264 
On Theories of Association 
Mr Yule's selection has been singularly favourable to his unjustified assumption 
that the points lie along a straight line ; actually the points are extremely steady 
between 29"85 and 30"3o, and tail off to the terminals, where the percentages in the 
smallest quadrant are much smaller. However, apart from the question of linearity 
of the distribution of r, there is little doubt that the values are in defect for low 
divisions and in excess for high divisions. The question therefore is, are these 
defects and excesses such as to invalidate the use of the method of tetrachoric 
functions applied even to variates as skew as the barometer data ? The only method 
of answering this question appears to us to consider the relation of the values found 
to their probable errors, and again the amount of stress which is likely in practice to 
be laid on the result deduced from a single fourfold table. We believe that practi- 
cally it is not necessary for the type of reasoning based on a fourfold table that the 
correlation found should be nearer than +'05. In the present case three coefficients 
exceed these limits, but if we proceed solely to two figures the first and last only 
lie outside these limits. Diagram XVI shows this result by the lines AA' and 
BE . The hatched part of the diagram corresponds to the region on either side 
the product-moment value of 2'5 times the probable error. Actually some 10% 
of the observations should lie outside these limits, i.e. 1^ observations instead 
of 3. But even these three have almost contact with the hatched area. It seems 
to us that in this very case of barometer data, which Mr Yule has chosen for 
its marked skewness to discredit tetrachoric r t , the coefficient defeats Mr Yule 
completely ! 
Now let us compare the results with Mr Yule's own coefficient of association. 
The difficulty in the comparison lies with the standard value of Q against which 
the other Q's are to be compared. Had Mr Yule studied the surface of constant Q 
— what we may term the association-surface — (see p. 184 above), then the Q corre- 
sponding to the best fitting association surface would have formed a standard Q. 
But we have at present no means of finding this standard Q, and Mr Yule tells us 
that he himself lays no stress on the diversity of values obtained for Q with different 
divisions. However, Mr Yule has himself taken the mean Q as a standard when 
he comes in a special case to deal with the relative stability of tetrachoric r t and 
association Q. Accordingly we follow him in taking a mean Q. But he has gone 
astray in simply taking the arithmetical average of his Q's or r t 's without regard 
to their probable errors. The proper means to take are weighted means, and the 
proper standard deviations are weighted standard deviations. Each value must 
be weighted with the inverse square of its probable error as the measure of its 
grade of accuracy. With this weighting we find : 
Barometric Heights. 
Weighted Mean 
Weighted 
Standard Deviation 
Tetrachoric /- ( 
Association Q 
•788G 
•9194 
•024.3 
■034<; 
