In mathematics, the total variation of a real-valued function f on the bounded interval [a, b] is

the supremum running over all partitions P = { x₁, ..., x_n } of the interval [a, b]. In effect, the total variation is the vertical component of the arc-length of the graph of f. The function f is said to be of bounded variation precisely if the total variation of f is finite.

Total variation distance in probability theory

In probability theory, the total variation distance between two probability measures P and Q on a sigma-algebra F is

Informally, this is the largest possible difference between the probabilities that the two probability distributions can assign to the same event.