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THE LIBRARY
OF
THE UNIVERSITY
OF CALIFORNIA
LOS ANGELES
A STATISTICAL INQURY
INTO
The Probability of Causes of the Production
of Sex iu Human Offspring
BY
SIMON .NEWCOMB
WASHINGTON, U. S. A.
Published bt the Carnegie Institution of Washington
June, 1904
CARNEGIE INSTITUTION OF WASHINGTON
Publication Ni >. 1 1
sTiio Box* gafHrnow cprece
nil ran I>1 \w M D ' OMP 1X1
BALI I MORS, Mi
&\cvned
no4
PREFATORY NOTE.
The present paper is an attempt to apply a rigorous theory of probable
inference to a question of genetic biology, taking statistical data as the
basis of the inquiry. If, in making such an investigation, the author
may seem to stray outside his professional field, he would reply that the
discussion of a biological question was by no means the sole object with
which the work has been undertaken. It has appeared to him that the
treatment of statistical data generally on a large scale, by the rigorous
methods of probable induction, leads one into a field the cultivation of
which promises important results to the science of the future; and he
hopes the work will show how it is possible by such methods to reach
conclusions on questions which elude all direct investigation.
The author has to acknowledge his indebtedness to the Trustees of the
Bache fund, who made him a grant to pay the expenses of the necessary
examination of genealogical data, and to the Census Bureau, through Mr.
W. C. Hunt, chief statistician for population, who supplied statistical
data relating to several thousand families culled from the Census records.
Washington, November, 1903.
89862
A STATISTICAL INQUIRY INTO THK PROBABILITY OF CAUSES
OF THE PRODUCTION OF SEX IN HUMAN OFFSPRING.
By Simon Newcomb.
1. Introductory 5
2. Preponderance of male births 6
3. Is the ratio of male to female births the same in all races? 7
4. Inquiry whether any unisexual tendency exists among parents 9
5. Unisexual tendency in multiple births 16
6. Processes suggested by the statistics of multiple births 21
7. Influence of the age of the parent on sex 22
8. Supposed influence of other conditions 26
9. Summary of conclusions 28
Appendix. Mathematical theory of the effect of unisexual tendency. ... 29
1. INTRODUCTORY.
The object of the present investigation is to make an application of
certain hitherto undeveloped statistical methods to a class of cases in
which the causes elude all direct inquiry. The subject of such methods,
although its general principles are well understood, is one which is sus-
ceptible of being worked out in detail to a far greater extent than has
yet been seriously attempted. The present paper may therefore be
regarded as having a double object: one to illustrate and apply certain
methods; the other to draw conclusions on a physiological question of
the widest scientific and human interest.
The idea that the sex of offspring may depend, to a greater or less
extent, on discoverable causes, perhaps even on causes within the control
of the parents, is a very natural one. Examples of what might be pos-
sible causes are the respective ages of the parents, their vigor or condition
of health, their conjugal habits, which may be infinitely varied, or the
relation of the time of conception to the periods of menstruation.
Besides this, it is possible that some parents, male or female, may be
endued with some special faculty for producing children of one sex
rather than another, owing to constitutional or other causes which, not
being apparent on the surface, might entirely elude direct investigation.
The method of investigating this tendency in the case of recognizable
possible causes is well known. The question whether parents having a
g STATISTICS OF SEX
certain characteristic, A. are more likely than parents of the class not-A
to have children of either Bes is determined l>> counting a sufficient num-
ber <>f the offspring <>l* parents of each class ami comparing the ratio of
male and female children in each class with the average ratio, if we
find thi- ratio to ho markedly differenl in the two classes, we conclude
that the characteristic A i- associated with some cause having a definite
effecl apon the production of Bex.
Although tli'' author proposes to apply this Bimple ami obvious method
to certain cases in the following investigations, his main objeci is t" go
farther, ami impure whether there are any causes or conditions what-
ever, known or unknown, which, in a decided degree, affecl the produc-
tion of sex. When we - me family consisting mainly or wholly of
male children, ami another consisting mainly or wholly of female chil-
dren, it is very natural to Busped that the excess, in each case, may be
due to some characteristic or faculty of the parents, or some peculiarity
of their constitution, which may or may not admit of discovery ami
investigation. The mere fad of this inequality, taken in Itself, does not,
however, prove anything, because it may he the natural resuH of those ac-
cident- which determine sex. hut of which we know nothing. Granting
the existence of constitutional or other tendencies of the kind supposed,
we may apply the term unisexual to them. We may then make the hypo-
thesis that there i- something in the constitution or habits of parents
which results in some having a unisexual tendency toward the production
of male children, and other- toward the production of female-, which
tendencies nevertheless elude investigation otherwise than by statistical
iin ;.,n of their effects. The desideratum i- to discover a criterion
by which we may distinguish between inequalities in the division of a
family between the two Bexes which are -imply the result of chance and
those which are the re-ult of a unisexual tendency on the pari of the
parent-. Such a criterion i- pointed out in the lir-t four Bections of the
presenl paper, and it- mathematical theory developed in the Appendix.
2. THE PREPONDERANCE OF MALE BIRTHS.
I rtain fact- preliminary to tin- inquiry may he Bel forth which will
•■■..• ;i- points of comparison in coordinating our conclusions. The
fir-t of these i- the well-known general fact that, in the entire Semitic
race, there i- a -mall hut well-marked preponderance of male over
female births. This preponderance is remarkably uniform in all Euro-
pean ami American countries where complete statistics of births are
available. Mulhall find- the ratio to he 1052 male to 1000 female births
PREPONDERANCE OF MALE BIRTHS 7
in Europe generally. For our presenl purpose it will be convenienl to
express the excess in a differenl form from this. The resull jusl cited
may be expressed by savin-- that, out of 2052 births, L052 are males
and 1000 females. We may also say thai 51.3 per cent arc male- and
1*. 7 per (-ciil arc females. Altogether, it seems to the writer thai the
besl expression for the sexes is the excess of male over female in 100
births. Since MitlhaH's conclusion implies that in 100 births 51.3 per
cent are males and 48.1 per cent are fcinalcs. the QumbeT expressing the
excess would be 2.6 per cent. The excess thus expressed is represented
by the symbol E- That is. we put Zi7 =tl xcess of male over
female births in a total of 100 births.
The slight variations in this excess found in different countries are no
greater than may be the result of accident. Although its value is so
nearly the same for different countries, there are still some circumstances
to be considered in connection with this definition. The most important
of these is that it is decidedly larger when still-births are included. In
France the male excess of such births is between 4 and •"> per cent of the
whole number, a proportion which is probably substantially the same in
most European countries. If we desire E m to express the physiological
probability of the production of a male child, still-births should be
included. We should then have, in France,
E"< = 2.93.
Although this expresses the proper physiological probability of the pro-
duction of male offspring, it will better conduce to our present operations
to consider only living births. For these we may take, as a normal
ratio, Mulhall's value, which will give
E in = 2.6
From the statistics of Massachusetts the ratios are found to be:
Living children, E m = 2.8 All births, E m = 3.3
It appears, therefore, that there is no material difference on the two
sides of the Atlantic.
3. IS THE RATIO OF MALE TO FEMALE BIRTHS THE SAME IN
ALL RACES?
So far as the author is aware the only races besides the Semitic for
which we have statistical data on which to base a conclusion are the
Mongolian race of Japan and the negro race in America. In the latter
we have only a limited registration of births, but sufficient to at least
point to a conclusion. The census of 1900 gives a registration record,
and a total record of births of children of the colored race as follows :
STATISTICS OF SEX
Bin Registration. Total.
Male 13,526 136,360
Female 13,244 136,626
The registration record shows ;i preponderance of male-- less than one-
half thai of tin' Semitic race; 1 1 1 * - census enumeration an actual prepon-
derance of female births.
The census tables also give the number <>t' negro children of each of
certain ages at a number of very earl] ag
Prom the census of L900 we find negro children:
Age. Male. ilo.
Under one month L0.200 L0.322
Under three months 32,840 33,353
Under one year 121,329 123.1 M
It is difficull to conceive of any cause why the data given to a census
taker should be systematically in error as to sex. The number under the
if one month can be nothing else than the number horn during that
month, less the deaths during the month. It', then, the preponderance
of male deaths is no greater in the negro than in the white race, it would
seem that an excess of male births does no1 characterize the negro face.
but rather the contrary. Moreover, the uniformity of the excess through-
oul the first year seems to confirm this conclusion. I find it also to be
confirmed by all the statistics of previous censuses since 1870. There is,
m general, in all these cases, a slighl excess of female negro or colored
children under one month of age, which is greater than would be due
to the excess of male over female deaths during the early period of life.
On the other hand, the recent Japanese census shows, in a number
of births exceeding one million, an excess of males practically the same
as in European countries. It would seem then-fore that there is no
difference in this respeel between the Semitic and the Mongolian race-.
A- the numbers relating to the negro race in America are not beyond
the possibility of doubt, and especially a- those of actual births regis-
tered -how a male excess, the suspicion, sometimes expressed, that this
-- may run through the whole order of mammals is at least worthy
..f examination. But the statistics of horses kept with much detail in
England and Germany .-how. on the whole, an almost equal division
between the sexes, the general tendency being toward a preponderance on
the female side.
It i- al-o a curious fact that in European countries where complete
statistics are available, the excess of male births is -mailer for illegiti-
mate than for legitimate children. The problem of explaining this
difference, which we can scarcely believe to he real, i- one which the
writer must leave to others.
UNISEXUAL TENDENCY 9
4. INQUIRY WHETHER ANY UNISEXUAL TENDENCY, PERMANENT
IN THE INDIVIDUAL IN EITHER DIRECTION, EXISTS AMONG
PARENTS.
The well-known fact being thai inequalities in the proportion of the
two sexes are almost the universal rule, some families consisting mainly
or entirely of male children, others mainly or entirely of female chil-
dren, the question hefore us is whether these inequalities are simply the
result of chance, or show unisexual tendencies on the part of the respective
parents. What we want is a criterion for distinguishing between these
two cases. To make clear the principle of the proposed criterion, we begin
with an illustration of its application. Let us suppose that we select at
random 100 families of two children each. Granting that this selection
corresponds to the general average, and leaving out of consideration the
small preponderance of male births, we shall have, in these families, 50
cases in which the first-born was a male, and 50 in which it was a
female. Of these two sets of 50 each, if there is no unisexual tendency,
the second child will be a male in one-half, or 25 cases, and a female in
the other cases. If there is no tendency of the kind sought, the final
result will be :
25 families of 2 females each;
25 families of 2 males each;
50 of 1 male and 1 female.
That is to say, in the great mass the number of families comprising
two children of the same sex will be the same as that of families com-
prising two children of opposite sexes.
Now let us suppose that, in consequence of some cause not known in
advance, some parents have a tendency toward the production of male
and others toward the production of female children. To fix the ideas,
let us suppose that in the case of one-half of the parents, which we call
Class A, there is a probability of three-fifths in favor of a male child,
and in the remaining half, called Class B, a similar preponderance in
favor of a female child. The method presupposes that we have no clue
to a decision as to which of these classes any given parents belong. The
first-born will still be a male in one-half and a female in the other half
of the cases. But, in case of the second child, the 50 parents of Class A
will have 30 male and only 20 female children. The 50 of Class B will
have 20 male and 30 female children. The probable distribution of
children between the two classes will be as follows: Of the 50 parents
of Class A. 30 will have male first-born children, and 20 will have female
first-born. The preponderance being the same in the cases of the second
10 STATISTICS OF SEX
child, the 30 males will be followed by 18 males and 12 Females. The
ales will be followed by 12 males and s females.
Tlic numbers will in- the same for Class B, only substituting female
for male preponderance. Tim- the total outcome will be:
i'Im-- \ Class B Total
:' malea l^ 8 26
Male-female L2 L2 24
Female-male L2 12 24
Miales 8 18 26
That i<. in 100 families of 2 children each, we shall have 52 with
children of • Bex, anil 18 with children i>f differenl sexes, instead of
50 "f each class.
The result will be yel more decisive when we consider families of a
ater number of children. Whatever the number of the family, tin'
theory of probabilities ami combinations gives a certain distribution of
male ami female children, which would he the mosi likely result of pure
chance For example, in families of ;;. ili,. most likely chance distri-
bution would he three cases in which the children were of differenl
Ben a f<>r one in which all three were of the Bame sex. But, were
tin-re any tendency among special parents to a production of chil-
dren of one sex, the proportion of families in which all three were of
tlie same ses would ho greater than that given by the law of chance
distribution. To Bhow the preponderance, let us suppose the tendency
to be the same as in the preceding example, and the number of families
of •'! children each to be 500, or 250 of each class. The result will be:
Class A Class li Total
3 males 54 16 7"
2 males, 1 female 108 72 isn
1 female. 2 males 72 lus 180
3 females 16 54 To
Assuming no unisexual tendency on the part of parent-, the probable
result wmild be a proportion of three families not all of the same Bex
one of the same sex. The results would then compare thus:
Families Actual Probable
3 children of 1 sex 140 125
3 children of 2 sexes 360 375
The re-ult is an excess of 1"» unisexual families.
Let us now pass mi to the genera] problem of which the preceding exam-
ples are Bpecial case.-. We have cited the well-known fact nf a amoral or
average unisexual tendency in the male direction among parent- of the
9 nitic race. The question before u- is whether this tendency of this i
- the Bame among all parent-. What we know to Btarl with is that, if
some parent- have a tendency greater than the normal to -produce male
children, then then- must be a corresponding tendency among other
6
UNISEXUAL TENDENCY 1 ]
parents to produce female children. It is this combined tendency toward
the production of children of one sex in some cases and tl ther sex in
other cases that, for the present purpose, I term unisexual.
The data for the investigation in question have been derived from two
sources. Mr. Hunt, chief of the Division of Population Statistics in
the Census Office, very courteously made for me a count of 2000 families
in which the parents were of various nationalities, enumerated in the
census of 1900. 1 have also had count,- made from a genealogy includ-
ing all the known families descended from one Andrew Newcomb, who
died about the year 1650, and winch served a valuable purpose as prob-
ably including a wider range of conditions than those all'ecting the fami-
lies (numerated in the Census. This was further extended to include a
great number of other family genealogies. The entire list may there-
fore be taken to include the widest possible range of ordinary conditions
which might affect the sex of offspring.
In the following summary of families, the first column of figures
gives numbers for the white families as supplied by the Census Office.
The second gives the corresponding numbers for families taken from
the genealogies. The third and fourth columns give the data for the
negro and Indian families, as supplied by the Census. The fifth gives
the sum for all the families.
This is followed by the probable numbers given by the theory of
chances, in case that there is no unisexual tendency among parents.
In each column the first line is the total number of families, the
second the number of those families of which the children are all of
the same sex, whether male or female. The following lines give the
number of bisexual families of each class, each division of the number
between the two classes being combined. For example, in families of
4 children, the line marked 3 and 1 gives the combined number of fami-
lies comprising 3 males and 1 female, together with those comprising
1 male and 3 females. The totality of the families enumerated is too
small to give any value to the separate enumeration of males and females.
A combination is therefore made in order to reduce the results to the
smallest number of distinct data. Thus the reader can see at a glance
to what extent, if any, a bisexual tendency can be found in the fami-
lies enumerated.
Families of 2 Children.
Class c ™ s G ™ Iogy Ne gro Indian Total Probable
Number of families 670 1051 56 6 1783
Same sex 322 547 30 2 901 892
Opposite sexes 348 504 26 4 882 892
12 STATISTICS OF SEX
In the case of the Census families, there are fewer pairs of children
of the Bame than of opposite sexes, which resull is the opposite of that of
a unisexual tendency. In the case of the genealogical families, the excess
is in the unisexual direction, bul is in pari dne to as excess of male
offspring recorded in the genealogies.
Families <<\ 3 Children.
«v.!m.s Guidon N,, « r " tod,M *°« 1,r " 1 '
Number 136 91 1 38 4 1392
Same sex 112 219 13 2 3HI 31s
Different sexes 324 695 26 2 1046 L044
In a chance distribution the unisexual families should be one-fourth
the entire Dumber. The actual Dumber is slightly below this, bo thai no
unisexual tendency is shown.
Families op i Children.
-vnius Geology »°S ro I "' 1 " 1 " ToM l>robab,e
Number 322 729 24 5 10S0
4 of same sex 10 110 1 L54 135
3 and 1 174 336 14 3 527 540
2 and 2 108 283 6 2 399 405
In a chance distribution the numbers of the three classes Bhould be
in tile proportion 1: 1:3. The actual Dumber of unisexual families in
the entire list is 19 in excess of the probable number. This excess is,
however, do greater than might well be the result of chance. The uum-
■ of families having 3 out of 4 children of the same Bes shows the
opposite of a unisexual tendency.
Families of 5 Children.
(J£m analogy Np 8 r " II " 1 " 1 " To,,u P»bable
Number of families 19] 620 18 3 832
5 of same sex 10 49 1 60 52
4 and 1 56 177 4 1 23s 260
3 and 2 126 394 L3 2 534 520
Here again there is an of 8 unisexual families over the oormal
Dumber of 52. The effect of a unisexual tendency would be to produce a
number -mailer than the probable one of families consisting of 3 children
of ■ . ami 2 of another, bul there is an excess in the ca.-i' <>f th.
families. We can. therefore, only attribute the deviation to chance.
UNISEXUAL TENDENCY
L3
Families of 6 Children.
Sua Geleafogy Ne G ro Ina,ftn T ''" Probable
Number of families 94 626 8 2 730
6 of same sex 6 16 22 23
5 and 1 18 141 1 2 162 137
4 and 2 43 287 6 ::::«, 342
3 and 3 27 182 1 210 228
The deviations from the norma] are in all cases rather less than we
might expect as a result of chance deviation.
Families of 7 Children.
CeneSa Genealogy Ne S ro Indlan ™ al Probable
Number of families 62 517 10 5S9
7 of one sex 1 11 12 9
6 and 1 4 59 63 65
5 and 2 23 165 3 191 193
4 and 3 34 2S2 7 323 322
Families of 8 Children.
Number of families 24 417 6 447
8 of one sex 3 1 4 3
7 and 1 1 27 28 28
6 and 2 7 85 3 95 98
5 and 3 12 185 1 198 196
4 and 4 4 117 1 122 122
Families of 9 Children.
Number of families 8 306 3 317
9 of one sex 1 1 1
8 and 1 1 9 10 11
7 and 2 52 1 53 45
6 and 3 3 99 102 104
5 and 4 4 145 2 151 156
Families of 10 Children.
Number of families 5 295 300
10 of 1 sex 0.6
9 and 1 7 7 5.9
8 and 2 1 37 38 26.3
7 and 3 3 67 70 70.3
6 and 4 117 117 123.0
5 and 5 1 67 68 73.9
! j STATISTICS OF SEX
Families oe i 1 ro 16 Chii dren,
809 families of 1 1 children: oei • Prob.
11 of one Bex 0.2
i" and i 6 2.2
9 and 2 16 11.2
v and :: 32
md i 68
tnd :■ ^T 94.2
L24 families of 12 children
12 of one Bex 0.1
ll and l 2 0.7
.ml 2 i" ll
9 and 3 18 13.3
v and 4 39 30.0
7 and ." 35 48.0
6 and 6 20 28.0
50 families ol 13 children:
13 of one Bex 0.0
13 and 1 0.2
11 and 2 1 0.9
10 and 3 » 3.4
9 and 4 9 8.7
8 and R 15 15.7
7 and 6 21 21.0
25 families of 14 children:
I 1 of one sex 0.0
13 and 1 0.0
12 and 2 0.3
11 and 3 3 1.1
10 and i 5 3.0
9 and 5 9 6.1
8 and 6 6 9.1
7 and 7 2 5.1
1 1 families of 1 5 children:
1 5 of one Bex (| 0.0
1 1 and l
13 and 2 0.1
12 and :: 0.3
II and 4 1 0.9
10 and 5 2 2.0
9 and 6 5 3.4
8 and 7 3 11
7 families of 16 children :
16 Of one sex 0.0
15 and 1 0.0
1 1 and 2 0.0
13 and :: 0.1
12 and 1 "I
1 1 and 5 1 1.0
10 and 6 1 1.7
9 and 7 :: 2.4
md 8 2 li
UNISEXUAL TENDENCY j-
hi the case of families of 7 or more, a unisexual tendency would be
shown by a deficiency in the numbers of families with a nearly equal
number of males and females. These are given in the last line of each
series of families. We lind such a delieienev to he actually shown. The
numbers are :
Actual: 101, 35, 4G, 19, 25, 4, 7, 2
Probable: L07, 39, 51, 23, 23, 7, 5, 1
Actual sum = 239
Probable sum — 256
Deficit 17
This deviation is not large enough to base a conclusion upon. It is
partly due to a deficit in the reports of female children in the genealogies,
the respective recorded numbers of male and female being:
Number of male children 3,339
Number of female children 3,020
Excess of male children 319
It thus appears that about 110 males are reported against 100 females;
in other words, 52.5 per cent of the whole number are males. This
excess over the normal is to be attributed to the greater difficulty of trac-
ing female than male children, owing to the greater liability of the former
to be omitted from a record, especially when they die young.
We shall therefore expect to see a slight unisexual indication in the
numbers from this cause alone, so that we may regard the deviation as
explained without supposing any actual tendency of the kind sought.
The general result of the count of 2838 families embracing 13,257
children is that the distribution of male and female offspring follows the
statistical laws of chance within the limits of probable deviation, the
actual deviations being as great in one direction as in the opposite one.
Consequently, there are no unisexual tendencies on the part of parents
sufficiently great to be of practical importance. We are not, however,
justified in concluding from these numbers alone that there can be abso-
lutely no unisexual differences in the human race, nor that no possible
conditions are productive of such a difference. Our conclusions only
preclude conditions affecting the sex of a child which may occur with a
certain frequency. For example, if one-tenth of the parents in the whole
list practiced any hain't, or possessed any characteristic, which would
lead to two-thirds of their children being of one sex, the effect would
show in the statistics. But if only a single pair of parents in the whole
],, STATISTICS OP SEX
li-t had ;i unisexual tendency, or if, in the great majority of cases, this
tendency was exceedingly small, the effect would not be shown.
AtS to thi> latter case, the Fact of any unisexual tendency, however
minute, would be of greal scientific Interest, bu1 of no practical im-
portance. It would hardly be worth while for any parent or any com-
munity to lay greal stress on any cause which would result only in
increasing the chances of male or Female offspring by 3 or I per cent.
It seems highly improbable thari an\ ver$ rare or highly artificial cause
would produce a unisexual tendency, if no ordinary cause produced it.
The absence of any strongly marked unisexual tendency in the Families
we have examined, therefore, justifies us in concluding, at Least with
a high degree of probability, that the causes of sex are beyond artificial
control.
Till: UNISEXUAL TENDENCY IN MULTIPLE MIRTHS.
We have next to consider 6 sort of family in which the condition- arc
peculiarly Favorable for drawing conclusions on the genera] question of
the cause of sex. The.-' are Families consisting of children of a single
birth in twins or triplets. Considering first the case of twins, we begin
with the effects which would result on two extreme hypotheses as to the
cans \.
Eypothe8I8 I. The distinction of male and female exists in original
germs, antecedent to conception, 'presumably supplied by the father.
In this case we should find the same random distribution of twin chil-
dren between the two aexes that we find to exist in Families of two. In
four births of twins we should have oi E two male-, one of two Females,
and two bisexual, only in two ways could this conclusion be avoided.
one i- by supposing that a germ of either sex is more likely than nol to
have one of the same sex in physical juxtaposition with it. This vie*
in- inadmissible because, even it' such juxtaposition did exist in any
case at any moment, it would not be permanent. The other supposition
i- that at certain periods there is an abnormal excess of male germs and
at other periods a similar excess of (''■male germs in the Bame father. In
the ale, -He.' of permanent unisexualism on the part of anj one Father,
which was shown in the preceding section, such an inequality can uot
be permanent. It therefore seems to me that this supposition is also
too artificial and unlikely to be considered. We may therefore consider
the statistical distribution of twin children between the sexes to afford a
test of the above hypothesis.
STATISTICS OF MULTI PLK IMIiTHS i;
Hypothesis II. Sex is entirely determined by the conditions to
which the germ is subject during Hi* 1 early stages of its develo/tmrni.
All these conditions are the same ab initio for the tun members of tin-
pair. The result of the hypothesis would therefore be thai twin children
would always be of the same sex.
The statistics show that neither of these hypotheses is correct taken
singly; the actual result being an intermediate one between those of the
two hypotheses. A child of one sex is more likely than not to have a
twin of the same sex ; but there is only a certain preponderance of proba-
bility for this.
The negation of the first hypothesis leads to the conclusion that the
original germs supplied by the father, if not completely asexual, can at
most have no other sexual quality than a slightly greater tendency to
develop into one sex than into the other. While the existence of such a
tendency is not out of the question, the more likely conclusion would
seem to be that the part of the father is completely asexual, and that the
determination of sexes is entirely the function of the mother. It is true
that this contravenes certain supposed conclusions from statistics which
will be considered in the next section. But these seem to me open to
misconstruction.
We pass next to the actual numbers shown by the statistics of France
and Germany. In the following table the first line shows the number of
births giving rise to two males; the second to the number of bisexual
births; the third of births of two females. In the next two lines are
the total number of male and female children resulting from all the
births. The preponderance of males is, on the whole, the normal one,
showing that, in the production of twins, there are no causes affecting sex
which act in any way differently from those in the ordinary cases. Then
is given the normal number of bisexual births as it would have been were
the determination of sex, in the case of the two children, completely inde-
pendent, as required by hypothesis I. The proportional deficiency of
the actual number of bisexual births over this probable number may be
used to define the unisexual tendency in the case of twins. The observed
fact may be set forth thus : The probable proportions of unisexual and
bisexual twins, if the sex of each child were determined independently of
the other, and the percentage of pairs of the several classes would be:
2 m., 0.260; m. and 1, 0.500; 2 f., 0.240,
while the actual proportions are:
2 m., 0.332; m. and f., 0.354; 2 f., 0.314.
18 STATISTICS OP SEX
Si \ OP Tw a IANOB \M' Bl 1:1 [N.
i Berlin Total
is:,:.
2 males 9 2,968 L2.505
cual 9,826 3,489 13.315
2 females 8,949 2,862 11,801
: males 28 9,426 38,826
Total females 27,724 9,193 36,917
M. to 1"" P 104.2 102.6 L03.8
2.1 L.2 1.9
ial bisexual 14,149 1,603 18,752
Deficil I U14 r,.437
The formal discussion, by algebraic methods, of the unisexual ten-
dency Implied in these uumbers will be found in the Appendix. Thi
methods are not, however, necessary to convey an idea of the principles
by which the results are t<> be explained. What makes a discussion of
these principles of interest is thai the numbers derived from the statis-
tics of twins may be applied to the case of triplets, and a comparison of
tlir actual statistics of triplets with those derived from the statistics of
twins will be of interest.
The processes which we presuppose are these: During an unknown
period of time, commencing with the momenl of conception, the two
■ms are exposed to a Beries of common influences, either in the male
or female direction, tending to make them of the Bame sex. As we can,
without appreciable error, make abstraction of the small normal prepon-
derance toward the male sex, we may say that, in the general average, this
unisexual tendency will be as often in one direction as in the other. Thus,
in one-half the Ca8e8, which we term group A. tin' common influen<
preponderate in the male direction, ami in other cases, which we call
group B, in the female direction.
Bui the-e preponderating influences do no1 completely determine the
There are accidental causes operating differently on the two grow-
ing organisms, which may result in their becoming of opposite sexes.
Numerically Btated, the conclusion to which we are led is that the
statistics of twins may he explained by supposing that, in Group \
there is ;i probability of 0.7*3 in favor of either of the organisms taken at
random becoming male, and therefore 0.23 in favor of its becoming
female; while iii -roup !'» there are similar probabilities in the oppo-
site direction. Thi-. I say, is a conclusion from the statistics of twin-.
raming that there i- no interaction between the two organisms tending
to make them of the same -e\. [ may remark, however, thai this assump-
tion in t l of twins would have no special significance. The com-
STATISTICS OF .Ml'LTII'LK MIRTHS 1!)
lunation of probabilities would lead I" the same resull whether we -up-
posed it or not. The main point is thai there is some preponderating
tendency of the pair of organisms towards one aes in some cases and the
opposite sex in the remaining cases. Stating probable results in per-
centages, tliev would be :
In group A, probability of 2 males 0.77 = 59.3 per cent.
In group A, probability of 2 females 0.23 = 5.3 per cent.
Total unisexual percentage G4.il
Bisexual 35.4
In group B the results would be the same, interchanging male and
female.
These numbers, it will be noticed, show the percentages actually given
by the statistics. The mathematical method developed in the Appendix
shows that the results may be accounted for by assuming a certain uni-
sexual tendency represented by a fraction t Inning the value
a = 0.27
This coefficient may be considered to express the efficiency of all tbe
causes tending to produce sex which are common to the two twin mem-
bers of the family. In other words, assuming this unisexual coefficient,
the result will be 77 per cent of twins of the same sex, and 23 per cent
of twins of different sexes, these being the actual results of observation.
A most interesting fact is that, by the methods developed in the Ap-
pendix, we may apply this coefficient « to determine how the sexes in
families of triplets should be divided. There are two ways of proceeding.
We may assume the unisexual tendency to be the same in triplets as we
have found it to be in twins, as naturally ought to be the case. From
this we can determine what proportion of triplets should be unisexual,
and compare the result with statistics. The other method consists in
determining the value of the unisexual tendency from the statistics of the
triplets in order to see how much it differs from that determined from
the statistics of twins. Adopting the first method the problem is: We
have three organisms subject to such conditions that, in the case of each,
there is a probability of 0.77 that it will prove of one sex and of 0.23
that it will prove of the other. What are the respective probabilities
that the three organisms will be unisexual ; that is, all three of the same
sex; and that one shall be of one sex and two of the opposite? These
probabilities are found in the x4.ppendix to be :
Unisexual: percentage, 46.9
Bisexual: " 53.1
20 STATISTICS OF SKX
To compare the statistics I have collected the ai see of triplet- found
in the French and German tables of births with the following results:
Krnuce l • -rlln T .
185:f 1 A l
:; males 342 32 374
8 Females 304 28 332
Total unisexual t',46 60 706
exuaJ tit;: 43 710
Observed proportion of unisexual triplets 49.9 per cent.
It appears from these numbers thai when we compare the probabilities
derived from the case of twins with the actual facts in the case of triplets,
there is a discrepancy. From the facts in the case '>!' twins, we Bhould
conclude thai 46.9 per cent of all triplets Bhould be unisexual; we actu-
ally find thai t9.9 per cenl is unisexual, an excess of •"> per cent.
The discrepancy may take the other form by determining the amounl of
unisexual preponderance in the case of triplets as we have done in the
case of twins. This preponderance is 0.79 instead of 0.77. That is to
say, grouping the triplets as we have the twins, there is a probability of
0.79 thai any one organism of a triplet of group A will develop into a
male, and thai one of group B will develop into a female. The coefficient
of unisexual tendency is, therefore, for triplets,
a = 0.29
Now, we Bhould suppose, a priori, thai the ratio of the unisexual pre-
ponderance to the effects of the accidental causes which finally determine
the sex would be the Bame with twins and triplets. It is true thai the
discrepancy between 0.2*3 and 0.29, or between L6.9 and 19.9 per cenl is
n<>t greater than mighl easily have been the result of fortuitous deviation.
Still it i- larger than we should expect, [f we may regard it as expressing
a real law. we may Buppose that, besides the independenl causes at action
tending toward one sex or tl ther, there i- an interaction between tin 1
two organisms, by which the Bex of one influences thai of the other in
it- own direction.
Apart from this, the general conclusions from triplet- confirm that
from twins i — there are not male and female germs. It would seem
that we have in this a practically conclusive negation of the theory of
completely determined Bex in the original germs and may provisionally
ept that of complete asexuality on the part of Buch germs, Bubject,
however, to farther statistical tests.
STATISTICS OF MULTIPLE BIRTHS
'.'I
6. PROCESSES IN THE DETERMINATION OF SEX SUGGESTED BY
THE STATISTICS OF MULTIPLE BIRTHS.
The view that, if the sex i> not completely determined in the original
format ion of a germ, it must he determined at some definite momenl of
development — that there can be no intermediate state between complete
asemality and complete sexuality — is one which, at first sight, seems
almost axiomatic. And yet, the preceding statistics of multiple births
seem to show that such is not the ease, and that there may he a series of
causes acting first in one direction and then in the other, each of which
tends to make one sex or the other more probable until, gradually, the
sex is definitely determined. An analogue to this determination by a suc-
cession of accidental causes may be constructed in the following way:
Let A lie a large pipe or aqueduct, from the mouth B of which a stream
flows into a gradually widening river V. At a certain distance below
the exit B the river is divided into two branches by a promontory P. On
one side of this promontory, which we may call the male side, the river
is slightly broader than on the other. Between the exit and the promon-
tory, the river flows over a rough bottom with many eddies, but the ulti-
mate result must be that every drop of water which comes from the
conduit ultimately passes on one side of the promontory or the other.
But the side on which it shall pass is not determined at any one moment.
As a drop, or, to give the analogy a more complete form, a small particle
suspended in the water, leaves the conduit, it is equally likely to pass into
one branch of the river or the other. If it chance to incline to the right
after leaving the conduit, there will be a greater probability of its passing
into the right branch, but this will be only a probability until a certain
point of the course is reached. A particle reaching the point M, for
example, will be likely to go into the female branch, but yet may be car-
ried by an eddy across to the opposite side before it reaches it. One at
E", although farther down, will still be uncertain ; possibly its course may
not be decided until it almost reaches P. A particle on one bank or the
other will be more and more likely to pass into the corresponding branch
the farther down it is found. When the particle once crosses one of the
dotted lines PE and PS the branch it will take will be completely deter-
mined.
STATISTICS OF SEX
\ case of twins or of triplets has its analogue in tin 1 case of two oc
three particles emerging from the conduit in contiguity. They are more
likely to keep together and enter the same eddies than if thej were widely
parated in the beginning. To -peak with uumerical exactness there is
a probability of 0.7*3 thai they will pass on the same Bide of the promon-
tory and of 0.23 thai they will separate. In the case of triplets the cor-
responding probability would be 0.79; bu1 these are only probabilities.
V any moment any two particles may widen their distance and be drawn
into differenl parts of the stream, never to reunite.
We may thus say thai the question, which branch of the river a particle,
emerging from the conduit, is to flow into, will be determined by a 31 1
of accidents tending in our direction or the other; and the mosl plausible
conclusion from the statistics of twins is thai Bex is determined in an
analogous way.
7. INFLUENCE OF THE AGE OF THE PARENT ON SEX.
The changes produced by age in the human system are such thai
may mosl plausibly look to them as causes affecting the ses of offspring.
The question of the influence of the age of the parenl has been Btudied
by several investigators, especially by Rosenfeld, Sadler and Bertillon. I
have not been able to refer to the original work of Bertillon and shall
therefore confine myself to citing, in its proper place, one of his conclu-
sions bearing on the case. Dr. Rosenfeld gives the follow ing table of the
sexes of re than thirty thousand births in Vienna, arranged according
to the age of the father. I add the percentage h' ,„ for each age:
Viian \ St \i ibth - op Births.
Age Of tathel Male children Female children LOO II 1 A.',,,
Under 26 873 767 113.7 6.5
to 30 <;.090 5,717 10G.5 3.3
30 35 11,987 11.291 106.2 3.1
1" 3,606 3,559 101.3 0.6
10 622 502 128.9 in.;
Over 50
The table Bhowe a decided pre] derance of male children in the <m-> v
of young and old fathers as compared with those in middle life. The
conclusion thence drawn is that male unisexuality is at its maximum in
young and old peopl<
from the statistics of Norway, reached the same conclusions
irds young father-. bu1 the opposite as regards <»hl ones. Hi-
numbers for the ratio of male to female births, arranged according to
the the father, are as follow - :
INFLUENCE OF THE AGE OF THE PARENTS •.<;;
Norwegian Statistu 8 of Bim as.
Age of father 100 U : W A,„
Below 20 117.0 7.9
20 to 25 101.5 0.7
25 30 109.0 4.4
30 35 105.9 2.9
35 40 102.6 1.3
40 45 104.6 2.2
45 50 103.8 1.8
50 55 98.4 —0.8
55 60 97.9 —1.0
Over 60 99.8 —0.1
Roscnfeld ^ivcs a similar classification, arranged according to the age
of the mother, as follows :
V i i:\.x.\ Statistics of Births.
Age of mother Male children Female children Urn M : F E m
Below 17 19 7 271.4 46.1
17 to 20 366 341 107.3 3.6
20 25 4,444 4,161 106.8 3.3
25 30 7,287 6,759 107.8 3.7
30 40 9,907 9,356 105.9 2.8
Over 40 1,412 1,396 101.1 0.5
The enormous preponderance of male births in the case of mothers
under 17 years of age is probably the result of accident and not expressive
of a general law, the births, 26 in number, being too few to base a deter-
minate conclusion upon. If we combine all the mothers under 20 years
of age, the result will be :
100 M :F = 110.6; E m =5.0
The numbers now show a marked preponderance of male children
borne by very young mothers, which drops to the normal at the age of
20 and falls below it at the age of 40.
It may be noted in this connection that the ratio of male and female
children is, in the general average, somewhat above the normal, possibly
indicating an imperfection of the record by not including all female chil-
dren. This, however, will not alter the conclusion.
All these conclusions as regards the age of the parent seem to me to
lack a solid foundation, from the fact that the ages of the two parents
are not completely distinguished. I shall discuss this difficulty after
setting forth the results of the genealogical statistics collected by myself.
The effect of difference of ag-e between father and mother was investi-
gated by Sadler, who laid clown the general law that the older parent has
2 | STATISTICS ()!•• si:x
a preponderating influence in the direction of determining children of
his or her own Bex. Bui Ahlfeldl reached the opposite conclusion, find-
ing that when the father was i v than l" years older than the mother,
there was a preponderance of female children instead of tin- normal excess
male children. Hi- oumbers are, however, i"<> few to base any con-
clusions upon; and the same is probably true of the statistics used by
Sad
I have nut attempted to investigate this subjeci by age because the
data are qoI at hand for the purpose. In-trad of < I < • i 1 1 u r this, I have taken
the order of progression of children in families as found in the gene-
alogies already cited. In each family the Bex of the several children
was tabulated in the order, firsl born, Becond born, third born, etc. Then
the total number of first-born children of each Bex, the Becond born, and
bo "ii. was taken. The results are Bhown in the Eollowing table in which
the firsl column gives the order of birth. This [g followed by the
respective number of male and female first-born children, in the same
line the Qumbers are given for the Becond hnrn. and so on. In families
i>r more than 1 I children, the fourteenth and those following are all
tabulated together, a- their separate numbers are too Bmal] to base a
conclusion upon.
The fourth column gives the total number of children; the fifth the
38 of males, and the sixth the percentage of this excess.
Comparison or Male lnd Female Children in the Obdeb of Birth
i\ American Families.
i irder
• •f
Birth
1 3,906
- 3.261
3 2.605
1 2,145
1 ,766
1,406
T 1,102
8 782
9 I
in 4"7
11 246
12 142
13 71
1 1 to 17 :, i
iles
::.265
2,987
2.532
2. "2 1
1,651
1,338
1,020
::,\
513
356
221
108
52
12
Sum
7.171
6,248
5,137
1,169
::.H7
2.7 11
2,122
L.536
1,105
763
167
250
123
93
Excess
of
M;ilf>8
274
73
121
115
68
82
28
7:.
51
25
34
19
9
1.1
1.4
3.0
3.4
2.5
3.9
1.8
7.3
6.7
5.3
L3.6
1 5. l
9.8
Corrected
&m
6.7
2.2
—0.8
0.8
i 2
0.3
1.7
—it 1
5.1
4.5
3.1
11.4
13.2
7.6
Total. ..IS
35,345
L619
4.6
INFLUENCE OF THE ,\(JE OF THE I'AKENTS 25
It will be seen thai the cm-i— of male births in the general average
markedly exceeds its normal value. We musl regard this divergence aE
unreal and attribute it to the greater liability of a female child to be
omitted from the record. As this omission would be probably about
equal in the case of all the successive children, w<> may assume that the
values of E m are all equally in error from this cause. The normal value
from the statistics of birth being about 2.4, while the count gives !.«'.,
we subtract the excess, 2.2, from each separate value of E m and thus
obtain corrected values of the percentage of excess, which are found in
the last column.
It will be seen from the numbers of this column that the excess of
males among first-born children exceeds G per cent. This shows that
there are about eight males to seven females of this class. But, in the
case of the second child, the percentage of excess drops to 2.2, which is
slightly blow the normal and, in the case of the third child, it becomes
negative, showing that, after we correct the supposed defect of the
record, there is actually a slight excess of female births.
The rapidity of the drop from 6.7 in the case of the first birth to 2.2
in the case of the second and then to a negative quantity in the case of
the third, seems to show quite conclusively that the excess of males in
the number of the first-born children is not attributable to the age of the
mother, but to the fact that it is a first child, irrespective of age. That
the fall is too rapid to be the effect of age is shown in the following way :
The difference of age at the birth of the first and third child is not
likely to have been more, in the general average, than three years. Xow
a drop of -A in the percentage in three years would imply a drop of
twice this amount between the ages of 17 and 24, which we may take
as the probable range in the case of a first child. The approach to
uniformity in the percentage in these cases where the marriage must have
been at such different ages, precludes the supposition that age is the main
factor in the case.
Continuing our study of the table, we find a remarkable uniformity
in the number of male and female births up to the eighth child. In
the case of the second child the excess is still fairly well marked. Thus
we may conclude that the tendency toward male excess, though
greatly diminished, is probably not wholly obliterated in the case of
the second child. But from the fourth to the eighth inclusive, the devia-
tions are so small that we may regard them as the effects of accident. In
the case of the six children from the third to the eighth, it would seem
that the birth of the two sexes is equally probable. Then, from the
o(j STATISTICS OP BEX
ninth child onward we find an excess of males which generally exceeds
the normal all through. Bui it L8 nol ai all certain that this arifi
from a onisexual tendency in the case of older parents. It Is quite
possible thai it may be attributed to first-born children after remar-
riage, the table having been constructed without any reference to the
mother ami giving only children in Families by the same father. It
must also he note,! that the- total numbers beyond the tenth child be-
come too -mall to predicate a very certain conclusion upon. A more
complete investigal >f the subjecl will therefore be necessary before
it can he Baid with certainty whether the result- derived by Rosenthal in
the Vienna statistics in the case of ''Id parents are correct, or whether
we here have to do with the first-born of second or third wives.
It mighl appear, at lir-t Bight, that these statistics do not decide
whether the variation in the proportion of male and female, as the
family advances in number, are due to the male or female parent. Hut
a consideration of the ratio between the number of acts on the part of
the two parents who are concerned in the case, decide- the probabilities
m favor of the mother.
A more conclusive Investigation than has as yet I n made i- m
-an to absolutely decide whether. as ha- been 31 d in this paper.
the part id' the father is completely asexual. To make this investiga-
tion, it i- necessary to compare the statistics of births by mothers of one
and the same class with father- id' different age8. Since the ratio of
male to female i> the same, at leasl from the third to the eighth birth,
the preferable method is to confine the investigations to those births
which may he grouped all together, SO far as the mother i- concerned.
We then compare the -e\ of each child of this cla8S with the age of it-
father ami. by a sufficienl accumulation of cases, ascertain whether
the ratio varies with that Bf
v EXAMINATION OP CERTAIN OTHER CONDITIONS WHICH HAVE
BEEN SUPPOSED TO INFLUENCE THE PRODUCTION OF SEX
It has sometimes been supposed thai the destruction of an important
fraction of the male population of a country by war. such a- ha- occa-
sionally been known in history, ha- resulted in a greater preponderate
male offspring in the country so affected. A very slighl analysis of
the supposed cause will -how that tin- proposition belongs to a class
which require very Btrong proof. Granting the truth of the proposi-
tion: since those who were killed in war could not subsequently have
SUPPOSED INFLUENCE OF OTHER CONDITIONS
taken part in the propagation of the race, it would follow that those who
returned in safety showed a unisexual tendency in the male direction.
Thai a tendency of this sort could he produced in our man by tin-
mere death of another is a notion that hardly Deeds to be refuted, [f
such an ell'eel is real, it would therefore have to be the resull of priva-
tions and other evils suffered in war, and ool of the mere destruction of
life, a process which Nature is carrying on all the time. The question
would then be whether privations and sufferings generally produce a
male unisexual tendency. This idea seems to be conclusively nega-
tived by the fact that the male preponderance is not shown to be a
function of the wealth of the country, or the condition of the great
mass of the population.
Nevertheless, in order that none of my conclusions might be based on
a priori reasoning, and in order to answer the objection that there may
be something peculiar in the effect of privations suffered in war, which
differentiates them from other privations, I have examined the popula-
tion statistics found in the New York census for 1865, and the United
States census for 1870, enumerating the sexes of children who, from
their ages, must have been born about the close of the civil war. In
the case of the United States census I confined the examination to the
Southern States, because there it was that the suffering and privations
were the greater. The result, comprising enumeration of the sex of
more than 100,000 children, showed that the male preponderance was
as nearly as possible the normal one, and that not the slightest influence
of the war could be detected.
It has also been maintained that the practice of polygamy has been
found productive of the unisexual tendency in the female direction.
The data for deciding this question are insufficient; but I find that, in
the only region of the United States where such an effect would be
likely to be observable, there is the usual preponderance of male births.
Analysis w T ill show that this proposition also belongs to the most im-
probable class. The only polygamous practices which could reasonably
be supposed to affect sex are so far from rare that any unisexual tendency
arising from it would be brought to light by a very slight examination.
The author believes that the preceding paper contains sufficient matter
to disprove the supposition, without the necessity of further inquiry.
STATISTICS OF SEX
9. SUMMARY OF CONCLUSIONS
1 do qoI present the following summary of conclusions as being, in
all cases, bo well established as nol to be worth] of farther investigation.
Whether well or ill established, they are those indicated by the statis-
j, and I earnestly hope thai other investigators, more especially con-
rned with the subject, will take it Qp with more extended data and
tesi each conclusion separately. With this proviso we maj Bay that the
following propositions arc indicated by the statistics with a greater <>r
less degree of probability.
I. The preponderance of male over female births probably varies with
the race. Although remarkably uniform in all branches of the Semitic
pace, it -mils tn be cither uon-existenl or quite small in the Negro race.
II. There are n<» important differences ;i- regards capacity for pro-
ducing children Of "lie ses rather than the other which are permanent in
the individual. All father- ami all mothers are equally likely to have
children of either sex, excepl for the slighl variation- that may he duo
t.. age. In view of the great variety of conditions on which this con-
clusion is based, it seems in the highesl degree unlikely that there i-
anv way by which a parent can affect the sex of his or her offspring.
III. The most natural inference from all the statistical data i-
that the functions of the father in generation are entirely asexual,
the sex being determined wholly by the mother. If so, it cannot he
-aid that one father i- more likely than another to have children of
either sex. This conclusion requires to be tested by making a classifi-
cation of the -e\ of third horn and following children according to the
of the father.
IV. The Bex is not absolutely determined at any one momenl or by
any one act. hut i- the product of a -eric- of accidental causes, BOme
acting in one direction and some in another, until a preponderance in
one direction finally determine- it. 'The statistics of twins ami triplets
- ; ow \,ry Btrongly thai these accident- occur after conception,
hut throw no lighl upon the question of the time which they occupy.
\ . The jir-t horn child of any mother i- more likely to he a male in
the proportion of aboul s to ', . There i- probably a -mailer preponder-
ance in the case of the Becond child. Bui there i- qo conclusive evi-
dence that, after a mother ha- had two children, there IS any change
in her tendencies.
VI. The observed preponderance of male births in the Semitic race is
due mainly to the unisexual tendency of the mother in the ca-e id' a lir-t
child.
MATHEMATICAL THKORY &g
APPENDIX.
\l \ in km LTIOA1 'I'll EOBY OP THE EFFECT OF A [JNI8EX1 LL TENDENCY.
The statistical theory on which the preceding research is based, being
presumably susceptible of other applications than that here made, will
now be developed. So far as generality is concerned, nothing will be
lost by taking the special problem, considered in section I V preceding,
as a basis of investigation. The data of the problem will be as follows:
1. An indefinite number of pairs of parents, each pair of which may
have an indefinite number of children of either sex. The treatment
of this subject will include the general ease of an indefinite number of
causes, each of which may, on each trial, be productive of one or the
other of two different effects.
2. Taking the general average of the whole mass of couples, there is
a certain normal probability, p, that a child, taken at random, will be
male, and the probability 1 — p that it will lie female.
.'!. It may bo that this probability is the same for every individual
couple of the whole mass.' But it may also be that, for some of the
couples, the probability is greater than p. In this case it will neces-
sarily follow that for certain other couples the probability is less than
p, the latter quantity being the average for the whole mass.
4. In order not to complicate the problem too greatly, we shall sup-
pose that each of the individual couples belongs to one of three classes;
a class for which the probability of having a male child has the normal
value p, another for which it is greater than p by an unknown quantity
a, and a third for which it is less than p by the same quantity. We
designate these classes by A, B and C ; A representing couples with
probability p-\-a; B, those with probability p; and C, those with the
probability p — a. The numbers of classes A and C are necessarily equal.
Let us put:
// . the fraction of the total belonging to the two equal classes A and C ;
/;', the fraction of the whole mass belonging to class B.
We shall then have
h + 7i' = l.
Proceeding according to the method of probabilities, we suppose a
parent couple taken at random from the mass. The respective proba-
bilities that this couple will belong to the classes A, B and C are
y 2 K 1i' and y 2 h.
The probabilities of a male child are, in these several classes :
For class A, p -\- a.
B, p. (1)
C, p — a.
30 STATISTICS OF SEX
Then by the princip the tl riea of probabilities, if a couple
taken at random from the whole mass, the respective combined probabili-
that thf couple will be of one of the classes, and the child a male,
will
lu class A. ' Ji ( ji -f- «■ ).
B, h'p. (2)
« . - ,/,(/,-„,.
of which the sum is />. as it should be.
The problem before us is to find a criterion foT deciding whether
the quantity <i. which we may consider as the unisexual factor, and which
-iall call the coefficient of unisexuality, is <>r is not of appreciable
oitude. Su.-h a criterion Ls afforded by a count of males and females
in families of two <>r more children. The theory requires that, in a
family of a giveu uumber of children, we express the probable respective
numbers of males and female- in terms of the factor a.
The problem now assumes the following form: A parent couple,
taken at random from the whole mass, had n children; what is the
probability thai s of these children will be male- and n — g females?
1 Ising the notation
["] =
_ n(n— 1) (to— 2) .. . ( n — s+ 1)
t . 2 . 3 . . . s
we have the well-known theorem that, if the probability of an event on
a Bingle trial is /<. the probability of its occurring s times on n trials is
P =[*]/(!-,»)-
Putting for /< the three value- of the probabilities given in (1) we find
that the probabilities in question are ;
For class A, I""!*/' + «)' (1— /' — «)""•
For class II, f" w "| p« < 1— p)"— (3)
For class C, PH (/> — ,,)' (l—p + a )— g
Multiplying these expressions, as in (2), by the respective factors
'_.//. h! and '_•/'. putting for brevity
n — s = r
1 — V = 7
MATHEMATICAL THEORY ;]
.mil taking the sum of the products, we find the probability thai a
family of n children taken at random from the whole ma-.-, will com-
prise s males and r females to be
•' ■:' = { *H(J> + a)'(q-ay + <j>-*y{q + a)') + h'p'q' } [*] ( i
This expression may now be developed in even powers of a, the coeffi-
cients of the odd powers all vanishing. In the form
P;:i ) = (A + A 2 a 2 + A 4 „'+ ..) pi
the values of the first two coefficients are
A =(fl + h' ) p'rf —pq r
A, = hp- y- (f [J] + cf- [|] - rapg )
= J A { n(?i — l)]r—2(n — l)sj> + s(8 — 1) };/ ~ V - -
For our present purpose these terms suffice. To investigate uni-
sexual deviations it will also lead to no appreciable error to suppose
p = q = i/ 2
The value of P£i , that is, the probability that a family of n children
will consist of s males and r females now becomes
We may use this formula to express the probability in question for
the case of a family of any number of children, distributed in any way
among the two sexes. We shall now form these expressions for fami-
lies of various numbers of children. In doing this families in which
the sexual distributions are the reverse of each other will be combined.
For example, the equal probabilities that a family of five will be wholly
male and wholly female will be added into one sum; as will the proba-
bilities of 4 of one sex and 1 of the other, whichever sex it may be.
The pair of probabilities thus combined would be rigorously equal
when, and only when, there is an equal probability of male and female
children. But not only is the error involved in the assumption of
inequality unimportant for the present purpose but, resulting as it does
in giving too small a probability for a preponderance of male and too
large for the preponderance of females, it is nearly self-compensatory
when we combine families of inversely distributed sexes.
STATISTICS OF SKX
The computation of the formula (5) is shown in the following table.
To enable the essential aumbers of this table to I"' understood without
the oecessity of going through all the mathematical formulas, 1 shall
state their significance and application. <>n the lefl are found certain
possible values of /'. the number of children in a family from 2 to L2
inclusive. Bach block of aumbers connected with a single value "f n
relates Bolely to families of thai number of children.
In the nr\t column are given all possible distributions between the
two Bexts which the family can have. Complementary families as re-
de B( \ are combined. For example, a family <>f three children tnusl
consist either of three children of one Bex, whether male or female, and
none of the other; or it comprises two of one sex, whichever it may be,
ami one of the other. 'Luc two lines correspond to these cases.
The three following columns contain numbers employed in comput-
ing the probabilities as found in the expression on the right. The
denominators of the fractions which enter into these probabilities are
written after the sign -j- of division, and, in each sel relating to one
value of n, the fractions are reduced to the lea.-t common denominator.
but not to their lowed term-. This form of expression is \\>ri] for
convenience in tracing the law of the numbers and continuing the table.
The probability is expressed as the sum of two terms: one a pure num-
ber; the other a coefficient of the factor ha. The purely numerical
term shows what the respective probabilities of the division of Bexes
found in the Becond column will he in case of no unisexual tendency.
For example, in a family of four children there will then he one chance
of all four being of one aex, three chances of one being of one sex and
one of the other, and three chances of an equal division, making eight
chance- in all. Hence, in a great mass of such families, we shall have
one-eighth all of the Bame sex. four-eighths, or (-half, with a pre-
I lerance '■> to l. and three-eighths with an equal division.
The next term shows how this probability is mollified in case of a
unisexual tendency. The Bymbol // expresses the fraction of the whole
number of parent- which have such a tendency. The tendency in one-
half of this fraction of cases will he in the male, in the other in the
female direction. The symbol a i- the unknown amount of this t«n-
den<
These expressions for the probability are rigorous when n i- 2 >>r ■'>.
But, when // ha- a greater value, terms in the higher powers of a
really exist, the highest power being //. or // — 1. according as // is even
or odd. hut. a- ,i must always \,r a rather -mall factor, these high powers
may he neglected.
MATHEMATICAL THEORY
33
Constriction of THE Nimi.kh \i. Fok.mil.k.
n
r, s
Li
2
2,
1
1, i
•>
3
3,
1
~\ 1
3
4
4,0
1
3, 1
4
2, 2
6
5
5,
1
4, 1
5
3,2
10
6
6,
1
5, 1
6
4,2
15
3, 3
20
7
7,
1
6, 1
7
5,2
21
4, 3
35
8
8,
1
7, 1
8
6, 2
28
5, 3
56
4,4
70
9
9,
1
8, 1
9
7, 2
36
6, 3
84
5, 4
126
10
10,
1
9, 1
10
8, 2
45
7, 3
120
6,4
210
5, 5
252
LI
11,
1
10, 1
11
9, 2
55
8, 3
165
7, 4
330
6, 5
462
12
12,
1
11, 1
12
10, 2
66
9, 3
220
8, 4
495
7, 5
792
6, 6
924
(s — /•)- — n
Probability
16
32
64
128
256
512
1024
2048
f
~^
2
1 s- 2
+
- 2
1 S- 2
—
2ha*
6
1 s- 4
+
8ha*
- 2
3 s- 4
—
Ma*
12
1 s- 8
+
SAa 1
4-r 8
- 4
3s- 8
—
3/m 2
20
1 S- 16
+
5fta* s- 2
4
5-f- 16
+
5/<a 2 s- 2
- 4
1 S- Hi
—
10/<a 2 s- 2
30
Is- 32
+
15/*a 2 s- 8
10
6 s- 32
+
30Aa- s- 8
- 2
15 s- 32
—
ISha 1 s- 8
- 6
10 s- 32
—
30Aa 5 s- 8
42
1 s- 64
+
21/ ( «-'s- 16
18
7 s- 64
+
63/m 2 s- 16
2
21 s- 64
+
21//«* s- 16
- 6
35 s- 64
—
105Aa 2 -*- 16
56
Is- 128
+
7ha* s- 8
28
8 s- 128
+
28/m 2 s- 8
8
28 s- 128
+
28/m s s- 8
- 4
56 s- 128
—
28/ia 2 s- 8
- 8
35 s- 128
—
S5Ac a s- 8
72
Is- 256
+
9/m 2 s- 16
40
9 s- 256
+
45/<a-' s- 16
16
36 s- 256
+
727*a 2 s- 16
84 s- 256
- 8
126 s- 256
—
126^« 2 s- 16
90
1 s- 512
+
45^o 2 s- 128
54
10 s- 512
+
270Aa 2 s- 128
26
45 s- 512
+
585/m 2 s- 128
6
120 s- 512
+
360 ha 2 s- 128
- 6
210 s- 512
—
680Aa« s- 128
-10
126 s- 512
—
630//o 5 s- 128
110
1 s- 1024
+
55ha* s- 256
70
11 s- 1024
+
385Aa« s- 256
38
55 s- 1024
+
1045Aa 2 s- 256
14
165 s- 1024
+
11557m 2 s- 256
- 2
330 s- 1024
—
3307*a 2 s- 256
- 10
462 s- 1024
—
2310Aa 2 s- 256
132
1 s- 2048
+
33 Aa 2 s- 256
88
12 s- 2048
+
264/ia 2 s- 256
52
66 s- 2048
+
858ha 2 s- 256
24
220 s- 2048
+
1320/ta 2 s- 256
4
495 s- 2048
+
495Aa 2 s- 256
- 8
792 s- 2048
—
1584/ia 2 s- 256
-12
462 s- 2048
—
1386Aa 2 s- 256
;;1 STATISTICS OF SEX
The method of using the numbers is, Erom the statistics for each
value of //. to form conditional equations haying /< and u as unknown
quantities. These unknowns are i" be determj 1 by a ><>lution of the
equations. It will be seen thai A and a cannol be determined sepa-
rately, lint only the combination ha. We may therefore Buppose h = 1
withoul any loss of generality bo Ear as these equations arc concerned.
\\.' now make a practical application of this theory by determining
the numerical value of the unisexual tendency, a, in the respective cases
twins and triplets, as enumerated in section 5 preceding. The sta-
tistics of twins there cited >h<>w that, of such pairs, 0.646 are unisexual
and 0.354 bisexual. Equating these percentages to the expressions for
the probability we find
y 2 + 2ha= 0.646
y 2 — 2ha= 0.354
Subtracting these Erom each other we find 4Jia = 0.292, and hence,
supposing h = 1,
<r = 0.073
a = o.v;
We may now consider the case of triplets in two ways. Proceeding,
;n the case of twins, by equating each probability to the fraction indi-
cating the proportional number of the families to which ii relates, we
have the equations :
I | f 3fco =0.499 :; , — 3ha = 0.501
Solving these we derive, after putting h = 1,
0' = 0.0831 o = 0.29
We may also proceed in another way by substituting in the expres-
sions for the respective probabilities of unisexual and bisexual triplets
the value of ha derived Erom the case of twins. This will L r i\<-. as
has already 1 n stated, the percentage t6.9 for unisexual triplets
instead of 19.9, as has been found from observation. It may be added
thai this relation i- nol changed by changing the value of h; it is there-
fore indifferent what value we assign to //.
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