322 
MR. R. A. FISHER ON THE MATHEMATICAL 
bution will be a symmetrical one (Type VII.) having its centre at, x — m (fig. 1). It is 
therefore a perfectly definite problem to estimate the value of m (to find the best value of 
m) from a random sample of values of x. We have stated the problem in its simplest 
possible form : only one parameter is required, the middle point of the distribution. 
1 -s 2 
B . d, = — 7- e 4 
2 \/t r 
By the method of moments, this should be given by the first moment, that is by the 
mean of the observations : such would seem to be at least a good estimate. It is, 
however, entirely valueless. The distribution of the mean of such samples is in fact the 
same, identically, as that of a single observation. In taking the mean of 100 values of 
x, we are no nearer obtaining the value of m than if we had chosen any value of x out 
of the 100. The problem, however, is not in the least an impracticable one : clearly 
from a large sample we ought to be able to estimate the centre of the distribution with 
some precision ; the mean, however, is an entirely useless statistic for the purpose. 
By taking the median of a large sample, a fair approximation is obtained, for the standard 
error of the median of a large sample of n is ——, which, alone, is enough to show that 
2 \/n 
by adopting adequate statistical methods it must be possible to estimate the value for 
m, with increasing accuracy, as the size of the sample is increased. 
This example serves also to illustrate the practical difficulty which observers often 
find, that a few extreme observations appear to dominate the value of the mean. In 
these cases the rejection of extreme values is often advocated, and it may often happen 
that gross errors are thus rejected. As a statistical measure, however, the rejection of 
observations is too crude to be defended : and unless there are other reasons for rejec¬ 
tion than mere divergence from the majority, it would be more philosophical to accept 
these extreme values, not as gross errors, but as indications that the distribution of 
errors is not normal. As we shall show, the only Pearsonian curve for which the mean 
