528 
Standard Deviations of Small Samples 
Further by (xx) 
Thus 
Our results for and /X4, although expressed in other notation, are in 
accordance with " Student's " {loc. cit. p. 9), so also are our results (xv) and (xvi) 
although reached by a different method of approximation. We do not agree with 
his approximate values for /u,^, /j,^ or /3i and /S^. 
The calculations to find %/a, a-s,j(a-/s/'2n), ^1 and /S2 presented some trouble. 
In order to be correct to the four figures of decimals in the tabled results, tables 
of ten-figure logarithms had to be used in the logarithmic part of the work. 
Formulae (xvii) and (xviii) of the ftn. p. 526 were adopted, using Degen's Tables of 
the Logarithms of Factorials. M2' was calculated to nine figures, and even then, 
as n became large, the determination of the antilogarithms presented consider- 
able difficulty. Further the powers of 1 — a^j{a'^l'2,n) gave rise to trouble. The 
numerical work was undertaken by Ethel M. Elderton and Beatrice M. Cave, 
to whom very hearty thanks are due. We think the results may be depended 
on to the figures tabulated. 
It will be seen that by the time n = 50 the mode is as close to the mean as we 
should expect to find in any random sample of normal material ; the average 
mean S is only l"5°/„ from the usually adopted value a, and the average standard 
deviation only 0"3% from its customary value a/'\/2n. Further and /3a are 
•0105 and 3 0003 respectively, or for all practical purposes have reached their 
normal values. We think it must be concluded that for samples of 50 the usual 
theory of the probable error of the standard deviation holds satisfactorily, and 
that to apply it for the case of n = 25 would not lead to any error which would be 
of importance in the majority of statistical problems. 
On the other hand, if a small sample, n < 20 say, of a population be taken, the 
value of the standard deviation found from it will be usually less than the standard 
deviation of the true population. If we take the most probable value, X, as that 
