Winsorized mean

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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,^[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.

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.

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₎).

This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. Please help improve this article by introducing more precise citations. (March 2012) (Learn how and when to remove this message)

• 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.