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trimmean

Mean, excluding outliers

Syntax

m = trimmean(X,percent)
m = trimmean(X,percent,flag)
m = trimmean(___,'all')
m = trimmean(___,dim)
m = trimmean(___,vecdim)

Description

example

m = trimmean(X,percent) returns the mean of values of X, computed after removing the outliers of X. For example, if X is a vector that has n values, m is the mean of X excluding the highest and lowest k data values, where k = n*(percent/100)/2.

  • If X is a vector, then trimmean(X,percent) is the mean of all the values of X, computed after removing the outliers.

  • If X is a matrix, then trimmean(X,percent) is a row vector of column means, computed after removing the outliers.

  • If X is a multidimensional array, then trimmean operates along the first nonsingleton dimension of X.

example

m = trimmean(X,percent,flag) specifies how to trim when k (half the number of outliers) is not an integer.

example

m = trimmean(___,'all') returns the trimmed mean of all the values in X using any of the input argument combinations in the previous syntaxes.

example

m = trimmean(___,dim) returns the trimmed mean along the operating dimension dim of X.

example

m = trimmean(___,vecdim) returns the trimmed mean over the dimensions specified in the vector vecdim. For example, if X is a 2-by-3-by-4 array, then trimmean(X,10,[1 2]) returns a 1-by-1-by-4 array. Each value of the output array is the mean of the middle 90% of the values on the corresponding page of X.

Examples

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Find the relative efficiency of the 10% trimmed mean to the sample mean for a given data set.

Generate a 100-by-100 matrix of random numbers from the standard normal distribution. This matrix represents 100 samples, each containing 100 data points.

rng default;  % For reproducibility
X = normrnd(0,1,100,100);

Compute the sample mean and the 10% trimmed mean for each column of the data matrix.

m = mean(X); % Sample mean
trim = trimmean(X,10); % Trimmed mean

Compute the relative efficiency of the trimmed mean to the sample mean. The relative efficiency is the variance of the sample mean divided by the variance of the trimmed mean.

vm = var(m) % Variance of the sample mean
vm = 0.0094
vtrim = var(trim) % Variance of the trimmed mean
vtrim = 0.0097
efficiency = vm/vtrim % Relative efficiency of the trimmed mean to the sample mean
efficiency = 0.9663

The sample mean has a smaller variance than the trimmed mean (efficiency < 1). Therefore, the trimmed mean is less efficient than the sample mean.

Control the trimming for a distribution with outliers when k (half the number of outliers to be trimmed) is not an integer.

Generate a vector of random numbers from the Student's t distribution with degrees of freedom equal to 1. The Student's t distribution tends to have outliers.

rng default;  % For reproducibility
nu = 1; % Degrees of freedom
n = 60; % Number of rows
m = 1;  % Number of columns
x = trnd(nu,n,m); % Vector 

Visualize the distribution using a normal probability plot.

probplot(x)

Although the distribution is symmetric around zero, several outliers affect the mean.

Find the mean of the data.

mn = mean(x)
mn = 1.6452

Find the 33% trimmed mean of the data.

trim = trimmean(x,33)
trim = 0.4940

The 33% trimmed mean is closer to zero, which is more representative of the data. For the 33% trimmed mean, k is not an integer (k = 60*(33/100)/2 gives a value of 9.9). Therefore, trimmean rounds k to the nearest integer (10) by default.

Control trimming by rounding k down to the next smaller integer (9). Specify the control for trimming to 'floor'.

trim = trimmean(x,33,'floor')
trim = 0.4933

Find the trimmed mean along different dimensions for a matrix.

Generate a matrix of random numbers from the Student's t distribution. The Student's t distribution tends to have outliers.

rng('default')
nu = 1; % Degrees of freedom
n = 2; % Number of rows
m = 100;  % Number of columns
X = trnd(nu,n,m);

Visualize the distribution for each row of X using a normal probability plot.

for i = 1:n
    figure()
    probplot(X(i,:))
end

Find the mean for each row of X.

mn = mean(X,2)
mn = 2×1

   -2.7379
    2.0087

Find the 30% trimmed mean for each row of X. Specify dim = 2 as the operating dimension.

trim = trimmean(X,30,2)
trim = 2×1

   -0.0868
    0.1115

The 30% trimmed mean of each row is closer to zero, which is more representative of the data.

Calculate the trimmed mean over multiple dimensions by using the 'all' and vecdim input arguments.

Create a 5-by-4-by-2 array with some outlier values.

X = reshape(1:40,[5 4 2]);
X([3 37]) = -100
X = 
X(:,:,1) =

     1     6    11    16
     2     7    12    17
  -100     8    13    18
     4     9    14    19
     5    10    15    20


X(:,:,2) =

    21    26    31    36
    22    27    32  -100
    23    28    33    38
    24    29    34    39
    25    30    35    40

Find the 10% trimmed mean of X.

mall = trimmean(X,10,'all')
mall = 19.4722

mall is the mean of the middle 90% of the values in X.

Find the 10% trimmed mean for each page of X.

mpage = trimmean(X,10,[1 2])
mpage = 
mpage(:,:,1) =

   10.3889


mpage(:,:,2) =

   29.6111

For example, mpage(1,1,2) is the mean of the middle 90% of the values in X(:,:,2).

Input Arguments

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Input data that represents a sample from a population, specified as a vector, matrix, or multidimensional array.

  • If X is a vector, then trimmean(X,percent) is the mean of all the values of X, computed after removing the outliers.

  • If X is a matrix, then trimmean(X,percent) is a row vector of column means, computed after removing the outliers.

  • If X is a multidimensional array, then trimmean operates along the first nonsingleton dimension of X.

To specify the operating dimension when X is a matrix or an array, use the dim input argument.

trimmean treats NaN values in X as missing values and removes them.

Data Types: single | double

Percentage of input data to be trimmed, specified as a scalar between 0 and 100.

trimmean uses the value of percent to determine the number of outliers (highest and lowest k values in X) to remove from X before computing the mean. For X with n values, k = n*(percent/100)/2.

Data Types: single | double

Control for trimming when k (half the number of outliers) is not an integer, specified as one of the values in this table.

ValueDescription
'round'Round k to the nearest integer (round to a smaller integer if k is a half integer). This value is the default.
'floor'Round k down to the next smaller integer.
'weighted'If k = i + f, where i is an integer and f is a fraction, compute a weighted mean with weight (1 – f) for the (i + 1)th and (n – i)th values, and full weight for the values between them.

Data Types: char | string

Dimension along which to operate, specified as a positive integer scalar. If you do not specify a value, then the default value is the first array dimension of X whose size does not equal 1.

Consider a two-dimensional array X:

  • If dim is equal to 1, then trimmean(X,percent,1) returns a row vector containing the trimmed mean for each column in X.

  • If dim is equal to 2, then trimmean(X,percent,2) returns a column vector containing the trimmed mean for each row in X.

If dim is greater than ndims(X) or if size(X,dim) is 1, then trimmean returns X.

Data Types: single | double

Vector of dimensions, specified as a positive integer vector. Each element of vecdim represents a dimension of the input array X. The output m has length 1 in the specified operating dimensions. The other dimension lengths are the same for X and m.

For example, if X is a 2-by-3-by-3 array, then trimmean(X,10,[1 2]) returns a 1-by-1-by-3 array. Each element of the output is the mean of the middle 90% of the values on the corresponding page of X.

Data Types: single | double

Output Arguments

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Trimmed mean values, returned as a scalar, vector, matrix, or multidimensional array.

Tips

  • The trimmed mean is a robust estimate of the location of a data sample. If the data contains outliers, then the trimmed mean represents the center of the data better than the sample mean. However, if all the data is from the same probability distribution, then the trimmed mean is less efficient than the sample mean as an estimator of the data location.

Extended Capabilities

See Also

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Introduced before R2006a