340 THE FACTORS OF THE MIND

Instead of comparing each residual with the corresponding error
(as in the two preceding methods), the authors of many research
theses are content to compare the absolute mean of the residuals
(or their root-mean-square) with the mean deviation (or the
standard error) to be expected by chance. Usually, the latter is
computed from the probable or standard error either of the mean
correlation x or (less usually) of a zero correlation,2 with a sample of
the size observed. But I agree with Guilford that, for exact
purposes, " this criterion is too crude." 3 Not infrequently it leads
to the extraction of too few factors. The standard deviation of the
residuals may be only equal to, or even less than, the standard error
of the mean correlation ; yet several isolated residuals may be more
than twice the standard error of the corresponding correlations.

The Measurement of the Hierarchical Tendency.—The
relative importance of the several factors (a) in each of the
tests is specified by the squares of saturation coefficients and
(Z>) in the matrix as a whole is specified by the factor-
variances, i.e. by the sums of those squares. In an earlier
paper I have shown that, by repeated self-multiplication or
' squaring,' any matrix, if not already hierarchical, will be
reduced sooner or later to a hierarchical form as nearly
perfect as we desire ([102], p. 186). When this stage is
ultimately reached, the figures in the leading diagonal will
be proportional to the squares of the saturation coefficients
for the first factor. The speed with which a hierarchical
pattern is thus approached depends essentially on the
amount of separation between the first latent root and the

even when due to no common factor : for the errors of correlations are
themselves correlated; and, if some of the errors are large, the consistency
conditions that must necessarily obtain may of themselves produce some slight
hierarchical tendency. To take this into account would require an elaborate
correction of the ordinary formula ; but in most cases the correction is of an
order that does not seriously affect a broad determination along the simpler
lines. See M. Davies, * The General Factor among Persons' (unpublished
appendix to thesis): cf. also Pearson and Filon, < On the Probable Errors of
Frequency Constants and the Influence of Random Selection on Correlation,*
Phil. Trans., CXCI, A, pp. 229-311 ; and Pearson * On Lines and Planes
of Closest Fit,' Phil Mag., II, p. 559 et seq. (which deals more particularly
with the fitting of principal axes in the case of more than two variables).
* Cf. [84], p. 147.

2 Cf. [107], p. 45.

3 Loc. cit.9 p. 495.