= Solution
A <chi-squared test> compares discrepancies with their expected random size. For independent Gaussian observations with known <standard deviations>, a fitted model gives
$$
\chi^2=\sum_{i=1}^n\frac{[y_i-f(x_i;\widehat\theta)]^2}{\sigma_i^2}.
$$
Under a correct regular model with $p$ fitted parameters, its reference <statistical degrees of freedom> are usually $\nu=n-p$. Correlated errors instead require $\chi^2=\mathbf r^TC^{-1}\mathbf r$ with the correct <covariance matrix>. The reduced statistic has mean about one and <standard deviation> $\sqrt{2/\nu}$, so it should not be forced to equal exactly one.
For binned counts, Pearson's statistic is $\sum_i(O_i-E_i)^2/E_i$. With a fixed total and $k$ populated categories, fitting $p$ distribution parameters generally gives $\nu=k-1-p$. The approximation needs adequately populated expected bins and the appropriate dependence assumptions. Sparse data may require a Poisson <likelihood>, exact calculation or simulated reference distribution rather than a casual chi-squared approximation.
The upper-tail <probability> $p_{\rm value}=P(\chi^2_\nu\geq\chi^2_{\rm obs})$ measures how unusual the discrepancy would be under the null model; it is not the <probability> that the model is true. A large statistic can indicate a wrong model, underestimated noise or unmodelled systematic effects; an unusually small one can indicate overestimated errors or correlations. Residual plots and checks of the noise model are needed alongside the numerical test.
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