276 THE FACTORS OF THE MIND

ments Lave the form of ranks, it affords the simplest and speediest
way of estimating the predominance of a single general factor. It
takes but a fraction of the time required for a formal analysis
carried out in the usual way.1

Let us now suppose that the correlations between the
several pairs of tests are separately calculated in the usual
way, and that a full factor-analysis is then attempted.
For simplicity, we may keep to the method of simple
summation, as adopted by Thurstone and others, and
assume that the test-measurements, as usual, are in standard
measure.2 The calculated self-correlations will then be
unity; to simplify the proof, we can retain these values
since, as Thurstone observes, " the diagonal entries (in the
correlation matrix) may be given any value between zero
and unity without affecting the results markedly, especially
when the number of variables is large " ([122], p. 108).
With the simple-summation method, we begin by calculating
the sum of all the (inter-class) correlations, including the
assumed self-correlations. Let us write, for the average,
of the complete set of inter-class correlations calculated in

this way, rc = L 2>w, (i9 i' = i, . . . jfe). Then, the average

K

saturation coefficient for the general factor will be

PP» I3^~7 ; Thomson and Bailes, Forum of Education, IV, 1926, pp. 85-96).
The substitution of the average correlation for a correlation with an average
is (as we shall see later on) merely a corollary to the important theorem that
the covariance between sums is equal to the sum of covariances.

1 Burt, lot. cit.9 p. 17. It was used with ranks by Felling, Dewar, Wood,
and myself in preliminary studies of the general aesthetic factor by correlating
persons : with ranks the denominator for rjz can, of course, be directly deter-
mined by the formula used in the calculation of a rank correlation, viz.

12

2 If we remove these two limitations, and covariate test-measurements with
a diferential weighting, extrinsically or intrinsically deduced, and then
employ the method of weighted summation, the general argument will be
the same : but somewhat elaborate complications will be introduced owing
to the new rektions set up between what at the outset were independent
variables.