Midrange

For loudspeakers, see midrange speaker.
In statistics, the midrange or midextreme of a set of statistical data values is the arithmetic mean of the maximum and minimum values in a data set,^{[1]} or:
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 nonrobust statistic (having a breakdown point of 0, meaning that a single observation can change it arbitrarily), and it is rarely used in statistical analysis.
The midhinge is the 25% trimmed midrange, and is more robust, having a breakdown point of 25%.
Contents
Comparison with other measures
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.
For example, for a continuous uniform distribution with unknown maximum and minimum, the midrange is the UMVU estimator for the mean. The sample maximum and sample minimum, together with sample size, are a sufficient statistic for the population maximum and minimum – the distribution of other samples, conditional on a given maximum and minimum, is just the uniform distribution between the maximum and minimum and thus add no information. Thus the midrange, which is an unbiased and sufficient estimator of the population mean, is in fact the UMVU: using the sample mean just adds noise based on the uninformative distribution of points within this range.
Conversely, for the normal distribution, the sample mean is the UMVU estimator of the mean. Thus for platykurtic distributions, which can often be thought of as between a uniform distribution and a normal distribution, the informativeness of the middle sample points versus the extrema values varies from "equal" for normal to "uninformative" for uniform, and for different distributions, one or the other (or some combination thereof) may be most efficient.
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,^{[original research?]} where γ_{2} is the coefficient of excess kurtosis, defined as γ_{2} = (μ_{4}/(μ_{2})²) − 3.
Kurtosis (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 efficient measure of central tendency for values of γ_{2} form 2.0 to 6.0 as well as from −0.8 to 2.0.
Sampling properties
For a sample of size n from the standard normal distribution, the midrange M is unbiased, and has a variance given^{[2]} by
For a sample of size n from the standard Laplace distribution, the midrange M is unbiased, and has a variance given^{[3]} by
and, in particular, the variance does not decrease to zero as the sample size grows.
For a sample of size n from a zerocentred uniform distribution, the midrange M is unbiased, nM has an asymptotic distribution which is a Laplace distribution.^{[4]}
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 ): it is a solution to a variational problem.
See also
References
 ^ Dodge, Y. (2003) The Oxford dictionary of Statistical Terms, OUP. ISBN 0199206139
 ^ Kendall, M.G. & Stuart, A. (1969) The Advanced Theory of Statistics, Volume 1, Griffin. ISBN 0852641419 (Example 14.4)
 ^ Kendall, M.G. & Stuart, A. (1969) The Advanced Theory of Statistics, Volume 1, Griffin. ISBN 0852641419 (Example 14.5)
 ^ Kendall, M.G. & Stuart, A. (1969) The Advanced Theory of Statistics, Volume 1, Griffin. ISBN 0852641419 (Exercise 14.12)
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