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Many quadratic and likelihood-ratio statistics converge under a regular null hypothesis to a chi-squared distribution. Using its quantiles for a finite sample is accurate when the sample is large relative to the model dimension and expected category counts are not too small.

With fixed inverse-variance weights and weighted mean $\hat{\mu }$, $Q={\sum }_i{w}_i(y_i-\hat{\mu }{)}^2$. Under a common-effect normal model with $m$ independent estimates and known variances it has a chi-squared distribution with $m-1$ degrees of freedom. Estimated variances make this calibration approximate.

An ellipsoidal confidence region defined by a positive-definite matrix $A$ and a calibrated threshold $c$. For a known-covariance multivariate normal distribution sample, $A=n{\Sigma }^{-1}$ and the $1-\alpha$ quantile of the chi-squared distribution with $d$ degrees of freedom give exact coverage for the $d$-dimensional mean.

For a suitably regular generalized linear model, a residual deviance $D$ can be compared approximately with a chi-squared distribution on $n-p$ degrees of freedom to assess fit relative to the saturated model. Sparse or very small expected cell counts can invalidate this reference distribution. A large p-value means no detected lack of fit, not proof of correctness; dependence, wrong mean structure or overdispersion still require diagnostics.

For covariance eigenpairs $({\lambda }_k,{\varphi }_k)$, the $K$-coordinate FPCA mean statistic is $n{\sum }_{k=1}^K⟨\overline{X},{\varphi }_k{⟩}^2/{\lambda }_k$. Under a zero-mean null and standard estimation conditions, its plug-in version converges to a chi-squared distribution with $K$ degrees of freedom.

For nested null and alternative parameter spaces, the likelihood-ratio test statistic isUnder the regularity conditions of Wilks theorem, its null limit is a chi-squared distribution whose degrees of freedom equal the difference in model dimensions.

A sum of the squares of $n$ independent random variables with the standard normal distribution has the chi-squared distribution with $n$ degrees of freedom. A scaled Gaussian integral and factorization by independence give its moment-generating function $(1-2t{)}^{-n/2}$ for $t<1/2$. For positive $n$, this expectation diverges when $t\ge 1/2$.

The vector of category counts is the sum of independent categorical indicator vectors. Their means are $p$ and covariance is $\mathrm{diag}p-p{p}^T$, so the multivariate central limit theorem gives the displayed limit. At uniform probabilities on $k$ categories, Pearson-standardized deviations have covariance equal to the orthogonal projection onto the $(k-1)$-dimensional subspace of zero-sum vectors. Their squared norm has an asymptotic chi-squared distribution with $k-1$ degrees of freedom.

If $Z_1,\dots ,Z_{\nu }$ are independent standard normal random variables and $\lambda ={\sum }_j{\mu }_j^2$, then ${\sum }_j(Z_j+{\mu }_j{)}^2$ has a noncentral chi-squared distribution with $\nu$ degrees of freedom and noncentrality $\lambda$. For $\lambda =0$ it reduces to the chi-squared distribution. Squaring a shifted signed normal Wald statistic gives the case $\nu =1$.