Error calculations in practice

We are now in a position to estimate the standard errors for a large class of experiments. Let us briefly recapitulate. The final quantity Z is a function of the primary quantities A, B, C, … which are either measured directly or are the slopes or intercepts of straight lines drawn through points representing directly measured quantities.

If the quantity is measured directly, we take the mean of several values to be the best value and obtain its standard error by the method given in chapter 3. (During the present chapter we shall drop the word ‘standard’ in ‘standard error’. We shall not be considering the actual error in a measured quantity, and the word ‘error’ will refer to the standard error, i.e. the standard deviation of the distribution of which the quantity is a member.) If the quantity is the slope or intercept in a straight line, its value and error are obtained either from the method of least squares or from the method of taking the points in pairs.

The best value of Z is calculated from the best values of the primary quantities, and its error is obtained from their errors by the rules given in Table 4.1, or in general from (4.17) and (4.18).

There are often a large number of primary quantities to be measured, and it might be thought that the calculation of the error in each one and the subsequent calculation of the error in Z would be a laborious process.

Arithmetic is important

The object of an experiment is to obtain a number, and the correct working out of that number is just as important as the taking of the measurements. Many experiments performed by students, containing sensible measurements, are ruined by mistakes in calculating the results.

The following devices are available for calculations:

They are listed in order of decreasing expense and increasing availability. Choose the one appropriate to the job.

Computers

There are experiments where large computers are necessary, for example in the processing of radio and optical images studied in astrophysics, or in the determination of complicated biological structures by the analysis of X-ray diffraction patterns. However, for the type of experiment we are concerned with, the small computers available in the laboratory and in many homes are entirely adequate.

The great majority of the calculations that require a computer are best done with a spreadsheet. A spreadsheet such as Excel® is extremely versatile; it has a large range of functions – arithmetical, trigonometric, statistical, and logical – and a variety of features for controlling the appearance of the output, which includes the number of digits shown for a number. You can put comments and labels throughout the output, and you are strongly advised to do this liberally, so that when you look at it at a later date you will follow what has been done.

In the present chapter we are going to consider some common-sense aspects of doing experiments. They apply to all experiments, from the most elementary and simple to the most advanced and elaborate.

Preliminary experiment

In a real experiment, as opposed to an exercise, one nearly always does a trial experiment first. This serves several purposes.

(a) The experimenters ‘learns’ the experiment. Every experiment has its own techniques and routines, and the experimenter needs some training or practice in them. It is nearly always true that the first few measurements in an experiment are not as reliable or useful as the later ones, and it is usually more economical in time to have an initial period for finding out the best way of making the measurements and recording the results.

(b) The various pieces of apparatus are checked to see that they are working properly.

(c) A suitable range for each variable in the experiment is found.

(d) The errors in the different quantities can be estimated. As we have seen, this influences the strategy of the experiment proper, in the sense that more attention is given to those quantities whose errors give the major contributions to the final error.

Points (c) and (d) really add up to saying that any serious experiment must be planned, and that a few trial measurements provide a better basis for a plan than a lot of theory.

This book is intended to help you to do practical physics at college or university: its aim is to make the laboratory work more useful and profitable. We may start by asking what is the object of laboratory work in a university physics course. There are several possible objects. Laboratory work may serve

1. (a) to demonstrate theoretical ideas in physics,

2. (b) to provide a familiarity with apparatus,

3. (c) to provide a training in how to do experiments.

Let us consider each of these in turn.

Seeing something demonstrated in practice is often a great help in understanding it. For example, interference in light is not an intuitive concept. The idea that two beams of light can cancel each other and give darkness takes a little swallowing, and most people find it helpful to be given a visual demonstration. A demonstration is useful for another reason – it gives an idea of orders of magnitude. The interference fringes are in general close together, which indicates that the wavelength of light is small compared with everyday objects. But the demonstration is no substitute for a proper explanation, which goes into the details of geometry and phase relations. So the first object, the demonstration of theoretical ideas, has a definite but limited usefulness.

Experimental physics has occupied some of the finest intellects in the history of man, but the fascination of the subject is not always apparent in an undergraduate course of practical work. This book is about experimental physics and it is intended for undergraduates, but it does not describe a systematic course of experiments, nor is it a handbook of experimental techniques. Instead, it sets out to demonstrate a certain outlook or approach to experimental work. It is intended as a companion to a general course of practical work. My aim is to make the student more critical of what he does and more aware of what can be done, and in this way to make the course more interesting and meaningful.

The book is in three parts. The first is on the statistical treatment of data. I have tried to give the statistical theory not as an exercise in mathematics but rather as a tool for experimental work. This is perhaps the most difficult part of the book, and the student should not worry if he does not grasp all the mathematical details at first. He should read through the chapters to get a general understanding – and then go ahead and use the results. He can always return and master the proofs at a later stage. The second part is on experimental methods. I discuss a selection of instruments, methods, and experiments with a view to showing the craft of the experimenter.

Introduction

The communication of ideas, theories, and experimental results is an important part of scientific work. Vast quantities of scientific literature are pouring out into the world, and if you take up a scientific career of any kind you are almost certain to add to the flood. If you can achieve a good standard of writing, two benefits will accrue – one to yourself when people take note of what you have to say, and the other to the rest of the world who – strange to say – prefer their reading matter to be clear and interesting rather than obscure and dull.

We are going to consider some elementary features of good scientific writing in the present chapter. To make the discussion specific we shall confine it to a paper on some experimental work in physics, but much of what we have to say applies to scientific writing in general.

Title

The title serves to identify the paper. It should be brief – not more than about 10 words. You should bear in mind that the title will ultimately appear in a subject index. The compiler of an index relies heavily on the words in the title in deciding where it should appear. So if there are one or two key words which help to classify the work, try to put them in the title.

Abstract

Every paper should have an abstract of about 100 words or so, giving positive information about its contents.

In the present chapter we consider some examples of experimental techniques. They have been chosen because they contain many ingenious features and show that the same principles of good experimentation apply whether the experiment be advanced or elementary. They come from different branches of physics – optics, electricity, mechanics, and atomic physics – and are illustrated in different contexts – either as an instrument, a complete experiment, or an application. This is the hardest chapter in the book, as some of the physics may be unfamiliar to you. But each section stands alone, so you can omit any one of them – or indeed the whole chapter – at first reading, and carry on with the rest of the book. On the other hand, the experimental ideas contain so many instructive features that you will find a little perseverance well rewarded.

Rayleigh refractometer

(a) Description of instrument. The Rayleigh refractometer is an instrument devised to measure the refractive indices of gases and also small changes in the refractive indices of solids and liquids.

Monochromatic light from a vertical slit S (Fig. 7.1) is collimated by an achromatic lens L1 and falls on two vertical slits S1 and S2. The two beams pass through tubes T1 and T2 of equal length t, lying in the same horizontal plane. The beams then recombine to form vertical interference fringes in the focal plane of the lens L2. The fringes are viewed by a small cylindrical lens L3.

Introduction

In any experiment it is essential to keep a running record of everything that is done.

The record should be clear – and economical. On the one hand, you do not want to have to spend time subsequently searching pages of numbers without headings to find a particular set of results, or puzzling out from some meagre clues just what the conditions were when you made a certain set of measurements. On the other hand, to produce a record that is so clear that it may be followed with absolute ease by someone else is itself a time-consuming operation and is hardly necessary. You should aim at a record that you yourself will be able to interpret without too much difficulty after an interval of, say, a year.

In this chapter some suggestions for keeping the record are given. The important thing is not that you should regard them as a set of rules to be followed blindly, but rather that you should understand the spirit behind them, which is to produce a record – accurate, complete, and clear – with a minimum of effort.

Bound notebook versus loose-leaf

Some experimenters prefer a bound notebook; others use loose sheets. The advantage of a single bound book is that one knows where everything is – in the book. There are no loose bits of paper to be mislaid. The main disadvantage is that in an experiment of even moderate complexity one often goes from one part to another, and it is tiresome to have the various parts split into fragments in the record.

The present edition retains the basic outlook of the book, namely to demonstrate the purposive and critical approach which should be made to all experimental work in physics. But I have made a number of changes and additions in response to new experimental methods and the widespread use of computers, which I hope will add to its usefulness.

Substantial changes have been made in chapter 7, in which a selection of techniques are analysed to show the art and craft of the experimenter. I have added a section on the measurement of time and frequency, which includes an account of the caesium atomic clock and the present universal time scale. This is followed by a description of the Global Positioning System, which, based on atomic clocks, enables position on the surface of the Earth to be determined to very high precision. Timing techniques have a number of elegant and ingenious features, which, combined with the importance of their practical applications, make them instructive and interesting, both at the elementary and advanced level.

I have added an appendix on the χ2 distribution. The goodness of fit test based on this distribution finds widespread application in the physical, biological, medical, and social sciences. Many students have had an introduction to the topic at school and have learnt to apply the formulae of the test, but I feel that valuable insight is gained from seeing a derivation of the distribution, which I have given in a not too formal manner.