The statistics of the highest E value

In a previous publication, the Gumbel-Fisher-Tippett (GFT) extreme-value analysis has been applied to investigate the statistics of the intensity of the strongest reflection in a thin resolution shell. Here, a similar approach is applied to study the distribution, expectation value and standard deviation of the highest normalized structure-factor amplitude (E value). As before, acentric and centric reflections are treated separately, a random arrangement of scattering atoms is assumed, and E-value correlations are neglected. Under these assumptions, it is deduced that the highest E value is GFT distributed to a good approximation. Moreover, it is shown that the root of the expectation value of the highest ;normalized' intensity is not only an upper limit for the expectation value of the highest E value but also a very good estimate. Qualitatively, this can be attributed to the sharpness of the distribution of the highest E value. Although the formulas were derived with various simplifying assumptions and approximations, they turn out to be useful also for real small-molecule and protein crystal structures, for both thin and thick resolution shells. The only limitation is that low-resolution data (below 2.5 A) have to be excluded from the analysis. These results have implications for the identification of outliers in experimental diffraction data.