For an engineer or a physical scientist, the first necessary skill in doing theoretical analysis is to describe a problem in mathematical terms. To begin with, one must make use of the basic laws that elements of the problem. In continuum mechanics, these are the conservation laws for mass, momentum (Newton's second law), energy, etc. In addition, empirical constitutive laws are often needed to relate certain unknown variables; examples are Hooke's law between stress and strain, Fourier's law between heat flux and temperature, and Darcy's law between seepage velocity and pore pressure.

To derive the conservation law one may consider an infinitesimal element (a line segment, area, or volume element), yielding a differential equation directly. Alternatively, one may consider a control volume (or area, or line segment) of arbitrary size in the medium of interest. The law is first obtained in integral form; a differential equation is then derived by using the arbitrariness of the control volume. The two approaches are completely equivalent.

Let us first demonstrate the differential approach.

Transverse vibration of a taut string

Referring to Figure 1.1, we consider a taut string stretched between two fixed points at x = 0 and x = L and displaced laterally by a distribution of external force. Conservation of transverse momentum requires that the total lateral force on the string element be balanced by its inertia.

This short book is intended to be a practical guide, providing sets of rules that will help you to analyse the data you collect in your regular experimental sessions in the laboratory. Even more important, explanations and examples are provided to help you understand the ideas behind the formulae. Emphasis is also placed on thinking about the answers that you obtain, and on helping you get a feeling for whether they are sensible.

In contrast, this does not set out to be a text on statistics, and certainly not to be a complete course on the subject. Also, no attempt is made to provide rigorous mathematical proofs of many of the required formulae. These are important, and if required can be consulted in any standard textbook on the subject.

I believe that it will be necessary to read this material more than once. You really need to have understood the ideas involved before you do your first practical; but on the other hand, it would be much easier to absorb the material after you have actually done a couple of experiments and grappled with problems of trying to do the analysis yourself. Thus it is a good idea to read the book quickly, so that you at least discover what topics are covered and where to find them again when you need them. At this stage, you need not worry if not everything is entirely comprehensible.

Why estimate errors?

When performing experiments at school, we usually considered that the job was over once we obtained a numerical value for the quantity we were trying to measure. At university, and even more so in everyday situations in the laboratory, we are concerned not only with the answer but also with its accuracy. This accuracy is expressed by quoting an experimental error on the quantity of interest. Thus a determination of the acceleration due to gravity in our laboratory might yield an answer

g = (9.70 ± 0.15) m/s2.

In Section 1.4, we will say more specifically what we mean by the error of ±0.15. At this stage it is sufficient to state that the more accurate the experiment the smaller the error; and that the numerical value of the error gives an indication of how far from the true answer this particular experiment may be.

The reason we are so insistent on every measurement including an error estimate is as follows. Scientists are rarely interested in measurement for its own sake, but more often will use it to test a theory, to compare with other experiments measuring the same quantity, to use this parameter to help predict the result of a different experiment, and so on. Then the numerical value of the error becomes crucial in the interpretation of the result.

What are we trying to do?

In this chapter we are going to discuss the problem of obtaining the best description of our data in terms of some theory, which involves parameters whose values are initially unknown. Thus we could have data on the number of road accidents per year over the last decade; or we could have measured the length of a piece of metal at different temperatures. In either of these cases, we may be interested to see (i) whether the data lie on a straight line, and if so (ii) what are its gradient and intercept (see Fig. 2.1).

These two questions correspond to the statistics subjects known as Hypothesis Testing and Parameter Fitting. Logically, hypothesis testing precedes parameter fitting, since if our hypothesis is incorrect, then there is no point in determining the values of the free parameters (i.e. the gradient and intercept) contained within the hypothesis. In fact, we will deal with parameter fitting first, since it is easier to understand. In practice, one often does parameter fitting first anyway; it may be impossible to perform a sensible test of the hypothesis before its free parameters have been set at their optimum values.

Various methods exist for parameter determination. The one we discuss here is known as least squares. In order to fix our ideas, we shall assume that we have been presented with data of the form shown in Fig. 2.1, and that it corresponds to some measurements of the length of our bar yiobs at various known temperatures xi.

In Section 1.5 we discussed the Gaussian distribution. In this and the next appendix, we describe the binomial and Poisson distributions.

Let us imagine that we throw an unbiassed die 12 times. Since the probability of obtaining a 6 on a single throw is 1/6, we would expect on average to end up with two 6's. However, we would not be surprised if in fact we obtained 6 once or three times (or even not at all, or four times). In general, we could calculate how likely we are to end up with any number of 6's, from none to the very improbable 12.

These possibilities are given by the binomial distribution. It applies to any situation where we have a fixed number N of independent trials, in each of which there are only two possible outcomes, success which occurs with probability p, or failure for which the probability is 1 — p. Thus, in the example of the previous paragraph, the independent trials were the separate throws of the die of which there were N = 12, success consisted of throwing a 6 for which the probability p =1/6, while failure was obtaining any other number with probability 5/6.

The requirement that the trials are independent means that the outcome of any given trial is independent of the outcome of any of the others. This is true for a die because what happens on the next throw is completely unrelated to what came up on any previous one.