SIMPLE AND WEIGHTED SUMMATION METHODS 325

continue the iterative process. He can easily extrapolate by sum-
ming the geometrical progressions.1

The relations between the various results will become
obvious even to the non-mathematical if we glance at a
concrete example. I choose the table of correlations
factorized in two different ways by Kelley ([85], p. 58,
Table IX) to demonstrate the * irreconcilability ' which he
believes to exist between Thurstone's results and his own.

In Table V the figures at the foot of the three columns show the
results obtained by a first summation of the correlation matrix R
taken just as it stands, i.e. they form the vector w0R. Now, adopting
the table-by-column method, we take these sums (wl3 say) as weights
for multiplying the original correlations, row by row, and then make a
second summation to obtain the vector w^R = zv0Rz (cf. second line
of Table VI). To render the comparisons visible to the eye, the
weights are reduced at each stage to fractions of unity: the first
reduction is shown in the last line of Table V.

TABLE V

ist Multipliers.
Correlations.
Totals.

I
1-00
•70
•26
1-96

I
I
•JO •26
75 •45
•45
•35
1-90 i -06

4-92)1-96
1-90
1-06
4-92

•39^374
•386179
•215447
I-OOOOOO

1 The summation of the geometrical progression to shorten the labour of
successive approximation was briefly described at the March meeting of the
British Psychological Society (1937) : full working instructions are given in
Notes on Factor-Analysis (1936) obtainable from the laboratory, from which
both the foregoing proof and the following example are quoted (cL also
Appendix I, below). In the Notes it was not made sufficiently clear that, in
general, these elaborate methods of computation are only necessary when
figures are required to several significant places (e.g. for comparing results
by different methods, as here or in [114], or again for analysing a table of
covariances as distinct from correlations). When the variances are unknown,
and estimated communalities have to be inserted in their place, it would be
absurd to employ so refined and laborious a procedure. In my own experi-
ence, the resulting improvement is, as a rule, comparatively small ([93],
p. 294, [128], p. 63). Davies, however, has recently found that, in several
tables of correlations between persons, the simple summation method may
leave residuals which the investigator judges to be significant, whereas, if the
least-squares method is used, the residuals are at times considerably reduced,
so that nothing but a general factor can be validly demonstrated (cf. [130]).