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 second- 
order 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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