U KARL PEARSON 



As a matter of fact J^ may be treated as infinite compared with J'imr for values 

 of r = 0"2 and less, in which case 



li^{j2m) = {2n-\){2n-2,){2n-b) Ix -5. 



It will be found as a rule unnecessary to go beyond the terms in ii*. 



From (xvi) and (xvii) Table I has been constructed; this gives the value 

 of -logP for each value of r and „o-^. In other words it expresses the probability 

 that with a given probable error of a zero coefficient {i.e. "67449 ocr,) a given value 

 of the correlation will arise from this uncorrelated material in a random sample 

 of definite size. The size of the sample and the nature of the process by which r is 

 obtained are indifferent, provided regard is paid to them in determining ,o-,. 



Table II is the required extension of Palin Elderton's Table of Goodness of 

 Fit (see Biometrika, Vol. i. p. 159). It gives the value of — logP for n' = 4, 

 i.e. for four groups, from ^=1 to x^ = 25,000. This enables us to ascertain the 

 improbability of a given x^ even when that improbability is only significant in the 

 5000th place of figures. 



Table III gives the x^ which would have the same improbability as the r of 

 Table I, and is obtained by simple interpolation from Table II. 



Table IV replaces x^ by logx^ and forms a reasonable working table. Given 

 logx'' from the fourfold table and ocr,, we can find the value of r which expresses the 

 same improbability. 



Thus far we have not even selected our scale of correlation, which is wholly 

 determined by the choice of oO-^. We might take jO-y simply equal to ijjn, but 

 this would not be a really satisfactory scale of correlation improbabihties. The 

 reason is obvious ; it supposes a knowledge never conveyed by a fourfold table, 

 i.e. the knowledge involved in our having the material in a large number of equal- 

 ranged cells. Very naturally, therefore, we avoid this scale, for it certainly would 

 not give at all comparable values of r for those cases of fourfold table where the 

 material is known, or may legitimately be supposed, to be Gaussian. Accordingly we 

 adopt for „cr,. the value as given by a fourfold Gaussian table, i.e. 



1 , .... 



oO-r = ^XXa,XXa, (xVUl), 



where v and Xa ^^^ respectively 



v/i(l+a,)i(l-a,) s/^(l+a,)i(l-aj 



— and ^ , 



a table of which function has been recently published by me, and is reproduced here 

 as Table V. It enables us at once to determine ocr, . 



Finally, Table IV has been converted into an " abac," upon which the value 

 of r — the " equal improbability correlation " — can be read off as soon as log -^^ and ^a-r 



* For example for r = Q\ and oO-,.= -03, -logP=3-072 by (xvi), it equals 3-076 from (xvii) and 3-066 

 from the Gaussian, or probability integral. The most troublesome values were those for „o-^=-08, r = 0-2 

 and 0-3. They were finally determined as 1-924 and 3-903, but they cannot be guaranteed to a unit in 

 the last figure. 



