436 THE FACTORS OF THE MIND

The peculiarities of the index of measurement will be perceived
more clearly if we compare the actual frequencies with the normal.
From the ordinary table of the probability integral, we can deter-
mine precisely what frequencies would be required in theory by a
normal distribution having the same mean and the same standard
deviation as the present group. These theoretical percentages are
printed in Table VI immediately beneath the percentages actually
obtained. Since they have been calculated by taking the correla-
tions just as they stand, I have labelled them * -/-method.' Evi-
dently the fit is far from good. In particular the theoretical
distribution, reconstructed in this way, inevitably yields frequencies
beyond the values of + POO and — roo. But a coefficient of
correlation, of course, cannot go beyond either of these limits,
Hence, if our index of measurement is to be expressed in the form of a
correlation, true normality is precluded from the outset; and we
discover at once an obvious reason for the bunching at the tails,

We may, however, remove this limitation by a simple device.
We may adopt for our conventional measure, not the correlation
coefficient itself, but the number having this coefficient for its
hyperbolic tangent. The range of the numbers so derived is un-
limited in both directions ; and, as is well known, their sampling
distribution conforms almost exactly with the normal curve
(cf. [no], p. 451 and refs.).

Let us then apply this conversion. We take for our adjusted
measure x

T_~I 11 i + r . r3 , r5 , r7 ,

z = tanh x r = \ log, —•— = r-)------1------1------\- ...

i —' 357

where r is the correlation (or saturation coefficient) and z the
substituted measurement. In terms of z, the mean for the entire
group is now *o6 and the standard deviation -62. The size of the
standard deviation is alone sufficient to dispose of any notion that
the original assessments may have been assigned practically at
random: for, in that case, with correlations based on n =» 11

1 I give various equivalent expressions, since different students may prefer
to adopt different methods of making the transformation. Full tables of the
hyperbolic tangent will be found in Smithsonian Mathematical Tables;
Hyperbolic Functions, pp. 86 et seg.; or the calculation may be made from any
sixpenny set of mathematical tables that includes * natural, hyperbolic, or
Napierian logarithms' or the * exponential function.' Table VB in Fisher's
StatisticalMethods for Research Workers (p. 197) may also be used, if available :
for such work as the present, however, this involves interpolation ; hence we
find it more convenient to employ a specially compiled table giving z values
for r instead of vice versa.