THEORY OF STATISTICS.
Example iii.—In Example i. above, what would be the number
of af5’s, 4 and B being independent ?
(a)=1024 — 144 =880
(B)=1024 — 384 = 640
; _ 880 x 640
C0) (af) = 1000 = 550.
4. Finally, the criterion of independence may be expressed in
yet a third form, viz. in terms of the second-order frequencies
alone. If 4 and B are independent, it follows at once from the
preceding section that—
A)(B)(a
(4B)(af3) a (4)( a
And evidently (aB)(4p) is equal to the same fraction.
Therefore—
(AB)(aB) = (eB)(4B) (a))
(LB CO
(aB) (a8) (3)
AB B
2 ) rs (aB) (©)]
(4B) (of3)
The equation (b) may be read “The ratio of A’s to a’s amongst
the B’s is equal to the ratio of A’s to o’s amongst the 5's,” and
(c) similarly.
This form of criterion is a convenient one if all the four secondorder
 frequencies are given, enabling one to recognise almost at a
glance whether or not the two attributes are independent.
Example iv.—If the second-order frequencies have the following
values, are A and B independent or not?
(4B)=110 (eB) =90 (46) =290 {=f3)= 510.
Clearly (4B)(af3 > (a.B)(AB),
so A and B are not independent.
5. Suppose now that 4 and B are not independent, but related
in some way or other, however complicated.
Then! (45-0)
A and B are said to be positively associated, or sometimes simply
associated. If, on the other hand,
Ln < (B)
4B) <AUB)
(A By <2ts

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