216 
On Theories of Association 
modifying the frequencies. This is the ground we have had for applying <f) to 
theoretical but not to practical Mendelism*. 
A further illustration of this limitation of </> for given marginal frequencies of 
the fourfold is provided in the accompanying Diagram V (p. 217). Here the range of 
values possible for the Boas-Yulean is given for the special case where one variate 
has a median division, and the percentage at which the dichotomy of the other 
variate takes place is given on the horizontal line; for example, for a 10°/ o dichotomy 
(p must lie between + "3333. We do not ourselves lay stress on this limitation of 
the range of values in the Boas-Yulean, but if it be a defect of the coefficient 
of mean square contingency that for a fourfold table its value cannot exceed 
•707, it is also a defect of the coefficient recommended by Mr Yule that it also has 
a limited range for given marginal frequencies, a limitation not shared by the 
tetrachoric coefficient or even Mr Yule's coefficient of association. 
(10) The Coefficient of Contingency. 
We do not propose to take up at great length a defence of this coefficient 
because one of us has had for some years a memoir on the subject in hand which 
will soon see the light of day. But Mr Yule's criticisms arise from two sources, 
(i) from his disregard of corrections which practice has taught us were needful and 
which have been known for some time, (ii) from his obvious want of that con- 
fidence iu the method which arises from long experience of its applicability. 
The corrections needed are (a) those due to number of cells, and (b) the 
correction for class-index. If k — number of rows, A, = number of columns, then on 
the average of many random samples the correction for number of cells is 
* We have the following results for the small-pox data : 
Possible range of 0 
Boas-Yulean <p for given frequencies 
Sheffield -531 + -9181 to --1221 
Leicester -249 +-2806 to --2228 
Homerton-Fulham -423 +-8101 to - -2301 
How would Mr Yule compare these values of <p with each other or with those of ?• from continuous 
frequencies, which can range from - 1 to +1, or again with a Boas-Yulean <j> from such tables as 
499,200 
800 
500,000 
499,988 
12 
500,000 
498.306 
1694 
500,000 
499,987 
13 
500,000 
997,506 
2494 
1,000,000 
999,975 
25 
1,000,000 
0= + -O2, <£=+-0003, 
Possible range +-05 to --05, Possible range 4- -005 to - -005 '? 
The bulk of the mental defect and blindness data considered by Mr Yule has for <f> total possible 
ranges varying from -4 to -6 on the positive side and - 006 to '004 on the negative side. How can 
the resulting coefficients be intercomparable ? 
