IEC TR 62461:2015

IEC TR 62461 ®
Edition 2.0 2015-01
TECHNICAL
REPORT
colour
inside
Radiation protection instrumentation – Determination of uncertainty in
measurement
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 IEC or IEC's member National Committee in the country of the requester. If you have any questions about IEC
copyright or have an enquiry about obtaining additional rights to this publication, please contact the address below or
your local IEC member National Committee for further information.

IEC Central Office Tel.: +41 22 919 02 11
3, rue de Varembé Fax: +41 22 919 03 00
CH-1211 Geneva 20 info@iec.ch
Switzerland www.iec.ch
About the IEC
The International Electrotechnical Commission (IEC) is the leading global organization that prepares and publishes
International Standards for all electrical, electronic and related technologies.

About IEC publications
The technical content of IEC publications is kept under constant review by the IEC. Please make sure that you have the
latest edition, a corrigenda or an amendment might have been published.

IEC Catalogue - webstore.iec.ch/catalogue Electropedia - www.electropedia.org
The stand-alone application for consulting the entire The world's leading online dictionary of electronic and
bibliographical information on IEC International Standards, electrical terms containing more than 30 000 terms and
Technical Specifications, Technical Reports and other definitions in English and French, with equivalent terms in 15
documents. Available for PC, Mac OS, Android Tablets and additional languages. Also known as the International
iPad. Electrotechnical Vocabulary (IEV) online.

IEC publications search - www.iec.ch/searchpub IEC Glossary - std.iec.ch/glossary
The advanced search enables to find IEC publications by a More than 60 000 electrotechnical terminology entries in
variety of criteria (reference number, text, technical English and French extracted from the Terms and Definitions
committee,…). It also gives information on projects, replaced clause of IEC publications issued since 2002. Some entries
and withdrawn publications. have been collected from earlier publications of IEC TC 37,

77, 86 and CISPR.
IEC Just Published - webstore.iec.ch/justpublished

Stay up to date on all new IEC publications. Just Published IEC Customer Service Centre - webstore.iec.ch/csc
details all new publications released. Available online and If you wish to give us your feedback on this publication or
also once a month by email. need further assistance, please contact the Customer Service
Centre: csc@iec.ch.
IEC TR 62461 ®
Edition 2.0 2015-01
TECHNICAL
REPORT
colour
inside
Radiation protection instrumentation – Determination of uncertainty in

measurement
INTERNATIONAL
ELECTROTECHNICAL
COMMISSION
ICS 13.280 ISBN 978-2-8322-2216-4

– 2 – IEC TR 62461:2015 © IEC 2015
CONTENTS
FOREWORD . 5
INTRODUCTION . 7
1 Scope . 8
2 Normative references . 8
3 Terms and definitions . 9
4 List of symbols . 12
5 The GUM and the GUM S1 concept . 14
5.1 General concept of uncertainty determination . 14
5.1.1 Overview in four steps . 14
5.1.2 Summary of the analytical method for steps 3 and 4 . 15
5.1.3 Summary of the Monte Carlo method for steps 3 and 4 . 15
5.1.4 Which method to use: Analytical or Monte Carlo? . 16
5.2 Example of a model function . 16
5.3 Collection of data and existing knowledge for the example . 18
5.3.1 General . 18
5.3.2 Calibration factor for the example . 19
5.3.3 Zero reading for the example . 20
5.3.4 Reading for the example . 21
5.3.5 Relative response or correction factor for the example . 21
5.3.6 Comparison of probability density distributions for input quantities . 23
5.4 Calculation of the result of a measurement and its standard uncertainty
(uncertainty budget) . 25
5.4.1 General . 25
5.4.2 Analytical method . 25
5.4.3 Monte Carlo method . 26
5.4.4 Uncertainty budgets . 26
5.5 Statement of the measurement result and its expanded uncertainty . 27
5.5.1 General . 27
5.5.2 Analytical method . 28
5.5.3 Monte Carlo method . 28
5.5.4 Representation of the output distribution function in a simple form
(Monte Carlo method) . 31
6 Results below the decision threshold of the measuring device . 31
7 Overview of the annexes . 32
Annex A (informative) Example of an uncertainty analysis for a measurement with an
electronic ambient dose equivalent rate meter according to IEC 60846-1:2009 . 33
A.1 General . 33
A.2 Model function . 33
A.3 Calculation of the complete result of the measurement (measured value,
probability density distribution, associated standard uncertainty, and the
coverage interval) . 34
A.3.1 General . 34
A.3.2 Low level of consideration of measuring conditions . 35
A.3.3 High level of consideration of measuring conditions . 37
Annex B (informative) Example of an uncertainty analysis for a measurement with a
passive integrating dosimetry system according to IEC 62387:2012 . 40

B.1 General . 40
B.2 Model function . 40
B.3 Calculation of the complete result of the measurement (measured value,
probability density distribution, associated standard uncertainty, and the
coverage interval) . 41
B.3.1 General . 41
B.3.2 Low level of consideration of workplace conditions . 41
B.3.3 High level of consideration of workplace conditions . 43
Annex C (informative) Example of an uncertainty analysis for a measurement with an
electronic direct reading neutron ambient dose equivalent meter according to
IEC 61005:2003 . 46
C.1 General . 46
C.2 Model function . 46
C.3 Calculation of the complete result of the measurement (measured value,
probability density distribution, associated standard uncertainty, and the
coverage interval) . 47
C.3.1 General . 47
C.3.2 Analytical method . 47
C.3.3 Monte Carlo method . 48
C.3.4 Comparison of the result of the analytical and the Monte Carlo method . 49
Annex D (informative) Example of an uncertainty analysis for a calibration of radon
activity monitor according to the IEC 61577 series . 51
D.1 General . 51
D.2 Model function . 51
D.3 Calculation of the complete result of the measurement (measured value,
probability density distribution, associated standard uncertainty, and the
coverage interval) . 51
Annex E (informative) Example of an uncertainty analysis for a measurement of
surface emission rate with a contamination meter according to IEC 60325:2002 . 54
E.1 General . 54
E.2 Model function . 54
E.3 Calculation of the complete result of the measurement (measured value,
probability density distribution, associated standard uncertainty, and the
coverage interval) . 54
E.3.1 General . 54
E.3.2 Effects of distance . 55
E.3.3 Contamination non-uniformity . 55
E.3.4 Surface absorption . 56
E.3.5 Other influence quantities . 56
E.3.6 Uncertainty budget . 56
Bibliography . 59

Figure 1 – Triangular probability density distribution of possible values n for the
calibration factor N . 20
Figure 2 – Rectangular probability density distribution of possible values g for the
zero reading G . 21
Figure 3 – Gaussian probability density distribution of possible values g for the
reading G . 21
Figure 4 – Comparison of different probability density distributions of possible values:
rectangular (broken line), triangular (dotted line) and Gaussian (solid line) distribution . 24
Figure 5 – Distribution function Q of the measured value . 29

– 4 – IEC TR 62461:2015 © IEC 2015
Figure 6 – Probability density distribution (PDF) of the measured value . 30
Figure C.1 – Results of the analytical (red dashed lines) and the Monte Carlo method

(grey histogram and blue dotted and solid lines) for H * (10) . 50
Figure D.1 – Result of the analytical (red dashed lines) and the Monte Carlo method
(grey histogram and blue dotted lines) for K . 53
T
Table 1 – Symbols (and abbreviated terms) used in the main text (excluding annexes) . 12
Table 2 – Standard uncertainty and method to compute the probability density
distributions shown in Figure 4 . 24
Table 3 – Example of an uncertainty budget for a measurement with an electronic
dosemeter using the model function M = N K (G – G ) and low level of consideration
of the workplace conditions, see 5.3.5.2 . 27
Table 4 – Example of an uncertainty budget for a measurement with an electronic
dosemeter using the model function M = N K (G – G ) and high level of consideration
of the workplace conditions, see 5.3.5.3 . 27
Table A.1 – Example of an uncertainty budget for a dose rate measurement according
to IEC 60846-1:2009 with an instrument having a logarithmic scale and low level of
consideration of the measuring conditions, see text for details . 36
Table A.2 – Example of an uncertainty budget for a dose rate measurement according
to IEC 60846-1:2009 with an instrument having a logarithmic scale and high level of

consideration of the measuring conditions, see text for details . 38
Table B.1 – Example of an uncertainty budget for a photon dose measurement with a
passive dosimetry system according to IEC 62387-1:2007 and low level of
consideration of the workplace conditions, see text for details . 42
Table B.2 – Example of an uncertainty budget for a photon dose measurement with a
passive dosimetry system according to IEC 62387-1:2007 and high level of
consideration of the measuring conditions, see text for details . 44
Table C.1 – Example of an uncertainty budget for a neutron dose measurement
according to IEC 61005:2003 using the analytical method. 48
Table C.2 – Example of an uncertainty budget for a neutron dose rate measurement

according to IEC 61005:2003 using the Monte Carlo method . 49
Table C.3 – Results of the analytical and the Monte Carlo method . 50
Table D.1 – List of quantities used in formula (D.1) . 51
Table D.2 – List of data available for the input quantities of formula (D.1) . 52
Table D.3 – Example of an uncertainty budget for the calibration of a radon monitor
according to IEC 61577, see text for details . 52
Table E.1 – Example of an uncertainty budget for a surface emission rate
measurement according to IEC 60325:2002, see text for details . 57
Table E.2 – Example of an uncertainty budget for a surface emission rate
measurement according to IEC 60325:2002 for the determination of the uncertainty at
a measured value of zero . 58

INTERNATIONAL ELECTROTECHNICAL COMMISSION
____________
RADIATION PROTECTION INSTRUMENTATION –
DETERMINATION OF UNCERTAINTY IN MEASUREMENT

FOREWORD
1) The International Electrotechnical Commission (IEC) is a worldwide organization for standardization comprising
all national electrotechnical committees (IEC National Committees). The object of IEC is to promote
international co-operation on all questions concerning standardization in the electrical and electronic fields. To
this end and in addition to other activities, IEC publishes International Standards, Technical Specifications,
Technical Reports, Publicly Available Specifications (PAS) and Guides (hereafter referred to as “IEC
Publication(s)”). Their preparation is entrusted to technical committees; any IEC National Committee interested
in the subject dealt with may participate in this preparatory work. International, governmental and non-
governmental organizations liaising with the IEC also participate in this preparation. IEC collaborates closely
with the International Organization for Standardization (ISO) in accordance with conditions determined by
agreement between the two organizations.
2) The formal decisions or agreements of IEC on technical matters express, as nearly as possible, an international
consensus of opinion on the relevant subjects since each technical committee has representation from all
interested IEC National Committees.
3) IEC Publications have the form of recommendations for international use and are accepted by IEC National
Committees in that sense. While all reasonable efforts are made to ensure that the technical content of IEC
Publications is accurate, IEC cannot be held responsible for the way in which they are used or for any
misinterpretation by any end user.
4) In order to promote international uniformity, IEC National Committees undertake to apply IEC Publications
transparently to the maximum extent possible in their national and regional publications. Any divergence
between any IEC Publication and the corresponding national or regional publication shall be clearly indicated in
the latter.
5) IEC itself does not provide any attestation of conformity. Independent certification bodies provide conformity
assessment services and, in some areas, access to IEC marks of conformity. IEC is not responsible for any
services carried out by independent certification bodies.
6) All users should ensure that they have the latest edition of this publication.
7) No liability shall attach to IEC or its directors, employees, servants or agents including individual experts and
members of its technical committees and IEC National Committees for any personal injury, property damage or
other damage of any nature whatsoever, whether direct or indirect, or for costs (including legal fees) and
expenses arising out of the publication, use of, or reliance upon, this IEC Publication or any other IEC
Publications.
8) Attention is drawn to the Normative references cited in this publication. Use of the referenced publications is
indispensable for the correct application of this publication.
9) Attention is drawn to the possibility that some of the elements of this IEC Publication may be the subject of
patent rights. IEC shall not be held responsible for identifying any or all such patent rights.
The main task of IEC technical committees is to prepare International Standards. However, a
technical committee may propose the publication of a technical report when it has collected
data of a different kind from that which is normally published as an International Standard, for
example "state of the art".
IEC 62461, which is a technical report, has been prepared by subcommittee 45B: Radiation
protection instrumentation, of IEC technical committee 45: Nuclear instrumentation.
This second edition of IEC TR 62461 cancels and replaces the first edition, published in 2006,
and constitutes a technical revision. The main changes with respect to the previous edition
are as follows:
– add to the analytical method for the determination of uncertainty the Monte Carlo method
for the determination of uncertainty according to supplement 1 of the Guide to the
Expression of uncertainty in measurement (GUM S1), and
– add a very simple method to judge whether a measured result is significantly different from
zero or not based on ISO 11929.

– 6 – IEC TR 62461:2015 © IEC 2015
The text of this technical report is based on the following documents:
Enquiry draft Report on voting
45B/783/DTR 45B/813/RVD
Full information on the voting for the approval of this technical report can be found in the
report on voting indicated in the above table.
This publication has been drafted in accordance with the ISO/IEC Directives, Part 2.
The committee has decided that the contents of this publication will remain unchanged until
the stability date indicated on the IEC website under "http://webstore.iec.ch" in the data
related to the specific publication. At this date, the publication will be
• reconfirmed,
• withdrawn,
• replaced by a revised edition, or
• amended.
A bilingual version of this publication may be issued at a later date.

IMPORTANT – The 'colour inside' logo on the cover page of this publication indicates
that it contains colours which are considered to be useful for the correct
understanding of its contents. Users should therefore print this document using a
colour printer.
INTRODUCTION
The ISO/IEC Guide 98-3:2008, Uncertainty of measurement – Part 3: Guide to the expression
of uncertainty in measurement (GUM:1995) as well as its Supplement 1:2008, Propagation of
distributions using a Monte Carlo method (GUM S1), are general guides to assess the
uncertainty in measurement. This Technical Report lays emphasis on their application in the
area of radiation protection and serves as a practical introduction to the GUM and its
supplement 1 (GUM S1).
The process of determining the uncertainty delivers not only a numerical value of the
uncertainty; in addition it produces the best estimate of the quantity to be measured which
may differ from the indication of the instrument. Thus, it can also improve the result of the
measurement by using information beyond the indicated value of the instrument, e.g. the
energy dependence of the instrument.

– 8 – IEC TR 62461:2015 © IEC 2015
RADIATION PROTECTION INSTRUMENTATION –
DETERMINATION OF UNCERTAINTY IN MEASUREMENT

1 Scope
This Technical Report gives guidelines for the application of the uncertainty analysis accord-
ing to ISO/IEC Guide 98-3:2008 (GUM describing an analytical method for the uncertainty
determination) and its Supplement 1:2008 (GUM S1 describing a Monte Carlo method for the
uncertainty determination) for measurements covered by standards of IEC Subcommittee 45B.
It does not include the uncertainty associated with the concept of the measuring quantity,
e. g., the difference between H (10) on the ISO water slab phantom and on the person.
p
This Technical Report explains the principles of the ISO/IEC Guide 98-3:2008 (GUM),its
Supplement 1:2008 (GUM S1) and the special considerations necessary for radiation
protection at an example taken from individual dosimetry of external radiation. In the
informative annexes, several examples are given for the application on instruments, for which
SC 45B has developed standards.
This Technical Report is supposed to assist the understanding of the ISO/IEC Guide 98-
3:2008 (GUM), its Supplement 1: 2008 (GUM S1), and other papers on uncertainty analysis. It
cannot replace these papers nor can it provide the background and justification of the
arguments leading to the concept of the ISO/IEC Guide 98-3:2008 (GUM) and its Supplement
1:2008 (GUM S1).
Finally, this Technical Report gives a very simple method to judge whether a measured result
is significantly different from zero or not based on ISO 11929.
For better readability the correct terms are not always used throughout this technical report.
For example, instead of “random variables of a quantity” only the “quantity” itself is stated.
2 Normative references
The following documents, in whole or in part, are normatively referenced in this document and
are indispensable for its application. For dated references, only the edition cited applies. For
undated references, the latest edition of the referenced document (including any
amendments) applies.
IEC 60050 (all parts): International Electrotechnical Vocabulary (available at
http://www.electropedia.org)
ISO/IEC Guide 98-3:2008, Uncertainty of measurement – Part 3: Guide to the expression of
uncertainty in measurement (GUM:1995)
ISO/IEC Guide 98-3, Supplement 1:2008, Uncertainty of measurement – Part 3: Guide to the
expression of uncertainty in measurement (GUM:1995) – Propagation of distributions using a
Monte Carlo method
3 Terms and definitions
For the purposes of this document, the technical terms of IEC 60050-151 [1], and
IEC 60050-311 [2] as well as the following definitions taken from the ISO/IEC Guide 98-
3:2008 (GUM), and its Supplement 1:2008 (GUM S1) apply .
3.1
calibration factor
N
quotient of the true value of a quantity and the indicated value for a specified reference
radiation under specified reference conditions
3.2
conformity test
test for conformity evaluation
[SOURCE: IEC 60050-151:2001,151-16-15]
3.3
complete result of a measurement
set of values attributed to a measurand, including a value, the corresponding uncertainty and
the unit of measurement
Note 1 to entry: The central value of the whole (set of values) can be selected as measured value and a
parameter characterising the dispersion as uncertainty.
Note 2 to entry: The result of a measurement is related to the indication given by the instrument and to the values
of correction obtained by calibration and by the use of a model.
Note 3 to entry: In this Technical Report, the “measured value”, see Note 1 above, is abbreviated by M.
Note 4 to entry: In this Technical Report, the “indication given by the instrument”, see Note 2 above, is
abbreviated by G, and called “indicated value”.
Note 5 to entry: In this Technical Report, the “model”, see Note 2 above, is called “model function”, see 3.10 and
5.2.
[SOURCE: IEC 60050-311:2001, 311-01-01, modified]
3.4
correction factor
K
factor to the indicated value to correct for deviation of measurement conditions from calibra-
tion conditions
3.5
coverage factor
k
cov
numerical factor used as a multiplier of the (combined) standard uncertainty in order to obtain
an expanded uncertainty
Note 1 to entry: A coverage factor k is typically in the range of 2 to 3.
cov
[SOURCE: GUM:2008, 2.3.6]
___________
Numbers in square brackets refer to the bibliography.

– 10 – IEC TR 62461:2015 © IEC 2015
3.6
decision threshold
m*
value of the estimator of the measurand, which when exceeded by the result of an actual
measurement using a given measurement procedure of a measurand quantifying a physical
effect, one decides that the physical effect is present
Note 1 to entry: The decision threshold is defined such that in cases where the measurement result, m, exceeds
the decision threshold, m*, the probability that the true value of the measurand is zero is less or equal to a chosen
probability, α.
Note 2 to entry: If the result, m, is below the decision threshold, m*, the result cannot be attributed to the physical
effect; nevertheless it cannot be concluded that it is absent.
[SOURCE: ISO 11929:2010]
3.7
deviation
D
difference between the indicated values for the same value of the measurand of an indicating
measuring instrument, or the values of a material measure, when an influence quantity
assumes, successively, two different values
Note 1 to entry: This definition is applicable to all measuring instruments and influence quantities, but it should
mainly be used in those cases, where this deviation is independent of the indicated value.
[SOURCE: IEC 60050-311:2001, 311-07-03, modified ]
3.8
distribution function
F(x)
a function giving, for every value x, the probability that the random variable X be less than or
equal to x: F(x) = Pr(X ≤ x)
[SOURCE: GUM:2008, C.2.4; GUM S1:2008, 3.2]
3.9
expanded uncertainty
U
quantity defining an interval about the result of a measurement that may be expected to
encompass a large fraction of the distribution of values that could reasonably be attributed to
the measurand
Note 1 to entry: The expanded uncertainty is obtained by multiplying the (combined) standard uncertainty by a
coverage factor.
[SOURCE: GUM:2008, 2.3.5]
3.10
indicated value
G
quantity value provided by a measuring instrument or a measuring system
Note 1 to entry: An indication is often given by the position of a pointer on the display for analogue outputs, a
displayed or printed number for digital outputs, a code pattern for code outputs, or an assigned quantity value for
material measures.
3.11
influence quantity
quantity that is not the measurand but that effects the result of the measurement
___________
Original term “variation (due to an influence quantity)”.

Note 1 to entry: For example, temperature of a micrometer used to measure length.
[SOURCE: GUM:2008, B.2.10]
3.12
measured value
M
value determined from the indicated value, G, by applying the model function for the meas-
urement
Note 1 to entry: An example of a model function is given below. The calibration factor N, a deviation D, and a
correction factor K are applied:
M = N × K × (G – D)
The calculations according to this model function are not always performed. One main purpose of this model func-
tion of the measurement is, that it is necessary for any determination of the uncertainty according to the GUM (see
GUM, 3.1.6, 3.4.1 and 4.1; see also 5.2 of this Technical Report).
Note 2 to entry: In the GUM the measured value is called value of the measurand.
3.13
probability density function
f(x)
the derivative (when it exists) of the distribution function: f(x)=dF(x)/dx
b
Note 1 to entry: f(x)·dx is the “probability element”: f(x)·dx=Pr(x ∫
a
[SOURCE: GUM:2008, C.2.5; GUM S1:2008, 3.3, modified by adding “in general”]
3.14
reference conditions
set of specified values and/or ranges of values of influence quantities under which the uncer-
tainties, or limits of error, admissible for a measuring instrument are the smallest
[SOURCE: IEC 60050-311:2001, 311-06-02]
3.15
reference response
R
ref
response of the assembly under reference conditions to unit reference dose (rate) or activity
and is expressed as:
G
R =
ref
M
c
where G is the indicated value of the equipment or assembly under test and M is the true
c
value of the reference source
3.16
relative response
R
rel
quotient of the response and the reference response under specified conditions
Note 1 to entry: For the specified reference conditions, the response is the reciprocal of the calibration factor.

– 12 – IEC TR 62461:2015 © IEC 2015
3.17
response
R
ratio of the quantity measured under specified conditions by the equipment or assembly under
test and the true value of this quantity
3.18
standard uncertainty
standard deviation associated with the measurement result or an input quantity
Note 1 to entry: See GUM:2008, 2.3.4.
Note 2 to entry: The standard uncertainty of the measurement result is sometimes called “combined standard
uncertainty”.
Note 3 to entry: The quotient of the standard uncertainty and the measurement result is called “relative standard
uncertainty” and sometimes given as percentage.
3.19
type test
conformity test made on one or more items representative of the production
[SOURCE: IEC 60050-151:2001, 151-16-16]
3.20
uncertainty
uncertainty of measurement
parameter, associated with the result of a measurement, that characterises the dispersion of
the values that could reasonably be attributed to the measurand
Note 1 to entry: The parameter may be, for example, a standard deviation (or a given multiple of it), or the half-
width of an interval having a stated level of confidence (coverage probability).
[SOURCE: GUM:2008, 2.2.3]
4 List of symbols
Table 1 gives a list of the symbols (and abbreviated terms) used in the main text of this
Technical Report (excluding annexes).
Table 1 – Symbols (and abbreviated terms) used
in the main text (excluding annexes)
Unit (dose
Symbol Meaning
measurement)
a Half-width of an interval for possible values of a quantity As quantity
a Lower limit of an interval for possible values of a quantity As quantity
–
a Upper limit of an interval for possible values of a quantity As quantity
+
α Probability to detect an effect (state a result above zero) although in reality no –
effect is present (the true value is zero) also called “probability of false positive
decision”
c Sensitivity coefficient for the input quantity K
Sv
k
c Sensitivity coefficient for the input quantity M –
m
c Sensitivity coefficient for the input quantity M –
m 0
c Sensitivity coefficient for the input quantity N Sv
n
F(x)
Distribution function –
Unit (dose
Symbol Meaning
measurement)
f(x) Probability density function (for a continuous random variable) PDF Inverse of
quantity
G Indicated value, for example, reading of the dosemeter in units of H (10) Sv
p
ĝ Best estimate of G Sv
g Possible value (estimate) of G Sv
G Zero reading Sv
ĝ Best estimate of G Sv
0 0
g Possible value (estimate) of G Sv
0 0
h(x) Model function, see Note 1 to 3.12 As output
quantity
H (10)
Sv
Personal dose equivalent at a depth 10 mm
p
i
Running index (integer) –
j Running index (integer) –
K Correction factor, for example, for energy and angle of radiation incidence –
k̂ Best estimate of K –
k Possible value (estimate) of K –
–
k quantile of the standardized normal distribution for a given probability α
1–α
k Coverage factor –
cov
L Number of Monte Carlo trials –
M Measured value, for example, personal dose equivalent H (10) Sv
p
M
True value of a reference source Sv
c
m̂ Best estimate of M Sv
m Possible value (estimate) of M Sv
m* Decision threshold of M Sv
N
Calibration factor –
n̂ Best estimate of N –
n Possible value (estimate) of N –
p Coverage probability –
Q Distribution function for the output quantity –
q Arbitrary integer –
R Absolute response –
abs
R
Relative response –
rel
s Standard deviation of the distribution of the g-values Sv
ĝ
s Standard deviation of the distribution of the g -values Sv
ĝ 0
s Standard deviation of the distribution of the k-values –
k̂
s Standard deviation of the distribution of the n-values –
n̂
T Number of input quantities –
U Expanded uncertainty Sv
u(m̂)
Standard uncertainty associated with the best estimate of the measurement Sv
result, m̂
u (m̂) Uncertainty contribution to u of the input quantity G associated with the best
Sv
g
estimate of the measurement result, m̂
u (m̂) Uncertainty contribution to u of the input quantity G associated with the best Sv
g 0
estimate of the measurement result, m̂
u (m̂) Uncertainty contribution to u of the input quantity K associated with the best Sv
k
estimate of the measurement result, m̂

– 14 – IEC TR 62461:2015 © IEC 2015
Unit (dose
Symbol Meaning
measurement)
u (m̂) Uncertainty contribution to u of the input quantity N associated with the best Sv
n
estimate of the measurement result, m̂
X A non-specified quantity As quantity
x̂ Best estimate of X As quantity
x Possible value (estimate) of X As quantity
y
Random number from the standard Gaussian distribution –
z Random number out of the interval 0 . 1 (rectangular distribution) –

5 The GUM and the GUM S1 concept
5.1 General concept of uncertainty determination
5.1.1 Overview in four steps
The GUM:2008 and its supplement 1, GUM S1:2008:
– consider available quantities influencing the measurement, e.g. the experience of the
person performing the measurement,
– are partly based on the Bayes statistics (especially the GUM S1),
– are internationally accepted.
NOTE The methods of the GUM and the GUM S1 are described and explained in many papers [3] to [11].
The application of the GUM (analytical method) and GUM S1 (Monte Carlo method), not the
justification or the mathematics behind it, will be described in a simplified example in the
following subclauses. Further details can be found in the literature.
The following four steps are necessary for the propagation (determination) of uncertainty.
Especially, for the first two steps, the expertise of the evaluator is essential.
– Step 1: A mathematical model function (or an algorithm) has to be stated describing the
relation of the input quantities X and the output quantity M
i
M = h(X ,…, X ) (1)
1 T
where
T is the number of input quantities;
X is an input quantity;
i
M is the output quantity.
The model function should contain every quantity, including all corrections and correction
factors that can contribute a significant component of uncertainty to the result of the
measurement; details are given in 5.2.
– Step 2: The available information for the input quantities X has to be collected; details are
i
given in 5.3.
– Step 3: The standard uncertainty u(m̂) of the output quantity has to be calculated using
either the analytical method (explained in 5.1.2) or the Monte Carlo method (explained in
5.1.3). For this step, only the application of mathematics is required. This task can,
therefore, be performed completely by a computer program, for example, the software
“GUM Workbench” [12] or “UncertRadio” [13]; details are given in 5.4.
– Step 4: The expanded uncertainty U(m̂) and or the corresponding coverage interval have
to be stated; details are given in 5.5.

5.1.2 Summary of the analytical method for steps 3 and 4
In this subclause, a short summary is given in the following to illustrate the analytical method:
a) Firstly, for each input quantity X , i = 1.T, the best estimate x̂ and its standard uncertainty
i i
s(x̂) have to be obtained;
i
b) Secondly, the sensitivity coefficient, i.e. the partial derivative of the output quantity with
respect to each input quantity, has to be calculated: c = ∂h/∂x ; this is the slope of the
i i
model function h(x ). The larger it is the stronger is the impact of the corresponding input
i
quantity to the output quantity, thus, it is the “lever arm” or “impact” of the corresponding
input quantity.
c) Thirdly, the uncertainty contribution to the output quantity due to each input quantity has
to be calculated by multiplying the sensitivity coefficient and the standard uncertainty:
u (m̂) = |c | · s(x̂).
i i i
d) Fourthly, the combined standard uncertainty for the output quantity is computed as the
n
square root of the squared uncertainty contributions: u (mˆ ) = {u (mˆ )} ; in case some
c ∑ i
i=1
(random variables expressing the state of knowledge about the according) input quantities
are correlated with one another (i.e. they depend on each other), further terms need to be
added to the sum under the square root sign, as detailed in 5.2 of the GUM:2008.
e) Finally, the expanded uncertainty for the output quantity has to be calculated by
multiplying the standard uncertainty with the appropriated coverage factor (usually k = 2):
U (m̂) = 2 · u (m̂); if the probability distribution of the output quantity is not approximately
c c
Gaussian (or normal), the coverage factor may have another value, see 6.3 of the
GUM:2008 .
5.1.3 Summary of the Monte Carlo method for steps 3 and 4
In this subclause, a short summary, taken from the introduction and from 5.9.6 of the
GUM S1:2008, is given in the following to illustrate the Monte Carlo method:
This Supplement to the GUM is concerned with the propagation of probability distributions
through the mathematical model of measurement [GUM:1995, 3.1.6] as a basis for the
evaluation of uncertainty of measurement, and its implementation by a Monte Carlo method.
The treatment applies to a model having any number of input quantities, and a single output
quantity. The described Monte Carlo method is a practical alternative to the GUM uncertainty
framework [GUM:1995, 3.4.8]. It has value when
a) linearization of the model provides an inadequate representation or
b) the probability density function (PDF) for the output quantity departs appreciably from a
Gaussian distribution or a scaled and shifted t-distribution, e.g. due to marked asymmetry
of dominating influence quantities (i.e. those with large uncertainties) or due to a model
function with only very few influence quantities which are, in addition, non-Gaussian
distributed.
The Monte Carlo method can be stated as a step-by-step procedure, see 5.9.6 of the
GUM S1:2008:
a) select the number L of Monte Carlo trials to be made;
b) generate L vectors, by sampling from the assigned PDFs, as realizations of the (set of
i = 1.T) input quantities X ;
i
c) for each such vector, form the corresponding model value of M = h(X ), yielding L model
i
values M with j = 1.L;
j
d) sort these L model values into increasing order, using the sorted model values to provide
the distribution function for the output quantity Q;
e) calculate the average of M , …, M which is an estimate m̂ of M, and calculate their
1 L
standard deviation which is an evaluation of the standard measurement uncertainty u(m̂)
associated with m̂, see 5.4.3 d);

– 16 – IEC TR 62461:2015 © IEC 2015
...