A Student's Guide to Data and Error Analysis

239 Pages·2011·3.416 MB·English

by  Herman J. C. Berendsen

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A Student’s Guide to Data and Error Analysis
All students taking laboratory courses within the physical sciences and
engineering will benefit from this book, whilst researchers will find it an
invaluable reference. This concise, practical guide brings the reader up to
speedontheproperhandlingandpresentationofscientificdataanditsinac-
curacies. It covers all the vital topics with practical guidelines, computer
programs (in Python), and recipes for handling experimental errors and
reporting experimental data. In addition to the essentials, it also provides
further background material for advanced readers who want to understand
howthemethodswork.Plentyofexamples,exercises,andsolutionsarepro-
videdtoaidandtestunderstanding,whilstusefuldata,tables,andformulas
arecompiledinahandysectionforeasyreference.
HERMAN J. C. BERENDSENisEmeritusProfessorofPhysicalChemistryat
theUniversityofGroningen,theNetherlands.Hisresearchstartedinnuclear
magneticresonance,butfocusedlateronmoleculardynamicssimulationson
systemsofbiologicalinterest.Heisoneofthepioneersinthisfieldand,with
over37000citations,isoneofthemostquotedauthorsinphysicsandchem-
istry.Hehastaughtcoursesinmolecularmodelingworldwideandauthored
thebookSimulatingthePhysicalWorld(CambridgeUniversityPress,2007).

A Student’s Guide to
Data and Error Analysis
HERMAN J. C. BERENDSEN
EmeritusProfessorofPhysicalChemistry,
UniversityofGroningen,theNetherlands
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ISBN978-0-521-11940-5(Hardback)–ISBN978-0-521-13492-7(pbk.)
1. Erroranalysis(Mathematics) I. Title.
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Contents
Preface pagexi
Part I Data and error analysis 1
1 Introduction 3
2 Thepresentationofphysicalquantitieswiththeirinaccuracies 5
2.1 Howtoreportaseriesofmeasurements 5
2.2 Howtorepresentnumbers 9
2.3 Howtoexpressinaccuracies 10
2.4 Reportingunits 13
2.5 Graphicalpresentationofexperimentaldata 14
3 Errors:classificationandpropagation 18
3.1 Classificationoferrors 18
3.2 Errorpropagation 19
4 Probabilitydistributions 27
4.1 Introduction 27
4.2 Propertiesofprobabilitydistributions 29
4.3 Thebinomialdistribution 32
4.4 ThePoissondistribution 36
4.5 Thenormaldistribution 37
4.6 Thecentrallimittheorem 41
4.7 Otherdistributions 42
5 Processingofexperimentaldata 53
5.1 Thedistributionfunctionofadataseries 54
5.2 Theaverageandthemeansquareddeviation
ofadataseries 57
5.3 Estimatesformeanandvariance 58
5.4 AccuracyofmeanandStudent’st-distribution 59
vii
viii CONTENTS
5.5 Accuracyofvariance 60
5.6 Handlingdatawithunequalweights 61
5.7 Robustestimates 63
6 Graphicalhandlingofdatawitherrors 71
6.1 Introduction 71
6.2 Linearizationoffunctions 73
6.3 Graphicalestimatesoftheaccuracyofparameters 77
6.4 Usingcalibration 78
7 Fittingfunctionstodata 84
7.1 Introduction 84
7.2 Linearregression 87
7.3 Generalleast-squaresfit 92
7.4 Thechi-squaredtest 95
7.5 Accuracyoftheparameters 98
7.6 F-testonsignificanceofthefit 106
8 BacktoBayes:knowledgeasaprobabilitydistribution 111
8.1 Directandinverseprobabilities 111
8.2 EnterBayes 112
8.3 Choosingtheprior 114
8.4 ThreeexamplesofBayesianinference 114
8.5 Conclusion 121
References 123
Answers to exercises 125
Part II Appendices 133
A1 Combininguncertainties 135
A2 Systematicdeviationsduetorandomerrors 138
A3 Characteristicfunction 141
A4 Frombinomialtonormaldistributions 143
A4.1 Thebinomialdistribution 143
A4.2 Themultinomialdistribution 144
A4.3 ThePoissondistribution 145
A4.4 Thenormaldistribution 146