= Trimmed mean
{wiki}
For $0\leq\alpha<1/2$, an $\alpha$-<trimmed mean> sorts $n$ observations, removes the smallest and largest $k=\lfloor\alpha n\rfloor$, and averages the remaining $n-2k$ values. It limits the effect of a small number of extreme observations while still averaging a central portion of the data.
For $(1,2,3,4,100)$ and $\alpha=1/5$, removing $1$ and $100$ gives <trimmed mean> $(2+3+4)/3=3$, compared with <sample mean> $22$. Replacing $100$ by a larger value leaves this <trimmed mean> unchanged. Replacing enough observations can still move the retained central values arbitrarily far.
For a <probability distribution> with <quantile function> $F^{-1}$, the population <statistical functional> is
$$
T(F)=\frac{1}{1-2\alpha}\int_\alpha^{1-\alpha}F^{-1}(s)\,ds.
$$
The normalizing denominator is the retained fraction of <probability>. At a regular symmetric reference <probability distribution>, the <influence function of a trimmed mean> is a clipped version of the observation, scaled by this fraction. Centering and symmetry assumptions matter when using that simpler expression.
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