P-, Q-, AND R-TECHNIQUES 179

" futile," and to " entail a vast amount of unnecessary labour,"
because the general abilities revealed and " your * subjective' and
4 objective ' types " can all be reached, " as you have already shown,
by the direct and well tried route of correlating tests or traits.
As Spearman himself has pointed out, the vital thing in studying
types is simply to determine what trait goes with what " (cf. [56],

P. 54)- .

Now in many instances, as I readily own, to correlate the large
number of persons instead of the small number of traits may entail
not only a lengthy but a needless detour. But for the two
purposes I have mentioned this course seems all but indispensable.
Since I hold that a multiplicity of factors is also reached by correlat-
ing traits, the mere multiplicity of factors that results from correlat-
ing persons does not shake my faith in the validity of the method.
Yet it would be a mistake to imagine (as so many others have also
done) that more factors must necessarily emerge just because there
are more correlations : the rank of a correlation matrix derived from
the one and same set of measurements remains the same whether
we correlate by persons or by traits. As for the disparity of units,
that can be met, as we have just seen, either by standardizing the
measurements for each trait (e.g. when dealing with traits that are
normally distributed among the persons correlated) or by standard-
izing or ranking the measurements for each person (e.g. when dealing
with items that have an approximately normal or linear distribution
in the sample judged, such as essays by an unselected batch of
candidates or a series of specially selected test items spaced out at
approximately equal intervals).

3. Confusion between Variable and Population.—A third
criticism, perhaps the commonest of all, is that the proposed
method of analysis " confuses the distinction between vari-
able and population." This has been urged, not only by
Stephenson, but by several other critics.1 It is, however,
not an objection that should be brought against the pro-
cedure itself, but only against the selection of measurements
to which the procedure is applied.

If we take the term * population' literally, to denote a human
population (as, of course, it did in the early days when the statistical
terminology was taking shape), the objection may seem plausible.
The * population ' from which the sample figures are drawn is con-

1 Stephenson, he. tit. sup., cf. [136], p. 20, [138], p. 275.