CHAPTER V. 
MANIFOLD CLASSIFICATION. 
1. The general principle of a manifold classification—2-4. The table of 
double-entry or contingency table and its treatment by fundamental 
methods—5-8. The coefficient of contingency—9-10. Analysis of 
a contingency table by tetrads—11-13. Isotropic and anisotropic 
distributions—14-15. Homogeneity of the classifications dealt with 
in this and the preceding chapters: heterogeneous classifications. 
1. CrassiricaTiON by dichotomy is, as was briefly pointed out in 
Chap. I § 5, a simpler form of classification than usually occurs 
in the tabulation of practical statistics. It may be regarded as 
a special case of a more general form in which the individuals or 
objects observed are first divided under, say, s heads, 4; 4, . . .. 
A, each of the classes so obtained then subdivided under ¢ heads, 
B,, B,....B, each of these under heads, C,, Cy ..... . C,, and 
so on, thus giving rise to s. ¢. . . . . . ultimate classes altogether. 
2. The general theory of such a manifold as distinct from a 
twofold or dichotomous classification, in the case of n attributes 
or characters ABC .... XN, would be extremely complex: in the 
present chapter the discussion will be confined to the case of two 
characters, 4 and B, only. If the classification of the 4’s be s- 
fold and of the B’s t-fold, the frequencies of the st classes of the 
second order may be most simply given by forming a table with 
s columns headed 4, to 4, and ¢ rows headed B; to B. The 
number of the objects or individuals possessing any combination 
of the two characters, say 4,, and B,, ¢.e. the frequency of the 
class 4,,B,, is entered in the compartment common to the mth 
column and the mth row, the st compartments thus giving all 
the second-order frequencies. The totals at the ends of rows 
and the feet of columns give the first-order frequencies, <.e. the 
numbers of 4,’s and B,’s, and finally the grand total at the 
right-hand bottom corner gives the whole number of observations. 
Tables I. and II. below will serve as illustrations of such tables 
of double-entry or contingency tables, as they have been termed 
by Professor Pearson (ref. 1). 
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