Reduced $\chi^2$

$\chi^2$ 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 reduced $\chi^2$ 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 reduced $\chi^2$. 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.

    Rule of thumb for $\chi^2$
  • 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.
  • 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.