VARIANCE, COVARIANCE, AND CORRELATION  287
is then seen to be a very ancient one, most familiar from its occur-
rence in solving normal equations deduced from the requirement of
least squares (cf. p. 148). A matrix of order n X N, such as is
obtained by testing N persons with n tests, has a rank of n only, if
n < N9 and is therefore deducible from n factors, since any
(N — n) of its columns are linearly dependent on the rest. Self-
multiplication is the obvious device for reducing such a matrix to a
simpler and symmetrical shape, with an order equal to its rank;
and correlation is essentially a process of self-multiplication, which
(as we shall see) is reapplied in factor-analysis. But in correlating
tests, before multiplying the rows, (i) we first subtract their aver-
ages : this yields, after cross-multiplication, a matrix of unaveraged
covariances, but eliminates one important source of variance—
roughly identifiable with the * general factor for persons.' (Alter-
natively, if we start by correlating persons, we virtually eliminate
the * general factor for tests.') (ii) Secondly, we divide the product-
sums by the appropriate standard deviations, which, means pre-
multiplying and post-multiplying the product-matrix by a non-
singular diagonal matrix. This leaves the rank and the number of
factors unchanged,1 but alters the weighting of each, with the result
that the relative size of one or more of the factors may at times be
so diminished that it appears in effect suppressed. But, however
great the apparent change, a -factorial matrix jilting the cor-
relations can be always derived from that obtained on analysing the
covariances (or vice versa) by simply prefixing the diagonal matrix
(or its inverse) ([102], p. 193, [114], p. 174). Analogous changes, it
may be added, are also introduced by selection, e.g. by taking either
a group homogeneous as regards the general factor, or a group
differing in heterogeneity for the different correlated traits ; evi-
dently this will have the effect either of reducing the differences
between the averages or of altering the magnitudes of the standard
deviations.
There has been considerable disagreement between psychologists
about the need for correcting correlations for attenuation before
their factorization is attempted. Spearman and Hotelling, for
example, would first correct; Thurstone, Kelley, and I myself2 use
1 Unless a further modification is introduced, it also puts the variance of
each test (its self-correlation or * reliability coefficient *) equal to l-oo. The
result is to throw part of the specific and error factors into the general factors.
If, however, we substitute the amount of variance attributable to tjie
significant factors only, the rank of the matrix is at once reduced (cf. pp. 154 f.).
1 In my view, if the reliability coefficient is so low that the raw correlation
requires correction, that is a reason, not for factorizing corrected correlations
but for improving the experimental technique.