318 
MR. R. A. FISHER ON THE MATHEMATICAL 
and it will illustrate this use of the criterion of consistency if we derive formulae for 
this correction without approximation. 
Let £ be the value of the variate at the mid point of any group, a the interval of 
grouping, and x the true value of the variate at any point, then the k th moment of an 
infinite grouped sample is 
p = °° y+ia 
X I £*/(*) dx, 
in which of f (x) dx is the frequency, in any element dx, of the ungrouped population, and 
£ = (p+ir) a > 
p being any integer. 
Evidently the k th moment is periodic in 6, we will therefore equate it to 
A 0 + Aj sin 6 + A 2 sin 26... 
+ Bj cos 6 + Bv cos 26 _ 
Then 
But 
therefore 
n "• y+ia 
X 
| <16 
£ k f (x) dx 
7T 
it = — » 
J 0 
Jf-ia 
P = 00 
p Tr . 
X 
sin 
s6 <16 1 g f{x)dx. 
V - - * ' 
J 0 
■L-ia 
p=cc 
,'2ir 
,r , 1 _ 
I ( +2& 
X 
cos 
s6 d6 \ £ k f(x)dx. 
P = — 00 
Jo 
Jt-ia 
2 x 
6 
a 
£-2 7 rp. 
y+ia 
-jti 
d6 = — 
a 
d£, 
sin 50 = sin 
2tt 
a 
cos s6 — cos 
2 IT 
a 
£ k f (x) dx — 
1 
a . 
' x-}a 
hence 
