VARIANCE, COVARIANCE, AND CORRELATION 277

The relation between the two average correlations (with
and without self-correlations) is shown by

• = r?

It will be remarked that in this case the use of the correla-
tion ratio is equivalent to assuming known self-correlations.
It is for this reason that the intra-class correlation or its
equivalent, which (as we have seen) does not involve these
latter, may be regarded as an improvement on, or correction
of, the bare correlation ratio. Evidently the critical ratio,
given above for testing significance, if computed from the
saturation coefficients, could be expressed as follows :

(JV), ' V - (Sr,,r

Generalizing this result, we see that factor-analysis, like
analysis of variance, may be regarded as a division of the total
test-variance (or of a simple function of it) into two parts :
one part due to a common source of variation and the other
a residual variance. And our conclusions may be summed
up as follows. Under the simple conditions assumed, (i) the
average saturation coefficient, calculated from the completed
correlation table, is identical with the correlation ratio,
i.e. with the ratio of the observed standard deviation of
the means to the maximum standard deviation; (ii) the
average intercorrelation (i.e. excluding the assumed self-
correlations) is identical with the intra-class correlation,
which may be regarded (by those who prefer to think in
terms of the more familiar expressions) as a kind of corrected
correlation ratio; (iii) in factor-analysis the chief com-
parisons turn on the mean of the squares of the saturation
coefficients ; in analysis of variance it turns on the square
of the mean of the saturation coefficients.

So far we have assumed only one criterion of classification. We
have considered the various test-measurements as grouped by persons
only. In such a case we inquire whether the persons differ signific-
antly as regards their average performances in each of the k tests;
and each person's average is taken as indicating his ' factor-
measurement * in the ' general factor for the tests/ i.e. in the general