Chemistry Reference
In-Depth Information
The breadth of a distribution will reflect the dispersion of the measured quan-
tities about their mean. Simple summing of the deviation of each quantity from
the mean will yield a total of zero, since the mean is defined such that the sums
of negative and positive deviations from its value are balanced. The obvious
expedient then is to square the difference between each quantity and the mean of
the distribution and add the squared terms. This produces a parameter, called the
variance of the distribution, which reflects the spread of the observed values
about their mean and is independent of the direction of this spread. The positive
square root of the variance is called the standard deviation of the distribution. Its
units are the same as those of the mean.
The standard deviation is calculated from a moment about the mean rather
than about zero. The difference between M
i
, the molecular weight of any species
i, and the mean molecular weight A is M
i
2
A, and the jth moment of the normal-
ized distribution about the mean is
X
f
i
ð
M
i
2
A
Þ
j
U
j
5
(2-27)
The absence of a prime superscript on U indicates that the moment is taken
with reference to the arithmetic mean.
Since the arithmetic mean is the center of balance of the frequencies in the
distribution, the first moment of these frequencies about the mean must be zero:
X
x
i
ð
M
i
2
M
n
Þ 5
w
U
1
X
w
i
ð
M
i
2
M
w
Þ 5
n
U
1
5
0
(2-28)
The second moment about the mean is the variance of the distribution:
X
x
i
ð
M
i
2
M
n
Þ
2
2
n
U
2
5
5 ð
s
n
Þ
(2-29)
X
w
i
ð
M
i
2
M
w
Þ
2
2
w
U
2
5
5 ð
s
w
Þ
(2-30)
where s
n
and s
w
are the standard deviations of the number and weight distribu-
tions, respectively. Thus the standard deviation of the distribution is the square
root of the second moment about its arithmetic mean:
0
:
5
s
5 ð
U
2
Þ
(2-31)
It remains now to convert U
2
into terms of M
w
and M
n
:
From
Eq. (2-29)
,
n
U
2
5
P
x
i
ð
M
i
2
M
n
Þ
5
P
x
i
ð
M
i
2
2M
i
M
n
1
M
n
Þ
2
2M
n
P
x
i
M
i
1
M
n
X
x
i
5
P
x
i
M
i
2
2M
n
M
n
1
M
2
n
n
U
2
5
n
U
0
2
2
M
n
5
M
w
M
n
2
M
n
5
n
U
0
2
2
(2-32)
s
n
5 ð
M
w
M
n
2
M
n
Þ
0
:
5
(2-33)
s
n
=
M
n
5
M
w
M
n
2
1
(2-34)
