396 THE FACTORS OF THE MIND

figures may be attained without calculating the inter-
correlations between the several persons or applying any
factor-analysis at all.

The detailed proof (omitted in my Memorandum in its
published form) has been given in the preceding paper.
The following principle results. With the summation
method and with unity in the diagonal cells, the factor-
measurements are given by the unweighted sums of the
measurements that have been correlated. Hence, for any
given person, " the sum of his correlations is proportional
to his correlation with the sums " ; consequently, to obtain
his saturation coefficient for the first (or ' general ') factor,
we have merely to correlate his marks with the sums or
averages of all the persons' marks ([115], p. 164 : it is, of
course, assumed that all marks or measurements are in
standard measure).1

The same principle may be extended to other factors.
Thus, to obtain a person's saturation coefficient for the
second (or ' type ') factor we have merely to subtract the
averages from the original marks of the several persons,
standardize the residuals, and then correlate that person's
residuals with the sums or averages of all the residuals,
first reversing the signs of half the persons .to make summa-
tion possible. In abstract terms the procedure may sound
a little complicated ; but the arithmetic, it will be seen, is

1 This is the principle underlying the rough-and-ready method that I have
termed 'factor-analysis by simple averaging.' Suppose, for example, tnij
represents fs performance in the zth test, measured, let us say, in terms of
speed. Then the persons' factor-measurements for the general factor com-
mon to all the tests (i.e. each person's general or average speed) may be taken
as JStmJn, £mi2/n, . . . etc. ; similarly, the factor-measurements for the

general factor common to all the persons (i.e. the general or average difficulty of
each test) may be taken as Zm^/N, Sm^jN, etc. Saturation coefficients for

/ j

tests (or persons) can be calculated by correlating the observed measurements
for each row (or column) with the averages for all the rows (or columns).
For the observed table of measurements, a matrix of rank one can be fitted

from the marginal totals by taking the estimated *»21 as = ^i* ^2J';

2j2imij

and similarly for the other estimated w's. A table of residuals can be
obtained by subtraction ; and (after reversing the signs where necessary)
may be analysed for secondary * bipolar ' factors in the same way.