14 The FtuuJamental Prohleni of Practical Statistics 
Now the true standard deviation of the curve is not cr,,*, but is given by 
Vx +1 
(T = a, 
X 
= a,^ neai'ly, 
if in be fairly large and neither Q nor P small. 
In both cases the (standard deviation)- of the curve exceeds that of the series 
by a term dejjending on the square of the distance between mean and mode. 
Unless either P or Q are small, this will be a small difference, a fraction of the 
square of the plotting unit c. As the curves are not reached by equating moments, 
but from a geometrical property common to the polygon of frequency and to the 
curve, this deviation is to be anticipated. It is the more to be anticipated as in 
the case of the curve we are dealing with the moments of continuous areas, and 
in the case of the series with concentrated lumps*]". 
* Similarly for the limited range curve of p. 11 : 
<7o- = ("H- 1) 1 + j c- = »PV (e— i) c2. 
But the true standard deviation cr is given by 
1 
1 
1 + ^ 
^4 H^fH. 
where d = distance from mean to mode. 
t The true {standard deviation)- of the binomial is a{^ = iiiPQc'-. Thus 
a- = { mPQ + PQ + 1 ( y - P)^ = a^' + |c-, 
or the (standard deviation)- of the binomial series is less than that of curve by Jc-. 
The true standard deviation of the series when n is not indefinitely great is given by 
m + 2\ /, 1\ /, 111 + 2 
mPQ 1 + —— ^ M +- 1 + 
II \ n 
n \ It J \ " I 
— cr- - j:C- . (See preceding footnote.) 
If- 
n 
Thus the (standard deviation)- of the series is less than that of the curve by a term of the order 
