te THEORY OF STATISTICS. 
B were independent in the universe of (’s, and the corresponding 
differences for the frequencies (4B), (AC), and (BC). The four 
quantities in the brackets on the right represent, say, the four 
known associations, the bracket on the left the unknown association. 
Clearly, the relation is not of such a simple kind that the term on 
the left can be, in general, mentally evaluated. Hence in con- 
sidering the choice and number of associations to be actually 
tabulated, regard must be had to practical considerations rather 
than to theoretical relations. 
13. The particular case in which all the 2” —n +1 given associa- 
tions are zero is worth some special investigation. 
It follows, in the first place, that all other possible associations 
must be zero, z.e. that a state of complete independence, as we 
may term it, exists. Suppose, for instance, that we are given— 
(4)(B) _(4)(©) 
(4B) = 7 (40) = v 
_(B)C) _(4OXBO) _ (4)(B)(C) 
(BC) aN (4B0C) Tw (0) Ce 
Then it follows at once that we have also— 
AB)(BC) (4B)(AC) 
A450) = EENBC) (ABYC, 
ents a 
t.e. 4 and C are independent in the universe of B’s, and B and C 
in the universe of 4’s. Again, 
= (AE) _ DB) _ (A)XB)C) 
(4By)=(4B) - (4BC) = Sa 
_ AB) _ 4y)(By) 
re (r) 
Therefore 4 and B are independent in the universe of fs. 
Similarly, it may be shown that 4 and C' are independent in the 
universe of 8’s, B and C in the universe of a’s. 
In the next place it is evident from the above that relations of 
the general form (to write the equation symmetrically) 
“BO _(1) (#) (©) = 
N x. aN : 2 
must hold for every class-frequency. This relation is the general 
form of the equation of independence, (2) (d), Chap. III. (p. 26). 
14. It must be noted, however, that (8) is not a criterion for the 
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