# 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 winsorizing -- 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.

The winsorized mean is a useful estimator because by retaining the outliers without taking them too literally, it is less sensitive to observations at the extremes than the straightforward mean, and will still generate 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 x(1), the smallest, to x(10) the largest; order statistic notation) 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 x(2) and x(9): the extras substitute for the original values x(1) and x(10) which have been discarded and replaced.
This is equivalent to a weighted average of 0.1 times the 5th percentile (x(2)), 0.8 times the 10% trimmed mean, and 0.1 times the 95th percentile (x(9)).

## Notes

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