ON A NOVEL METHOD OF REGAKDING ASSOCIATION 5 



with the relative frequency of the two pairs of groups into which our categories 

 divide the variates. This has no relation at all to the assumption of a Gaussian 

 frequency. If a group contain Wi individuals, its probable error is 



■67449 N/wi(l-n7iV), 

 whether the variate be Gaussian or not, and the ratio of probable error to the number 

 in the group contains the factor '1/7 Wj, and increases rapidly as rii becomes small. 

 Any categories which contain only small percentages of the total in a fourfold 

 division, even if the variates be purely categorical and not quantitatively measurable 

 {e.g. divorced women and married women), involve increased probable error of our 

 conclusions, and no scale of r will be satisfactory which does not recognise this ; 

 (ii) beyond this when we suppose our fourfold table to represent Gaussian material 

 the value of r obtained by our selected scale of correlation deduced from " equality 

 of improbability" ought to be reasonably close to the r given by the usual process 

 on a Gaussian fourfold table. 



The probable error of the coefficient of correlation for a fourfold table on the 

 supposition that it is a random sample from uncorrelated material of Gaussian 

 distribution is 



■67449 



JNHK 



I {a + h) {a + c){d + h) {d + c) ,.... 



where H=-^c-i'^\ K=^c-i^\ 



h and h corresponding to the ratios of the distances of the means from the dividing 

 lines of the categories to their respective standard deviations. 



It is clear that this value increases rapidly with h or h, since 

 1 Ha + c) {h + d) 1 Ua + h) (c + d) 



increase to infinite values with h and k. 



Now the value of r from a Gaussian fourfold table has to be determined from an 

 equation of the form 



where v„_i and w„_i are known converging factors in A and ^ respectively*. 



It follows that the ratio of the standard deviation of r on the supposition 

 that it is truly zero to its observed value is approximately 



or approximately: 



„l^- A^±b) {a + c){d + b ){ d + c ) , . 



"'"'' jN{ad-bc)~ ^ ^' 



^~ {a + h)(a + c)(d + b){d + c) „o-/ ^ ^• 



* Phil Trans. Vol. 195, p. 6. 



