Dealing with Uncertainties (eBook)

Preface 6
Preface to the First Edition 7
Prolog. Seven Myths in Error Analysis 9
Contents 11
Introduction 13
The Exactness of Science 14
Falsification vs. Verification 16
Data without Uncertainty 16
Uncertainties in Every-Day Life 17
Basics on Data 19
Measurement Data 19
The Best Estimate 20
Directly Measured Data 21
Indirectly Measured Data 21
Counting vs. Measuring 22
The Basic Counting Model 23
Indirect Counting 24
Analog vs. Digital 24
``Analog'' Measurements 25
Dealing with Data (Numerals) 26
Valid Digits 26
Truncation of Numbers 26
Rounding 26
Basics on Uncertainties 29
General Characteristics of Uncertainties 31
Shape of Uncertainty Distributions 32
Definitions 34
Terminology 34
Necessary Requirements 34
Deviations 35
Random Uncertainties 37
Maximum Uncertainties (Tolerances) 41
Limits 42
Outliers (Flyers) 43
Uncertainty of Data Depending on One Variable 45
Length 45
Circular Area 45
Multiple Uncertainty Components (Quadratic Sum) 47
Properties of the ``Quadratic'' Sum 49
Subtraction in Quadrature 51
Combined Standard Uncertainty 51
Uncertainty Evaluations (Error Analysis) 52
Inductive Method 52
Deductive Method 53
Experimental Uncertainty 53
Counting and Measurement Uncertainties 54
Parameter Uncertainties 56
Model Uncertainties 56
Additional Experimental Uncertainties 60
Radioactive Decay, a Model for Random Events 61
Time Interval Distribution of Radioactive Events 61
Prescaling 63
Counting Loss (Dead Time) 64
Direct Correction of Dead Time 65
Direct Correction for Lost Counts 67
Loss Correction Using a Pulse Generator 67
Inductive Approach to Uncertainty (Example) 69
Properties of Data Sets and Arrays 69
Reproducibility within Data Sets 76
Linear Regression (Least-Squares Method) 77
Frequency and Probability Distributions 80
Frequency Distribution (Spectrum) 80
Characteristics of Distributions 83
Effect of Data Uncertainty on the Distribution 85
Probability Distributions 88
Binomial Distribution 88
Poisson Distribution 89
Normal (or Gaussian) Distribution 91
Finite Distributions 95
Convolution of Uncertainty Distributions 97
Statistical Confidence 98
Dealing with Probabilities 99
Deductive Approach to Uncertainty 104
Theoretical Situation 104
Practical Situation 104
Best Estimates Using Internal Uncertainties 106
Deductive vs. Inductive Uncertainties 109
The Sign of an Uncertainty 110
Benefits of Repeated Measurements 111
Regression Analysis (Least-Squares Method) 116
Weighted Mean 118
Weighted Linear Regression 119
General Regression Analysis 121
Benefits of Regression Analysis 122
Data Consistency within Data Sets 123
Criterion of Chauvenet 123
Discarding Data with Internal Uncertainties 125
Correlation 126
Introduction 126
Measure of Relation 127
Correlated (Systematic) Uncertainties 128
Sign of a ``Systematic'' Uncertainty 129
Differentiation from Uncorrelated Uncertainties 130
More Examples of Correlated Uncertainties 137
External Scale Uncertainties? 141
Differentiation from ``Systematic Errors'' 141
Gross Mistakes 142
Corrections 142
Correlation in Cases of Linear Regression 148
Weighted Linear Regression (Example) 148
Linear Regression without Weighting (Example) 149
Consistency among Data Sets 152
Contradictory Data Sets 154
Dependent (Correlated) Data Sets 156
Target Shooting as a Model for Uncertainties 158
Accuracy vs. Precision 160
Dealing with Internal Uncertainties 161
Calculations with Both Types of Uncertainties 164
Uncertainty of a Sum 165
Uncertainty of a Difference 166
Uncertainty of a Product 169
Uncertainty of a Ratio 170
Uncertainty of a Power (Root) 171
Uncertainty of More Exotic Functions 172
Total Uncertainty 173
Adding Correlated to Uncorrelated Uncertainties 173
Combined Uncertainty of a Single Best Estimate 174
Combined Uncertainty of Data Sets 174
Using Internal Uncertainties for Diagnosis 176
Quality Assurance 176
Analogy to Bayes' Principle 178
Chi-Squared Test 179
Presentation and Estimation of Uncertainties 183
Graphic Presentation, Also of Uncertainties 183
Uncertainty Bars (Error Bars) 184
Charts 185
Transformations 185
Correct Presentation of Uncertainties 190
Finding the Size of Internal Uncertainties 192
Ideal Situation in Measurements 193
Pragmatic Solution for Measurements 193
Estimating the Size of Uncertainties 194
Finding Upper Limits 195
Finding Lower Limits 197
Feedback of Uncertainties on Experiment Design 198
Optimizing Experiments 198
Reasons For and Against Optimization 199
Prevalent Design Criteria 200
Optimizing Background Measurements 201
Optimized Simple Background Measurement 201
Optimized Complex Background Measurement 202
Optimizing with Respect to Dead Time 204
Optimizing in View of the Mathematical Presentation 206
Optimizing Flat Dependences 207
Optimizing Linear Dependences 207
Optimum Angles for Cross Section Measurements 208
Achieving the Smallest Overall Uncertainty 209
The Ratio Method 209
Solutions 218
Index 231