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).