ISO/PRF 5725-5

ISO/TC 69/SC 6
Secretariat: JISC
Date: 2025-07-14xx
Accuracy (trueness and precision) of measurement methods and
results —
Part 5:
Alternative methods for the determination of the precision of a
standard measurement method
Exactitude (justesse et fidélité) des résultats et méthodes de mesure —
Partie 5: Méthodes alternatives pour la détermination de la fidélité d'une méthode de mesure normalisée
PROOF
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ii
Contents
Foreword . iv
Introduction . v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Symbols and abbreviations . 1
5 Robust methods for data analysis: Algorithms A and S. 3
5.1 Applications of robust methods of data analysis . 3
5.2 Robust analysis: Algorithm A . 5
5.3 Robust analysis: Algorithm S . 6
5.4 Formulae: robust analysis for a particular level of a uniform-level design . 8
5.5 Formulae: robust analysis for a particular level of a split-level design . 9
5.6 Formulae: robust analysis for a particular level of an experiment on a heterogeneous
material . 9
6 Robust methods for data analysis: Q method and Hampel estimator . 10
6.1 Rationale for computationally intensive estimators . 10
7 Robust statistical analysis of results by means of the Q/Hampel method in a one-way
replicated design . 10
7.1 Introduction to the Q/Hampel method . 10
7.2 Determination of the robust reproducibility standard deviation 𝑠𝑅 using the Q method . 10
7.3 Determination of the robust repeatability standard deviation 𝑠𝑟 using the Q method . 12
7.4 Determination of the robust mean 𝑥 ∗ using the Hampel estimator . 13
8 Robust statistical analysis of results by means of the Q/Hampel method in a staggered
nested design with two factors . 14
8.1 Data layout and nomenclature . 14
8.2 Determination of the robust reproducibility standard deviation 𝑠𝑅 using the Q method . 14
8.3 Determination of the robust intermediate standard deviation using the Q method . 15
8.4 Determination of the robust repeatability standard deviation 𝑠𝑟 using the Q method . 15
8.5 Determination of the robust mean 𝑥* using the Hampel estimator . 16
Annex A (normative) Determination of the robust mean using the Hampel estimator . 18
Annex B (informative) Derivations . 21
Annex C (informative) Examples . 25
Bibliography . 42

iii
Foreword
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This document was prepared by Technical Committee ISO/TC 69, Applications of statistical methods,
Subcommittee SC 6, Measurement methods and results.
This second edition cancels and replaces the first edition (ISO 5725-5:1998), which has been technically
revised. It also incorporates the Technical Corrigendum ISO 5725-5:1998/Cor.1:2005.
The main changes are as follows:
— — alternative experimental designs (split-level and design for heterogeneous material) have been
transferred to ISO 5725-3;
— — an additional robust approach, the Q method, which has improved breakdown properties, has been
added.
A list of all parts in the ISO 5725 series can be found on the ISO website.
Any feedback or questions on this document should be directed to the user’s national standards body. A
complete listing of these bodies can be found at www.iso.org/members.html.
iv
Introduction
This document uses two terms, trueness and precision, to describe the accuracy of a measurement method.
Trueness refers to the closeness of agreement between the average value of a large number of test results and
the true or accepted reference value. Precision refers to the closeness of agreement between test results.
General consideration of these quantities is given in ISO 5725-1 and so is not repeated here. This document
should be read in conjunction with ISO 5725-1 because the underlying definitions and general principles are
given there.
ISO 5725-2 is concerned with estimating, by means of interlaboratory experiments, standard measures of
precision, namely the repeatability standard deviation and the reproducibility standard deviation. It gives a
basic method for doing this, including methods of calculation. This document describes alternative calculation
methods to this basic method:
— — The basic method requires the preparation of a number of identical samples of the material for use in
the experiment. With heterogeneous materials this may not be possible, so that the use of the basic method
then gives estimates of the reproducibility standard deviation that are inflated by the variation between
the samples. The design for a heterogeneous material given in this document yields information about the
variability between samples which is not obtainable from the basic method; it may be used to calculate an
estimate of reproducibility from which the between-sample variation has been removed.
— — The basic method requires tests for outliers to be used to identify data that should be excluded from
the calculation of the repeatability and reproducibility standard deviations. Excluding outliers can
sometimes have a large effect on the estimates of repeatability and reproducibility standard deviations,
but in practice, when applying the outlier tests, the statistical expert may have to use judgement to decide
which data to exclude. This document describes robust methods of data analysis that may be used to
calculate repeatability and reproducibility standard deviations from data containing outliers without
using tests for outliers to exclude data, so that the results are no longer affected by the statistical expert’s
judgement.
v
Accuracy (trueness and precision) of measurement methods and
results — —
Part 5:
Alternative methods for the determination of the precision of a
standard measurement method
1 Scope
This document describes the use of robust methods for analysing the results of precision experiments without
using outlier tests to exclude data from the calculations, and in particular, the detailed use of several such
methods. The robust methods described in this document allow the data to be analysed in such a way that it
is not required to make decisions about outliers that affect the results of the calculations.
2 Normative references
The following documents are referred to in the text in such a way that some or all of their content constitutes
requirements 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, Statistics — Vocabulary and symbols — Part 1: General statistical terms and terms used in
probability
ISO 5725--1, Accuracy (trueness and precision) of measurement methods and results — Part 1: General
principles and definitions
3 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO 3534-1 and in ISO 5725-1 apply.
ISO and IEC maintain terminology databases for use in standardization at the following addresses:
— — ISO Online browsing platform: available at https://www.iso.org/obp
— — IEC Electropedia: available at https://www.electropedia.org/
4 Symbols and abbreviations
𝑏 Correction factor used for reproducibility standard deviation in the Q method
𝑝
𝑐 Correction factor used for repeatability standard deviation in the Q method
𝑝
𝐷 Within-cell difference between measurement results (used with subscripts as required).
¯
𝐷 Average of within-cell differences
𝑑 … 𝑑 Interpolation nodes for each value 𝑦 (Hampel estimator, non-iterative)
1 6 𝑖
𝐸{} Expectation of a statistical variable
𝐺 (𝑥) Interpolation function (Q method, robust reproducibility standard deviation)
−1
Function returning a given quantile of 𝐺 (𝑥) (Q method, robust reproducibility standard
𝐺 ()
1 1
deviation)
𝐺 (𝑥 ) Function of individual discontinuity points (Q method, robust reproducibility standard deviation)
1 𝑖
𝐺 (𝑥) Interpolation function (Q method, robust repeatability standard deviation)
−1
𝐺 () Function returning a given quantile of 𝐺 (𝑥) (Q method, robust repeatability standard deviation)
2 2
𝐺 (𝑥 ) Function of individual discontinuity points (Q method, robust repeatability standard deviation)
2 𝑖
𝐻 (𝑥) Cumulative distribution function of absolute between-laboratory differences (Q method)
𝐻 (𝑥) Cumulative distribution function of within-laboratory differences (Q method)
I{} Indicator function returning 1 when a condition (in {}) is met and 0 otherwise
i Identifier for a particular laboratory
j Identifier for a particular replicate observation
n Number of test results obtained in one laboratory at one level (i.e. per cell)
𝑃 Probability
𝑃 Probability used in calculation of adjustment factor for Algorithm S
𝜉
p Number of laboratories participating in the interlaboratory experiment
𝑝 Check value for solutions in non-iterative Hampel estimator
𝑚
𝑝′ Number of laboratories participating in an interlaboratory experiment (heterogeneous material)
SS Sum of squares
𝑆𝑆 Robust estimate of between-sample sum of squares (heterogeneous material)
H
𝑆𝑆 Robust estimate of repea
...