338 THE FACTORS OF THE MIND

can claim statistical significance ? and, secondly, what is the
relative importance of each one ?

The Significance of the Factors.—The first question is most
simply answered by testing the significance of the individual
correlations or residuals on which each factor is based. If a
residual is three times the probable error, it is customary to
assume that it is significant of an additional factor. There
are, however, a number of familiar pitfalls attending this
criterion.

First, the probable error should be that of the observed correlation :
in many investigations the residual is treated as itself an observed
correlation, and its probable error taken direct from the usual
table ; in others it is calculated as a probable error for the difference
between two observed correlations.1 Secondly, the number of
residuals should be taken into account : if there is only one residual
of the size required in a table for 7 tests, or if there are half a dozen
of that size in a table for 16 tests, we must not thereupon conclude
that they are significant merely because each reaches the conven-
tional level: for with chance distributions these are just about the
numbers we should expect. On the other hand, if there is a large
number of residuals somewhat less than three times the probable error,
we must not thereupon conclude that there is no further factor :
we cannot even conclude that a further factor is improbable : for,
if we are dealing with a single coefficient and that coefficient 13,
say, only twice the probable error, the chances are still more than
4 to i against so large a figure arising as a result of random sampling ;
and if half the figures in the table are of this order, their cumulative
evidence will be strongly in favour of a common factor, even though
none of them is up to the conventional level.3

1 The residual is the difference between an observed and a given hypo-
thetical correlation, and the hypothetical correlation is not to be treated as
if it were another observed correlation subject to sampling errors. The
proper procedure is that prescribed for testing " the significance of the
deviation of an observed coefficient from the expected value " (e.g. Fisher,
[50], pp. l89-o,o5>ex. 31).

2 In the early investigations with mental and scholastic tests, some of us,
who found certain striking deviations from the expected hierarchical value
occurring in table after table, claimed them as at least highly suggestive of
group-factors. Our critics, on the other hand, observing that the residual
or * specific' correlations were seldom equal to three times the probable error,
cited the same figures as disproving the existence of such group-factors, even
when the odds were in fact definitely against such figures having arisen as