The next statistical quantity, the Reduced \(\chi ^2\) is given by \[\chi ^2 = \frac {\sum w_j(y_{j,obs}-y_{j,calc})^2}{N-p} \] or equivalently, reduced \(\chi ^2 = (R_{wp}/R_{exp})^2\). Note that we previously (§8.3) defined this and the goodness-of-fit (GOF) as \(GOF^2 = \mathrm {reduced\ \chi ^2}\).
Note that the Reduced \(\chi ^2\) and \(R_{wp}\) values indicate the overall quality of the fit, which includes the fitting of the peak positions, shapes, intensities and the background. The \(R_{wp}\) value is not on any type of absolute scale and only has meaning in comparison to other models fit to the same data. The Reduced \(\chi ^2\) is on an absolute scale, but is very sensitive to errors in our fitting that may have negligible implications for the quality of our crystallographic model. In §13.3, I discuss some ideas that can help discern this.
Since one should never be able to improve the fit beyond the statistical scatter expected in our data points, the best that \(R_{wp}\) should ever be is \(R_{exp}\). Equivalently, the reduced \(\chi ^2\) (or GOF) should never drop below 1. If the reduced \(\chi ^2\) does drop below 1, then we have introduced too many parameters, and the fit is now adjusting to compensate for statistical scatter (sometimes people call this fitting noise) or what is more likely is that the uncertainties associated with the data points are overestimates, which can happen when actions such as pixel-splitting introduce some smoothing into the data.