THEORY OF STATISTICS. 
is therefore given in full in the following section. The advanced 
student should refer to the original memoir (ref. 1) for a completer 
treatment of the theory of the coefficient, and of its relation to 
the theory of variables. 
6. Generalising slightly the notation of the preceding chapters, 
let the frequency of 4,’s be denoted by (4,), the frequency of 
Bs by (B,), and the frequency of objects or individuals possessing 
both characters by (4,B,). Then, if the A’s and B’s be com- 
pletely independent in the universe at large, we must have for all 
values of m and n— 
An Bn 
(4,.B,) = ad =(4dnBn)y - : +35{1) 
If, however, 4 and B are not completely independent, (4,.B,) and 
(A,.B,), will not be identical for all values of m and n. Let 
the difference be given by 
On = (4B) TF (4B) - (2) 
A coefficient such as we are seeking may evidently be based in 
some way on these values of 8. It will not do, however, simply to 
add them together, for the sum of all the values of d, some of 
which are negative and others positive, must be zero in any case, 
the sum of both the (4B)’s and the (4B),’s being equal to the 
whole number of observations XV. It is necessary, therefore, to 
get rid of the signs, and this may be done in two simple ways: (1) 
by neglecting them and forming the arithmetical instead of the 
algebraical sum of the differences 3, or (2) by squaring the differ- 
ences and then summing the squares. The first process is the 
shorter, but the second the better, as it leads to a coefficient 
easily treated by algebraical methods, which the first process 
does mot: as the student will see later, squaring is very 
usefully and very frequently employed for the purpose of elimin- 
ating algebraical signs. Suppose, then, that every 0 is calculated, 
and also the ratio of its square to the corresponding value of 
(4B), and that the sum of all such ratios is, say, x2; or, in 
symbols, using 2 to denote “the sum of all quantities like ” :— 
Pel 3 
elie a 49 
Being the sum of a series of squares, x2 is necessarily positive, 
and if 4 and B be independent it is zero, because every 6 is zero. 
If, then, we form a coefficient C' given by the relation 
= x “A 
# VF + x2 ) 
64 
or