FAQ8: What is the difference between the Chi Square and RMS Figure-of-Merit?

Given a spectral envelope {s_i} and a synthetic model for the data {m_i}, then the chi-square is defined to be

. The RMS is defined to be .

The chi-square is computed assuming that noise within the data can be approximated by the square root of the signal recorded in each channel. The consequence of using a chi-square goodness-of-fit metric is to reduce the influence of the peak maximum when determining the parameters for the synthetic model. Notably, for high resolution data the use of chi-square to optimize the model may broaden the resulting FWHM compared to the same model optimized using the RMS.

Different goodness-of-fit metrics offer alternative views from which an optimum for a model can be established, however neither statistic is a measure for how well a model itself relates to the true chemical states present within the data.