While the mean of a set of values minimizes the sum of squares of deviations and the [[median]] minimizes the [[average absolute deviation]], the midrange minimizes the maximum deviation.

While the mean of a set of values minimizes the sum of squares of deviations and the [[median]] minimizes the [[average absolute deviation]], the midrange minimizes the maximum deviation.

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[[Category:Means]]

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[[Category:Statistics]]

Latest revision as of 01:05, November 4, 2007

In statistics, the mid-range or mid-extreme of a set of statistical data values is the arithmetic mean of the maximum and minimum values in a data set, or:

As such it is a measure of central tendency. It is highly sensitive to outliers and ignores all but two data points; therefore it is rarely used in statistical analysis.

A limited amount of experimental work on the efficiency of measures of central tendency for small samples by William D. Vinson reveals the following facts, where γ2 is the coefficient of kurtosis:

Values of γ_{2} Most efficient estimator of μ
-1.2 to -0.8 Midrange
-0.8 to 2.0 Arithmetic mean
2.0 to 6.0 Modified mean

This generalization holds for sample sizes (n) from 4 to 20. When n = 3, there can be no modified mean, and the mean is the most effiient measure of central tendency for values of γ_{2} form 2.0 to 6.0 as well as from −0.8 to 2.0, where γ_{2} = (μ_{4}/(μ_{2})²) − 3.

Furthermore, the midrange is an inefficient estimator of μ when the population is normal. However, for a sufficiently platykurtic distribution, the midrange is by far the most efficient estimator.

While the mean of a set of values minimizes the sum of squares of deviations and the median minimizes the average absolute deviation, the midrange minimizes the maximum deviation.