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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 is usually given as a percentage, but may 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 first and last values.  
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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==
In some regions of [[Central Europe]] it is also known as a '''Windsor mean''', 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 management statistics.<ref>Arulmozhi, G.; Statistics For Management, 2nd Edition, Tata McGraw-Hill Education, 2009, p. [http://books.google.com/books?id=2qcyNld-bHwC&pg=PA458&lpg=PA458&dq=Modified+mean 458]</ref>
 
==Interpolation==
When the percentage of points to discard does not yield a whole number, the trimmed mean may be defined by interpolation, generally linear interpolation, between the nearest whole numbers. For example, if you need to calculate the 15% trimmed mean of a sample containing 10 entries, strictly this would mean discarding 1 point from each end (equivalent to the 10% trimmed mean). If interpolating, one would instead compute the 10% trimmed mean (discarding 1 point from each end) and the 20% trimmed mean (discarding 2 points from each end), and then interpolating, in this case averaging these two values. Similarly, if interpolating the 12% trimmed mean, one would take the [[weighted average]]: weight the 10% trimmed mean by 0.8 and the 20% trimmed mean by 0.2.
 
==Advantages==
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]].
 
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}}</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}}</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}}</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.
 
==Examples==
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''.
 
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 <math>4/18 \approx 22%</math>).
 
==See also==
*[[Trimean]]
*[[Interquartile mean]]
 
==References==
{{Reflist}}
 
{{DEFAULTSORT:Truncated Mean}}
[[Category:Means]]
 
[[de:Mittelwert#Winsorisiertes oder gestutztes Mittel]]
[[fr:Moyenne#Moyenne tronquée (ou "réduite")]]

Revision as of 21:17, 6 February 2014

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