In probability theory, the total variation distance is a statistical distance between probability distributions, and is sometimes called the statistical distance, statistical difference or variational distance.

Consider a measurable space ${\displaystyle (\Omega ,{\mathcal {F}})}$ and probability measures ${\displaystyle P}$ and ${\displaystyle Q}$ defined on ${\displaystyle (\Omega ,{\mathcal {F}})}$. The total variation distance between ${\displaystyle P}$ and ${\displaystyle Q}$ is defined as

This is the largest absolute difference between the probabilities that the two probability distributions assign to the same event.

The total variation distance is an f-divergence and an integral probability metric.

The total variation distance is related to the Kullback–Leibler divergence by Pinsker’s inequality:

One also has the following inequality, due to Bretagnolle and Huber (see also ), which has the advantage of providing a non-vacuous bound even when ${\displaystyle \textstyle D_{\mathrm {KL} }(P\parallel Q)>2\colon }$

The total variation distance is half of the L¹ distance between the probability functions: on discrete domains, this is the distance between the probability mass functions

and when the distributions have standard probability density functions p and q,