366 
MR. R. A. FISHER ON THE MATHEMATICAL 
For any dilution the variance in the number of sterile plates is 
spq ,. 
and as the several dilutions represent independent samples, the total variance is 
6 -S (pq), 
,. = fl ^S( pq ). 
hence 
log ) 
Now S (pq) has an average value , therefore taking a = 2, 
b log a 
and 
(log a) 2 — ‘480453, 
S (pq) = 1 
being very nearly constant and within a small fraction of unity ; whence the efficiency 
of the method of counting the sterile plates is 
- 7 -r—- = 8771 per cent., 
7 r log 2 
a remarkably high efficiency, considering the simplicity of the method, the efficiency 
being independent of the dilution ratio. 
13. Summary. 
During the rapid development of practical statistics in the past few decades, the 
theoretical foundations of the subject have been involved in great obscurity. Adequate 
distinction has seldom been drawn between the sample recorded and the hypothetical 
population from which it is regarded as drawn. This obscurity is centred in the so-called 
“ inverse methods. 
On the bases that the purpose of the statistical reduction of data is to obtain statistics 
which shall contain as much as possible, ideally the whole, of the relevant information 
contained in the sample, and that the function of Theoretical Statistics is to show how 
such adequate statistics may be calculated, and how much and of what kind is the 
information contained in them, an attempt is made to formulate distinctly the types 
of problems which arise in statistical practice. 
Of these, problems of Specification are found to be dominated by considerations which 
may change rapidly during the progress of Statistical Science. In problems of Distri¬ 
bution relatively little progress has hitherto been made, these problems still affording 
a field for valuable enquiry by highly trained mathematicians. The principal purpose 
of this paper is to put forward a general solution of problems of Estimation. 
