Karl Pearson 
139 
Thus the corrected contingency for roughly equal frequencies and equal sub- 
ranges gives respectively "52 and "51, but the formula (xvii) gives for equal ranges 
52 and for roughly equal frequencies '57. It seems to me therefore that corrected 
contingency can be more safely applied than formula (xvii). 
Conclusions. 
A further discussion of the corrections needful when using the method of 
mean square contingency will shortly be published, but the present paper seems 
to indicate that it can in the bulk of cases for 3 x 3-fold, 4 x 3-fold or 4 x 4-fold 
tables be used effectively. I am not aware that any other effective method has 
been proposed for such tables. The assumption of the Gaussian distribution to 
determine the correlation of the variate with its class index need not be made, 
if the material thrown into broad classes has had a sample quantitatively 
determined. But we have seen in this paper that the Gaussian assumption to 
fix the means of the broad classes is in many cases amply sufficient to give good 
results even in non-Gaussian frequencies. Of course a control series to determine 
r xC x > where practicable, may be advantageously sought*, but until some better 
method can be suggested, the present seems to me the best available for dealing 
with the correlation of variates classed in a few broad categories. 
With contingency tables of 5 x 5 or 6x6, the correcting factors will generally 
be small, but corrections must certainly be made for 4x4 and 3x3 tables. 
* For example, I took Southampton frequency, and calculated from the original table the means of 
the three groups 892, 1970-5, 59-5, corresponding to the table on p. 137; they were -1-1006, + -4123, 
and +2-1746, leading to r r =-8056, as against the Gaussian value -7972. This is for a sensibly 
x v x 
non -Gaussian frequency, and a much worse agreement is permissible in actual practice, where the 
nature of the argument rarely turns on correlation differences of less than -05. 
18—2 
