Goodness of fit

Goodness of fit describes how closely a fitted statistical model agrees with the observed data in aspects relevant to the model assumptions.

The reduced chi-squared divides a weighted squared-residual statistic by its degrees of freedom. For an identifiable Gaussian linear model with $N$ independent observations, known correct errors and $p$ fitted coefficients, $\nu =N-p$ and its expectation is one. An observed value close to one is compatible with the model but does not prove it. Unknown systematic errors, dependent residuals or fitted noise scales change this interpretation.

A goodness-of-fit test compares a discrepancy statistic with its distribution under a fitted null model. A small p-value indicates that the observed discrepancy would be unusual if that model were correct.