178 
On Theories of Association 
<f> would be better than this, for it is equal to 33 with a probable error of '15 
which makes it for practical purposes unreliable. Or again, we repeat our experi- 
ment and find 
Shilling. 
•A 
a 
Head 
Ta.il 
Totals 
Head 
2 
0 
2 
Tail 
0 
2 
2 
Totals ... 
2 
2 
4 
Here Q is again unity or there is absolute association, and the probable 
error is zero or this association according to Mr Yule is also absolutely reliable. 
Further 0 = 1, and its apparent probable error is zero, but this is only apparent 
because the calculation of the probable error (as indeed of that of Mr Yule's Q, 
although he has not noticed it) is incorrect for such a case. 
These simple illustrations seem to indicate that there is nothing in the nature of 
things which necessarily demands that the association shall be unity when either 
b or c or both are zero. On the other hand the probability P (as derived from <f>, 
the mean square contingency) that the heads and tails of shilling and penny are 
independent is more than "90, and in the second case more than *25. It seems to 
us therefore that this mean square contingency method which gives reasonably 
satisfactory results, where the Boas-Yulean goes hopelessly astray, is far more 
likely to be preferable in the case of medical treatment to which Mr Yule proposes 
to apply his coefficient. 
There is another point also to be considered. Why should the range of a 
good coefficient of association lie for any given number of cells between + 1 and 
— 1 ? Let us examine the following table : 
(A) 
a 
b 
c 
d 
e 
/ 
9 
h 
i 
3 
Totals 
a 
40 
0 
0 
0 
0 
0 
0 
0 
0 
0 
40 
ft 
o 
0 
200 
0 
0 
10 
110 
10 
.30 
0 
360 
y 
0 
0 
0 
0 
0 
10 
440 
0 
120 
0 
570 
S 
0 
0 
0 
0 
0 
10 
20 
0 
0 
0 
30 
e 
0 
0 
0 
20 
10 
0 
0 
0 
0 
0 
30 
c 
0 
120 
0 
440 
10 
0 
0 
0 
0 
0 
570 
n 
0 
30 
10 
110 
10 
0 
0 
200 
0 
0 
360 
e 
0 
0 
o 
0 
0 
0 
0 
0 
0 
40 
40 
Totals 
40 
150 
210 
570 
30 
30 
570 
210 
150 
40 
2000 
If we treat each sub-range here as unity we find the correlation negative and 
equal to - T120. If therefore we assume this to be the ultimate distribution, this 
is the correlation coefficient. 
