* 
MISCELLANEA. 
I. Tables for estimating the Probability that the Mean of a unique 
Sample of Observations lies between — 05 and any given Distance of 
the Mean of the Population from which the Sample is drawn. 
By "STUDENT." 
In the last number of Biomefrika Mr Young completes the table given in Vol. x. p. 522 of 
the standard deviation frequency curves for small samples by working out the cases where the 
numbers in the sample are as low as two and three. 
In tlie course of his note he writes "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 probable error of an experimental result is deduced from a set of two or of 
three observations." 
Further on he states "it is evident that the probable error determined from a set of three 
observations is very untrustworthy and that when there are only two observations it is very 
much worse." 
Now in my original paper {Biometrika, Vol. vi. p. 1) I stopped at to = 4 because I had not 
realised that anyone would be foolish enough to work with probable errors deduced from a smaller 
number of observations, but now I too will complete my tables which will I think emphasise the 
moderation of the second quotation from Mr Young's note. 
Generally speaking there are two objects in determining the standard deviation of a set of 
observations, namely (1) to compare it with the standard deviation of similar sets of observations, 
and (2) to estimate the accuracy with which the mean of the observations represents the mean of 
the population from which the sample is drawn. 
The former purpose is served by the table which Mr Young was engaged in completing, the 
latter, which is by far the most common use of the S.D., by the table which I gave in my original 
paper and which I now propose to complete downwards by including n = 2 and n = 3 and to 
extend upwards as far as 11 = 30. 
In the tables the probability is given (to four places of decimals) that the mean of a unique 
sample shall lie between - 00 and a distance z from the mean of the population, z being measured 
in terms of the s.D. (5) of the sample. 
[By unique I mean to say that all the information which we have (or at all events intend to 
use) about the distribution of the population is given by the sample in question.] 
To compare with the last column of the table {n = 30) I have given the corresponding prob- 
ability calculated fi'om the nearest normal curve, namely the one with s.D. . (not 
Vn - 3 Vn - 1 
as is usually given) and this shows I think that for ordinary purposes Sheppard's tables may be 
used with n >30, 
