# The Uncertainty in Physical Measurements: An Introduction to Data Analysis in the Physics Laboratory

Springer Science & Business Media, Sep 18, 2008 - Science - 289 pages
The scienti c method is based on the measurement of di erent physical qu- tities and the search for relations between their values. All measured values of physical quantities are, however, a ected by uncertainty. Understanding the origin of uncertainty, evaluating its extent, and suitably taking it into account in data analysis, are fundamental steps for assessing the global accuracy of physical laws and the degree of reliability of their technological applications. The introduction to uncertainty evaluation and data analysis procedures is generally made in laboratory courses for freshmen. During my long-lasting teaching experience, I had the feeling of some sort of gap between the ava- able tutorial textbooks, and the specialized monographs. The present work aims at lling this gap, and has been tested and modi ed through a feedback interaction with my students for several years. I have tried to maintain as much as possible a tutorial approach, that, starting from a phenomenolo- cal introduction, progressively leads to an accurate de nition of uncertainty and to some of the most common procedures of data analysis, facilitating the access to advanced monographs. This book is mainly addressed to - dergraduate students, but can be a useful reference for researchers and for secondary school teachers. The book is divided into three parts and a series of appendices. Part I is devoted to a phenomenological introduction to measurement and uncertainty. In Chap.

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### Contents

 Physical Quantities 3 12 Physical Quantities 5 13 Direct and Indirect Measurement 6 14 Time Dependence of Physical Quantities 8 15 Counting of Random Events 10 16 Operative Deﬁnition of Physical Quantities 11 17 The Experimental Method 12 Measurement Units 13
 84 Nonindependent Quantities 163 85 Summary 167 Problems 168 Conﬁdence Levels 169 92 The Student Distribution 173 93 Applications of the Conﬁdence Level 174 Problems 176 Correlation of Physical Quantities 177

 22 Measurement Standards 14 23 The International System of Units SI 15 24 Other Systems of Units 18 25 Dimensional Analysis 20 Problems 24 Measuring Instruments 26 32 Classiﬁcations of Instruments 29 33 Static Characteristics of Instruments 31 34 Accuracy of an Instrument 35 35 Dynamical Behavior of Instruments 37 36 Counters 43 Uncertainty in Direct Measurements 45 42 Measurement Resolution 46 43 Random Fluctuations 48 44 Systematic Errors 61 45 Summary and Comparisons 68 Problems 74 Basic Probability Concepts 77 52 Sample Space Events 81 53 Probability of an Event 82 54 Addition and Multiplication of Events 87 55 Probability of the Sum of Events 89 56 Probability of the Product of Events 91 57 Combinatorial Calculus 94 Problems 96 Distributions of Random Variables 99 62 Random Variables and Distribution Laws 104 63 Numerical Characteristics of Distributions 108 64 Poisson Distribution 115 65 Normal Distribution 121 66 Meaning of the Normal Distribution 126 67 The CauchyLorentz Distribution 130 68 Multivariate Distributions 132 Problems 136 Statistical Tools 139 72 Sample Means and Sample Variances 143 73 Estimation of Parameters 147 Problems 152 Uncertainty in Indirect Measurements 154 82 Independent Quantities Linear Functions 156 83 Independent Quantities Nonlinear Functions 159
 102 Linear Correlation Coefficient 179 103 Linear Relations Between Two Quantities 181 104 The Least Squares Method 186 Problems 192 The Chi Square Test 193 112 Deﬁnition of Chi Square 194 113 The Chi Square Distribution 198 114 Interpretation of the Chi Square 202 Problems 203 Presentation of Experimental Data 207 A2 Tables 210 A3 Graphs 212 A4 Histograms 216 Systems of Units 218 B2 Units Not Accepted by the SI 223 B3 British Units 224 B4 NonSI Units Currently Used in Physics 225 B5 Gauss cgs Units 226 C Tables 227 C3 Integrals of the Standard Normal Distribution 229 C4 Integrals of the Student Distribution 233 C5 Integrals of the Chi Square Distribution 235 C6 Integrals of the Linear Correlation Coefficient Distribution 237 D Mathematical Complements 239 D2 Transformed Functions of Distributions 243 D3 Moments of the Binomial Distribution 245 D4 Moments of the Uniform Distribution 246 D5 Moments of the Poisson Distribution 247 D6 Moments of the Normal Distribution 248 D7 Parameters of the Cauchy Distribution 252 D8 Theorems on Means and Variances 253 Experiments 255 Measurement of Period 258 Elastic Constant 261 Oscillations 266 Dependence of Period on Length 269 Inﬂuence of Mass and Amplitude 274 E7 Time Response of a Thermometer 277 Suggested Reading 283 Index 285 Copyright