VARIANCE, COVARIANCE, AND CORRELATION 279

When persons are correlated instead of tests, much the same
argument can be used. But a further problem then arises, because,
though we may nearly always assume a normal distribution for
persons, we cannot always do so for tests or test-items. Very fre-
quently—for instance, in the investigations on esthetic appreciation
already mentioned—the test-items are chosen so that there are
approximately equal differences between successive pairs, and are
ranked by the persons in serial order.1 In such cases it would
appear, from a comparison of calculations actually carried out, that,
unless N and k are small, the same equation may be still used for F
(with the necessary modifications in the degrees of freedom) and
(for approximate estimates) the same criterion may still be employed.
When N and k are small, the obvious alternative is to calculate the
actual distribution and use y? or some related function.2

The advantage of using analysis of variance lies not only
in the more precise tests of significance that the method
allows (a point of special importance with small samples),
but also in the further possibilities which the method opens
up for the effective planning of experiments. On the other
hand, factor-analysis, if somewhat more laborious to carry
out, claims to yield additional information which could not
be reached by an analysis of variance alone—e.g. the detailed
factor-saturations and weighted factor-measurements.
Moreover, in applying analysis of variance to psychological
data, there are, it will be seen, two assumptions involved
which factor-analysis may at first sight seem to avoid: first,
analysis of variance assumes (or appears to assume) that the

items in the columns (e,g. in agricultural experiments, the interaction of
different types or treatments of soil with different types or varieties of
plants). There may, however, also be an interaction between the row items
(or the column items) among themselves (e.g. of the different treatments
with each other). In a psychological research, this would take the form of
interactions between the different persons assessed, or again between the
different traits assessed. Here there seems to be a new field of research which
hitherto has been almost entirely neglected and yet which is of supreme
importance in a complex subject like psychology, where almost every measure-
able characteristic is apt actively to influence every other.

1 Cf. Dewar [118], p. 33. A fuller comparison of the results of factor-
analysis and analysis of variance for problems in which persons rather than
tests are correlated will be found in the sequel to B. M. D. Cast's paper [134].

2 Friedmann,i Use of Ranks to Avoid Assumption of Normality Implicit in
the Analysis of Variance/ J. Amer. Stat. Assoc., XXXII, 1937, pp. 675