# Truncated mean

**trimmed meanmodified mean**

A truncated mean or trimmed mean is a statistical measure of central tendency, much like the mean and median.wikipedia

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### Interquartile mean

**interquartile**

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.

The interquartile mean (IQM) (or midmean) is a statistical measure of central tendency based on the truncated mean of the interquartile range.

### Mean

**mean valuepopulation meanaverage**

A truncated mean or trimmed mean is a statistical measure of central tendency, much like the mean and median.

In this case, one can use a truncated mean.

### Trimmed estimator

**trimmedtrimming**

As with other trimmed estimators, the main advantage of the trimmed mean is robustness and higher 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 instance, the 5% trimmed mean is obtained by taking the mean of the 5% to 95% range.

### Average

**Rushing averageReceiving averagemean**

A truncated mean or trimmed mean is a statistical measure of central tendency, much like the mean and median.

### ISU Judging System

**ISU best scoresWRCode of Points**

This is also known as the (for example in US agriculture, like the Average Crop Revenue Election), due to its use in Olympic events, such as the ISU Judging System in figure skating, to make the score robust to a single outlier judge.

The GOE values from the nine judges are averaged using the "trimmed mean" procedure, where the highest and lowest values are discarded and an average is calculated from the remaining seven values.

### Winsorized mean

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.

A winsorized mean is a winsorized statistical measure of central tendency, much like the mean and median, and even more similar to the truncated mean.

### Average Crop Revenue Election

This is also known as the (for example in US agriculture, like the Average Crop Revenue Election), due to its use in Olympic events, such as the ISU Judging System in figure skating, to make the score robust to a single outlier judge.

The ACRE alternative provides eligible producers a state-level revenue guarantee, based on the 5-year state Olympic average yield and the 2-year national average price.

### Robust statistics

**robustbreakdown pointrobustness**

In this regard it is referred to as a robust estimator.

Panels (c) and (d) of the plot show the bootstrap distribution of the mean (c) and the 10% trimmed mean (d).

### Cauchy distribution

**LorentzianCauchyLorentzian profile**

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.

Other, more precise and robust methods have been developed For example, the truncated mean of the middle 24% of the sample order statistics produces an estimate for x_0 that is more efficient than using either the sample median or the full sample mean.

### Libor

**LIBOR tenorLondon Interbank Offered RateBBA LIBOR**

The Libor benchmark interest rate is 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 4/18 ≈ 22%).

The BBA throws out the highest 4 and lowest 4 responses, and averages the remaining middle 10, yielding a 22% trimmed mean.

### Winsorizing

**Winsorised estimatorswinsorized**

This example can be compared with the one using the Winsorising procedure.

Thus a winsorized mean is not the same as a truncated mean.

### Trimean

Trimean

Truncated mean

### Statistics

**statisticalstatistical analysisstatistician**

A truncated mean or trimmed mean is a statistical measure of central tendency, much like the mean and median.

### Median

**averagesample medianmedian-unbiased estimator**

### Probability distribution

**distributioncontinuous probability distributiondiscrete probability distribution**

It involves the calculation of the mean after discarding given parts of a probability distribution or sample at the high and low end, and typically discarding an equal amount of both.

### Sampling (statistics)

**samplingrandom samplesample**

It involves the calculation of the mean after discarding given parts of a probability distribution or sample at the high and low end, and typically discarding an equal amount of both.

### Efficiency (statistics)

**efficientefficiencyinefficient**

As with other trimmed estimators, the main advantage of the trimmed mean is robustness and higher 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).

### Central Europe

**CentralCentral Europeanmiddle Europe**

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.

### Figure skating

**figure skaterfigure skatersskating**

This is also known as the (for example in US agriculture, like the Average Crop Revenue Election), due to its use in Olympic events, such as the ISU Judging System in figure skating, to make the score robust to a single outlier judge.

### Weighted arithmetic mean

**averageaverage ratingweighted average**

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.

### Outlier

**outliersconservative estimateirregularities**

The truncated mean is a useful estimator because it is less sensitive to outliers than the mean but will still give a reasonable estimate of central tendency or mean for many statistical models.

### Location parameter

**locationlocation modelshift parameter**

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.

### Normal distribution

**normally distributednormalGaussian**

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.

### Order statistic

**order statisticsorderedth-smallest of items**

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.

### Maximum likelihood estimation

**maximum likelihoodmaximum likelihood estimatormaximum likelihood estimate**

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.