[Race]

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

necessarily.

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

classes.

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.