Mathematical Methods for Physics and Engineering - Matematica.NET

31StatisticsIn this chapter, we turn to the study of statistics, which is concerned withthe analysis of experimental data. In a book of this nature we cannot hopeto do justice to such a large subject; indeed, many would argue that statisticsbelongs to the realm of experimental science rather than in a mathematicstextbook. Nevertheless, physical scientists and engineers are regularly called uponto perform a statistical analysis of their data and to present their results in astatistical context. Therefore, we will concentrate on this aspect of a much moreextensive subject. § 31.1 Experiments, samples and populationsWe may regard the product of any experiment as a set of N measurements of somequantity x or set of quantities x,y,...,z. This set of measurements constitutes thedata. Each measurement (or data item) consists accordingly of a single number x ior a set of numbers (x i ,y i ,...,,z i ), where i =1,...,,N. For the moment, we willassume that each data item is a single number, although our discussion can beextended to the more general case.As a result of inaccuracies in the measurement process, or because of intrinsicvariability in the quantity x being measured, one would expect the N measuredvalues x 1 ,x 2 ,...,x N to be different each time the experiment is performed. We may§ There are, in fact, two separate schools of thought concerning statistics: the frequentist approachand the Bayesian approach. Indeed, which of these approaches is the more fundamental is still amatter of heated debate. Here we shall concentrate primarily on the more traditional frequentistapproach (despite the preference of some of the authors for the Bayesian viewpoint!). For a fullerdiscussion of the frequentist approach one could refer to, for example, A. Stuart and K. Ord,Kendall’s Advanced Theory of Statistics, vol. 1 (London: Edward Arnold, 1994) or J. F. Kenneyand E. S. Keeping, Mathematics of Statistics (New York: Van Nostrand, 1954). For a discussionof the Bayesian approach one might consult, for example, D. S. Sivia, Data Analysis: A BayesianTutorial (Oxford: Oxford University Press, 1996).1221