HIERARCHICAL CRITERIA 335

that they display simple and regular patterns. An extreme
example is the covariance matrix whose pattern is hier-
archical. Since in such a case the variance of the one
* general' factor is equal to the total variance and that of all
the others equal to zero, the standard deviation is a maxi-
mum. According to Spearman, a hierarchical pattern of
this type is the special feature of every correlation table
obtained with cognitive tests.

Nevertheless, that is not what we should have anticipated.
Many psychologists—Thorndike, for example—apparently
expected the mind to be composed of a number of inde-
pendent c fundamental abilities' or * faculties/ whose
influence was approximately equal.1 Indeed, this, it
would seem, is the assumption which underlies the hypo-
thesis of * simple structure.' But on this assumption
(provided the tests for each ability were equal in number
and equally efficient) we should find, even with the method
of weighted summation, that the factor-variances were all
of much the same size. In that case both their range and
their standard deviation would be approximately nil.

The £ Principal Tetrad-difference ' Criterion.—In practice
what we usually discover is a set of factor-variances whose
standard deviation lies between these two extremes, some-
times inclining more towards the maximum value, more
rarely tending in the direction of zero. In the early days
of factor-psychology it was assumed by many writers that
any departure from the hierarchical ideal was attributable
to errors of sampling; and Spearman's examination of all
the available correlation tables [24] provoked a sharp con-
troversy as to whether a single general factor was or was not
sufficient to explain the entire amount of observed correla-
tion, and in particular whether his criterion—the calcula-
tion of the intercolumnar correlations—provided a safe
and adequate test. Later, he himself suggested a second
and still more elaborate criterion, namely, the calculation of
all the c tetrad differences? and their sampling errors,
with a view to showing that, within the margin of error,
the * tetrad differences' are all zero [52]. Our first task,
therefore, will be to examine these two criteria, and to see
1 Educational Psychology, 1903, p. 39.