ON A NOVEL METHOD OF EEGARDING ASSOCIATION 21 



Suppose r=-995, r' = -990025, 



l-r' = -009975 and X= -01007, 



Vtt 



V239-94 6 (•009975)"''-^^^ / _ -01 -0003 



238-946 -995 \ 240-946 241x242 



It is clear that for such high values we may treat the series factor as unity. 

 We find 



- log P= 240-360. 



Now putting r=-99, we have 



-logP = 204-525, 

 and thence by interpolation 



-logP = 215-74o, 

 where r= -992. 



The correlation is thus very high, but not perfect, and this seems reasonable 

 because we are not really making the assumption of a Gaussian frequency, and the 

 value of P if very small is still not zero. 



Conclusions. Without laying too much stress on a short series of numerical 

 illustrations, which were merely taken at random for various values of the correlation 

 and for various divisions of the categories*, we may, I think, conclude that our 

 correlation scale of the improbability of independence in variates gives quite reason- 

 able results when tested on fourfold tables treated as or really representing Gaussian 

 distributions. In these cases we shall rarely get a divergence amounting to twice 

 the probable error, and usually a result well within the probable error. In the 

 following table I have put together the chief results for the present series of 

 illustrations. In the first column we have the value of ^ ; in the second column the 

 resulting probability of independence, P ; in the third column we have <^'^ the mean 

 square contingency ; in the fourth column Gj, the coefficient of mean square con- 

 tingency ; in the fifth column rjj^ = ^; in the sixth column Yule's coefficient of 

 association, Q^ ; in the seventh column the coefficient of correlation, r^, as found by 

 a fourfold table on the assumption of Gaussian distribution of frequency ; and lastly, 

 in the eighth column, the value of the coefficient of correlation, Vp-, on the equal 

 improbability scale discussed in the present memoir. 



Several interesting results are at once manifest : 



(i) The reader will at once appreciate the difficulty of mental apprehension 

 of the relative probabilities involved in the column P. 



(ii) The order of the probabilities is not the same as that of the coefficients 

 of correlation r^ found by assuming a Gaussian frequency. Nor should we anticipate 

 that they would be ; for by increasing the total population, and still distributing it 



* I have worked out numerous other examples since the illustrations here given, and on the basis of 

 them might have very reasonably supposed the divergences in I and II to be due to arithmetical errors. 

 But I have not found such errors. For example take, to compare with II, the Table given by Macdonell 

 (Biometrika, I. p. 222) for stature and left m. finger length (Table XII of his paper) : it gives -68 as 

 against Macdonell's '6631 -013, as found by Gaussian methods. 



3—3 



