ISO/TS 21749:2005

Measurement uncertainty for metrological applications — Repeated measurements and nested experiments

ISO/TS 21749:2005 follows the approach taken in the Guide to the expression of the uncertainty of measurement (GUM) and establishes the basic structure for stating and combining components of uncertainty. To this basic structure, it adds a statistical framework using the analysis of variance (ANOVA) for estimating individual components, particularly those classified as Type A evaluations of uncertainty, i.e. based on the use of statistical methods. A short description of Type B evaluations of uncertainty (non-statistical) is included for completeness. ISO/TS 21749:2005 covers experimental situations where the components of uncertainty can be estimated from statistical analysis of repeated measurements, instruments, test items or check standards. It provides methods for obtaining uncertainties from single-, two- and three-level nested designs only. More complicated experimental situations where, for example, there is interaction between operator effects and instrument effects or a cross effect, are not covered. ISO/TS 21749:2005 is not applicable to measurements that cannot be replicated, such as destructive measurements or measurements on dynamically varying systems (such as fluid flow, electronic currents or telecommunications systems). It is not particularly directed to the certification of reference materials (particularly chemical substances) and to calibrations where artefacts are compared using a scheme known as a "weighing design". For certification of reference materials, see ISO Guide 35. When results from interlaboratory studies can be used, techniques are presented in the companion guide ISO/TS 21748. The main difference between ISO/TS 21748 and this Technical Specification is that the ISO/TS 21748 is concerned with reproducibility data (with the inevitable repeatability effects), whereas this Technical Specification concentrates on repeatability data and the use of the analysis of variance for its treatment. ISO/TS 21749:2005 is applicable to a wide variety of measurements, for example, lengths, angles, voltages, resistances, masses and densities.

Incertitude de mesure pour les applications en métrologie — Mesures répétées et expériences emboîtées

General Information

• Status: Published
• Publication Date: 03-Mar-2005
• ICS: 17.020 - Metrology and measurement in general
• Technical Committee: ISO/TC 69/SC 6 - Measurement methods and results
• Drafting Committee: ISO/TC 69/SC 6/WG 7 - Statistical methods to support measurement uncertainty evaluation
• Current Stage: 9093 - International Standard confirmed
• Completion Date: 06-Oct-2022

Standards Content (Sample)

TECHNICAL ISO/TS
SPECIFICATION 21749
First edition
2005-02-15
Corrected version
2005-07-15

Measurement uncertainty for
metrological applications — Repeated
measurements and nested experiments
Incertitude de mesure pour les applications en métrologie — Mesures
répétées et expériences emboîtées




Reference number
ISO/TS 21749:2005(E)
©
ISO 2005

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ISO/TS 21749:2005(E)
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ISO/TS 21749:2005(E)
Contents Page
Foreword. iv
Introduction . v
1 Scope. 1
2 Normative references . 1
3 Terms and definitions. 2
4 Statistical methods of uncertainty evaluation .3
4.1 Approach of the Guide to the expression of uncertainty of measurement . 3
4.2 Check standards . 4
4.3 Steps in uncertainty evaluation. 5
4.4 Examples in this Technical Specification. 6
5 Type A evaluation of uncertainty . 6
5.1 General. 6
5.2 Role of time in Type A evaluation of uncertainty . 7
5.3 Measurement configuration. 14
5.4 Material inhomogeneity. 16
5.5 Bias due to measurement configurations . 17
6 Type B evaluation of uncertainty . 26
7 Propagation of uncertainty . 27
7.1 General. 27
7.2 Formulae for functions of a single variable .28
7.3 Formulae for functions of two variables. 28
8 Example — Type A evaluation of uncertainty from a gauge study . 30
8.1 Purpose and background. 30
8.2 Data collection and check standards. 30
8.3 Analysis of repeatability, day-to-day and long-term effects. 31
8.4 Probe bias. 31
8.5 Wiring bias. 33
8.6 Uncertainty calculation. 35
Annex A (normative) Symbols . 37
Bibliography . 38

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ISO/TS 21749:2005(E)
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies
(ISO member bodies). The work of preparing International Standards is normally carried out through ISO
technical committees. Each member body interested in a subject for which a technical committee has been
established has the right to be represented on that committee. International organizations, governmental and
non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the
International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International Standards
adopted by the technical committees are circulated to the member bodies for voting. Publication as an
International Standard requires approval by at least 75 % of the member bodies casting a vote.
In other circumstances, particularly when there is an urgent market requirement for such documents, a
technical committee may decide to publish other types of normative document:
— an ISO Publicly Available Specification (ISO/PAS) represents an agreement between technical experts in
an ISO working group and is accepted for publication if it is approved by more than 50 % of the members
of the parent committee casting a vote;
— an ISO Technical Specification (ISO/TS) represents an agreement between the members of a technical
committee and is accepted for publication if it is approved by 2/3 of the members of the committee casting
a vote.
An ISO/PAS or ISO/TS is reviewed after three years in order to decide whether it will be confirmed for a
further three years, revised to become an International Standard, or withdrawn. If the ISO/PAS or ISO/TS is
confirmed, it is reviewed again after a further three years, at which time it must either be transformed into an
International Standard or be withdrawn.
Attention is drawn to the possibility that some of the elements of this document may be the subject of patent
rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO/TS 21749 was prepared by Technical Committee ISO/TC 69, Applications of statistical methods,
Subcommittee SC 6, Measurement methods and results.
This corrected version of ISO/TS 21749:2005 incorporates the correction of the title.

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ISO/TS 21749:2005(E)
Introduction
Test, calibration and other laboratories are frequently required to report the results of measurements and the
associated uncertainties. Evaluation of uncertainty is an on-going process that can consume time and
resources. In particular, there are many tests and other operations carried out by laboratories where two or
three sources of uncertainty are involved. Following the approach in the Guide to the expression of uncertainty
of measurement (GUM) to combining components of uncertainty, this document focuses on using the analysis
of variance (ANOVA) for estimating individual components, particularly those based on Type A (statistical)
evaluations.
An experiment is designed by the laboratory to enable an adequate number of measurements to be made, the
analysis of which will permit the separation of the uncertainty components. The experiment, in terms of design
and execution, and the subsequent analysis and uncertainty evaluation, require familiarity with data analysis
techniques, particularly statistical analysis. Therefore, it is important for laboratory personnel to be aware of
the resources required and to plan the necessary data collection and analysis.
In this Technical Specification, the uncertainty components based on Type A evaluations can be estimated
from statistical analysis of repeated measurements, from instruments, test items or check standards.
A purpose of this Technical Specification is to provide guidance on the evaluation of the uncertainties
associated with the measurement of test items, for instance as part of ongoing manufacturing inspection.
Such uncertainties contain contributions from the measurement process itself and from the variability of the
manufacturing process. Both types of contribution include those from operators, environmental conditions and
other effects. In order to assist in separating the effects of the measurement process and manufacturing
variability, measurements of check standards are used to provide data on the measurement process itself.
Such measurements are nominally identical to those made on the test items. In particular, measurements on
check standards are used to help identify time-dependent effects, so that such effects can be evaluated and
contrasted with a database of check standard measurements. These standards are also useful in helping to
control the bias and long-term drift of the process once a baseline for these quantities has been established
from historical data.
Clause 4 briefly describes the statistical methods of uncertainty evaluation including the approach
recommended in the GUM, the use of check standards, the steps in uncertainty evaluation and the examples
in this Technical Specification. Clause 5, the main part of this Technical Specification, discusses the Type A
evaluations. Nested designs in ANOVA are used in dealing with time-dependent sources of uncertainty. Other
sources such as those from the measurement configuration, material inhomogeneity, and the bias due to
measurement configurations and related uncertainty analyses are discussed. Type B (non-statistical)
evaluations of uncertainty are discussed for completeness in Clause 6. The law of propagation of uncertainty
described in the GUM has been widely used. Clause 7 provides formulae obtained by applying this law to
certain functions of one and two variables. In Clause 8, as an example, a Type A evaluation of uncertainty for
a gauge study is discussed, where uncertainty components from various sources are obtained. Annex A lists
the statistical symbols used in this Technical Specification.
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TECHNICAL SPECIFICATION ISO/TS 21749:2005(E)

Measurement uncertainty for metrological applications —
Repeated measurements and nested experiments
1 Scope
This Technical Specification follows the approach taken in the Guide to the expression of the uncertainty of
measurement (GUM) and establishes the basic structure for stating and combining components of
uncertainty. To this basic structure, it adds a statistical framework using the analysis of variance (ANOVA) for
estimating individual components, particularly those classified as Type A evaluations of uncertainty, i.e. based
on the use of statistical methods. A short description of Type B evaluations of uncertainty (non-statistical) is
included for completeness.
This Technical Specification covers experimental situations where the components of uncertainty can be
estimated from statistical analysis of repeated measurements, instruments, test items or check standards.
It provides methods for obtaining uncertainties from single-, two- and three-level nested designs only. More
complicated experimental situations where, for example, there is interaction between operator effects and
instrument effects or a cross effect, are not covered.
This Technical Specification is not applicable to measurements that cannot be replicated, such as destructive
measurements or measurements on dynamically varying systems (such as fluid flow, electronic currents or
telecommunications systems). It is not particularly directed to the certification of reference materials
(particularly chemical substances) and to calibrations where artefacts are compared using a scheme known
[14]
as a “weighing design”. For certification of reference materials, see ISO Guide 35 .
When results from interlaboratory studies can be used, techniques are presented in the companion guide
[15]
ISO/TS 21748 . The main difference between ISO/TS 21748 and this Technical Specification is that the
ISO/TS 21748 is concerned with reproducibility data (with the inevitable repeatability effects), whereas this
Technical Specification concentrates on repeatability data and the use of the analysis of variance for its
treatment.
This Technical Specification is applicable to a wide variety of measurements, for example, lengths, angles,
voltages, resistances, masses and densities.
2 Normative references
The following referenced documents are indispensable for the application of this document. For dated
references, only the edition cited applies. For undated references, the latest edition of the referenced
document (including any amendments) applies.
ISO 3534-1:1993, Statistics — Vocabulary and symbols — Part 1: Probability and general statistical terms
ISO 3534-3:1999, Statistics — Vocabulary and symbols — Part 3: Design of experiments
ISO 5725-1, Accuracy (trueness and precision) of measurement methods and results — Part 1: General
principles and definitions
ISO 5725-2, Accuracy (trueness and precision) of measurement methods and results — Part 2: Basic method
for the determination of repeatability and reproducibility of a standard measurement method
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ISO/TS 21749:2005(E)
ISO 5725-3, Accuracy (trueness and precision) of measurement methods and results — Part 3: Intermediate
measures of the precision of a standard measurement method
ISO 5725-4, Accuracy (trueness and precision) of measurement methods and results — Part 4: Basic
methods for the determination of the trueness of a standard measurement method
ISO 5725-5, Accuracy (trueness and precision) of measurement methods and results — Part 5: Alternative
methods for the determination of the precision of a standard measurement method
ISO 5725-6, Accuracy (trueness and precision) of measurement methods and results — Part 6: Use in
practice of accuracy values
Guide to the expression of uncertainty in measurement (GUM), BIPM, IEC, IFCC, ISO, IUPAC, IUPAP, OIML,
1993, corrected and reprinted in 1995
3 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO 3534-1, ISO 3534-3, ISO 5725 (all
parts) and the following apply.
3.1
measurand
well-defined physical quantity that is to be measured and can be characterized by an essentially unique value
3.2
uncertainty of measurement
parameter or an estimate of the parameter, associated with the result of a measurement, that characterizes
the dispersion of the values that could reasonably be attributed to the quantity being measured
3.3
Type A evaluation
method of evaluation of uncertainty by using statistical methods
3.4
Type B evaluation
method of evaluation of uncertainty by means other than statistical methods
3.5
standard uncertainty
uncertainty expressed as a standard deviation associated with a single component of uncertainty
3.6
combined standard uncertainty
standard deviation associated with the result of a particular measurement or series of measurements that
takes into account one or more components of uncertainty
3.7
expanded uncertainty
combined standard uncertainty multiplied by a coverage factor which usually is an appropriate critical value
from the t-distribution which depends upon the degrees of freedom in the combined standard uncertainty and
the desired level of coverage
3.8
effective degrees of freedom
degrees of freedom associated with a standard deviation composed of two or more components of variance
NOTE The effective degrees of freedom can be computed using the Welch-Satterthwaite approximation (see GUM,
G.4).
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ISO/TS 21749:2005(E)
3.9
nested design
experimental design in which each level (i.e. each potential setting, value or assignment of a factor) of a given
factor appears in only a single level of any other factor
NOTE 1 Adapted from ISO 3534-3:1999, definition 2.6.
NOTE 2 See ISO 3434-3:1999, 1.6, for the definition of level.
3.10
fixed effects
〈factors〉 effects resulting from the preselection of levels of each factor over the range of values of the factors
3.11
random effects
〈factors〉 effects resulting from the sampling at each level of each factor from the population of levels of each
factor
3.12
balanced nested design
nested design experiment in which the number of levels of the nested factors is constant
[ISO 3534-3:1999, definition 2.6.1]
3.13
mean square for random errors
sum of squared error divided by the corresponding degrees of freedom
NOTE See ISO 3534-1:1993, 2.85 for the definition of the degrees of freedom.
4 Statistical methods of uncertainty evaluation
4.1 Approach of the Guide to the expression of uncertainty of measurement
The Guide to the expression of uncertainty of measurement (GUM) recommends that the result of
measurement be corrected for all recognized significant systematic effects, that the result accordingly be the
best (or at least unbiased) estimate of the measurand and that a complete model of the measurement system
exists. The model provides a functional relationship between a set of input quantities (upon which the
measurand depends) and the measurand (output quantities). The objective of uncertainty evaluation is to
determine an interval that can be expected to encompass a large fraction of the distribution of values that
could reasonably be attributed to the measurand. Since a bias cannot be quantified exactly, when a result of
measurement is corrected for bias, the correction has an associated uncertainty.
The general approach, beginning from the modelling process, is the following.
NOTE The approach here relates to input quantities that are mutually independent. It is capable of a further
generalization to mutually dependent input quantities (see the GUM, 5.2).
a) Develop a mathematical model (functional relationship) of the measurement process or measurement
system that relates the model input quantities (including influence quantities) to the model output quantity
(measurand). In many cases, this model is the formula (or formulae) used to calculate the measurement
result, augmented if necessary by random, environmental and other effects such as bias correction that
may affect the measurement result.
b) Assign best estimates and the associated standard uncertainties (uncertainties expressed as standard
deviations) to the model input quantities.
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ISO/TS 21749:2005(E)
c) Evaluate the contribution to the standard uncertainty associated with the measurement result that is
attributable to each input quantity. These contributions shall take into account uncertainties associated
with both random and systematic effects relating to the input quantities, and may themselves involve
more detailed uncertainty evaluations.
d) Aggregate these standard uncertainties to obtain the (combined) standard uncertainty associated with the
measurement result. This evaluation of uncertainty is carried out, according to GUM, using the law of
propagation of uncertainty, or by more general analytical or numerical methods when the conditions for
the law of propagation of uncertainty do not apply or it is not known whether they apply.
e) Where appropriate, multiply the standard uncertainty associated with the measurement result by a
coverage factor to obtain an expanded uncertainty and hence a coverage interval for the measurand at a
prescribed level of confidence. The GUM provides an approach that can be used to calculate the
coverage factor. If the degrees of freedom for the standard uncertainties of all the input quantities are
infinite, the coverage factor is determined from the normal distribution. Otherwise, the (effective) degrees
of freedom for the combined standard uncertainty is estimated from the degrees of freedom for the
standard uncertainties associated with the best estimates of the input quantities using the Welch-
Satterthwaite formula.
The GUM permits the evaluation of standard uncertainties by any appropriate means. It distinguishes the
evaluation by the statistical treatment of repeated observations as a Type A evaluation of uncertainty, and the
evaluation by any other means as a Type B evaluation of uncertainty. In evaluating the combined standard
uncertainty, both types of evaluation are to be characterized by variances (squared standard uncertainties)
and treated in the same way.
Full details of this procedure and the additional assumptions on which it is based are given in the GUM.
The purpose of this Technical Specification is to provide additional detail on the evaluation of uncertainty by
statistical means, concentrating on b) above, whether obtained by repeated measurement of the input
quantities or of the entire measurement.
In this Technical Specification the term “artefact” is often used in the context of measurement. This usage is to
be given a general interpretation in that the measurement may also relate to a bulk or chemical item, etc.
4.2 Check standards
A check standard is a standard required to have the following properties.
a) It shall be capable of being measured periodically.
b) It shall be close in material content and geometry to the production items.
c) It shall be a stable artefact.
d) It shall be available to the measurement process at all times.
Subject to its having these properties, an ideal check standard is an artefact selected at random from the
production items, if appropriate, and reserved for this purpose.
Examples of the use of check standards include
 measurements on a stable artefact, and
 differences between values of two reference standards as estimated from a calibration experiment.
Methods for analysing check standard measurements are treated in 5.2.3.
In this Technical Specification, the term “check standard” is to be given a general interpretation. For instance,
a bulk or chemical item may be used.
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ISO/TS 21749:2005(E)
4.3 Steps in uncertainty evaluation
4.3.1 The first step in the uncertainty evaluation is the definition of the measurand for which a measurement
result is to be reported for the test item. Special care should be taken to provide an unambiguous definition of
the measurand, because the resulting uncertainty will depend on this definition. Possibilities include
 quantity at an instant in time at a point in space,
 quantity at an instant in time averaged over a specified spatial region,
 quantity at a point in space averaged over a time period.
For instance, the measurands corresponding to the hardness of a specimen of a ceramic material are (very)
different
a) at a specified point in the specimen, or
b) averaged over the specimen.
4.3.2 If the value of the measurand can be measured directly, the evaluation of the standard uncertainty
depends on the number of repeated measurements and the environmental and operational conditions over
which the repetitions are made. It also depends on other sources of uncertainty that cannot be observed
under the conditions selected to repeat the measurements, such as calibration uncertainties for reference
standards. On the other hand, if the value of the measurand cannot be measured directly, but is to be
calculated from measurements of secondary quantities, the model (or functional relationship) for combining
the various quantities must be defined. The standard uncertainties associated with best estimates of the
secondary quantities are then needed to evaluate the standard uncertainty associated with the value of the
measurand.
The steps to be followed in an uncertainty evaluation are outlined as follows.
a) Type A evaluations:
1) If the output quantity is represented by Y, and measurements of Y can be replicated, use an ANOVA
model to provide estimates of the variance components, associated with Y, for random effects from
 replicated results for the test item,
 measurements on a check standard,
 measurements made according to a designed experiment.
2) If measurements of Y cannot be replicated directly, and the model
Y = f (X , X , ., X )
1 2 n
is known, and the input quantities X can be replicated, evaluate the uncertainties associated with the
i
best estimates x of X ; then the law of propagation of uncertainty can be used.
i i
3) If measurements of Y or X cannot be replicated, refer to Type B evaluations.
i
b) Type B evaluations: evaluate a standard uncertainty associated with the best estimate of each input
quantity.
c) Aggregate the standard uncertainties from the Type A and Type B evaluations to provide a standard
uncertainty associated with the measurement result.
d) Compute an expanded uncertainty.
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ISO/TS 21749:2005(E)
4.4 Examples in this Technical Specification
The purpose of the examples in various clauses of this Technical Specification and the more detailed case
study in Clause 8 is to demonstrate the evaluation of uncertainty associated with measurement processes
having several sources of uncertainty. The reader should be able to generalize the principles illustrated in
these sections to particular applications. The examples treat the effect of both random effects and systematic
effects in the form of bias on the measurement result. There is an emphasis on quantifying uncertainties
observed over time, such as those for time intervals defined as short-term (repeatability) and for intermediate
measures of precision such as day-to-day or run-to-run, as well as for reproducibility. For the reader's
purpose, the time intervals should be defined in a way that makes sense for the measurement process in
question.
To illustrate strategies for dealing with several sources of uncertainty, data from the Electronics and Electrical
Engineering Laboratory of the National Institute of Standards and Technology (NIST), USA, are featured. The
measurements in question are volume resistivities (Ω⋅cm) of silicon wafers. These data were chosen for
illustrative purposes because of the inherent difficulties in measuring resistivity by probing the surface of the
wafer and because the measurand is defined by an ASTM test method and cannot be defined independently
of the method.
The intent of the experiment is to evaluate the uncertainty associated with the resistivity measurements of
silicon wafers at various levels of resistivity (Ω·cm), which were certified using a four-point probe wired in a
specific configuration. The test method is ASTM Method F84. The reported resistivity for each wafer is the
average of six short-term repetitions made at the centre of the wafer.
5 Type A evaluation of uncertainty
5.1 General
5.1.1 Generally speaking, any observation that can be repeated (see GUM, 3.1.4 to 3.1.6) can provide data
suitable for a T
...