XXXI"-*, LI-*] Illtl'lHlllHini, 



(a) To find the "most likely" value p in a Small Sample of Size n. 

 The equation (xxxvi) may be solved in two ways: 



(i) By a series expressing p in powers of with rocUici.-nis which an- 



7t "~ 1 



powers of r, so that p changes sign with r. 

 The aeries as far as l/(n I) 8 is as follows : 



* Xi(r) X 2 (r) X,(r) 



O 7^ ... . i vvvt/n 1 



n-1 (71 -I) 2 (n-1) 8 



where Mr) = ir(l -**), X 1 (r) = ir(5r 1 - 1)(1 -r 1 ), 



and X 3 (r) = T i ?r r(l7r 4 -87- 2 -l)(l-r a ). 



The values of the first three Vs are given in Table LI*. 



Illustration. Suppose r = '6, what is the " most likely" value of p, the correla- 

 tion in the parent population for a sample of 25 1 

 Applying formula (xxxvii) we have 



For a non-tabled value of r, it is better to interpolate between the two values 

 of p for the two nearest values of r, than to interpolate for each separate valm- <>f 

 the Vs. 



Owing to the relative smallness of the X's in (xxxvii) compared with the v's in 

 (xxx) on p. clxx, three values in the former will usually suffice unless n be very small. 



(ii) By a method similar to that by which we have approximated to the mode. 



Suppose p! 2 = pr has been obtained through a first approximation to p, as by 

 the method given above, then let E be found from the quadratic 



(n-l)(l-p l *)E 2 -(2n-3)p 1 2 E-(n-2) = ......... (xxxviii), 



and its value substituted in 



...... (xxxix). 



The expression pi 2 + e will be a closer approximation to pr. The approximation 

 may, if it appears needful, be repeated. 



Illustration (i). Test the case of the accuracy of the previous illustration by 

 taking 



= -591,936, ^r = -3551,6l60=p 1 2 



as a first approximation, r being '6 and n = 25. Here 



pi* = '1261,3976, l-pi 4 = -8738,6024, and r 2 - Pl * = '2338,6024, 

 and the quadratic becomes 



20-9726,4576 E 2 - 16-6925,9520^ - 23 = 0. 

 This leads to E= 1-5182,4679. 



