360 Distribution of Correlation Coefficient in Small Samples 
Here p = -30, in (1 — p^) = .Ql, and n = 25. = rp = -Id. We will assume 
as a first approximation to p, p = -40, hence p^^ = -20. Equation (Ixxiii) for 
n = n — 1 gives 
{n - 1) X -QQE^ = (2w - 3) x -20 + {n - 2)jE„_.,. 
Put n = 25, and E' = = E^-i, and we have to find E', 
23-04^:'2 - 9-4:E' - 23 = 0, 
which gives E' = 1-223,7367 ; from this we deduce E„+i = 1-225,5025, and 
e = - -029,9238, 
leading to rp = -170,0762, or p = -34015. 
Starting again with p^^ = -17008, we find 
23-305,7472£:'2 - 7-99376^' - 23 = 0, 
giving E' = 1-179,6109 and E„+i = 1-181,3579. 
Whence we deduce e = + -003,999, 
and accordingly rp = -174,079 and p = -34816, a close enough approximation. 
But if we had "equally distributed our ignorance" we should have found* 
p = -49217. 
These results seem extremely suggestive. If we were to observe the correlation 
of length and breadth of skull in a new sample of 25 skulls, then an observed 
value of -50 wovild give a "most likely value" on the equal distribution of igno- 
rance of •4922. 
But no biometrician would admit absolute ignorance in such a case; the 
correlation has been determined rather vaguely and not very adequately so that 
results range from something like zero to -6. But this d priori knowledge leads 
on precisely the same basis as Bayes' Theorem to the value p = -3482 — a result 
very much closer to previous experience of the mean value, than to the observed 
result. And there are relatively few cases in which some such, if only vague, 
d jyriori experience does not exist. 
In the light of the above illustrations we consider it justifiable to assert that the 
results deduced from the principle of the "equal distribution of ignorance" have 
academic rather than practical value, and we hold that to apply it without con- 
sideration of its basis to the problem of finding the most likely values of the statistical 
constants of a sampled population from the values observed in a small sample may 
lead to results very wide from the truth. 
(9) S'pecial Cases of Frequency for n small. 
We shall now discuss individually the lowest sample sizes. 
(i) Samples of Two, n = 2. Here 
sin~^p 
r = -^—^ , 
* EqnntiDii (Ixvii) would a;ivc f> = 49204, nearly as good practically as (Ixviii) 
