FOUNDATIONS OF THEORETICAL STATISTICS. 
361 
By interpolation, 
i = 0’441624 
or 
a = 2-26437. 
We may therefore summarise these results as follows 
Uncorrected estimate of a- . 2 • 28254 
Sheppard’s correction. —0-01833 
Correction for maximum likelihood. —0*01817 
“ Correction ” for minimum + +0*07332 
Far from shaking our faith, therefore, in the adequacy of Sheppard’s correction, 
when small, for normal data, this example provides a striking instance of its effective¬ 
ness, while the approximate nature of the + test renders it unsuitable for improving a 
method which is already very accurate. 
It will be useful before leaving the subject of grouped normal data to calculate the 
actual loss of efficiency caused by grouping, and the additional loss due to the small 
discrepancy between moments with Sheppard’s correction and the optimum solution. 
To calculate the loss of efficiency involved in the process of grouping normal data, let 
when do- is the group interval, then 
v =/(£) + ~ f" {£) + 
a 
q s 
1920322,560'^ ^ + 
-fid 1 1 + L <r _1 ) + TU777^ t-6 U + 3 ) + 
a 
q u 
l 
1920 
322,560 
(+-15f 4 + 45f-15) + .., 
whence 
log v = log / + fj (f - 1) - ( f + - 2) + (? + 6t + 3f-1) - 
and 
^2 
0 
dnr 
> log *--? + M T5 - Wo {3f+2) + *sko (5 #‘ +12 ^ +1 > 
1 J 1 _ a/_ a 4 q 6 
+1 12 144 ~ 4320 ' ’' J ’ 
of which the mean value is 
