By Student 
3 
Section I. 
Samples of n individuals are drawn out of a population distributed normally, 
to find an equation which shall represent the frequency of the standard deviations 
of these samples. 
If s be the standard deviation found from a sample XiW^-.-Xn (all these being 
measured from the mean of the population), then 
)} \ n / n n- n- 
Summing for all samples and dividing by the number of samples we get the 
mean value of s- which we will write s-, 
_., _ n^l.i nfi.j _ ytt2 (ra — 1) 
n n- n 
where fi., is the second moment coefficient in the original normal distribution of x : 
since x-^, X2, etc., are not correlated and the distribution is normal, products in- 
volving odd powers of x^ vanish on summing, so that |y equal to 0. 
If Mj^ represent the -R"' moment coefficient of the distribution of s- about the 
end of the range where s- — 0, 
{n - 1) 
Again ^^ = {^^-(^^)7 
S {x^^)\- 2S (x,^) fS (x,)Y _^ fS (x,) 
8 {x,') IS jx-'x.i) _ 2S{x^) _ {x^-xj) 8(x,*) 
11- n- 
QS(x-x-) 
-{ + other terms involving odd powers of x^ etc., 
which will vanish on summation. 
Now S(x\*) has H terms but S{xi-X2^) has ^n{n — l), hence summing for all 
samples and dividing by the number of samples we get 
ilf/ = ^ + _ _ , {n--l) ^ ^ (n-lj 
n n n- ir ^' if 
= f^{n^-2n-l} +^0,-l)j„._2H + 3}. 
Now since the distribution of .r is normal, /j,^ = 3/^2', hence 
M.' = {Sn - 3 + If- - 2a + 3} = OL^lK^iill) . 
1—2 
