# trimmean

trimmed mean of a vector or a matrix

### Syntax

m=trimmean(x[,discard [,flag [,verbose]]])

### Arguments

- x
real or complex vector or matrix

- discard
Optional real value between 0 and 100 representing the part of the data to discard. It discard is not in the [0,100] range, an error is generated. Default value for discard=50.

- flag
Optional string or real parameter which controls the behaviour when x is a matrix. Available values for flag are : "all", 1, 2, r or c (default is flag="all"). The two values flag=r and flag=1 are equivalent. The two values flag=c and flag=2 are equivalent.

- verbose
Optional integer. If set to 1, then enables verbose logging. Default is 0.

### Description

A trimmed mean is calculated by discarding a certain percentage of the lowest and the highest scores and then computing the mean of the remaining scores. For example, a mean trimmed 50% is computed by discarding the lower and higher 25% of the scores and taking the mean of the remaining scores.

The median is the mean trimmed 100% and the arithmetic mean is the mean trimmed 0%.

A trimmed mean is obviously less susceptible to the effects of extreme scores than is the arithmetic mean. It is therefore less susceptible to sampling fluctuation than the mean for extremely skewed distributions. The efficiency of a statistic is the degree to which the statistic is stable from sample to sample. That is, the less subject to sampling fluctuation a statistic is, the more efficient it is. The efficiency of statistics is measured relative to the efficiency of other statistics and is therefore often called the relative efficiency. If statistic A has a smaller standard error than statistic B, then statistic A is more efficient than statistic B. The relative efficiency of two statistics may depend on the distribution involved. For instance, the mean is more efficient than the median for normal distributions but not for some extremely skewed distributions. The efficiency of a statistic can also be thought of as the precision of the estimate: The more efficient the statistic, the more precise the statistic is as an estimator of the parameter. The trimmed mean is less efficient than the mean for normal distributions.

For a vector or matrix `x`

,
`t=trimmean(x,discard)`

returns in scalar
`t`

the mean of all the entries of `x`

,
after discarding `discard/2`

highest values and
`discard/2`

lowest values.

`t=trimmean(x,discard,'r')`

(or, equivalently,
`t=trimmean(x,discard,1)`

) returns in each entry of the
row vector `t`

the trimmed mean of each column of
`x`

.

`t=trimmean(x,discard,'c')`

(or, equivalently,
`t=trimmean(x,discard,2)`

) returns in each entry of the
column vector `t`

the trimmed mean of each row of
`x`

.

This function computes the trimmed mean of a vector or matrix
`x`

.

For a vector or matrix `x`

, `m=trimmean(x,discard)`

returns in scalar`m`

the trimmed mean of all the entries of `x`

.

`m=trimmean(x,'r')`

(or, equivalently, `m=trimmean(x,1)`

)returns in each entry of the row vector
`m`

the trimmed mean of each column of `x`

.

`m=trimmean(x,'c')`

(or, equivalently, `m=trimmean(x,2)`

)returns in each entry of the column vector
`m`

the trimmed mean of each row of `x`

.

### Example with x as vector

In the following example, one computes the trimmed mean of one data vector, with the default discard value equal to 50 and verbose logging. The data is made of 9 entries. The algorithms sorts the vector and keeps only indices from 3 to 7, skipping indices 1, 2, 8 and 9. The value 4000, which is much larger than the others is not taken into account. The computed trimmed mean is therefore 50.

data = [10, 20, 30, 40, 50, 60, 70, 80, 4000]; computed = trimmean(data,verbose=1);

### Example with x as matrix

In the following example, one computes the trimmed mean of one data matrix. The chosen discard value is 50. The orientation is "r", which means that the data is sorted row by row. For each column of the matrix, one computes a trimmed mean. The trimmed mean is the line vector [25 25 25 25].

data = [10 10 10 10 20 20 20 20 30 30 30 30 4000 4000 4000 4000]; computed = trimmean(data,50,orien="r");

### References

Luis Angel Garcia-Escudero and Alfonso Gordaliza, Robustness Properties of Means and Trimmed Means, JASA, Volume 94, Number 447, Sept 1999, pp956-969

Trimmed Mean, http://davidmlane.com/hyperstat/A11971.html

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