By Student 
5 
The first moment coefficient about the end of the ranoe will therefore be 
00 n-l nx 
C X 2 e -"^ -^d x G 
I 
G 
c\ n -1 nx- 
.v = x 
n 
x = 0 
I 
+ 
n-3 nx 
X e ^''2 dx 
The first part vanishes at each limit and the second is equal to 
n — 1 J- 
?) 1) 
I n 
and we see that the higher moment coefficients will be formed by mnltiplying 
successively by /Uo, /^.j, etc., just as appeared to be the law of formation 
'}Z '} t 
of M,,', if/, M/, etc. 
Hence it is probable that the curve found represents the theoretical distribu- 
tion of s- ; so that although we have no actual proof we shall assume it to do so in 
what follows. 
The distribution of s may be found from this, since the frequency of s is equal 
to tliat of 5^ and all that we must do is to compress the base line suitably. 
Now if <f){s-) be the frequency curve of s- 
and y.^ = f(s) „ „ „ „ „ s, 
then 
or 
Hence 
is the distribution of s. 
This reduces to 
y,d{s-) = ij,ds, 
y.,ds =2yisds, 
y, =2sy,. 
n - 3 ns- 
y, = 2Gs{s-f^ e~^^ 
ns'' 
7/2 = 2 Cs'^--e~ 27;. 
Hence y = Ax'^~-e will give the frequency distribution of standard devia- 
tions of samples of n, taken out of a population distributed normally with standard 
deviation a. The constant A may be found by equating the area of the curve as 
follows : — 
Area = ^| x''^~'^e '^"^'dx. (Let /^^ represent 1 x^e '^"'dx 
•2 J , nx'- . 
Then = - x^-'~ -e ^Adx 
n dx V / 
n 
+ — (p-l) I xi'-"e ^"-dx 
.v = 0 « ' 
since the first part vanishes at both limits. 
