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, ..., xn } 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.