ISO/IEC Guide 98-3:2008/Suppl 1:2008

GUIDE 98-3/Suppl.1
Uncertainty of measurement
Part 3:
Guide to the expression of
uncertainty in measurement
(GUM:1995)
Supplement 1:
Propagation of distributions
using a Monte Carlo method
First edition 2008
©
ISO/IEC 2008
ISO/IEC GUIDE 98-3/Suppl.1:2008(E)
PDF disclaimer
This PDF file may contain embedded typefaces. In accordance with Adobe's licensing policy, this file may be printed or viewed but
shall not be edited unless the typefaces which are embedded are licensed to and installed on the computer performing the editing. In
downloading this file, parties accept therein the responsibility of not infringing Adobe's licensing policy. The ISO Central Secretariat
accepts no liability in this area.
Adobe is a trademark of Adobe Systems Incorporated.
Details of the software products used to create this PDF file can be found in the General Info relative to the file; the PDF-creation
parameters were optimized for printing. Every care has been taken to ensure that the file is suitable for use by ISO member bodies. In
the unlikely event that a problem relating to it is found, please inform the Central Secretariat at the address given below.

©  ISO/IEC 2008
All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized in any form or by any means,
electronic or mechanical, including photocopying and microfilm, without permission in writing from either ISO at the address below or
ISO's member body in the country of the requester.
ISO copyright office
Case postale 56 • CH-1211 Geneva 20
Tel. + 41 22 749 01 11
Fax + 41 22 749 09 47
E-mail copyright@iso.org
Web www.iso.org
Published in Switzerland
ii © ISO/IEC 2008 – All rights reserved

ISO/IEC GUIDE 98-3/Suppl.1:2008(E)
Contents Page
Foreword .v
Introduction.vi
1 Scope.1
2 Normative references.2
3 Terms and definitions .2
4 Conventions and notation .6
5 Basic principles .8
5.1 Main stages of uncertainty evaluation .8
5.2 Propagation of distributions .9
5.3 Obtaining summary information.9
5.4 Implementations of the propagation of distributions.10
5.5 Reporting the results .11
5.6 GUM uncertainty framework .12
5.7 Conditions for valid application of the GUM uncertainty framework for linear models .13
5.8 Conditions for valid application of the GUM uncertainty framework for non-linear models .14
5.9 Monte Carlo approach to the propagation and summarizing stages .15
5.10 Conditions for the valid application of the described Monte Carlo method .16
5.11 Comparison of the GUM uncertainty framework and the described Monte Carlo method .17
6 Probability density functions for the input quantities.18
6.1 General .18
6.2 Bayes’ theorem.19
6.3 Principle of maximum entropy.19
6.4 Probability density function assignment for some common circumstances .20
6.4.1 General .20
6.4.2 Rectangular distributions.20
6.4.3 Rectangular distributions with inexactly prescribed limits .20
6.4.4 Trapezoidal distributions.22
6.4.5 Triangular distributions .23
6.4.6 Arc sine (U-shaped) distributions.24
6.4.7 Gaussian distributions.25
6.4.8 Multivariate Gaussian distributions .25
6.4.9 t-distributions.26
6.4.10 Exponential distributions .28
6.4.11 Gamma distributions.28
6.5 Probability distributions from previous uncertainty calculations .29
7 Implementation of a Monte Carlo method.29
7.1 General .29
7.2 Number of Monte Carlo trials .29
7.3 Sampling from probability distributions.29
7.4 Evaluation of the model.30
7.5 Discrete representation of the distribution function for the output quantity.30
7.6 Estimate of the output quantity and the associated standard uncertainty.31
7.7 Coverage interval for the output quantity.31
7.8 Computation time.32
7.9 Adaptive Monte Carlo procedure.32
7.9.1 General .32
7.9.2 Numerical tolerance associated with a numerical value.32
7.9.3 Objective of adaptive procedure.33

© ISO/IEC 2008 – All rights reserved iii

ISO/IEC GUIDE 98-3/Suppl.1:2008(E)
7.9.4 Adaptive procedure .33
8 Validation of results .35
8.1 Validation of the GUM uncertainty framework using a Monte Carlo method.35
8.2 Obtaining results from a Monte Carlo method for validation purposes .35
9 Examples .36
9.1 Illustrations of aspects of this Supplement .36
9.2 Additive model .37
9.2.1 Formulation .37
9.2.2 Normally distributed input quantities.37
9.2.3 Rectangularly distributed input quantities with the same width.39
9.2.4 Rectangularly distributed input quantities with different widths .41
9.3 Mass calibration.42
9.3.1 Formulation .42
9.3.2 Propagation and summarizing .43
9.4 Comparison loss in microwave power meter calibration.45
9.4.1 Formulation .45
9.4.2 Propagation and summarizing: zero covariance.46
9.4.3 Propagation and summarizing: non-zero covariance.51
9.5 Gauge block calibration .53
9.5.1 Formulation: model .53
9.5.2 Formulation: assignment of PDFs .55
9.5.3 Propagation and summarizing .58
9.5.4 Results .59
Annex A Historical perspective.61
Annex B Sensitivity coefficients and uncertainty budgets .62
Annex C Sampling from probability distributions.63
C.1 General.63
C.2 General distributions.63
C.3 Rectangular distribution .64
C.4 Gaussian distribution.65
C.5 Multivariate Gaussian distribution.66
C.6 t-distribution.67
Annex D Continuous approximation to the distribution function for the output quantity.69
Annex E Coverage interval for the four-fold convolution of a rectangular distribution .72
Annex F Comparison loss problem .74
F.1 Expectation and standard deviation obtained analytically .74
F.2 Analytic solution for zero estimate of the voltage reflection coefficient having associated
zero covariance.75
F.3 GUM uncertainty framework applied to the comparison loss problem .76
Annex G Glossary of principal symbols.78
Bibliography .83
Alphabetical index .86

iv © ISO/IEC 2008 – All rights reserved

ISO/IEC GUIDE 98-3/Suppl.1:2008(E
...