# Winsorized mean

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Statistical measure of central tendency

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A **winsorized mean** is a [winsorized](/source/Winsorising) [statistical](/source/Statistical) [measure of central tendency](/source/Measure_of_central_tendency), much like the [mean](/source/Mean) and [median](/source/Median), and even more similar to the [truncated mean](/source/Truncated_mean). It involves the calculation of the mean after [winsorizing](/source/Winsorizing) — replacing given parts of a [probability distribution](/source/Probability_distribution) or [sample](/source/Sampling_(statistics)) 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](/source/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](/source/Outlier) 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](/source/Robust_estimator).

## Drawbacks

The winsorized mean uses more information from the distribution or sample than the [median](/source/Median). However, unless the underlying distribution is [symmetric](/source/Symmetric_probability_distribution), the winsorized mean of a sample is unlikely to produce an [unbiased estimator](/source/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](/source/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](/source/Truncated_mean), and 0.1 times the 95th percentile (*x*(9)).

## Notes

1. **[^](#cite_ref-1)** [Dodge, Y](/source/Yadolah_Dodge) (2003) *The Oxford Dictionary of Statistical Terms*, OUP. [ISBN](/source/ISBN_(identifier)) [0-19-920613-9](https://en.wikipedia.org/wiki/Special:BookSources/0-19-920613-9) (entry for "winsorized estimation")

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## References

- Wilcox, R.R.; Keselman, H.J. (2003). "Modern robust data analysis methods: Measures of central tendency". *Psychological Methods*. **8** (3): 254–274. [doi](/source/Doi_(identifier)):[10.1037/1082-989X.8.3.254](https://doi.org/10.1037%2F1082-989X.8.3.254). [PMID](/source/PMID_(identifier)) [14596490](https://pubmed.ncbi.nlm.nih.gov/14596490).

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