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 ofNmeasurements of some

quantityxor set of quantitiesx,y,...,z. This set of measurements constitutes the

data. Each measurement (ordata item) consists accordingly of a single numberxi

or a set of numbers (xi,yi,...,,zi), wherei=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 quantityxbeing measured, one would expect theNmeasured

valuesx 1 ,x 2 ,...,xNto 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).