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| In [[statistics]] the '''trimean (TM)''', or '''Tukey's trimean''', is a measure of a [[probability distribution]]'s [[average|location]] defined as a [[weighted average]] of the distribution's [[median]] and its two [[quartiles]]:
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| : <math>TM= \frac{Q_1 + 2Q_2 + Q_3}{4}</math>
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| This is equivalent to the average of the [[median]] and the [[midhinge]]:
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| : <math>TM= \frac{1}{2}\left(Q_2 + \frac{Q_1 + Q_3}{2}\right)</math>
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| The foundations of the trimean were part of [[Arthur Bowley]]'s teachings, and later popularized by statistician [[John Tukey]] in his 1977 book<ref>{{cite book |last=Tukey |first=John Wilder |authorlink= |coauthors= |editor= |others= |title=Exploratory Data Analysis |origdate= |origyear= |origmonth= |url= |format= |accessdate= |edition= |date= |year=1977 |month= |publisher=Addison-Wesley |location= |language= |isbn= 0-201-07616-0 |doi = |pages= |chapter= |chapterurl= |quote = }}
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| </ref> which has given its name to a set of techniques called [[Exploratory data analysis]].
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| Like the median and the midhinge, but unlike the [[sample mean]], it is a [[statistically resistant]] [[L-estimator]] with a [[Robust statistics#Breakdown point|breakdown point]] of 25%. This beneficial property has been described as follows:
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| {{quote|An advantage of the trimean as a measure of the center (of a distribution) is that it combines the [[median]]'s emphasis on center values with the [[midhinge]]'s attention to the extremes.|Herbert F. Weisberg|''Central Tendency and Variability''<ref>Weisberg, H. F. (1992). ''Central Tendency and Variability''. Sage University. ISBN 0-8039-4007-6 ([http://books.google.com/books?id=tPw6b1VXfkoC&pg=PA39&ots=XMd1BObzZr&dq=trimean&sig=WMtwB-23lw44mmnnrrvz4lPHFKQ#PPA39,M1 p. 39])</ref>}}
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| ==Efficiency==
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| Despite its simplicity, the trimean is a remarkably [[Efficiency (statistics)|efficient]] estimator of population mean. More precisely, for a large data set (over 100 points) from a symmetric population, the average of the 20th, 50th, and 80th percentile is the most efficient 3 point L-estimator, with 88% efficiency.{{sfn|Evans|1955|loc=Appendix G: Inefficient statistics, pp. [http://archive.org/stream/atomicnucleus032805mbp#page/n925/mode/2up 902–904]}} For context, the best 1 point estimate by L-estimators is the median, with an efficiency of 64% or better (for all ''n''), while using 2 points (for a large data set of over 100 points from a symmetric population), the most efficient estimate is the 29% [[midsummary]] (mean of 29th and 71st percentiles), which has an efficiency of about 81%. Using quartiles, these optimal estimators can be approximated by the midhinge and the trimean. Using further points yield higher efficiency, though it is notable that only 3 points are needed for very high efficiency.
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| ==See also==
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| *[[Truncated mean]]
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| *[[Interquartile mean]]
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| ==References==
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| {{reflist}}
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| {{refbegin}}
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| *{{cite isbn|0898744148}}
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| {{refend}}
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| ==External links==
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| *[http://mathworld.wolfram.com/Trimean.html Trimean] at [[MathWorld]]
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| [[Category:Summary statistics]]
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| [[Category:Means]]
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| [[Category:Robust statistics]]
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| [[Category:Exploratory data analysis]]
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