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The Winsorized Mean
Winsorization can be used to estimate the mean and standard deviation of a distribution although the
data set has a few missing or unreliable values at either or both ends of the distribution.

Example 15.3

Again using the data in Example 15.1, replace the four censored values by the next largest value,
which is 6.1. Replace the four largest values by the next smallest value, which is 8.9. The replaced
values are shown in italics. This gives the Winsorized sample (n = 27) below:

6.1 6.1 6.1 6.1 6.1 6.3 6.5 6.7 6.9
7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 8.0
8.1 8.3 8.5 8.7 8.9 8.9 8.9 8.9 8.9

Compute the sample mean (y) and standard deviation (s) of the resulting Winsorized sample
in the usual way:

y = 7.53 µg/L s = 1.022 µg/L

The Winsorized mean (y w ) is the mean of the Winsorized sample:

y w = y = 7.53 µg/L

The Winsorized standard deviation (s w ) is an approximately unbiased estimator of s and is
computed from the standard deviation of the Winsorized sample as:

(
sn 1) 1.02 27 1)
(
–
–
s w = ------------------- = ------------------------------ = 1.48 µg/L
ν 1 19 1
–
–
where n is the total number of observations and ν is the number of observations not replaced
during Winsorization. ν = 27 – 4 – 4 = 19 because the four “less-than” values and the four largest
values have been replaced.
If the data are from a normal distribution, the upper and lower limits of a two-sided 100(1 α)%
confidence interval of the mean η are:

y w ± t ν−1,α/2 -------
s w
n

where t ν−1,α/2 cuts 100(α/2)% off each tail of the t distribution with ν – 1 degrees of freedom. Note that
the degrees of freedom are ν – 1 instead of the usual n – 1, and s w replaces s.
The trimmed mean and the Winsorized mean are best used on symmetric data sets that have either
missing (censored) or unreliable data on the tail of the distribution. If the distribution is symmetric, they
give unbiased estimates of the true mean.
© 2002 By CRC Press LLC