MISCELLANEA. 
Note on the Standard Deviations of Samples of Two or Three. 
By ANDREW W. YOUNG, M.A. 
In an "Editorial" contained in Vol. x of Bionietrika* , there is a discussion of the distribution 
of the values of the standard deviation of a population which are deduced from small samples of 
the population. It is there shown how the distribution approaches normahty as the number, n, 
in the sample increases, a table of the characteristic constants of the frequency curves for various 
values of n being given. The smallest sample considered is that of » = 4, but samples of two and 
three are of occasional occurrence especially in physical work and now and again a value of the 
proltable error of an experimental result is deduced from a set of two or of three observations. 
A knowledge of the theoretical distribution of the standard deviations for such small samples 
will give us some idea of the reliabiUty of this procedm-e and it is the object of this note to supply 
this omission from the former paper. 
"Student's" formula for the distribution of samples of standard deviation is 
where o- is the standard deviation of the whole population and 2 is the standard deviation given 
by a sample of size n. Thus the distribution for samples of two is 
y = , 
extending from 2 = 0 to 2 = oo , i.e. the distribution, is simply half of a normal curve; and the 
distribution for samples of three is 
y = Ih^e 
also, of course, extending from 2 = 0 to 2 = co . 
It is easy to find by direct integration the moment coefficients of these ciu-ves. 
Case of Samples of Two. 
If we denote by N the total nvimber of samples in the assumed distribution, 
-J TT 
■ 
N = yo e <^-(Z2 = 2/0"- 
/ 0 
Taking the first four moment coefficients about the origin to be /x/, fi./, /xj', fi^', and the moment 
coefficients about the mean to be, as usual, /^tj, JLI3, ju.4, we have 
u/ = Mean value of 2 = 2 = |? f °°2e~°^-f/2 = = 4^ = -56420-. 
" N n 2Af n/tt 
0 
u.' = 2^6 "-rfS = — — = - 
N J 0 N 4: 2 
2 ' 
giving: = 0-2 - = -18170-2, 
and the standard deviation of 2 = o-^ = -42630-. 
* "On the Distribution of the Standard Deviations of Small Samples." Appendix I to papers by 
"Student" and R. A. Fisher. Biomeirika, Vol. x. p. 522, 1915. 
