Experiments samples and populations

31
Statistics
In this chapter, we turn to the study of statistics, which is concerned with
the analysis of experimental data. In a book of this nature we cannot hope
to do justice to such a large subject; indeed, many would argue that statistics
belongs to the realm of experimental science rather than in a mathematics
textbook. Nevertheless, physical scientists and engineers are regularly called upon
to perform a statistical analysis of their data and to present their results in a
statistical context. Therefore, we will concentrate on this aspect of a much more
extensive subject.§
31.1 Experiments, samples and populations
We may regard the product of any experiment as a set of N measurements of some
quantity x or set of quantities x, y, . . . , z. This set of measurements constitutes the
data. Each measurement (or data item) consists accordingly of a single number xi
or a set of numbers (xi , yi , . . . , , zi ), where i = 1, . . . , , N. For the moment, we will
assume that each data item is a single number, although our discussion can be
extended to the more general case.
As a result of inaccuracies in the measurement process, or because of intrinsic
variability in the quantity x being measured, one would expect the N measured
values x1 , x2 , . . . , xN to be different each time the experiment is performed. We may
§
There are, in fact, two separate schools of thought concerning statistics: the frequentist approach
and the Bayesian approach. Indeed, which of these approaches is the more fundamental is still a
matter of heated debate. Here we shall concentrate primarily on the more traditional frequentist
approach (despite the preference of some of the authors for the Bayesian viewpoint!). For a fuller
discussion 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. Kenney
and E. S. Keeping, Mathematics of Statistics (New York: Van Nostrand, 1954). For a discussion
of the Bayesian approach one might consult, for example, D. S. Sivia, Data Analysis: A Bayesian
Tutorial (Oxford: Oxford University Press, 1996).
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