1、 NASA HANDBOOK Measurement Uncertainty Analysis Principles and Methods NASA Measurement Quality Assurance Handbook ANNEX 3 Measurement System Identification: Metric July 2010 National Aeronautics and Space Administration Washington DC 20546 NASA-HDBK-8739.19-3Approved: 2010-07-13 Baseline APPROVED F
2、OR PUBLIC RELEASE DISTRIBUTION UNLIMITED Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-ii This page intentionally left blank. Provided by IHSNot for R
3、esaleNo reproduction or networking permitted without license from IHS-,-,-iii DOCUMENT HISTORY LOG Status Document Revision Approval Date Description Baseline 2010-07-13 Initial Release (JWL4)This document is subject to reviews per Office of Management and Budget Circular A-119, Federal Participatio
4、n in the Development and Use of Voluntary Standards (02/10/1998) and NPD 8070.6, Technical Standards (Paragraph 1.k). Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Provided by IHSNot for ResaleNo reproduction or networking permitted without license
5、 from IHS-,-,-v This page intentionally left blank. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-vi TABLE OF CONTENTS FOREWORD IV TABLE OF CONTENTS VI LIST OF FIGURES XIV LIST OF TABLES XV ACRONYMS AND ABBREVIATIONS XVII PREFACE XVIII ACKNOWLEDGEM
6、ENTS XIX EXECUTIVE SUMMARY XIX CHAPTER 1: INTRODUCTION 1 1.1 PURPOSE . 1 1.2 SCOPE 1 1.3 BACKGROUND 2 1.4 APPLICATION 2 CHAPTER 2: BASIC CONCEPTS AND METHODS 8 2.1 DEFINE THE MEASUREMENT PROCESS 8 2.2 DEVELOP THE ERROR MODEL 9 2.3 IDENTIFY MEASUREMENT ERRORS AND DISTRIBUTIONS . 9 2.3.1 Error Distrib
7、utions 12 2.3.2 Choosing the Appropriate Distribution 13 2.4 ESTIMATE UNCERTAINTIES 14 2.4.1 Distribution Variance . 14 2.5 COMBINE UNCERTAINTIES 16 2.5.1 Variance Addition Rule Direct Measurements . 16 2.5.2 The Variance Addition - General Model 16 2.5.3 Error Correlations Direct Measurements 17 2.
8、5.4 Cross-Correlations between Error Components . 18 2.5.5 Combined Uncertainty . 19 2.5.6 Establishing Correlations . 19 2.5.7 Degrees of Freedom . 20 2.6 REPORT THE ANALYSIS RESULTS . 21 2.6.1 Confidence Limits and Expanded Uncertainty 21 CHAPTER 3: ESTIMATING UNCERTAINTY 24 Provided by IHSNot for
9、 ResaleNo reproduction or networking permitted without license from IHS-,-,-vii 3.1 TYPE A ESTIMATES 24 3.1.1 Statistics for Sampled Values . 24 3.2 TYPE B ESTIMATES 27 3.2.1 Normal Distribution . 27 3.2.2 Lognormal Distribution 28 3.2.3 Type B Degrees of Freedom 28 3.2.4 Students t Distribution 29
10、3.2.5 Other Distributions . 29 CHAPTER 4: INTERPRETING AND APPLYING EQUIPMENT SPECIFICATIONS 31 4.1 MEASURING AND TEST EQUIPMENT 31 4.1.1 Artifacts 31 4.1.2 Instruments . 31 4.1.3 Sensors and Transducers 31 4.1.4 Signal Conditioners 32 4.1.5 Data Acquisition . 33 4.1.6 Data Processors 33 4.1.7 Outpu
11、t Displays 33 4.2 PERFORMANCE CHARACTERISTICS . 33 4.2.1 Static Characteristics 34 4.2.2 Dynamic Characteristics 34 4.2.3 Other Characteristics 34 4.3 OBTAINING SPECIFICATIONS 34 4.4 INTERPRETING SPECIFICATIONS . 34 4.4.1 Terms, Definitions and Abbreviations 35 4.4.2 Qualifications, Stipulations and
12、 Warnings . 36 4.4.3 Specification Units . 36 4.5 APPLYING SPECIFICATIONS 37 4.5.1 Confidence Levels 37 4.5.2 Error Distributions 39 4.5.3 Combining Specifications 40 CHAPTER 5: DIRECT MEASUREMENTS 43 5.1 DEFINE THE MEASUREMENT PROCESS 43 5.1.1 Gage Block Specifications . 44 5.1.2 Micrometer Specifi
13、cations . 44 5.1.3 Environmental Temperature Specifications . 44 5.2 DEFINE THE ERROR MODEL 45 5.3 IDENTIFY ERROR SOURCES AND DISTRIBUTIONS 45 5.4 ESTIMATE UNCERTAINTIES 46 5.4.1 Gage Block Bias . 46 5.4.2 Repeatability (Random Error) 48 Provided by IHSNot for ResaleNo reproduction or networking per
14、mitted without license from IHS-,-,-viii 5.4.3 Resolution Error . 49 5.4.4 Operator Bias 49 5.4.5 Environmental Factors Error 50 5.5 COMBINE UNCERTAINTIES 53 5.6 REPORT ANALYSIS RESULTS 54 5.6.1 Average Measured Value and Combined Uncertainty . 54 5.6.2 Measurement Process Errors and Uncertainties .
15、 54 5.6.3 Confidence Limits 55 5.6.4 In-tolerance Probability 55 CHAPTER 6: MULTIVARIATE MEASUREMENTS 58 6.1 DEVELOP THE PARAMETER VALUE EQUATION 58 6.1.1 Cylinder Volume Example . 58 6.2 DEVELOP THE ERROR MODEL 59 6.2.1 Cylinder Volume Example . 59 6.3 DEVELOP THE UNCERTAINTY MODEL . 61 6.3.1 Cylin
16、der Volume Example . 61 6.4 IDENTIFY MEASUREMENT PROCESS ERRORS 61 6.4.1 Cylinder Volume Example . 61 6.5 ESTIMATE MEASUREMENT PROCESS UNCERTAINTIES 62 6.5.1 Measurement Bias Uncertainty 63 6.5.2 Repeatability Uncertainty . 63 6.5.3 Resolution Uncertainty . 66 6.5.4 Operator Bias Uncertainty 66 6.5.
17、5 Environmental Factors Uncertainty 66 6.6 COMPUTE UNCERTAINTY COMPONENTS 69 6.7 ACCOUNT FOR CROSS-CORRELATIONS . 70 6.7.1 Measurement Biases . 71 6.7.2 Operator Biases 71 6.7.3 Environmental Factors Errors 72 6.8 COMBINE UNCERTAINTY COMPONENTS. 72 6.9 REPORT ANALYSIS RESULTS 73 6.9.1 Cylinder Volum
18、e and Combined Uncertainty 73 6.9.2 Measurement Process Errors and Uncertainties . 74 6.9.3 Confidence Limits 75 CHAPTER 7: MEASUREMENT SYSTEMS 78 7.0 GENERAL 78 7.1 SYSTEM ANALYSIS PROCEDURE . 78 7.2 DEVELOP THE SYSTEM MODEL 79 7.2.1 Load Cell Measurement System. 79 7.3 DEFINE THE SYSTEM INPUT 80 P
19、rovided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-ix 7.3.1 Load Cell Measurement System. 80 7.4 DEFINE THE SYSTEM MODULES 80 7.4.1 Load Cell Module (M1) 80 7.4.2 Amplifier Module (M2) 81 7.4.3 Digital Multimeter Module (M3) 81 7.5 IDENTIFY MODULE ERROR
20、SOURCES . 81 7.5.1 Load Cell Module (M1) 81 7.5.2 Amplifier Module (M2) 83 7.5.3 Digital Multimeter Module (M3) 85 7.6 DEVELOP MODULE ERROR MODELS 86 7.6.1 Load Cell Module (M1) 87 7.6.2 Amplifier Module (M2) 88 7.6.3 Digital Multimeter Module (M3) 89 7.7 DEVELOP MODULE UNCERTAINTY MODELS . 89 7.7.1
21、 Load Cell Module (M1) 90 7.7.2 Amplifier Module (M2) 90 7.7.3 Digital Multimeter Module (M3) 90 7.8 ESTIMATE MODULE UNCERTAINTIES 91 7.8.1 Load Cell Module (M1) 91 7.8.2 Amplifier Module (M2) 93 7.8.3 Multimeter Module (M3) 94 7.9 COMPUTE SYSTEM OUTPUT UNCERTAINTY 96 7.10 REPORT ANALYSIS RESULTS 97
22、 7.10.1 Confidence Limits 98 CHAPTER 8: UNCERTAINTY ANALYSIS FOR ALTERNATIVE CALIBRATION SCENARIOS 99 8.1 CALIBRATION SCENARIOS OVERVIEW 99 8.1.1 Special Notation . 100 8.1.2 Measurement Error Sources . 100 8.1.3 Calibration Error and Measurement Error 101 8.1.4 UUT Attribute Bias 101 8.1.5 MTE Attr
23、ibute Bias 102 8.2 SCENARIO 1: THE MTE MEASURES THE UUT ATTRIBUTE VALUE 102 8.3 SCENARIO 2 : THE UUT MEASURES THE MTE ATTRIBUTE VALUE . 104 8.4 SCENARIO 3: THE MTE AND UUT ATTRIBUTE VALUES ARE COMPARED . 106 8.5 SCENARIO 4: THE MTE AND UUT MEASURE A COMMON ARTIFACT . 107 8.5.1 Special Cases for Scen
24、ario 4 109 8.6 UNCERTAINTY ANALYSIS EXAMPLES . 109 8.6.1 Scenario 1: The MTE Measures the UUT Attribute Value 109 8.6.2 Scenario 2: The UUT Measures the MTE Attribute Value . 113 8.6.3 Scenario 3: MTE and UUT Attribute Values are Compared 115 Provided by IHSNot for ResaleNo reproduction or networkin
25、g permitted without license from IHS-,-,-x 8.6.4 Scenario 4: The MTE and UUT Measure a Common Artifact 118 CHAPTER 9: UNCERTAINTY GROWTH 121 9.1 BASIC METHODOLOGY . 121 9.2 PROJECTED UNCERTAINTY . 123 9.2.1 Two-Sided Tolerances 123 9.2.2 Two-Sided Symmetric Tolerances . 125 9.2.3 Single-Sided Tolera
26、nces . 126 9.3 RELIABILITY MODELS 127 9.3.1 Exponential Model . 127 9.3.2 Mixed Exponential Model 128 9.3.3 Weibull Model 129 9.3.4 Gamma Model 129 9.3.5 Mortality Drift Model . 130 9.3.6 Warranty Model . 131 9.3.7 Random Walk Model . 131 9.3.8 Restricted Random Walk Model 132 9.4 ANALYSIS EXAMPLE .
27、 133 APPENDIX A TERMS AND DEFINITIONS 135 APPENDIX B PROBABILITY DISTRIBUTIONS 147 B.1 NORMAL DISTRIBUTION 147 B.2 LOGNORMAL DISTRIBUTION 149 B.3 EXPONENTIAL DISTRIBUTION 150 B.4 QUADRATIC DISTRIBUTION 152 B.5 COSINE DISTRIBUTION . 153 B.6 U-SHAPED DISTRIBUTION 154 B.7 UNIFORM (RECTANGULAR) DISTRIBU
28、TION 155 B.7.1 Round-off Uniform Distribution 155 B.7.2 Truncation Uniform Distribution . 156 B.8 TRIANGULAR DISTRIBUTION 157 B.9 TRAPEZOIDAL DISTRIBUTION 159 B.10 STUDENTS T DISTRIBUTION 161 B.11 THE UTILITY DISTRIBUTION . 162 B.12 APPLYING THE UNIFORM DISTRIBUTION 166 B.12.1 Criteria for Selecting
29、 the Uniform Distribution 166 B.12.2 Cases where the Uniform Distribution is Applicable 167 B.12.3 Incorrect Application of the Uniform Distribution . 168 APPENDIX C STATISTICAL SAMPLE ANALYSIS 170 C.1 SAMPLED VALUES . 170 C.1.1 Example 1 AC Voltage Measurements . 171 Provided by IHSNot for ResaleNo
30、 reproduction or networking permitted without license from IHS-,-,-xi C.1.2 Example 2 Temperature Measurements 172 C.2 SAMPLED CELLS . 173 C.2.1 Example 1 AC Voltage Measurements . 174 C.3 SAMPLED MEAN VALUES 175 C.3.1 Example 1 Pressure Measurements 177 C.4 OUTLIER TESTING 178 C.4.1 Background 179
31、C.4.2 Chauvenets Criterion 179 C.5 NORMALITY TESTING . 180 C.5.1 Background 180 C.5.2 Skewness and Kurtosis Tests (n 10) . 181 C.5.3 Chi-square (2) Test (n 50) . 181 C.5.4 Shapiro-Wilk Test (20 n 50) 182 C.6 SAMPLE SIZE EVALUATION 182 C.6.1 Methodology 183 C.6.2 Example 1 Evaluation using a Sample S
32、tandard Deviation . 184 C.6.3 Example 2 Evaluation using a Population Standard Deviation . 185 APPENDIX D ESTIMATING TYPE B DEGREES OF FREEDOM 187 D.1 METHODOLOGY . 187 D.2 ANALYSIS FORMATS . 189 D.2.1 Format 1: % of Values . 189 D.2.2 Format 2: X out of N . 190 D.2.3 Format 3: % of Cases . 191 D.2.
33、4 Format 4: % Range . 191 APPENDIX E BAYESIAN ANALYSIS 192 E.1 BAYES THEOREM . 192 E.1.1 Joint Probability . 192 E.1.2 Conditional Probability 193 E.2 RISK ANALYSIS FOR A MEASURED VARIABLE 194 E.3 A PRIORI KNOWLEDGE 194 E.4 POST-TEST KNOWLEDGE . 194 E.5 BIAS ESTIMATES 195 E.5.1 Refinement of the UUT
34、 Bias Estimate . 195 E.5.2 Refinement of the MTE Bias Estimate . 198 E.6 IN-TOLERANCE PROBABILITIES . 199 E.6.1 UUT Attribute In-Tolerance Probability 199 E.6.2 MTE Attribute In-Tolerance Probability 199 APPENDIX F FORCE GAUGE ANALYSIS EXAMPLE 201 F.1 MEASUREMENT PROCESS OVERVIEW . 201 Provided by I
35、HSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-xii F.2 UNCERTAINTY ANALYSIS PROCEDURE. 203 F.2.1 Calibration Weight (m) 203 F.2.2 Conversion Factor (cf) . 204 F.2.3 Digital Resolution (res) 204 F.2.4 Repeatability (rep) 204 F.3 IN-TOLERANCE PROBABILITY 208 APPEN
36、DIX G SPECTRUM ANALYZER ANALYSIS EXAMPLE 210 G.1 MEASUREMENT PROCESS OVERVIEW . 210 G.2 UNCERTAINTY ANALYSIS PROCEDURE. 212 G.2.1 UUT Resolution Error (UUT,res) 213 G.2.2 Channel A Power Sensor Bias (APS ) 214 G.2.3 Channel B Power Sensor Bias (BPS ) . 214 G.2.4 Power Meter Error (,PMb and,PMres ) .
37、 214 G.3 IN-TOLERANCE PROBABILITY 218 APPENDIX H ROTAMETER ANALYSIS EXAMPLE 222 H.1 MEASUREMENT PROCESS OVERVIEW . 222 H.1.1 Regression Analysis . 223 H.2 UNCERTAINTY ANALYSIS PROCEDURE. 224 H.2.1 Measurement Reference Flow Rate (,Rref b and ,Rref r ) 225 H.2.2 Outlet Gas Pressure (,out bP and ,out
38、rP ) 225 H.2.3 Outlet Gas Temperature (,out bT and ,out rT ) 226 H.2.4 Uncertainty in Rcorr. 226 H.3 PREDICTED FLOW RATE UNCERTAINTY . 230 APPENDIX I WINGBOOM AOA ANALYSIS EXAMPLE 234 I.1 MEASUREMENT PROCESS OVERVIEW . 234 I.2 SYSTEM MODEL 235 I.3 SYSTEM INPUT 235 I.3.1 Calibrator Fixture Bias . 236
39、 I.3.2 Measurement Repeatability 236 I.4 SYSTEM MODULES 236 I.4.1 Potentiometer Module (M1) 236 I.4.2 Signal Conditioner Module (M2) 240 I.4.3 Data Processor Module (M3) 241 I.5 MODULE ERROR MODELS 243 I.5.1 Potentiometer Module (M1) 243 I.5.2 Signal Conditioner Module 243 Provided by IHSNot for Res
40、aleNo reproduction or networking permitted without license from IHS-,-,-xiii I.5.3 Data Processor Module 243 I.6 MODULE UNCERTAINTY MODELS. 244 I.6.1 Potentiometer Module 244 I.6.2 Signal Conditioner Module 244 I.6.3 Data Processor Module 245 I.7 ESTIMATE MODULE UNCERTAINTIES 245 I.7.1 Potentiometer
41、 Module 245 I.7.2 Signal Conditioner Module 246 I.7.3 Data Processor Module 247 I.8 SYSTEM OUTPUT AND UNCERTAINTY . 248 REFERENCES 249 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-xiv LIST OF FIGURES Figure 3-1. Repeatability Distribution .24 Figu
42、re 5-1. Right-handed Lognormal Distribution 47 Figure 5-2. Pareto Chart for Micrometer Calibration 55 Figure 5-3. Micrometer Bias Distribution .57 Figure 6-1. Pareto Chart for Cylinder Volume Measurement .75 Figure 7-1. Block Diagram for Example System .78 Figure 7-2. Load Cell Calibration Setup 79
43、Figure 7-3. Block Diagram of Load Cell Measurement System .80 Figure 7-4. Pareto Chart for Load Cell Module .97 Figure 7-5. Pareto Chart for Amplifier Module .98 Figure 7-6. Pareto Chart for Digital Multimeter Module 98 Figure 9-1. Measurement Uncertainty Growth 121 Figure 9-2. Measurement Reliabili
44、ty versus Time 121 Figure 9-3. Parameter Bias Distribution 122 Figure 9-4. In-tolerance Probability versus Time Exponential Model .128 Figure 9-5. In-tolerance Probability versus Time Mixed Exponential Model 128 Figure 9-6. In-tolerance Probability versus Time Weibull Model 129 Figure 9-7. In-tolera
45、nce Probability versus Time Gamma Model. .130 Figure 9-8. In-tolerance Probability versus Time Mortality Drift Model .130 Figure 9-9. In-tolerance Probability versus Time Warranty Model .131 Figure 9-10. In-tolerance Probability versus Time Random Walk Model .132 Figure 9-11. In-tolerance Probabilit
46、y versus Time Restricted Random Walk Model 132 Figure B-1. Normal Distribution 148 Figure B-2. Right-handed Lognormal Distribution .149 Figure B-3. Right-handed Exponential Distribution 150 Figure B-4. Left-Handed Exponential Distribution .150 Figure B-5. Quadratic Distribution 152 Figure B-6. Cosin
47、e Distribution .153 Figure B-7. U-Shaped Distribution 154 Figure B-8. Round-off Uniform Distribution 156 Figure B-9. Truncation Uniform Distribution .157 Figure B-10. Triangular Distribution .158 Figure B-11. Trapezoidal Distribution .159 Figure B-12. Students t Distribution .161 Figure B-13. Utilit
48、y Distribution .163 Figure F-1. Pareto Chart for Force Gauge Calibration Uncertainty 207 Figure F-2. Force Gauge Bias Distribution209 Figure G-1. Spectrum Analyzer Calibration Setup 210 Figure G-2. Power Splitter Characterization Setup .211 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,
