K. Pearson 
283 
This is the fundamental formula for finding the true moments of frequency 
distributions from the grouped frequencies. The rule is clear. In order to evaluate 
I' 
Zx^~'^dx, since we know the value of Zx^ ^ for a.' = ajo, 
. . x.p, we have to 
find the area of a curve of which we are given j3 + 1 ordinates; we have accordingly 
to use the best available quadrature formulse, taking care that the exactness 
of the formula corresponds to the degree of the moment investigated. 
For practical working, since Zf^ = N \s, large, it is convenient to take = 0, and 
our formula then becomes 
N 
' Zx''-^dx (viii). 
Here we must be very careful to notice that our origin is the start of the base- 
element in which the frequency begins, that Z(, = N is the total frequency, and Zp 
is zero, and that Xp is measured to the end of the last base-element h for which we 
are considering the frequency. Thus a length Xp + h, aud not Xp, would be the total 
range we should obtain by plotting the frequencies z as if they were ordinates 
at the middle of the elements. This process therefore tends to exaggerate the 
range. As a rule it is convenient in frequency distributions to determine the /x's 
about the mean. In this case they may be found from the /i"s about any other 
line by the formulae 
H-s = Ms 
Should the frequency 
(ix). 
observations we are dealing with cover a complete distribu- 
tion we can proceed somewhat differently. Let y = f{x) be the frequency distribu- 
tion and let it be absolutely confined within the range I of x. If we take ^■ = 0, 
at one end of this range we have for the integral curve Z= ydx. Now, whatever 
be the form ot the frequency distribution, whether it gives a curve of high contact 
or not at a.' = 0 and x = I, it must follow, if the range be absolutely limited, that 
Z=Q for a; = I, and ^2^= constant = iV at x = {). But usually there is contact of a 
high order at one or both ends of the range. I shall therefore work out the 
modifications which must be made in the moments when there is high contact at 
one end at least, say SiX x = l. 
Thus for X = 1, we have 
Z = 0, 
dZ 
dx 
d'j 
dx- 
= 0, 
d?Z 
dx^ 
0, etc., 
28—2 
