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

Consider a random variable X whose probability distribution belongs to a parametric model P_θ parametrized by θ.

Say T is a statistic; that is, the composition of a measurable function with a random sample X₁,...,X_n.

The statistic T is said to be complete for the distribution of X if, for every measurable function g,

The statistic T is said to be boundedly complete for the distribution of X if this implication holds for every measurable function g that is also bounded.

The Bernoulli model admits a complete statistic. Let X be a random sample of size n such that each X_i has the same Bernoulli distribution with parameter p. Let T be the number of 1s observed in the sample, i.e. ${\displaystyle \textstyle T=\sum _{i=1}^{n}X_{i}}$. T is a statistic of X which has a binomial distribution with parameters (n,p). If the parameter space for p is (0,1), then T is a complete statistic. To see this, note that

Observe also that neither p nor 1 − p can be 0. Hence ${\displaystyle E_{p}(g(T))=0}$ if and only if:

On denoting p/(1 − p) by r, one gets: