Enter your own $(x, y)$ data with uncertainties $\sigma_x$ and $\sigma_y$, fit a model, and see whether the fit is actually consistent with your data via a $\chi^2$ goodness-of-fit test — the same workflow you'd use to check a lab measurement against a theoretical model.
| x | y | σy | σx |
|---|
Leave σx / σy blank or 0 for "no known uncertainty" (that column's error bars and its contribution to the effective variance are then switched off for that point).
$f(x)$ is the fitted polynomial with $p$ free parameters, and $\nu = N-p$ is the number of degrees of freedom. Parameters are found by minimizing $\chi^2$ (weighted least squares), and the $p$-value is the probability that $\chi^2$ this large (or larger) would arise from a correct model just by random chance.
Ordinary least squares only weighs $y$-uncertainty. When a point also has an $x$-uncertainty, this sim uses the effective-variance method: it projects $\sigma_x$ onto the $y$-axis using the fit's local slope, $\sigma_{\mathrm{eff}}^2 = \sigma_y^2 + \left(\frac{df}{dx}\right)^2\sigma_x^2$, then refits and repeats until the parameters stop changing.
Rule of thumb for the reduced chi-square $\chi^2_\nu$: values near 1 mean the scatter in your data matches your stated uncertainties. $\chi^2_\nu \gg 1$ means the model doesn't fit well or your error bars are too small; $\chi^2_\nu \ll 1$ usually means the error bars are overestimated (or the model has more free parameters than the data can justify).
The $p$-value turns that into a single probability: conventionally $p<0.05$ is called a "bad fit" (reject the model at 95% confidence), while $p>0.05$ means the data don't give you a reason to reject it — that is not the same as proving the model is correct, only that it isn't contradicted by this data.