[Race]

The Measurement of Differences between

Variable Quantities 11

In biological statistics it is often necessary to study the differences between

two variable types. The problem may be exemplified by a

consideration of the differences between the types represented by populations

of various countries, as, for instance, between the populations of

Sweden, Switzerland, and Central Africa. It is obvious that the type

of Sweden differs less from that of Switzerland than the latter differs

from the type of Central Africa. Nevertheless, it is difficult to say just

what is meant by greater or lesser difference in type. The attempt to

establish and describe varieties of races according to characteristic features

that are considered as significant from a morphological point of

view suffers, therefore, from a lack of clarity of concept.

The differences between the averages of types have been utilized for

the purpose of segregating subtypes of human races, as, for instance, in

the classification of the local types into which the European race may

be divided. Pigmentation, stature, form of hair, head, face, and nose

have been so utilized. For example, local types have been described by

Deniker 22 by assigning to each group peoples among whom certain

average values of measurements are found. All those that have average

statures, head indices, facial forms, nose forms, and pigmentation falling

within certain limits that may be expressed numerically were assigned

by him to a certain subrace. Although it is possible to give in

this manner a definite description of local types, the biological significance

of the observed differences remains undetermined. Obviously the

classification obtained by the method here indicated will vary according

to the limits set for each division. If we call tall those populations whose

average stature is more than 170 cm., their assignment to a subdivision

will not be the same as the one obtained when we call those tall whose

average stature is more than 172 cm. If no valid reason can be given for

181the choice of one or the other limit, then the subtype so established can

have only a conventional descriptive meaning. If we wish to establish

a biologically significant classification we should have to prove that the

descriptive features selected are morphologically significant. Furthermore,

it would be necessary to distinguish between environmental and

hereditary influences that determine the particular features which are

made the basis of the classification. As a matter of fact, this study has

never been made; and since the lines of demarcation between classes

are arbitrary, these classes will be only a convenient schematic review

of the distribution of certain selected combinations of descriptive features. 13

In the following pages I wish to discuss the question whether a valid

method of comparing closely allied forms can be found, so that arbitrary

classifications may be avoided and measurable differences between two

types established.

It may seem that maps showing the distribution of a single feature or

of combined features would give this information. Retzius' maps of

Sweden, Livi's maps of Italy, Virchow's map of hair color in Germany,

anthropological maps of France, England, and Spain, all illustrate the

distribution of forms of the body, either by showing the areas in which

the same average value of a measurement occurs or by showing areas

in which certain selected values occur with equal frequency. The

maps are intended to convey the impression that sameness of

average values or of frequencies indicates the occurrence of the same

racial forms. They also indicate that the differences between types

are equivalent whenever the differences between averages or between

frequencies of occurrence of selected values are the same. This

has often led to the interpretation that the values whose frequency is

shown represent separate racial types. Thus, the frequency of long-headedness

in an area is often said to mean that a long-headed race

forms a certain proportion of the population, although no biological

basis can be given for the claim that the arbitrarily selected values

represent a separate racial type.

The essential difficulty of our problem may be made clearer by the

following considerations. Each racial type is variable. When we study

the distribution of any particular feature, let us say of the cephalic index

among European types, we find that the forms which occur in each

182area are variable, and the individuals composing the populations of different

areas show in part the same numerical values of the measurement.

The distribution of forms in each population is such that the types overlap.

The average cephalic index in Sweden may be 77; in Bavaria 85.

Nevertheless, there will be many individuals that have the index 81,

both in Bavaria and in Sweden; and according to this particular feature

individuals may belong to either group. We know that if two regions

are not too far apart, in most cases it is quite impossible to assign with

certainty a single individual to either of them.

If we should assume for the moment the variability found both in

Sweden and in Bavaria to be very low, so that the highest cephalic index

occurring among Swedes would be not more than 80 and the lowest

occurring in Bavaria not less than 82, then the two series would appear

to us entirely distinct. It would be quite inadmissible to claim that the

differences between the pair of groups were the same in the cases of

greater variability (which has actually been observed) and the lesser

variability (which has here been assumed), although in both cases the

averages show the same differences. In the latter case we judge that the

difference is greater, or perhaps better, more fundamental.

Obviously our judgment is influenced by the degree of variability;

still more, by the degree of overlapping of the two series. Only if we

assume quite arbitrarily that the individuals that show the average

values of the measurement in question — or some other selected value —

were the true representatives of the whole population, and that all others

were present only as foreign, intrusive elements, or if their occurrence

represented modifications of the typical form due to extraneous causes

— only under these conditions could we say that the difference between

the selected values represents the difference between the types. A concept

of variability like the one involved in these assumptions is, however,

quite inadmissible. The group must be considered as a *class* and

its variability determined by the definition of the class in question.

Our detailed study of the class will always be directed toward the discovery

of new principles of classification by means of which subclasses

are formed whose variability will be less than that of the original class.

In this way we try to define the newly formed subclasses more sharply

than the original class, and the advance in our knowledge consists in

the discovery of the factors that make the subclass more determinate. It

would be quite arbitrary to select one particular individual as the type,

and to claim either that all others are not really members of the class or

183that they are modified forms of the type. This method of procedure

would contravene the fundamental concept of variability, for a variable

comprises all the representatives of a class, the individual components of

which are only defined in so far as they are members of the class — this

in contradistinction to constants which are assumed to be completely

defined and must therefore be the same in every case.

As soon as these principles are held clearly in mind, it appears that

the ordinary definition of arithmetical difference is not applicable in

our case. The term “difference” as applied to variables does not mean

the same as the term “difference” applied to constants. Variables cannot

be brought into a measurable series by the same means that we use

for constants which may be compared by means of an arbitrary standard

that is also constant.

The problem before us is how to overcome these difficulties — how to

give a definite meaning to the differences between variables and make

these differences measurable.

The question has been treated by G. H. Mollison 14 and by J.

Czekanowski. 25 Mollison has discussed particularly the problem of differences

between two types, and he gives an arbitrary formula which

later on was modified by St. Poniatowski. 36

In the following pages I shall discuss some possible approaches to

the problem.

What we call difference in this case is not by any means an arithmetical

difference; it is a judgment of the degree of dissimilarity of two

series. If two series are so far apart that notwithstanding their variability

they do not overlap, they are entirely dissimilar. If they do overlap they

will be the more dissimilar, the less the amount of overlapping. In this

sense we may say that two pairs of series in which the amount and character

of overlapping are equal will be equally dissimilar. While we may

thus determine equality of dissimilarity we are not in a position to

determine quantitatively the degree of dissimilarity.

In treating this problem we may first of all explain the meaning of

similarity and dissimilarity by means of a few examples. Let us assume

that a pure Negro and a pure White population are to be compared.

The types are so distinct in all their features that in comparing them

we should emphasize simply their dissimilarities. Now let us assume

184that a third community is added, consisting perhaps of baboons. It appears

at once that our point of view would be shifted from a consideration

of dissimilarities between Negroes and Whites to the similarities

which they have in common as compared with the baboon, and their

similarities will appear to us now under a new angle and as of different

value.

When we compare a group of blond, blue-eyed North Europeans

with dark complexioned, brown-eyed South Europeans, their dissimilarities

are the most striking feature. If we add a Negro community to

these two groups the similarities between the North and South Europeans

would be much more prominently in our minds. We may observe

the same changing attitude when we speak of family resemblances, or

similarities. When we consider the children of a family, entirely by

themselves, without any reference to any other family, they will appear

to us as dissimilar. If the family has a particular characteristic feature,

let us say, for instance, a long narrow nose, which all the children have

to a greater or less extent, this will become the feature which makes

them similar as compared to the rest of the population.

It is, therefore, clear that the concept of the degree of similarity depends

upon the characteristics of all the groups that are under consideration

and will change with the groups that are being compared.

In investigations on heredity it has been customary to determine the

degree of similarity by means of the coefficient of correlation. When,

for instance, parents and offspring are compared, the coefficient of correlation

between the two will indicate the degree of their similarity.

There is a biological relation between parent and offspring. The average

form of the offspring is determined by the degree to which the parent

differs from the average of the population to which he belongs. In

marriage we may have selective mating through which the forms of two

parents may be correlated. When the husband differs from the average

of the population by a certain amount his wife may differ by a correlated

amount. In both of these cases there is a functional relation between the

two values. The distinguishing feature of fraternal correlation is that

we are dealing with a natural group in which there is no true functional

relation between the members. In a very large fraternity, disregarding

the fraternity as part of a population, the bodily form of one

member does not influence in any way either the average body form of

the rest of the fraternity or the distribution of the individual forms.

This is due to the fact that the members of the fraternity are all members

185of the same variable class, while in all the other cases previously

noted we are dealing with relations between different classes. Fraternal

correlation originates only in a population in which the fraternities

represent different types. If all the families had the same average value

there would be no correlation and no similarity between brothers.

The greater the heterogeneity of the family lines, the greater will be the

correlation and similarity between members of a fraternity.

Exactly the same considerations may be made for racial types. A local

variety may be considered as a fraternal group. The coefficient of correlations

between the local groups will then be a measure of their heterogeneity

or of their dissimilarity.

The problem of the definition of similarities has been treated fully

in experimental psychology. Weber's law is actually based on the observation

that the differences between two pairs of sensations are judged to

be equal. In this case the basis of empirical determination of similarity

is the probability of mistaking one difference for another. It is not, as

was originally assumed, a measure of quantitative value of the sensation

itself. This concept of similarity holds good not only in the case of simple

sensations but also in the field of more complex experience. We may

speak of similarity, or of the probability of failing to differentiate, for

the most diverse kinds and the most complex forms of mental experience.

The problem that we are discussing here has suggested itself in

every comparative study of mental processes.

In an analogous manner we may define the degree of similarity as

the probability of mistaking an individual who belongs to one group for

a member of any of the other groups concerned. The degree of dissimilarity

may then be determined by the probability of recognizing an individual

as belonging to his own group.

The same measurement will occur with varying frequency in the

groups forming the aggregate of groups that is being investigated. Each

individual may belong to any one of these groups and the probability

of his belonging to a particular group will be determined by the ratio

between the frequency of the measurement identifying the individual

as a member of his group and of its frequency in the aggregate. Thus

the probability of the correct assignment of a single individual or of all

individuals of the group having the trait in question can be determined.

When each series is compared with the aggregate of all the series and the

degrees of diversity are established these may be subtracted from one

186another, and in this manner differences in the degree of similarity may

be determined.

When three series are compared in this manner in pairs, the resultant

values are not additive. If only series (1) and series (2), then series (1)

and (3), then series (2) and (3) are considered, the sum of the difference

between (1) and (2) plus that between (2) and (3) will not

be equal to the difference between (1) and (3). This is another expression

of the observation made before that the meaning of similarity

changes with the aggregate of the series that is being considered.

It might also seem possible to arrange the single series in the order

of their averages and to determine their dissimilarities step by step.

Here the difficulty may arise that two succeeding averages may be

nearly the same, while their variabilities may be quite different. Whenever

this occurs quite an erroneous impression of the differences will

be given. The reason for this difficulty lies in the fact that the difference

as here defined depends upon the averages and variabilities of the

single series, and that certain combinations of these two values result

in the same degree of dissimilarity.

In the case treated here the various series enter into the aggregate

according to the number of individuals representing each series. It

might be, for instance, that a large mass of material has been accumulated

for one group and that another group is known through the study

of a very few individuals only. Our expression contains, therefore,

a weighting according to number which obscures the more general theoretical

question. If the groups were known perfectly, then all would

have equal weight, i.e., we should have to assume them to be represented

by equal numbers.

Whether this point of view or the other should be taken depends

upon the clarity of our concept of the characteristics of each group.

If we assume each group as thoroughly studied and therefore known in

all its characteristics, then equal numbers will represent the conditions

adequately. On the other hand, if we are impressed by the unclassified

series as a whole, without detailed study of each group, and if we try to

determine the similarities and dissimilarities on this basis, the actual

numerical frequency of each group will correspond to the conditions

of the investigation. If subjective elements are to be eliminated as far

as possible, we must try to adjust conditions so that equal numbers can

be applied. As a matter of fact, our judgment of similarity in all cases

187of this type is fluctuating; sometimes one group, sometimes another, is

most prominently in our minds, and the actual assignments are therefore

different from the two extreme forms discussed here and may lie

somewhere in between, or they may change with changing mental conditions.

The more thorough our knowledge of each series, the closer

will be the approach to the treatment of all classes as equal in number.

The method here discussed presents the inconvenience that the values

obtained for similarity are the smaller, the larger the number of series

forming the aggregate, so that when the number of similar series is very

great the values of their similarities will be exceedingly small.

In the final results it may appear that some of these series have the

same degree of dissimilarity. If the averages and variabilities of these

series are also indicative of identity, the series should be combined.

It must be remembered that it is possible for a number of different

distributions to result in the same amount of dissimilarity. Since every

distribution depends at least upon two constants, average and standard

deviation, there are whole sets of functions which will give us the same

value for the total probability of mistaking a member of one series for a

member of the rest of the aggregate. However, owing to the general

likeness of forms of distribution, the occurrence of this event is improbable.

On the other hand, dissimilarity can occur only when distributions

are unlike. The minimum amount of dissimilarity is found

when all the series are identical. If there are *n* series, the value of dissimilarity,

in other words the probability of assigning any one individual

to its proper series, will be 1/*n*.

In applying the fundamental thought underlying our considerations

to the classification of mankind, we might ask ourselves which are the

series for which the similarity or the probability of a misjudgment becomes

zero, and these might be considered as the present fundamental

human types. A satisfactory solution of this problem must not be based

on the consideration of a few standardized measurements, but the features

to be studied must be selected after a careful investigation of what

is most characteristic of each group.

It is also feasible to find in this manner outstanding types of a definite

area and to arrange them according to the degrees of their similarity.

The interpretation of the similarity, whether due to mixture, environment,

or other causes, is of course a purely biological problem for which

the statistical inquiry furnishes the material but which cannot be solved

by statistical methods.188

We have seen that, in an attempt to analyze a mixed series according

to types, the individuals of a definite bodily form are not all assigned

by us to the group to which they belong. The impression which

we receive of characteristic forms of a particular series depends upon

the distribution and the forms of individuals whom we assign to it, and

for this reason our impression of the general characteristic form of the

series is expressed by the average of individuals whom we assign to it.

This value is obtained by averaging all those individuals who, according

to our judgment, are assigned to the local type, leaving out the others

that are placed erroneously. This consideration shows that we receive

an exaggerated impression of the characteristics of a series, because

individuals that are similar to other series are assigned to them according

to their appearance and are merged in the general background represented

by the aggregate. Our impression, however, does not correspond

to an actual type. This proves that the attempts to analyze a series into

a number of subtypes according to similarities of individuals is methodologically

not admissible, and that all subdivisions must be based on

the study of the series as a whole, not upon selected types.

The chief difficulty in the practical application of the method outlined

in the preceding pages is due to the facts that the degree of similarity

depends upon the aggregate treated, and that there is no relation

between the numerical values obtained for different aggregates. Not

even the equality of differences between several given series need persist

if new members are added to the aggregate or are taken away from it.

In cases of continuous changes of a type from one extreme form to

another, an artificial classification of the aggregate is unavoidable. By

means of repeated adjustment equal degrees of similarity might be

found according to the method outlined here, but the actual carrying

out of such a plan offers serious difficulties. In such cases each series

might be considered as a specialized form of the general aggregate

and compared with it. The aggregate itself may, however, be established

in two different ways. We may disregard the number of existing

individuals, considering each morphological type contained in the aggregate

as a unit. The units would then be given equal weight (i.e.,

equal numbers of cases). Or we may take the whole series as it exists at

the present time, counting the total number of individuals that it contains,

regardless of local types that may represent the same morphological

form. By either of these methods we ascertain how dissimilar each

morphological type is from the aggregate, but these values cannot be

189used to determine the mutual dissimilarities of the single series contained

in the aggregate. When the types are combined according to the present

actual number of individuals representing them, the most numerous type

will appear least distinct from the average, merely on account of the

large number of its members. This difficulty can hardly be avoided by

comparing each series with the aggregate of the remaining series, because

by this method the standard of comparison is changing. On the

other hand, the formation of the aggregate by giving equal weight to

each morphological type entails the difficulty that we tried to avoid,

namely, an arbitrary classification of the groups as a number of morphological

types.

The problem may be approached in another manner. We may determine

the frequency distribution of the differences between individuals

belonging to one series and those belonging to all the series of the aggregate

including the one selected for study. In this inquiry we have to

determine the average difference between the representatives of one

series and those of all the series, and the variability of this difference.

When the series are arranged in pairs, the differences between the averages

are additive, but the variabilities are not comparable. The interrelations

between the series can be determined only when we consider

any one series in relation to the whole series.

The problems take a slightly different form when populations are

compared with regard to features that occur in a certain percentage of

individuals and are absent in the rest. If, for instance, one population

consists of 15 per cent Negroes and 85 per cent Whites, another one of

30 per cent Negroes and 70 per cent Whites, it might seem that the

difference could be stated simply as a difference of 15 per cent, but

obviously the dissimilarity of these two types of population would not

be the same as in another pair in which we have 40 per cent Negroes

and 60 per cent Whites in one and 55 per cent Negroes and 45 per cent

Whites in the other. In the latter case the populations would seem more

alike to us than in the former case. The difficulty is still more pronounced

if there are present not merely two types but a larger number

in varying proportions. In all these cases we may apply the same

methods which we used for the determination of similarity of measurable

quantities.190

11 *Quarterly Publication of the American Statistical Association* (December,

1922), pp. 425-445.

22 *The Races of Man* (London, 1900).

31 See also St. Poniatowski, “Ueber den Wert der Index Klassifikation,” *Archiv für Anthropologie*, N.F. vol. 10 (1911), p. 50.

41 *Morphologisches Jahrbuch*, vol. 42, p. 79.

52 *Korrespondenz-Blatt der Deutschen anthropologischen Gesellschaft*, vol. 40.

63 *Archiv für Anthropologie*, N.F. vol. 10 (1911), p. 274.