THEORY OF STATISTICS. 
If § denote the excess of (AB) over (4B), then, in order tc keep 
the totals of rows and columns constant, the general table 
(¢f. the table for the case of independence on p. 27) must 
be of the form 
Attribute. 
Attribute. Total. 
5 8 
he (AB), +3 (AB)y—-d - (4) 
a (aB)y— 3 (aB)y +0 (a) 
Total “= (B) (R) | v 
Therefore, quite generally we have— 
(4B) - (4B), = (af) = (aB)y = (4B), = (48) = (@B), = (aB). 
12. The value of this common difference 8 may be expressed 
in a form that is useful to note. We have by definition — 
5= (4B) - (4B), = (4B) - Lh 
Bring the terms on the right to a common denominator, and 
express all the frequencies of the numerator in terms of those of 
the second order ; then we have— 
rl (ABIAB) + (B+ 48) + 0] 
\ -[(4B) + (4B)][(45) + («B)] 
= 1 {4B)oP) - (B)(4B) | 
That is to say, the common difference is equal to 1/Nth of the 
difference of the cross products” (4.B)(af) and (aB)(4f). 
It is evident that the difference of the cross-products may be 
very large if IV be large, although 8 is really very small. In 
using the difference of the cross-products to test mentally the 
sign of the association in a case where all the four second-order 
frequencies are given, this should be remembered : the difference 
should be compared with , or it will be liable to suggest a higher 
degree of association than actually exists. 
Example ix.—The following data were observed for hybrids of 
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