xliv Tables for Statisticians and Biometricians [IV, V VII 



Here N = 57, n' = 3, but as there are four variates, we enter the Table with n = 4, 

 and find: 



J2 2 =-0536, o-jj, = -0418, and \ d = ('4431 - -0536)/-0418 = 9'32. 

 This is almost three times as great as Xi=3'20, which gives Pi = *011; there 

 is therefore a very significant correlation of an order about '67 between rainfall 

 and the three variates longitude, latitude and altitude. 



Users of the table must bear in mind its limitations. The theory on which it 

 is based depends upon our sampling being made from an indefinitely large normal 

 population ; the argument is based, in the case of both if and R z , on the im- 

 probability (or probability) of the observed result, supposing the variates are 

 uncorrelated. The table tells us the probability of association, but if we conclude 

 that the variates are correlated, it really throws no light on the closeness with 

 which the value deduced from the sample approaches the actual value in the 

 parent population. We do not know at present the distribution of if for a normal 

 surface, when correlation actually exists, and we cannot overlook the point that 

 the chief value of the correlation ratio arises from cases in which we have at least 

 grave doubts as to the nature of the regression being linear, i.e. from cases in which 

 the variates are not normally correlated. 



One fact may, however, be borne in mind for the normal case, the standard error 

 of rj z will be a maximum for the case of no association. In other cases it will be 

 reduced by a factor of the form 1 if, or some function of this factor, which 

 vanishes when 77 2 =1. Accordingly ay sets a limit to the standard error of if, 

 when correlation actually exists, and since twice the standard error gives a 

 reasonable limit to the difference between the if of the parent population and 

 that of the sample, we may obtain from ay some appreciation of the range probable 

 in the sample ?? 2 . 



For example, in Illustration (iii) we have 2ay = '019,556 and rfp^ = '525,045 ; 

 thus we may reasonably expect the r) z of the parent population to lie between 

 505,489 and 544,601, or its correlation ratio between '71 and '74. 



Again in Illustration (vi), 2a-jj = '0836 and R z = '4431, and we may reasonably 

 expect the R 2 of the parent population to lie between '3595 and '5267, or the R 

 of the population between '60 and *73, the greater range in this latter case being 



due to the paucity of observations. 







TABLES V VII. (For use in computing Tetrachoric Functions.) 



Table of the First Twenty Tetrachoric Functions to Seven Decimal Places. 

 (Alice Lee, D.Sc., Biometrika, Vol. xvn. pp. 343 354.) 



Table XXIX of Part I of this work gives the values of the tetrachoric functions 

 TI > T"2 , T 3 , r 4 , TS and TO at intervals of one-thousandth in the argument TO = (1 A). 

 The value of h is also given. The range of T O is from '001 to '500, and of h from 

 3-090 to -000. 



