Reduced χ²

χ² is the sum of the normalized and squared residuals:

χ² = Σ_i((I_model,i-I_experiment,i)/σ_i)²

And the reduced χ² is χ² divided 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:

Reduced χ² = χ²/(N-K).

It is always the reduced χ² that is reported in fitting programs (not χ²), and usually just denoted χ² ("reduced" is implicitely assumed)... This is also the case in SasView. So when you see χ², it is usually the reduced χ².

When fitting a model, χ² is minimized. For a perfect model, one would expect to get a reduced χ² ~ 1, as the difference between model and data is typically about the same size as the experimental errors.

Rule of thumb:
• If you get a reduced χ² much larger than unity, then the model is not perfect. May need to be improved (depending on the scientific question).
• If you get a reduced χ² much smaller than unity, then the model has too much freedom (too many parameters) or the errrors (σ_i) of the data may be overestimated.

More thorough and precise introductions can be found elsewhere:
• General introduction on Wikipedia.
• Specific for small-angle scattering (SAXS and SANS): Larsen and Pedersen 2021