After fitting a measurement model, the residuals should be compared with the noise model that justified the fit. For independent Gaussian measurement uncertainties $\sigma_i$, the statistic $$\chi^2=\sum_{i=1}^n\frac{[y_i-f(x_i;\hat\theta)]^2}{\sigma_i^2}$$ measures residual size in units of expected measurement scatter.

If $n$ data points are fitted with $p$ effective free parameters, the number of residual degrees of freedom is approximately $$\nu=n-p.$$ When the model and uncertainty estimates are reasonable, $\chi^2$ is typically of order $\nu$, so the reduced chi-square $$\chi^2_\nu=\frac{\chi^2}{\nu}$$ is often near one.

This is a diagnostic, not a mechanical pass/fail rule. Suppose ten measurements are fitted with two parameters, giving $\nu=8$. A value $\chi^2=7.6$ is quite compatible with the expected scatter. A value $\chi^2=80$ suggests that residuals are much larger than the stated uncertainties predict: the model may omit physics, the uncertainties may be underestimated, or observations may not be independent. Conversely, an extremely small value can indicate overestimated uncertainties, strong correlations, or excessive model flexibility.

The pattern of residuals matters as much as the scalar statistic. Residuals that oscillate systematically, drift with $x$, or grow with signal size reveal structure that one number can hide.

Chi-square diagnostics therefore connect theory, measurement uncertainty, and model criticism. They do not prove that a model is true; they test whether observed discrepancies look plausible under the assumptions used to analyze the experiment.