# Midhinge

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In [statistics](/source/Statistics), the **midhinge** (MH) is the average of the first and third [quartiles](/source/Quartile) and is thus a measure of [location](/source/Location_parameter). Equivalently, it is the 25% [trimmed](/source/Trimmed_estimator) [mid-range](/source/Mid-range) or 25% [midsummary](/source/Midsummary); it is an [L-estimator](/source/L-estimator). The midhinge MH is defined as MH ⁡ ( X ) = Q 1 , 3 ( X ) ¯ = Q 1 ( X ) + Q 3 ( X ) 2 = P 25 ( X ) + P 75 ( X ) 2 = M 25 ( X ) . {\displaystyle {\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](/source/Interquartile_range) (IQR), the difference of the third and first [quartiles](/source/Quartile) (i.e. IQR = *Q*3 − *Q*1), which is a measure of [statistical dispersion](/source/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](/source/John_Tukey)'s work on [exploratory data analysis](/source/Exploratory_data_analysis) in the late 1970s,[1] and *midhinge* is a fairly modern term dating from around that time. The midhinge is slightly simpler to calculate than the [trimean](/source/Trimean) (TM), which originated in the same context and equals the average of the [median](/source/Median) (*~X* = *Q*2 = *P*50) and the midhinge: MH ⁡ ( X ) = 2 TM ⁡ ( X ) − med ⁡ ( X ) = 2 Q 1 + 2 Q 2 + Q 3 4 − Q 2 . {\displaystyle {\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}}}

## See also

- [Interquartile mean](/source/Interquartile_mean)

- [L-estimator](/source/L-estimator)

## References

1. **[^](#cite_ref-1)** Tukey, J. W. (1977) *Exploratory Data Analysis*, Addison-Wesley. [ISBN](/source/ISBN_(identifier)) [0-201-07616-0](https://en.wikipedia.org/wiki/Special:BookSources/0-201-07616-0)

## External links

- [H-spread](http://mathworld.wolfram.com/H-Spread.html) at [MathWorld](/source/MathWorld)

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