_348	THE  FACTORS OF THE  MIND
assign an exact rank is hardly practicable. We know that the
majority of the factors have no statistical significance ; but, until
we have applied a criterion of significance, we cannot say how many
of them can be treated as non-existent. We must, therefore, fall
back on a less summary method of determining the ratio involved.
According to the nature of the precise characteristic we are pro-
posing to test, we may take the ratio between the observed and the
maximum values for the standard deviations (or variances) of
(i) the initial correlations or their residuals, (ii) the saturation
coefficients derived from these correlations or residuals, (iii) the
factor-variances derived from these saturation coefficients, or
(iv) yet higher ^c moments' derived from the factor-variances by
carrying the same process to a further stage.
(i) The Ratio of Residuals.—Let us begin by examining the
means and variances of the residual correlations, since this form of
comparison has been adopted by Thurstone in his last, most interest-
ing research. Consider the matrices of residual correlations,
obtained after eliminating s, s-{- i, ... factors from an empirical
n X n correlation matrix. The relative importance of two successive
factors may be expressed by comparing either the mean deviations
or the standard deviations (or variances) of the two successive sets of
residuals. If (as usual) the residual matrices are bipolar, we can
take the ratio of the variances to be
Let us first put jt = o.    We then have, for the first of these ratios,
n
Ev*
——.   This ratio will certainly vary with the size of f_1? and
vj + Z v*
2
will therefore give some indication of the hierarchical tendency of the
initial matrix. But as a measure of that tendency it is not cast in
so convenient a form as the ratios already proposed.
Let us now put s = number of significant factors. Since Thur-
stone deals with a reduced correlation matrix of minimum rank, we
may designate that minimum rank (n — i). Then the rank of the
residual matrix will be (say) u ±= n — s — t. Hence (s + t) of its n
latent roots (or fact or-variances, as we call them, in dealing with a
correlation table) will be zero. But if all the significant factors
have already been extracted, then we should expect the u remaining
non-zero factor-variances to be approximately equal. Then
n—r
SSrf = Zv* = uv^z.     After   extracting   one   more   factor,   the
s+r