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metric, and harmonic. Given a set of n real numbers (x1, x2, …, xn), the three means are defined
as follows:
Arithmetic mean
+⋯…………………+
AM = = ∑
=
Geometric mean
GM =
Harmonic mean
=
It can be shown that the following inequality holds:
AM … GM … HM
The values are equal only if x1 = x2 = c xn. We can get a useful insight into these alternative
calculations by defining the functional mean. Let f(x) be a continuous monotonic function defined
in the inter val 0 … y 6 ∞. The functional mean with respect to the function f(x) for n positive real
numbers (x1, x2, …, xn) is defined as
=
where f-1(x) is the inverse off(x). The mean values defined in Equations (2.1) through (2.3) are
special cases of the functional mean, as follows:
■ AM is the FM with respect to f(x) = x
■ GM is the FM with respect to f(x) = ln x
■ HM is the FM with respect to f(x) = 1/x
Example
2.3 Figure 2.6 illustrates the three means applied to various data sets, each of which has eleven
data points and a maximum data point value of 11. The median value is also included in the chart.
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