By Student 
303 
R, aod (2) the d priori probability that R for the popiihition lies between any given 
limits. Now (2) can hardly ever be known, so that some arbitrary assumption 
must in general be made; when we know (1) it will be time enough to discuss 
what will be the best assumption to make, but meanwhile I may suggest two 
more or less obvious distributions. The first is that any value is etjually likely 
between + 1 and — 1, and the second that the probability that x is the value is 
proportional to \ — x": this I think is more in accordance with ordinary experi- 
ence : the distribution of d j^viori distribution would then be expressed by the 
equation y = \{\ — x-). 
But whatever assumption be made, it will be necessary to know (1), so that 
the solution really turns on the distribution of r for samples drawn from the same 
population. Now this has been determined for large samples with as much accuracy 
as is required, for Pearson and Filon {Phil. Trans. Vol. 191 A., p. 229 et seq.) showed 
1 — ... 
that the standard deviation is — and of course for large samples the distribution 
is sure to be practically normal unless r is very close to unity. But their method 
involves approximations which are not legitimate when the sample is small. 
Besides this the distribution is not then normal, so that even if we had the standard 
deviation a great deal would still remain unknown. 
In order to throw some light on this question I took a correlation table* 
containing 3000 cases of stature and length of left middle finger of criminals, 
and proceeded to draw samples of four from this population -|-. This gave me 
750 values of r for a population whose real correlation was "66. By taking the 
statures of one sample with the middle finger lengths of the next sample I was 
enabled to get 750 values of r for a population whose real correlation was zero. 
Next I combined each of the samples of four with the tenth sample before it and 
with the tenth sample after it, thus obtaining two sets of 750 1 values from samples 
of 8, with real correlation "66 and zero. 
Besides this empirical work it is possible to calculate d priori the distribution 
for samples of two as follows. 
For clearly the only values possible are + 1 and — 1, since two points must 
always lie on the regression line which joins them§. 
Next consider the correlation between the ditference between the values of one 
character in two successive individuals, and the difference between the values of 
the other character in the same individuals. It is well known to be the same as 
that between the values themselves, if the individuals be in random order. 
* Biometrika, Vol. i. p. 219. W. K. Macdonnell. t Biometrika, Vol. vi. p. 13. Student. 
X Not strictly independent, but practicallj' sufficiently nearly so. This method was adopted in order 
to save arithmetic. 
§ There are of course indeterminate cases when the values are the same for one character, but they 
become rarer as we decrease the unit of grouping until with an infinitesimal unit of grouping the 
statement in the text is true. 
