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Correlation of length and breadth of head in American Negroes.

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CORRELATION OF LENGTH AND BREADTH O F H E A D
IN AMERICAN NEGROES
MELVILLE J. HERSKOVITS
The problem of homogeneity or heterogeneity of a population is one
to which too small attention has been paid by anthropologists in the
course of their analyses of existing populations. It is, however, one
which sooner or later must be studied in detail, and it is believed that
research into the processes which make for homogeneity or heterogeneity, quite aside from racial background, will give ultimately a far
clearer idea of the developmental processes which hunian groups undergo
than will be obtained from studies exclusively designed to determine
racial purity. The extent to which all human groups have intermingled
is too often overlooked, to the resdting confusion of the findings, and
it would seem that the general principle of strict induction from existing
populations is preferable to analyses of populations into racial types.
The variability of the American Negro, and a study of the homogeneity of this group, has already been approached by the writer in
two papers.l In these, a statistical analysis of the variability of the
averages of fraternities of Negro children from New York City was
utilised, and the results were such as seemed to justify the view that the
American Negro has become homogeneous to a surprising degree,--surprising, that is, when the amount of mixture he represents is considered.
At the same time, it was pointed out that homogeneity and racial purity
are not to be confused, but that the former is the result of inbreeding of
a group, the ancestry of which may be as varied ethnically as is that of
the American Negro, or that of the Rehobother Bastards of South Africa. Of course, homogeneity does not preclude unity of racial origin
in a human group, t o a greater or less extent, for there can be the same
inbreeding leading to homogeneity among the descendants of persons of
one racial type, such as has been demonstrated for the Tennessee
mountaineers.2
It has been shown by Boas3 that the correlation of the length and
breadth of head offers another indication of the extent to which a population may be regarded as honiogeneous. Commenting on the coeffi'Proc. X X I s t Int. Cong. Amer. The Hague, August, 1924. J . Am. Stat. dss.,1925,
XX,380-389.
%abel Gordon, Reduction of Variability in an Inbred Population (MS), Doctorat
Diss., Columbia University, 1925.
'Franz Boas, The Cephalic Index, American Anthropologist (n. s.), vol. 1, (1899),
pp. 448461.
AN. 3.
PHYS.
ANTHROP..1926,Vol. IX.No. 1.
87
MELVILLE J. HERSKOVITS
'88
cients of correlation which he presents Boas remarks, "It appears . . .
that the degree of correlation between length and breadth of the head is
very slight, and that its values differ considerably among the various
races. . . . The coefficient of conelation in the Parisians, for instance, is
exceedingly low. . . . The niost plausible explanation of this phenomenon lies in the efiect of mixture of types upon thecoefficientof correlation.
. . . I think this effect of mixture is a sufficient explanation of the low
value of the coefficient of correlation for Paris, where we find a very
heterogeneous population. . . ."4 The figures for the populations for
which the coefficients of correlation between length and breadth of
head have been computed will be given below; it will be well, however,
to consider first the data from which these figures are drawn for the
American Negro groups studied, and the variabiiities which are involved
in the coefficients of correlation later coinputed from them.
The present paper is a result of a portion of the measurements in a
general study of Variability under Racial Crossing, with special reference to the American Negro-White population. The data represented
in it were gathered in New York City and Washington, D. C. Prom the
former city come the 1211 boys of Public School 89, and 63 male adults,
the data for which are tabulated with those for the 477 male adults
measured at Howard University in Tl'ashingt~n.~The measurements
were made with spreading calipers of the type designed by Martin, and
TABLE1. AGE ME-ZNSFOR LENGTH.\Nil BREADTH
COLORED
BOYS.
Age
'
5-6
6-7
7-8
8-9
9-10
10-11
11-12
12-13
13-14
14-15
15-16
16-17
17-18
Pia.
22
55
101
124
130
135
3 40
126
140
117
81
30
12
OF
HEAD,N E W Y O R K
Length of head (rnrn.)
Mean
&fl
Breadth of head (nun.)
Mean
f U
174.61
178.35
178.49
179.67
181.49
181.58
183.75
184.40
186.18
187.98
190.06
190.30
190.66
139.46
140.90
139.62
141.56
141.75
142.19
142.85
143.66
144.03
146.48
147.19
146.63
147.66
k6.43
h6.46
~k6.46
16.18
15.52
h6.67
15.71
f6.42
55-76
16.48
16.02
16.03
-
CITY
f4.17
f5.87
15.00
2k4.93
f5.28
15.39
14.51
14.83
15.71
b4.77
h5.44
&4.69
-
'1. c., pp. 453-455.
6This study is being carried on by the writer as Fellow of the Board of Fellowships
in the Biological Sciences, National Research Council, and the data gathered at
Howard University were taken with the assistance of a grant from the Committee
on Human Migrations of the National Research Council. The writer wishes to
express his thanks to Dr. Jacob M. Ross, Principal of Public School 89, Manhattan,
and his teaching staff, for their cooperation while measurements were being taken
there, and to the President and Faculty of Howard University for their permission to
work there, and for their numerous aids and hearty cooperation.
CORRELATION OF HEAD I N AMERICAN NEGROES
89
consist of the niasinium glabellar length of the head, and the maximum
width, the indices being computed from these two figures. In the case
of the New 'I'ork boys, the factor of growth has had to be reckoned with,
and the computation for means and standard deviations were first made
within age groups, the results of which are shown in Table 1. For the
age means for cephalic index for these boys, the reader is referred to the
writer's paper presented at the XXIst Congress of Americanists, already referred to.
The male adult material gathered at Howard University and in New
TATiLE
2.
TABLEOF LENGTHAND BREADTHOF HEADAND
INDEX
FOR 539 MALENEGROADULTS.
DISTRIBUTION
Length of kead
Mm.
No.
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
19s
199
200
20 1
202
203
204
205
206
207
208
209
210
211
212
213
214
315
216
217
218
Width of head
Mm.
No.
CEPHALIC
Cephalic index
Index
No.
2
6
7
-7
6
10
13
15
15
10
25
9
35
29
80
31
31
36
33
I8
29
20
36
16
15
14
10
10
8
5
1
1
1
2
1
1
-
1
131
132
133
134
135
138
137
138
139
140
141
142
143
144
145
146
147
148
149
I50
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
1
1
2
1
1
2
5
2
2
10
11
21
8
25
33
24
47
28
28
49
5s
34
15
32
21
25
15
6
8
5
3
5
4
3
2
1
1
62.5
63.5
64.5
65.5
66.5
67.5
68.5
69.5
70.5
71.5
72.5
73.5
74.5
75.5
76.5
77.5
78.5
79.G
80.5
81.5
82.5
83.5
84.5
85.5
86.5
87.5
8S.5
89.5
90.5
1
-
1
1
1
7
6
14
29
30
47
75
67
55
66
39
38
21
12
12
8
4
2
1
1
1
90
MELVILLE J. HERSKOVITS
York City were not arranged by age, as the individuals are from 17 years
of age up, ranging mainly about the decade 17-27. The data as tabulated, for length and breadth of head and cephalic index are shown in
Table 2.
It was felt,however,that before the Public School data could be utilised
for comparative purposes, corrections should be made for age changes,
and therefore the figures for all ages were lumped and averaged, with
the following results, assuming sigma as constant for each year, which is
practically the case, as can be seen from the table.
Sigma
h7.24
Length of head. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Breadth of head. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
~5.57
Cephalic Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
h3.5 1
However, the standard deviation, computed from data thus thrown
together cannot be utilised in this fashion, since the factor of growth
makes for heterogeneous data and overlapping of the distributions:
this was corrected by the formula
where 8 is the mean square variability of the entire group, uncorrected,
n the number of cases in each age group, d the deviation of the mean of
each age group from the mean for the total series, and N the total number of cases. The results, therefore, were
Length of head. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Breadth of head. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cephalic Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
~6.15
h5.11
~3.47
It is these results which may be compared with those for the adult
Negro group from Howard University and New York City of 539
individuals :
Mean
*cr
Length of head. .........................
196.98
~6.77
Breadth of head. . . . . . . . . . . . . . . . . . . . . . . . .
151.58
k5.74
77.05
k3.49
Cephalic Index. . . . . . . . . . . . . . . . . . . . . . . . . .
We find a close correspondence between the standard deviations for
both traits and for the cephalic index, and these correspondences, when
tested statistically, are found to be due not to chance sampling, but are
significant. The means for absolute length and breadth are, of course,
not available for comparison, because of the factor of growth involved
in the New York boys' series, which tends materially t o Iower them.
CORRELATION OF HEAD IN AMERICAN XEGROES
91
If we compare the mean square variability of the cephalic index of
the individuals studied here with the data gathered for numerous other
populations, which are given in Table 3, we find that the variability
is less than that for the skeIetal collectioiis of Western Reserve University, both Negro and White, than for the modern French inhabitants
TABLE3. COMPARATIVE LENGTES.BREADTHS. AND CEPHALIC INDICES OF VARIOUS
POPULATIONS ( M E A N S AND STANDARD
Population
NO.
Length (mm.)
DEVIATIONS).
Breadth (mm.)
Index
New York Negro boys
78.08dZ3.47
1211
(183.44)f6.15 (143.12)A5.11
Howard Univ. and N.Y .
77.05f3.49
151.5815.74
196.98f6.77
male adult Negroes
,539
W. R. U. White'
144.28f5.67 *80.69+4.74
181.42f8.19
167
*75.89%3.13
139.30f5.66
186.20f6.51
W. R. U. Negro1
87
78.02rt2.92
Oxford Students2
152.84dZ4.92
959
196.05f6.23
77.23 f2.74
191.66rt6.05
English Criminals*
150.44dZ5.01
3000
78.33f2.90
193.5156.16
Cambridge men3
153.96i5.05
1000
*73.99rt2.80
185.13rt5.75
134.87f4.60
134
Naqada skulls4
177.87f7.05
*78.99f2.58
138.52f5.00
50
Batatelas
*76.48+2.77
179.48f5.61
135.48f3.38
50
Gaboon (1864)5
*75.34E3.26
189.06f6.27
Whitechapel English6
140.67f5.28
137
75.52dZ2.91
190.52f5.90
Cairo-born males'
144.45f4.67
802
185.82f5.94
Aino (Koganai)s
141.23f3.90 "77.50rt2.39
87
180.58+6.09
Germans (Ranke)*
150.47+5.85 *84.30&3.50
100
181.853~5.84 144.9315.21 *80.82+3.79
Parisians (Broca)s
?
143.51f5.42
*80.78I3.84
179.93f6.30
French peasants9
57
142.11A4.95 *76.99dZ2.30
185.85f5.35
Tenerife males1°
245-9
81.30 f3.53
190.6015.69
Samoan maless'
154.80 f4.46
68
*Anaddition of 1%has been made to the mean indices marked thus since they are
cranial populations, so that comparison with series of living may readily be made.
Cf. R. Martin, Lehrb. d. Anthrop., 421.
IT.Wingate Todd, Cranial Capacity and Linear Dimensions in White and Negro.
Am. J. Ph.ys. Avtlh., 1923, vi, 97 ff.
'JE. Shuster, First Results from theOxford AnthropologicalLaboratories. Biometriha, 1911-13, viii, 40 ff.
aW. R. Macdonnell, On Criminal Anthropometry and the Identification of Criminals. Biometrika. 1901-2. i. 177 ff.
4Fawcett and Lee, A Second Study of the Variation and Correlation of the Human
Skull. Biometrika, 1901-2, i, 408 ff.
ER.Crewdson Bennington, A Study of the Negro Skull. Bionzetrika, 1911-12, viii,
ono LT
A V L rt.
eW. R. Macdonnell, A Study of the Variation and the Correlation of the Human
Skull. Biumetrika, 1904, iii, 191 ff.
7Myer M. Orensteen, Correlation of Anthropometrical Measurements in Cairoborn Natives. Biometrika, 1915-17, xi, 67 ff.
8Lee and Pearson, Data for the Problem of the Evolution in Man. VI. Pkil.
Trans. Roy. Sac., "A", cxcvi, 225 ff.
gKarl Pearson, Mathematical Contribution to the Theory of Evolution. Phil.
Trans. Roy. SOC.,"A", 1896, clxxxvii, 253 ff.
10E. A. Hooton, The Ancient Inhabitants of the Canary Islands. Harvard Afr.
Studies,1925, vii, 89, 91, 96.
"L. R. Sullivan, A Contribution to Soman Somatology. Mem. Bernice Pauahi
Bishop Mus., 1921, viii, no. 2.
and for French peasants, and than for Samoan males. On the other
hand, it is greater than the figures for the various groups of English
studied, both living and cranial collections, and more than the variation
9%
MELVILLE J. HERSKOVITS
for most of the primitive and prehistoric groups represented, such as
the Egyptian Naqada skulls, the Batatela and Gaboon crania, the Aino
and the Cairo-born natives, and the Tenerife male crania collected by
Hooton, while the same variability, approximately, is to be seen between these Kegroes and the 100 Germans measured by Ranke. Altho
the figures for the means for absolute length and breadth are included
in the table, as well as the variabilities, the former cannot be used in
making comparisons, since many of them represent cranial collections,
the others being measurements on the living; and for the further reason
that comparison of indices is more desirable, since the ratios represented
are not so easily affected by dserences between measurements on the
living and on crania. The inclusion of all the results given is made,
however, because they are necessary in the computation of the coeEcients of correlation, the mean indices from cranial series being increased by 1G/& Particular attention should be paid to the lowness of the
standard deviations for all three traits for both Negro groups studied
by the writer and for the Negro crania in the Western Reserve collection, when these are compared with the similar variabilities for the
Whites in the Western Reserve collection. These last represent an unselected group {except insofar as there is an economic selection a t work)
and the differences between the Negro groups and the figure for the
White group are highly significant if the relative homogeneities are considered.
The comparison of standard deviations for length, breadth, and
cephalic index is of interest also in the light of the theories of heredity
which are prevalent today. If Mendelian segregation of traits were
operative, then the variability of the American Negro group, being of
niixed racial origin, should be greater than that of every population
which has contributed to its ancestry. If head length be considered
first, it is seen that all three samples of the American Negroes are less
variable than American Whites, than the Batatela, but more so than
the English. For breadth, the American Whites are about the same
variability as the American Negroes in the W. R. U. collection but
greater than any of the other populations with the exception of the
Germans, altho the difference here is not great between any of the
populations. The variability of cephalic index has already been discussed, but for convenience it may again be noted that the highest
variability of the entire series is that of the W. R. U. White population,
while the Germans, French peasants, and Whitechapel English series
fall either well within the range of variability of the American Negro
CORRELATION OF HE-4D IN AMERICAN NEGROES
93
groups or are larger in this respect than they are. The operation of the
Mendelian principle, if it is operative, cannot, it goes without saying, be
observed adequately in any rough fashion such as this since actual
parent-and-child material is most desirable. But on the other hand, the
principles which have been laid down are so broad that if they were
working at all, certain statistical results should be observable, as explained above. And these do not appear. It seems to argue a blending
of type rather than a segregation, but the entire problem must be laid
aside for further and more detailed consideration, both from other
aspects of these data, and thru the measurement of actual family groups,
both of which, it is hoped, will be done.6
Since these groups of American Negroes represent descent from West
African and European groups in the main (altho there is also an appreciable amount of American Indian ancestry present, it is believed) it is
of interest to compare the means for cephalic index for them with those
for the populations allied to them. Thus, they are seen to be quite
near the figure for the English group, as might be expected, since there
is undoubtedly, thru the White ancestry, much English blood represented in these Negroes. On the other hand, this comparison does not
hold if we use the figure for the Western Reserve Negroes. The closeness to the Batatela mean index would be quite striking if it were not
contrasted by the difference between the American Negro and the Gaboon indices. Comparisons on the basis of one trait or one index are,
in the main, unsafe, and therefore there will be no attempt to draw
relationships on the basis of the figures given in this paper.
The computation of the coefficient of correlation between length and
breadth of the head may be carried out in two ways, both of which are
represented in this paper. The first is that of the Pearson productmoment formula, and the correlation between the two traits for the
New York Negro children was performed in this way, there being used,
because of the complicating growth factor, the deviations of the individual measurements from the respective age means rather than the
raw data. However, Pearson has given a formula7 by which this correlation may be carried out with less computation, an important matter
in time-saving. This formula employs the coefficients of variation of
the Iength and breadth of the head and of the cephalic index,
6Boas (in The Head-Forms of the Italians as Influenced by Heredity and Environment. Amer. Xnth., (n. s.) vol. XV, 1913, pp. 163-188) has shoirixi that the Central
Italians, who are very mixed, have higher variability than either the north or south
Italians. The fact that African and White indices are so close, however, makes the
applicability of this study to the present one difficult.
'Karl Pearson, 011. cit., pp. 279-SO.
MELVILLE J. HERSKOVITS
94
V equalling
in the formula
r=
100 sigma
Mean
v,”+v: -v:
2vevb
where r is the coefficient of correlation, V, the coefficient of variation
for length of head, V, that for the width, and Vi that for the cephalic
index. To test the utilisability of the formula, it was used after the
co&cient of correlation had been computed on the same data for New
York boys by the long method, and the two results are
Y (by the product-moment method) = +.201
r (by coefficient of variability method) = .216
It will be seen that tho the latter tends to change the coefficient of
+
TABLE
4. COEFFICIENTS
OF CORRELATION
BETWEEN LENGTHAND BREADTH
OF
HEADFOR VARIOUSPOPULATIONS*
NO.
Population
Coefficient of correlation (r)
Tenerife males (Hooton)
+.586
Batatela (Bennington)
+.586
Male adult Eskimo skulls2
f.47
Aino (Koganai-Lee and Pearson)
.4316
English Criminals (Macdonnell)
+.4016
W. R. U. Negro males (Todd)
+.3789
1000 Cambridge men (Macdonnell)
+.3448
134 Naqada (Fawcett and Lee)
+.344
- Australians8
+.313
212 University of Aberdeen students*
.3007
100 Germans (Rank-Lee
and Pearson)
$2861
137 Whitechapel English (Macdonnell)
280
- Ancient Egyptians (Flinders-Petrie)s
.2705
802 Cairo-born natives (Ornsteen)
244
243 Sioux Living Males2
.24
3.216
539 Howard Univ. and AT. Y. C. Adult Male Negroes
1211 New York Negro children
+.201
50 Gaboon, 1864 (Bennington)
+.187
959 Oxford students (Shuster)
.143
Bagdi caste of Bengal:!
+.13
57 Modem French Peasants (Pearson)
+.1263
- Baden male skulls2
Adult male Indians, British Columbia
.08
Modern French Parisians (Pearson)
.0474
- Adult male Sushwap Indians2
.04
167 W. R. U. White males (Todd)
--.0045
68 Samoan males (Sullivan)
--.058
‘The coefficients for populations for which references have been given in Table 3
may be looked up in the original publications by reference to the titles appended to
that table. Such papers only as were not cited in Table 3 are noted here.
ZF.Boas, The Cephalic Index, loc. cit., p. 453.
3From table in R. Martin, Lehrb. d. Anthrop., p. 706.
‘Reid and Mulligan, Communication from the Anthropological Laboratory of the
University of Aberdeen. J. Roy. Anthrop. Inst., 1925, liv, 287 ff. The figure given
here is for the 17-year-old group. I t is lower for the other three groups which were
studied in this paper.
&KarlPearson, Phil. Trans. Roy. SOL.,clxxxvi, 280.
245-9
50
47
87
3000
87
+
+
++
+
+
-
-
+
++
+
+
CORRELATION OF HEAD IN AMERICAN NEGROES
95
correlation somewhat, there is not a serious difference, and therefore
both methods are represented without separate desipnation in the
following Table 4.
In surveying the coefficients of correlation for the populations under
consideration, it will be seen that the Negro populations studied in
America are relatively honiogeneous. The correlations for the three
figures given for American Negro groups,-Western Reserve crania, the
New York boys and the Howard University data,-are strikingly higher
than the extremely low correlation for the Western Reserve White
crania. It is to be regretted that no available data were a t hand from
which correlations for other American White representative populations
might have been computed, as this would throw further light on the
relative homogeneities. However, it seems likely, from the comparison
made, that the American Negro is more homogeneous than the White
population of this country, taken a t large. That the correlation cornputed here seems to be an indication of homogeneity would seem to show
in a consideration of the other populations represented in Table 4. The
higher figures represent popiilations which one might expect to show
relatively large homogeneity of type,-the inhabitants of Tenerife,
the Batatela, a Congo tribe, the Eskimo, the Aino, various groups of
English, the Naqada crania, and the like. On the other hand, a t the
lower end of the scale we find the Parisians, the mixed Indians of British
Columbia, the inhabitants of Gaboon, where there is much mingling,
and others. It is not assumed that the list of relative degrees of correlation gives an adequate statement of homogeneity nor can it be relied on
to show of itself the extent to which this exists or not. Because, if the
conelations were indicators of this, we should not find that the coefficient for the Oxford students and for the Samoans would be so Iow,
since the first, at least, should be as homogeneous as the other English
groups studied. It is quite possible that if we had two traits always
paired and thus inherited, the result of crossing would be a high correlation, even tho there were heterogeneity; however, it is not without
significance that we find the American Negroes so relatively high in the
list in view of the comparative variabilities of Table 3 and their
standing in it.
If it is true, then, that the American Negro is tending toward a homogeneity of type, it is well to inquire into the cause of this, and to see
whether there niay be observed any socially selective factor operative
to bring about this consolidation of type. If we examine into the case
historically, we find that the amount of primary crossing which has
96
MELVILLE J. HERSKOVITS
occurred between Whites and Negroes has materially lessened since
the Civil War. From the 539 genealogies collected in the course of this
study from the adults measured, there were only two the writers of
which claimed to be primary crosses. While it is possible that this is
far too low a proportion, the fact that there is today social pressure on
the Negro as well as on the White side against relations with members
of the other race would tend to naturally lower the amount of direct
crossing. At the same time, the crossing which has taken place is making itself felt in the Negro population thru the intermarriage of the
earlier mixed individuals and their descendants with those who represent no mixture, and the percentage of pure Negroes, it is believed, is
being steadily lessened. Indeed, only 20% of the total population
represented in these genealogies gave themselves as without some mixture, altho this figure may be somewhat low due to the fact that a
majority of these genealogies come from University students.
However, if the above phenomena be granted, there is the further
problem of the mechanism of selection which is working to bring the
lighter and darker elements among the Negro population together, and
thus make for the consolidation of type which seems to be foreshadowed.
It was felt that a clue might be obtained thru a consideration of the
invidious nature of color distinctions,-lightness being favored, thus
causing the dark men to seek the lighter women as wives, while the
lighter women marry the darker inen because of the advantageous
position in the match afforded by their color. A test of this was made
with 380 of the Howard students who were measured, and they were
asked, each in turn “Who is the lighter, your father or your mother?”
The replies, when tabulated, were enlightening. There are, of course,
three possible answers ; father lighter, mother lighter, and parents
about the same. The results are as follows: Father lighter, 115 (29Oj,):
same color, 50 (13%); mother lighter, 215 (%yo).
This represents, of course, a definite trend. altho it is not claimed that
it is more than ’just that. A t the same time, it means that there is
something of a consolidation of type occurring among the American
Negro population, which would easily account for the relatively low
variability which traits show, and for the relatively high figures which
the computations of coefficients of correlation for American Negro
groups show.
In summary, therefore, we may say that the conclusions from the
study described in this paper are as follows:
CORWLATION OF HEAD IN AMERICAN NEGROES
97
1. The study of the problem of homogeneity of existing types should
constitute an excellent means of determining the processes of changes
among human groups.
2. The cephalic index alone cannot be utilised as criterion for relating the American Negro generically to any of the other groups represented in the paper.
3. The variability of the cephalic index seems to bear out the hypothesis of relative homogeneity of the American Negro population.
4. This is further borne out by the computed coefficients of correlation between length and breadth of head for the groups studied, when
this is taken in connection with the result mentioned immediately
above.
5. This consolidation of type foreshadowed by the figures resulting
from such analyses of the American Negro might be accounted for by
social selection on the basis of color.
6. As compared to other populations, the variabilities of head length,
head breadth, and cephalic index are not sufliciently high to allow us to
see the operation of Mendelian principles of heredity, but rather to
question whether this result might have been obtained if they had been
operative.
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