Essential Mathematical Methods for the Physical Sciences

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. This justifies the inclusion of the subject in a book such as this, but we will concentrate on those aspects of direct relevance to the presentation of experimental data.

Experiments, samples and populations

We may regard the product of any experiment as a set of N measurements of some quantity x or of some 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 xi, x2, …, xN to be different each time the experiment is performed. We may therefore consider the xi as a set of N random variables.