152 THE FACTORS OF THE MIND

" the i expected ' values can be calculated from the marginal totals,
so that the total of the £ expected' values agrees with the corres-
ponding marginal total" (Yule, [25], ch. v), and the same criteria
can be applied to test divergences. Accordingly, in my first paper,
I proposed that we should take for each test " the sum of its co-
efficients as measuring its general tendency to correlate " ; and I
indicated that, for fitting the observed coefficients with an ideal
hierarchy, " theoretical values might be obtained by various
mathematical formulae." * Later, where the best possible fit was
desired (i.e. one conforming with the requirement of' least squares'),
weighted summation was substituted for unweighted. But in
either case the final values were calculated by applying the product
theorem,

To reduce the analysis to the lowest possible terms, one
further step remains to be taken. Just as we have nor-

class, then the requirement is that the several traits or tests shall yield a
sub-classification of his performances according to one and the same general
principle, e.g. the tests or traits must all be aspects of general intelligence, but
otherwise independent (uninfluenced by any further i overlapping specific '),
" Tests of homogeneity are mathematically identical with tests of in-
dependence " (Yule, loc, cit.)* This principle, however, usually causes some
surprise to the non-mathematical student. The explanation is to be found
in the technical meaning attached to the words * independence ' and' associa-
tion.' As Yule points out, " the student should carefully note that in
statistics the word * association' has a technical meaning different from the
one current in ordinary speech." If all our tests measure performances
belonging to the same general class, they appear to be more or less correlated,
But the * association * contemplated by the criteria is a specialized association
existing over and above that due to their * homogeneity ' as members of the
same general class or * universe ' ([25], p, 28 : for symmetric contingency
tables resulting in this way, cf. Tables A and B, p. 74, and Table IV, p. 70).
1 Brit. J. PsychoL, III, 1909, pp, 160, 163. It will be seen that this
account assumes the presence of self-correlation, raa, r^, . . ., fpp, rq^ fitting
the hierarchical matrix. With my own formulae these values had to be
supplied by inter- and extra-polation. This procedure aroused some
criticism ; and an alternative formula was kindly suggested by Prof. Spear-
man. However, I now consider that the simple summation formula still
yields the quickest and (if successive approximation is employed) the most
accurate procedure, apart from the more elaborate method of weighted
summation. Although, on theoretical grounds, the relative merits of the
different formulae have been much disputed, the actual differences between
the figures obtained are usually all but negligible ([93], p. 294)* With a
perfect hierarchy all the procedures commonly suggested yield precisely the
same results; in particular, the results of weighted summation, are
identical with those of simple summation.