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

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Springer Science & Business Media, Sep 18, 2008 - Science - 289 pages
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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
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About the author (2008)

Since 1978, the author has worked on x-ray spectroscopy (XAFS) with Synchrotron Radiation. He participated in the realization of the PWA laboratories at Adone (Frascati) and of the italian beamline BM08-Gilda at ESRF (Grenoble). He regularly performs XAFS measurements at international synchrotron facilities, including Adone (Frascati), LURE (Orsay), ESRF (Grenoble).  Prof. Fornasini has also developed a software package for the analysis of EXAFS, specifically tailored for the study of anharmonic thermal effects.

Prof. Fornasini has been a member of the International Advisory Committee on Synchrotron Radiation in Materials Science since 1996 and given several invited lectures and seminars in Chester (UK), Bonn, Regensburg, Bad Honnef, Dakar, Warsaw, Frascati, Bologna, Corvallis (Oregon), Stanford University, Hirosaki (Japan), Chiba (Japan).  Since 1997, he is a lecturer at the Italian School of Synchrotron Radiation, organized every two years by the "Societą Italiana di Luce di Sincrotrone". He has given lectures on x-ray techniques in various Italian and international schools (Torino 1999, Rostock 2000, Varsaw 2004, Planneralm Austria 2005, Dakar 2005).

Prof. Fornasini is the author or co-author of over 100 papers published in international journals and referees of the journals: Physical Review Letters, Physical Review B, Physics Letters, and Journal of Synchrotron Radiation.

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