94 On the Probable Error of the Correlation Coefficient 
The mean values of p 10 ', p. 20 ' etc. in all samples are p 10 , p 20 etc., since the 
moments are crude and simple averages of individual values. Let dp 10 , dp 20 , 
dp 0l , dp 02 , dp n be the deviations of p l0 ',p20, Poi> P<a>Pn from their means p 10 , p w , 
Poi>Po<2, Pn- The mean value of r we have called f. Let dr be the deviation of r 
from its mean r, then (xv) becomes 
- + dr = Pn + dp n - (p l0 + dp l0 ) (p 01 + dp 0l ) 
*J{p 20 + dp w - (p w + dp 10 ) 2 } x Vboa + dp 0 . 2 - (p 01 + dp 0l ) 2 } 
Choose the fixed origin of measurement of each character to be the mean 
of that character in the whole population, then p w =p 01 = 0, and (xvi) becomes 
j + d r = Pn + dp u -dp w dp 01 , ..s 
V|i>2o + dp 20 - {dp 10 )-\ x V|j^o2 + dpw - (dp 01 y\ 
If the distributions and correlations of the deviations of the moments in 
samples of n are known this is the equation for determining the distribution of 
the values of the correlation coefficient. The average value of the right-hand 
side of (xvii) will be r. The average values of the square, cube etc. of the 
right-hand expression will give the crude second, third etc. moments of r from 
which the moments of deviations from mean value of the correlation coefficient 
can be derived. 
Now if (xvii) be expanded in powers and products of the deviations it may 
be anticipated that the average values of terms of higher order in the deviations 
are of higher order in 1/n, and that there is a limit to the number of terms needed 
to give a required approximation. The approximation sought in the crude moments 
of r is to terms in 1/n 2 only, in order that the moments from the mean may be to 
terms in ljn 2 , and so that ay 5 for instance, which is known* to have the value 
(1 — p 2 ) 2 /n to the first approximation for normal frequency, may be further carried 
to a term in Ijn 2 . 
Thus the process for determining r is to expand (xvii) and to find and insert 
in it the average values in samples of n of the various powers and products of the 
deviations of the moments involved, carrying the process on as far as is necessary 
to gather in all significant terms as defined above : and a similar process applied 
to the squares and cubes etc. of (xvii) determines the higher moments of r. 
Were the samples sufficiently large these deviations would approximate, as has 
been shown, to normal distributions, and the known properties of such distributions 
could be utilised in evaluating the complicated mean, but we are dealing with 
small samples where the deviations are not so distributed, and it is necessary, 
in the first place, to evaluate these moments of deviations in terms of the higher 
moments of the whole distribution without making any assumptions or any 
approximations within the limits assigned. After this is done the distribution 
of the two characters will be assumed normal and the results expressed in terms 
of p, the coefficient of correlation of the material examined, and n, the number 
in the sample, only. 
* Biometrika, Vol. ix. p. 5 (if /3 2 = j3 2 ' = 3). 
