= Solution
A <chi-squared test> compares residuals with the random scatter predicted by a model. For independent measurements $y_i$ with known Gaussian <standard deviations> $\sigma_i$ and predictions $\mu_i(\theta)$, form
$$
Q=\sum_{i=1}^{N}\frac{[y_i-\mu_i(\widehat\theta)]^2}{\sigma_i^2}.
$$
If the mean model and error assumptions are correct, a full-rank linear fit of $p$ parameters gives a <chi-squared distribution> with $\nu=N-p$ <statistical degrees of freedom>. With a fully specified prediction and no fitted parameters, $\nu=N$. Nonlinear fits usually make this an approximation. Independent external calibration is not the same as fitting a parameter to these test data.
Under the reference distribution, the mean statistic is $\nu$ and its <variance> is $2\nu$. The upper-tail <p-value> $\Pr(\chi_\nu^2\geq Q_{\mathrm{obs}})$ quantifies how unusual a discrepancy at least this large would be under the assumed model. A small value suggests model failure or incorrect uncertainties; it is not the <probability> that the model is true. An unusually small statistic can also reveal inflated uncertainties or correlations. A rejection threshold should be specified in advance.
For independent <Poisson distribution> counts, the model <variance> is its expected count $\mu_i$. The <Pearson chi-squared statistic> is $\sum(y_i-\mu_i)^2/\mu_i$. The estimated-error alternative $\sigma_i\simeq\sqrt{y_i}$ is common for large counts but can differ substantially when counts are small. In that regime the <Poisson deviance> or a direct <likelihood> calculation is preferable, with simulation if an accurately calibrated goodness-of-fit <probability> is required.
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