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The halfspace depth of $x\in {\mathbb{R}}^d$ for a probability measure $P$ is the least mass of any closed half-space containing $x$:For an empirical distribution, it is the smallest fraction of observations in such a closed half-space. The boundary is included, so observations exactly on it count.

In one dimension,For observations $0,1,2$, the depth at $1$ is $2/3$, and the depth at either endpoint is $1/3$. Atoms can therefore give depth greater than $1/2$. An invertible affine map preserves halfspace depth, because it carries the family of closed half-spaces bijectively to itself and preserves their associated probabilities. A Tukey median maximizes this depth; the maximizing point need not be unique.

A Tukey median maximizes halfspace depth. For a probability measure $P$ on ${\mathbb{R}}^d$, the depth of a point $x$ isFor a sample, $P$ is replaced by its empirical distribution, so $P(H)$ is the fraction of observations in the closed half-space $H$. Every direction is considered; choosing a single coordinate direction would give a different procedure.

In one dimension, the depth is the smaller of the masses in the closed rays to the left and right of $x$. For the sample $(0,1,2,3,100)$, point $2$ has depth $3/5$, while $100$ has depth $1/5$. The maximizing point is the sample median. In higher dimensions this gives a geometric extension of a central location that depends on how many observations lie on each side, rather than on their squared distances. Maximizers need not be unique, so a single-valued estimator needs a selection rule.