= Solution
For independent Gaussian measurements with correctly specified <standard deviations>, model residuals give the <chi-squared test> statistic
$$
X^2=\sum_{i=1}^n\frac{[y_i-f_i(\widehat\theta)]^2}{\sigma_i^2}.
$$
For a linear fit of $p$ identifiable parameters under the null model, it has a <chi-squared distribution> with $\nu=n-p$ <degrees of freedom>. The upper-tail <p-value> $P(\chi_\nu^2\geq X_{\rm obs}^2)$ tests unusually large residuals; a nonlinear fit generally uses this distribution only approximately. Correlated Gaussian errors instead require $X^2=\mathbf r^{\mathsf T}\Sigma^{-1}\mathbf r$.
For counts, <Pearson's chi-squared test> uses $X^2=\sum_j(O_j-E_j)^2/E_j$, with $E_j$ calculated under the <null hypothesis>. For $k$ multinomial categories and $p$ fitted parameters the usual asymptotic <degrees of freedom> are $k-1-p$; the subtraction of one reflects the fixed total count. Sparse expected counts can invalidate this approximation, requiring a suitable exact or simulation-based test. Contingency-table independence gives $(r-1)(c-1)$ <degrees of freedom>.
\b[A small <p-value> indicates tension with the specified model and error assumptions; it is not the <probability> that the null is true.] Excessive $X^2$ may reflect a wrong model, underestimated errors, correlations or outliers. An implausibly small value may suggest overestimated errors or hidden constraints. For an adequate fitted model $X^2/\nu$ is typically near one, with fluctuations, rather than required to equal one.
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