FOUNDATIONS OF THEORETICAL STATISTICS. 
329 
hence 
0oc 
Now this factor is constant for all samples which have the same value of 61 , hence 
the variation of with respect to f) is represented by the same factor, and conse¬ 
quently 
log 4> = C' + inb S-e"; 
whence 
where 
h = J^ lo g'/(^)> 
0 1 being the optimum value of 6 . 
The formula 
1 c~ 
- — = gf 
(Tg Ctj 
supplies the most direct way known to me of finding the probable errors of statistics. 
It may be seen that the above proof applies only to statistics obtained by the method 
of maximum likelihood.* 
For example, to find the standard deviation of 
* A similar method of obtaining the standard deviations and correlations of statistics derived from 
large samples was developed by Pearson and Filon in 1898 (16). It is unfortunate that in this memoir 
no sufficient distinction is drawn between the population and the sample, in consequence of which the 
formulae obtained indicate that the likelihood is always a maximum (for continuous distributions) when 
the mean of each variate in the sample is equated to the corresponding mean in the population (16, p. 232, 
“ A r = 0 ”). If this were so the mean would always be a sufficient statistic for location ; but as we have 
already seen, and will see later in more detail, this is far from being the case. The same argument, indeed, 
is applied to all statistics, as to which nothing but their consistency can be truly affirmed. 
The probable errors obtained in this way are those appropriate to the method of maximum likelihood, 
but not in other cases to statistics obtained by the method of moments, by which method the examples 
given were fitted. In the ‘ Tables for Statisticians and Biometricians ’ (1914), the probable errors of the 
constants of the Pearsonian curves are those proper to the method of moments ; no mention is there made 
of this change of practice, nor is the publication of 1898 referred to. 
It would appear that shortly before 1898 the process which leads to the correct value, of the probable 
errors of optimum statistics, was hit upon and found to agree with the probable errors of statistics found 
by the method of moments for normal curves and surfaces ; without further enquiry it would appear to 
have been assumed that this process was valid in all cases, its directness and simplicity being peculiarly 
attractive. The mistake was at that time, perhaps, a natural one ; but that it should have been discovered 
and corrected without revealing the inefficiency of the method of moments is a very remarkable circumstance. 
In 1903 the correct formulae for the probable errors of statistics found by the method of moments are 
given in ‘ Biometrika ’ (19) ; references are there given to Sheppard (20), whose method is employed, as 
well as to Pearson and Filon (16), although both the method and the results differ from those of the latter. 
