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# Winsorized mean

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. It involves the calculation of the mean after replacing given parts of a probability distribution or sample at the high and low end with the most extreme remaining values,[1] typically doing so for an equal amount of both extremes; often 10 to 25 percent of the ends are replaced. The winsorized mean can equivalently be expressed as a weighted average of the truncated mean and the quantiles at which it is limited, which corresponds to replacing parts with the corresponding quantiles.

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• ✪ Dealing with an outlier - Winsorize
• ✪ Trimmed Means
• ✪ 3-3 Trimmed Mean Video

## Contents

The winsorized 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 almost all statistical models. In this regard it is referred to as a robust estimator.

## Drawbacks

The winsorized mean uses more information from the distribution or sample than the median. However, unless the underlying distribution is symmetric, the winsorized mean of a sample is unlikely to produce an unbiased estimator for either the mean or the median.

## Example

• For a sample of 10 numbers (from x1, the smallest, to x10 the largest) the 10% winsorized mean is
${\displaystyle {\frac {\overbrace {x_{2}+x_{2}} +x_{3}+x_{4}+x_{5}+x_{6}+x_{7}+x_{8}+\overbrace {x_{9}+x_{9}} }{10}}.\,}$
The key is in the repetition of x2 and x9: the extras substitute for the original values x1 and x10 which have been discarded and replaced.
This is equivalent to a weighted average of 0.1 times the 5th percentile (x2), 0.8 times the 10% trimmed mean, and 0.1 times the 95th percentile (x9).

## Notes

1. ^ Dodge, Y (2003) The Oxford Dictionary of Statistical Terms, OUP. ISBN 0-19-920613-9 (entry for "winsorized estimation")