THEORY OF STATISTICS. 
that is (aD) i) (ap) 
(B)  (B)’ 
and the other two identities may be similarly deduced. 
The student may find it easier to grasp the nature of the rela- 
tions stated if the frequencies are supposed grouped into a table 
with two rows and two columns, thus: — 
Attribute. 
Attribute. — Total. 
B B 
2 (45) (48) (1) 
a (aB) (aB) i (a) 
CAT —— | CEE TEST ceo rm er c—— 
Total (B) 8) N 
Equation (1) states a certain equality for the columns; if this 
holds good, the corresponding equation 
(4B) (eB) 
4) (a 
must hold for the rows, and so on. 
2. The criterion may, however, be put into a somewhat 
different and theoretically more convenient form. The equation 
(1) expresses (AB) in terms of (B), (5), and a second-order fre- 
quency (4); eliminating this second-order frequency we have— 
(45) (ABYyL(AR) (4) 
(By (By Hy 
s.e. in words, “the proportion of 4’s amongst the B’s is the same 
as in the universe at large.” The student should learn to recog- 
nise this equation at sight in any of the forms— 
HY, 
(B) "& 
(4B) = (8) ) 
“am 2 
AB 
AB)=2"t"1 
(am) -E3 
4B) 4) B) 4 
Eh 
The equation (d) gives the important fundamental rule : If the attre- 
butes A and B are independent, the proportion of AB's tn the universe 
ts equal to the proportion of A’s multiplied by the proportion of B’s. 
26 
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