Ixxx 



Tables for Statisticians and Biometricians [X XI 1 



for n = 50, p = '6 we have Fig. (i), where the dots show the ordinates calculated from 

 a Type VI curve. 



For n below 25 when the correlation is high, we still get good fits, as in 

 Fig. (ii), for n = 22 and p = '9. 



But for values of n below 25 when the correlation is low, the Pearson curves 

 diverge too much from the true distribution as in Figs, (iii) and (iv). 



If N be the number of samples the actual curve of frequency of v is 



y, = N(l-f)U n -V*Tt n _ 1 (v) (vi), 



where Ti n _ 1 (v) = T m (v) is the function tabled in Table X for n = 2 to 25, m = to 

 11 '5. This function is independent of p. 



The reader must be cautious not to confuse pu/vicrz = Qu> say, with r the 

 correlation in the sample ; &i and <r 2 are the standard deviations in the parent 

 population, and if Si, 2 be those in a sample with correlation coefficient r, 



_ 



and the variation of Si, 2% with the sample gives wholly different distributions for 

 r and Qu, even when the sample becomes large. 



Fig. (v) illustrates the difference between the distributions of r and Qn=pul<ri<r2 

 in the case of samples of 20 for p = 0'6 in the parent population. 



That the student may appreciate the great difference in sampling between the 



distribution of r and of Q u = 



= = - 



O"i (72 



r, Tables XP and XI 6 are provided. It 



will be seen from Table XP that with samples of 100 even, for considerable cor- 

 relation in the present population the means are only approaching equality, and 

 the standard deviations still remain widely apart. Table XP indicates by the 

 series of values of /?i and /3 2 for various samples, how different are the forms of 

 the two curves. By mere inspection or by rough interpolation from these tables 

 the student may appreciate in the case of any given sample the grossness of the 

 error which arises when^u/oio^ is substituted for r, for example in dealing with 

 the standard error of a ' tetrad' in investigations as to factors of general intelligence 

 in psychology*. 



* It may be suggested that the divergence of the two curves would be less if we took much larger 

 samples. That this is not so, the following values of the standard deviations in the case of p=Q-5 

 sufficiently indicate : 



In each case the standard deviation of ^- is about 50 per cent, increase on that of r. The ^- curve 



<r,<r 2 0i0- 2 



tends more rapidly, even when p is large, to approach the normal form. 



