--- title: "TrimmedMean" language: "en" type: "Symbol" summary: "TrimmedMean[list, f] gives the mean of the elements in list after dropping a fraction f of the smallest and largest elements. TrimmedMean[list, {f1, f2}] gives the mean when a fraction f1 of the smallest elements and a fraction f2 of the largest elements are removed. TrimmedMean[list] gives the 5% trimmed mean TrimmedMean[list, 0.05]. TrimmedMean[dist, ...] gives the trimmed mean of a univariate distribution dist." keywords: - location statistic - modified arithmetic mean - modified mean - robust estimate - robust location estimate - truncated mean canonical_url: "https://reference.wolfram.com/language/ref/TrimmedMean.html" source: "Wolfram Language Documentation" related_guides: - title: "Descriptive Statistics" link: "https://reference.wolfram.com/language/guide/DescriptiveStatistics.en.md" - title: "Statistical Moments and Generating Functions" link: "https://reference.wolfram.com/language/guide/StatisticalMomentsAndGeneratingFunctions.en.md" - title: "Robust Descriptive Statistics" link: "https://reference.wolfram.com/language/guide/RobustDescriptiveStatistics.en.md" - title: "Date & Time" link: "https://reference.wolfram.com/language/guide/DateAndTime.en.md" related_functions: - title: "Mean" link: "https://reference.wolfram.com/language/ref/Mean.en.md" - title: "Quantile" link: "https://reference.wolfram.com/language/ref/Quantile.en.md" - title: "TrimmedVariance" link: "https://reference.wolfram.com/language/ref/TrimmedVariance.en.md" - title: "WinsorizedMean" link: "https://reference.wolfram.com/language/ref/WinsorizedMean.en.md" - title: "Median" link: "https://reference.wolfram.com/language/ref/Median.en.md" - title: "BiweightLocation" link: "https://reference.wolfram.com/language/ref/BiweightLocation.en.md" - title: "WinsorizedVariance" link: "https://reference.wolfram.com/language/ref/WinsorizedVariance.en.md" - title: "MeanShiftFilter" link: "https://reference.wolfram.com/language/ref/MeanShiftFilter.en.md" related_tutorials: - title: "Descriptive Statistics" link: "https://reference.wolfram.com/language/tutorial/NumericalOperationsOnData.en.md#7135" --- # TrimmedMean TrimmedMean[list, f] gives the mean of the elements in list after dropping a fraction f of the smallest and largest elements. TrimmedMean[list, {f1, f2}] gives the mean when a fraction f1 of the smallest elements and a fraction f2 of the largest elements are removed. TrimmedMean[list] gives the 5% trimmed mean TrimmedMean[list, 0.05]. TrimmedMean[dist, …] gives the trimmed mean of a univariate distribution dist. ## Details * ``TrimmedMean`` gives a robust estimate of the mean by excluding extreme values. * The trimming fraction is determined by the parameters Subscript[``f``, 1] and Subscript[``f``, 2], which indicate the fraction ``f1`` of the smallest elements and the fraction ``f2`` of the largest elements to be removed. * ``TrimmedMean[list, {f1, f2}]`` gives the mean of ``Sort[list, Less][[1 + ⌊f1 n⌋ ;; n - ⌊f2 n⌋]]``, where ``n`` equals the length of ``list``. ```wl [image] [image] [image] ``` * ``TrimmedMean[{{x1, y1, …}, {x2, y2, …}, …}, f]`` gives ``{TrimmedMean[{x1, x2, …}, f], TrimmedMean[{y1, y2, …}, f], …}``. * ``TrimmedMean[dist, {f1, f2}]`` gives ``Mean[TruncatedDistribution[Quantile[dist, {f1, 1 - f2}], dist]]`` for a univariate distribution ``dist``. --- ## Examples (23) ### Basic Examples (4) Trimmed mean after removing extreme values: ```wl In[1]:= TrimmedMean[{-10, 1, 1, 1, 1, 20}, 0.2] Out[1]= 1 ``` --- Trimmed mean after removing the smallest extreme values: ```wl In[1]:= TrimmedMean[{-10, 1, 1, 1, 1, 20}, {0.2, 0}] Out[1]= (24/5) ``` --- Trimmed mean of a list of dates: ```wl In[1]:= RandomDate[6] Out[1]= {DateObject[{2024, 4, 27, 0, 38, 46.2511}, "Instant", "Gregorian", -5.], DateObject[{2024, 10, 20, 22, 2, 27.031}, "Instant", "Gregorian", -5.], DateObject[{2024, 3, 4, 22, 39, 17.0503}, "Instant", "Gregorian", -5.], DateObject[{2024, 12, 25, 22, 58, 29.8844}, "Instant", "Gregorian", -5.], DateObject[{2024, 1, 5, 21, 48, 15.1744}, "Instant", "Gregorian", -5.], DateObject[{2024, 12, 26, 0, 51, 44.2641}, "Instant", "Gregorian", -5.]} In[2]:= TrimmedMean[%] Out[2]= DateObject[{2024, 7, 18, 23, 9, 49.9426}, "Instant", "Gregorian", -5.] ``` --- Trimmed mean of a symbolic distribution: ```wl In[1]:= TrimmedMean[ExponentialDistribution[λ]] Out[1]= (18 - 19 Log[19] + 18 Log[20]/18 λ) ``` ### Scope (10) #### Data (9) Exact input yields exact output: ```wl In[1]:= TrimmedMean[{1, 20, 3, 4}, 2 / 5] Out[1]= (7/2) In[2]:= TrimmedMean[{Sqrt[2], E, Pi, Pi ^ 2, 1, 2, 3}, 1 / 4] Out[2]= (1/5) (5 + Sqrt[2] + E + π) ``` --- Approximate input yields approximate output: ```wl In[1]:= TrimmedMean[{5., 10., 4., 25., 2., 1.}, 0.2] Out[1]= 5.25 In[2]:= TrimmedMean[N[{5, 10, 4, 25, 2, 1}, 30], 0.2] Out[2]= 5.25000000000000000000000000000 ``` --- ``TrimmedMean`` for a matrix gives columnwise means: ```wl In[1]:= TrimmedMean[RandomReal[1, {50, 2}], .1] Out[1]= {0.505479, 0.380264} ``` --- Trimmed mean works with large arrays: ```wl In[1]:= TrimmedMean[RandomReal[1, 10 ^ 6], 1 / 100] Out[1]= 0.500205 In[2]:= TrimmedMean[RandomReal[1, {10 ^ 5, 5}]] Out[2]= {0.500179, 0.499209, 0.500229, 0.500484, 0.499184} ``` --- ``SparseArray`` data can be used just like dense arrays: ```wl In[1]:= sp = SparseArray[{{i_, i_} :> i, {i_, j_} /; j == i + 1 :> i - 1}, {100, 10}, 1]; In[2]:= TrimmedMean[sp] Out[2]= {1, 1, 1, 1, 1, 1, 1, 1, 1, 1} ``` --- Trimmed mean of a ``TimeSeries`` : ```wl In[1]:= ts = TemporalData[TimeSeries, {{{3, 8, 4, 11, 9, 2}}, {{{1, 3, 5, 7, 8, 10}}}, 1, {"Continuous", 1}, {"Discrete", 1}, 1, {ResamplingMethod -> {"Interpolation", InterpolationOrder -> 1}}}, False, 10.1]; In[2]:= TrimmedMean[ts, 1 / 6]//N Out[2]= 6. ``` Trimmed mean depends only on the values: ```wl In[3]:= TrimmedMean[ts["Values"], 1 / 6]//N Out[3]= 6. ``` --- Trimmed mean works with data involving quantities: ```wl In[1]:= data = Quantity[RandomReal[1, 6], "Meters"] Out[1]= {Quantity[0.569715, "Meters"], Quantity[0.165186, "Meters"], Quantity[0.952671, "Meters"], Quantity[0.855585, "Meters"], Quantity[0.981888, "Meters"], Quantity[0.0222484, "Meters"]} In[2]:= TrimmedMean[data] Out[2]= Quantity[0.591216, "Meters"] ``` --- Compute trimmed mean of dates: ```wl In[1]:= dates = WolframLanguageData[All, "DateIntroduced"]; In[2]:= DateHistogram[dates] Out[2]= [image] In[3]:= TrimmedMean[dates] Out[3]= DateObject[{2009, 9, 25}, "Day"] ``` --- Compute trimmed mean of times: ```wl In[1]:= RandomTime[3] Out[1]= {TimeObject[{0, 23, 47.6431}, "Instant"], TimeObject[{6, 41, 35.8222}, "Instant"], TimeObject[{22, 56, 48.0332}, "Instant"]} In[2]:= TrimmedMean[%] Out[2]= TimeObject[{10, 0, 43.8328}, "Instant", -5.] ``` List of times with different time zone specifications: ```wl In[3]:= {TimeObject[{12}, TimeZone -> 0], TimeObject[{12}, TimeZone -> 2], TimeObject[{12}, TimeZone -> "Asia/Tokyo"]} Out[3]= {TimeObject[{12}, "Hour", 0.], TimeObject[{12}, "Hour", 2.], TimeObject[{12}, "Hour", "Asia/Tokyo"]} In[4]:= TrimmedMean[%] Out[4]= TimeObject[{1, 20, 0}, "Instant", "Asia/Tokyo"] In[5]:= %["TimeZone"] Out[5]= "Asia/Tokyo" ``` #### Distributions (1) The trimmed mean for a univariate distribution: ```wl In[1]:= TrimmedMean[NormalDistribution[1.3, .2]] Out[1]= 1.3 In[2]:= TrimmedMean[BinomialDistribution[10, 1 / 2]] Out[2]= (1640/319) In[3]:= TrimmedMean[WeibullDistribution[α, β]] Out[3]= (10/9) (Piecewise[{{β*(Gamma[1 + 1/α, Log[20/19]] - Gamma[1 + 1/α, Log[20]]), Log[20/19]^(1/α) > 0 && Log[20/19]^(1/α) - Log[20]^(1/α) < 0 && β > 0}}, 0]) ``` ### Applications (3) Obtain a robust estimate of location when outliers are present: ```wl In[1]:= TrimmedMean[{1, 5, 2, 6, 10, 10 ^ 5, 5, 4, -200, 5}, .1]//N Out[1]= 4.75 ``` Extreme values have a large influence on the ``Mean`` : ```wl In[2]:= Mean[{1, 5, 2, 6, 10, 10 ^ 5, 5, 4, -200, 5}]//N Out[2]= 9983.8 ``` --- Simulate a trajectory with heavy-tailed measurement noise: ```wl In[1]:= n[] := RandomVariate[CauchyDistribution[0, 1.2]] f[] := {u Cos[u] + n[], u Sin[u] + n[]}; data = Table[f[], {u, 0, 6π, 1 / 100}]; ``` The underlying signal and simulated path with noise: ```wl In[2]:= {ParametricPlot[{u Cos[u], u Sin[u]}, {u, 0, 6π}], pd = ListPlot[data, AspectRatio -> 1]} Out[2]= {[image], [image]} ``` Smooth the trajectory using a moving ``TrimmedMean`` : ```wl In[3]:= td = TimeSeries[data, {0., 6π}, ValueDimensions -> 2]; In[4]:= smooth[r_] := MovingMap[TrimmedMean, td, {Quantity[r, "Events"], Center}, "ReflectedDifferences"] ``` Increasing the block size gives a smoother trajectory: ```wl In[5]:= Quiet@Table[ParametricPlot[Evaluate[smooth[r]["PathFunction"][t]], {t, 0, 6π}], {r, {25, 50, 100}}] Out[5]= {[image], [image], [image]} ``` --- Find a trimmed mean for the heights of children in a class: ```wl In[1]:= heights = Quantity[{134, 143, 131, 140, 145, 136, 131, 136, 143, 136, 133, 145, 147, 150, 150, 146, 137, 143, 132, 142, 145, 136, 144, 135, 141}, "Centimeters"]; In[2]:= ListPlot[heights, Filling -> Axis, AxesLabel -> Automatic] Out[2]= [image] In[3]:= TrimmedMean[heights]//N Out[3]= Quantity[140., "Centimeters"] ``` Compare a few trimmed means: ```wl In[4]:= (tmeans = Table[TrimmedMean[heights, f], {f, {.05, .2, .45}}])//N Out[4]= {Quantity[140., "Centimeters"], Quantity[140.13333333333333, "Centimeters"], Quantity[141., "Centimeters"]} In[5]:= n = Length[heights]; ListPlot[Join[{heights}, Table[{{0, m}, {n, m}}, {m, tmeans}]], Joined -> {False, True, True, True}, Filling -> {1 -> 0}, AxesLabel -> Automatic, PlotLegends -> {"heights", .05, .2, .45}] Out[5]= [image] ``` Plot the trimmed mean as a function of trimmed fraction: ```wl In[6]:= trimMean[f_ ? NumericQ] /; 0 ≤ f < 0.5 := TrimmedMean[heights, f] In[7]:= Plot[trimMean[f], {f, 0, 0.49}, AxesLabel -> {f}] Out[7]= [image] ``` ### Properties & Relations (5) A 0% ``TrimmedMean`` is equivalent to ``Mean`` : ```wl In[1]:= TrimmedMean[Range[10], 0] Out[1]= (11/2) In[2]:= Mean[Range[10]] Out[2]= (11/2) ``` --- ``TrimmedMean`` approaches ``Median`` as ``f`` approaches 1/2: ```wl In[1]:= data = RandomVariate[CauchyDistribution[0, 1], 1000]; In[2]:= md = Median[data] Out[2]= -0.0112343 In[3]:= Plot[{TrimmedMean[data, f], md}, {f, 0.1, .499}, AxesLabel -> {f}] Out[3]= [image] ``` --- ``TrimmedMean`` of a distribution is the mean of its ``TruncatedDistribution`` : ```wl In[1]:= \[ScriptCapitalD] = NormalDistribution[a, b]; In[2]:= TrimmedMean[\[ScriptCapitalD], {.5, 0}] Out[2]= 0.56419 (1.77245 a + 1.41421 b) ``` Mean of the ``TruncatedDistribution`` with appropriate bounds: ```wl In[3]:= td = TruncatedDistribution[Quantile[\[ScriptCapitalD], {.5, 1}], \[ScriptCapitalD]]; In[4]:= Mean[td] Out[4]= 0.56419 (1.77245 a + 1.41421 b) ``` --- ``TrimmedMean`` of a sample gives an estimate of the mean of a truncated distribution: ```wl In[1]:= \[ScriptCapitalD] = NormalDistribution[]; data = RandomVariate[\[ScriptCapitalD], 10 ^ 6]; In[2]:= TrimmedMean[data, {.5, 0}] Out[2]= 0.799639 ``` Mean of the ``TruncatedDistribution`` with appropriate bounds: ```wl In[3]:= td = TruncatedDistribution[Quantile[\[ScriptCapitalD], {.5, 1}], \[ScriptCapitalD]]; In[4]:= Mean[td] Out[4]= 0.797885 ``` --- ``TrimmedMean`` drops the data beyond a certain quantile level, then computes the sample mean: ```wl In[1]:= len = 100; f = 1 / 10; data = RandomReal[1, len]; In[2]:= data1 = Part[Sort[data], 1 + Floor[len f] ;; len - Floor[len f]]; In[3]:= TrimmedMean[data, f] == Mean[data1] Out[3]= True ``` ``WinsorizedMean`` clips the data beyond a certain quantile level, then computes the sample mean: ```wl In[4]:= data2 = Clip[data, {RankedMin[data, 1 + Floor[len f]], RankedMax[data, 1 + Floor[len f]]}]; In[5]:= WinsorizedMean[data, f] == Mean[data2] Out[5]= True ``` Plot the sorted data against the sample with elements removed and the clipped sample: ```wl In[6]:= d0 = Transpose[{Range[1, len], Sort@data}]; d1 = Transpose[{Range[1 + Floor[len f], len - Floor[len f]], data1}]; d2 = Transpose[{Range[1, len], Sort@data2}]; In[7]:= ListPlot[{d0, d1, d2}, Filling -> {2 -> 0}, PlotLegends -> {"data", "data1", "data2"}] Out[7]= [image] ``` ### Possible Issues (1) ``TrimmedMean`` requires numeric values: ```wl In[1]:= TrimmedMean[{a, b, c}, .1] ``` TrimmedMean::arg1: The first argument {a,b,c} is expected to be a numeric vector or matrix. ```wl Out[1]= TrimmedMean[{a, b, c}, 0.1] ``` ## See Also * [`Mean`](https://reference.wolfram.com/language/ref/Mean.en.md) * [`Quantile`](https://reference.wolfram.com/language/ref/Quantile.en.md) * [`TrimmedVariance`](https://reference.wolfram.com/language/ref/TrimmedVariance.en.md) * [`WinsorizedMean`](https://reference.wolfram.com/language/ref/WinsorizedMean.en.md) * [`Median`](https://reference.wolfram.com/language/ref/Median.en.md) * [`BiweightLocation`](https://reference.wolfram.com/language/ref/BiweightLocation.en.md) * [`WinsorizedVariance`](https://reference.wolfram.com/language/ref/WinsorizedVariance.en.md) * [`MeanShiftFilter`](https://reference.wolfram.com/language/ref/MeanShiftFilter.en.md) ## Tech Notes * [Descriptive Statistics](https://reference.wolfram.com/language/tutorial/NumericalOperationsOnData.en.md#7135) ## Related Guides * [Descriptive Statistics](https://reference.wolfram.com/language/guide/DescriptiveStatistics.en.md) * [Statistical Moments and Generating Functions](https://reference.wolfram.com/language/guide/StatisticalMomentsAndGeneratingFunctions.en.md) * [Robust Descriptive Statistics](https://reference.wolfram.com/language/guide/RobustDescriptiveStatistics.en.md) * [Date & Time](https://reference.wolfram.com/language/guide/DateAndTime.en.md) ## History * [Introduced in 2007 (6.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn60.en.md) \| [Updated in 2017 (11.1)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn111.en.md) ▪ [2024 (14.1)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn141.en.md)