Trimmed Means
RAND R. WILCOX
Volume 4, pp. 2066–2067
in
Encyclopedia of Statistics in Behavioral Science
ISBN-13: 978-0-470-86080-9
ISBN-10: 0-470-86080-4
Editors
Brian S. Everitt & David C. Howell
 John Wiley & Sons, Ltd, Chichester, 2005
Trimmed Means
A trimmed mean is computed by removing a proportion of the largest and smallest observations and
averaging the values that remain. Included as a special case are the usual sample mean (no trimming)
and the median. As a simple illustration, consider the
11 values 6, 2, 10, 14, 9, 8, 22, 15, 13, 82, and 11. To
compute a 10% trimmed mean, multiply the sample
size by 0.1 and round the result down to the nearest integer. In the example, this yields g = l. Then,
remove the g smallest values, as well as the g largest,
and average the values that remain. In the illustration,
this yields 12. In contrast, the sample mean is 17.45.
To compute a 20% trimmed mean, proceed as before;
only, now g is 0.2 times the sample sizes rounded
down to the nearest integer. Some researchers have
considered a more general type of trimmed mean [2],
but the description just given is the one most commonly used.
Why trim observations, and if one does trim, why
not use the median? Consider the goal of achieving a relatively low standard error. Under normality,
the optimal amount of trimming is zero. That is, use
the untrimmed mean. But under very small departures from normality, the mean is no longer optimal
and can perform rather poorly (e.g., [1], [3], [4],
[8]). As we move toward situations in which outliers are common, the median will have a smaller
standard error than the mean, but under normality,
the median’s standard error is relatively high. So, the
idea behind trimmed means is to use a compromise
amount of trimming with the goal of achieving a relatively small standard error under both normal and
nonnormal distributions. (For an alternative approach,
see M Estimators of Location). Trimming observations with the goal of obtaining a more accurate
estimator might seem counterintuitive, but this result
has been known for over two centuries. For a nontechnical explanation, see [6].
Another motivation for trimming arises when
sampling from a skewed distribution and testing some
hypothesis. Skewness adversely affects control over
the probability of a type I error and power when
using methods based on means (e.g., [5], [7]). As
the amount of trimming increases, these problems are
reduced, but if too much trimming is used, power can
be low. So, in particular, using a median to deal with
a skewed distribution might make it less likely to
reject when in fact the null hypothesis is false.
Testing hypotheses on the basis of trimmed means
is possible, but theoretically sound methods are not
immediately obvious. These issues are easily addressed, however, and easy-to-use software is available
as well, some of which is free [7, 8].
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
Hampel, F.R., Ronchetti, E.M., Rousseeuw, P.J., & Stahel, W.A. (1986). Robust Statistics, Wiley, New York.
Hogg, R.V. (1974). Adaptive robust procedures: a partial
review and some suggestions for future applications and
theory, Journal of the American Statistical Association 69,
909–922.
Huber, P.J. (1981). Robust Statistics, Wiley, New York.
Staudte, R.G. & Sheather, S.J. (1990). Robust Estimation
and Testing, Wiley, New York.
Westfall, P.H. & Young, S.S. (1993). Resampling Based
Multiple Testing, Wiley, New York.
Wilcox, R.R. (2001). Fundamentals of Modern Statistical
Methods: Substantially Increasing Power and Accuracy,
Springer, New York.
Wilcox, R.R. (2003). Applying Conventional Statistical
Techniques, Academic Press, San Diego.
Wilcox, R.R. (2004). (in press). Introduction to Robust
Estimation and Hypothesis Testing, 2nd Edition, Academic Press, San Diego.
Further Reading
Tukey, J.W. (1960). A survey of sampling from contaminated
normal distributions, in Contributions to Probability and
Statistics, I. Olkin, S. Ghurye, W. Hoeffding, W. Madow
& H. Mann, eds, Stanford University Press, Stanford.
RAND R. WILCOX