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.

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

${\frac {\overbrace {x_{(2)}+x_{(2)}} +x_{(3)}+x_{(4)}+x_{(5)}+x_{(6)}+x_{(7)}+x_{(8)}+\overbrace {x_{(9)}+x_{(9)}} }{10}}.\,$

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

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

References

Wilcox, R.R.; Keselman, H.J. (2003). "Modern robust data analysis methods: Measures of central tendency". Psychological Methods. 8 (3): 254–274. doi:10.1037/1082-989X.8.3.254. PMID 14596490.

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

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