This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (February 2012) (Learn how and when to remove this template message)

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: "Statistical distance" – news · newspapers · books · scholar · JSTOR (December 2020) (Learn how and when to remove this template message)

In statistics, probability theory, and information theory, a **statistical distance** quantifies the distance between two statistical objects, which can be two random variables, or two probability distributions or samples, or the distance can be between an individual sample point and a population or a wider sample of points.

A distance between populations can be interpreted as measuring the distance between two probability distributions and hence they are essentially measures of distances between probability measures. Where statistical distance measures relate to the differences between random variables, these may have statistical dependence,^{[1]} and hence these distances are not directly related to measures of distances between probability measures. Again, a measure of distance between random variables may relate to the extent of dependence between them, rather than to their individual values.

Statistical distance measures are not typically metrics, and they need not be symmetric. Some types of distance measures, which generalize *squared* distance, are referred to as (statistical) *divergences*.

Many terms are used to refer to various notions of distance; these are often confusingly similar, and may be used inconsistently between authors and over time, either loosely or with precise technical meaning. In addition to "distance", similar terms include deviance, deviation, discrepancy, discrimination, and divergence, as well as others such as contrast function and metric. Terms from information theory include cross entropy, relative entropy, discrimination information, and information gain.

A **metric** on a set *X* is a function (called the *distance function* or simply **distance**) *d* : *X* × *X* → **R**^{+}
(where **R**^{+} is the set of non-negative real numbers). For all *x*, *y*, *z* in *X*, this function is required to satisfy the following conditions:

*d*(*x*,*y*) ≥ 0 (*non-negativity*)*d*(*x*,*y*) = 0 if and only if*x*=*y*(*identity of indiscernibles*. Note that condition 1 and 2 together produce*positive definiteness*)*d*(*x*,*y*) =*d*(*y*,*x*) (*symmetry*)*d*(*x*,*z*) ≤*d*(*x*,*y*) +*d*(*y*,*z*) (*subadditivity*/*triangle inequality*).

Many statistical distances are not metrics, because they lack one or more properties of proper metrics. For example, pseudometrics violate property (2), identity of indiscernibles; quasimetrics violate property (3), symmetry; and semimetrics violate property (4), the triangle inequality. Statistical distances that satisfy (1) and (2) are referred to as divergences.

- Total variation distance (sometimes just called "the" statistical distance)
- Hellinger distance
- Lévy–Prokhorov metric
- Wasserstein metric: also known as the Kantorovich metric, or earth mover's distance
- Mahalanobis distance

- Kullback–Leibler divergence
- Rényi's divergence
- Jensen–Shannon divergence
- Bhattacharyya distance (despite its name it is not a distance, as it violates the triangle inequality)
- f-divergence: generalizes several distances and divergences
- Discriminability index, specifically the Bayes discriminability index is a positive-definite symmetric measure of the overlap of two distributions.