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Boas, Franz. Race, Language and Culture – T18


The Analysis of Anthropometrical Series 11

The criticisms of my investigations relating to the bodily forms of
descendants of immigrants in New York in comparison with those
of their European-born parents are based largely upon the current
method of subdividing anthropometric series in a number of arbitrary
groups and to describe the whole series by the percentual frequencies
of these groups. For instance, it is said that a certain population contains
such and such percentages of short, medium and tall individuals, or such
and such percentages of individuals with elongated, medium and
rounded head forms; or a larger number of groups are distinguished
or characterized by various combinations of forms.

Since the applicability of this method, particularly the interpretation
of the frequencies of these groups, is of fundamental importance for
the formulation of many problems a consideration of the theoretical
basis of this procedure seems desirable.

In support of this method the statement is always made that averages
have no meaning, that it is necessary to determine the distributions of
individual values. That is true. Two series may have the same average
and still be quite distinct; but two series cannot have different averages
and be nevertheless identical. The average is of great value as a discriminating
criterion, particularly because it can be determined with
greater accuracy than any other value that depends upon the distribution
of individual values of the series. For this purpose it is indispensable.
Although we ascribe to the average no more than this discriminating
value, it should not be neglected.

The average of the series may be the same but the distribution of individual
values may be quite different. The attempt is made to overcome
this difficulty by the establishment of groups and the determination
of their percentual frequencies. In this manner we learn more about the
series than by a statement of the average.

The solution offered by this method is not satisfactory, because it
gives a very inadequate picture of the distribution of frequencies. It
176may be asked whether a better method may not be found. This problem
has been solved by the introduction of the mean square variability
as a measure of the scattering of individuals in the series. Experience
has shown that in many series average and mean square variability permit
us to determine with an adequate degree of accuracy the frequency
of any selected group. As an example I give Livi's observations on the
cephalic index of 7,760 enlisted soldiers from Palermo. The average and
its mean square variability are 79.1 ± 3.66. The distribution of cephalic
indices according to observation and theory is as follows:

tableau Index | Observation | Theory

The theoretical values may be obtained from any table of the probability
integral. 12

In a short series we must be satisfied with the mean square variability
as the best attainable index of the character of the series because the
distribution of the individual values is too much affected by chance.

In order to explain the reasons that compel us to adopt this method
of presentation a more fundamental consideration of the character of
variable quantities seems desirable.

We must define the difference between a constant and a variable.
An example will illustrate this. It is obvious that the two statements: a
cubic centimeter of pure water at greatest density at a given place
weighs 1 gram; and the stature of Scotchmen is 175 cm., do not mean
the same formally. If I should extend the term “water” to include
water of any temperature and any kind of impurity the two statements
would be formally of the same kind. The essential difference in
the first case is that the term “water” is assumed to be completely defined,
as opposed to the incomplete definition of what is a Scotchman
or what is impure water. A constant is the measure of a completely
defined object, a variable the series of measures of all the incompletely
defined individuals composing a class. Only if we know all the influences
that determine each member of the class completely could they
also become constants. The class itself is completely defined, not the
individual representatives of the class. Variability is not a specifically
177biological problem but an expression of the fact that the individuals of
a class are subject to unknown influences.

This point of view is of the greatest importance for a logical treatment
of variables. It shows that every member of a class has all the
essential traits that characterize the class, but modified by unknown
factors. Therefore, if I want to describe the class — in our case an anthropometric
series — I must try to express both the essential class character
and the influence of the unknown factors. When we segregate a
particular group characterized by certain metric values out of the whole
class, we do not only unnecessarily restrict the material that is being
discussed, but — and this is more important — we segregate certain combinations
of unknown factors and thus introduce a subjective element
that has no relation whatever to the series itself. The series is a unit that
cannot be broken and that must be described as a unit. The average
and mean square variability fulfil these conditions because they consider
each individual of the series as of equal value. They have the added
advantage that in many cases they describe the distribution of individuals
with sufficient accuracy. We have seen that two series having different
averages cannot be equal. Two series with different mean square
variabilities also cannot be equal. When two series have the same
average and variabilities they may be equal, but this does not follow

It follows from what has been said that when two averages are different
and the variability remains the same, changes in the percentual
frequencies of selected groups will follow. If the variabilities are also
different the same frequencies for selected groups may result, although
the series are distinct.

I will now turn to the fundamental question as to what may be inferred
from the description of a series by means of average and mean
square variability. A comparison between the description of constants
and variables shows that the distribution of variants, however they may
be expressed, are solely a description of the class. From the fact that a
cubic centimeter of pure water of greatest density weighs i gram I cannot
infer why this is the case; so also in a variable the observed values
have solely a descriptive value. The fact that water and mercury have
different specific weights does not tell me why they differ in specific
weight. In the same way, if I have one variable expressed by the measure
183 ± 3 and another expressed by 184 ± 4, I know only that they
are different, and, if the distribution of individual values is of the usual
type, the numerical values would not tell me why they are different.
178It may be that each corresponding individual grew on the average by
one unit and that the growth itself was variable. This would give the
observed result. It might also be that each value was somehow changed
in its frequency of occurrence, which would also account for the observed
changes. It might also be that new elements were introduced
so that the two series would not be comparable. Even the most intense
study of the observed numerical values will throw no light upon the
causal factors that bring about the change in both average and variability.
The constantly repeated attempts to interpret descriptive features
without further data do not prove that this method is acceptable.

As an example I chose a discussion by Hans Fehlinger 13 of the gradual
decrease of the average cephalic index with increasing age, from birth
until the adult stage is reached. Fehlinger concludes that the only possible
explanation of this phenomenon is selective mortality, because in
the series of children of various ages the frequency of round-headed individuals
gradually decreases. If it were not the cephalic index, but
stature that is under consideration, nobody would imagine that the
gradual disappearance of those of short stature is due to selection, for we
know that it is due to individual growth. The changes of head index
are also due to growth. The development of the frontal sinuses and of
the muscular attachments at the occiput bring it about that the antero-posterior
diameter of the head increases more rapidly than the transversal
one. Therefore the cephalic index decreases with increasing age,
not on account of an elimination of the round heads. It seems hardly
plausible that very few children who have a certain cephalic index
should die, let us say, between 8 and 11 years of age, while of those
who have another index, one-third would die. Still, this would be required
as a general phenomenon of the development of population if
the universal decrease of the number of round-headed individuals in
all populations were to be explained by selection.

I have discussed the whole subject somewhat fully, in order to show
that the statistical data are purely descriptive, that the interpretation
must be based on biological considerations. It follows that all attempts
to derive conclusions solely from the statistical data are futile. 24179

We have to return to our previous remarks which showed that every
individual belonging to a series or class has the essential characteristics
of the class modified by unknown causes. From this point of view a
fundamental error made in the comparison of subgroups determined by
selected measurements becomes apparent. It consists in the grouping
together of individuals that happen to have the same measurements
but belong to different classes. Thus Fehlinger equates the long-headed
boys of from 4 to 6 years with adults with long heads, although they are
from a biological point of view not equivalent and belong to different

It is easy to show that the critique of these concepts is not unnecessary
dialectic refinement, for real differences in such cases are the rule, not
exceptions. Thus I found that, when in two populations individuals
whose head index is 80 are selected the head forms of the children of
this part of the population are not determined by the selected index of
the parents alone, but also by the average index of the population to
which the parents belong. If the average index of the population is
76 the children of the group with index 80 will have an index of about
78.4; if the average index of the population is 84, the index of the
children of the group with index 80 would be 81.6.

Therefore it is wrong to speak of the “blond,” “round-headed” or
“tall” groups in various parts of Europe as though they were identical
from a biological point of view. Blond Italians are Italians, tall Sicilians,
Sicilians, and round-headed Swedes, Swedes. Maps showing the distribution
of long-headedness, tallness, etc., do not give trustworthy information
regarding the distribution of types.

It follows, that, if we wish to understand the character of a variable
which is defined by certain known characteristics of the class and by
many unknown factors influencing the individuals, we must not segregate
a small part of the class and assume that we have segregated a
factor or factors causing the variability. All we have done is to segregate
individuals who have the same measure which, however, may be
due to the most diverse unknown influences. No conclusion can be
drawn from such a procedure. The class has to be treated as a whole
and every attempted analysis must be based on the study of single factors
within the whole series.180

11 Archiv für Rassen- und Gesellschafts-Biologie, vol. 10 (1913), pp. 290 et seq.

21 See for example W. F. Sheppard, Biometrika, vol. 2 (1902-1903), pp. 174 et seq.

31 Petermann's Mitteilungen, vol. 59 (1913), pp. 19 et seq.

42 Exceptions are certain forms of distribution which imply the presence of disturbing
factors. Such are unusually high or low variability, presence of decided
multiple maxima, forms of asymmetry. Additional observations or the study of
interrelations between series may supply materials for further analysis. The point
made here shows only that the ordinary descriptive features of a normal series
gives no clue that allows us to interpret its origin.