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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