VARIANCE, COVARIANCE, AND CORRELATION 281

" variance in psychology is purely an arbitrary matter,
depending solely on the whim of the psychologist."
" Factor-analysis/' he contends, " is impossible unless all
measurements are first reduced to standard measure;
and this is automatically effected by the usual product-
moment correlation " : in short, " the only safe assumption
is that the true variances are all equal for all abilities "
([117], p. 423; cf. [96], p. 199).

That mental measurement is to a large extent arbitrary I
should never wish to deny : indeed, my proposal was
accompanied by an explicit reservation, added on that very
ground ([117], p. 4-I9)-1 But arbitrariness is not fatal.
The units employed for physical measurement were once
as arbitrary and variable as those now employed in mental
testing ; and most of them are still defined by an artificial
convention. The foot, the span, the cubit, and the ell
were based on the notion that each man could be trusted
to provide his own standard, namely, the length of some
member of his body : Henry I of England sought to
eliminate variation by stretching out his own arm : (the
modern physicist prefers a bar of platinum, but even he has
to specify its temperature) ; David of Scotland came
nearer to the psychologist's procedure, when he ordained
that an inch or * thumb ' should be an average of three
thumbs—those of " an merkle man, an man of measurable
stature, and a lytell man." Variance depends in part upon
the way we select the population to be measured : let us

1 Thus, in dealing with marks for the same set of scripts, where the
variances seemed to depend very largely c on the whim of the examiners/ I
expressly usedc correlations between persons' instead ofc covariances between
persons ' ([93], pp. 267-9). Here my procedure was criticized by another
colleague on the opposite ground, namely, that the differences in the
standard deviation ought not to be disregarded. Once more I agree with
the reason advanced, but not with the conclusion drawn. In such cases my
proposal is not (as was wrongly supposed) that the standard deviations should
be ignored altogether, but that their implications should be considered as
a separate problem. Let me add that I take mye reciprocity principle * and
' symmetry criterion* to apply primarily to covariance matrices, or to correla-
tion matrices regarded as covariance matrices. If we start by assuming
those principles to be correct, and seek diagonal elements that conform to
them, each principle yields an additional method for discovering or checking
the most appropriate values for the variances of the items * correlated/