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 considering
 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 associations
 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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