# Statistical descriptors in crystallography

## Basic notions

Problems arising from the interface between the mathematical theory of probability and statistics and the practising experimentalist are not confined to procedural issues. One cannot tackle such problems without realizing their important philosophical component originating from the need to justify the lack of rigour and the unavoidable approximations in the treatment of experimental data. For this reason, the attention of the non-specialist is drawn to the two main interpretations of probability used by statisticians.

In the frequentist point of view, the probability of an event is taken to be equal to the limit of the relative frequency of the chosen event with respect to all possible events as the number of trials goes to infinity. The appeal of the frequentist approach for physical scientists lies in the apparent objectivity of its treatment of data. Almost all textbooks of statistics written for physical scientists follow this approach (e.g. Hamilton, 1964).

On the other hand, the Bayesian approach extends the interpretation of probability to include degrees of belief or knowledge in propositions. We pass from the probability of events (frequentist) to the probability of propositions (Bayesian). Nevertheless the axioms used to define the mathematical properties of probability remain unchanged. Consequently many of the statistical procedures of the two approaches are identical, apart from some changes of emphasis. The Bayesian approach is very scantily mentioned in textbooks for physical scientists (see however, French, 1978; Box & Tiao, 1973).

The frequentist school reproaches the Bayesians for their apparent lack of objectivity. The Bayesians consider that objectivity in statistics is illusory, noting that everything is interpreted through the use of preconceived models. Most physical scientists argue in terms of frequentist concepts but implicitly use a more subjective or Bayesian touch in dealing with their experimental data. Often encountered-expressions like 'the probable value of a parameter' are meaningful only in the Bayesian framework (French & Oatley, 1982).

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* Copies of NIST Technical Note 1297 may be obtained from the NIST Calibration Program or from Dr. B.N. Taylor or Dr. C.E. Kuyatt, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA.
© 1989, 1995 International Union of Crystallography
Updated 18th Sept. 1996

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