FREQUENCY DISTRIBUTION OF THE VALUES OF THE 
CORRELATION COEFFICIENT IN SAMPLES FROM 
AN INDEFINITELY LARGE POPULATION. 
By R. a. fisher. 
1. My attention was drawn to the problem of the frequency distribution of the 
correlation coefficient by an article published by Mr H. E. Soper* in 1913. Seeing 
that the problem might be attacked by means of geometrical ideas, which I had 
previously found helpful in the consideration of samples, I have examined the two 
articles by " Studentf ," upon which Mr Soper's more elaborate work was based, 
with a view to checking and verifying the conclusions there attained. 
"Student," if I do not mistake his intention, desiring primarily to obtain 
a just estimate of the accuracy to be ascribed to the mean of a small sample, 
found it necessary to allow for the fact that the mean square error of such a 
sample is not generally equal to the standard deviation of the normal population 
from which it is drawn. He was led, in fact, to study the frequency distribution 
of the mean square error. He calculated algebraically the first four moments of 
this frequency curve, both about the zero point, and about its mean, observed 
a simple law to connect the successive moments, and discovered a frequency curve, 
which fitted his moments, and gave the required law. 
Thus if the members of a sample, 
71X = Xl -\- X2 ... + y 
and iifi^ = («i — xy + {x2 — x)- + . + — xf, 
the frequency with which the mean square error lies in the range d/u, is propor- 
tional to 
This result, although arrived at by empirical methods, was established almost 
beyond reasonable doubt in the first of "Student's" papers. It is, however, of 
interest to notice that the form establishes itself instantly, when the distribution 
of the sample is viewed geometrically. 
* Biometrika, Vol. ix. p. 91. t Ibid. Vol. vi. pp. 1 and 302. 
Biometrika x 65 
