8 



KARL PEARSON 



neglected in order to obtain it. But the relation pointed out between the correlation 

 of the variates and the correlation of their means to x suggests a possible and not 

 unreasonable scale of reducing contingency probabilities of independence to corre- 

 lation probabilities of independence. In order to think on a readily apprehensible 

 scale — which that of excessive iinprobabilities is not — find ^ the correlation which 

 would give the same improbability as the fcontingency does on, the hypothesis that 

 the two variates are unassociated. The correlation is thus used — not as a measure 

 of regression, of which we know nothing in the case of a purely categorical fourfold 

 division, but as a standard of relative improbability. 



It will be at once obvious that if we are to carry out this suggestion of a cor- 

 relation measure of improbability we want a very considerable extension of existing 

 tables. We need 



(i) A large extension of Palin Elderton's table for goodness of fit, shewing the 

 values of P, or better log P, for such high values of ^ as occur in contingency tables 

 with populations running from a few hundred to two or three thousand. 



(ii) A nearly new table indicating the probability that in uncorrelated material 

 a value of r will be reached which is many times its standard deviation; at present 

 we only know these improbabilities up to about 5 or 6, on the hypothesis that the 

 values of r follow a Gaussian curve, and this is inadmissible. 



(iii) A table giving on some reasonable hypothesis the values of ^a-^ for various 

 relative firequencies of the variates in the fourfold classification. 



Before we consider such tables it is well to note some points which arise in 

 dealing with the epistemological side of contingency. If we are dealing with a table 



Contingency Table. 



of the following type, we find that, the frequency occurring only in the diagonal 

 column and there being mxm cells, the coeflficient, C„, of mean-square contingency 



^4 



m— 1 



m 



