Introduction to Statistics in Metrology

Preface
Contents
About the Authors
Chapter 1: Introduction
	1.1 Measurement Uncertainty: Why Do We Care?
	1.2 The History of Measurement
	1.3 Measurement Science and Technological Development
	1.4 Allegations of Deflated Footballs (``Deflategate´´)
	1.5 Fatality Rates During a Pandemic
	1.6 Summary
	1.7 Related Reading
	References
Chapter 2: Basic Measurement Concepts
	2.1 Introduction
	2.2 Measurement Terminology
		2.2.1 General Measurement Terminology
			2.2.1.1 Measurand
			2.2.1.2 True Value (True Value of a Quantity)
			2.2.1.3 Measurement Accuracy
			2.2.1.4 Measurement Precision
			2.2.1.5 Resolution
			2.2.1.6 Measurement Repeatability
			2.2.1.7 Measurement Reproducibility
			2.2.1.8 Independence of Measurements
		2.2.2 Error Approach Terminology
			2.2.2.1 Measurement Error
			2.2.2.2 Systematic Measurement Error
			2.2.2.3 Random Measurement Error
		2.2.3 Uncertainty Approach Terminology
			2.2.3.1 Measurement Uncertainty
			2.2.3.2 Level of Confidence (Coverage Probability)
			2.2.3.3 Coverage Interval
			2.2.3.4 Measurement Model
		2.2.4 Terminology of Calibration
			2.2.4.1 Measuring and Test Equipment (M&TE)
			2.2.4.2 Metrological Traceability
			2.2.4.3 Calibration
			2.2.4.4 Tolerance Test
			2.2.4.5 Certification Uncertainty
	2.3 Types of Measurements
		2.3.1 Physical Measurements
		2.3.2 Electrical Measurements
		2.3.3 Other Types of Measurements
	2.4 Sources of Uncertainty
		2.4.1 Evaluating Sources of Uncertainty
	2.5 Summary
	2.6 Related Reading
	2.7 Exercises
	References
Chapter 3: The International System of Units, Traceability, and Calibration
	3.1 History of the SI and Base Units
		3.1.1 SI Constants
		3.1.2 Time: Second (s)
		3.1.3 Length: Meter (m)
		3.1.4 Mass: Kilogram (kg)
		3.1.5 Electric Current: Ampere (A)
		3.1.6 Temperature: Kelvin (K)
		3.1.7 Quantity of Substance: Mole (mol)
		3.1.8 Luminous Intensity: Candela (cd)
	3.2 Derived Units
	3.3 Unit Realizations
		3.3.1 Gauge Block Interferometer
		3.3.2 Josephson Volt
	3.4 Advancements in Unit Definitions
		3.4.1 Kibble (Watt) Balance
		3.4.2 Intrinsic Pressure Standard
	3.5 Metrological Traceability
	3.6 Measurement Standards
		3.6.1 Certified Reference Materials
		3.6.2 Check Standards
	3.7 Calibration
		3.7.1 The Calibration Cycle
		3.7.2 Legal Aspects of Calibration
		3.7.3 Technical Aspects of Calibration
		3.7.4 Calibration Policies and Requirements
			3.7.4.1 ISO 17025
			3.7.4.2 ANSI Z540.1 and ANSI/NCSL Z540.3:2006
	3.8 Summary
	3.9 Related Reading
	3.10 Exercises
	References
Chapter 4: Introduction to Statistics and Probability
	4.1 Introduction
	4.2 Types of Data
	4.3 Exploratory Data Analysis
		4.3.1 Calculating Summary Statistics
			4.3.1.1 Summary Statistics for Continuous Data
			4.3.1.2 Summary Statistics for Discrete Data
		4.3.2 Graphical Displays of Data
			4.3.2.1 Graphical Displays for Continuous Data
			4.3.2.2 Graphical Displays for Discrete Data
	4.4 Probability Distributions
		4.4.1 Identification of Probability Distributions
			4.4.1.1 Continuous Distributions
			4.4.1.2 Discrete Distributions
		4.4.2 Estimating Distribution Parameters
		4.4.3 Assessing Distributional Fit
	4.5 Related Reading
	4.6 Exercises
	References
Chapter 5: Measurement Uncertainty in Decision Making
	5.1 Introduction
	5.2 Measurement Uncertainty and Risk
		5.2.1 Measurement Uncertainty and Risk in Manufacturing
			5.2.1.1 Test Uncertainty Ratio
			5.2.1.2 Measurement Decisions
			5.2.1.3 False Accept and False Reject Risks
			5.2.1.4 Guardbanding
			5.2.1.5 Risk with Biased Measurements
		5.2.2 Measurement Uncertainty and Risk in Calibration
			5.2.2.1 Decision Rules in Calibration
	5.3 Summary
	5.4 Related Reading
	5.5 Exercises
	References
Chapter 6: The Measurement Model and Uncertainty
	6.1 Introduction
	6.2 Uncertainty Analysis Framework
		6.2.1 Standard Uncertainty
		6.2.2 Type A Uncertainty Evaluation
		6.2.3 Type B Uncertainty Evaluation
		6.2.4 Combined Standard Uncertainty
		6.2.5 Confidence Level and Expanded Uncertainty
	6.3 Direct Measurements and the Basic Measurement Model
		6.3.1 Case Study: Voltage Measurement
		6.3.2 Discussion
	6.4 Indirect Measurements and the Indirect Measurement Model
		6.4.1 Case Study: Neutron Yield Measurement
		6.4.2 Discussion
	6.5 Related Reading
	6.6 Exercises
	References
Chapter 7: Analytical Methods for the Propagation of Uncertainties
	7.1 Introduction
	7.2 Mathematical Basis
	7.3 The Simple Case: First-Order Terms with Uncorrelated Inputs
		7.3.1 Measurement Examples
	7.4 First-Order Terms with Correlated Inputs
		7.4.1 Covariance, Correlation, and Effect on Uncertainty
		7.4.2 Measurement Examples
	7.5 Higher-Order Terms with Uncorrelated Inputs
		7.5.1 Measurement Examples
	7.6 Multiple Output Quantities
	7.7 Limitations of the Analytical Approach
	7.8 Related Reading
	7.9 Exercises
	References
Chapter 8: Monte Carlo Methods for the Propagation of Uncertainties
	8.1 Introduction to Monte Carlo Methods
		8.1.1 Random Sampling Techniques and Random Number Generation
			8.1.1.1 Sampling from Normal and Non-Normal Distributions
			8.1.1.2 Generating Correlated Random Samples (Normal Distribution)
		8.1.2 Generation of Probability Density Functions Using Random Data
		8.1.3 Computational Approaches
			8.1.3.1 Linear Congruential Generator
			8.1.3.2 Better PRNG Algorithms
	8.2 Standard Monte Carlo for Uncertainty Propagation
		8.2.1 Monte Carlo Techniques
			8.2.1.1 Case Study: Calculating Density
			8.2.1.2 Sensitivity Coefficients
			8.2.1.3 Convergence Plots and Adaptive Sampling
	8.3 Comparison to the GUM
		8.3.1 Quantitative GUM Validity Test
	8.4 Monte Carlo Case Studies
		8.4.1 Case Study: Neutron Yield Measurement
		8.4.2 Case Study: RC Circuit
	8.5 Summary
	8.6 Related Reading
	8.7 Exercises
	References
Chapter 9: Design of Experiments in Metrology
	9.1 Introduction
	9.2 Factorial Experiments in Metrology
		9.2.1 Defining the Measurand and Objective of the Experiment
		9.2.2 Selecting Factors to Incorporate in the Experiment
		9.2.3 Selecting Factor Levels and Design Pattern
		9.2.4 Analysis of CMM Errors via Design of Experiments (24 Full Factorial)
		9.2.5 Finite Element Method (FEM) Uncertainty Analysis via Design of Experiments (27-3 Fractional Factorial)
		9.2.6 Summary of Factorial DOEx Method
	9.3 ANOVA Models in Metrology
		9.3.1 Random Effects Models
		9.3.2 Mixed Effects Models
		9.3.3 Underlying ANOVA Assumptions
		9.3.4 Gauge R&R Study (Random Effects Model)
		9.3.5 Voltage Standard Uncertainty Analysis (Mixed Effects Model)
		9.3.6 Summary of ANOVA Method
	9.4 Related Reading
	9.5 Exercises
	References
Chapter 10: Determining Uncertainties in Fitted Curves
	10.1 The Purpose of Fitting Curves to Experimental Data
		10.1.1 Resistance vs. Temperature Data
		10.1.2 Considerations When Fitting Models to Data
	10.2 Methods for Fitting Curves to Experimental Data
		10.2.1 Linear Least Squares
		10.2.2 Uncertainty in Fitting Parameters
		10.2.3 Weighted Least Squares: Non-constant u(y)
		10.2.4 Weighted Least Squares: Uncertainty in Both x and y
	10.3 Uncertainty of a Regression Line
		10.3.1 Uncertainty of Fitting Parameters
		10.3.2 Confidence Bands
		10.3.3 Prediction Bands
	10.4 How Good Is the Model?
		10.4.1 Residual Analysis
		10.4.2 Slope Test
		10.4.3 Quantitative Residual Analysis
	10.5 Uncertainty in Nonlinear Regression
		10.5.1 Nonlinear Least Squares
		10.5.2 Orthogonal Distance Regression
		10.5.3 Confidence and Prediction Bands in Nonlinear Regression
	10.6 Using Monte Carlo for Evaluating Uncertainties in Curve Fitting
		10.6.1 Monte Carlo Approach
		10.6.2 Markov-Chain Monte Carlo Approach
	10.7 Case Study: Contact Resistance
	10.8 Drift and Predicting Future Values
		10.8.1 Uncertainty During Use
		10.8.2 Validating Drift Uncertainty
			10.8.2.1 Type B Uncertainty
			10.8.2.2 Type A Measurement Uncertainty
			10.8.2.3 Drift Uncertainty
			10.8.2.4 Expanded Uncertainty
	10.9 Calibration Interval Analysis
	10.10 Summary
	10.11 Related Reading
	10.12 Exercises
	References
Chapter 11: Special Topics in Metrology
	11.1 Introduction
	11.2 Statistical Process Control (SPC)
		11.2.1 Case Study: Battery Tester Uncertainty and Monitoring Via SPC
		11.2.2 Discussion
	11.3 Binary Measurement Systems (BMS)
		11.3.1 BMS Overview
		11.3.2 BMS Case Study Introduced
		11.3.3 Evaluation of a BMS
			11.3.3.1 Within-Operator Agreement
			11.3.3.2 Between-Operator Agreement
			11.3.3.3 Assessing BMS Correctness
		11.3.4 Sample Sizes for a BMS Study
	11.4 Measurement System Analysis with Destructive Testing
	11.5 Sample Size and Allocation of Samples in Metrology Experiments
	11.6 Summary of Sample Size Recommendations
	11.7 Bayesian Analysis in Metrology
	11.8 Related Reading
	11.9 Exercises
	References
Appendix A: Acronyms and Abbreviations
Appendix B: Guidelines for Valid Measurements
	Related Reading: Electrical Measurements
	Related Reading: Time and Frequency Measurements
	Related Reading: Physical Measurements
	Related Reading: Temperature Measurement
	Related Reading: Radiation
	Related Reading: General Measurement and Instrumentation Techniques
Appendix C: Uncertainty Budget Case Study: CMM Length Measurements
	Coordinate Measuring Machine (CMM) Measurements
		Product Acceptance Uncertainty: Dimensional Part Inspection with a CMM
		Radius of Curvature of a Spherical Mirror
		The Measurement Model
		Measurement Considerations
			Surface Form of the Mirror
			CMM Probing Force
		ROC Measurement
		Uncertainty Analysis
			CMM Measurement Process Uncertainty
			CMM Positioning Error (Standards)
			Fit Uncertainty
			Combined Standard Uncertainty: u(R)
			Final Results
	Related Reading
Appendix D: Uncertainty Quick Reference
	GUM Method for Measurement Uncertainty
	Percentage Points of the t Distribution
	Guardbanding
		Symmetric Specification Limits
		Asymmetric Specification Limits
		One-Sided Specification Limits
	Metrology Reference Table
Appendix E: R for Metrology
	Introduction
		Installation of R
		R Packages
	R for Metrology
	Summary
References
Index