Half-range

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, defined as:

The mid-range is the midpoint of the range; as such, it is a measure of central tendency.

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, 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 ${\displaystyle x/2,}$ while it changes the sample mean, which also has breakdown point of 0, by only ${\displaystyle x/n.}$ It is thus of little use in practical statistics, unless outliers are already handled.

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