Midhinge

In statistics, the midhinge is the average of the first and third quartiles and is thus a measure of location.wikipedia
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Quartile

quartileslower quartilelower and upper quartiles
In statistics, the midhinge is the average of the first and third quartiles and is thus a measure of location. The midhinge is complemented by the H-spread, or interquartile range, which is the difference of the third and first quartiles and which is a measure of statistical dispersion, in sense that if one knows the midhinge and the interquartile range, one can find the first and third quartiles.
The values found by this method are also known as "Tukey's hinges"; see also midhinge.

Trimmed estimator

trimmedtrimming
Equivalently, it is the 25% trimmed mid-range or 25% midsummary; it is an L-estimator.
Midhinge, the 25% trimmed mid-range

Mid-range

midsummarymidrangehalf-range
Equivalently, it is the 25% trimmed mid-range or 25% midsummary; it is an L-estimator.
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.

L-estimator

L-estimation
Equivalently, it is the 25% trimmed mid-range or 25% midsummary; it is an L-estimator.
A more detailed list of examples includes: with a single point, the maximum, the minimum, or any single order statistic or quantile; with one or two points, the median; with two points, the mid-range, the range, the midsummary (trimmed mid-range, including the midhinge), and the trimmed range (including the interquartile range and interdecile range); with three points, the trimean; with a fixed fraction of the points, the trimmed mean (including interquartile mean) and the Winsorized mean; with all points, the mean.

Interquartile range

inter-quartile rangebelowinterquartile
The midhinge is complemented by the H-spread, or interquartile range, which is the difference of the third and first quartiles and which is a measure of statistical dispersion, in sense that if one knows the midhinge and the interquartile range, one can find the first and third quartiles.
For a symmetric distribution (where the median equals the midhinge, the average of the first and third quartiles), half the IQR equals the median absolute deviation (MAD).

Trimean

The midhinge is slightly simpler to calculate than the trimean, which originated in the same context and equals the average of the median and the midhinge.
This is equivalent to the average of the median and the midhinge:

Interquartile mean

interquartile
Interquartile mean
Midhinge

Statistics

statisticalstatistical analysisstatistician
In statistics, the midhinge is the average of the first and third quartiles and is thus a measure of location.

Location parameter

locationlocation modelshift parameter
In statistics, the midhinge is the average of the first and third quartiles and is thus a measure of location.

Statistical dispersion

dispersionvariabilityspread
The midhinge is complemented by the H-spread, or interquartile range, which is the difference of the third and first quartiles and which is a measure of statistical dispersion, in sense that if one knows the midhinge and the interquartile range, one can find the first and third quartiles.

John Tukey

TukeyTukey, JohnJohn W. Tukey
The use of the term "hinge" for the lower or upper quartiles derives from John Tukey's work on exploratory data analysis, and "midhinge" is a fairly modern term dating from around that time.

Exploratory data analysis

explorative data analysisexploratorydata analysis
The use of the term "hinge" for the lower or upper quartiles derives from John Tukey's work on exploratory data analysis, and "midhinge" is a fairly modern term dating from around that time.

Median

averagesample medianmedian-unbiased estimator
The midhinge is slightly simpler to calculate than the trimean, which originated in the same context and equals the average of the median and the midhinge.

MathWorld

CRC Press sues authorWolfram Mathworld
* H-spread at MathWorld

Coefficient of variation

CVrelative standard deviationcoefficients of variation
A more robust possibility is the quartile coefficient of dispersion, half the interquartile range divided by the average of the quartiles (the midhinge),.

Box plot

boxplotbox and whisker plotadjusted boxplots
In addition to the points themselves, they allow one to visually estimate various L-estimators, notably the interquartile range, midhinge, range, mid-range, and trimean.

Central tendency

Localitycentral locationcentral point
Midhinge: the arithmetic mean of the two quartiles.

Five-number summary

five-number summaries
In addition to the points themselves, many L-estimators can be computed from the five-number summary, including interquartile range, midhinge, range, mid-range, and trimean.

Average absolute deviation

mean absolute deviationmean deviationMAD
trimmed L ∞ norm statistics: for example, the midhinge (average of first and third quartiles) which minimizes the median absolute deviation of the whole distribution, also minimizes the maximum absolute deviation of the distribution after the top and bottom 25% have been trimmed off..

Interdecile range

Analogous measures of location are given by the median, midhinge, and trimean (or statistics based on nearby points).