35(5 Dish'ihvtion of Correlation Coefficient in Snudl Samples 
numerator of (Ixv) equated to zero, and (Ixvi) with E' for and E^-x as simultaneous 
equations to find E' and p^, and so obtain a good approximation straight ofl, 
when n is of the order 25, or a fair first approximation when n is smaller. 
Applying this we have from (Ixv) 
Hence 
{n - 1) (1 - po") Po" - (2n - 3) p,^ (r^ _ p„4) _ - 2) (r^ - p,^)^ = 0, 
2 , . *^-2 
or Po ^ 
n — 1 
^ _ 2 
and therefore p = r x ^ ^ (Ixvii). 
Thus it will be seen that on the hypothesis of the equal distribution of ignorance 
for n = 100, the ratio of p to r will differ less than -99402 from unity. On the other 
hand if n be 5, the ratio of p to r may differ from unity by as much as -8660 does. 
For example, if n = 5, then p = -26278 if r = -3. But for a sample of five the 
standard deviation of a value of p between -2 and -3 is of the order -18 to -20, so 
that there is little to be gained by treating the observed -30 as corresponding to 
a sampled population of -26. 
We shall now proceed to determine an expansion for p. If R be the ratio of 
Eqn. (liii), we find from (Ixiv) 
p = r(l-p2)4/4_^. 
or p = — \/ ^ 
^ 1 — pr V w 
I- pr \ n-lj V ^■■■J' 
whence on substituting 
^ _ 1 LzA' , 1 (1 - rp) (1 - p^) I (l + rp-2f^p ^) (l- p^) \ 
2n-l"^8 (n-l)2 ^16 {n - 1)^ 
and after inversion 
- _ /, _ 1 1 -r^ (1 - 5r^) (1 - r^) (1 + 8f^ - ITr^) (1 - r^) \ 
P ^'V ~ 2 n-\^ 8(%-l)2 "^^ 16(^-1)3 + "V 
(Ixviii). 
This result gives us a measure of the correctness of (Ixvii), for that equation may 
be written 
% — 1 
n — ly 
(1 - r^) _ (1 + 3r^) (1 - r^) _ (1 + 2f2 + 5f*) (1 - r^) 
2 (w - 1) 8 (w - 1)2 16 (w - 1)3 
+ ... (Ixix). 
