# Total variation

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

[itex]\sup_P \sum_i | f(x_{i+1})-f(x_i) | [itex]

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

[itex]\sup\left\{\,\left|P(A)-Q(A)\right| : A\in F\,\right\}.[itex]

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

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