HIERARCHICAL CRITERIA 359

efficients that he has tabulated, the factor-variances can be readily
calculated for his 12 factors. The proportionate values are shown
in the last four columns of Table VII. Since n is much higher than
usual, the values will be exceptionally low if we treat n as the
denominator (column 3). For the present comparison, therefore,
it will be better to take as the divisor what Thurstone calls " the
significant common factor variance," i.e. the total factor variance
for the first 12 factors (Ehf in his notation), namely, 39-6,1 in short,
to use 07)^ instead of -^-j2. It will be seen that, except for certain
explicable peculiarities in the tail, the form of the curve is very
similar to that already indicated by the theoretical values in the
first column.

One or two points call for a passing comment, (i) The increase in
the factor-variances at the loth and nth factors is evidently due to
Thurstone's change of procedure : " at the loth factor a refinement
was introduced . . . that increases the amount of the total variance
that is accounted for by each new factor." The anomalous swelling
towards the end of the tail is thus at once explained, (ii) With
tests applied to 240 persons we should expect coefficients represent-
ing a true correlation of zero to be distributed about zero with a
standard error of ^ *°6" an<^ a probable error of ^ -04 : these figures
seem to agree with the standard deviations given by Thurstone for
his residuals. Accordingly, it is only natural to find the factor-
variances diminishing far more slowly when they reach this level.
That explains the further peculiarities towards the end of the table,
(iii) Since, however the tetrachoric correlation was used, Thurstone
tells us that their standard error should be placed at a higher
figure than usual, namely, about -09 (p. 61), Now the standard
deviation of the first set of residuals gives a figure of only -127 and
of the second set -098. After all, therefore, it may be doubted
whether more than one or two factors at most can be considered as
definitely established, and whether the evidence for more than two
or three others is even suggestive. On external grounds I certainly
suspect that at least half a dozen factors are probably operative :
but the intrinsic evidence by itself can hardly be cited as disproving
Spearman's doctrine of a single * general factor.'

(b) Second-moment Criteria.—But, as I have already argued, the
real issue is not to prove or disprove the existence of such a single

1 Thurstone, of course, does not profess to analyse the total amount of
variance (i.e. », since his tests are presumed to be in standard measure), but
only that part of it that is attributable to common factors : this amount in
turn is made as small as possible by keeping the number of common factors as
small as possible.