By Student 
riow u-s- = — ^ — ' ' 
11 \ II / \ n 
— other terms of odd order which will vanish on summation. 
Summing for all values and dividing by the number of cases we get 
R^^a.^a,, + m,M, = ^ + ,1.? ^ - '^^ - S/x,^ ^ , 
where Rxa^-z is the correlation between u" and s'-. 
7, ,(«-!) o("-l)ro ,("-!) 
Hence Ru-i^iaxc^cTsi = 0 or there is no correlation between and s". 
Section III. 
To find the equation representing the frequency distribution of the means 
of samples of n drawn from a normal population, the mean being expressed in 
terms of the standard deviation of the sample. 
(J K^^ 
We have y = —^■^s''^'^ e as the equation representing the distribution of s, 
the standard deviation of a sample of n, when the samples arc drawn from a 
normal population with standard deviation a. 
Now the means of these samples of n are distributed according to the equation 
V 'lira 
and we have shown that there is no correlation between x, the distance of the 
mean of the sample, and s, the standard deviation of the sample. 
Now let us suppose x measured in terms of s, i.e. let us find the distribution 
01 z = - . 
s 
If we have yi = 4> (*') ^-i^tl y^ — ^ {z) as the equations representing the frequency 
of X and of z respectively, then 
, , dx 
y,dx = y,dz = y, ~, 
•■• y2 = sy,. 
* Airy, Theory of Errors of Observations, Part 11. § 6. 
