Tukey depth

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In statistics and computational geometry, the Tukey depth ^[1] is a measure of the depth of a point in a fixed set of points. The concept is named after its inventor, John Tukey. Given a set of n points ${\displaystyle {\mathcal {X}}_{n}=\{X_{1},\dots ,X_{n}\}}$ in d-dimensional space, Tukey's depth of a point x is the smallest fraction (or number) of points in any closed halfspace that contains x.

Tukey's depth measures how extreme a point is with respect to a point cloud. It is used to define the bagplot, a bivariate generalization of the boxplot.

For example, for any extreme point of the convex hull there is always a (closed) halfspace that contains only that point, and hence its Tukey depth as a fraction is 1/n.