NASA-HDBK-8739.19-3 Approved: 2010-07-13 Baseline 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 APPROVED FOR PUBLIC RELEASE – DISTRIBUTION UNLIMITED This page intentionally left blank. ii DOCUMENT HISTORY LOG Document Status Approval Date Description Revision Baseline 2010-07-13 Initial Release (JWL4) This document is subject to reviews per Office of Management and Budget Circular A-119, Federal Participation in the Development and Use of Voluntary Standards (02/10/1998) and NPD 8070.6, Technical Standards (Paragraph 1.k). iii This page intentionally left blank. v TABLE OF CONTENTS FOREWORD IV TABLE OF CONTENTS VI LIST OF FIGURES XIV LIST OF TABLES XV ACRONYMS AND ABBREVIATIONS XVII PREFACE XVIII ACKNOWLEDGEMENTS 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 Distributions ........................................................................................................................ 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 2.5.4 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 vi 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 Student’s t Distribution ................................................................................................................ 29 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 Output 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 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 Specifications ........................................................................................................... 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 vii 5.4.3 Resolution Error ........................................................................................................................... 49 5.4.4 Operator Bias................................................................................................................................ 49 5.4.5 Environmental Factors Error ........................................................................................................ 50 5.5 COMBINE UNCERTAINTIES.......................................................................................