362 
MR. R. A. FISHER ON THE MATHEMATICAL 
neglecting the periodic terms ; and consequently 
2 <t f 1 aj~ a 
** ~ n \ 12 2880 
Now for the mean of ungrouped data 
2 (T 
= ~ 
n 
so that the loss of efficiency due to grouping is nearly 
a 
12 
The further loss caused by using the mean of the grouped data is very small, for 
a _ v 2 _ * n , a \ 
- ~ — — 1 + 77> > 
n n \ 12/ 
neglecting the periodic terms ; the loss of efficiency by using v\ therefore is only 
a 
2880 
Similarly for the efficiency for scaling, 
i 1 3f 2 1 [a 3 A - 2 0 \ 
a 
26^+110^+36^-7) 
a 
360 
a 
30,240 
of which the mean value is 
1,814,400 
(9f + 21f 2 -5) 
(51f+ 315^ 6 +351f 4 -55f 2 +9) + ...}, 
_ 2_ J _ af a 4 ad 83cd 
6 + 40 270 + 129,600 
neglecting the periodic terms ; and consequently 
a 
2 C 2 4 
_ a = *_i 1 + 2 l + JL_ _ 
2n\ 6 360 10,800 
For ungrouped data 
so that the loss of efficiency in scaling due to grouping is nearly —. This may be made 
as low as 1 per cent, by keeping a less than j. 
The further loss of efficiency produced by using the grouped second moment with 
Sheppard's correction is again very small, for 
2 _ V4 — V2 _ 2(F / A or r( i 
V ‘ n n 6 360 
neglecting the periodic terms. 
