330 THE FACTORS OF THE MIND

On the other hand, with the centroid method there are
more factors than tests ; and hence the factor-measure-
ments can only be estimated approximately : the calcu-
lations for Thurstone's factors necessarily involve the
same kind of indeterminacy as has been shown to be involved
in the estimation of Spearman's g—and for precisely the
same reason.

There is, however, a still more important difference. In
my Memorandum I briefly stated that the simple summa-
tion method virtually treats the first or dominant factor
(' the general factor ') as the simple sum or average of the
several test scores (just as it virtually treats the factor
loadings as the sum or average of the correlations),1 while
the least-squares method treats it as given by a weighted
sum or average of the several test-scores, determining
the weights on the lines of the ordinary multiple regression
equation. In the same way, the secondary or subdominant
factors are treated as unweighted and weighted averages of
the deviations.

These corollaries may perhaps best be demonstrated as follows.
As before, let M be the matrix of measurements, so standardized
that MM' = R> the matrix of correlations or standardized co-
variances. Let F be the ' factorial matrix ' containing the saturation
coefficients, and P the e population matrix * containing the factor-
measurements. Then M = FP and P^F'R^M ([101], p. 80).
For simplicity consider only the first or * general * factor (g), i.e. the
first column of F (/j) and the first row of P (^>a). Then, by the
centroid formula :

go on to treat the group studied as a sample only, the factor-measurements
deduced exactly for this group may be regarded as approximate estimates for
the entire population. If the group has been tested twice, so that the
4 reliability' of each test is known, that can easily be taken into account in
weighting the tests : we have only to multiply each test (weighted as above)
by the square root of its reliability coefficient.

1 [93]> P- 287 and pp. 247-8, 300 ; cf. [102], p. 176. As noted in the
latter paper, Kelley in [85] has since independently reached much the same
conclusion : in particular he shows that, if a correlation matrix for two
variables only be analysed by Thurstone's method, the first factor proves to
be merely an average of the initial variables : he has " not undertaken an
algebraic analysis of Thurstone components for 3, 4, or a larger number of
variables" ; but in three specific cases his arithmetical solution manifestly
leads to a similar result ([85], pp. 58-61).