48 THE FACTORS OF THE MIND

(3) In factor-analysis-, however, we deal not with attri-
butes defining discrete classes, but with variables differing
in degree. This brings us to arguments of a third type.
What we have to infer is not the probability that i will be
intelligent or win a scholarship ; that is still too crude a
formulation : what we require is an estimation of the most
probable amount by which i will deviate above or below
the average, first, as regards intelligence, and, secondly, in
the marks at the scholarship examination.

By the eductive method we should calculate at a single
step the correlation between the intelligence test (which we
may call test i) and the scholarship examination (which we
may call test 2)—or rather the * regression ' of the latter
on the former. If the marks for both are in standard
measure, we have

m2i = r21 . mu

where m^ is z'?s mark in test I, m2i is the estimate of his
mark in test 2, and rai the coefficient of correlation. Here
we may assume that, if i is not in the batch tested and
followed up, his most probable mark in test 2 will be the
average of those examinees in the tested sample who ob-
tained the same mark as he in test I.

By the inductive-deductive method we should proceed
as follows. For simplicity we may continue to suppose
that only one test has been used to measure intelligence,
and that only one common factor, namely, g, is affecting
both the measurements in the intelligence test and the
marks in the scholarship examination.1 Having first
factorial composition oŁ the same set of tests is not only different for different
individuals, but different at different ages.

1 The student, I hope, will not assume that, if we had a number of tests
incorporated into the intelligence and scholarship examinations respectively
(instead of only one in each), that would of itself improve the relative trust-
worthiness of inferences based on the factor g ; or that, if we also had several
factors instead of one, that would necessarily give the indirect predictions a
higher value: for, with matrix notation, all the arguments in the text can
be generalized for as many variables in each set as we please, I may add
that, if we want to find the best possible predictive coefficients enabling
us to infer from one multiple test to another multiple test, we ought strictly
to employ an entirely different procedure, which I have called bi-multiple
correlation; but that procedure would again short-circuit the factorial
deduction.