On the cause of hierarchical order among the correlation coefficients of a number of variates taken in pairs

The object of this paper is to investigate the cause of a phenomenon in the realm of chance, which has become technically known as hierarchical order among correlation coefficients, and has been held to prove the existence of a general factor running through the correlated varieties, and the absence of group factors running through some but not through all of them. The question arose in the science of experimental psychology, but it is here, after the introductory paragraphs, considered as a general question in probability. When mental tests are applied to a number of subjects, and the correlations between the marks are calculated for every possible pair of tests, the correlation coefficients obtained show, as a rule, a tendency to arrange themselves in hierarchical order. By this is meant that the order of sequence of the mental tests, according to the size of the correlation of each with a fixed one of their number, proves to be largely independent of the choice of this latter. If the mental tests, which we may call by the namesx 1 ,x 2 ,x 3 ,..., have been arranged in order according to the total correlation of each with all the others, and if a square table such as the following be formed:— then the hierarchical order shows itself in the fact that there is a tendency for each correlation coefficientr to be greater than its neighbour on the right and its neighbour below it. A method, which has been used for measuring the degree of perfection of the hierarchical order, is to take the correlation of each pair of columns of the above table. Clearly, if hierarchical order is present, all these correlations will be high, and in the most perfect case will become unity.