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48 HANDBOOK OF PORTFOLIO MATHEMATICS
Thehalf-widthis an even more frequently used measure of dispersion.
Here, we take the height of a distribution at its peak, the mode. If we find
the point halfway up this vertical measure and run a horizontal line through
it perpendicular to the vertical line, the horizontal line will touch the dis-
tribution at one point to the left and one point to the right. The distance
between these two points is called the half-width.
Next, themean absolute deviationormean deviationis the arithmetic
average of the absolute value of the difference between the data points
and the arithmetic average of the data points. In other words, as its name
implies, it is the average distance that a data point is from the mean. Ex-
pressed mathematically:M= 1 /N
∑N
i= 1ABS(Xi−A) (2.06)where: M=The mean absolute deviation.
N=The total number of data points.
Xi=The ith data point.
A=The arithmetic average of the data points.
ABS( )=The absolute value function.Equation (2.06) gives us what is known as thepopulationmean abso-
lute deviation. You should know that the mean absolute deviation can also
be calculated as what is known as thesamplemean absolute deviation.
To calculate the sample mean absolute deviation, replace the term 1/N in
Equation (2.06) with 1/(N−1). You use the sample version when you are
making judgments about the population based on a sample of that popula-
tion.
The next two measures of dispersion, variance and standard deviation,
are the two most commonly used. Both are used extensively, so we cannot
say that one is more common than the other; suffice to say they are both the
most common. Like the mean absolute deviation, they can be calculated
two different ways, for a population as well as a sample. The population
version is shown, and again it can readily be altered to the sample version
by replacing the term 1/N with 1/(N−1).
Thevarianceis the same thing as the mean absolute deviation except
that we square each difference between a data point and the average of
the data points. As a result, we do not need to take the absolute value of
each difference, since multiplying each difference by itself makes the re-
sult positive whether the difference was positive or negative. Further, since
each distance is squared, extreme outliers will have a stronger effect on the