H. E. BoPER, A. W. YouNCx, B. M. Cave, A. Lee, K. Pearson 307 
For example if iV = 10, for all values of p between + -957 and — -957, the 120 
triplets will give a better result than the 10 individuals. Even 50 triplets would 
be better than 10 individuals for all values of p between + -894 and — -894. But 
the question arises whether we can consider the 120 triplets from a sample of 
10 individuals as much a random sample as 120 triplets from an indefinitely large 
population, and this can hardly be the case. It may be, however, that 50 triplets 
out of the 120 would be sufficiently independent to give a better result than 10 
individuals. It is very desirable that a full study should be made of such restricted 
samphng, for without such study it is not possible to assert how far the probable 
errors of doublet or triplet procedure are greater than those of the product 
moment method. 
Of course in such a case as that referred to, the labour of the triplet process will 
be considerably greater, for we have to determine the sign of the correlation in 50 
or 120 cases, instead of applying the product moment process to 10 individuals, 
and the labour rapidly increases with increase in the size of the set (doublet, triplet, 
etc.) and the size of the sample. Still the labour may be worth while in the case 
of small populations, where the best result is of considerable importance. We 
have not endeavoured to extend the theory to quadruplets or quintettes, because 
the labour of determining the sign of the correlation in these cases is very con- 
siderable. 
In the case of triplets, we require the sign of the product moment 
^lyj + ^2^2 + ^ 3^3 _ i^l + ^2 + ^s) iVl + ^2 + 
3 9 
or the sign of {x^ - x^) {tj^ ~ ]j) + (xg - Xj) [i/^ - y). 
Now suppose the triplet arranged according to the character x in ascending 
order x^, x^, Xg, then x^, — x^ and Xg — x^ will always be positive, and accordingly 
if either both y^ — y ^■iid y.^ — y are positive, or both negative the sign of the 
correlation is obvious. On the other hand if y^ — y and y.^ — y are of opposite 
sign, the matter has got to be a little more carefully considered. But if y has been 
found and the above differences determined, in most cases it is not needful to 
actually multiply out, in order to realise the sign. A graphic process depending 
on the plotting of the triplet triangle seemed on the whole more laborious than 
the above. 
Of course in samples of three the U-shaped distributions give a minimum where 
dy/dr = 0, and we have therefore an aniimode, not a mode. The values of this 
antimode are recorded in Table II, p. 368. It will be seen that all the antimodes 
are negative for positive correlations in the sampled popidation. The antimode 
asymptotes to the value — '613,9616, which it reaches when p = + 1. In this 
case the value actually fails at p = + 1, for the equation dyjdr = 0 is satisfied 
w-l 
then for all values of r, owing to the presence of the factor (1 — 2 . ggg 
Equation (xxxvii). Nevertheless the antimode curve goes right up to the point 
indicated above and this value must be used for the purpose of interpolation 
