Iviii Tables for Statisticians and Biometricians [XXXIV 



total frequency ia the gth row, and N the whole population, then if the two 

 variates are independent, the frequency to be expected in the p, qth cell will be 



N N ■ N ' 



and the observed excess over this, i.e. iipq ^^^^ , is termed the ' contingency ' in 



this cell. The total contingency must be of course zero, i.e. the sum of all the 

 cell contingencies. If, however, we take only the positive excess contingencies and 



divide them by N, i.e. ilr = -^ S+ ( n^j V^)' ^^® obtain the so-called 'mean 



contingency.' On the assumption of normal frequency distribution it is possible 

 to ded\ice the actual correlation from i/r, provided that the cells are sufficiently 

 small for summation to replace integration. As in practice our cells are hardly 

 likely to exceed 8x8, and may be smaller and unequal in area, we shall generally 

 find a value below that of the true correlation, even if the system be accurately 

 normal. A corrective factor corresponding to the class-index correlation has not 

 yet been theoretically deduced. But experience seems to show that to add half i\\e 

 correction due to class-index correlations gives good results. That is to say, that, 

 if rj, be the correlation found from the Abac, p. 65, and r^Q and r^^ be the class- 

 index correlations for x and y, we should take for the true correlation : 



r = r^ + \ 



= i 



rj,+ 





''xCjyCy 



.(xlviii). 



It is clear that this is the same thing as taking the mean of the crude mean 

 c(mtingency correlation and its value as corrected for the class-index correlations. 

 The following illustrations may indicate the method of procedure. 



Illustration (i). Find the correlation from the table on p. lix by mean 

 contingency. The first number in each cell is the frequency reduced to 1000, the 

 second number is that to be expected on the basis of independent probability, and 

 the third is the mean contingency of the cell. 



The sum of the positive contingencies is 94136, hence the mean contingency 

 is '094. Entering the diagram with "094 on the base scale, we pass up the vertical 

 to the curve, and then along the horizontal to the left hand scale and find r^ = ■28.5. 



The class-index correlation for the vertical marginal frequency is ?•(; ="9645, 

 and that for the horizontal marginal frequency is '9624*. Hence 



and > = H'307 -h -28.5) = -296. 



The table is actually a true Gaussian distribution with correlation equal 



to -300. 



* Bioiiicliiku, Vol. ix. p. 21S. 



