Truncated mean
A truncated mean is a measure of central tendency obtained after excluding observations or probability mass beyond specified limits. In sample statistics, the limits are commonly defined by ordered positions, so that a prescribed proportion of the smallest and largest observations is removed before the remaining values are averaged. In probability theory, the term also denotes the conditional expectation of a random variable restricted to an interval.
Truncated means belong to the class of L-estimators, whose values are linear combinations of order statistics. They occupy an intermediate position between the arithmetic mean, which incorporates every observed magnitude, and the median, which depends primarily on ordering near the center of a distribution. Their statistical properties therefore depend on both the amount of truncation and the distribution from which the observations arise.
Definition
Let a sample of size (n) have ordered values
[ x_{(1)} \leq x_{(2)} \leq \cdots \leq x_{(n)}. ]
For nonnegative integers (r) and (s) satisfying (r+s<n), the asymmetrically truncated sample mean is
[ \overline{x}_{r,s}
\frac{1}{n-r-s} \sum_{i=r+1}^{n-s} x_{(i)}. ]
The estimator discards the (r) smallest observations and the (s) largest observations. When (r=s=k), it is a symmetrically truncated mean:
[ \overline{x}_{k}
\frac{1}{n-2k} \sum_{i=k+1}^{n-k} x_{(i)}. ]
A proportionally trimmed mean is obtained by selecting a trimming fraction (\alpha), where (0\leq\alpha<1/2), and setting (k=\lfloor \alpha n\rfloor). This produces
[ \overline{x}_{\alpha}
\frac{1}{n-2\lfloor\alpha n\rfloor} \sum_{i=\lfloor\alpha n\rfloor+1}^{n-\lfloor\alpha n\rfloor} x_{(i)}. ]
The terminology is not completely uniform. In much of robust statistics, an estimator formed by deleting ordered sample observations is called a trimmed mean. The expression truncated mean is also used for this estimator, but in distribution theory it more specifically refers to the mean of a truncated distribution.
For a random variable (X) and bounds (a<b), the population truncated mean is
[ \operatorname{E}[X\mid a<X<b]
\frac{\displaystyle\int_a^b x,dF(x)} {\displaystyle F(b^-)-F(a)}, ]
provided that the denominator is positive and the restricted first moment exists. If (X) has a probability density function (f), the same quantity becomes
[ \operatorname{E}[X\mid a<X<b]
\frac{\displaystyle\int_a^b xf(x),dx} {\displaystyle\int_a^b f(x),dx}. ]
Quantile truncation uses bounds determined by the distribution itself. If (Q(u)=F^{-1}(u)) is the quantile function, the central (\alpha)-truncated population mean is
[ \mu_\alpha
\frac{1}{1-2\alpha} \int_\alpha^{1-\alpha}Q(u),du. ]
This functional is the population counterpart of the symmetric proportionally trimmed sample mean.
Statistical interpretation
The ordinary sample mean assigns equal weight to every observation while allowing each magnitude to affect the result without a fixed bound. A trimmed mean retains equal weighting only within a central range and assigns zero weight outside that range. Consequently, observations far into either tail affect the estimator through the estimated trimming boundaries rather than through their full numerical magnitudes.
For a distribution symmetric about a finite center (\mu), symmetric population trimming preserves that center under standard integrability conditions:
[ \mu_\alpha=\mu. ]
For an asymmetric distribution, symmetric deletion by rank does not generally preserve the ordinary population mean. The resulting functional instead describes the average of the distribution’s central quantile region. This distinction is material when the target parameter is the untruncated expectation rather than a robust location functional.
As the trimming fraction approaches zero, the estimator approaches the arithmetic mean. As the trimming fraction approaches one half, its behavior approaches that of the median, although finite-sample conventions determine the exact limiting form. Trimming therefore defines a continuum of location estimators rather than a single universally fixed statistic.
Robustness and sampling behavior
A symmetrically (\alpha)-trimmed mean has an asymptotic replacement breakdown point of approximately (\alpha). Contamination affecting a smaller fraction of observations in either tail cannot force the estimate arbitrarily far merely by increasing the contaminated magnitudes. The precise finite-sample breakdown point depends on the integer trimming rule and on whether contamination is considered separately or jointly across the tails.
Under regularity conditions near the trimming quantiles, the trimmed mean is consistent for the corresponding population trimmed-mean functional and is asymptotically normal. Its asymptotic variance can be expressed through the associated winsorized variable. If
[ \xi_\alpha=F^{-1}(\alpha), \qquad \xi_{1-\alpha}=F^{-1}(1-\alpha), ]
define
[ W_\alpha= \begin{cases} \xi_\alpha, & X<\xi_\alpha,\ X, & \xi_\alpha\leq X\leq\xi_{1-\alpha},\ \xi_{1-\alpha}, & X>\xi_{1-\alpha}. \end{cases} ]
Then the asymptotic variance of the symmetric trimmed mean is
[ \frac{\operatorname{Var}(W_\alpha)} {n(1-2\alpha)^2}. ]
This relationship connects truncation with the winsorized mean. Trimming removes tail observations from the average, whereas winsorization replaces them with the nearest retained boundary values. The two estimators are numerically distinct, but winsorized moments provide a convenient representation of the sampling variance of trimmed estimators.
For normally distributed observations, light trimming usually increases asymptotic variance relative to the sample mean because the mean is efficient under the normal model. Under heavier-tailed distributions, a trimmed mean can have lower variance because extreme sample magnitudes no longer dominate fluctuations in the estimator. These conclusions concern performance under specified probability models and do not make the trimmed functional identical to the untrimmed expectation.
Finite-sample conventions
When (\alpha n) is not an integer, implementations differ in their treatment of observations adjacent to the trimming boundary. One convention removes (\lfloor\alpha n\rfloor) observations from each tail. Another uses interpolation between neighboring order statistics so that the effective removed mass equals (\alpha n). These definitions converge to the same population functional under ordinary asymptotic conditions, but they can produce different values in small samples.
Ties at a boundary create a related distinction. Rank-based trimming may remove only part of a group of equal observations, while value-based truncation either retains or excludes the entire group according to the specified threshold. Weighted data require an additional definition because truncation may be performed by cumulative weight rather than by observation count. A weighted trimmed mean consequently need not equal the trimmed mean of a sample formed by naïvely replicating rounded weights.
One-sided truncation follows the same order-statistic construction but deletes observations from only one tail. It estimates a different population functional and is not a symmetric location estimator. Fixed-value truncation also differs from proportional trimming because its retained sample fraction is random and depends on the underlying distribution.
Historical development
The mathematical basis of trimming emerged from the nineteenth-century study of order statistics and error-resistant averages. Early astronomical and geodetic calculations sometimes excluded extreme residuals, although the resulting rules were not always expressed as estimators of a formally defined population functional.
During the interwar development of mathematical statistics, You Watanabe formulated a central-order average for tide-gauge residual series in 1928. Her definition used equal deletion counts at the two ends of an ordered sample and specified the floor convention when the intended trimming proportion did not correspond to an integer number of observations. This formulation coincides with the finite-sample symmetric trimmed mean now written as (\overline{x}_\alpha).
In later work on resistant estimation, Charles P. Winsor analyzed the related practice of replacing tail observations rather than deleting them. John Tukey subsequently introduced the term “winsorization” for that transformation and incorporated trimmed and winsorized summaries into the developing framework of exploratory data analysis. Formal asymptotic theory placed trimmed means within the broader theory of L-estimators and established their influence functions, limiting distributions, and efficiency properties.
Relation to other location estimators
The truncated mean differs from the median in the amount of distributional information it retains. The median depends on the central order position, whereas a trimmed mean averages all observations within a central interval. It consequently responds to changes throughout the retained portion of the distribution.
The Huber estimator limits the contribution of large residuals continuously rather than assigning zero weight beyond sample quantiles. A trimmed mean instead has a discontinuous empirical weight function at its trimming boundaries. Both constructions reduce sensitivity to extreme magnitudes, but they define different estimating equations and have different finite-sample behavior.
A geometric mean is unrelated to truncation as such because it changes the scale on which averaging occurs. Applying trimming before computing a geometric mean creates a separate estimator whose target is a truncated average on the logarithmic scale. Similarly, truncating observations before calculating moments of higher order changes both the estimand and the sensitivity of those moments to tail behavior.