clxiv Tables for Statisticians and Biometricians [XXXI XXXIV 



The ordinates of the frequency curve calculated by aid of Table XXXIII were : 



These are not adequate for the construction of the curve, but are sufficient 

 to indicate, as will be shown in a later section, that Pearson's appropriate curve, 

 Type VI, for the above values of 01, /Si, f$% gives an excellent fit. 



See the diagram (p. clxix), where the normal curve to mean = '7991, er = '028,868, 

 and total = 10,000 samples is also drawn. 



It will be clear that, when p is as great as '8, with a sample as large even as 160 

 the normal curve gives a poor law of distribution, but the Pearson skew curve, 

 even when based on the interpolated, not too accurately computed, constants gives 

 satisfactorily the skew distribution of r in samples. 



TABLES XXXI-, LI*-* 



Special Cases of Frequency for n very small. 

 (i) Samples of Two, n = 2. 

 Here we have 



_ _ sin" 1 p 



.(viii), 



and for a group of .M-samples of two the distribution consists of two "lumps" 



M cos" 1 ( p) i M cos- 1 p 



-t=m (+) at r = + I and - *- = m (-) at r = - 1. Thus 



p = COS 7T 



m(-) 



with a standard error of TT 



'/ 



M 



if the number of pairs M be considerable. We can thus, by extracting pairs from 

 a normal population, determine by the ratio of the number of positive to the number 

 of negative correlations the true value of p, the correlation in the parent population. 

 If x i> y\ be the values of the variates in the first, x%, y z in the second member of the 

 pair, then if #1 be taken > # 2 , the correlation will be positive if yi be > y 2 , and 

 negative if y\ be < 3/2- It is therefore possible by inspection to determine in each 

 pair the sign of the correlation. Further constants of the distribution are 



/u 2 =l-r 2 , ^=-2^(1-7*), ^ = (l-r")(l + 3r"),) 

 and thus ft = 4r 2 /(l - r 2 ), /S 2 = (1 + 3r 2 )/(l - r 2 ) j 



sin 



m 



* The factor TT has been dropped from this formula in Biometrika, Vol. xi. p. 361, ftn. 



