344 Distribution of Correlation Coefficient in Small Samples 
and 
(n-l)(l-po*)^„=(2n-3)po2 
w-2 
.(xliv). 
But even this would be laborious had we to find successive values of from 
(xhv). Actually, if n be moderately large, and E„_i tend to equahty fairly 
rapidly. For example the following are the values of E^ for = -6 : 
Ii = 2-078,4173, = 1-873,8688 and therefore E^ = -901,5845. 
^2 
■901,5845 
^9 
1-470,8511 
Eu 
1-511,0031 
E23 
1-527,2571 
1-257,5588 
Eio 
1-475,5703 
En 
1-514,1874 
1-528,7778 
1-183,5106 
Eu 
1-486,5966 
Eis 
1-516,9985 
E25 
1-530,1728 
1-451,8703 
Eli 
1-492,1848 
Eis 
1-519,5018 
E26 
1-531,4570 
E, 
1-377,5430 
El3 
1-498,5199 
E20 
1-521,7436 
E' 
1-529,7263 
E, 
1-453,2879 
Eli 
1-503,1022 
En 
1-523,7636 
E" 
1-531,0459 
Es 
1-445,7342 
Eli 
1-507,3770 
E22 
1-525,5928 
E'" 
1-530,1488 
Clearly £'.„ and E^-i approach equality. Now put £^25 = -^24 w = 25 in 
(xliv) and we have for po = "6 
20- 8896£:'2 - 16-92.B' - 23 = 0, ■ 
which gives for the root required 
E' = 1-529,7263. 
But we might also have made E^^ = £^26 ^^^^1 so reached 
21- 7600^;"2 - 17-64^;" - 24 = 0, 
which gives E" = 1-531,0459. 
It is better therefore in finding En to equate E^ and E„_-^ than E,^ and E^+i. 
A still closer approximation may be found by noting that 
E^-E' = En+, - E", nearly, 
where e is very small. Hence since 
n (1 - po') En^xEn - (2'« - 1) Po^E,, - {n - 1) = 0, 
we have 
(n - 1) + {2n - 1) po^E' - n (1 - po*) E' E" 
or. 
E'" = E' + €== --- 
n(l-p,^) {E' + E")-{2n-l)p,^ 
n - 1) -I- w (1 - po*) E'^ 
.(xlv). 
n{\-p,^){E' + E")-{2n-\)p,^ 
For the case of p^ = -6 and n = 25 we find 
E'" = 1-530,1488, 
and £"25 — E'" = -000,0240, a close agreement. As a matter of fact as we only use 
En in a small term the approximation E' is generally quite sufficient. 
In the above method all turns on finding a good value of po^, i.e. a first 
approximation to the value of the product of p and r. This may be obtained in 
either of the following ways : 
