Karl Pearson and David Heron 
279 
The reader may ask what are our actual grounds for neglecting the results 
obtained from extreme divisions in the case of tetrachoric r t ? The answer is that 
it is not only experience of how slight skewness at extreme divisions produces large 
changes in the value of r t , but our knowledge that the weights of these outlying 
divisions are in the case of r t (to a lesser extent in the case of <£) insignificant 
as compared with the weights of the central divisions. No such grave differences 
exist in the case of Q or (to a slightly lesser extent) in the case of to. The 
following table gives the weights, treating the weights of the division at 20 
as unity. 
Division 
Q 
0} 
4> 
20 
l 
1 
1 
1 
25 
69 
2-7 
12-5 
24-5 
30 
370 
6-1 
27-4 
97-9 
35 
772 
10-9 
36-2 
171-9 
Ifi 
977 
14-3 
40-1 
187-7 
45 
1276 
16-7 
44-7 
153-1 
50 
977 
15-8 
40-1 
145-6 
55 
625 
15-0 
31-4 
97-1 
60 
319 
11-9 
25-2 
55-4 
65 
142 
10-0 
18-5 
22-6 
70 
54 
7-7 
12-5 
10-6 
75 
16 
5-9 
6-9 
3-8 
80 
1-5 
3 7 
3-0 
1-1 
85 
1-1 
2-5 
0-9 
0-3 
It will be clear: first that it is idle to measure the variations of these quantities 
from anything but a weighted mean, and secondly that no one would after seeing 
such weights dream of determining tetrachoric r t from extreme divisions. If we 
omit the first two and the last two values of r t , noting their slight weight, then 
the remaining values in no case differ by *04 from the true correlation and the 
mean divergence is - 015 only. Mr Yule's Q must be steadier here than r t , because 
its range is limited by the nature of the case to about "05, while there is no limit 
to the range of the latter. 
(H) Association in a typical table of Ordinary Statistical Practice. 
Thus far we have dealt with the relative stability of the coefficient of 
association and tetrachoric r t on material especially selected by Mr Yule to 
exhibit the variable character of tetrachoric r t . But in concluding this branch 
of our discussion we should like to exhibit the relationship of the two coefficients 
for such surfaces as occur in ordinary statistical practice. For this purpose we 
will make no selection ourselves, but illustrate the matter on the correlation table 
chosen by Mr Yule himself in his first memoir on Association*. We take the 
table as it is given without any knowledge of its degree of approach to the 
Gaussian or of the material with which it deals. Its coefficient of correlation by 
the product-moment method = '677. Now we commenced by leaving out all the 
* See Phil. Trans. Vol. 194 A, p. 277. 
