358 Distribution of Correlation. Coefficient in Small Samples 
and writing = Tj^n-i} we have after some transformations 
+ 0 ■ 
(1 - r2p2) (^1 + - {n - 1) (1 ~p) + rp {{n +!)(!- r^p') - {2n - l)r^ (1 - p^)} E\ 
(Ixx), 
where («. — 1) (1 — r^p^) E\ = (to — 3) rp + f-; — (Ixxi). 
The method is now straightforward, at least for n moderately large. We put 
E„ = = E' in (Ixxi) and substitute the resulting value of E' for E„ in (Ixx), 
and thus reach the small correction on p. 
Illustration. In a sample of 25 pairs only of parent and child the correlation 
for a certain character was found to be -6. What is the most reasonable value to 
give to p in the sampled population ? 
If we distributed our ignorance equally the result would be that stated on 
p. 357, i.e. 
p = -59194. 
But, in applying Bayes' Theorem to this case, to what result of experience do we 
appeal? Clearly the only result of experience by which we could justify this 
"equal distribution of ignorance" would be the accumulative experience that in 
past series the correlation of parent and child had taken with equal frequency of 
occurrence every value from — 1 to + 1. To appeal to such a result is absurd; 
Bayes' Theorem ought only to be used where we have in past experience, as for 
example in the case of probabiUties and other statistical ratios, met with every 
admissible value with roughly equal frequency. There is no such experience in 
this case. On the contrary the mean value of p for very long series of frequencies 
of 1000 and upwards is known to be + -46 and the range is hardly more than -40 to 
•52. We may accordingly take p = -46 and 7)i (1 — p^) = = about -0004, whence 
m = •0004/-7884 = -000,507 say. 
Thus — 7^^^ — =^ = 82-1828 and the term containing it is the dominating term in 
m {n — 1) 
Equation (Ixiii). Thus p will differ little from p. We find 
- _ - -4 37,006 - -4 24,959 
^ " ^ ^'^ 70-034,491 + 2-790,925 C ' 
from (Ixx). 
We next determine En = E' from (Ixxi), i.e. 
24 X -923,824l"2 - 47 x -276^ - 23 = 0, 
which gives us E' = 1-352,2185, thus ijj = -00225 and 
p = -46225, 
a totally different "most likely value" from that obtained by "equally distributing 
our ignorance." 
