Mid-range
In
The mid-range is closely related to the range, a measure of statistical dispersion defined as the difference between maximum and minimum values. The two measures are complementary in sense that if one knows the mid-range and the range, one can find the sample maximum and minimum values.
The mid-range is rarely used in practical statistical analysis, as it lacks efficiency as an estimator for most distributions of interest, because it ignores all intermediate points, and lacks robustness, as outliers change it significantly. Indeed, for many distributions it is one of the least efficient and least robust statistics. However, it finds some use in special cases: it is the maximally efficient estimator for the center of a uniform distribution, trimmed mid-ranges address robustness, and as an L-estimator, it is simple to understand and compute.
Robustness
The midrange is highly sensitive to outliers and ignores all but two data points. It is therefore a very non-
A
These trimmed midranges are also of interest as descriptive statistics or as L-estimators of central location or skewness: differences of midsummaries, such as midhinge minus the median, give measures of skewness at different points in the tail.[2]
Efficiency
Despite its drawbacks, in some cases it is useful: the midrange is a highly
For example, for a
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 robust analog is the trimean, which averages the midhinge (25% trimmed mid-range) and median.
Small samples
For small sample sizes (n from 4 to 20) drawn from a sufficiently platykurtic distribution (negative
Excess kurtosis (γ2) | Most efficient estimator of μ |
---|---|
−1.2 to −0.8 | Midrange |
−0.8 to 2.0 | Mean |
2.0 to 6.0 | Modified mean |
For n = 1 or 2, the midrange and the mean are equal (and coincide with the median), and are most efficient for all distributions. For n = 3, the modified mean is the median, and instead the mean is the most efficient measure of central tendency for values of γ2 from 2.0 to 6.0 as well as from −0.8 to 2.0.
Sampling properties
For a sample of size n from the
For a sample of size n from the standard Laplace distribution, the mid-range M is unbiased, and has a variance given by:[6]
and, in particular, the variance does not decrease to zero as the sample size grows.
For a sample of size n from a zero-centred
Deviation
While the mean of a set of values minimizes the sum of squares of
See also
References
- ^ Dodge 2003.
- ^ Velleman & Hoaglin 1981.
- ^ Vinson, William Daniel (1951). An Investigation of Measures of Central Tendency Used in Quality Control (Master's). University of North Carolina at Chapel Hill. Table (4.1), pp. 32–34.
- ^ Cowden, Dudley Johnstone (1957). Statistical methods in quality control. Prentice-Hall. pp. 67–68.
- ^ Kendall & Stuart 1969, Example 14.4.
- ^ Kendall & Stuart 1969, Example 14.5.
- ^ Kendall & Stuart 1969, Example 14.12.
- Dodge, Y. (2003). The Oxford dictionary of Statistical Terms. Oxford University Press. ISBN 0-19-920613-9.
- Kendall, M.G.; Stuart, A. (1969). The Advanced Theory of Statistics, Volume 1. Griffin. ISBN 0-85264-141-9.
- Velleman, P. F.; Hoaglin, D. C. (1981). Applications, Basics and Computing of Exploratory Data Analysis. Duxbury Press. ISBN 0-87150-409-X.