AReference numberISO 5725-5:1998(E)INTERNATIONALSTANDARDISO5725-5First edition1998-07-15Accuracy (trueness and precision) ofmeasurement methods and results —Part 5:Alternative methods for the determination ofthe precision of a standard measurementmethodExactitude (justesse et fidélité) des résultats et méthodes de mesure —Partie 5: Méthodes alternatives pour la détermination de la fidélité d uneméthode de mesure normaliséeISO 5725-5:1998(E)© ISO 1998All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized in any form or by any means, electronicor mechanical, including photocopying and microfilm, without permission in writing from the publisher.International Organization for StandardizationCase postale 56 • CH-1211 Genève 20 • SwitzerlandInternet

[email protected] in SwitzerlandiiContents Page1 Scope . 12 Normative references 13 Definitions 24 Split-level design 24.1 Applications of the split-level design 24.2 Layout of the split-level design 24.3 Organization of a split-level experiment 34.4 Statistical model 44.5 Statistical analysis of the data from a split-level experiment . 54.6 Scrutiny of the data for consistency and outliers . 64.7 Reporting the results of a split-level experiment 74.8 Example 1: A split-level experiment — Determination of protein 75 A design for a heterogeneous material 135.1 Applications of the design for a heterogeneous material . 135.2 Layout of the design for a heterogeneous material . 145.3 Organization of an experiment with a heterogeneous material . 155.4 Statistical model for an experiment with a heterogeneous material 165.5 Statistical analysis of the data from an experiment with a heterogeneous material 175.6 Scrutiny of the data for consistency and outliers . 205.7 Reporting the results of an experiment on a heterogeneous material . 215.8 Example 2: An experiment on a heterogeneous material 215.9 General formulae for calculations with the design for a heterogeneous material 295.10 Example 3: An application of the general formulae . 306 Robust methods for data analysis . 336.1 Applications of robust methods of data analysis . 336.2 Robust analysis: Algorithm A . 356.3 Robust analysis: Algorithm S . 366.4 Formulae: Robust analysis for a particular level of a uniform-level design . 386.5 Example 4: Robust analysis for a particular level of a uniform-level design . 386.6 Formulae: Robust analysis for a particular level of a split-level design . 426.7 Example 5: Robust analysis for a particular level of a split-level design . 426.8 Formulae: Robust analysis for a particular level of an experiment on a heterogeneous material . 456.9 Example 6: Robust analysis for a particular level of an experiment on a heterogeneous material . 45AnnexesA (normative) Symbols and abbreviations used in ISO 5725 50B (informative) Derivation of the factors used in algorithms A and S 53C (informative) Derivation of equations used for robust analysis 55D (informative) Bibliography . 56© ISO ISO 5725-5:1998(E)iiiForewordISO (the International Organization for Standardization) is a world-wide federation of national standards bodies (ISOmember bodies). The work of preparing International Standards is normally carried out through ISO technicalcommittees. Each member body interested in a subject for which a technical committee has been established hasthe right to be represented on that committee. International organisations, governmental and non-governmental, inliaison with ISO, also take part in the work. ISO collaborates closely with the International ElectrotechnicalCommission (IEC) on all matters of electrotechnical standardization.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.ISO 5725-5 was prepared by Technical Committee ISO/TC 69, Applications of statistical methods, SubcommitteeSC 6, Measurement methods and results.ISO 5725 consists of the following parts, under the general title Accuracy (trueness and precision) of measurementmethods and results:— Part 1: General principles and definitions— Part 2: Basic method for the determination of repeatability and reproducibility of a standard measurementmethod— Part 3: Intermediate measures of the precision of a standard measurement method— Part 4: Basic methods for the determination of the trueness of a standard measurement method— Part 5: Alternative methods for the determination of the precision of a standard measurement method— Part 6: Use in practice of accuracy valuesParts 1 to 6 of ISO 5725 together cancel and replace ISO 5725:1986, which has been extended to cover trueness(in addition to precision) and intermediate precision conditions (in addition to repeatability conditions andreproducibility conditions).Annex A forms an integral part of this part of ISO 5725. Annexes B, C and D are for information only.ISO 5725-5:1998(E)© ISOivIntroduction0.1 This part of ISO 5725 uses two terms trueness and precision to describe the accuracy of a measurementmethod. Trueness refers to the closeness of agreement between the average value of a large number of test resultsand the true or accepted reference value. Precision refers to the closeness of agreement between test results.0.2 General consideration of these quantities is given in ISO 5725-1 and so is not repeated here. This part ofISO 5725 should be read in conjunction with ISO 5725-1 because the underlying definitions and general principlesare given there.0.3 ISO 5725-2 is concerned with estimating, by means of interlaboratory experiments, standard measures ofprecision, namely the repeatability standard deviation and the reproducibility standard deviation. It gives a basicmethod for doing this using the uniform-level design. This part of ISO 5725 describes alternative methods to thisbasic method.a) With the basic method there is a risk that an operator may allow the result of a measurement on one sample toinfluence the result of a subsequent measurement on another sample of the same material, causing theestimates of the repeatability and reproducibility standard deviations to be biased. When this risk is consideredto be serious, the split-level design described in this part of ISO 5725 may be preferred as it reduces this risk.b) The basic method requires the preparation of a number of identical samples of the material for use in theexperiment. With heterogeneous materials this may not be possible, so that the use of the basic method thengives estimates of the reproducibility standard deviation that are inflated by the variation between the samples.The design for a heterogeneous material given in this part of ISO 5725 yields information about the variabilitybetween samples which is not obtainable from the basic method; it may be used to calculate an estimate ofreproducibility from which the between-sample variation has been removed.c) The basic method requires tests for outliers to be used to identify data that should be excluded from thecalculation of the repeatability and reproducibility standard deviations. Excluding outliers can sometimes have alarge effect on the estimates of repeatability and reproducibility standard deviations, but in practice, whenapplying the outlier tests, the data analyst may have to use judgement to decide which data to exclude. Thispart of ISO 5725 describes robust methods of data analysis that may be used to calculate repeatability andreproducibility standard deviations from data containing outliers without using tests for outliers to exclude data,so that the results are no longer affected by the data analyst’s judgement.INTERNATIONAL STANDARD © ISO ISO 5725-5:1998(E)1Accuracy (trueness and precision) of measurement methods andresults —Part 5:Alternative methods for the determination of the precision of a standardmeasurement method1 ScopeThis part of ISO 5725— provides detailed descriptions of alternatives to the basic method for determining the repeatability andreproducibility standard deviations of a standard measurement method, namely the split-level design and adesign for heterogeneous materials;— describes the use of robust methods for analysing the results of precision experiments without using outliertests to exclude data from the calculations, and in particular, the detailed use of one such method.This part of ISO 5725 complements ISO 5725-2 by providing alternative designs that may be of more value in somesituations than the basic design given in ISO 5725-2, and by providing a robust method of analysis that givesestimates of the repeatability and reproducibility standard deviations that are less dependent on the data analyst sjudgement than those given by the methods described in ISO 5725-2.2 Normative referencesThe following standards contain provisions which, through reference in this text, constitute provisions of this part ofISO 5725. At the time of publication, the editions indicated were valid. All standards are subject to revision, andparties to agreements based on this part of ISO 5725 are encouraged to investigate the possibility of applying themost recent editions of the standards indicated below. Members of IEC and ISO maintain registers of currently validInternational Standards.ISO 3534-1:1993, Statistics — Vocabulary and symbols — Part 1: Probability and general statistical terms.ISO 3534-3:1985, Statistics — Vocabulary and symbols — Part 3: Design of experiments.ISO 5725-1:1994, Accuracy (trueness and precision) of measurement methods and results — Part 1: Generalprinciples and definitions.ISO 5725-2:1994, Accuracy (trueness and precision) of measurement methods and results — Part 2: Basic methodfor the determination of repeatability and reproducibility of a standard measurement method.ISO 5725-5:1998(E)© ISO23 DefinitionsFor the purposes of this part of ISO 5725, the definitions given in ISO 3534-1 and in ISO 5725-1 apply.The symbols used in ISO 5725 are given in annex A.4 Split-level design4.1 Applications of the split-level design4.1.1 The uniform level design described in ISO 5725-2 requires two or more identical samples of a material to betested in each participating laboratory and at each level of the experiment. With this design there is a risk that anoperator may allow the result of a measurement on one sample to influence the result of a subsequentmeasurement on another sample of the same material. If this happens, the results of the precision experiment willbe distorted: estimates of the repeatability standard deviation srwill be decreased and estimates of the between-laboratory standard deviation sLwill be increased. In the split-level design, each participating laboratory is providedwith a sample of each of two similar materials, at each level of the experiment, and the operators are told that thesamples are not identical, but they are not told by how much the materials differ. The split-level design thus providesa method of determining the repeatability and reproducibility standard deviations of a standard measurementmethod in a way that reduces the risk that a test result obtained on one sample will influence a test result onanother sample in the experiment.4.1.2 The data obtained at a level of a split-level experiment may be used to draw a graph in which the data forone material are plotted against the data for the other, similar, material. An example is given in figure 1. Suchgraphs can help identify those laboratories that have the largest biases relative to the other laboratories. This isuseful when it is possible to investigate the causes of the largest laboratory biases with the aim of taking correctiveaction.4.1.3 It is common for the repeatability and reproducibility standard deviations of a measurement method todepend on the level of the material. For example, when the test result is the proportion of an element obtained bychemical analysis, the repeatability and reproducibility standard deviations usually increase as the proportion of theelement increases. It is necessary, for a split-level experiment, that the two similar materials used at a level of theexperiment are so similar that they can be expected to give the same repeatability and reproducibility standarddeviations. For the purposes of the split-level design, it is acceptable if the two materials used for a level of theexperiment give almost the same level of measurement results, and nothing is to be gained by arranging that theydiffer substantially.In many chemical analysis methods, the matrix containing the constituent of interest can influence the precision, sofor a split-level experiment two materials with similar matrices are required at each level of the experiment. Asufficiently similar material can sometimes be prepared by spiking a material with a small addition of the constituentof interest. When the material is a natural or manufactured product, it can be difficult to find two products that aresufficiently similar for the purposes of a split-level experiment: a possible solution may be to use two batches of thesame product. It should be remembered that the object of choosing the materials for the split-level design is toprovide the operators with samples that they do not expect to be identical.4.2 Layout of the split-level design4.2.1 The layout of the split-level design is shown in table 1.The p participating laboratories each test two samples at q levels.The two samples within a level are denoted a and b, where a represents a sample of one material, and b representsa sample of the other, similar, material.© ISO ISO 5725-5:1998(E)34.2.2 The data from a split-level experiment are represented by:yijkwheresubscript i represents the laboratory (i = 1, 2, ., p);subscript j represents the level (j = 1, 2, ., q);subscript k represents the sample (k = a or b).4.3 Organization of a split-level experiment4.3.1 Follow the guidance given in clause 6 of ISO 5725-1:1994 when planning a split-level experiment.Subclause 6.3 of ISO 5725-1:1994 contains a number of formulae (involving a quantity denoted generally by A) thatare used to help decide how many laboratories to include in the experiment. The corresponding formulae for thesplit-level experiment are set out below.NOTE — These formulae have been derived by the method described in NOTE 24 of ISO 5725-1:1994.To assess the uncertainties of the estimates of the repeatability and reproducibility standard deviations, calculatethe following quantities.For repeatability()[]Apr=−196 1 2 1, (1)For reproducibility()[]()[]ApR=+−+−196 1 2 1 1 8 1224, gg (2)with g = sR/sr.If the number n of replicates is taken as two in equations (9) and (10) of ISO 5725-1:1994, then it can be seen thatequations (9) and (10) of ISO 5725-1:1994 are the same as equations (1) and (2) above, except that sometimesp - 1 appears here in place of p in ISO 5725-1:1994. This is a small difference, so table 1 and figures