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 relations
 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 frequency
 (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 recognise
 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 attrebutes
 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.

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