524 
Standard Deviations of Small Samples 
O P Q 2 
OP = mode = S, OQ = mean = S. 
To obtain this approximation to (v) let us assume 2 = 2 + e, and suppose e 
small. Expanding log y we find : 
log ij = log 2/0 + {n - 2) log S - in {"Ljaf + [l - j e 
ne^ /_ n-2 , 
- J— 1 H XT + terms in e . 
2a^ \ n -^2] 
Hence since % is at our choice we will take it so that 
(^i) 
and thus : 
?/ = yo2"-2e ' e ' <^'/(2«) + etc. 
Or. if e be small compared with a, the distribution is the normal curve : 
y^y^e (vii), 
yn — ^, 
— a and standard deviation cr/V 2n. If n as usual be 
considerable, this agrees with the ordinary result, i.e. X = cr and a-^ = al^lln, the 
distribution being treated as normal. 
We will now deal with the full result (v). We have : 
i//- ijt^d^^yo t^'+P'-e ' dt (viii), 
Jo Jo 
and clearly Mp depends on a knowledge of 
Lg= r u'^e-i'''du (ix), 
Jo 
for we have : 
