An Introduction to Mechanics grew out of a one-semester course at the Massachusetts Institute of Technology—Physics 8.012—intended for students who seek to understand physics more deeply than the usual freshman level. In the four decades since this text was written physics has moved forward on many fronts but mechanics continues to be a bedrock for concepts such as inertia, momentum, and energy; fluency in the physicist's approach to problem-solving—an underlying theme of this book—remains priceless. The positive comments we have received over the years from students, some of whom are now well advanced in their careers, as well as from faculty at M.I.T. and elsewhere, reassures us that the approach of the text is fundamentally sound. We have received many suggestions from colleagues and we have taken this opportunity to incorporate their ideas and to update some of the discussions.

We assume that our readers know enough elementary calculus to differentiate and integrate simple polynomials and trigonometric functions. We do not assume any familiarity with differential equations. Our experience is that the principal challenge for most students is not with understanding mathematical concepts but in learning how to apply them to physical problems. This comes with practice and there is no substitute for solving challenging problems. Consequently problem-solving takes high priority. We have provided numerous worked examples to help provide guidance. Where possible we try to tie the examples to interesting physical phenomena but we are unapologetic about totally pedagogical problems.

This appendix discusses some of the datasets that are used as examples throughout the text. They are all available online from www.cambridge.org/9781107607590.

Although R can load files directly from the web, it is good practice to download the individual files to a local directory so that they can be used off-line. The files needed for this chapter are rutherford.dat, reynolds.txt, hipparcos.txt.gz and pedroni.dat.

Michelson's speed of light data

A. A. Michelson – known to students of physics for the famous Michelson-Morley experiment – made great advances in precision optical measurements, particularly the measurement of the speed of light. Here we shall use a set of 100 measurements of the speed of light in air taken in summer 1879 originally published by Michelson (1882) and reproduced by Stigler (1977).

The first few data values are shown in Table B.1. Each of the numbers represents the speed recorded in one ‘run’ of the apparatus, in units of km s−1. Each run was in fact an average of several individual measurements. The 100 numbers, which together form a sample, are divided into five groups of 20, each group labelled an ‘experiment’. The speed measurements are in principle continuous, but Michelson's data have been rounded to the nearest 10 km s−1. Stigler (1977) applied Michelson's own corrections to the modern value of c to give a value of 299 734.5 km s−1 for the speed of light in air.

How do we know if the model fitted to our data is actually a good match to the data? And how do we quantify the uncertainty on the estimates of the model's parameters? The first question can be addressed by significance testing, and the second can be answered using confidence intervals.

A thought experiment

We shall return to the thought experiment begun in Chapter 5, drawing from a bag containing sweets of two colours, red and green. But now let us imagine that we do not know the proportions of red and green sweets. Instead, we are allowed to draw 10 times from the bag, with replacement. A simple hypothesis is that the bag contains equal numbers of red and green. What do we say about this hypothesis if we get eight greens from our 10 draws?

Let's assume the bag contains equal proportions, and find the probability for getting data like ours.

Science is not about certainty, it is about dealing rigorously with uncertainty. The tools for this are statistical. Statistics and data analysis are therefore an essential part of the scientific method and modern scientific practice, yet most students of physical science get little explicit training in statistical practice beyond basic error handling. The aim of this book is to provide the student with both the knowledge and the practical experience to begin analysing new scientific data, to allow progress to more advanced methods and to gain a more statistically literate approach to interpreting the constant flow of data provided by modern life.

More specifically, if you work through the book you should be able to accomplish the following.

1. • Explain aspects of the scientific method, types of logical reasoning and data analysis, and be able to critically analyse statistical and scientific arguments.

2. • Calculate and interpret common quantitative and graphical statistical summaries.

3. • Use and interpret the results of common statistical tests for difference and association, and straight line fitting.

4. • Use the calculus of probability to manipulate basic probability functions.

5. • Apply and interpret model fitting, using e.g. least squares, maximum likelihood.

6. • Evaluate and interpret confidence intervals and significance tests.

Students have asked me whether this is a book about statistics or data analysis or statistical computing. My answer is that they are so closely connected it is difficult to untangle them, and so this book covers areas of all three.

This book was written because I could not find a suitable textbook to use as the basis of an undergraduate course on scientific inference, statistics and data analysis. Although there are good books on different aspects of introductory statistics, those intended for physicists seem to target a post-graduate audience and cover either too much material or too much detail for an undergraduate-level first course. By contrast, the ‘Intro to stats’ books aimed at a broader audience (e.g. biologists, social scientists, medics) tend to cover topics that are not so directly applicable for physical scientists. And the books aimed at mathematics students are usually written in a style that is inaccessible to most physics students, or in a recipe-book style (aimed at science students) that provides ready-made solutions to common problems but develops little understanding along the way.

This book is different. It focuses on explaining and developing the practice and understanding of basic statistical analysis, concentrating on a few core ideas that underpin statistical and data analysis, such as the visual display of information, modelling using the likelihood function, and simulating random data. Key concepts are developed using several approaches: verbal exposition in the main text, graphical explanations, case studies drawn from some of history's great physics experiments, and example computer code to perform the necessary calculations. The result is that, after following all these approaches, the student should both understand the ideas behind statistical methods and have experience in applying them in practice.

addition rule The rule for combining probabilities of events Pr(A ∪ B) = Pr(A) + Pr(B) − Pr(A ∩ B)

Bayes' theorem In its simplest form Bayes' theorem states Pr(A∣B) = Pr(B∣A) Pr(A)/Pr(B) and can be used to transpose the conditionals, i.e. relate A∣B to B∣A

bias Quantifies how far the average statistic lies from the parameter it is estimating

binomial The distribution of the number of successes obtained in a series of n independent trials, when the probability of success (per trial) is p

bootstrap A resampling scheme allowing randomised datasets to be produced by resampling from the existing data

box plot Graphical representation of numerical data based on Tukey's five-number summary

bubble plot A type of scatter plot using circles as the plotting symbols, with the areas of the circles representing a third variable

central limit theorem States that the mean of n random variables tends to variable with a Gaussian distribution as n → ∞ (under quite general conditions)

central moment The Nth-order central moment is the mean (expectation) of the Nth power of the deviation of a random variable from its mean

chi-square In general the sum of squares of ν independent normal variables has a chisquare distribution with ν degrees of freedom

chi-square statistic Test statistic formed from the (weighted) sum of squares of differences between data and prediction (model)

chi-square test A goodness-of-fit test that compares the chi-square statistic to a chi-square reference distribution

How should you summarise a dataset? This is what descriptive statistics and statistical graphics are for. A statistic is just a number computed from a data sample. Descriptive statistics provide a means for summarising the properties of a sample of data (many numbers or values) so that the most important results can be communicated effectively (using few numbers). Numerical and graphical methods, including descriptive statistics, are used in exploratory data analysis (EDA) to simplify the uninteresting and reveal the exceptional or unexpected in data.

Plotting data

One of the basic principles of good data analysis is: always plot the data. The brain–eye system is incredibly good at recognising patterns, identifying outliers and seeing the structure in data. Visualisation is an important part of data analysis, and when confronted with a new dataset the first step in the analysis should be to plot the data. There is a wide array of different types of statistical plot useful in data analysis, and it is important to use a plot type appropriate to the data type. Graphics are usually produced for screen or paper and so are inherently two dimensional, even if the data are not.

The variables can often be classified as explanatory or response.

We can use what we have learnt to start making some inferences about data. Maybe we have collected measurements of a quantity and wish to see if these are consistent with some theoretical expectation. We don't just want to compute the sample mean but to compare it with something else. Perhaps we have two samples, taken under different conditions (such as a ‘treatment’ and ‘control’ group) and wish to see if their mean responses differ. Another very common situation is that we have measurements of some response (y) taken at different values of some explanatory variable (x) and wish to quantify the way that y responds. We can go some way to getting useful inferences out of such data using numerical and graphical summaries (Chapter 2). These can be refined once we have studied some probability theory (Chapters 4 and 5).

Inference about the mean of a sample

We take repeated measurements of a single quantity, or measure the same quantity for each member of a finite sample, and wish to discover whether these data are consistent with a predetermined theoretical value. We want to know if our sample is consistent with being randomly drawn from a theoretical population, with some particular population mean. As an example, let's consider the first ‘experiment’ (batch of 20 runs) of Michelson's dataset (see Appendix B, section B.1).

Dotted throughout the book are extracts of computer code that show how to perform the calculations under discussion. The examples are based on specific problems discussed in the text, but should be clear enough that they can also be used, with very little effort, for ‘real life’ data analysis problems. The computer codes are written in the R environment, which is introduced in this appendix.

What is R?

R is an environment for statistical computation and data analysis. You can think of it as a suite of software for manipulating data, producing plots and performing calculations, with a very wide range of powerful statistical tools. But it is also a programming language, so you can construct your own analyses with a little programming effort. It is one of the standard packages used by statisticians (professional and academic). To install R visit www.r-project.org/.

A first R session

First of all, start R, either by typing R at the command prompt (e.g. Linux) or double-clicking on the relevant icon (e.g. Windows).

A typical R session involves typing some commands into a ‘console’ window, and viewing the text and/or graphical output (which may appear in a pop-out window). The prompt is usually a ‘>’ sign, but can be changed if desired. At the prompt you can enter commands to execute.Virtually all commands in R have a command(arguments) format, where the name of the command is followed by some arguments enclosed in brackets (if there are no arguments the brackets are still present but empty).

Why should a scientist bother with statistics? Because science is about dealing rigorously with uncertainty, and the tools to accomplish this are statistical. Statistics and data analysis are an indispensable part of modern science.

In scientific work we look for relationships between phenomena, and try to uncover the underlying patterns or laws. But science is not just an ‘armchair’ activity where we can make progress by pure thought. Our ideas about the workings of the world must somehow be connected to what actually goes on in the world. Scientists perform experiments and make observations to look for new connections, test ideas, estimate quantities or identify qualities of phenomena. However, experimental data are never perfect. Statistical data analysis is the set of tools that helps scientists handle the limitations and uncertainties that always come with data. The purpose of statistical data analysis is insight not just numbers. (That's why the book is called Scientific Inference and not something more like Statistics for Physics.)

Scientific method

Broadly speaking, science is the investigation of the physical world and its phenomena by experimentation. There are different schools of thought about the philosophy of science and the scientific method, but there are some elements that almost everyone agrees are components of the scientific method.