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
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.