Sufficient Statistic:

Suppose X is a random vector with probability distribution (or density) P(X | V), where V is a vector of parameters, and X_o is a realization of X. A statistic T(X) is called a sufficient statistic if the conditional probability (density) P(X | T(X_o); V) does not depend upon V for any possible X_o.

In other words, the observed value T(X_o) of a sufficient statistic T bears all the information (about the vector V of parameters) the data X_o contain.

A trivial example of a sufficient statistic is T(X) = X, but such a statistic is useless. Practically interesting cases of sufficient statistics are those when T has (much) smaller dimension than the vector X itself. Such statistics allow you to reduce all the observed data values to a smaller set of values preserving all the information about the parameter V.

See also Complete statistic

If the symbols do not display properly, try
the graphic version of this page