set-valued interpretation as stated in the Definition and not further elaborate on disambiguation strategies. We will still colloquially speak of “the” median even in this case.

**An extrema-stripping procedure to determine the median.**

An (albeit inefficient) way to determine the median of a multiset can be based directly on this rank-order definition:

\[def-med-extr\] Let a finite multiset $\mathcal{X}$ of real numbers be given. Delete its smallest and greatest element, then again the remaining smallest and greatest element etc., until an empty set is obtained. The one or two points that were deleted in the very last step of this sequence are medians, with the entire median set given as their convex hull.

**The median as $1/2$-quantile.**

The median is also the $1/2$-quantile of the given set, i.e., the smallest number $\mu$ for which at least half of the given data are less or equal $\mu$. In the following definition, $\#S$ denotes the cardinality of a finite multiset $S$.

Let $\mathcal{X}$ be a finite multiset of real numbers. The median of $\mathcal{X}$ is defined as $$\begin{aligned}
\operatorname{med}(\mathcal{X}) &:=
\min \left\{\mu\in\mathbb{R}~|~
\#(\mathcal{X}\cap(-\infty,\mu])\ge\tfrac12\#\mathcal{X}\right\} \;.
\label{med-lhalf}\end{aligned}$$

Note that the ambiguity in the case of even $\#\mathcal{X}=N$ is resolved here by taking the minimum. A 