FINAL CONCLUSIONS 437

assessments, we should have expected the standard deviation of z

to be in the neighbourhood of ~~^== — —— = 0-35, whereas

V n — 3 v <*
actually it is nearly twice that figure.

Accordingly, assuming a normal distribution, having the mean and
the standard deviation specified above, we can now calculate what
percentages of the whole group should fall between the successive
values of r given in the table. These percentages are inserted in the
last line but one. It is, however, almost equally interesting to
inquire what frequencies we should expect with a normal distribu-
tion varying symmetrically about zero (the perfect temperament)
taken as the mean. Accordingly, we may smooth away the
asymmetry of the observed distribution by averaging the percent-
ages on either side of zero and recalculate the standard deviation
(it remains practically the same) and once more deduce the
theoretical percentages fitting a normal curve. These further
figures are appended in the last line of Table VI.

The effect of the hyperbolic transformation is vividly shown by
plotting the frequencies above an r-scale and a z-scale respectively.
With the f-scale, taking, that is, the correlations as they stand
(Fig. 2), the distribution appears anything but normal: yet when
expanded to a z-scale they show a reasonably close resemblance to
a normal distribution (Fig. 3). When the theoretical percentages for
a strictly normal distribution (Fig. 3, dotted line) are converted
back to an f-scale (Fig. 2, dotted line) all resemblance to a normal
curve is lost: and it is scarcely surprising if such flattened distribu-
tions have been supposed to put the hypothesis of normality entirely
out of court.

The degree to which the various theoretical percentages fit the
observed may be tested more precisely by the usual y? method
(i.e. dividing the squares of the discrepancies by the theoretical
values and summing the ratios). The last of the hypotheses men-
tioned above—that the true distribution is symmetrical about the
intermediate type (measured by zero)—yields, as was to be ex-
pected, the poorest fit of all: (x* = 58-09 ; P—the probability
that the divergences from the hypothetical distribution are due
solely to errors of sampling—therefore lies, as Pearson's Tables1
show, between I in 10,000 and I in 100,000). The figures based on
the correlations as they stand are not much better (x2 = 54'47;
P is therefore less than I in 1,000). The figures given by the

1 Tables for Statisticians and Biometricians, Table XII (' Tables for Testing
Goodness of Fit/ Enter the table with nf = one more than the number of
degrees of freedom: cf. [no], p. 418).