Ixxxvi Tables for Statisticians and Biometricians [X XP 



(6) To draw the curve for m = \n 1, we have to plot y for each value of v in 

 (vi). In the first place ^(1 p 2 )^ 71 " 1 ) is a numerical factor which does not vary 

 with v\ T^ n _ 1 (v) is given in the first column of each subsection of our Table X 

 The value of & v may be found from Newman and Glaisher's Tables of (? and e~ x in 

 the Cambridge Philosophical Transactions (Vol. xm. 1883, pp. 145 272). Thus 

 the work can be carried out without the use of logarithms on the arithmometer. 

 But this process involves interpolating into the above tables for x = pv. The 

 alternative process is to use logarithms and take 



log y v = log N + (m + ) log (1 - p 2 ) + (-434,2945p) v + log T m (v) . . .(vii), 

 so that for the given p 



log y = C +C l v + log T m (v), 



where GO and G\ are constants. For this method log T m (v) is provided in the second 

 column of each subsection of Table X. The labour here lies in finding the anti- 

 logarithms. In this application the two methods are of about equal length, when 

 we do not use more decimal places than are needful for practical statistical work. 



A most important point must be kept in mind : we have to plot y for v negative 

 as well as positive. ef v changes from & v to e~ pv with change in sign of v, but T m (v) 

 does not change sign with v ; we have always T m ( v) = T m (v). 



A matter of interest as well as of practical value in the plotting of the curve 

 lies in the determination of the position of the mode v. It may be shown that the 

 equation to determine the modal abscissa v is 



v T,n-i(v) , .... 



'-- (vm) - 



It is not feasible to solve this equation and find v directly. But if the right-hand 

 side be tabled for each value of m(= \n 1) and each value of v, we obtain a series 

 of values of p, for which, with the given value of m, v would be the modal value v of 

 that sampling series. These values of p are given in the third column of each 

 subsection of the table. 



Illustration (i). In samples of 20 from a parent population of correlation p = 0'6, 

 find the modal, i.e. most probable value of p u , the product moment coefficient. 



Here we. look under the subsection n = 20 (m = 9"0) and find for 

 fl = 15-5, p = -59990, and v=16O, p = -60864. 

 Hence for p = '60000, we have by linear interpolation, 



v = 15-5 + ^ ? x 0-5 = 15-505721. 



"UUo/ < * 



This is accordingly the required value of v, the mode. But 



n pn n ~ 



1 p 2 <TI & % 1 p z 



Thus fl u = (1 ~ P % = -496,183, 



