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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]]: | |||
: <math>TM= \frac{Q_1 + 2Q_2 + Q_3}{4}</math> | |||
This is equivalent to the average of the [[median]] and the [[midhinge]]: | |||
: <math>TM= \frac{1}{2}\left(Q_2 + \frac{Q_1 + Q_3}{2}\right)</math> | |||
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 = }} | |||
</ref> which has given its name to a set of techniques called [[Exploratory data analysis]]. | |||
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: | |||
{{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>}} | |||
==Efficiency== | |||
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. | |||
==See also== | |||
*[[Truncated mean]] | |||
*[[Interquartile mean]] | |||
==References== | |||
{{reflist}} | |||
{{refbegin}} | |||
*{{cite isbn|0898744148}} | |||
{{refend}} | |||
==External links== | |||
*[http://mathworld.wolfram.com/Trimean.html Trimean] at [[MathWorld]] | |||
[[Category:Summary statistics]] | |||
[[Category:Means]] | |||
[[Category:Robust statistics]] | |||
[[Category:Exploratory data analysis]] | |||
Revision as of 13:22, 25 February 2013
In statistics the trimean (TM), or Tukey's trimean, is a measure of a probability distribution's location defined as a weighted average of the distribution's median and its two quartiles:
This is equivalent to the average of the median and the midhinge:
The foundations of the trimean were part of Arthur Bowley's teachings, and later popularized by statistician John Tukey in his 1977 book[1] which has given its name to a set of techniques called Exploratory data analysis.
Like the median and the midhinge, but unlike the sample mean, it is a statistically resistant L-estimator with a breakdown point of 25%. This beneficial property has been described as follows: 31 year-old Systems Analyst Bud from Deep River, spends time with pursuits for instance r/c cars, property developers new condo in singapore singapore and books. Last month just traveled to Orkhon Valley Cultural Landscape.
Efficiency
Despite its simplicity, the trimean is a remarkably 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.Template:Sfn 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.
See also
References
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External links
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