MEASUREMENT
Part I General Problems 1 Measurability 3 11 What Can Be Measured? 3 12 Counting and Measuring 4 13 Physical Measurement 7 14 Psychophysical Measurement 10 15 The Debate at the British Association for the Advancement of Science 12 16 A Turning Point: Stevens’s Twofold Contribution 13 161 Direct Measurement of Percepts 13 162 Classification of Measurement Scales 14 17 The Representational Theory 17 18 The Role of the Measuring System 18 19 The Proposed Approach 19 References 21 2 Uncertainty 23 21 Why are Measurement Results not Certain? 23 22 Historical Background 24 221 Gauss, Laplace and the Early Theory of Errors 24 222 Fechner and Thurstone: The Uncertainty of Observed Relations 26 223 Campbell: Errors of Consistency and Errors of Methods 30 224 The Contribution of Orthodox Statistics 31 225 Uncertainty Relations in Quantum Mechanics 32 226 The Debate on Uncertainty at the End of the Twentieth Century 34 23 The Proposed Approach 36 231 Uncertainty Related to the Measurement Scale and to Empirical Relations 37 232 Uncertainty Related to the Measurement Process and the Measuring System 39 233 Information Flux Between the Objects(s) and the Observer 40 References 41 Part II The Theory 3 The Measurement Scale: Deterministic Framework 45 31 What is the Meaning of Measurement? 45 32 The General Framework 47 321 Overview 47 322 Some Formal Statements 48 323 Overview of the Main Types of Scales 49 33 Ordinal Scales 49 331 Motivations for Dealing with Ordinal Scales 49 332 Serialising and Numbering Objects 50 333 Representation for Order Structures 52 34 Interval Scales 55 341 Dealing with Intervals 55 342 Measuring Differences 56 343 Representation for Difference Structures 59 35 Ratio Scales for Intensive Structures 65 351 Is Empirical Addition Necessary for Establishing a Ratio Scale? 65 352 Extensive and Intensive Quantities 66 353 Scaling Intensities 67 354 Representation for Intensive Structures 70 36 Ratio Scales for Extensive Structures 73 361 The Role of Additivity in Measurement 73 362 Representation for Extensive Structures 76 37 Derived Scales 80 371 Derived Versus Fundamental Scales 80 372 Representation for Derived Scales 81 373 Systems of Quantities 85 374 The International System of Metrology 86 38 Summary 88 References 90 4 The Measurement Scale: Probabilistic Approach 93 41 Working with Probability 93 411 The Nature of Probability 93 412 The Rules of Probability 94 413 An Illustrative Example 97 414 Probability as a Logic 100 415 Probabilistic Variables 100 416 Probabilistic Functions 102 417 Probabilistic Relations 103 418 Continuity 105 419 Non-probabilistic Approaches to Measurement Uncertainty 106 42 Probabilistic Representations 107 43 Probabilistic Fundamental Scales 108 431 Order Structures 108 432 Difference Structures 110 433 Intensive Structures 110 434 Extensive Structures 111 44 Probabilistic Derived Scales 111 441 An Introductory Example 111 442 Probabilistic Cross-Order Structures 113 443 Probabilistic Cross-Difference Structures 115 45 Summary 115 References 116 5 The Measurement Process 117 51 How Can We Measure? 117 52 Deterministic Model of the Measurement Process 121 53 Probabilistic Model of the Measurement Process 124 54 Probability Space of the Measurement Process 127 541 From Numbers to Numbers 128 542 From Things to Numbers 132 55 Systematic Effects 137 56 Continuous Versus Discrete Representations 142 57 Overall Probabilistic Framework and Generalisations 144 References 145 6 Inference in Measurement 147 61 How Can We Learn from Data? 147 62 Probabilistic Models and Inferences 148 621 The Bernoullian Model 148 622 A Classification of Probabilistic Inferences 149 63 Measurement Evaluation 155 64 Measurement Verification 160 65 Summary 161 References 162 7 Multidimensional Measurement 163 71 What Happens when Moving from One to Two Dimensions 163 72 Distances and Metrics 165 73 Nominal and Distance Structures 167 731 Nominal Structures 167 732 Distance Structures 168 74 Probabilistic Representation for Nominal and Metric Structures 172 75 Additional Notes on Multidimensional Measurement 174 References 175 Part III Applications 8 Perceptual Measurement 179 81 Measuring the Impossible 179 82 Measuring the Intensity of a Sensation 181 821 Premise: Some Acoustic Quantities 181 822 Loudness of Pure Tones 182 823 Loudness of Pink Noise 188 824 Direct Measurement of Loudness: Master Scaling 190 825 Direct Measurement of Loudness: Robust Magnitude Estimation 193 826 Indirect Measurement: Loudness Model 198 83 State of the Art, Perspective and Challenges 199 References 203 9 The Evaluation of Measurement Uncertainty 205 91 How to Develop a Mathematical Model of the Measurement Process 205 911 Statement of the Problem 205 912 Linear Models 206 913 Systematic Effects and Random Variations 207 914 Observability 209 915 Low-Resolution Measurement 210 916 Practical Guidelines 213 917 Hysteresis Phenomena 214 918 Indirect Measurement 216 92 Measurement Software 217 93 A Working Example 219 References 221 10 Inter-Comparisons and Calibration 223 101 A Worldwide Quality Assurance System for Measurement 223 102 A Probabilistic Framework for Comparisons 224 1021 How Key Comparisons Work 224 1022 Checking the Individual Results 224 1023 The Paradigm of the Probabilistic Scale 226 1024 Summary of the Proposed Approach 230 1025 A Working Example 231 103 Calibration 232 References 236 11 Measurement-Based Decisions 237 111 The Inferential Process in Conformance Assessment 237 112 A Probabilistic Framework for Risk Analysis 238 1121 Insight into Conformance Assessment 238 1122 Probabilistic Framework 242 1123 Illustrative Example 243 113 Software for Risk Analysis 246 114 Chemical Analysis 246 115 Legal Metrology 248 116 A Working Example 249 References 251 12 Dynamic Measurement 253 121 Dynamic Measurement: An Introduction 253 122 Direct Dynamic Measurement 254 1221 A Probabilistic Framework for Direct Dynamic Measurement 254 1222 Evaluation of the Uncertainty Generated by Dynamic Effects in Instrumentation 261 123 Indirect Dynamic Measurement: Spectrum Measurement 265 References 271