Introduction

In chapter 3 we considered the normal distribution largely due to its similarityto the distribution of data observed in many experiments involving repeatmeasurements of a quantity. In particular, the normal distribution is useful fordescribing the spread of values when continuous quantities such as temperatureor time interval are measured.

Another important category of experiment involves counting. As examples, we maycount the number of electrons scattered by a gas, the number of charge carriersthermally generated in an electronic device, or the number of beta particlesemitted by a radioactive source. In these situations, distributions thatdescribe discrete quantities must be considered. In this chapter we consider twosuch distributions important in science: the binomial and Poissondistributions.

The binomial distribution

One type of experiment involving discrete variables entails removing an objectfrom a population and classifying that object in one of a finite number of ways.For example, we might test an electrical component and classify it‘within specification’ or ‘outside specification’.Owing to the underlying (and possibly unknown) processes causing components tofail to meet the specification, we can only give a probability that anyparticular component tested will satisfy the specification. Whenn objects are removed from a population and tested, or whena coin is tossed n times, we speak of performingn trials. The result of a test (e.g. ‘pass’)or the result of a coin toss (e.g. ‘head’) is referred to as anoutcome.

Preface to the second edition

I thank Cambridge University Press, and in particular Simon Capelin, for the opportunity to revisit Data Analysis with Excel. I have revised sections of the book to include topics of contemporary relevance to undergraduate students, particularly in the area of uncertainty in measurement. I hope the book will continue to assist in developing the quantitative skills of students destined to graduate in the physical sciences. There is little doubt that the demand for such skills will continue to grow in society in general and particularly within industry, research, education and commerce.

This edition builds on the first with a new chapter added and others undergoing major or minor modifications (for example, to remedy mistakes, update references or include more end of chapter exercises).

Introduction

We may believe that the ‘laws of chance’ that apply when tossing a coin or rolling dice have little to do with experiments carried out in a laboratory. Rolling dice and tossing coins are the stuff of games. Surely, well planned and executed experiments provide precise and reliable data, immune from the laws of chance. Not so. Chance, or what we refer to more formally as probability, has rather a large role to play in every experiment. This is true whether an experiment involves counting the number of beta particles detected by nuclear counting apparatus in one minute, measuring the time a ball takes to fall a distance through a liquid or determining the values of resistance of 100 resistors supplied by a component manufacturer. Because it is not possible to predict with certainty what value will emerge when a measurement is made of a quantity, say of the time for a ball to fall a fixed distance through liquid, we are in a similar position to a person throwing several dice, who cannot know in advance which numbers will appear ‘face up’. If we are not to give up in frustration at our inability to discover the ‘exact’ value of a quantity experimentally, we need to find out more about probability and how it can assist rather than hamper our experimental studies.

In many situations a characteristic pattern or distribution emerges in data gathered when repeat measurements are made of a quantity. A distribution of values indicates that there is a probability associated with the occurrence of any particular value. Related to any distribution of ‘real’ data there is a probability distribution which allows us to calculate the probability of the occurrence of any particular value. Real probability distributions can often be approximated by a ‘theoretical’ probability distribution. Though it is possible to devise many theoretical probability distributions, it is the so called ‘normal’ probability distribution (also referred to as the Gaussian distribution) that is most widely used. This is because histograms of data obtained in many experiments have shapes that are very similar to that of the normal distribution. An attraction of the normal and other distributions is that they provide a way of describing data in a quantitative manner which complements and extends visual representations of data such as the histogram. Using the properties of the normal distribution we are usually able to summarise a whole data set, which may consist of many values, by one or two carefully chosen numbers.

Introduction

Establishing and understanding the relationship between quantities are principal goals in the physical sciences. As examples, we might be keen to know how the:

• size of a crystal depends on the growth time of the crystal;

• output intensity of a light emitting diode varies with the emission wavelength;

• amount of light absorbed by a chemical species depends on the species concentration;

• electrical power supplied by a solar cell varies with optical power incident on the cell;

• viscosity of an engine oil depends upon the temperature of the oil;

• rate of flow of a fluid through a hollow tube depends on the internal diameter of the tube.

Once an experiment is complete and the data presented in the form of an x–y graph, an examination of the data assists in answering important qualitative questions such as: Is there evidence of a clear trend in the data? If so, is that trend linear, and do any of the data conflict with the general trend? A qualitative analysis often suggests which quantitative methods of analysis to apply.

There are many situations in the physical sciences in which prior knowledge or experience suggests a relationship between measured quantities. Perhaps we are already aware of an equation which predicts how one quantity depends on another. Our goal in this situation might be to discover how well the equation can be made to ‘fit’ the data.