TO THE THEORY OF EVOLUTION. 
15 
Now consider what happens in the case of perfect correlation, i.e. , all the observa¬ 
tions fall into a straight line. Hence if ad > bo, either b or c is zero, for a straight 
line cannot cut all four compartments, and a and d are obviously positive. Thus c 
and b can only he zero if 77 = (c + d)[a + c)f N 2 or (a + b)(b -f- d)/ N 3 . In order 
that b should he zero, it is needful that h and Jc, as given by (iv.) and (v.), should he 
positive or a + c > b -j- d, a -{- b > c + d, and the mean fall under the 45 line 
through the vertical and horizontal lines dividing the table into four compartments, 
i.e., Ji > l\ These conditions would be satisfied if ad >bc and a > d, c > b. Now 
suppose our four-compartment table arranged so that 
ad >bc, a>d, c>b, 
and consider the function 
or 
Qi = sin 7 
7r 
V 
(a + b)(b + d)/W 
„ . 7 r ad — be 
Qi = sm - 
(a + b)(b + d) 
. . . . (li.), 
. . . . (lii.). 
This function vanishes if r) = 0, and it further — unity if b = 0. Thus it agrees 
at the limits 0 and 1 with the value of the correlation coefficient. Again, when h 
and Jc are both zero, a = d, b = c, and Qi = sin y- a + ~ | , is thus r by (xx.). Hence 
we have found a function which vanishes with r and equals unity with r, while it is 
also equal to r if the divisions of the table be taken through the medians. 
Now, I take it that these are very good conditions to make for any function oi 
a, b, c, d which is to vanish with the “ transfer,” and to serve as a measure of the 
degree of dependent variability, or what Mr. Yule has termed the degree of 
“ association.” Mr. Yule has selected for his coefficient of association the expression 
_ ad — be 
= ad + be 
(liii.). 
This vanishes with the transfer, equals unity if b or c be zero, and minus unity if a 
or d be zero. The latter is, of course, unnecessary if we agree to arrange a , b, c, d 
so that ad is always greater than be. Now it is clear that Q 3 possesses a great 
advantage over Q L in rapidity of calculation, but the coefficient of correlation is also 
a coefficient which measures the association, and it is a great advantage to select one 
which agrees to the closest extent with the correlation, for then it enables us to 
determine other important features of the system. 
If we do not make all the above conditions, we easily obtain a number of coeffi¬ 
cients which would vanish with the transfer. Thus for example the coil elation of 
Equation (xxxviii.) is such an expression.* It has the advantage of a symmetrical 
form, and has a concise physical meaning. It does not, however, become unity when 
* In fact (xxxvii.) gives us « = o'hP'k'i'hk- 
