8(5(5 Distribution of Correlation Coefficient in Small Samples 
oi' p = {mp - m,„)/{mj, + m„) (Ixxxiii), 
a very simple formula if m.p or has been found. 
Clearly p = {2m.p - N) N, 
and 8p = 28)nJN, 
m\ N 
Thus the probable error of p found in this manner is 
•67449 ^ /m,: 
and can easily be evaluated, for it gives : 
Probable Error of p = ^^TTpi (Ixxxv). 
We see it is larger by the factor — ^, which is greater than unity, than the 
usual value for the product moment process of the correlation. But a new point 
arises : N is the number of triplets in the present process, and N the number of 
individuals in the product moment process. If we take M triplets and N individuals, 
we have to compare 
•67449 a/1 - p^lV'M with -67449 (1 - p^)/VN, 
and these probable errors will be equal if 
M = N/{1 - 
If the number of triplets be > N/{1 — p^) the triplet process will be more accurate 
than the product moment method. The number of triplets required for equality 
of probable errors are for the various values of p : 
p = 
0 
M = 
N, 
p = -5 
M = 
l-333iV, 
p = 
•1 
M = 
1-OlOiV, 
^ = •6 
M = 
l-563iV, 
p = 
•2 
M = 
l-042iV, 
p = -7 
M = 
l-923iV, 
p = 
•3 
M = 
l-099iV, 
p = -8 
M = 
2-n8N, 
p = 
•4 
M = 
M90iV, 
p = -9 
M = 
5-26SN. 
This series would seem to suggest, since a triplet contains three individuals, 
that to use the triplet process with equal exactness with the product moment 
process, in the case, say, of p = -5, we should need a population of 4iV. But this 
assumes that each triplet is based upon three independent individuals. Actually 
a population of N provides -^.N {N — 1) (N — 2) triplets and if these could be 
considered as an independent sample of M triplets, we should have a less value of 
the probable error of p by the triplet process using all possible sets than by the 
product moment process, provided 
^ (iV - 1) (iV - 2) 
