Table Of ContentUndergraduate Lecture Notes in Physics
Costas Christodoulides
George Christodoulides
Analysis and
Presentation of
Experimental
Results
With Examples, Problems and
Programs
Undergraduate Lecture Notes in Physics
Undergraduate Lecture Notes in Physics (ULNP) publishes authoritative texts covering
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Physics Program, Bard College, Annandale-on-Hudson, NY, USA
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SUNY Suffolk County Community College, Long Island, NY, USA
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Humboldt University, Oldenburg, Niedersachsen, Germany
More information about this series at http://www.springer.com/series/8917
Costas Christodoulides George Christodoulides
(cid:129)
Analysis and Presentation
of Experimental Results
With Examples, Problems and Programs
123
Costas Christodoulides George Christodoulides
Department ofPhysics, Schoolof Applied CentreValibel, Institute forLanguage
MathematicalandPhysical Sciences andcommunication
National Technical University of Athens Universitécatholique deLouvain
Zografou,Athens GR15780 Louvain-la-Neuve
Greece Belgium
Code and examples for the book “Analysis and Presentation of Experimental Results”:
https://github.com/aperbook/APER.
ISSN 2192-4791 ISSN 2192-4805 (electronic)
Undergraduate Lecture Notesin Physics
ISBN978-3-319-53344-5 ISBN978-3-319-53345-2 (eBook)
DOI 10.1007/978-3-319-53345-2
LibraryofCongressControlNumber:2017932001
Translation from the Greek language edition: Amάktrη jai Paqotrίarη Peiqalasijώm
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Preface
Theimportanceoftheanalysisandthepresentationofexperimentaldatacannotbe
overstated.Goodexperimentalresultsmayberendereduselessbyfailingtokeepto
certainrulesintheirpresentation,either toanaudienceorinwrittenform.Itisour
intention to present in this book these methods at an introductory university level.
Those working in the experimental sciences, but also anyone involved in the
analysis of numerical data, may find the book useful.
The book is intended to be used as a textbook and this has determined its
characteristics:Thetheoreticalproofsaregiveninconsiderabledetail,manyfigures
are used, as well as a large number of examples and problems to be solved by the
reader.Thevastmajorityoftheexamplesaresolvedusingfoursoftwarepackages:
Excel®, Origin®, Python and R. Most of the problems may also be solved using
these programs. Excel® is used due to its wide availability as a program for data
analysis, Origin® because it is an excellent program for creating graphical pre-
sentations of data. Python and R are used because they are free, open-source
programming languages, widely usedin data science. Reference to these programs
is made using the symbols [E], [O], [P] and [R], respectively. The same symbols
also indicate that a certain problem may be solved using the corresponding
program.
ThebookmaybeusedasatextbookforanintroductorycourseonDataAnalysis
and Presentation. It is hoped that it will provide a useful addition to the existing
literature.
Athens, Greece Costas Christodoulides
January 2017 George Christodoulides
v
Contents
1 Basic Statistical Concepts.. ..... .... .... .... .... .... ..... .. 1
1.1 Tables, Distributions and Histograms.. .... .... .... ..... .. 1
1.2 Probability Density... ..... .... .... .... .... .... ..... .. 14
1.3 Mean Value.... .... ..... .... .... .... .... .... ..... .. 15
1.4 Measures of Dispersion .... .... .... .... .... .... ..... .. 18
1.4.1 Range.. .... ..... .... .... .... .... .... ..... .. 18
1.4.2 Deviation from the Mean and Mean
Absolute Deviation from the Mean. .... .... ..... .. 19
1.4.3 Standard Deviation. .... .... .... .... .... ..... .. 20
2 Measurement Errors . .... ..... .... .... .... .... .... ..... .. 39
2.1 Errors of Measurements.... .... .... .... .... .... ..... .. 39
2.1.1 Accidental or Random Errors. .... .... .... ..... .. 39
2.1.2 Systematic Errors .. .... .... .... .... .... ..... .. 40
2.1.3 Personal Errors.... .... .... .... .... .... ..... .. 42
2.1.4 Occasional Errors .. .... .... .... .... .... ..... .. 42
2.1.5 The Errors in Reading the Indications
of Instruments..... .... .... .... .... .... ..... .. 43
2.2 Errors in Compound Quantities .. .... .... .... .... ..... .. 45
2.2.1 Error in a Sum or a Difference.... .... .... ..... .. 46
2.2.2 Error in a Product.. .... .... .... .... .... ..... .. 48
2.2.3 Error in a Power... .... .... .... .... .... ..... .. 48
2.2.4 Error in a Quotient. .... .... .... .... .... ..... .. 49
2.2.5 The Use of Differentials . .... .... .... .... ..... .. 50
3 A Thought Experiment.... ..... .... .... .... .... .... ..... .. 57
3.1 The Thought Experiment... .... .... .... .... .... ..... .. 57
Reference... .... .... .... ..... .... .... .... .... .... ..... .. 75
4 The Statistical Analysis of Experimental Results .... .... ..... .. 77
4.1 The Mean and the Dispersion of the Results
of Measurements .... ..... .... .... .... .... .... ..... .. 77
vii
viii Contents
4.2 The Standard Deviations ... .... .... .... .... .... ..... .. 80
4.2.1 The Standard Deviation of the Measurements. ..... .. 80
4.2.2 The Standard Deviation of the Mean ... .... ..... .. 90
4.2.3 The Relationship Between r and r .... .... ..... .. 93
x
4.2.4 The Relationship Between sx and r and rx... ..... .. 94
4.3 The Standard Deviation of the Standard Deviation
of the Mean.... .... ..... .... .... .... .... .... ..... .. 98
4.4 Information Derived from the Measurement of x and sx .... .. 100
4.4.1 The Mean Value of the Results
of the Measurements and Its Standard Deviation ... .. 100
4.4.2 The Statistical Distribution of the Results
of the Measurements.... .... .... .... .... ..... .. 103
4.4.3 Statistical Estimates for the Mean.. .... .... ..... .. 115
4.4.4 Summary of the Method of Analysis of the Results. .. 116
References .. .... .... .... ..... .... .... .... .... .... ..... .. 121
5 The Presentation of Numerical Results.... .... .... .... ..... .. 123
5.1 Significant Figures and Rounding of Numbers... .... ..... .. 123
5.2 The Presentation of a Numerical Result of a Series of
Measurements .. .... ..... .... .... .... .... .... ..... .. 124
5.3 The Number of Significant Figures Used in the Presentation
of Numerical Results . ..... .... .... .... .... .... ..... .. 126
5.4 The International System of Units (S.I.) and the Rules
of Its Use.. .... .... ..... .... .... .... .... .... ..... .. 129
5.5 Recommendations on the Notation Used for Mathematical
Constants, Algebraic Parameters, Variables, Indices,
Mathematical Functions, Operators, Physical Units,
Elementary Particles and Isotopes .... .... .... .... ..... .. 134
Reference... .... .... .... ..... .... .... .... .... .... ..... .. 136
6 The Propagation of Errors. ..... .... .... .... .... .... ..... .. 137
6.1 The Combination of Two Series of Measurements
of the Same Physical Magnitude . .... .... .... .... ..... .. 137
6.1.1 The Mean x of All the Measurements... .... ..... .. 138
6.1.2 The Standard Deviation sx of All the Measurements. .. 138
6.1.3 The Standard Deviation of the Mean r
x
of All the Measurements. .... .... .... .... ..... .. 139
6.2 The Mean and the Standard Deviation of a Function
of Measured Quantities .... .... .... .... .... .... ..... .. 141
6.2.1 The Mean and the Standard Deviations
of a Function of One Variable .... .... .... ..... .. 141
6.2.2 The Mean and the Standard Deviation
of an Algebraic Sum.... .... .... .... .... ..... .. 146
Contents ix
6.2.3 The Mean and the Standard Deviations
of a General Function of Many Variables.... ..... .. 149
6.2.4 Another Approach to the Evaluation of the Mean
and the Standard Deviation of a Compound
Quantity .... ..... .... .... .... .... .... ..... .. 155
6.3 The Error in Qðx; y; z;...Þ Due to the Errors in x; y; z;..... .. 157
6.3.1 The Case of Asymmetrical Errors.. .... .... ..... .. 158
7 The Three Basic Probability Distributions . .... .... .... ..... .. 163
7.1 Histograms of Experimental Probabilities... .... .... ..... .. 163
7.2 The Binomial or Bernoulli Distribution .... .... .... ..... .. 166
7.2.1 The Normalization of the Binomial Distribution.... .. 178
7.2.2 The Mean Value of x for the Binomial Distribution. .. 178
7.2.3 The Standard Deviation of x from the Mean
for a Binomial Distribution... .... .... .... ..... .. 179
7.3 The Poisson Distribution ... .... .... .... .... .... ..... .. 180
7.3.1 The Normalization of the Poisson Distribution..... .. 188
7.3.2 The Mean Value of x for the Poisson Distribution.. .. 188
7.3.3 The Standard Deviation from the Mean of x
for the Poisson Distribution .. .... .... .... ..... .. 188
7.4 The Normal or Gaussian Distribution.. .... .... .... ..... .. 192
7.4.1 The Normalization of the Gaussian Distribution.... .. 200
7.4.2 The Mean Value of x for the Gaussian Distribution. .. 200
7.4.3 The Standard Deviation of x from the Mean
for the Gaussian Distribution . .... .... .... ..... .. 200
7.4.4 Testing Whether a Set of Data Has a Gaussian
Distribution.. ..... .... .... .... .... .... ..... .. 201
7.4.5 The Gaussian Distribution and the Phenomenon
of Diffusion . ..... .... .... .... .... .... ..... .. 209
References .. .... .... .... ..... .... .... .... .... .... ..... .. 214
8 The Statistics of Radioactivity... .... .... .... .... .... ..... .. 215
8.1 The Behavior of Large Samples. The Law of Radioactivity.. .. 215
8.2 Nuclear Disintegrations and the Binomial Distribution. ..... .. 218
8.3 Radioactivity and the Poisson Distribution.. .... .... ..... .. 231
8.4 The Counting Rate of Nuclear Disintegrations and Its Error . .. 234
Reference... .... .... .... ..... .... .... .... .... .... ..... .. 238
9 Elements from the Theory of Errors.. .... .... .... .... ..... .. 239
9.1 The Normal or Gaussian Law of Errors.... .... .... ..... .. 239
9.2 The Lyapunov Central Limit Theorem. .... .... .... ..... .. 244
9.3 The Best Estimate that May Be Made for the Real Value
of a Magnitude, Based on the Results
of N Measurements of It ... .... .... .... .... .... ..... .. 253
x Contents
9.4 The Weighting of Values... .... .... .... .... .... ..... .. 255
9.5 The Joint Probability Density for Two Random Variables... .. 272
9.6 The Probability Density of the Sum
of Two Random Variables.. .... .... .... .... .... ..... .. 277
9.6.1 TheProbabilityDensityoftheSumofTwoNormally
Distributed Random Variables .... .... .... ..... .. 280
References .. .... .... .... ..... .... .... .... .... .... ..... .. 289
10 Comparison and Rejection of Measurements... .... .... ..... .. 291
10.1 The Problem of the Rejection of Measurements.. .... ..... .. 291
10.2 Chauvenet’s Criterion. ..... .... .... .... .... .... ..... .. 293
10.3 Comments Concerning the Rejection of Measurements ..... .. 296
10.4 Comparison of the Means of Two Series of Measurements
of the Same Quantity. ..... .... .... .... .... .... ..... .. 298
11 The Method of Least Squares ... .... .... .... .... .... ..... .. 301
11.1 Introduction .... .... ..... .... .... .... .... .... ..... .. 301
11.2 The Theoretical Foundation of the Method of Least Squares. .. 302
11.3 The Fitting of Curves to Experimental Points ... .... ..... .. 305
11.3.1 Straight Line. ..... .... .... .... .... .... ..... .. 305
11.3.2 Polynomial .. ..... .... .... .... .... .... ..... .. 327
11.3.3 Other Curves. ..... .... .... .... .... .... ..... .. 340
11.3.4 The Reduction of Non-linear Relations to Linear... .. 342
11.4 The Choice of the Optimum Function Fitted
to a Set of Experimental Results . .... .... .... .... ..... .. 346
11.5 The Fractional Absolute Deviation of the Experimental
Values from the Values of the Curve.. .... .... .... ..... .. 346
11.6 Smoothing . .... .... ..... .... .... .... .... .... ..... .. 348
11.7 The Error in a Value Read off a Smoothed Curve.... ..... .. 354
11.8 The Regression Line and the Coefficient of Correlation..... .. 357
11.9 The Use of the Method of Least Squares in the Solution
of a System of Overdetermined Linear Equations .... ..... .. 362
11.9.1 Equations in Two Variables .. .... .... .... ..... .. 363
11.9.2 Equations in Three Variables . .... .... .... ..... .. 368
References .. .... .... .... ..... .... .... .... .... .... ..... .. 375
12 Graphs. .... .... .... .... ..... .... .... .... .... .... ..... .. 377
12.1 Introduction .... .... ..... .... .... .... .... .... ..... .. 377
12.2 The Axes.. .... .... ..... .... .... .... .... .... ..... .. 378
12.2.1 Linear Scales ..... .... .... .... .... .... ..... .. 378
12.2.2 Logarithmic Scales. .... .... .... .... .... ..... .. 380
12.2.3 Polar Diagrams.... .... .... .... .... .... ..... .. 384
12.2.4 Other Matters Relating to the Axes and the Scales
of Graphs ... ..... .... .... .... .... .... ..... .. 387
12.2.5 Legends of the Figure, Labels of the Axes
and the Units of Physical Magnitudes... .... ..... .. 391
