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Winsorized mean

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Statistical measure of central tendency
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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, 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.

Advantages

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

x ( 2 ) + x ( 2 ) ⏞ + x ( 3 ) + x ( 4 ) + x ( 5 ) + x ( 6 ) + x ( 7 ) + x ( 8 ) + x ( 9 ) + x ( 9 ) ⏞ 10 . {\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")
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