DERIVATION OF CHIEF THEORIES 147

are proportional to one another, can always be expressed in
terms of a single * factor/ that is, as the product of a single
column of c saturation coefficients' (one for each of the n
tests) post-multiplied by a single row of ' factor-measure-
ments ? (one for each of the N persons) " ; or, in matrix
notation, M = £. g, where g is the row-vector denoting the
hypothetical measurements of the N testees in the ' general
factor/ It will usually be convenient to express these
hypothetical measurements in unitary standard measure.
We may then write the equation M = $. p, where p now
denotes the normalized row-vector of factor-measurements
for the first and only factor.1

Actually, of course, no empirical table of measurements
will ever have such a rank : the figures furnished by actual
tests will never be exactly proportional for all the persons
tested. The first step, therefore, will be to seek a hypothetical
set of measurements of rank one, which will yield the * best
fit' to the data experimentally obtained.

For the sake of argument let us adopt the simple notion suggested
in my previous memorandum [93] that marks given to the rth
examinee for the /th test may be regarded as the result of counting
up the number of correct answers he has given. We can express
this number as a percentage (or better as a decimal fraction) of the
total number of marks awarded. Thus, the marginal totals, at
bottom and at the side of the mark-sheet, may be taken to indicate
(i) the general mark for each child in the ability tested by all the
tests, and (ii) the proportion of marks contributed by each of the
tests. We might now treat the mark-sheet as a kind ofnxN
contingency table, and examine the data along the lines used for
testing ' homogeneity ' or independence in c manifold classifica-
tions/ On multiplying the marginal totals, each with each in the
usual way (Yule's equation I [25], p. 64), we shall obtain a matrix
of rank one, which may serve to indicate the c expected' marks—
i.e. the marks we should expect each examinee to get on the hypo-
thesis that there was a single general factor only.2 To test this

1 Here and elsewhere capital letters denote matrices (e.g. M) and (where
confusion is likely) heavy block letters (e.g. f, g) denote vectors, i.e. matrices
containing a single row or column only.

2 This is the procedure I have used in what I have termed * factor-
analysis by simple averaging' (see Notes on Factor-analysis. II. Physical
Measurements).