Mid-range

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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:

$\frac{\max x + \min x}{2}.$

As such it is a measure of central tendency.

The midrange is highly sensitive to outliers and ignores all but two data points. It is therefore a very non-robust statistic, and it is rarely used in statistical analysis.

[edit] Comparison with other measures

[edit] Efficiency

Despite its drawbacks, in some cases it is useful: the midrange is a highly efficient estimator of μ, given a small sample of a sufficiently platykurtic distribution, but it is inefficient for mesokurtic distributions, such as the normal.

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 excess kurtosis, defined as γ₂ = (μ₄/(μ₂)²) − 3.

Kurtosis (γ₂)	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 efficient measure of central tendency for values of γ₂ form 2.0 to 6.0 as well as from −0.8 to 2.0.

[edit] 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 (defined as $\max\left|x_i-m\right|$ ): it is a solution to a variational problem.