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Concept
Mid-range
Summary
In statistics, the mid-range or mid-extreme is a measure of central tendency of a sample defined as the arithmetic mean of the maximum and minimum values of the data set:
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.
The midrange is highly sensitive to outliers and ignores all but two data points. It is therefore a very non-robust statistic, having a breakdown point of 0, meaning that a single observation can change it arbitrarily. Further, it is highly influenced by outliers: increasing the sample maximum or decreasing the sample minimum by x changes the mid-range by while it changes the sample mean, which also has breakdown point of 0, by only It is thus of little use in practical statistics, unless outliers are already handled.
A trimmed midrange is known as a – the n% trimmed midrange is the average of the n% and (100−n)% percentiles, and is more robust, having a breakdown point of n%. In the middle of these is the midhinge, which is the 25% midsummary. The median can be interpreted as the fully trimmed (50%) mid-range; this accords with the convention that the median of an even number of points is the mean of the two middle points.
Official source
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