SIMPLE AND WEIGHTED SUMMATION METHODS 331

f _ ^Tik - f _. W°R

-\/£S TiTi V^o R Wo

where w0 as usual denotes the simple summation operator.
Accordingly,p-L=f\R ~*M = - °. ==- = °

where the numerator is merely the unweighted sum of each person's
measurements and the denominator reduces these sums to unitary
standard measure.

The foregoing argument implies that the same correlation matrix,
R = MM', is used throughout, whether we are calculating p or/.
Now for the former purpose—(the appraisal of abilities * (p]—
Thurstone takes an R having diagonal elements of unity ([84],
p. 227) ; for the latter purpose—the calculation of factor-saturations
(f)—he proposes to substitute a different R, containing in its diagonal
the highest correlation in each column (pp. 89, 108). In that case
the expression RR~i) in the final equation f or ;£>, cannot be taken as
precisely equivalent to the unit matrix, I; the change, however, only
involves a slight difference in weighting.1 Thurstone himself

1 In the unabridged version of my Memorandum I gave the proof in a
more general and a somewhat more elaborate form, which shows that the
result is not limited to unweighted summation nor to a correlation table
with the self-correlations taken as unity. With the various criticisms
of my conclusion I shall briefly deal in Part III. Here I limit the
proof to the simplest possible case, not only because the underlying principle
will then be clearer and, I hope, beyond all controversy, but also because this
simpler proof alone is needed to justify my use of this procedure in calculating
such factor-correlations in the past and in deducing certain results in the
chapters that follow. The tables in Part III (pp. 391, 398-9) will serve to
illustrate the present argument by a concrete instance ; Kelley's first example
(loc. cit.y p. 58, Table IX) will serve to illustrate the extension of the same
conclusion to cases where communalities are inserted in the leading diagonal
instead of unity.

In general, however, if the correlation table is small, and if the hierarchical
tendency is steeply graded (and therefore not overlaid by marked group-
factors), the saturation coefficients obtained respectively " with unity " and
" with communalities in the diagonal cells " differ in the way illustrated in
Table IIIA and IIIB : cf. [93], p. 307. Thus we may convert the n X i
factorial matrix given in Thurstone's Table 5 ([84], p. 101) into a * prop-
ladder pattern' by appending the coefficients for the specific factors as
deduced from Spearman's two-factor theorem; the resulting n X (n -f- l)
factorial matrix will reproduce his Table I quite as well as the n X n matrix
shown in his Table 2. The factorial matrix reached by Godfrey Thomson
(J. Ed. Psych., XXV, p. 367) on applying Hotelling's original method to a