WORKING METHODS FOR COMPUTERS 449

termed it the * summation formula,' to distinguish, it from others
that were tried at the same time, more particularly from the
c product formula/ which was based on the geometric instead of on
the arithmetic mean. As in ordinary forms of averaging, so here,
we may take either weighted or unweighted sums. The method
of unweighted summation can be regarded as a simplification of,
and a first approximation to, that of weighted summation.

L SINGLE-FACTOR ANALYSIS
A. SIMPLE-SUMMATION METHOD

Formula : Saturation coefficient rag =—~t

The basic principle can best be understood if we begin with the
simplest case of all, namely, that in which only one factor is
involved, and in which, therefore, the correlations are assumed to
form a perfect hierarchy (except perhaps for minor observational
errors). Let us suppose that the observed correlations are those
printed in the body of Table I below. The figures are fictitious,
and have been artificially derived by multiplying each of the satura-
tion coefficients (shown along the top and left-hand margins) by
every other, in accordance with the product-equation. These
saturation coefficients are presumed to be unknown ; and our
object is to rediscover them.

The working procedure is as follows:

I. Find the total intercorrelation of each test by adding each
column of observed correlations : (-72 + '63 + . . . -J- -4.5 = 2'34>
etc.).

matrix (i.e. with a real factorial matrix) ; and (ii) such a set of diagonal elements
yields (in general) a completed matrix of lowest possible rank. When weighted
summation is used, the same completed matrix with the same rank and the same
diagonal elements is ultimately reached. Hence, for the latter procedure, the diagonal
values can first be calculated by simple summation so as to save labour.

In practice, however, I do not attach importance to the precise minimal rank as
such. I find it difficult to conceive that an empirical correlation matrix, any more
than an empirical covariance matrix, can have a definitely assignable minimal rank.
Moreover, I should, argue that the relative sizes of the variances ought not to be
arbitrarily limited by intrinsic mathematical considerations. Indeed, it would seem
defensible on theoretical grounds to assume slightly larger variances than the apparent
minimum. These minor points make but little difference to the general procedure or
to the results actually obtained: but they save the computer from laboriously
struggling after an unwarranted precision in his final figures.

I may add that the more elaborate variants of the primary formula were originally
devised for certain specialized problems arising out of my investigations for the
London County Council, and were given (with other derivatives) in various Reports,
usually without full proof. Here only those that are of frequent utility are included.
The general line of proof has been indicated in the text. I am indebted to the Council
for permission to incorporate material from their Reports.

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