Truncated mean - Revision history

Unruffled haslett: remove 'Drawbacks' section, as it is redundant and silly as per discussion on talk page.

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remove 'Drawbacks' section, as it is redundant and silly as per discussion on talk page.

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	One situation in which it can be advantageous to use a truncated mean is when estimating the [[location parameter]] of a [[Cauchy distribution]], a bell shaped probability distribution with (much) fatter tails than a [[normal distribution]]. It can be shown that the truncated mean of the middle 24% sample [[order statistics]] (i.e., truncate the sample by 38% at each end) produces an estimate for the population location parameter that is more efficient than using either the sample median or the full sample mean.<ref name=rothenberg>{{cite journal\|last1=Rothenberg\|first1=Thomas J.\|last2=Fisher\|first2=Franklin, M.\|last3=Tilanus\|first3=C.B.\|year=1964\|volume=59\|issue=306\|journal=Journal of the American Statistical Association\|title=A note on estimation from a cauchy sample\|pages=460–463\|doi=10.1080/01621459.1964.10482170}}</ref><ref name=bloch>{{cite journal\|last1=Bloch\|first1=Daniel\|year=1966\|volume=61\|issue=316\|journal=Journal of the American Statistical Association\|title=A note on the estimation of the location parameters of the Cauchy distribution\|pages=852–855\|jstor=2282794\|doi=10.1080/01621459.1966.10480912}}</ref> However, due to the fat tails of the Cauchy distribution, the efficiency of the estimator decreases as more of the sample gets used in the estimate.<ref name=rothenberg/><ref name=bloch/> Note that for the Cauchy distribution, neither the truncated mean, full sample mean or sample median represents a [[maximum likelihood]] estimator, nor are any as asymptotically efficient as the maximum likelihood estimator; however, the maximum likelihood estimate is more difficult to compute, leaving the truncated mean as a useful alternative.<ref name=bloch/><ref name=ferguson>{{cite journal\|last1=Ferguson\|first1=Thomas S.\|author-link= Thomas S. Ferguson \|year=1978\|journal=Journal of the American Statistical Association \|volume=73\|issue=361\|title=Maximum Likelihood Estimates of the Parameters of the Cauchy Distribution for Samples of Size 3 and 4\|pages=211–213\|jstor=2286549\|doi=10.1080/01621459.1978.10480031}}</ref>		One situation in which it can be advantageous to use a truncated mean is when estimating the [[location parameter]] of a [[Cauchy distribution]], a bell shaped probability distribution with (much) fatter tails than a [[normal distribution]]. It can be shown that the truncated mean of the middle 24% sample [[order statistics]] (i.e., truncate the sample by 38% at each end) produces an estimate for the population location parameter that is more efficient than using either the sample median or the full sample mean.<ref name=rothenberg>{{cite journal\|last1=Rothenberg\|first1=Thomas J.\|last2=Fisher\|first2=Franklin, M.\|last3=Tilanus\|first3=C.B.\|year=1964\|volume=59\|issue=306\|journal=Journal of the American Statistical Association\|title=A note on estimation from a cauchy sample\|pages=460–463\|doi=10.1080/01621459.1964.10482170}}</ref><ref name=bloch>{{cite journal\|last1=Bloch\|first1=Daniel\|year=1966\|volume=61\|issue=316\|journal=Journal of the American Statistical Association\|title=A note on the estimation of the location parameters of the Cauchy distribution\|pages=852–855\|jstor=2282794\|doi=10.1080/01621459.1966.10480912}}</ref> However, due to the fat tails of the Cauchy distribution, the efficiency of the estimator decreases as more of the sample gets used in the estimate.<ref name=rothenberg/><ref name=bloch/> Note that for the Cauchy distribution, neither the truncated mean, full sample mean or sample median represents a [[maximum likelihood]] estimator, nor are any as asymptotically efficient as the maximum likelihood estimator; however, the maximum likelihood estimate is more difficult to compute, leaving the truncated mean as a useful alternative.<ref name=bloch/><ref name=ferguson>{{cite journal\|last1=Ferguson\|first1=Thomas S.\|author-link= Thomas S. Ferguson \|year=1978\|journal=Journal of the American Statistical Association \|volume=73\|issue=361\|title=Maximum Likelihood Estimates of the Parameters of the Cauchy Distribution for Samples of Size 3 and 4\|pages=211–213\|jstor=2286549\|doi=10.1080/01621459.1978.10480031}}</ref>

	==Drawbacks==
	The truncated mean uses more information from the distribution or [[Sample (statistics)\|sample]] than the [[median]], but unless the underlying distribution is [[Symmetry\|symmetric]], the truncated mean of a sample is unlikely to produce an [[Bias of an estimator\|unbiased estimator]] for either the mean or the median.

	==Statistical tests==		==Statistical tests==

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	For most statistical applications, 5 to 25 percent of the ends are discarded. For example, given a set of 8 points, trimming by 12.5% would discard the minimum and maximum value in the sample: the smallest and largest values, and would compute the mean of the remaining 6 points. The 25% trimmed mean (when the lowest 25% and the highest 25% are discarded) is known as the [[interquartile mean]].		For most statistical applications, 5 to 25 percent of the ends are discarded. For example, given a set of 8 points, trimming by 12.5% would discard the minimum and maximum value in the sample: the smallest and largest values, and would compute the mean of the remaining 6 points. The 25% trimmed mean (when the lowest 25% and the highest 25% are discarded) is known as the [[interquartile mean]].

	The median can be regarded as a fully truncated mean and is most robust. As with other [[trimmed estimator]]s, the main advantage of the trimmed mean is robustness and higher [[Efficiency (statistics)\|efficiency]] for mixed distributions and heavy-tailed distribution (like the [[Cauchy distribution]]), at the cost of lower efficiency for some other less heavily-tailed distributions (such as the normal distribution). For intermediate distributions the differences between the efficiency of the mean and the median are not very big, e.g. for the student-t distribution with 2 degrees of freedom the variances for mean and median are nearly equal.		The median can be regarded as a fully truncated mean and is most robust. As with other [[trimmed estimator]]s, the main advantage of the trimmed mean is robustness and higher [[Efficiency (statistics)\|efficiency]] for mixed distributions and heavy-tailed distribution (like the [[Cauchy distribution]]), at the cost of lower efficiency for some other less heavily tailed distributions (such as the normal distribution). For intermediate distributions the differences between the efficiency of the mean and the median are not very big, e.g. for the student-t distribution with 2 degrees of freedom the variances for mean and median are nearly equal.

	==Terminology==		==Terminology==

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2022-01-21T16:16:45Z

Examples

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	The scoring method used in many [[sport]]s that are evaluated by a panel of judges is a truncated mean: ''discard the lowest and the highest scores; calculate the mean value of the remaining scores''.<ref name="wsj-sport">{{cite web \|url=https://www.wsj.com/articles/SB10000872396390443477104577551253521597214 \|title=Removing Judges' Bias Is Olympic-Size Challenge \|last1=Bialik \|first1=Carl\|date=27 July 2012 \|website=The Wall Street Journal \| accessdate=7 September 2014}}</ref>		The scoring method used in many [[sport]]s that are evaluated by a panel of judges is a truncated mean: ''discard the lowest and the highest scores; calculate the mean value of the remaining scores''.<ref name="wsj-sport">{{cite web \|url=https://www.wsj.com/articles/SB10000872396390443477104577551253521597214 \|title=Removing Judges' Bias Is Olympic-Size Challenge \|last1=Bialik \|first1=Carl\|date=27 July 2012 \|website=The Wall Street Journal \| accessdate=7 September 2014}}</ref>

	The [[Libor]] benchmark interest rate is [[Libor#Calculation\|calculated]] as a trimmed mean: given 18 ~~response~~, the top 4 and bottom 4 are discarded, and the remaining 10 are averaged (yielding trim factor of 4/18 ≈ 22%).<ref>{{Cite web\|url=http://www.bbalibor.com/explained/the-basics\|title=bbalibor: The Basics\|publisher=The British Bankers' Association}}</ref>		The [[Libor]] benchmark interest rate is [[Libor#Calculation\|calculated]] as a trimmed mean: given 18 responses, the top 4 and bottom 4 are discarded, and the remaining 10 are averaged (yielding trim factor of 4/18 ≈ 22%).<ref>{{Cite web\|url=http://www.bbalibor.com/explained/the-basics\|title=bbalibor: The Basics\|publisher=The British Bankers' Association}}</ref>

	Consider the data set consisting of:		Consider the data set consisting of:

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	The truncated mean is a useful estimator because it is less sensitive to [[outlier]]s than the mean but will still give a reasonable estimate of central tendency or mean for many statistical models. In this regard it is referred to as a [[Robust statistics\|robust estimator]]. For example, in its use in Olympic judging, truncating the maximum and minimum prevents a single judge from increasing or lowering the overall score by giving an exceptionally high or low score.		The truncated mean is a useful estimator because it is less sensitive to [[outlier]]s than the mean but will still give a reasonable estimate of central tendency or mean for many statistical models. In this regard it is referred to as a [[Robust statistics\|robust estimator]]. For example, in its use in Olympic judging, truncating the maximum and minimum prevents a single judge from increasing or lowering the overall score by giving an exceptionally high or low score.

	One situation in which it can be advantageous to use a truncated mean is when estimating the [[location parameter]] of a [[Cauchy distribution]], a bell shaped probability distribution with (much) fatter tails than a [[normal distribution]]. It can be shown that the truncated mean of the middle 24% sample [[order statistics]] (i.e., truncate the sample by 38% at each end) produces an estimate for the population location parameter that is more efficient than using either the sample median or the full sample mean.<ref name=rothenberg>{{cite journal\|last1=Rothenberg\|first1=Thomas J.\|last2=Fisher\|first2=Franklin, M.\|last3=Tilanus\|first3=C.B.\|year=1964\|volume=59\|issue=306\|journal=Journal of the American Statistical Association\|title=A note on estimation from a cauchy sample\|pages=460–463\|doi=10.1080/01621459.1964.10482170}}</ref><ref name=bloch>{{cite journal\|last1=Bloch\|first1=Daniel\|year=1966\|volume=61\|issue=316\|journal=Journal of the American Statistical Association\|title=A note on the estimation of the location parameters of the Cauchy distribution\|pages=852–855\|jstor=2282794\|doi=10.1080/01621459.1966.10480912}}</ref> However, due to the fat tails of the Cauchy distribution, the efficiency of the estimator decreases as more of the sample gets used in the estimate.<ref name=rothenberg/><ref name=bloch/> Note that for the Cauchy distribution, neither the truncated mean, full sample mean or sample median represents a [[maximum likelihood]] estimator, nor are any as asymptotically efficient as the maximum likelihood estimator; however, the maximum likelihood estimate is more difficult to compute, leaving the truncated mean as a useful alternative.<ref name=bloch/><ref name=ferguson>{{cite journal\|last1=Ferguson\|first1=Thomas S.\|author-link= Thomas S. Ferguson \|year=1978\|journal=Journal of the American Statistical Association \|volume=73\|issue=361\|title=Maximum Likelihood Estimates of the Parameters of the Cauchy Distribution for Samples of Size 3 and 4\|~~page~~=~~211~~\|jstor=2286549\|doi=10.1080/01621459.1978.10480031}}</ref>		One situation in which it can be advantageous to use a truncated mean is when estimating the [[location parameter]] of a [[Cauchy distribution]], a bell shaped probability distribution with (much) fatter tails than a [[normal distribution]]. It can be shown that the truncated mean of the middle 24% sample [[order statistics]] (i.e., truncate the sample by 38% at each end) produces an estimate for the population location parameter that is more efficient than using either the sample median or the full sample mean.<ref name=rothenberg>{{cite journal\|last1=Rothenberg\|first1=Thomas J.\|last2=Fisher\|first2=Franklin, M.\|last3=Tilanus\|first3=C.B.\|year=1964\|volume=59\|issue=306\|journal=Journal of the American Statistical Association\|title=A note on estimation from a cauchy sample\|pages=460–463\|doi=10.1080/01621459.1964.10482170}}</ref><ref name=bloch>{{cite journal\|last1=Bloch\|first1=Daniel\|year=1966\|volume=61\|issue=316\|journal=Journal of the American Statistical Association\|title=A note on the estimation of the location parameters of the Cauchy distribution\|pages=852–855\|jstor=2282794\|doi=10.1080/01621459.1966.10480912}}</ref> However, due to the fat tails of the Cauchy distribution, the efficiency of the estimator decreases as more of the sample gets used in the estimate.<ref name=rothenberg/><ref name=bloch/> Note that for the Cauchy distribution, neither the truncated mean, full sample mean or sample median represents a [[maximum likelihood]] estimator, nor are any as asymptotically efficient as the maximum likelihood estimator; however, the maximum likelihood estimate is more difficult to compute, leaving the truncated mean as a useful alternative.<ref name=bloch/><ref name=ferguson>{{cite journal\|last1=Ferguson\|first1=Thomas S.\|author-link= Thomas S. Ferguson \|year=1978\|journal=Journal of the American Statistical Association \|volume=73\|issue=361\|title=Maximum Likelihood Estimates of the Parameters of the Cauchy Distribution for Samples of Size 3 and 4\|pages=211–213\|jstor=2286549\|doi=10.1080/01621459.1978.10480031}}</ref>

	==Drawbacks==		==Drawbacks==
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	==Statistical tests==		==Statistical tests==

	It is possible to perform a [[Student's t-test]] based on the truncated mean, which is called Yuen's t-test,<ref>Yuen, K.K. (1974) The two-sample trimmed t for unequal population variances. Biometrika, 61, 165-170.</ref><ref>Wilcox, R.R. (2005). Introduction to robust estimation and hypothesis testing. Academic Press.</ref> which also has several implementations in [[R (programming language)\|R]].<ref>https://cran.r-project.org/web/packages/WRS2/</ref><ref>https://cran.r-project.org/web/packages/DescTools/</ref>		It is possible to perform a [[Student's t-test]] based on the truncated mean, which is called Yuen's t-test,<ref>Yuen, K.K. (1974) The two-sample trimmed t for unequal population variances. Biometrika, 61, 165-170.</ref><ref>Wilcox, R.R. (2005). Introduction to robust estimation and hypothesis testing. Academic Press.</ref> which also has several implementations in [[R (programming language)\|R]].<ref>{{Cite web\|url=https://cran.r-project.org/web/packages/WRS2/\|title=WRS2: A Collection of Robust Statistical Methods\|date=20 July 2021}}</ref><ref>{{Cite web\|url=https://cran.r-project.org/web/packages/DescTools/\|title = DescTools: Tools for Descriptive Statistics\|date = 9 September 2021}}</ref>

	==Examples==		==Examples==

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References after punctuation per WP:REFPUNCT, WP:CITEFOOT, WP:PAIC + other fixes

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	{{short description\|Statistical measure of central tendency}}		{{short description\|Statistical measure of central tendency}}
	{{~~Refimprove~~\|date=July 2010}}		{{More citations needed\|date=July 2010}}
	A '''truncated mean''' or '''trimmed mean''' is a [[Statistics\|statistical]] [[Average\|measure of central tendency]], much like the [[mean]] and [[median]]. It involves the calculation of the mean after discarding given parts of a [[probability distribution]] or [[Sampling (statistics)\|sample]] at the high and low end, and typically discarding an equal amount of both. This number of points to be discarded is usually given as a percentage of the total number of points, but may also be given as a fixed number of points.		A '''truncated mean''' or '''trimmed mean''' is a [[Statistics\|statistical]] [[Average\|measure of central tendency]], much like the [[mean]] and [[median]]. It involves the calculation of the mean after discarding given parts of a [[probability distribution]] or [[Sampling (statistics)\|sample]] at the high and low end, and typically discarding an equal amount of both. This number of points to be discarded is usually given as a percentage of the total number of points, but may also be given as a fixed number of points.

	For most statistical applications, 5 to 25 percent of the ends are discarded. For example, given a set of 8 points, trimming by 12.5% would discard the minimum and maximum value in the sample: the smallest and largest values, and would compute the mean of the remaining 6 points. The 25% trimmed mean (when the lowest 25% and the highest 25% are discarded) is known as the [[interquartile mean]].		For most statistical applications, 5 to 25 percent of the ends are discarded. For example, given a set of 8 points, trimming by 12.5% would discard the minimum and maximum value in the sample: the smallest and largest values, and would compute the mean of the remaining 6 points. The 25% trimmed mean (when the lowest 25% and the highest 25% are discarded) is known as the [[interquartile mean]].

	The median can be regarded as a fully truncated mean and is most robust. As with other [[trimmed estimator]]s, the main advantage of the trimmed mean is robustness and higher [[Efficiency (statistics)\|efficiency]] for mixed distributions and heavy-tailed distribution (like the [[Cauchy distribution]]), at the cost of lower efficiency for some other less heavily-tailed distributions (such as the normal distribution). For intermediate distributions the differences between the efficiency of the mean and the median are not very big, e.g. for the student-t distribution with 2 degrees of freedom the variances for mean and median are nearly equal.		The median can be regarded as a fully truncated mean and is most robust. As with other [[trimmed estimator]]s, the main advantage of the trimmed mean is robustness and higher [[Efficiency (statistics)\|efficiency]] for mixed distributions and heavy-tailed distribution (like the [[Cauchy distribution]]), at the cost of lower efficiency for some other less heavily-tailed distributions (such as the normal distribution). For intermediate distributions the differences between the efficiency of the mean and the median are not very big, e.g. for the student-t distribution with 2 degrees of freedom the variances for mean and median are nearly equal.

	==Terminology==		==Terminology==
	In some regions of [[Central Europe]] it is also known as a '''Windsor mean''',{{~~fact~~\|date=October 2016}} but this name should not be confused with the [[Winsorized mean]]: in the latter, the observations that the trimmed mean would discard are instead replaced by the largest/smallest of the remaining values.		In some regions of [[Central Europe]] it is also known as a '''Windsor mean''',{{citation needed\|date=October 2016}} but this name should not be confused with the [[Winsorized mean]]: in the latter, the observations that the trimmed mean would discard are instead replaced by the largest/smallest of the remaining values.

	Discarding only the maximum and minimum is known as the '''{{visible anchor\|modified mean}}''', particularly in management statistics.<ref>Arulmozhi, G.; Statistics For Management, 2nd Edition, Tata McGraw-Hill Education, 2009, p. [https://books.google.com/books?id=2qcyNld-bHwC&pg=PA458&lpg=PA458&dq=Modified+mean 458]</ref> This is also known as the '''{{visible anchor\|Olympic average}}''' (for example in US agriculture, like the [[Average Crop Revenue Election]]), due to its use in Olympic events, such as the [[ISU Judging System]] in [[figure skating]], to make the score robust to a single outlier judge.<ref>{{cite web \|url=http://farmdocdaily.illinois.edu/2012/08/lessons_from_libor.html \|title=Lessons from LIBOR \|author=Paul E. Peterson \|date=August 3, 2012 \|quote=Once the quotes are compiled, LIBOR uses a trimmed mean process, in which the highest and lowest values are thrown out and the remaining values are averaged. This is sometimes called an "Olympic average" from its use in the Olympics to eliminate the impact of a biased judge on an athlete's final score.}}</ref>		Discarding only the maximum and minimum is known as the '''{{visible anchor\|modified mean}}''', particularly in management statistics.<ref>Arulmozhi, G.; Statistics For Management, 2nd Edition, Tata McGraw-Hill Education, 2009, p. [https://books.google.com/books?id=2qcyNld-bHwC&pg=PA458&lpg=PA458&dq=Modified+mean 458]</ref> This is also known as the '''{{visible anchor\|Olympic average}}''' (for example in US agriculture, like the [[Average Crop Revenue Election]]), due to its use in Olympic events, such as the [[ISU Judging System]] in [[figure skating]], to make the score robust to a single outlier judge.<ref>{{cite web \|url=http://farmdocdaily.illinois.edu/2012/08/lessons_from_libor.html \|title=Lessons from LIBOR \|author=Paul E. Peterson \|date=August 3, 2012 \|quote=Once the quotes are compiled, LIBOR uses a trimmed mean process, in which the highest and lowest values are thrown out and the remaining values are averaged. This is sometimes called an "Olympic average" from its use in the Olympics to eliminate the impact of a biased judge on an athlete's final score.}}</ref>
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	The truncated mean is a useful estimator because it is less sensitive to [[outlier]]s than the mean but will still give a reasonable estimate of central tendency or mean for many statistical models. In this regard it is referred to as a [[Robust statistics\|robust estimator]]. For example, in its use in Olympic judging, truncating the maximum and minimum prevents a single judge from increasing or lowering the overall score by giving an exceptionally high or low score.		The truncated mean is a useful estimator because it is less sensitive to [[outlier]]s than the mean but will still give a reasonable estimate of central tendency or mean for many statistical models. In this regard it is referred to as a [[Robust statistics\|robust estimator]]. For example, in its use in Olympic judging, truncating the maximum and minimum prevents a single judge from increasing or lowering the overall score by giving an exceptionally high or low score.

	One situation in which it can be advantageous to use a truncated mean is when estimating the [[location parameter]] of a [[Cauchy distribution]], a bell shaped probability distribution with (much) fatter tails than a [[normal distribution]]. It can be shown that the truncated mean of the middle 24% sample [[order statistics]] (i.e., truncate the sample by 38% at each end) produces an estimate for the population location parameter that is more efficient than using either the sample median or the full sample mean.<ref name=rothenberg>{{cite journal\|last1=Rothenberg\|first1=Thomas J.\|last2=Fisher\|first2=Franklin, M.\|last3=Tilanus\|first3=C.B.\|year=1964\|volume=59\|issue=306\|journal=Journal of the American Statistical Association\|title=A note on estimation from a cauchy sample\|pages=460–463\|doi=10.1080/01621459.1964.10482170}}</ref><ref name=bloch>{{cite journal\|last1=Bloch\|first1=Daniel\|year=1966\|volume=61\|issue=316\|journal=Journal of the American Statistical Association\|title=A note on the estimation of the location parameters of the Cauchy distribution\|pages=852–855\|jstor=2282794\|doi=10.1080/01621459.1966.10480912}}</ref> However, due to the fat tails of the Cauchy distribution, the efficiency of the estimator decreases as more of the sample gets used in the estimate.<ref name=rothenberg/><ref name=bloch/> Note that for the Cauchy distribution, neither the truncated mean, full sample mean or sample median represents a [[maximum likelihood]] estimator, nor are any as asymptotically efficient as the maximum likelihood estimator; however, the maximum likelihood estimate is more difficult to compute, leaving the truncated mean as a useful alternative.<ref name=bloch/><ref name=ferguson>{{cite journal\|last1=Ferguson\|first1=Thomas S.\|~~authorlink~~= Thomas S. Ferguson \|year=1978\|journal=Journal of the American Statistical Association \|volume=73\|issue=361\|title=Maximum Likelihood Estimates of the Parameters of the Cauchy Distribution for Samples of Size 3 and 4\|page=211\|jstor=2286549\|doi=10.1080/01621459.1978.10480031}}</ref>		One situation in which it can be advantageous to use a truncated mean is when estimating the [[location parameter]] of a [[Cauchy distribution]], a bell shaped probability distribution with (much) fatter tails than a [[normal distribution]]. It can be shown that the truncated mean of the middle 24% sample [[order statistics]] (i.e., truncate the sample by 38% at each end) produces an estimate for the population location parameter that is more efficient than using either the sample median or the full sample mean.<ref name=rothenberg>{{cite journal\|last1=Rothenberg\|first1=Thomas J.\|last2=Fisher\|first2=Franklin, M.\|last3=Tilanus\|first3=C.B.\|year=1964\|volume=59\|issue=306\|journal=Journal of the American Statistical Association\|title=A note on estimation from a cauchy sample\|pages=460–463\|doi=10.1080/01621459.1964.10482170}}</ref><ref name=bloch>{{cite journal\|last1=Bloch\|first1=Daniel\|year=1966\|volume=61\|issue=316\|journal=Journal of the American Statistical Association\|title=A note on the estimation of the location parameters of the Cauchy distribution\|pages=852–855\|jstor=2282794\|doi=10.1080/01621459.1966.10480912}}</ref> However, due to the fat tails of the Cauchy distribution, the efficiency of the estimator decreases as more of the sample gets used in the estimate.<ref name=rothenberg/><ref name=bloch/> Note that for the Cauchy distribution, neither the truncated mean, full sample mean or sample median represents a [[maximum likelihood]] estimator, nor are any as asymptotically efficient as the maximum likelihood estimator; however, the maximum likelihood estimate is more difficult to compute, leaving the truncated mean as a useful alternative.<ref name=bloch/><ref name=ferguson>{{cite journal\|last1=Ferguson\|first1=Thomas S.\|author-link= Thomas S. Ferguson \|year=1978\|journal=Journal of the American Statistical Association \|volume=73\|issue=361\|title=Maximum Likelihood Estimates of the Parameters of the Cauchy Distribution for Samples of Size 3 and 4\|page=211\|jstor=2286549\|doi=10.1080/01621459.1978.10480031}}</ref>

	==Drawbacks==		==Drawbacks==
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	==Statistical tests==		==Statistical tests==

	It is possible to perform a [[Student's t-test]] based on the truncated mean, which is called Yuen's t-test <ref>Yuen, K.K. (1974) The two-sample trimmed t for unequal population variances. Biometrika, 61, 165-170.</ref><ref>Wilcox, R.R. (2005). Introduction to robust estimation and hypothesis testing. Academic Press.</ref>, which also has several implementations in [[R (programming language)\|R]]. <ref>https://cran.r-project.org/web/packages/WRS2/</ref><ref>https://cran.r-project.org/web/packages/DescTools/</ref>		It is possible to perform a [[Student's t-test]] based on the truncated mean, which is called Yuen's t-test,<ref>Yuen, K.K. (1974) The two-sample trimmed t for unequal population variances. Biometrika, 61, 165-170.</ref><ref>Wilcox, R.R. (2005). Introduction to robust estimation and hypothesis testing. Academic Press.</ref> which also has several implementations in [[R (programming language)\|R]].<ref>https://cran.r-project.org/web/packages/WRS2/</ref><ref>https://cran.r-project.org/web/packages/DescTools/</ref>

	==Examples==		==Examples==
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	*[[Trimean]]		*[[Trimean]]
	*[[Interquartile mean]]		*[[Interquartile mean]]
	*[[~~Winsorized_mean~~]]		*[[Winsorized mean]]

	==References==		==References==

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2020-10-15T09:13:22Z

208.53.111.247: /* Advantages */ minor clarification for unfamiliar readers

2020-07-19T16:00:56Z

Advantages: minor clarification for unfamiliar readers

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	The truncated mean is a useful estimator because it is less sensitive to [[outlier]]s than the mean but will still give a reasonable estimate of central tendency or mean for many statistical models. In this regard it is referred to as a [[Robust statistics\|robust estimator]]. For example, in its use in Olympic judging, truncating the maximum and minimum prevents a single judge from increasing or lowering the overall score by giving an exceptionally high or low score.		The truncated mean is a useful estimator because it is less sensitive to [[outlier]]s than the mean but will still give a reasonable estimate of central tendency or mean for many statistical models. In this regard it is referred to as a [[Robust statistics\|robust estimator]]. For example, in its use in Olympic judging, truncating the maximum and minimum prevents a single judge from increasing or lowering the overall score by giving an exceptionally high or low score.

	One situation in which it can be advantageous to use a truncated mean is when estimating the [[location parameter]] of a [[Cauchy distribution]], a bell shaped probability distribution with (much) fatter tails than a [[normal distribution]]. It can be shown that the truncated mean of the middle 24% sample [[order statistics]] (i.e., truncate the sample by 38%) produces an estimate for the population location parameter that is more efficient than using either the sample median or the full sample mean.<ref name=rothenberg>{{cite journal\|last1=Rothenberg\|first1=Thomas J.\|last2=Fisher\|first2=Franklin, M.\|last3=Tilanus\|first3=C.B.\|year=1964\|volume=59\|issue=306\|journal=Journal of the American Statistical Association\|title=A note on estimation from a cauchy sample\|pages=460–463\|doi=10.1080/01621459.1964.10482170}}</ref><ref name=bloch>{{cite journal\|last1=Bloch\|first1=Daniel\|year=1966\|volume=61\|issue=316\|journal=Journal of the American Statistical Association\|title=A note on the estimation of the location parameters of the Cauchy distribution\|pages=852–855\|jstor=2282794\|doi=10.1080/01621459.1966.10480912}}</ref> However, due to the fat tails of the Cauchy distribution, the efficiency of the estimator decreases as more of the sample gets used in the estimate.<ref name=rothenberg/><ref name=bloch/> Note that for the Cauchy distribution, neither the truncated mean, full sample mean or sample median represents a [[maximum likelihood]] estimator, nor are any as asymptotically efficient as the maximum likelihood estimator; however, the maximum likelihood estimate is more difficult to compute, leaving the truncated mean as a useful alternative.<ref name=bloch/><ref name=ferguson>{{cite journal\|last1=Ferguson\|first1=Thomas S.\|authorlink= Thomas S. Ferguson \|year=1978\|journal=Journal of the American Statistical Association \|volume=73\|issue=361\|title=Maximum Likelihood Estimates of the Parameters of the Cauchy Distribution for Samples of Size 3 and 4\|page=211\|jstor=2286549\|doi=10.1080/01621459.1978.10480031}}</ref>		One situation in which it can be advantageous to use a truncated mean is when estimating the [[location parameter]] of a [[Cauchy distribution]], a bell shaped probability distribution with (much) fatter tails than a [[normal distribution]]. It can be shown that the truncated mean of the middle 24% sample [[order statistics]] (i.e., truncate the sample by 38% at each end) produces an estimate for the population location parameter that is more efficient than using either the sample median or the full sample mean.<ref name=rothenberg>{{cite journal\|last1=Rothenberg\|first1=Thomas J.\|last2=Fisher\|first2=Franklin, M.\|last3=Tilanus\|first3=C.B.\|year=1964\|volume=59\|issue=306\|journal=Journal of the American Statistical Association\|title=A note on estimation from a cauchy sample\|pages=460–463\|doi=10.1080/01621459.1964.10482170}}</ref><ref name=bloch>{{cite journal\|last1=Bloch\|first1=Daniel\|year=1966\|volume=61\|issue=316\|journal=Journal of the American Statistical Association\|title=A note on the estimation of the location parameters of the Cauchy distribution\|pages=852–855\|jstor=2282794\|doi=10.1080/01621459.1966.10480912}}</ref> However, due to the fat tails of the Cauchy distribution, the efficiency of the estimator decreases as more of the sample gets used in the estimate.<ref name=rothenberg/><ref name=bloch/> Note that for the Cauchy distribution, neither the truncated mean, full sample mean or sample median represents a [[maximum likelihood]] estimator, nor are any as asymptotically efficient as the maximum likelihood estimator; however, the maximum likelihood estimate is more difficult to compute, leaving the truncated mean as a useful alternative.<ref name=bloch/><ref name=ferguson>{{cite journal\|last1=Ferguson\|first1=Thomas S.\|authorlink= Thomas S. Ferguson \|year=1978\|journal=Journal of the American Statistical Association \|volume=73\|issue=361\|title=Maximum Likelihood Estimates of the Parameters of the Cauchy Distribution for Samples of Size 3 and 4\|page=211\|jstor=2286549\|doi=10.1080/01621459.1978.10480031}}</ref>

	==Drawbacks==		==Drawbacks==

GuccizBud: /* Statistical tests */ Copy edit (minor) ▸ Grammar ▸ Run-on sentence.

2020-06-14T05:54:57Z

Statistical tests: Copy edit (minor) ▸ Grammar ▸ Run-on sentence.

← Previous revision		Revision as of 05:54, 14 June 2020
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	==Statistical tests==		==Statistical tests==

	It is possible to perform a [[Student's t-test]] based on the truncated mean, ~~this~~ is called Yuen's t-test <ref>Yuen, K.K. (1974) The two-sample trimmed t for unequal population variances. Biometrika, 61, 165-170.</ref><ref>Wilcox, R.R. (2005). Introduction to robust estimation and hypothesis testing. Academic Press.</ref>, which also has several implementations in [[R (programming language)\|R]] <ref>https://cran.r-project.org/web/packages/WRS2/</ref><ref>https://cran.r-project.org/web/packages/DescTools/</ref>		It is possible to perform a [[Student's t-test]] based on the truncated mean, which is called Yuen's t-test <ref>Yuen, K.K. (1974) The two-sample trimmed t for unequal population variances. Biometrika, 61, 165-170.</ref><ref>Wilcox, R.R. (2005). Introduction to robust estimation and hypothesis testing. Academic Press.</ref>, which also has several implementations in [[R (programming language)\|R]]. <ref>https://cran.r-project.org/web/packages/WRS2/</ref><ref>https://cran.r-project.org/web/packages/DescTools/</ref>

	==Examples==		==Examples==

David Eppstein: Thomas S. Ferguson

2020-06-06T07:03:30Z

Thomas S. Ferguson

← Previous revision		Revision as of 07:03, 6 June 2020
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	The truncated mean is a useful estimator because it is less sensitive to [[outlier]]s than the mean but will still give a reasonable estimate of central tendency or mean for many statistical models. In this regard it is referred to as a [[Robust statistics\|robust estimator]]. For example, in its use in Olympic judging, truncating the maximum and minimum prevents a single judge from increasing or lowering the overall score by giving an exceptionally high or low score.		The truncated mean is a useful estimator because it is less sensitive to [[outlier]]s than the mean but will still give a reasonable estimate of central tendency or mean for many statistical models. In this regard it is referred to as a [[Robust statistics\|robust estimator]]. For example, in its use in Olympic judging, truncating the maximum and minimum prevents a single judge from increasing or lowering the overall score by giving an exceptionally high or low score.

	One situation in which it can be advantageous to use a truncated mean is when estimating the [[location parameter]] of a [[Cauchy distribution]], a bell shaped probability distribution with (much) fatter tails than a [[normal distribution]]. It can be shown that the truncated mean of the middle 24% sample [[order statistics]] (i.e., truncate the sample by 38%) produces an estimate for the population location parameter that is more efficient than using either the sample median or the full sample mean.<ref name=rothenberg>{{cite journal\|last1=Rothenberg\|first1=Thomas J.\|last2=Fisher\|first2=Franklin, M.\|last3=Tilanus\|first3=C.B.\|year=1964\|volume=59\|issue=306\|journal=Journal of the American Statistical Association\|title=A note on estimation from a cauchy sample\|pages=460–463\|doi=10.1080/01621459.1964.10482170}}</ref><ref name=bloch>{{cite journal\|last1=Bloch\|first1=Daniel\|year=1966\|volume=61\|issue=316\|journal=Journal of the American Statistical Association\|title=A note on the estimation of the location parameters of the Cauchy distribution\|pages=852–855\|jstor=2282794\|doi=10.1080/01621459.1966.10480912}}</ref> However, due to the fat tails of the Cauchy distribution, the efficiency of the estimator decreases as more of the sample gets used in the estimate.<ref name=rothenberg/><ref name=bloch/> Note that for the Cauchy distribution, neither the truncated mean, full sample mean or sample median represents a [[maximum likelihood]] estimator, nor are any as asymptotically efficient as the maximum likelihood estimator; however, the maximum likelihood estimate is more difficult to compute, leaving the truncated mean as a useful alternative.<ref name=bloch/><ref name=ferguson>{{cite journal\|last1=Ferguson\|first1=Thomas S.\|year=1978\|journal=Journal of the American Statistical Association \|volume=73\|issue=361\|title=Maximum Likelihood Estimates of the Parameters of the Cauchy Distribution for Samples of Size 3 and 4\|page=211\|jstor=2286549\|doi=10.1080/01621459.1978.10480031}}</ref>		One situation in which it can be advantageous to use a truncated mean is when estimating the [[location parameter]] of a [[Cauchy distribution]], a bell shaped probability distribution with (much) fatter tails than a [[normal distribution]]. It can be shown that the truncated mean of the middle 24% sample [[order statistics]] (i.e., truncate the sample by 38%) produces an estimate for the population location parameter that is more efficient than using either the sample median or the full sample mean.<ref name=rothenberg>{{cite journal\|last1=Rothenberg\|first1=Thomas J.\|last2=Fisher\|first2=Franklin, M.\|last3=Tilanus\|first3=C.B.\|year=1964\|volume=59\|issue=306\|journal=Journal of the American Statistical Association\|title=A note on estimation from a cauchy sample\|pages=460–463\|doi=10.1080/01621459.1964.10482170}}</ref><ref name=bloch>{{cite journal\|last1=Bloch\|first1=Daniel\|year=1966\|volume=61\|issue=316\|journal=Journal of the American Statistical Association\|title=A note on the estimation of the location parameters of the Cauchy distribution\|pages=852–855\|jstor=2282794\|doi=10.1080/01621459.1966.10480912}}</ref> However, due to the fat tails of the Cauchy distribution, the efficiency of the estimator decreases as more of the sample gets used in the estimate.<ref name=rothenberg/><ref name=bloch/> Note that for the Cauchy distribution, neither the truncated mean, full sample mean or sample median represents a [[maximum likelihood]] estimator, nor are any as asymptotically efficient as the maximum likelihood estimator; however, the maximum likelihood estimate is more difficult to compute, leaving the truncated mean as a useful alternative.<ref name=bloch/><ref name=ferguson>{{cite journal\|last1=Ferguson\|first1=Thomas S.\|authorlink= Thomas S. Ferguson \|year=1978\|journal=Journal of the American Statistical Association \|volume=73\|issue=361\|title=Maximum Likelihood Estimates of the Parameters of the Cauchy Distribution for Samples of Size 3 and 4\|page=211\|jstor=2286549\|doi=10.1080/01621459.1978.10480031}}</ref>

	==Drawbacks==		==Drawbacks==

Samlin413 at 19:19, 16 January 2020

2020-01-16T19:19:09Z

← Previous revision		Revision as of 19:19, 16 January 2020
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	A '''truncated mean''' or '''trimmed mean''' is a [[Statistics\|statistical]] [[Average\|measure of central tendency]], much like the [[mean]] and [[median]]. It involves the calculation of the mean after discarding given parts of a [[probability distribution]] or [[Sampling (statistics)\|sample]] at the high and low end, and typically discarding an equal amount of both. This number of points to be discarded is usually given as a percentage of the total number of points, but may also be given as a fixed number of points.		A '''truncated mean''' or '''trimmed mean''' is a [[Statistics\|statistical]] [[Average\|measure of central tendency]], much like the [[mean]] and [[median]]. It involves the calculation of the mean after discarding given parts of a [[probability distribution]] or [[Sampling (statistics)\|sample]] at the high and low end, and typically discarding an equal amount of both. This number of points to be discarded is usually given as a percentage of the total number of points, but may also be given as a fixed number of points.

	For most statistical applications, 5 to 25 percent of the ends are discarded~~; the 25% trimmed mean (when the lowest 25% and the highest 25% are discarded) is known as the [[interquartile mean]]~~. For example, given a set of 8 points, trimming by 12.5% would discard the minimum and maximum value in the sample: the smallest and largest values, and would compute the mean of the remaining 6 points.		For most statistical applications, 5 to 25 percent of the ends are discarded. For example, given a set of 8 points, trimming by 12.5% would discard the minimum and maximum value in the sample: the smallest and largest values, and would compute the mean of the remaining 6 points. The 25% trimmed mean (when the lowest 25% and the highest 25% are discarded) is known as the [[interquartile mean]].

	The median can be regarded as a fully truncated mean and is most robust. As with other [[trimmed estimator]]s, the main advantage of the trimmed mean is robustness and higher [[Efficiency (statistics)\|efficiency]] for mixed distributions and heavy-tailed distribution (like the [[Cauchy distribution]]), at the cost of lower efficiency for some other less heavily-tailed distributions (such as the normal distribution). For intermediate distributions the differences between the efficiency of the mean and the median are not very big, e.g. for the student-t distribution with 2 degrees of freedom the variances for mean and median are nearly equal.		The median can be regarded as a fully truncated mean and is most robust. As with other [[trimmed estimator]]s, the main advantage of the trimmed mean is robustness and higher [[Efficiency (statistics)\|efficiency]] for mixed distributions and heavy-tailed distribution (like the [[Cauchy distribution]]), at the cost of lower efficiency for some other less heavily-tailed distributions (such as the normal distribution). For intermediate distributions the differences between the efficiency of the mean and the median are not very big, e.g. for the student-t distribution with 2 degrees of freedom the variances for mean and median are nearly equal.

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	*[[Trimean]]		*[[Trimean]]
	*[[Interquartile mean]]		*[[Interquartile mean]]
			*[[Winsorized_mean]]

	==References==		==References==