J. A. Harris 
447 
as compared with random pairs from the population (or universe) which they 
constitute. Thus fraternal correlations are intra-class and also intra-racial, as 
are also homotypic coefficients. In one case, the " class " is fixed by ancestry 
and family environment, in the other, by ancestry and individual environment. 
Avuncular correlations — in which all nephews (nieces) of the descendant genera- 
tion are compared with all the uncles (aunts) of the ascendant generation — are 
illustrations of inter-class, but also intra-racial, correlation. 
The most familiar illustration of intra-class con'elations are the fraternal and 
homotypic ; practically the only inter-class relationships hitherto considered are 
the avuncular and the cousinship in heredity. But these correlations might be 
powerful research tools in morphology and physiology as well as in heredity 
and sociology. 
The reason for their limited use in the past is in part technical. When the 
number of observations is as large as is desirable, the formation and verification 
of the ordinary correlation table (where each individual of the N used is entered 
only once) is an irksome task, but when each measurement is compared with 
a number of others, the purely clerical labour involved in tabulation becomes 
onerous in the extreme. If n be the number of individuals in any class, the 
number of combinations within the class is a« (h — 1), while if the tables be 
rendered symmetrical* as is generally desirable~f-, the number of combinations 
for each class is n(n — l), which gives for the m classes constituting the popu- 
lation S[n(n — 1)] or, where n is constant, m [n (n — 1 )] entries in the intra-class 
correlation surface. With n as low as 20 and in only 250, this gives 95,000 
combinations. In inter-class correlation if p be the number in the first and 
q the number of observations in the second (associated) class, the permutations 
for a class are pq, and the entries for the population S (pq), or iiipq if both 
p and q are invariable. Obviously, neither m, p, nor q need be large to give 
very heavy tables. 
The labour involved in constructing such tables and making sure that they 
are free from errors is very great indeed. To meet this difficulty in extreme 
cases, Pearsonj has shown how formulae may be deduced for obtaining the 
coefficient of correlation between any grades of kindred from the means of 
arrays into which the kindred may be grouped. The method suggested here 
reduces the work to still simpler terms, obviating entirely the necessity for the 
* For discussions of the advantages of the symmetrical table sec K. Pearson and others, Phil. 
Trans. Roy. Soc. Lond. A, Vol. cxcvn. pp. 285—379, 1901 ; K. Pearl, Biometrika, Vol. v. pp. 219—251, 
1907; K. Pearson and A. Barrington, Euyenics Laboratory Memoirs, No. V., 1909; J. Arthur Harris, 
Biometrika, Vol. vn. pp. 214—218, 1909. 
f Jennings [American Naturalist, Vol. xlv. pp. 123 — 128, 1911) has suggested a method of calculating 
the symmetrical table constant without actually rendering the table symmetrical. I believe his method 
is not serviceable where more than a single pair of characters are involved. 
J Pearson, K., Pliil. Trans. Roy. Soc. Lond. A, Vol. cxcn. pp. 271—274, 1899. For other methods 
of dealing with problems involving many measurements see Biometrika, Vol. n. pp. 09 — 71, 77 — 78, 
1902. 
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