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.

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 therefore consider the xi as a set of N random variables.