Normal probability plot analysis of error in measured and derived quantities and standard deviations

Abstract

Normal probability plot analysis is applied to independent sets of crystallographic structure factor measurements (F) and the derived coordinates (p). Differences between corresponding pairs of structure factors (ΔF) in the two sets are examined in terms of their pooled standard deviations (σF) by plotting the ordered statistic δm = ΔF/σF against the expected normal distribution. Differences between pairs of coordinates (Δp) are similarly examined in a δp = Δp/σp half-normal probability plot. Both plots result in linear arrays of unit slope and zero intercept, for normal error distribution in the experiment and the model and correctly assigned standard deviations. Analysis of departures from this ideal, especially when both plots are considered together, provides detailed information of the kinds of error in δm and in δp. By inference, the kinds of error in F and σF as well as in p and σp can be deduced. The normal probability plot δR = |Fmeas| − |Fcalc|/σFmeas should ideally also be linear, with unit slope and zero intercept. Deviations from ideal provide considerably more information than the conventional R values. Analysis of δR in combination with δm plots allows further specification of the error distribution. Examples using these plots are given and discussed, based both on real and on simulated data.