ON A NOVEL METHOD OF EEGAEDING ASSOCIATION 7 



Hence ^^^Ja + h){a + cXd + b){d+.c) ^^..^^ 



and as a result 



Thus tlie probable error of r^^ for material with zero association takes the simple 

 form of "67449 -y=, precisely the value of the probable error of a zero r found 



from product moment formula, and quite independent of the division between the 

 categories. 



We have accordingly 



r',, ' Njad-hcf 



„(/, {a + 'b){a + c){d + h){d + c) 



.(ix). 



hh 



Thus X which gives the probability of the observed mean square* contingency is 

 actually equal to the ratio r^j/ocr, and approximately equal to r/jO-,; in other words, 



for the simple case of a fourfold table \ which determines the probability that the 

 system is a random sample from unassociated material is really the ratio of an 

 observed correlation on the Gaussian hypothesis of distribution to its probable error 

 on the assumption of unassociated variates*. 



It will be clear, however, that x = i''l<P'r will only give a very rough approximation 

 to the value of r, since all terms but the first in the series for r in (iv) must be 



* If <l? be the mean square contingency, then it is easy to show that o<^,t, i-e. the standard deviation 

 of <^, if there were no association of variates = l/ViV^. Thus we have <^/o(r^=x, or the ratio of <^ to its 

 standard deviation is also x- In the same manner the standard deviation of a Yule " coefficient of associ- 

 ation," Q = {ad- bc)l{ad + be) is 



„ _(l-<2^) /I i + l + l 

 1 /H i i a _\ In 



a^ad «"■«- 2 V 6c "^ 6 ^ c "^ 6c ~ 2 V 6c ' 



but as before 6 = (6 + d) (a + V)\N, c = (a-vc)(c + d)IN, hence 



1 / N^ 



»'^«-2 V {a + c){c + d){h + d){a + b) 



_1 N^ ^{a^ c) {<i + d){b + d){a+ 5) 

 ~2 (a + e){c + d){b + d){a + b) 



1 1 >J{a + e)(c + d) (6 + d){a + b) 

 ~^ JN be 



1 ij(a + c){c + d) (6 + d){a + 6) 

 ~ JW ad + bc 



Hence if we use the constants of the observed material 



Q ^N{ad-bc) .,^^^.„._ ^^y 



oO'q V (a + c){c + d) (6 + d) (a + 6) 

 Or the improbability of the sample as a sample of uncorrelated material, is the same when found from Q 

 as from r^ or from contingency, all are expressible in terms of x- 



