Volume VI 
MARCH, 1908 
No. 1 
BIOMETKIKA. 
THE PKOBABLE ERROR OF A MEAN. 
By student. 
Introduction. 
•'. Any experiment may be regarded as forming an individual of a "population" 
of experiments which might be performed under the same conditions. A series 
of experiments is a sample drawn from this population. 
Now any series of experiments is only of value in so far as it enables us to form 
a judgment as to the statistical constants of the population to which the experi- 
ments belong. In a great number of cases the question finally turns on the value 
of a mean, either directly, or as the mean difference between the two quantities. 
If the number of experiments be very large, we may have precise information 
as to the value of the mean, but if our sample be small, we have two sources of 
uncertainty: — (1) owing to the "error of random sampling" the mean of our series 
of experiments deviates more or less widely from the mean of the population, and 
(2) the sample is not sufficiently large to determine what is the law of distribution 
of individuals. It is usual, however, to assume a normal distribution, because, in 
a very large number of cases, this gives an approximation so close that a small 
sample will give no real information as to the manner in which the population 
deviates from normality : since some law of distribution must be assumed it is 
better to work with a curve whose area and ordinates are tabled, and whose 
properties are well known. This assumption is accordingly made in the present 
paper, so that its conclusions are not strictly applicable to populations known not 
to be normally distributed ; yet it appears probable that the deviation from 
normality must be very extreme to lead to serious error. We are concerned here 
solely with the first of these two sources of uncertainty. 
The usual method of determining the probability that the mean of the popula- 
tion lies within a given distance of the mean of the sample, is to assume a normal 
distribution about the mean of the sample with a standard deviation equal to 
s/V?i, where s is the standard deviation of the sample, and to use the tables of 
the probability integral. 
Biometiika vi 1 
