354 Distribution of Correlation Coefficient in Small Samples 
the correlation of the characters in the population samples, and desire to ascertain 
whether a small sample of some population similar to a knoivn population confirms 
our experience. 
For example, we may have twenty pairs of brothers recorded for some special 
character. Our a priori knowledge is certainly not that all correlations between 
pairs of brothers from — 1 to + 1 are equally likely to occur! On the contrary 
we anticipate a value which will not be very far from 0-5. And this a priori 
conviction is so great, that if the small sample did not give a value which con- 
sidering the size of the sample was compatible with the correlation in the sampled 
population being near 0-5, we should suspect errors in the measurement or some 
form of disturbing selection. In such cases, and something like them appears 
to us most frequent in biometric practice, it is we think erroneous to apply Bayes' 
Theorem. All it seems possible to do is to assume that we have drawn a value 
near the mode of our distribution, for our sampled population is much more likely 
to have a single value, that of our a jyriori experience, than every possible value 
from — 1 to + 1. If Bayes' Theorem confirms this value — so much the better; 
if it does not, its fundamental hypothesis is usually so unjustified that it seems 
most unreasonable to assert that it must give the most likely value of the 
correlation in the sampled population. 
The fuller solution of Eqn. (Ixi) thus appears to have academic rather than 
practical value. Still certain points of theoretical interest arise in the discussion 
of both (Ix) and (Ixi). Let us suppose that our a priori knowledge consists in the 
distribution of p about a mean p with a standard deviation k. It is convenient to 
take = m {1 — p'^), where m is an arbitrary constant. Probably k = 0, whenever 
p = 1, and this suggested this form ; but since m is quite arbitrary we lay no 
stress on this point. The equation to determine the most likely value of p now 
becomes 
n-1 (p-pf 
d r (1 -p2) 2 g 2m(l-pr 
dp ' 0 (cosh z — pr] 
dz = 0, 
m (w — 1) (1 — p"^) 
Now this equation cannot in general be solved unless we know the order of the 
product m (n — 1). Certain cases, however, can be considered. If m be very 
large, i.e. if there be very considerable scatter in our past experience of p, then 
/c)/„_i= (1 - p^)rln (Ixiv), 
problems is very narrowly limited. For example we measure twenty individuals of a population for 
stature, and seek the best value of the variability of the sampled population from the result. Would 
it not be unreasonable to suppose that d j^riori this variability may be equally likely to have any value 
from 0 to 00 ? Our h priori knowledge is that it is somewhere between 2"-5 and 3"-0 and very far from 
equally likely even between these values. To justify the equal distribution of our ignorance, we should 
have to assume that we neither knew the exact character measured, nor the unit in which it was measured, 
and such ignorance can only be very exceptional in the present state of our knowledge. 
