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 proportions
are arbitrary. A form possessing the same property but
certain marked advantages over @ is suggested in ref. (3).
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