Precision, Accuracy and the Calculation of Error

The question of precision and accuracy in QPA via XRD is a difficult one. It is simple enough to calculate errors on the basis of replication or precision in the mathematical fit. However, determination of the actual accuracy of the analysis is no trivial task in a standardless method. In fact, it cannot be achieved without recourse to some other measure of the sample that does incorporate standards. Too often, analysts will report Rietveld errors (see Appendix A) calculated during refinement as the errors in the final quantification. These numbers relate purely to the mathematical fit of the model and have no bearing on the accuracy or otherwise of the quantification itself. 

Consider, for example, a three-phase mixture of corundum, magnetite and zircon. Such a sample was presented as Sample 4 in the lUCr CPD round robin on quantitative phase analysis. Its components were chosen with the deliberate aim of creating a sample in which severe microabsorption occurs. Table 11.3 shows the weighed amounts of each component and the results of replicate analyses of three different sub-samples of this material. 

In this context, the Rietveld error represents the uncertainty in the mathematical fit between the observed and calculated patterns and is the value most often quoted as the error in the phase abundance. Contrasting with this is the standard deviation of the mean abundances, which represents the expected precision in the analysis and is 3 to 4 times greater than the Rietveld derived errors. The good level of fit achieved in conducting these analyses (evidenced in the low i -factors) could lead the analyst to conclude that the mean value the standard deviation of the mean is an adequate measure of the phase abundances and their errors. However, the Rietveld errors and the replication errors are at least an order of magnitude smaller than the bias (measured - weighed). The bias, due to the presence of severe microabsorption, represents the true accuracy that can be achieved in this system if the analyst takes no further steps to identify the cause and minimize the effect of absorption contrast or other aberrations which may affect accuracy. 

In the example above, the phases are such that the chemistry is unambiguous and the phase quantification could have been derived by normative calculation from bulk elemental analysis (XRF). This is not often the case, but it is frequently possible to establish the composition of each phase within a system via electron probe microanalysis or similar and conduct the inverse of a normative calculation to derive the bulk chemistry from the XRD QPA. This can then be compared with the results of a standards based technique such as XRF to obtain a measure of the accuracy of the XRD analysis. Examples of such calculations are given later in the sections dealing with application in mineralogical and industrial situations. Where this is not possible or practical, it is better to consider XRD QPA as a semi-quantitative technique at best. 

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