344 THEORY OF STATISTICS. 
to test whether we may treat the distribution as approximately 
normal (cf. also § 16 below). 
(As regards the theory of sampling for the median and per- 
centiles generally, cf. ref. 15, Laplace, Supplement II. (standard 
error of the median), Edgeworth, refs. 5, 6, 7, and Sheppard, ref. 
27: the preceding sections have been based on the work of 
Edgeworth and Sheppard.) 
10. Standard Error of the Arithmetic Mean.—Let us now pass 
to a fresh problem, and determine the standard error of the 
arithmetic mean. 
This is very readily obtained. Suppose we note separately at 
each drawing the value recorded on the first, second, third . . . . 
and nth card of our sample. The standard-deviation of the values 
on each separate card will tend in the long run to be the same, 
and identical with the standard-deviation o of « in an indefinitely 
large sample, drawn under the same conditions. Further, the 
value recorded on each card is (as we assume) uncorrelated with 
that on every other. The standard-deviation of the sum of the 
values recorded on the nm cards is therefore a/n.oc, and the 
standard-deviation of the mean of the sample is consequently 
1/nth of this; or, 
o 
On ="In . (5) 
This is a most important and frequently cited formula, and the 
student should note that it has been obtained without any 
reference to the size of the sample or to the form of the frequency- 
distribution. It is therefore of perfectly general application, if 
oc be known. We can verify it against our formula for the 
standard-deviation of sampling in the case of attributes. The 
standard-deviation of the number of successes in a sample of m 
observations is a/m.pg: the standard-deviation of the total 
number of successes in n samples of m observations each is there- 
fore a/mm.pq: dividing by n we have the standard-deviation of 
the mean number of successes in the = samples, viz. mpg [n/n 
agreeing with equation (5). 
11. For a normal curve the standard error of the mean is to 
the standard error of the median approximately as 100 to 125 
(¢f. § 4), and in general the standard errors of the two stand in 
a somewhat similar ratio for a distribution not differing largely 
from the normal form. For the distribution of statures used as 
an illustration in § 6 the standard error of the median was found 
to be 0:0349: the standard error of the mean is only 0-0277. 
The distribution being very approximately normal, the ratio of