330 
MR. R. A. FISHER ON THE MATHEMATICAL 
in samples from an infinite population of which the true value is p, 
log/= lo gp + y log ( l - p ), 
o 
cp 
JL\ og f=X- 
% _ y_ 
P 1 - p ' 
o2 
C 
x y 
ap log/ 
Now the mean value of x in pn , and of y is (l— p ) n , hence the mean value of 
a? log/i9 
therefore 
\p 1 — p ! 
2 _ p{ i-y>) 
cr« = 
n 
the well-known formula for the standard error of p. 
7. Satisfaction of the Criterion of Sufficiency. 
That the criterion of sufficiency is generally satisfied by the solution obtained by 
the method of maximum likelihood appears from the following considerations. 
If the individual values of any sample of data are regarded as co-ordinates in 
hyperspace, then any sample may be represented by a single point, and the frequency 
distribution of an infinite number of random samples is represented by a density 
distribution in hvperspace. If any set of statistics be chosen to be calculated from 
the samples, certain regions will provide identical sets of statistics ; these may be called 
isostatistical regions. For any particular space element, corresponding to an actual 
sample, there will be a particular set of parameters for which the frequency in that 
element is a maximum ; this will be the optimum set of parameters for that element. 
If now the set of statistics chosen are those which give the optimum values of the 
parameters, then all the elements of any part of the same isostatistical region will 
contain the greatest possible frequency for the same set of values of the parameters, 
and therefore any region which lies wholly within an isostatistical region will contain 
its maximum frequency for that set of values. 
Now let 6 be the value of any parameter, 0 the statistic calculated by the method of 
maximum likelihood, and 6i any other statistic designed to estimate the value of 6, 
then for a sample of given size, we may take 
f{e, e, e 1 )dede 1 
to represent the frequency with which 6 and 9 i lie in the assigned ranges dd and dOi . 
