282 On the Systematic Fitting of Curves 
takes a curve which has the least root mean square deviation from all the points 
of some smooth curve with a moment system determined by the 11 points. 
Hence it is (juite possible that the method of moments may actually give better 
results than the method of least squares in such a case as the above, if after the 
determination of the curve it becomes necessary to compare experience and obser- 
vation at other points than the eleven used in the first determination. 
Fig. 1 gives the theoretical curves and the points of observation in cases I, II, 
IV, V, and VI. 
(4) Case (ii). The frequency z of individuals falling within p elementary 
ranges of a total range ph is observed or measured, to determine the true mean 
and moments of the system. 
Let y = f{x) be the curve giving the frequency distribution, and z,. the frequency 
observed within the range of the variable x from x = x^-i to x =Xr. Then what we 
actually observe are 
pi p2 p/) 
Zi — I ydx, Z2 = I ydx, ... Zp= \ ydx. 
Let N be the total number of observations, or 
N = Zi + Z2+ ... + Zp. 
For the nth. moment about a line through the origin perpendicular to the range 
we require 
p> 
Nfin = I x^ydx. 
J x„ 
Now let Z = j'' ydx, 
J X 
i.e. be all the frequency from x = x to above the value x. Then 
p> pp pp 
Z^ = I ydx, Zi = I ydx, ... Zp= I ydx 
J X„ J X, J Xp 
are known and given by 
N, Z2 + Z3 + ... + Zp, Z3 + z,+ ... + Zp, z^-\- z^+ ... + Zp, Zp, 0. 
Since dZjdx = — y, we have 
Thus 
fXp 
J Xo 
dZ . 
~y- ax 
dx 
-Zx^ 
Xp p 
+ n\ 
Xo J X 
x'^-'Zdx 
0 
fXp 
= ZoXo" + n Zx'' 
J Xo 
-'dx. 
H'n ^0" "1" ^7- 
j Zx^- 
'dx 
.(vii). 
