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reduced_chi2.html
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<!DOCTYPE HTML>
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<title>Reduced chi-square</title>
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<h1>Reduced $\chi^2$</h1>
<p>
$\chi^2$ is the sum of the normalized and squared residuals:
$$\chi^2 = \sum_i{\left(\frac{I_\mathrm{model,i}-I_\mathrm{experiment,i}}{\sigma_i}\right)^2}$$
The <em>reduced</em> $\chi^2$ is calculated by dividing $\chi^2$ by the number of degrees of freedom ($DoF$). In most cases, $DoF = N-K$, where $N$ is the number of data points and $K$ is the number of fitted parameters in the model:
$$\chi^2_\mathrm{reduced} = \chi^2/(N-K).$$
Fitting programs always report the reduced $\chi^2$, donoting it simply as $\chi^2$ ("reduced" is implicitely assumed). This is also the case in SasView. So when you see $\chi^2$, it is usually the <em>reduced</em> $\chi^2$.
When fitting a model, $\chi^2$ is minimized. For a perfect model, one would expect to get a reduced $\chi^2\sim 1$, as the difference between model and data is typically about the same size as the experimental errors.<br>
<ul style="border-width:3px; border-style:solid;border-color:#697698; padding: 1em; padding-left: 40px"><b>Rule of thumb for $\chi^2$</b>
<li>If you get a reduced $\chi^2$ much larger than unity, then the model is not perfect and can be improved. However, depending on the scientific question, an approximate model may be fine.</li>
<li>If you get a reduced $\chi^2$ much smaller than unity, then the model may have too much freedom (too many parameters) or the errors ($\sigma_i$) of the data may be overestimated.</li>
</ul>
<ul><b>Further reading:</b>
<li>General introduction on <a href="https://en.wikipedia.org/wiki/Reduced_chi-squared_statistic" target="_blank">Wikipedia</a>. </li>
<li>Specific for small-angle scattering (SAXS and SANS): <a href="https://doi.org/10.1107/S1600576721006877" target="_blank">Larsen and Pedersen, 2021</a></li>
</ul>
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