\ XT'J l,itru<l,n-li.ni 1 \\.\\ii 



ami tin- modal \alue of tin- product liniment coellicirni in 



/>ii = 'MM, I W x o-jo-f 

 and its mean value is 



11 = (1 -- J /3o- 1 <r 2 = '57000 x ai<7j. 



The actual value y/ a lor tin- pan-nta! population is 



jta 0*6010'* 



This indicates that the most probable value of p n in samples of twenty taken 

 from a population with correlation O'O is about ^ less than the parental value. 



' Illustration (ii). Find the modal value in samples of 5 drawn from a parent 

 population of correlation p = 0*9. 



In the subsection of Table X for n=5, the first entry in the third column for 

 v = 9-0 is p = -90000. It follows therefore that v = '90000, and accordingly 



1-p 2 



n= v o-i(72 = '342 x 

 n 



Further pu = (l -- Jpffi<r 2 = '720 x <7jcr 2 , 



while for the parent population 



P\\ = p 0"l <7 2 = '9000 X CTj CT 2 . 



Thus the value of p n most likely to arise in the sampling is 62 / less than the 

 value in the sampled population, while the mean value in sampling is 20 / less 

 than that in the parent population. 



These illustrations will suffice to indicate how cautious one must be in accepting 

 a value of pu found from a small sample as approaching in value that of the parent 

 population. 



Two remarks may be made here as to the plotting of frequency curves of v or 

 of pu/<ri<r z = Q 1 i. 



In the first place the numbers standing in brackets in the first column, that for 

 T m , in each subsection indicate the number of zeros between the decimal place and 

 the first figure recorded. These values, though so small, are of much importance 

 as the factor ef v can become very great when p and v are considerable. Thus for 

 n = 22, p = 0'9, to draw the frequency curve for v we must go as far as v = 240'0, 

 i.e. beyond the limits of our table which goes only to v=120'0; but in such 

 cases of high p and n > 20, a Pearson curve tits well and we have accordingly only 

 published our table as far as v= 120-0. 



The second remark to be made here is that the reader must settle before he 

 draws his diagram whether he wishes to plot a curve of the frequency distribution 



of Qn =PU/<TI<TZ, or of v = _ - a Q n . It is of little importance which is selected, but 



of great importance to see that the scales are appropriate. In Figs, (i) (iv) above 

 the frequency distributions are for v, and we use equation (vi) above in plotting 

 y v to v. 



