III. —ASSOCIATION. 9 
The coefficient is only mentioned here to direct the attention 
of the student to the possibility of forming such a measure of 
association, a measure which serves a similar purpose in the case 
of attributes to that served by certain other coefficients in the 
cases of manifold classification (cf. Chap. V.) and of variables 
(¢f. Chap. IX., and the references to Chaps. X. and XVI). For 
further illustrations of the use of this coefficient the reader is 
referred to the reference (1) at the end of this chapter; for the 
modified form of the coefficient, possessing the same properties 
but certain advantages, to ref. (3); and for a mode of deducing 
another coefficient, based on theorems in the theory of variables, 
which has come into more general use, though in the opinion of 
the present writer its use is of doubtful advantage, to ref. (4). 
Reference should also be made to the coefficient described in § 10 
of Chap. XI. The question of the best coefficient to use as a 
measure of association is still the subject of controversy: for a 
discussion the student is referred to refs. (3), (5), and (6). 
14. In concluding this chapter, it may be well to repeat, for the 
sake of emphasis, that (cf. § 5) the mere fact of 80, 90, or 99 per 
cent. of A’s being B implies nothing as to the association of 4 
with B; in the absence of information, we can but assume that 
80, 90, or 99 per cent. of a’s may also be B. In order to apply 
the criterion of independence for two attributes 4 and B, it is 
necessary to have information concerning a’s and A’s as well as 
A’s and B’s, or concerning a universe that includes both a’s and 
A’s, B's and B’s. Hence an investigation as to the causal 
relations of an attribute 4 must not be confined to 4’s, but must 
be extended to a’s (unless, of course, the necessary information 
as to a's is already obtainable): no comparison is otherwise 
possible. It would be no use to obtain with great pains the 
result (¢f. Example vi.) that 296 per thousand of deaf-mutes 
were imbecile unless we knew that the proportion of imbeciles 
in the whole population was only 1'5 per thousand ; nor would 
it contribute anything to our know dge of the heredity of deaf- 
mutism to find out the proportion of deaf-mutes amongst the 
offspring of deaf-mutes unless the proportions amongst the off- 
spring of normal individuals were also investigated or known. 
REFERENCES. 
(1) YuLe, G. U., “On the Association of Attributes in Statistics,” Phil. 
Trans. Roy. Soc., Series A, vol. cxciv., 1900, p. 257. (Deals fully 
with the theory of association : the association coefficient of § 13 
suggested.) 
(2) Yur, G. U., ‘““Notes on the Theory of Association of Attributes in 
Statistics,” Biometrika, vol. ii., 1903, p. 121. (Contains an abstract 
of the principal portions of (1) and other matter.) 
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