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

3 | |

5 | |

6 | |

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 |

285 | |

### Other editions - View all

Arbeits-Und Dienstrecht Der Krankenhausartze Von A-Z H. P. Lippert,B. R. Kern No preview available - 1993 |

The Uncertainty in Physical Measurements: An Introduction to Data Analysis ... Paolo Fornasini No preview available - 2010 |

The Uncertainty in Physical Measurements: An Introduction to Data Analysis ... Paolo Fornasini No preview available - 2008 |

### Common terms and phrases

amplitude Appendix approximate behavior binomial distribution calculated Chap coefficient continuous random variable correlation corresponding coverage factor Data Analysis defined degrees of freedom depends dimensional directly measured discrete random variable display resolution distribution of sample due to random elastic constant equation estimate evaluate Example experiment experimental points expressed finite frequency function graph histogram independent input quantity instrument integral interval length Let us consider limiting distribution linear regression mass maximum mean and variance mercury-in-glass thermometer normal distribution obtained oscillation outcomes parameters parent distribution parent population pendulum period physical quantity Poisson distribution procedure quantity G random fluctuations relation relative uncertainty respectively sample means sample space sample variance significant digits single measures Springer Science+Business Media square test standard deviation standard normal standard uncertainty statistical statistically independent systematic errors Table temperature tossed uncertainty due unit standard verify weighted average width zero