328 THE FACTORS OF THE MIND

difficulty, however, is by no means peculiar to methods of factor-
analysis as such, but to all attempts to deduce an average from a
small set of data. If I calculate an average and a mean deviation
from only a dozen persons, and then add a I3th and recalculate, the
recalculated figures are bound to differ, unless that 13th person is
very carefully picked. Similarly, if I calculate a set of factors from
only a dozen tests, those factors, being nothing but an average and a
set of mean deviations derived from those tests, are almost bound to
alter when a I3th test is added.

In making these comparisons I have so far assumed that the
same complete matrix is factorized in either case. What I have
said above needs a little modification when applied to the
best-known representatives of the two methods, since the different
writers complete an incomplete correlation matrix in different
ways. Hotelling's method inserts self-correlations of i-oo in the
leading diagonal: hence, with n tests, the correlation matrix
cannot be perfectly fitted without using n factors—even if the
intercorrelations by themselves are perfectly hierarchical. On the
other hand, Thurstone's method (in theory, though not it would
seem in practice) is applied to a c reduced correlation matrix' in
which c communalities ' are inserted; and these are so chosen as to
reduce the rank of the matrix artificially to the lowest obtainable
number. In such a case an exact fit can be obtained for the inter-
correlations with less than n factors; for Thurstone does not count
the specific factors, as they are not required to reconstruct the
reduced correlation matrix. His gain, however, seems purely
academic. It is true that with (say) a dozen tests the correlations
could always be perfectly fitted with 7 factors only. But in how
many tables are the probable errors so minute that we should look
for as many as 7 factors ? Moreover, by merely printing the correla-
tions to a different number of decimal places, we may in many cases
alter its rank far more drastically than by any change in the arbitrary
communalities. However, as I have so often insisted, what our factors
have to interpret is not so much the correlation matrix, R, as the
original set of measurements, M; the device of correlating is only
an incidental step, and not always the best. Now, Hotelling's
method, like the method of least squares, always yields a better fit
to the original measurements, even if we retain only a small number
of factors (e.g. significant factors only) ; with n factors the fit is
perfect. With the Thurstone method, on the other hand, as with
the summation method, we have to invoke the specific factors as
well before we can obtain so good a fit to the original measurements
([101], p. 80). And since both Tlmrstone and Spearman believe in