Distributionwide

Distributionwide is a coined term used in statistics and data science to describe measures that characterize the breadth or reach of a probability distribution across its domain. Unlike standard descriptors such as mean, variance, or interquartile range, distributionwide is intended to capture how widely a distribution extends and how much of the domain it effectively covers. The term is not widely adopted in formal mathematics, but it appears in some software documentation and methodological discussions as an umbrella for breadth-related metrics.

There are several common interpretations of distributionwide:

- Coverage at a density threshold: the proportion of the domain where the distribution assigns non-negligible mass. For a density f on a domain D, and a threshold t > 0, the distributionwide coverage is C(t) = ∫_{D_t} f(x) dx, where D_t = {x in D : f(x) > t}.

- Mass-based width: the width of the smallest interval containing a specified probability mass p (0 < p < 1). This distributionwide width w_p is defined as the length of the shortest interval I such that ∫_I f(x) dx = p. For symmetric unimodal f, I is often centered at the mode.

- Tail-breadth: the extent of the distribution into the tails, often characterized by quantile-based ranges such as the width of the interval between the α- and β-quantiles.

See also: probability distribution, spread, credible interval, interquantile range, tail behavior.