318 Miscellanea 
The second moment coefficient about the axis of y is 
k(z a +z a ,)J 0 J ' 4A + 3j 6 - z a +z a . ' 
The second moment coefficient about the mean is 
6'z,+z, 9 • (z 8 +z s y- is t («.+2,0 2 J 12 1 *U+%/J' 
Clearly when A is reasonably small (- — —J is a quantity of the second order and in 
\Zg + Zf/J 
x2 =ife ^ 
1 ' N 
1- * ^ h_ K - 
this case 
so that 
12oy7 { iV 
when the unit of grouping is uniform and small. 
(ii) When the unit of grouping is neither uniform nor small and there is no special know- 
ledge of the nature of the distribution, we must needs fall back on the Gaussian curve to give us 
a first approximation to z s and z a > for each group. 
In this case 
l-\*=Ns{ ( Zs ~ Zs ' )2 \ (ix)* 
and it is necessary to determine it, after fitting the frequency by means of Sheppard's tables. 
Finally, what correction, if any, is to be made for the grouping of y ? 
This will become more apparent from the alternative formula for rf, namely 
No 
2 
For the second moment of each array should be corrected by the subtraction of where k 
is the unit of grouping of y so that 
_ S ^-y s +ys- y? -S{y-y a f 
_ S(y - 2/ s) 2 +2S(y - y s ) (y s - y)+S(y a -y?-S(y-y a ) 2 
= ^{>hiy»-yf } 
N<r* 
since S(y a — y) 2 when summed for each individual becomes S {n s (y s -y)' 2 \ when summed for 
each array, and S (y — y s ) (y s — y) vanishes for each array. 
Hence there is no correction to be made for the y grouping except Sheppard's correction for 
the Standard Deviation of y. 
* The suggestion of this formula I owe to Professor Pearson. 
