450 
Intra-Class and Inter-Class Correlations 
purposes, (b) it is desirable to calculate several constants from the same 
material with modified grouping or weighting of some of the classes, (c) it is 
desired to obtain the correlation without seriating the measures for the classes 
at all*. 
Problem III. To Determine Direct and Cross Intra-class Coefficients of 
Correlation from the First Two Moments of the Individual Classes. 
Let x be the measure of a first and y the measure of a second character of 
an individual of one of the several classes forming the population N. Then if 
correlations for the population be determined, r X!/ , where x and y are measures 
upon the same individual, may be called an organic correlation, while r XlXl , r VxVi 
may be designated as direct and r XlVi as a cross intra-class correlation, where 
the subscripts 1 and 2 indicate that the measures are taken on a "first" and 
a "second" individual of the class respectively. 
A. Direct Intra-class Relationships. 
Let n be the number of individual measurements, e.g. of siblings or homo- 
types, in one of m classes. Let 2 indicate a summation within any class and 
S a summation of classes. For an individual class the rough moment coefficients 
around 0 as origin are S(cc')/n and %{x'-)jn, while the possible combinations 
of individuals are n(n — 1). Since the table is symmetrical m 1 = m 2 , a^^a.,, and 
the mean product moment for any class is 
{[%(x')Y-2(^)}/n(n-l) (v), 
or for the whole population of m classes 
S{[2(x')Y-Z(x*)}/S[n(n-l)] (vi), 
or when n is constant from class to class 
8 {[S (*■')? - S [n(n - 1)] (vii). 
To complete the calculation of r only the first and second moments for the 
population are necessaiy. Two cases are possible : (a) the n's are identical for all 
classes, and (b) n differs from class to class. In case (a) the moment coefficients 
for the whole weighted population of 8 [n(n— 1)] — m\n(n — 1)] individuals are 
unchanged by weighting, and are given by 
S[$(x')]/N, 8[X(^)]/N (viii). 
For case (b) the arithmetical routine is a little longer, the classes being weighted 
differently, but always in a (n — l)-fold manner. Thus the moment coefficients are 
S [(« -1)2 (x')]/S [n (n -1)1 8 [(* - 1) 2 W}/S [n (n - 1)] (ix). 
From this point the calculation of r is straightforward by (i). For clearness 
both case (a) and (b) will be illustrated. 
* In this case the data for the means and standard deviations are obtained by summing first and 
second powers of the numbers from the record sheets without classification. 
