THEORY OF STATISTICS. 
(B)= (4B), so that all A’s are B and also all B’s are 4. The 
three corresponding cases of complete disassociation are— 
(4) (7) (8) 
ol [tayias) Bryce cay 
| (aB) i . | ’ i ‘a) 
GES —— — a] te -— Cr — 
B® SI yaw 
It is required to devise some formula which shall give the value 
+1 in the first three cases, —1 in the second three, and shall 
also be zero where the attributes are independent. Many such 
formule may be devised, but perhaps the simplest possible (though 
not necessarily the most advantageous) is the expression— 
@=(4B)(aB) - (4B)(aB) 
(48)(af) + (4F)(eB) 
151 No 
(@B)ah) + (BYE) 
—where § is the symbol used in the two last sections for the 
difference (4B) —~ (4B),. It is evident that @ is zero when the 
attributes are independent, for then 6 is zero: it takes the value +1 
when there is complete association, for then the second term in 
both numerator and denominator of the first form of the expression 
is zero: similarly it is — 1 where there is complete disassociation, 
for then the first term in both numerator and denominator is 
zero. () may accordingly be termed a coefficient of association. 
As illustrations of the values it will take in certain cases, the 
association between deaf-mutizm and imbecility, on the basis of the 
English census figures (Example vi.) is +091 ; between light eye 
colour in father and in son (Ex _ ‘nla vii.) +066 ; between colour of 
flower and prickliness of fruit ia vatura (Example ix.) — 0°28, an 
association which, however, as already stated, is probably of no 
practical significance and due to mere fluctuations of sampling. 
The student should note that the value of @ for a given table 
is unaltered by multiplying either a row or a column by any 
arbitrary number, 7.c. the value is independent of the relative 
proportions of A’s and o’s included in the table. This property 
is of importance, and renders such a measure of association 
specially adapted to cases (e.g. experiments) in which the propor- 
tions are arbitrary. A form possessing the same property but 
certain marked advantages over @ is suggested in ref. (3). 
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