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Midhinge

Wikipedia contributorsAdapted for Fathomly · View authors
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Source updated January 20, 2025

In statistics, the midhinge (MH) is the average of the first and third quartiles and is thus a measure of location. Equivalently, it is the 25% trimmed mid-range or 25% midsummary; it is an L-estimator. The midhinge MH is defined as {\begin{aligned}\operatorname {MH} (X)&={\overline {Q_{1,3}(X)}}\\&={\frac {Q_{1}(X)+Q_{3}(X)}{2}}\\&={\frac {P_{25}(X)+P_{75}(X)}{2}}\\&=M_{25}(X).\end{aligned}}

The midhinge is related to the interquartile range (IQR), the difference of the third and first quartiles (i.e. IQR = Q3 − Q1), which is a measure of statistical dispersion. The two are complementary in sense that if one knows the midhinge and the IQR, one can find the first and third quartiles.

The use of the term hinge for the lower or upper quartiles derives from John Tukey's work on exploratory data analysis in the late 1970s, and midhinge is a fairly modern term dating from around that time. The midhinge is slightly simpler to calculate than the trimean (TM), which originated in the same context and equals the average of the median (~X = Q2 = P50) and the midhinge: {\begin{aligned}\operatorname {MH} (X)&=2\operatorname {TM} (X)-\operatorname {med} (X)\\&=2\;{\frac {Q_{1}+2Q_{2}+Q_{3}}{4}}-Q_{2}.\end{aligned}}

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This article is adapted from the Wikipedia article “Midhinge”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Interquartile mean

The interquartile mean, also called midmean, is a statistical measure of central tendency based on the truncated mean of the interquartile range. The IQM is very similar to the scoring method used in sports that are evaluated by a panel of judges: "discard the lowest and the highest scores; calculate the mean value of the remaining scores".

L-estimator

In statistics, an L-estimator is an estimator which is a linear combination of order statistics of the measurements. This can be as little as a single point, as in the median (of an odd number of values), or as many as all points, as in the mean.