-rate a
constant Attack-rate ee: Seuss
* See especially the illustrations of such ‘‘restricted” tables and their regression lines in a paper
by Waite on Finger-Prints: Biometrika, Vol. x, pp. 421—478.
Erne, M. EvpErRToN AND KARL PEARSON 561
Both of these values may be considered significant and negative, and hence
when the attack-rate is constant there is a sensible, if not very close relationship
between increased isolation and reduced general mortality from diphtheria.
This confirms the view already reached that while isolation is associated with
higher attack-rate its effect is to lessen the number of deaths whether they be
reckoned as case-mortality or general population death-rate.
(6) What is the meaning of the Association between Isolation and increasing
prevalency of Diphtheria? The analysis of this problem is more complicated.
The obvious answer of those who advocate increasing isolation would be that
it has been adopted in those districts where the disease is most prevalent, and
this of course may turn out to be correct. But we may ask in turn upon what
statistics they depend to demonstrate their view that isolation lessens the preval-
ence of the disease and is therefore advantageous, if our data demonstrate that
where there is more isolation, there there is more diphtheria? It can only be by
an analysis of no simple character that it is possible to deduce from such data
that the practice of isolation has lessened the amount of the disease.
There is, however, a preliminary problem to be dealt with. The isolation-rate
has been increasing very sensibly from 1904 to 1912, the attack-rate has lessened
although very slightly, the case-mortality has lessened and the mortality on the
population is considerably less. These facts are exhibited in the following table :
Means | Standard Deyiutions
Variate | Symbol |
| 1904-1908
1909-1912 | 1904-1908 | 1909-1912
Te! = oneal
Attack-rate per 1000 population | A |
be TERS IO pe MEIGS Fe) ‘657 | ‘639
Isolation-rate per 100 attacked | JI | 42:4 | 55°7 25°52 | 25-18
Mortality per 1000 population M 174 138 ‘080, ‘061
Mortality per 100 attacked m | 14:6 Ti Pa peeo air 501
Now it may well be, since the attack-rate has changed so little, that in the
towns with increasing attack-rate there has been increasing isolation, both
quantities changing with the time, but having no causal relation the one to the
other. It is of some interest therefore to consider the type of districts in which
isolation is most practised. In the first place we ask if any known bad social
conditions are associated with prevalence of diphtheria. We took as our measure
of sanitary conditions (1) the infant death-rate, or the deaths of children under one
year per 1000 births, (11) overcrowding, or the percentage of the population in
private families with more than two in a room. We found the following results :
First Period Second Period
Variates Correlated 1904-1908 1909-1912
Attack-rate and Infant Death-rate — "206+ -074 — °206 + ‘072
Attack-rate and Overcrowding ... — 1538+ ‘075 — 186+ ‘074
562 A Study of the Effects of Diphtheria Isolation
These are not very considerable, but they are consistent, and indicate, as
far as they go, that the incidence of diphtheria is not dependent upon such
measures as the above of unfavourable sanitary conditions.
If we now turn to the correlation between the mortality-rate on the population
and these measures of unfavourable sanitary conditions we find:
First Period Second Period
Variates Correlated 1904-1908 1909-1912
Death-rate from Diphtheria and Infant Death-rate +081 + 076 +°118+ ‘074
Death-rate from Diphtheria and Overcrowding ... +061 + 079 +°004+ :075
All these are indeed positive, but they are of no significance and if they were
significant would be so small as to be of no importance. The first indeed might
have been anticipated to show a higher value, for a certain number of deaths from
diphtheria must be deaths of infants. We can only conclude that as far as these
measures of unsanitary conditions are concerned they do not in any way determine
the diphtheria death-rate.
We now turn to the isolation-rate and find:
First Period Second Period
Variates Correlated 1904-1908 -1909-1912
Isolation-rate and Infant Deathrate ... — ‘414+ 064 - 375 + °065
Isolation-rate and Overcrowding Bh — 236+ 073 — 235+ :071
These are significant although not very large and we conclude that most
isolation is practised in those districts which have the lowest infant deathrate and
the least overcrowding; the correlations are sensible if not very large. In other
words the towns with better health conditions have adopted more extensively the
practice of isolating diphtheria cases.
It seemed further of interest to determine: (1) whether diphtheria and isolation
were more or less associated with urban conditions, and we took for this purpose
the number of persons per acre, and (ii) whether the well-to-do character of the
district, as measured by the number of domestic servants, indoor and outdoor,
male and female per 100 private families, has any influence on the incidence of
mortality from, or the isolation of diphtheria. We found:
First Period Second Period
Variates Correlated 1904-1908 1999-1912
Persons per Acre and Attack-rate ine +165 + °075 +043 + ‘075
Persons per Acre and Death-rate a +:°169+4°075 +:°115+:074
Persons per Acre and Isolation-rate ... + :073 + ‘076 + °053 + ‘075
Not one of these correlations is of any importance, if indeed any of them can be
considered significant. It is thus clear that the intensity of urban conditions has
very little to do with the prevalence of diphtheria, for if anything the suburban
conditions have the lesser death-rate; clearly isolation has no sensible relatfon
to number of persons per acre.
Erae, M. Evprerton AND KARL PEARSON 563
Turning now to our measure of the prosperity of the district, we find that it
has no influence on the attack-rate, that it sensibly, but not very intensely affects
the mortality, the higher death-rate occurring in the poorer districts, and that
isolation is associated quite significantly with the prosperity of the district, i.e. the
more well-to-do the district the more isolation is practised *.
First Period Second Period
Variates Correlated 1904-1908 1909-1912
Number of Domestic Servants and Attack-rate ... +095 + 076 + 024+ °075
Number of Domestic Servants and Death-rate ... —°219+°073 — +308 + :068
Number of Domestic Servants and Isolation-rate + °437 + ‘062 + °363 +065
We conclude therefore that the more prosperous and generally healthier
districts are associated with fuller isolation, and that the more prosperous, but
not necessarily the more healthy districts, have the less diphtheria death-rate.
On the other hand the incidence of the disease seems independent of the prosperity
or density of population of the district and to be somewhat greater in those towns
where the sanitary conditions as judged by infant death-rate and overcrowding are
better.
Thus as far as our measures go, we must conclude that diphtheria is not to be
considered as a disease of markedly urban districts, of overcrowded or of insanitary
districts. It would appear that the more prosperous and healthy districts have
the greater isolation and that these are subject to somewhat the greater incidence.
* Of course this may largely mean that the more prosperous towns introduce isolation to remove
the supposed danger of infection when servants of the families of the well-to-do are attacked.
+ In order to ascertain whether the variates persons per acre (p,) and overcrowding (O) were merely
measures of the size of the town population (P) we correlated P with p, and with O and found:
"pp, = +404 + -064 (1904-8), = + -402 + -063 (1909-12),
Tpo= +°091 + :076 (1904-8), = + :074 = -075 (1909-12).
Thus overcrowding has no relation to the size of the town, the larger towns do not show more over-
crowding. There is, however, a considerable association of persons per acre with total population,
the larger towns having more persons per acre without exhibiting any more significant overcrowding.
Making the population constant we find:
First Period 1904-1908 | Second Period 1909-1912
Total Correlation Partial Correlation | Total Correlation Partial Correlation
» |—— a =
Tap, =+°165£-075 | pray, = +1224 -076 "4p, = +1043 © 075 May, = +°023- 075
Tyo = —- "1534075 | pryg = —°1674%:075 |) typ = - 1186 4-074 go = — 1140+ 074
Thus correcting for population only makes the relation between persons per acre and incidence still
more insignificant, while the relation between incidence and less overcrowding becomes slightly greater,
without rising to any real importance.
+ This result must not offhand be extended to subdistricts of our towns, it is an inter-urban and not
intra-urban statement.
Biometrika x 72
564 A Study of the Effects of Diphtheria Tsolation
It will be seen at once that this conclusion opens up new problems: (i) Is the
greater isolation the outcome of greater incidence, the only remedy suggestable
for greater incidence being a more complete isolation? (ii) Is the greater
incidence in some manner a result of the greater isolation, and does it really tell
against isolation as a remedy against the spread of diphtheria? The association
of greater isolation with local prosperity would then be merely a measure of the
economic capacity of the district for carrying out the accepted sanitary code.
(iii) If (ii) is to be answered in the negative, then is there any factor in
prosperity which makes for a greater diphtheritic incidence? The final answers
to these problems can probably not be given on the basis of the present data.
The correlations under discussion although significant are not of such a marked
character as to provide more than provisional statements, or indeed more than
suggestions for further inquiry and tabulation.
(7) Does greater Isolation follow increasing Incidence, or greater Incidence
follow increasing Isolation? The problem is a much more subtle one than appears
at first sight. What we have established is that those towns with the higher
isolation-rate have the higher attack-rate. It does not follow from this that the
individual town which increases its isolation-rate will increase its attack-rate.
To determine whether this is so we took as our variates: increase in isolation-
rate ([) between the periods 1904-8 and 1909-12*, and the similar increase A of
the attack-rate. We found:
Ti = Zoot 012;
a value probably significant, although not quite so large as that found for the
inter-urban relation :
Y47=+°427 + 063 (for 1904-8),
= + ‘290 + 069 (for 1909-12).
We can, we think, therefore conclude that the towns which increase their isolation-
rate are those with increasing attack-rate, just as the towns with higher isolation-
rate are those with higher attack-rate.
But this does not answer the question as to which is “the cart” and which
“the horse”! Does the increased attack-rate precede or follow the increased
isolation-rate ? To answer this question we divided our material into three
periods each of three years, let us say 7,, T, and 7;. Then the attack-rate
increase between 7’, and 7, was correlated with the isolation-rate in 7, and the
isolation-rate increase between 7, and 7, with the attack rate in 7;. In other
words we asked whether towns with most rapid increase of attack-rate in the
* That is the total number treated in hospital x 100 and divided by the total number of attacks was
taken for the first period and for the second period, and their difference (second period—first period)
was treated as increase in isolation-rate. In the same way the sum of the totals attacked for the years
of the first period x 1000 and divided by the sum of the calculated intercensal populations for the same
years was treated as the attack-rate, and the difference of second and first period values taken as the
increase in the attack-rate.
Erne, M. ELpERTON AND KARL PEARSON 565
early periods had most isolation in the later period, and whether the towns with
most rapid increase of isolation-rate in the earlier periods had most incidence in
the later period. We found:
ES Hate 004 + ‘077,
rid, = t+ 085 + “077.
Thus there is no significant relation whatever between either increase of attack-
rate or increase of isolation-rate in the first periods, and the isolation or the
incidence in the following period.
As criticism of this result it might, perhaps, be suggested that the correlation
of A,_, and J, will be influenced by what has been the course of J in the periods
T, and T, and the nature of A in 7,; we have accordingly, in order to test this,
made the isolation-rate constant in the first two periods and the attack-rate
constant in the third period and find
Thy 4g" 44 I, ~ +147 ae ‘076.
This is still of no real significance, although the sign appears to indicate that
where the isolation-rate has been constant then increasing attack-rate in the
earlier period is followed by very slightly more isolation in the third period, even
if the attack-rate in that period be itself constant.
Similarly we determined :
UR OTE ply Rit ‘O77 + 077.
This coefficient shows that towns which have increased their isolation-rate during
a period of constant attack are not liable to sensibly heavier attack in the
following period.
It would thus seem that our first two problems are both to be answered in the
negative. Towns which increase their isolation are not those which in the fol-
lowing period have most incidence, nor are those which have increasing incidence
markedly those of most isolation in the following period. Attack and isolation
appear to have no causative relation, and the association we have found between
more isolation and more incidence seems to be contemporaneous rather than
successional. We are, it seems, compelled to search for something in the environ-
ment, which favours incidence and at the same time isolation. The only common
factor that we have been able to reach at present is the prosperity and general
healthy condition of the town. Under these circumstances there appears to be
economic possibility of greater isolation, but why should there be greater incidence ?
Is it possible that in the more prosperous towns there is greater consumption of
some easily contaminated commodities, which may act as carriers of the disease,
or more concourse of those of susceptible ages at places of public amusement or
instruction ?
(8) Test of the “organic” nature of the correlation of Isolation- and Attack-
rates by the method of Variate Differences. If the suggestion made at the end
of the last section be correct we should anticipate that by the use of the method
129
566 A Study of the Effects of Diphtheria Isolation
of variate differences we should free ourselves from the influence of the time
factor, if attack-rate and isolation-rate increase simultaneously in the more
prosperous towns, but without organic association. We have nine years’ returns,
but the epidemic nature of diphtheria in many cases does not give one great
confidence in applying the method of variate differences to individual years.
We considered that it would not be wise to deal with smaller intervals than
three-year periods, and should have preferred had the data been available to work
with five-year intervals. As it is, we cannot with three-year intervals for each
town go beyond the second differences. We have accordingly 228 isolation-rates
and attack-rates obtained from 76 towns for each of three three-year periods,
152 first differences, and 76 second differences. We may symbolise them as I’ and
A’, 6,1’ and 6,4’ and 6,J’ and 6,4’. We found the following results :
rpg = +°332 +040,
15,1'3,A! — + 936 + ‘052,
13,1'3,4/ = + 159 + 075.
The first of these results compares reasonably with the previous results for the
first and second periods on p. 552, i.e.
1904-1908: rzy = + 427 + 063,
1909-1912: rz4 = + 290 + 069,
with a mean value of +°358. And this is the more true because the values of
- rrq were found by the product moment method without grouping, while 77-4, was
obtained from grouping in a correlation table*.
Now the above values bring out very markedly that when we endeavour to
remove the influence of the time factor and to obtain a purely organic relationship
between J and A, we more than halve the correlation between them by proceeding
to the second difference only. If we might suppose that a hyperbola would give
the asymptotic value of 75 775 4’ from the above three known correlations we should
have
7084
Ne i 2
T3T8,4' = S105 a5 542,
which indicates, although no stress can be laid on actual numbers, that at about
the fifth difference rs 75 4, would tend to become negative. All we think it
possible to say would be that if the time factor be eliminated there is very little
positive organic association between high isolation-rate and high attack-rate to be
cleared up,—certainly not more than is indicated by the correlation on p. 565:
I, 43" Ay yl, ear piles ‘076,
* It may be noted that 7; 4/3 7 was also found from a correlation table, but "5,1 ,4' 88 having only
76 cases by product moment without grouping.
Erne, M. EvpERTON AND KARL PEARSON 567
which seems to suggest that other things being constant increasing incidence is to
some slight extent followed—probably as the only suggested remedy—by the
higher isolation rates*.
(9) Can any other Factors be determined which measure the Relation between
urban conditions and the Incidence of Diphtheria? It is worth while from this
standpoint to place the towns with which we have dealt in the order of incidence,
each town being credited with the mean of the three attack-rates for each of
three-year periods. Now an examination of the four columns of this table shows
that, with the exception of Oxford—which has a child incidence (‘89 as com-
pared to ‘70) considerably above the population incidence owing to relatively
few children—the towns with the least diphtheria are the Midland, and _parti-
cularly the Northern manufacturing towns. These constitute practically the
whole of the first column of 19 towns. The last column contains the big ports
and certain suburban metropolitan districts, indeed all these for which we have
data except Plymouth, Devonport and Tottenham fall into the second half of the
Seventy-siz Towns in order of their Diphtheria Incidence Rates 1904-1912.
OCOOBONIKDOP WHY
West Bromwich (‘40) | 20 Rotherham (91) | 39 Birkenhead (1:20) | 58 Brighton
Northampton (45) | 21 South Shields (92) | 40 Rhondda (1:24) | 59 Stockton
Wigan (48) | 22 Preston (93) | 41 Smethwick (1:25) | 60 Grimsby
Walsall (49) | 23 Wallasey (94) | 42 Barrow (1:25) | 61 Leyton
Stockport (53) | 24 Bath (95) | 43 Newport (1:25) | 62 West Ham
Oldham (59) | 25 Bootle (96) | 44 Wimbledon (1°30) | 63 Salford
Bolton (59) | 26 York (99) | 45 Great Yarmouth (1°31) | 64 Nottingham
Oxford (‘70) | 27 Blackpool (99) | 46 Southend-on-Sea (1°32) | 65 St Helens
Barnsley (71) | 28 Tynemouth (1:00) | 47 Birmingham (1°32) | 66 Walthamstow
Southport - (72) | 29° Tottenham (1:02) | 48 Gillingham (1:34) | 67 Ilford
Rochdale (73) | 30 Halifax (1:03) | 49 Ipswich (1°36) | 68 Southampton
Leicester (76) | 31 Sheffield (1:03) | 50 Liverpool (1:37) | 69 Cardiff
Manchester (‘79) | 32 Plymouth (1:07) | 51 Hornsey (1°39) | 70 Enfield
Bury (80) | 33. Coventry (1:11) | 52 Darlington (1°39) | 71 Hull
Blackburn (83) | 34 Warrington (1:11) | 53 Acton (1:43) | 72 Bristol
Wolverhampton (:86) | 35 Devonport (1:13) | 54 Newcastle (1:45) | 73 Croydon
Burnley (86) | 36 Sunderland (1:14) | 55 Burtonon Trent (1°48) | 74 Portsmouth
Huddersfield (89) | 37. Bournemouth (1°17) | 56 Bradford (1°56) | 75 Derby
Wakefield (91) | 38° Middlesbrough (1:20) | 57 Willesden (1°56) 76 Lincoln
* It is perhaps worth while putting on record the additional statistical constants obtained in
deducing the above correlations, as they are probably fairly reliable values and should be compared
with the two period constants on p. 561:
A’ =Mean Attack-rate 1:26 ; Standard Deviation, Attack-rate 655
I’ =Mean Isolation-rate 47-75 ; Standard Deviation, Isolation-rate 26°341
6,4’ =Mean Increase in Attack-rate —'086; Standard Deviation of change in Attack-rate 648
8,4’ =Mean Increase in Isolation-rate 9:03; Standard Deviation of Increase in Isolation-rate 1°05
Thus while most towns have been sensibly increasing their amount of isolation by 17 °/, to 18 °/, of
its mean value, the decrease in the attack-rate has only been 6°/, to 7°/, of the mean incidence, and
the correlations show that this decrease has not occurred in the towns with marked increase of
isolation.
568 A Study of the Effects of Diphtheria Isolation
table, and there can be little doubt that on the whole sea-port conditions and
the big new neighbourhoods round London favour, while manufacturing con-
ditions restrict, the incidence of diphtheria. We have not data, however, available
upon which we could test water and milk supply, or extent of consumption of
milk and fish in these towns. The results for Derby and Lincoln are remarkable,
but they are high for all three periods, and this notwithstanding the rapid
increase of isolation in those towns.
At first sight it seemed to us that the towns in the first column were markedly
those in which there had been a greatly restricted birthrate*, while those in the
last column were towns of greater fertility. Taking the births per 100 married
women from 15 to 45 (B) we found:
TAB = + ‘013 ar O75.
Thus there is no association between incidence and the well-to-do character of a
town as estimated by a low birthrate.
Again having regard to the character of the towns in our first column, it
occurred to us to test the incidence in relation to the employment of males in
manufacturing processes involving smoke. We took out of the 1911 census the
percentage (8) of males over 10 years of age, who fell under a rough test of
smoke-producing occupations, namely 1x. 1, x. 1-2, 5-8, x1v. 1, Xv. and XVIII.
1-6 of the Registrar-General’s classification, and we found :
TA a ae "180 ta 073.
This is possibly significant and would undoubtedly be emphasised had we
included as a factor the women engaged in textile industries. There seems
therefore some slight reason to suppose that the conditions favourable to smoke
production are unfavourable to the spread of diphtheria.
If the data could be procured, it would be worth while considering water and
milk supply and the extent of fish consumption in the towns we have dealt with.
If these were found to be of little influence, the road would certainly be clearer
for dealing with the chronic diphtheritic human carrier as the chief source of
the spread of the diphtheria bacillus.
(10) Conclusions.
(a) No influence of greater isolation in reducing the attack-rate from diph-
theria is discoverable. In fact there is a sensible, if not large, positive association
between the isolation-rate and attack-rate.
(b) The case mortality is somewhat less where there is more isolation. This
may very probably be accounted for by more cases coming under specialised medical
care.
* We had partially in view here also the possibility that restricted birthrate meant employment of
women and thus less breast-feeding and greater use of milk, so that cross-currents might be at work.
ib
Erue. M. Exuprerton AnD KARL PEARSON 569
(c) The attack-rate appears to be greater in the more prosperous towns and
in towns of somewhat better sanitary conditions. We have not found the pre-
valence of diphtheria associated with overcrowding or with the conditions leading
to high infant mortality.
(d) While a low birthrate, taken either as a measure of prosperity, or as
a measure of the employment of women and so of the prevalence of hand feeding,
appears to have no significance for the attack-rate of diphtheria, smoke-producing
manufactures are probably unfavourable to the prevalence of the disease, which
appears to attach itself in the main to the large ports and metropolitan suburban
districts.
(e) The association between the attack- and isolation-rates observed is not
very significant, and while it might, to a very small extent, be due to increased
isolation following or accompanying increased attack, it 1s more probably an
association due to the more prosperous towns practising more isolation, and also
to there being some element in prosperity which assists the spread of the disease.
Generally all the correlations are of a low order; they contain, however,
nothing to support the theory that isolation markedly limits the incidence of
diphtheria ; the disease itself does not appear where overcrowding is greatest nor
where the population is most dense; on the other hand isolation is most practised
in those towns where domestic servants are most common and which may be
supposed to be most prosperous. The chief argument for isolation—which can be
drawn from the present data—is a lessened case-mortality, but such mortality
might be obtained in all probability by specialised medical service as apart from
isolation.
MISCELLANEA.
I. On the Probable Error of a Coefficient of Mean Square
Contingency.
By KARL PEARSON, F.R.S.
Ler the sampled population be considered as to two variates and be represented by the total
M and the cell-frequency m,, for the pth row and gth column cell. Further let the vertical
marginal frequencies be given by m,, and the horizontal marginal frequencies by m,,, so that
Myq+ Moq+ ...+Mpgt...=M.q,
My + Mp. t+... + Myqg+...=Mp:-
Let the corresponding quantities for the sample be WV, 2pq, 2.q and np..
Then we know that the mean square contingency ¢? is given by
sg Hn: EEN
& (w uu Yn)
Pe ny eae ioe
NxN Wo
p=
summed for every cell.
Now in the great bulk of statistical phenomena we do not know more of the sampled popula-
Mm, Mp:
—* and —2 equal to
ou
tion than is given by the sample, and thus to determine ¢?2 we must put
n, Np: : ‘ :
and . Doing this we obtain the usual
the most probable values known to us*, namely, Wi V
value for the mean square contingency
2
P= (3% - Mn) |n Ra ltp } eee ebalt seeaeees -eceeeeReeee eres (ii).
Starting from (ii) Blakeman and Pearson have found + the probable error of the mean square
contingency. The process is admittedly very laborious and although it has now been used fairly
often, it must be confessed that its chief value is to obtain appreciation of the probable errors of
contingency coefficients in general, rather than in any usefulness in recording significant differences
between long series of individual coefficients.
But it has not been sufficiently recognised that the probable error thus found is that of the
approximate value of the mean square contingency (ii) and not that of its true value (i). It is
indeed the probable error of the expression actually used, but it is not the probable error of the
true value as given by (i). The latter is easy to find and deserves consideration. Let us write
Nm Mp | M?= ppg
ol: (Mg — 2 “ hey (3
then 2 __ § { LE LE | fe a eee
ie N Mpg N Hyg)”
* Pearson, Philosophical Magazine, Vol. u. 1900, pp. 164 et seq.
+ Biometrika, Vol. v. pp. 191 et seq.
Miscellanea 571
where we shall use ¢/2 and ¢,” for the true and approximate values of the mean square con-
tingency. Thus
Now for a sample of constant size pp, is constant and therefore representing small deviations
by differentials
én
5 =F 29 (72 mt)
as Ppq
Square, add for all samples and divide - the number of such samples and we have
4 (5 ) 8 Rg Up'q'
ees y a) pa!’p'q
Org 2= a7 A 5 Olan +3, 2 | —— Xon,o ly
72 2 p} Ry'q'’ Rngtp'q' }?
oe ON ie pq N' Mg Hp'q! pa ™p'q'” ™pa"p'a
where oy, is the standard deviation of n,q and r is the correlation of deviations in 7p and
pq ™p'q’
Np'q 3 S iS a summation for every cell and 3 for every pair of cells.
But it is well known*® that
Myq Mp'q'
r a loo a ee he
"pa"'a’ Mou”
where y is the factor 1—(N -1)/(M/—1), usually put unity, since M is as a rule large compared
with WV, and which will be here put unity for the remainder of the work.
4 n2,.m 4 n,m 8 NngNy'q MygMp'q
Hence ofa2=— 8 ( nar a) A Ss ( pg 2) on S ( pa!p'a’ “nq 2)
mee Mp ng N M? peng N Poghy'g AL?
4 N? 5M 4 Ng M 2 :
= s( men {8 ("4 ali Se ty A iv).
diy Mp ng N Mpyq w)
This is the standard deviation of the true value of the mean square contingency, and in most
cases will be of no service, for we do not know the true values of mpg and ppq-
Fig 7 rp/q’
If we put these equal to the values obtained from the actual sample under consideration we
obtain the approximate value of the standard deviation of the true mean square contingency,
which we may represent by the symbol («¢ 2) and compare with what Blakeman and Pearson
a
found, i.e. (%% 2). Thus our alternatives are
a /t
a? + 67449 (042.5
and pa’ t 67449 (og, 2),
The real thing is pi? + 67449 0g 2.
Shall we obtain a better insight into the variation of this by taking the approximate values of
both ¢? and Tg Ps OF by taking the probable error of $,2? The problem is a subtle one, and,
perhaps, only to be solved by experiment, not by theory. Of course when we take numerous
samples and calculate $,’, then oy 2 will measure their variability. But this is not what we seek.
We use ¢,? as an approximation to ¢,?, and it is the variability of the true value that we want.
Are we not right in choosing (og), as its best value? In short would not—on the average of a
great number of samples—(og), give us a closer result to THe than (79,2), !
Returning now to equation (iv) and putting in the observed values for 7pq, fypq We have
(o92)a= - $1 aes {sa Pa ene ere (v).
* The values here given are the true values before we approximate by putting mpq/M=Mpq/N, ete.
Biometrika x 73
572 Miscellanea
Or, after some reductions
2 1 :
(72), aN (hat dae = du'} Liana seariancersenre cocenodan aco (vi),
7 _ hiqp: a
where ve=S 4 (0 N ) SES4 aba ge cbs a sides e ROS (vil)
(2 .q%p.)*
(» = ares)
and Data ee Nh ee eas Se (viii).
N.gNp
Again we have ane 5
Var
(7$) a iN 7 yt = hath vancedenneccen ooseee eee eee (ix).
Now what we usually need is the ae error of the contingency coefficient
= p'(1+$%)
But TC,=o¢ A — C2)? =04/(1+ $2)?
Thus the probable error of the coefficient of mean square contingency
_ 67449 [a5/pa? +1 — a’) 2
67419 x 00,= SF | oo } ue (x).
This expression is much simpler than that for the probable error of the actually used value
as given by Blakeman and Pearson*. It is not, however, asserted that it possesses greater
theoretical validity. Those authors illustrate their formula by calculating the probable error of
the contingency coefficient in the case of the association of handwriting and general intelligence
in 1801 schoolgirls. They find
C= ‘2957 + 0192.
In the course of their work they deduce
a2 = 09580,
(o¢,,), = 03268,
Wa? = "14865.
Using these values we have from (ix)
1 14865
See | = 3.
(Tb)a Sian loosest 90420} 0369
It is clear therefore that (74,),, does not differ very substantially from (og, ),. Calculating
from (x) the probable error of C,, we find it=-0217, while the Blakeman-Pearson process gave
0192. The two values only differ by ‘0025, which is unlikely to be of importance in the case of
most inferences in practical statistics.
Beyond the knowledge of ¢,? only ,? is required by the present process.
N.glp:\” Fs 2 .qMp:
Py) ee eer oe ee GrgueDs
1 g he N ae N
SS :
N.qMp: N gp:
This may be written
In finding the mean square contingency ¢,7, however, the three expressions
2
(x _ Nqp:
a N N gp:
pq — “ar
’
N.qMp:
N
N.qMp:
and W
* loc. cit. p. 194.
eeeeEeeE—
Miscellanea 573
must have been written down for each cell and thus y,? can be readily calculated. We can also
treat W,? as given by
Ve=S Nibyg_\ _ 1-3¢,2
; (2 .qM:)” <
but the cubing of the often rather large cell frequencies is troublesome, just as it is rather more
troublesome to calculate
2
b2=S (= we )- 1
N.gNp:
N.qNy:\”
Rpg — >
1 (mm N )
¢ 2 as
than da WV S AS ;
owing to the largeness of the squares in the former expression.
II. Measurements of Medieval English Femora.
In a forthcoming memoir on the English Long Bones there will be a good deal to be said
about the conclusions reached by Dr Parsons in his recent paper on the Rothwell femora.
Meanwhile he has started an attack on the Biometric School in a Journal whose columns are
not open to adequate reply,—i.e. to a reply of not greater length than the published attack—
from members of that school. In his communication he suggested that I was unacquainted with
the condition of atfairs at Rothwell, and behind this charge tried to escape any answers to
the essential questions I asked him, and thus those questions still remain unanswered.
The communication I made ran as follows :
My informant who I hope is trustworthy speaks of (i) “the great mass of bones beneath the
church at Rothwell” and (ii) of “the great collection of human bones beneath the old parish
church at Rothwell” ; further (iii) “there are probably some 5000 or 6000 individuals represented
in the vault at Rothwell, either altogether or in part”; and again (iv) “The stack varies in
height and breadth, but is nowhere as high or broad as that at Hythe, although it is much
longer. I know that at Hythe there are the remains of rather over 4000 people,....... I think
that this collection contains more than this, partly because the stack is so much longer, partly
because the bones are so much more decomposed and have therefore settled more.”
Manouvrier after much piecing and mending while only able to measure the lengths of about
16 femora from the neolithic burial places of Montigny and Esbly, was yet able to determine the
pilastric index of 127, and the platymeric index in 127 bones, that is to say in eight times as
many bones as those for which he could obtain the maximum length. And had he dealt fully
with the head and neck and the popliteal region, the multiplying factor would probably have
been ten. Had piecing, mending and a maximum of care in handling been used, I can hardly
believe that what Manouvrier achieved at Montigny was not possible for Dr Parsons at Rothwell.
Dr Parsons writes: “If the remains of femurs, whether they are fit or unfit for measurement,
are counted it will be found that females are quite as numerous as males though measurable
male femurs from their stronger build are Jess liable to break in being extricated from the pile of
bones, and so there are more of them available for measurement.” The italics are mine.
Much depends on the method of ‘extrication,’ and if the capacity of a bone to stand a hole
being drilled in it with a bradawl be part of the necessary fitness for measurement then the
number might undoubtedly be limited. But trusting to what I know has been achieved by the
SY fe Miscellanea
French, I feel convinced that if Dr Parsons could measure 277 femoral heads where the femoral
length was measurable, he could easily have measured 2000 heads in all and thus have ascer-
tained, definitely, whether his Rothwell series is unique in showing a significant depression in
frequency between 45 and 47mm. Further he could on such material by dealing with numbers
8 to 10 times those he has provided have given definite answers to many of the problems con-
cerning platymery and the pilastric and popliteal indices, which other observers have been vainly
trying to solve on far less adequate and in many cases far more fragmentary material.
I would note that Dr Parsons gives no reply at all to my question of why he used Dwight’s
measurements as a criterion of sex when they referred to bones with the cartilages attached,
because without this reply his careful attention to ‘other points’ when the head fell between
45 and 47 mm. seems one-sided, and of no value in sexing the collection as a whole. He
further gives no reply whatever to my question of why it is the male end, not the dwarf end,
of his female distribution which is lacking, if absence of females be due to breakage.
I would also state (i) that I have not sexed the Rothwell bones and therefore cannot say how
far I should or should not agree with Dr Parsons. Dr Lee using the best available mathematical
process found 145 9s and 133 gs, while Dr Parsons has 1039s and 174¢s. How this shows any
agreement I fail to perceive ; (ii) that I have made no assertion about the bones being of the
13th and 14th centuries. I merely headed my letter with Dr Parsons’ heading ‘ Measurements
of Medieval English Femora,” and asked why, if Dr Parsons holds these bones to be such, he
considers them without cartilages comparable with the mixed results of a modern American
dissecting room plus the cartilages.
KP,
CAMBRIDGE : PRINTED BY JOHN CLAY, M.A. AT THE UNIVERSITY PRESS
_ ARTHUR THOMSON, ‘University of Oxford
ARTHUR ee Royal covet of Surgeons ci
2
( 2] Column with six ana a half Cervical and thirteen True TA
ebrx, Ww $80! HON Rte ‘of the Cervical Spinal Cord and Nerves. DoverasG. ~~ el aR
h.B. Edin., M.A. Trin. Coll, Camb. Appendices Epiploice and Pericolic Membranes. :
,M.B. The Topog? raphical and Applied Anatomy of the Gall-Bladder and Bile-Ducts. -
ALMSLE I.B. Observations on Certain Structural Détails of the Neck of the Femur.
LEY, oe i The Neck of the wes as a Static sole eat, _M. PF. aie Aes B. Ss.
| Ribs.
a
a
lithic Date. eran Plates XXIIT_XXV.). ‘By ©. G.
‘ion at Gibraltar in 1912. By W. L. H. Ducxworts, M.D.,
and Tribal Groups i in India. By W. Crooxz. The Orienta. ai, a
‘ Y. J. Perry. Lois de Croissance. By Dr Pavt Gopry. The: Ha, Se NER
slands and its Makers. (With Plates XXVI—XXXV.) By Hon. i
. Magic and Witchcraft on the Chota-Nagpur Plateau.—A Study - Ma
y Sanat Cuanpra Roy, M.A., B.L., Ranchi. Les Touareg du Sud. Be
By Fr. pe Zentner. Flint- finds in connection with Sand. By >
s me Recent Work on Later Quaternary Geology and Anthropology, dir Se oF
its be AG on the question of ‘*Pre-Boulder-Clay Man”. By A. Irvine, D.Sc., B.A. Notes on . Sia
at Hal- Saflieni, Malta: By Rev. H. J. DUKINFIELD Astury, M.A., Litt. D., F.R.Hist.S8. ee aad arte
Princess. By G, W. Murray. The Experimental Investigation of Flint Fracture and : Seine:
m to. Problems of Human aa eam eee ae a cahial ine ve ey 8. Hazzuepine j
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~ (i) On the Probable Error of a Cocfticient of ee Square Contingency. By Kant
(a) Measurements of Medieval English Femora.
« # ; Page a
The publication of a paper in Biometrika marks that in the Editor’s opinion it contains either in
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Biometrika appears about four times a year. ‘A volume containing about’ 500 pages, with plates and
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Association of Finger-Prints. By H. Warne M ae B. Se. “With Plate XX and
a Thirty Diagrams in the text) . . 0. Sete Ais ies eae
II. On the Problem of Sexing Osteometric Material. ‘By. eae Parson, F.R
(With One Diagram in the text) see COREL NS en ee
fe TTT Further Evidence of Natural Selection i in Man. By ETHEL M, Expertow and
Kany Pranson, FBS 36h VA ei bn lal 2 ae ee
‘IV. Frequency Distribution of the Tralee AG the Correlation Coefticient in samples
from an, indefinitely Large Population. By R. A. ‘FISHER : he .
V. On the Distribution of the Standard Deviations of Small Samples Append 3 vi
-to Papers by “Student” and R. A. FISHER. (Editorial. MAAC,
Vi. Tuberculosis and Segregation. By. AttcE LEE, D.Se. Ree ees ye, *
VII. The Influence of Isolation on the Diphtheria Attack- and Death- rates, By ie
Ernen M, Exprrton and Karu PEARSON, F.R.S. (With ‘Two oars in ae i
: the text) . . 5 bee Heer. Mages (aes : ass,
Miscellanea: Mek ee uray Vai era ne « if 7 :
PEARSON . sear itaa soe tome apees : 3 : ( Senn alee ae ie
By Kart Pearson. . .. .
tk i Rss , é : % Ne we
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