3o0 DiMribution of Correlation Coefficient in Small Samjdes 
This series has now to be inverted and leads after considerable algebra to 
_ / 5 
^2(n-l) ' 8{n-l)^ ' 16 (n- 1)3 
•••) (Iv). 
5 OjV) , (61 - 101p2) (1 -p^) (367 - 1480p2 + 1273p'') (1 - p^) 
2{n-l) 8{n-l)^ ^ 16 (n - 1)^ 
(17606 - 125727p^ + 246783p^ - U3782/)«) (1 - p^) 
256 (n - 1)* 
The above series is of very considerable interest from more than one standpoint*. 
In the first place it appears that .Soper's approximation {Biometrika, Vol. ix. p. 108) 
was not valid. He obtained 
= P 1 
3 , (41 + 2 3p^)(l-p^) I 
2 (n - 1) 8(n - 1)^ '"j " 
Thus for n = 25, p = -6, Soper's formula gives -62811, and (Iv) gives -64205, 
while the exact value is -64194. It is clear that the coefficient f in Soper's 
second term of the series can naver approach the ^ of the more exact expression. 
At first the difference was found very perplexing, especially when the algebra had 
been verified ; but the solution appears to lie in the consideration that the best 
fitting Pearson curve to the frequency is not one tied down to the range — 1 to + 1. 
That curve is fitted by two moments only, but if we fit a curve by the first four 
moments and use the general expression 
^2 (5^2- ¥1 - 9) 
we obtain r = -64192, 
a value close to the true value. In other words the use of the third and fourth 
moments to find the mode is far more important than fixing down the range to 
the theoretically possible values ; that process determines much more quickly the 
form of the frequency curve, but it does not give nearly such a good fit as allowing 
the Pearson curve freedom to adjust itself by means solely of the first four 
moments f. On the other hand a Pearson curve determined by the first four 
moments does describe fairly accurately the frequency distributions of r for 
n = 25 and upwards : see p. 337. 
(6) Equation for Modes and Antimodes {n = 3). 
Still another method of approaching the modal value has been found occasionally 
of service J. 
* We have used the expansion in terms of - 1) rather than as (« - 1) appears to arise more 
simply in all the formulae. The form in is 
1 I 5(1 - P-) ^ (81 - 101p^)(l - p ^) ^ (651 - 1884p^+ 1273p^)(l - p^) ^ _ \ 
f The Pearson curve determined from the range does not give good values of the frequency for 
n = 25, even when we use the true values of f and a,, and not Soper's approximations to these constants. 
+ It was used successfully in calculating the antimode in the case of samples of three, when the 
correlation in the sampled population was low. It gave a fairly good "jumping off point" even for 
higher values of the correlation. 
