FOUNDATIONS OF THEORETICAL STATISTICS. 
359 
We shall conclude with a few illustrations of important types of discontinuous 
distribution. 
1. The Poisson Series. 
m 2 
2P 
involves only the single parameter, and is of great importance in modern statistics. 
For the optimum value of m. 
whence 
or 
S 
(—m + x log m) 
dm 
= 0, 
m = x. 
The most likely value of m is therefore found by taking the first moment of the series. 
Differentiating a second time, 
so that 
as is well known. 
A = s(-A =-*. 
o-fo \ ml m 
3 m 
n 
2. Grouped Normal Data. 
In the case of the normal curve of distribution it is evident that the second moment 
is a sufficient statistic for estimating the standard deviation ; in investigating a sufficient 
solution for grouped normal data, we are therefore in reality finding the optimum 
correction for grouping ; the Sheppard correction having been proved only to satisfy 
the criterion of consistency. 
For grouped normal data we have 
Ps = 
1 
0"\/ 2-7T 
and the optimum values of m and a are obtained from the equations, 
3 d 2 
