V.—MANIFOLD CLASSIFICATION. 
this coefficient is zero if the characters 4 and B are completely 
independent, and approaches more and more nearly towards 
unity as x? increases. In general, no sign should be attached 
to the root, for the coefficient simply shows whether the two 
characters are or are not independent, and nothing more, but in 
some cases a conventional sign may be used. Thus in Table II. 
slight pigmentation of eyes and of hair appear to go together, 
and the contingency may be regarded as definitely positive. If 
slight pigmentation of eyes had been associated with marked 
pigmentation of hair, the contingency might have been regarded 
as negative. (is Professor Pearson’s mean square contingency 
coefficient! 
7. The coefficient, in the simple form (4), has one disadvantage, 
viz. that coeflicients calculated on different systems of classi- 
fication are not comparable with each other. It is clearly desir- 
able for practical purposes that two coefficients calculated from 
the same data classified in two different ways should be, at least 
approximately, identical. With the present coefficient this is not 
the case: if certain data be classified in, say, (1) 6x 6-fold, (2) 
3 x 3-fold form, the coefficient in the latter form tends to be the 
least. The greatest possible value of the coefficient is, in fact, 
only unity if the number of classes be infinitely great; for any 
finite number of classes the limiting value of € is the smaller the 
smaller the number of classes. This may be briefly illustrated as 
follows. Replacing §,,, in equation (3) by its value in terms of 
(dnB,) and (4,,B,), we have— 
4,B,.) 
ox {UBS , 
LU; 
and therefore, denoting the expression in brackets by S, 
S-N 
0-5" © 
Now suppose we have to deal with a ¢x #fold classification in 
which (4,,) = (B,,) for all values of m; and suppose, further, that 
the association between 4,, and B,, is perfect, so that (4,,B,)= 
(4) = (B,,) for all values of m, the remaining frequencies of the 
second order being zero; all the frequency is then concentrated 
in the diagonal compartments of the table, and each contributes 
1 Professor Pearson (ref. 1) terms 3a sub-contingency ; x2 the square contin- 
gency ; the ratio x%/N, which he denotes by ¢2, the mean square contingency ; 
and the sum of all the &’s of one sign only, on which a different coefficient can 
be based. the mean contingency. 
65 
(5, 
A