Dynamics, or the theory of motion, is the science of the temporal passage of spatial events. At least this is how this study has come to be defined. But is it really true that we can arrive at a clear understanding of dynamics using this definition?

The naïve understanding is satisfied with this explanation. Indeed, what is so clear and simple as space and time? Space is what we see with our eyes, and time is what we feel as everything is passing by, one thing always after the other. But is this true? Who has ever seen space? I mean that one can only see objects in space and that they stand in the particular relations we call “in front of,” “behind,” “to the right of” and “to the left of.” We coordinate every object with a place in space; but to speak of the space itself, we then have to mentally extract all of the objects. That is a very broad abstraction. How do I know that this space exists when all bodies are removed? Not through experience, since all observations refer to those real things and their respective distances can only be defined with respect to the things around them. It is therefore a peculiar construction in which we embed things such that it is attached to them but can never be observed, and, unlike forces or heat that will make them glow, has no effect on them, yet it dictates far-reaching laws.

Introduction

Although there is still resistance to the theory of relativity, it should be pointed out that this resistance is founded upon conceptual objections. It is beyond dispute that the theory is physically useful, that its assertions are well verified by observable phenomena. What opponents of the theory find problematic are the ideas upon which the theory is founded. On the other hand, it is precisely these ideas that the defenders of the theory hold to be its greatest achievement and in which they claim to find the true significance of Einstein's work. It is therefore of interest to study the formation of these ideas, their content and their significance.

We begin by rejecting two ineffective objections. The theory has been criticized for contradicting ordinary common sense. It is necessary to concede that this is true, but we refuse to see this as a criticism because a theory like this, which provides an analysis of the most profound abstract ideas, will necessarily contradict certain naïve intuitions from everyday life. We do not discount the value which is attached to this simplicity of understanding, but a mentality adapted to the practical needs of existence (and is ordinary common sense anything else?) cannot be required to possess the proper critical faculty of a theory of knowledge: “The chisels and hammers are fine for working a piece of wood, but to engrave you need an engraver's needle,” these words in Kant should not be far from view whenever one wants to contradict the theory of relativity with elementary objections.

Having placed the empirical foundation of the principles of the constancy of the speed of light in axiomatic form and recognizing that several of these axioms have not yet received conclusive experimental support, it will now be of interest to consider an experiment to confirm the light principle which I had not mentioned, but to which Born has referred.

The eclipses of the moons of Jupiter may be used for an astronomical measurement of the speed of light. It is well known that the delays in the eclipses of a moon will progressively increase over the course of a year; the resulting overall delay corresponds to the time that the light has taken to traverse the length of the axis of the Earth's orbit. Hence, we are measuring the speed of light in a one directional sense. Now Maxwell has already pointed out that the speed of light must be different depending upon whether it travels with or against the direction of the orbit of the Earth because the inertial system S, in which the sun and the elliptical orbits of the planets are at rest, will itself have a velocity V (which Born calls v) with respect to a preferred inertial system J, which according to the older theory is that in which the aether is at rest; the speed of light must therefore really be c + V in one direction and c − V in the other.

If it is reasonable to assume that a population consists of values that have a Gaussian distribution, then what will be the distribution of a property (a ‘statistic’) of a sample drawn from this Gaussian ‘parent’? The property might be the mean, variance or standard deviation of the sample. Each of these properties has a sampling distribution, which can be described as follows.

We imagine a very large or infinite population that has a Gaussian distribution with mean μand standard deviation ?. A sample consisting of n values is randomly drawn from this population. A property of the sample is calculated, in order to estimate the corresponding population parameter. We then draw another sample, also of size n, and calculate the same property for this second sample. The process is repeated many times. Next the distribution of that property is examined; the distribution becomes manifest as a result of taking a large number of repeated samples (all of size n). The distribution is the sampling distribution of the property in question. It is understood that, in any particular experimental situation, we do not actually need to draw a large number of samples; this process is a conceptual one that enables us to infer, from one actual sample, the variability (depicted by the shape of the sampling distribution) of our estimate of the population parameter. In section 9.1 we review the material already discussed in section 8.6.2.

The uncertainty that accompanies the best estimate of a measurand is usually based on fewer than 20 degrees of freedom, and sometimes fewer than 10. The reason is as follows.

For Type A evaluations of uncertainty, the number of degrees of freedom, v, is related to the sample size, n. Thus, when calculating the mean of a sample, v= n–1. Where measurements are made ‘manually’ (not under computer control), n and therefore ? are likely to be small. Where measurements are computer-controlled and the environment is sufficiently stable, it is easy to amass samples consisting of hundreds or even thousands of values from the same population. We might therefore think that the number of degrees of freedom associated with the uncertainty in the measurand is also very high. However, this is unlikely to be so, since there will probably exist systematic errors that can be corrected for but that will nevertheless leave a Type B uncertainty. Such an uncertainty is generally associated with fewer degrees of freedom. Admittedly, the estimation of a systematic error may also be based on a large number of repeated measurements. The calibration of the 312 -digit DMM by means of simultaneous measurements with an 812 -digit DMM in section 6.1.2 is a case in point. A large number of such measurements could in principle allow us to determine an uncertainty in the systematic error of the 312 -digit DMM that is associated with a large number of degrees of freedom. However, the readings of the 812 -digit DMM themselves have an uncertainty obtained from its calibration report that is likely to be based on fewer degrees of freedom.

In writing this book, we address several groups of readers who require an understanding of measurement, and of uncertainty in measurement, in science and technology.

Undergraduates in science, for example, should have texts that set out the concepts and terminology of measurement in a clear and consistent manner. At present, students often encounter texts that are mutually inconsistent in several aspects. For example, some texts use the terms error and uncertainty interchangeably, whilst others assign them distinctly different meanings. Such inconsistency is liable to confuse students, who are consequently unsure about how to interpret and communicate the results of their measurements.

Until recently, a similar lack of consistency affected those whose primary occupation includes measurement, the evaluation of uncertainty in measurement, instrument and artefact calibration and the maintenance of standards of measurement – that is, professional metrologists. International trade, for example, requires mutual agreement among nations on what uncertainty is, how it is calculated and how it should be communicated; for a global economy to work efficiently, lack of such agreement cannot be tolerated. In the mid 1990s, international bodies, charged with the definition, maintenance and development of technical standards and standards of measurement in a variety of fields, published and disseminated the Guide to the Expression of Uncertainty in Measurement – the ‘GUM’. These bodies included the Bureau International des Poids et Mesures (BIPM) or International Bureau of Weights and Measures, the International Standardisation Organisation (ISO) and the International Electrotechnical Commission (IEC).

For the newcomer, unfamiliarity with the specialist vocabulary of scientific disciplines like physics and chemistry can act as an obstacle to learning those disciplines. What can be even more challenging is that science employs many words such as force and energy that are used in various ways in everyday language. The science of measurement, in particular, has many terms, such as error, uncertainty and accuracy, that also occur in day-to-day use in contexts far removed from measurement. In this chapter we consider terms used in measurement, including those with an everyday or popular meaning such as error, and we clarify their meaning when used in the context of measurement.

Measurement and related terms

Measurement

Measurement is a process by which a value of a particular quantity such as the temperature of a water bath or the pH of a solution is obtained. In the case of length measurement, this might involve measuring the atomic-scale topography of a surface using an instrument such as an atomic-force microscope (AFM), or measuring the length of a pendulum using a metre rule. Values obtained through measurement form the foundation upon which we are able to

• test both new and established scientific theories;

• decide whether a component, such as a resistor, is within specification;

• compare values obtained by workers around the world of a particular quantity, such as the thickness of the ozone layer of the atmosphere;

• quantify the amount of a particular chemical species, such as the amount of steroid in a sample of urine taken from an athlete; and

• establish the proficiency of laboratories involved with the testing and calibration of equipment.

Random errors arise from uncontrollable small changes in the measurand, instrumentation or environment. These changes are evident as variations in the values obtained when we carry out repeat measurements. In this chapter we shall consider methods of quantifying these variations: that is, describing them numerically using statistical methods. Some basic statistical concepts will therefore be introduced and discussed.

Sampling from a population

In statistics, the term population refers to the number of possible, but not necessarily actual, measured values. In some situations a population consists of an infinite number of values. In practice, we can measure only a sample drawn from a population, since time and resources are always limited. We hope and expect that the sample is representative of the population. In almost every case of measurement we sample a population, and the quantities of interest obtained from the sample (sometimes called sample statistics) should reliably represent corresponding parameters in the population (the population parameters). An example of such a quantity of interest, which quantifies the amount of scatter in values, is the standard deviation of the values.

There are cases where a sample may, in fact, be the entire population. Thus the examination results of a class of 30 students can be analysed statistically in order to determine, for example, the mean mark and the range of marks, with no attempt at generalising. The teacher of the class may be interested simply in that particular class.

Random errors, evaluated using statistical methods, create a Type A uncertainty. A known systematic error in a measured value should be corrected for, and after the correction has been made, the uncertainty in the correction contributes to the uncertainty in that value. The uncertainty in the correction, and hence in the value, may be Type A or Type B, depending on how the uncertainty is evaluated. The finally reported uncertainty of a measurand, called the combined uncertainty, is likely to have both Type A and Type B components, but becomes wholly Type B when subsequent use is made of it.

In this chapter we consider how to evaluate the combined uncertainty of a measurand. The procedure to be described makes no distinction between Type A and Type B uncertainties. It may appear then as if we have gone to unnecessary trouble in assigning types to uncertainties, but this classification is desirable since it emphasises the different methods by which they are evaluated. It is also useful as a reminder that, whereas an ‘error’ can be random or systematic, ‘uncertainty’ is a separate concept whose two types are distinguished from each other by different names, ‘Type A’ and ‘Type B’. However, once uncertainties have been classified, Type A and Type B uncertainties are treated identically thereafter.

In this chapter we describe how consistency and clarity may be brought to the calculation and expression of uncertainty in measurement.

The goal of any measurement is to establish a numerical value for the measurand. Depending on the accuracy that we wish to claim for the numerical value, the procedure that gives us the value may be relatively simple and direct, involving no more than a tape-measure, for example. In other situations the process may be more complicated, with several intermediate stages requiring the resources of a well-equipped laboratory. Thus, if the measurand is the width of a table, the tapemeasure is all that is needed. On the other hand, if the measurand is the accurate mass of an object, we need to know the value of the buoyancy correction (since the weight of the object is less by an amount equal to the weight of the volume of air that it displaces). This in turn requires knowledge of the volume of the object and of the density of air (which is a function of temperature, pressure and composition) at the time of measurement.

There are three components of a measurement: the measurand itself; the measuring instrument (which can be a stand-alone instrument such as a thermometer, or a complex system that occupies a whole laboratory); and the environment (which includes the human operator). The environment will, in general, affect both the measurand and the measuring instrument.

In this chapter we present four case studies based on typical undergraduate experiments, involving the determination of best estimates of measurands, standard uncertainties, expanded uncertainties and coverage intervals. For completeness, we include a brief description of each experiment. The equipment required is inexpensive or can usually be found in an undergraduate science laboratory. The account of each experiment contains data obtained in an actual experiment.

We have not included a detailed introduction to each experiment, nor have we indicated how each might be improved or ‘finessed’. The account of each experiment is biased towards giving details of the data analysis such as the calculations of standard uncertainties and coverage intervals. A more detailed analysis would normally require consideration of the uncertainty in the calibration of instruments used. For many undergraduate experiments such information is not available, and therefore we have not included the contribution of the calibration uncertainty to the combined standard uncertainty. At the end of the account of each experiment we suggest practically based exercises related to the experiment.

Reporting measurement results

An account of an experiment, as presented in a formal report, may contain many sections with headings such as introduction, materials and methods, results, analysis and conclusion. With respect to the analysis of data, best estimates of particular quantities obtained through experiment and by other means should be communicated clearly, concisely, and in a manner that is useful to others.

After measurement, we assign an estimated value to a measurand as well as an accompanying uncertainty. The uncertainty is usually expressed as an interval around the estimated value. With any such interval we associate a probability that the actual or true value of the measurand falls within that interval. Measurands are usually continuous quantities such as temperature, voltage and time. However, when discussing probabilities in the context of measurement it is convenient first to consider ‘experiments’ in which the outcomes are discrete, for example tossing a coin, where the outcome is a head or a tail.

Distribution of scores when tossing coins or dice

A fair coin falls heads up with probability 1/2 and tails up also with probability 1/2. A fair coin is an idealised object (since all real coins have a slight bias towards either heads or tails) and presents the simplest case of a ‘uniform’ probability distribution. When a probability distribution is uniform, the possible outcomes of an experiment (tossing a coin in this case) occur with equal probability. We will show how non-uniform probabilities emerge as soon as two or more fair coins are considered. These non-uniformities tend to a characteristic pattern called a Gaussian (or ‘normal’) probability density distribution. For the sake of brevity we shall usually refer to the ‘Gaussian probability distribution’ as simply the ‘Gaussian distribution’. Likewise we shall usually refer to the ‘uniform probability density distribution’ as the ‘uniform distribution’.