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In [[statistics]], the '''midhinge''' ({{dfn|MH}}) is the average of the first and third [[quartile]]s and is thus a measure of [[location parameter|location]].
Equivalently, it is the 25% [[trimmed estimator|trimmed]] [[mid-range]] or 25% [[midsummary]]; it is an [[L-estimator]]. The midhinge {{math|MH}} is defined as <math display="block">\begin{align}
\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{align}</math>

The midhinge is related to the [[interquartile range]] ({{dfn|IQR}}), the difference of the third and first [[quartile]]s (i.e. {{math|1= IQR = ''Q''{{sub|3}} − ''Q''{{sub|1}}}}), which is a measure of [[statistical dispersion]]. The two are complementary in sense that if one knows the midhinge and the {{abbr|IQR|interquartile range}}, 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,<ref>Tukey, J. W. (1977) ''Exploratory Data Analysis'', Addison-Wesley. {{ISBN|0-201-07616-0}}</ref> and ''midhinge'' is a fairly modern term dating from around that time. The midhinge is slightly simpler to calculate than the [[trimean]] ({{dfn|TM}}), which originated in the same context and equals the average of the [[median]] ({{math|1= ''{{overset|lh=.7em|~|X}}'' = ''Q''{{sub|2}}}} {{math|1= = ''P''{{sub|50}}}}) and the midhinge:
<math display="block">\begin{align}
\operatorname{MH}(X) &= 2 \operatorname{TM}(X) - \operatorname{med}(X) \\
&= 2\; \frac{Q_1 + 2Q_2 + Q_3}{4} - Q_2.
\end{align}</math>

==See also==
* [[Interquartile mean]]
* [[L-estimator]]

==References==
{{reflist}}

==External links==
* [http://mathworld.wolfram.com/H-Spread.html H-spread] at [[MathWorld]]

[[Category:Means]]
[[Category:Exploratory data analysis]]