526 Standard Deviations of Small Samples 
These accurate values of %, the mean standard deviation of samples, were first 
given by " Student " (Zoc. cit. p. 8). Now by Wallis' Theorem 
TT ^2^^ + 1) — P'^od^^^'of even numbers up to 2n 
ri- 
1 
ll 
'n — 
2 
2 product of odd numbers up to 2n — 1 ' 
Thus (xivA) for n large tends to become 
Vn 
and (xivb) 2. = — 
'\lnsln — 1'\n 
These values, however, really only suffice to show the approach of 2 to cr, as 
they depend on the neglect of terms of the order - as compared to 1, and we should 
get absurd results for by subtracting the square of the above values of 2 from 
/Lt/ in (xi). All they really tell us is that for n large S = cr, but they give no true, 
approximation in - *. 
n 
If we use Stirling's Theorem up to the third termf , i.e. 
.! = V2^.^e-(l + ^+2,-y, 
we obtain S = > 
3 ^ \ /X 
139 
But we should be compelled to introduce the term — gj^g^Q^ i'^^o Stirling's 
expression to reach the second terms in /Ltg and yu.4. As we have indicated (p. 523, 
ftn.), such a term, even if used, will not lead to profitable results. It is better 
to work with the full formulae. It is desirable to find the full third and fourth 
moment coefficients in order to determine ^-^ and and so measure as n increases 
the rapidity of approach to the normal curve. 
* " Student" has used an extension of Wallis' Theorem, which will suffice for certain constants only, 
t We can write (xivA) 
S = -^^ Ji==J_ /f, (xvii), 
V« |n-2 V T ^ ' 
and (xivB) 
2 ^ /tt_ . .... 
s/w(2^("-3)|^(n-3))2V 2' 
and then apply Stirling's Theorem. 
