• theoretical residual values;
  • data-quality indicators;
  • fit-quality indicators;
  • quality indicators

The usual residual values are complemented by expectation values based solely on the experimental data and the number of model parameters. These theoretical R values serve as benchmark values when all of the basic assumptions for a least-squares refinement, i.e. no systematic errors and a fully adequate model capable of describing the data, are fulfilled. The prediction of R values as presented here is applicable to any field where model parameters are fitted to data with known precision. For crystallographic applications, F 2-based residual benchmark values are given. They depend on the first and second moments of variance, intensity and significance distributions, 〈σ2〉, 〈I o 2〉, 〈I o 22〉. Possible applications of the theoretical R values are, for example, as a data-quality measure or the detection of systematic deviations between experimental data and model predicted data, although the theoretical R values cannot identify the origin of these systematic deviations. The change in R values due to application of a weighting scheme is quantified with the theoretical R values.