272 THE FACTORS OF THE MIND

attempting to indicate what are the general relations be-
tween the two procedures. So far as I am aware this has
never yet been done.

The similarity both in object and in method is somewhat
obscured by the current terminology. What is usually
described as an * analysis of variance ? is really an analysis
of a matrix of measurements by means of its c square-
sums.' Similarly, what is commonly regarded as a factor-
analysis of correlations is, in my view at any rate, really an
analysis of the same matrix of measurements (or a standard-
ized version of it) by means of its < product-sums' ; and
the factorization consists essentially in c rotating axes'
until the product-sums are reduced to an equivalent set of
c square-sums ' ([101], p. 77). Under simple conditions the
relations between the two methods are elementary and
direct, and can be expressed in algebraic form.

Let us confine ourselves first of all to the analysis of
correlations or covariances between tests. We may suppose
that k tests have been applied to N persons x and that the
test-measurements are expressed as deviations about the
average for each test. The simplest case is that in which
(i) the initial measurements are in standard measure

inadequacy of the earlier statistical procedure. Mr. Eysenck has carried
this out at my suggestion; and has shown that, contrary to Wells' own
inferences, his table reveals a positive and significant general factor. This is
also confirmed by a formal analysis of variance. Such results appear to me to
be a striking illustration of the valuable advance in method made during the
last 30 years.)

1 It is not easy to find a notation which shall be consistent with that cus-
tomarily used in factor-analysis and in the analysis of variance; indeed, in
expositions of the ktter the usage of different writers varies in a way that is
most confusing to the student. I suggest that where (as is usually the case)
we are concerned with testing a classification by one criterion only (either by
persons or by tests) we follow Fisher and Yule, and write k for the number
in each of the ' classes* (or ' populations') whose means are to be compared,
retaining N or n to denote the number of ' classes,' according as the £ class'
denotes the set of measurements for the same test or the set of measurements for
the same person. I shall use Q (quadrata) to denote square-sums, P to denote
product-sums, and V to denote the mean square or variance obtained by
dividing the square-sums by the number of degrees of freedom. The
present use of the symbols-JP, Q, F, and F has no relation to their use to
denote certain matrices or axes in factorial theory.