464 
1 ntra-Class and Inter-Class Correlations 
first two moment coefficients of individual classes, [2 (x) 2 (y')]/pq the product 
moment coefficient for an associated pair of classes and 
8[Z(a>')X(y')]/8(pq), (xiii) 
the rough product moment coefficient for the whole population. 
For every first class of p individuals there is an associated second class of q 
individuals. Hence, the individuals of the first class are always weighted* in a 
g-fold manner and those of the second class in a p-fold manner. The population 
constants weighted for the calculation of r are given by the I'ough moment co- 
efficients about 0 as origin, 
S[qX(x')]/S(pq), Slqti^/Sipq), (xiv) 
8 [pZ (y')]/S (pq), 8 [pt (y'*)]/S(pq) (xv) 
Since x and y may be measures of the same or different characters it is clear 
that the formula is the same for direct or cross inter-class correlation. 
Illustration (IV) 10. Racial Heterogeneity anal Homogamy in Man. 
It has sometimes been suggested that homogamy for physical and mental 
characters in man is due to (a) the existence in the population of numerous local 
races, (b) the excessive frequency of intra-racial marriages, attributable to pro- 
pinquity merely, and (c) the pooling of records from many local races in the same 
correlation surface. 
A study of the literature of assortative mating convinces one that so simple an 
explanation is inadequate. Nevertheless, it is very important to give full weight 
to any factors tending to bring about a spurious correlation for assortative mating. 
One way of testing the question is to make random matings within the local 
races, pool the results for a series of districts, and determine the correlation f. The 
influence of the local differentiation can be read off in the inter-sexual correlation 
coefficient thus secured J. 
A convenient series of materials for stature is that gathered by Tocher§ for 
the counties of Scotland. To be sure, his measurements are on individuals who 
have become insane, but data for the normal population are lacking. 
We may work from the table of constants (Bionietrika, Vol. v. Supplement, 
Table XIX), taking populations of arbitrary size, say 100 males, m, and 100 
* When it is possible to collect data so that either p or q, or both, are constant throughout the 
m pairs or classes the work is obviously considerably simplified. Unfortunately such cases are rare in 
practice. 
t This was actually done for parish churchyards in the cooperative study of assortative mating for 
duration of life. Cf. Biometrika, Vol. n. p. 485, 1903. 
% Of course our problem has some similarity to but differs fundamentally from that considered 
by Tschepourkowsky in his study of the resemblance between man and woman. Cf. Biometrika, Vol. iv. 
pp. 161—168, 1905. 
§ Tocher, J. F., "The Anthropometric Characteristics of the Inmates of Asylums in Scotland," 
Biometrika, Vol. v. pp. 298—350, 1907. 
