There are many values that are sometimes reported in Rietveld as an indication of the fit quality and a handful of them are computed in GSAS-II. The ones that matter most in my opinion are weighted profile R-factor, \(R_{wp}\) or occasionally named in GSAS-II as wRp is given by \[R_{wp} = \sqrt { \frac {\sum w_j(y_{j,obs}-y_{j,calc})^2}{\sum w_jy_{j,obs}^2} }\] where \(y_j\) are the the points in the diffraction pattern and there are \(N\) points total. When the data points are statistically weighed, \(w_j = 1/\sigma _j^2\) , where \(\sigma _j\) is the standard uncertainty for point \(y_j\).
Since \(R_{wp}\) is statistically defined, we can ask what is the statistical expectation value for this quantity. To obtain that we note that the definition for the standard uncertainty is that it is the expectation value for how the \(y_j\) values differ from the model, meaning that \(\sigma _j^2 = <(y_{j,obs}-y_{j,calc})^2>\). Thus, we can create a quantity \(R_{exp}\) which is the statistically expected value for \(R_{wp}\), but is better thought of as the best possible value for \(R_{wp}\). With statistical weighting \(w_j = 1/\sigma _j^2\) so the expectation value for \(<\sum w_j(y_{j,obs}-y_{j,calc})^2>\) should be the number of observations, but since we are also fitting \(p\) variables, we should actually make an adjustment for that, so we can replace the sum with \(N-p\) . (Note that \(p << N\) so \(N-p \cong N\).) Thus, we obtain \[R_{exp} = \sqrt { \frac {N-p}{\sum w_jy_{j,obs}^2} }\]