XVIL.—NORMAL CORRELATION, ; 
this group of four frequencies is also of the same sign as r,, the 
ratio of the cross-products being unity, or the association zero, 
if 71, is zero. Ina normal distribution, the association is therefore 
of the same sign—the sign of rg—for every tetrad of frequencies 
in the compartments common to two rows and two columns ; that 
is to say, the distribution is isotropic. It follows that every 
grouping of a normal distribution is isotropic whether the class- 
intervals are equal or unequal, large or small, and the sign of the 
association for a normal distribution grouped down to 2- x 2-fold 
form must always be the same whatever the axes of division 
chosen. 
These theorems are of importance in the applications of the 
theory of normal correlation to the treatment of qualitative 
characters which are subjected to a manifold classification. The 
contingency tables for such characters are sometimes regarded as 
groupings of a normal distribution of frequency, and the coefficient 
of correlation is determined on this hypothesis by a rather lengthy 
procedure (ref. 14). Before applying this procedure it is well, 
therefore, to see whether the distribution of frequency may be 
regarded as approximately isotropic, or reducible to isotropic form 
by some alteration in the order of rows and columns (Chap. V. 
$3 9-10). If only reducible to isotropic form by some rearrange- 
ment, this rearrangement should be effected before grouping the 
table to 2-x 2-fold form for the calculation of the correlation 
coefficient by the process referred to. If the table is not reducible 
to isotropic form by any rearrangement, the process of calculating 
the coefficient of correlation on the assumption of normality is to 
be avoided. Clearly, even if the table be isotropic it need not be 
normal, but at least the test for isotropy affords a rapid and 
simple means for excluding certain distributions which are not 
even remotely normal. Table II. of Chap. V. might possibly be 
regarded as a grouping of normally distributed frequency if re- 
arranged as suggested in § 10 of the same chapter—it would be 
worth the investigator’s while to proceed further and compare 
the actual distribution with a fitted normal distribution—but 
Table IV. could not be regarded as normal, and could not be 
rearranged so as to give a grouping of normally distributed 
frequency. 
13. If the frequencies in a contingency-table be not large, and 
also if the contingency or correlation be small, the influence 
of casual irregularities due to fluctuations of sampling may 
render it difficult to say whether the distribution may be regarded 
as essentially isotropic or no. In such cases some further con- 
densation of the table by grouping together adjacent rows and 
columns, or some process of “smoothing” by averaging the 
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