  We introduce $k$-variance, a generalization of variance built on the
machinery of random bipartite matchings. $K$-variance measures the expected
cost of matching two sets of $k$ samples from a distribution to each other,
capturing local rather than global information about a measure as $k$
increases; it is easily approximated stochastically using sampling and linear
programming. In addition to defining $k$-variance and proving its basic
properties, we provide in-depth analysis of this quantity in several key cases,
including one-dimensional measures, clustered measures, and measures
concentrated on low-dimensional subsets of $\mathbb R^n$. We conclude with
experiments and open problems motivated by this new way to summarize
distributional shape.
