In statistics, a confidence region is a multi-dimensional generalization of a confidence interval. For a bivariate normal distribution, it is an ellipse, also known as the error ellipse. More generally, it is a set of points in an n-dimensional space, often represented as a hyperellipsoid around a point which is an estimated solution to a problem, although other shapes can occur.

The confidence region is calculated in such a way that if a set of measurements were repeated many times and a confidence region calculated in the same way on each set of measurements, then a certain percentage of the time (e.g. 95%) the confidence region would include the point representing the "true" values of the set of variables being estimated. However, unless certain assumptions about prior probabilities are made, it does not mean, when one confidence region has been calculated, that there is a 95% probability that the "true" values lie inside the region, since we do not assume any particular probability distribution of the "true" values and we may or may not have other information about where they are likely to lie.

Suppose we have found a solution ${\displaystyle {\boldsymbol {\beta }}}$ to the following overdetermined problem:

where Y is an n-dimensional column vector containing observed values of the dependent variable, X is an n-by-p matrix of observed values of independent variables (which can represent a physical model) which is assumed to be known exactly, ${\displaystyle {\boldsymbol {\beta }}}$ is a column vector containing the p parameters which are to be estimated, and ${\displaystyle {\boldsymbol {\varepsilon }}}$ is an n-dimensional column vector of errors which are assumed to be independently distributed with normal distributions with zero mean and each having the same unknown variance ${\displaystyle \sigma ^{2}}$.

A joint 100(1 − α) % confidence region for the elements of ${\displaystyle {\boldsymbol {\beta }}}$ is represented by the set of values of the vector b which satisfy the following inequality:

where the variable b represents any point in the confidence region, p is the number of parameters, i.e. number of elements of the vector ${\displaystyle {\boldsymbol {\beta }},}$ ${\displaystyle {\boldsymbol {\hat {\beta }}}}$ is the vector of estimated parameters, and s² is the reduced chi-squared, an unbiased estimate of ${\displaystyle \sigma ^{2}}$ equal to

Further, F is the quantile function of the F-distribution, with p and ${\displaystyle \nu =n-p}$ degrees of freedom, ${\displaystyle \alpha }$ is the statistical significance level, and the symbol ${\displaystyle X^{\operatorname {T} }}$ means the transpose of ${\displaystyle X}$.

The expression can be rewritten as: