Tables for
Volume F
Crystallography of biological macromolecules
Edited by E. Arnold, D. M. Himmel and M. G. Rossmann

International Tables for Crystallography (2012). Vol. F, ch. 18.6, pp. 513-514   | 1 | 2 |

Section 18.6.4. Statistical functions

A. T. Brunger,a* P. D. Adams,b W. L. DeLano,c P. Gros,d R. W. Grosse-Kunstleve,b J.-S. Jiang,e N. S. Pannu,f R. J. Read,g L. M. Riceh and T. Simonsoni

aHoward Hughes Medical Institute, and Departments of Molecular and Cellular Physiology, Neurology and Neurological Sciences, and Stanford Synchrotron Radiation Laboratory (SSRL), Stanford University, 1201 Welch Road, MSLS P210, Stanford, CA 94305, USA,bThe Howard Hughes Medical Institute and Department of Molecular Biophysics and Biochemistry, Yale University, New Haven, CT 06511, USA,cGraduate Group in Biophysics, Box 0448, University of California, San Francisco, CA 94143, USA,dCrystal and Structural Chemistry, Bijvoet Center for Biomolecular Research, Utrecht University, Padualaan 8, 3584 CH Utrecht, The Netherlands,eProtein Data Bank, Biology Department, Brookhaven National Laboratory, Upton, NY 11973–5000, USA,fDepartment of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1,gDepartment of Haematology, University of Cambridge, Wellcome Trust Centre for Molecular Mechanisms in Disease, CIMR, Wellcome Trust/MRC Building, Hills Road, Cambridge CB2 2XY, England,hDepartment of Molecular Biophysics and Biochemistry, Yale University, New Haven, CT 06511, USA, and iLaboratoire de Biologie Structurale (CNRS), IGBMC, 1 rue Laurent Fries, 67404 Illkirch (CU de Strasbourg), France
Correspondence e-mail:

18.6.4. Statistical functions

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The CNS language contains a number of statistical operations, such as binwise averages and summations. The resolution bins are defined by a central facility in CNS.

Fig.[link] shows how [\sigma_{A}], [\sigma_{\Delta}] and D (Read, 1986[link], 1990[link]) are computed from the observed structure factors (`fobs') and the calculated model structure factors (`fcalc') using the CNS statistical operations. The first five operations are performed for the reflections in the test set, while the last three operations expand the results to all reflections. The `norm' function computes normalized structure-factor amplitudes for the specified arguments. The `sigacv' function evaluates [\sigma_{A}] from the normalized structure factors. The `save' function computes the statistical average [\hbox{save}(\,f) = {\textstyle\sum_{hkl}\displaystyle f_{hkl} (w/\varepsilon) \over \textstyle\sum_{hkl}\displaystyle w}, \eqno(]where w is 1 and 2 for centric and acentric reflections, respectively, and [epsilon] is the statistical weight. The averages are computed binwise, and the result for a particular bin is stored in all selected reflections belonging to the bin.


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Example for statistical operations provided by the CNS language. `norm', `sigacv', `save' and `sum' are functions that are computed internally by the CNS program. Binwise operations are in italics (`sigacv', `save' and `sum'). The result for a particular bin is stored in all elements belonging to the bin. The [\sigma_{A}] (`sigmaA') parameters are computed in binwise resolution shells. The [\sigma_{\Delta}] (`sigmaD') and D parameters are then computed from [\sigma_{A}] and binwise averages involving [|{\bf F}_{o}|^{2}] and [|{\bf F}_{c}|^{2}]. The binwise results are expanded to all reflections by the last three statements. `test' is an array that is 1 for all reflections in the test set and 0 otherwise. `sum' is a binwise operation on all reflections with the same partitioning used for the test set.


Read, R. J. (1986). Improved Fourier coefficients for maps using phases from partial structures with errors. Acta Cryst. A42, 140–149.
Read, R. J. (1990). Structure-factor probabilities for related structures. Acta Cryst. A46, 900–912.

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