360 THE FACTORS OF THE MIND

* general factor,' but to show how completely the correlations as
they stand can be explained by the high variances, not of one factor
but of a very few. Let us therefore turn to a measurement of the
general hierarchical tendency, and from * first-moment' criteria to
c second.' As before we may begin by considering the theoretical
figures suggested by the general trend of the earlier correlation
tables previously published. Taking the higher of the theoretical
proportions in Table VII, we may say that the unadjusted standard
deviation of the factor-variances tends in most tables to rise towards
a figure of 9S^Vn' (With an indefinitely large number of variables,
therefore, the adjusted standard deviation will also tend towards
that figure, so long as the correlations remain preponderantly
positive : with a limited number of tests, the adjusted standard
deviation will be much lower.) If the total variance remained the
same, the corresponding standard deviation for a perfect hierarchy
would be '7SVn- Hence for the ordinary type of table the tetrad-

-ZW2

difference ratio 07]22 = 7^^ wu* ke in the neighbourhood of -50 ;
(2*Vt)

for a perfect hierarchy the figure would, as we have seen, soar to
I-oo. We may thus say that, as a general rule, a ratio appreciably
above -50 may be considered as indicating an unusually close
approach to the hierarchical pattern ; and a ratio appreciably
below -25 an unusually close approach to equal or equalized factor-
variances.

Let us compare this with actual results.1 I shall again choose
the two tables analysed by Thurstone, chiefly because his con-
clusions seem at first sight most strongly opposed to Spearman's
views on hierarchical tendencies. At the same time his figures
will enable us to see how far the theoretical criteria are applicable
to data which fall short of the requisite conditions in two or three
common respects : first, an approximate method of analysis—
simple summation instead of weighted—has been employed, so that
the factor-saturations are not strictly independent, and the factor-
variances are not exactly equal to the latent roots ; secondly, not all
the factors have been extracted ; thirdly, the high probable errors
have apparently increased the variances of the less dominant
factors.

Let us begin with Thurstone's illustrative analysis of Brigham's
table for 15 intelligence-tests ([15], pp. 108 el seq.) : this is typical
of the more usual table, and Thurstone has also appended an
analysis by weighted summation. With the latter method (p. 133)

1 Once again I am indebted to calculations made by Mr. Eysenck (for
Thurstone's work) and Mr. Barlow (for that of earlier investigators).