In statistics, sufficiency is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. A sufficient statistic contains all of the information that the dataset provides about the model parameters. It is closely related to the concepts of an ancillary statistic which contains no information about the model parameters, and of a complete statistic which only contains information about the parameters and no ancillary information.

A related concept is that of linear sufficiency, which is weaker than sufficiency but can be applied in some cases where there is no sufficient statistic, although it is restricted to linear estimators. The Kolmogorov structure function deals with individual finite data; the related notion there is the algorithmic sufficient statistic.

The concept is due to Sir Ronald Fisher in 1920. Stephen Stigler noted in 1973 that the concept of sufficiency had fallen out of favor in descriptive statistics because of the strong dependence on an assumption of the distributional form (see Pitman–Koopman–Darmois theorem below), but remained very important in theoretical work.

Roughly, given a set ${\displaystyle \mathbf {X} }$ of independent identically distributed data conditioned on an unknown parameter ${\displaystyle \theta }$, a sufficient statistic is a function ${\displaystyle T(\mathbf {X} )}$ whose value contains all the information needed to compute any estimate of the parameter (e.g. a maximum likelihood estimate). Due to the factorization theorem (see below), for a sufficient statistic ${\displaystyle T(\mathbf {X} )}$, the probability density can be written as ${\displaystyle f_{\mathbf {X} }(x;\theta )=h(x)\,g(\theta ,T(x))}$. From this factorization, it can easily be seen that the maximum likelihood estimate of ${\displaystyle \theta }$ will interact with ${\displaystyle \mathbf {X} }$ only through ${\displaystyle T(\mathbf {X} )}$. Typically, the sufficient statistic is a simple function of the data, e.g. the sum of all the data points.

More generally, the "unknown parameter" may represent a vector of unknown quantities or may represent everything about the model that is unknown or not fully specified. In such a case, the sufficient statistic may be a set of functions, called a jointly sufficient statistic. Typically, there are as many functions as there are parameters. For example, for a Gaussian distribution with unknown mean and variance, the jointly sufficient statistic, from which maximum likelihood estimates of both parameters can be estimated, consists of two functions, the sum of all data points and the sum of all squared data points (or equivalently, the sample mean and sample variance).

In other words, the joint probability distribution of the data is conditionally independent of the parameter given the value of the sufficient statistic for the parameter. Both the statistic and the underlying parameter can be vectors.

A statistic t = T(X) is sufficient for underlying parameter θ precisely if the conditional probability distribution of the data X, given the statistic t = T(X), does not depend on the parameter θ.

Alternatively, one can say the statistic T(X) is sufficient for θ if, for all prior distributions on θ, the mutual information between θ and T(X) equals the mutual information between θ and X. In other words, the data processing inequality becomes an equality: