Table of Contents

Foreword <br>Introduction<br>1 Scope<br>2 Normative references<br>3 Terms and definitions<br>4 Conventions and notation <br>5 Basic principles<br>&nbsp;&nbsp;5.1 Main stages of uncertainty evaluation <br>&nbsp;&nbsp;5.2 Propagation of distributions <br>&nbsp;&nbsp;5.3 Obtaining summary information<br>&nbsp;&nbsp;5.4 Implementations of the propagation of distributions<br>&nbsp;&nbsp;5.5 Reporting the results <br>&nbsp;&nbsp;5.6 GUM uncertainty framework <br>&nbsp;&nbsp;5.7 Conditions for valid application of the GUM uncertainty <br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;framework for linear models <br>&nbsp;&nbsp;5.8 Conditions for valid application of the GUM uncertainty <br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;framework for non-linear models <br>&nbsp;&nbsp;5.9 Monte Carlo approach to the propagation and summarizing <br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;stages <br>&nbsp;&nbsp;5.10 Conditions for the valid application of the described <br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Monte Carlo method<br>&nbsp;&nbsp;5.11 Comparison of the GUM uncertainty framework and the <br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;described Monte Carlo method<br>6 Probability density functions for the input quantities<br>&nbsp;&nbsp;6.1 General<br>&nbsp;&nbsp;6.2 Bayes' theorem<br>&nbsp;&nbsp;6.3 Principle of maximum entropy<br>&nbsp;&nbsp;6.4 Probability density function assignment for some <br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;common circumstances<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;6.4.1 General <br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;6.4.2 Rectangular distributions<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;6.4.3 Rectangular distributions with inexactly <br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;prescribed limits <br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;6.4.4 Trapezoidal distributions<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;6.4.5 Triangular distributions <br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;6.4.6 Arc sine (U-shaped) distributions<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;6.4.7 Gaussian distributions<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;6.4.8 Multivariate Gaussian distributions <br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;6.4.9 t-distributions<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;6.4.10 Exponential distributions<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;6.4.11 Gamma distributions<br>&nbsp;&nbsp;6.5 Probability distributions from previous uncertainty <br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;calculations<br>7 Implementation of a Monte Carlo method<br>&nbsp;&nbsp;7.1 General<br>&nbsp;&nbsp;7.2 Number of Monte Carlo trials<br>&nbsp;&nbsp;7.3 Sampling from probability distributions<br>&nbsp;&nbsp;7.4 Evaluation of the model<br>&nbsp;&nbsp;7.5 Discrete representation of the distribution function <br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;for the output quantity<br>&nbsp;&nbsp;7.6 Estimate of the output quantity and the associated <br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;standard uncertainty<br>&nbsp;&nbsp;7.7 Coverage interval for the output quantity<br>&nbsp;&nbsp;7.8 Computation time<br>&nbsp;&nbsp;7.9 Adaptive Monte Carlo procedure<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;7.9.1 General<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;7.9.2 Numerical tolerance associated with a numerical <br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;value<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;7.9.3 Objective of adaptive procedure<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;7.9.4 Adaptive procedure<br>8 Validation of results<br>&nbsp;&nbsp;8.1 Validation of the GUM uncertainty framework using a <br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Monte Carlo method<br>&nbsp;&nbsp;8.2 Obtaining results from a Monte Carlo method for <br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;validation purposes<br>9 Examples<br>&nbsp;&nbsp;9.1 Illustrations of aspects of this Supplement<br>&nbsp;&nbsp;9.2 Additive model<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;9.2.1 Formulation<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;9.2.2 Normally distributed input quantities<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;9.2.3 Rectangularly distributed input quantities with <br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the same width <br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;9.2.4 Rectangularly distributed input quantities with <br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;different widths<br>&nbsp;&nbsp;9.3 Mass calibration<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;9.3.1 Formulation<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;9.3.2 Propagation and summarizing<br>&nbsp;&nbsp;9.4 Comparison loss in microwave power meter calibration<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;9.4.1 Formulation<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;9.4.2 Propagation and summarizing: zero covariance<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;9.4.3 Propagation and summarizing: non-zero covariance<br>&nbsp;&nbsp;9.5 Gauge block calibration<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;9.5.1 Formulation: model<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;9.5.2 Formulation: assignment of PDFs<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;9.5.3 Propagation and summarizing<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;9.5.4 Results<br>Annex A - Historical perspective<br>Annex B - Sensitivity coefficients and uncertainty budgets<br>Annex C - Sampling from probability distributions<br>&nbsp;&nbsp;C.1 General<br>&nbsp;&nbsp;C.2 General distribution<br>&nbsp;&nbsp;C.3 Rectangular distribution<br>&nbsp;&nbsp;C.4 Gaussian distribution<br>&nbsp;&nbsp;C.5 Multivariate Gaussian distribution<br>&nbsp;&nbsp;C.6 t-distribution<br>Annex D - Continuous approximation to the distribution function <br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;for the output quantity<br>Annex E - Coverage interval for the four-fold convolution of <br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;a rectangular distribution<br>Annex F - Comparison loss problem<br>&nbsp;&nbsp;F.1 Expectation and standard deviation obtained analytically<br>&nbsp;&nbsp;F.2 Analytic solution for zero estimate of the voltage <br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;reflection coefficient having associated zero covariance<br>&nbsp;&nbsp;F.3 GUM uncertainty framework applied to the comparison <br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;loss problem<br>Annex G - Glossary of principal symbols<br>Bibliography<br>Alphabetical index

Abstract

Describes a general numerical approach, consistent with the broad principles of the GUM [ISO/IEC Guide 98-3:2008, G.1.5], for carrying out the calculations required as part of an evaluation of measurement uncertainty.

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Published
Document Type	Standard
Status	Current
Publisher	International Organization for Standardization
Pages
ISBN
Committee	TMBG