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है”ह”ह

IS 15393-3 (2003): Accuracy (Trueness and Precision) ofMeasurement Methods and Results, Part 3: IntermediateMeasures of the Precision of a Standard Measurement Method[PGD 25: Engineering Metrology]

IS 15393 (Part 3): 2003ISO 5725-3:1994

W-F?a%7m

m~amnmw-iiq?l

Wmkl-r( W-lia@m-f@G)

Indian Standard

ACCURACY ( TRUENESS AND PRECISION ) OFMEASUREMENT METHODS AND RESULTS

PART 3 INTERMEDIATE MEASURES OF THE PRECIStONOF A STANDARD MEASUREMENT METHOD

ICS 17.020; 03.120.30

@ BIS 2003

BUREAU OF INDIAN STANDARDSMANAK BHAVAN, 9 BAHADUR SHAH ZAFAR MARG

NEW DELHI 110002

September 2003 Price Group 10

Engineering Metrology Sectional Committee, BP 25

NATIONAL FOREWORD

This Indian Standard ( Part 3 ) which is identical with ISO 5725-3:1994 ‘Accuracy ( trueness andprecision ) of measurement methods and rfkults — Part 3: Intermediate measures of the precision ofa standard measurement method’ issued by the International Organization for Standardization ( ISO )was adopted by the Bureau 01 Indian Standard on the recommendations of the Engineering MetrologySectional Committee and approval of the Basic and Production Engineering Division Council,

This standard specifies the four intermediate precision measures due to changes in observation conditions( time, calibration, equipments and operator) within a laboratory.

Many different factors may contribute to the variability of results from a measurement method includingoperator, equipment used, calibration of equipment, environment ( temperature, humidity air-pollution,etc ); batch of a reagent and time elapsed between measurements.

The text of the ISO Standard has been approved as suitable for publication as an Indian Standardwithout deviations. In this adopted standard -certain conventions are, however, not identical to thoseused in Indian Standards. Attention is particularly drawn to the following:

a) Wherever the words ‘International Standard’ appear referring to this standard, they should beread as ‘Indian Standard’.

b) Comma ( , ) has been used as a decimal marker in the International Standards, while in IndianStandards, the current practice is to use a point ( . ) as the decimal marker.

In the adopted standard, reference appears to the following International Standards for which IndianStandards also exists. The corresponding Indian Standards which are to be substituted in their placeare listed below along with their degree of equivalence for the editions indicated:

International Standard Corresponding Indian Standard Degree of Equivalence

ISO 5725-1 : 1994 Accuracy IS 15393 ( Part 1 ) :2003 Accuracy Identical( trueness and precision ) of ( trueness and precision ) ofmeasurement methods and measurement methods andresults — Part 1 : General principles results : Part 1 General principlesand definitions and definitions

ISO 5725-2 : 1994 Accuracy IS 15393 ( Part 2 ) :2003 Accuracy( trueness and precision ) of ( trueness and precision ) ofmeasurement methods and measurement methods andresults — Part 2: Basic method for results : Part 2 Basic method for thethe determination of repeatability determination of repeatability andand reproducibility of a standard reproducibility of a standardmeasurement method measurement melhod

do

This standard ( Part 3 ) covers the intermediate measures of the precision of a standard measurementmethod. The other five parts of the standard are listed below:

IS No. Title

IS 15393 ( Part 1 ) : 2003/ Accuracy ( trueness and precision ) of measurement methods andISO 5725-1:1994 results : Part 1 General principles and definitions

IS 15393 ( Part 2 ) : 2003/ Accuracy ( trueness and precision ) of measurement methods andISO 5725-2:1994 results : Part 2 Basic method for the determination of repeatability and

reproducibility of a standard measurement method

( Continued on third cover)

IS 15393 (Part 3):2003ISO 5725-3:1994

Introduction

0.1 ISO 5725 uses two terms “trueness” and “precision” to describethe accuracy of a measurement method. “Trueness” refers to the close-ness of agreement between the average value of a large number of testresults and the true or accepted reference value. “Precision” refers to thecloseness of agreement between test results.

0.2 General consideration of these quantities is given in ISO 5725-1 andso is not repeated here. It is sttessed that ISO 5725-1 should be read inconjunction with all other parts of ISO 5725 because the underlying defi-nitions and general principles are given there.

0.3 Many different factors (apati from variations between supposedlyidentical specimens) may contribute to the variab~lity of results from ameasurement method, including:

a) the operator;

b) the equipment used;

c) fhe calibration of the equipment;

d) the environment (temperature, humidity, air pollution, etc.);

e) the batch of a reagent;

f) the time elapsed between measurements.

The variability between measurements performed by different operaforsand/or with different equipment will usually be greater than fhe variabilitybetween measurements carried out within a short interval of time by asingle operator using fhe same equipmenf.

0.4 Two conditions of precision, fermed repeatability and reproducibilityconditions, have been found necessary and, for many -pracfical eases,useful for describing the variability of a measurement method. Under re-peatability conditions, factors a) to f) in 0.3 are considered constanfs anddo not contribute to the variability, while under reproducibility conditionsthey vary and do contribute to the variability of the test results. Thus re-peafabilify and reproducibility conditions are the fwo extremes of pre-cision, the first describing the minimum and the second fhe maximumvariability in results. Intermediate conditions between these two extremeconditions of precision are also conceivable, when one or more of factors

I.,

IS 15393 (Part 3) :2003ISO 5725-3:1994

a) to f) are allowed to vary, and are used in certain specified circum-stances.

Precision is normally expressed in terms of standard deviations.

0.5 This part of ISO 5725 focuses on intermediate precision measuresof a measurement method. Such measures are called intermediate astheir magnitude lies between the two extreme measures of the precisionof a measurement method: repeatability and reproducibility standard de-viations.

To illustrate the need for such intermediate precision measures, considerthe operation of a present-day laboratory connected with a productionplant involving, for example, a three-shift working system wheremeasurements are made by different operators on different equipment.Operators and equipment are then some of the factors that contribute tothe variability in the test results. These factors need to be taken into ac-count when assessing the precision of the measurement method.

0.6 The intermediate precision measures defined in this part ofISO 5725 are primarily useful when their estimation is part of a procedurethat aims at developing, .standardizing, or controlling a measurementmethod within a Iaboratoty. These measures can also be estimated in aspecially designed interlaboratory study, but their interpretation and appli-cation then requires caution for reasons explained in 1.3 and 9.1.

0.7 The four factors most likely to influence the precision of ameasurement method are the following.

a)

b)

c)

d)

Time: whether the time interval between successive measurementsis short or long.

Calibration: whether the same equipment is or is not recalibratedbetween successive groups of measurements.

Operato~ whether the same or different operators carry out the suc-cessive measurements.

Equipment: whether the same or different equipment (or the sameor different batches of reagents) is used in the measurements.

0.8 It is, therefore, advantageous to introduce the following J4-factor-different intermediate precision conditions (44 = 1, 2, 3 or 4) to take ac-count of changes in measurement conditions (time, calibration, operatorand equipment) within a laboratory.

a) 14=1:

b) M=2:

C) M=3:

d) M=4:

only one of the four factors is different;

two of the four factors are different;

three of the four factors are different;

all four factors are different.

Different intermediate precision conditions lead to different intermediateprecision standard deviations denoted by S,(~, where the specific con-ditions are listed within the parentheses. For example, Sluo) is the inter-

ii

IS 15393 (Part 3) :2003ISO !5725-3 :1994

mediate precision standard deviation with different times (T) andoperators (0).

0.9 For measurements under intermediate precision conditions, one ormore of the factors listed in 0.7 is or are different. Under repeatabilityconditions, those factors are assumed to be constant.

The standard deviation of test results obtained under repeatability con-ditions is generally less than that obtained under the conditions for inter-mediate precision conditions. Generally in chemical analysis, the standarddeviation under intermediate precision conditions may be two or threetimes as large as that under repeatability conditions. It should not, ofcourse, exceed the reproducibility standard deviation.

As an example, in the determination of copper in copper ore, acollaborative experiment among 35 laboratories revealed that the standarddeviation under one-factordifferent intermediate precision conditions (op-erator and equipment the same but time different) was 1,5 times largerthan that under repeatability conditions, both for the electrolytic gravimetryand NazS20~ titration methods.

...M

IS 15393 (Part 3) :2003ISO 5725-3:1994

Indian Standard

ACCURACY ( TRUENESS AND PRECISION ) OFMEASUREMENT METHODS AND RESULTS

PART 3 INTERMEDIATE MEASURES OF THE PRECISIONOF A STANDARD MEASUREMENT METHOD

1 Scope

1.1 This part of ISO 5725 specifies four intermedi-ate precision measures due to changes in observationconditions (time, calibration, operator and equipment)

within a laboratory. These intermediate measures canbe established by an experiment within a specificlaboratory or by an interlaboratory experiment.

Furthermore, this part of ISO 5725

a)

b)

c)

d)

discusses the implications of the definitions of in-termediate precision measures;

presents guidance on the interpretation and appli-cation of the estimates of intermediate precisionmeasures in practical situations;

does not provide any measure of the errors inestimating intermediate precision measures;

does not concern itself with determining thetrueness of the measurement method itse~, butdoes discuss the connections between truenessand measurement conditions.

1.2 This part of ISO 5725 is concerned exclusivelywith measurement methods which yield measure-ments on a continuous scale and give a single valueas the test result, although the single value may be

the outcome of a calculation from a set of obser-vations.

1.3 The essence of the determination of these in-termediate precision measures is that they measurethe ability of the measurement method to repeat testresults under the defined conditions.

1.4 The statistical methods developed in this partof ISO 5725 rely on the premise that one can poolinformation from “similar” measurement conditions

to obtain more accurate information on the inter-

mediate precision measures. This premise is apowerful one as long as what is claimed as “similar”is indeed “similar”. But it is very difficult for thispremise to hold when intermediate precision meas-ures are estimated from an interlaboratory study. Forexample, controlling the effect of “time” or of “oper-ator” across laboratories in such a way that they are“similar”, so that pooling information from differentlaboratories makes sense, is very difficult. Thus, usingresults from interlaboratory studies on intermediateprecision measures requires caution. Within-Iaborator-y studies also rely on this premise, but insuch studies it is more likely to be realistic, becausethe control and knowledge of the actual effect of afactor is then more within reach of the analyst.

1.5 There exist other techniques besides the onesdescribed in this part of ISO 5725 to estimate and toverify intermediate precision measures within a lab-

1

IS 15393 (Part 3) :2003ISO 5725-3:1994

oratory, for example, control charts (see ISO 5725-6).This part of ISO 5725 does not claim to describe theonly approach to the estimation of intermediate pre-cision measures within a specific laboratory.

NOTE 1 This part of ISO 5725 refers to designs of ex-periments such as nested designs. Some basic informationis given in annexes B and C. Other references in this areaare given in annex E.

2 Normative references

The following standards contain provisions which,through reference in this text, constitute provisionsof this part of ISO 5725. At the time of publication, theeditions indicated were valid. All standards are subjectto revision, and parties to agreements based on thispart of ISO 5725 are encouraged to investigate thepossibility of applying the most recent editions of thestandards indicated below. Members of IEC and ISOmaintain registers of currently valid InternationalStandards.

ISO 3534-1:1993, Statistics — Vocabulary and sym-bols — Part 1: Probability and general statisticalterms.

ISO 5725-1:1994, Accuracy (trueness and precision)of measurement methods and results — Part 1:General principles and definitions.

ISO 5725-2:1994, Accuracy (trueness and precision)of measurement methods and results — Part 2.” Basicmethod for the determination of repeatability andreproducibility of a standard measurement method.

ISO Guide 33:1989, Uses of certified reference ma-terials.

ISO Guide 35:1989, Certification of reference ma-terials — General and statistical principles.

3 Definitions

For the purposes of this part of ISO 5725, the defi-nitions given in ISO 3534-1 and ISO 5725-1 apply.

The symbols used in ISO 5725 are given in annex A.

4 General requirement

In order that the measurements are made in the sameway, the measurement method shall have beenstandardized. All measurements forming part of anexperiment within a specific laborato~ or of an inter-Iaboratoty experiment shall be carried out accordingto that standard.

5 Important factors

5.1 Four factors (time, calibration, operator andequipment) in the measurement conditions within alaboratory are considered to make the main contri-butions to the variability of measurements (seetable 1).

5.2 “Measurements made at the same time” in-clude those conducted in as short a time as feasiblein order to minimize changes in conditions, such asenvironmental conditions, which cannot always beguaranteed constant. “Measurements made at differ-ent times”, that is those carried out at long intervalsof time, may include effects due to changes in en-vironmental conditions.

Table 1 — Four imDortant factors and their states

i

Factor

Time

Calibration

Operator

Equipment

I

Measurementconditionswithina laboratory

State1 (same) I State2 (dtierent)Measurements made at the sametime

No calibration between measurements

Same operator

Same equipment without recali-bration

Measurements made at differenttimes

Calibration carried out betweenmeasurements

Different operators

Different equipment

2

5.3 “Calibration” does not refer here to any cali-bration required as an integral part of obtaining a testresult by the measurement method. It refers to thecalibration process that takes place at regular intervalsbetween groups of measurements within a labora-tory.

5.4 In some operations, the “operator” may be, infact, a team of operators, each of whom performssome specific part of the procedu~e. In such a case,the team should be regarded as the operator, and anychange in membership or in the allotment of dutieswithin the team should be regarded as providing adifferent “operator”.

5.5 “Equipment” is often, in fact, sets of equip-ment, and any change in any significant componentshould be regarded as providing different equipment.As to what constitutes a significant component,common sense must prevail. A change ofthermometer would be considered a significant ‘com-ponent, but using a slightly different vessel to containa water bath would be considered trivial. A change ofa batch of a reagent should be considered a significantcomponent. It can lead to different “equipment” or toa recalibration if such a change is followed by cali-bration.

5.6 Under repeatability conditions, all four factorsare at state 1 of table 1. For intermediate precisionconditions, one or more factors are at state 2 oftable 1, and are specified as “precision conditionswith M factor(s) different”, where Al is the numberof factors at state 2. Under reproducibility conditions,results are obtained by different laboratories, so thatnot only are all four factors at state 2 but also thereare additional effects due to the differences betweenlaboratories in management and maintenance of thelaboratories, general training levels of operators, andin stability and checking of test results, etc.

5.7 Under intermediate precision conditions with Mfactor(s) different, it is necessary to specify whichfactors are at state 2 of table 1 by means of suffixes,for example:

— time-different intermediate precision standard de-viation, Sip);

— calibration-different intermediate precision stan-dard deviation, s,(c);

—

—

—

—

6

6.1

I

I

operatordiffere nt~eviation, s,(o);

IS 15393 {Part 3) :2003ISO 5725-3:1994

intermediate precision standard

[time + operator]-different intermediate precisionstandard deviation, quo);

[time + operator + equipment]-different inter-mediate precision standard deviation, slfloE);

md many others in a similar fashion.

Statisticalmodel

Basicmodel

For estimating the accuracy (trueness and precision)of a measurement method, it is useful to assume thatevery test result, y, is the sum of three components:

y=m+B+e . . . (1)

where, for the particular material tested,

m is the general mean (expectation);

B is the laboratory component of bias under re-peatability conditions;

e is the random error occurring in everymeasurement under repeatability conditions.

A discussion of each of these components, and ofextensions of this basic model, follows.

6.2 General mean, m

6.2.1 The general mean, m, is the overall mean ofthe test results. The value of m obtained in acollaborative study (see ISO 5725-2) depends solelyon the “true value” and the measurement method,and does not depend on the laboratory, equipment,operator or time by or at which any test result hasbeen obtained. The general mean of the particularmaterial measured is called the “level of the test”; forexample, specimens of different purities of a chemicalor different materials (e.g. different types of steel) willcorrespond’to different levels.

In many situations, the concept of a true value p holdsgood, such as the true concentration of a solutionwhich is being titrated. The level m is not usually equalto the true value p; the difference (m – p) is called the“bias of the measurement method”.

3

IS 15393 (Part 3):2003ISO 5725-3:1994

In some situations, the level of the test is exclusivelydefined by the measurement method, and the con-cept of an independent true value does not apply; forexample, the Vicker’s hardness of steel and theMicum indices of coke belong to this category. How-ever, in general, the bias is denoted by 6 (6 = O whereno true value exists), then the general mean m is

NOTE 2 Discussion of the bias term d and a descriptionof trueness experiments are given in ISO 5725-4.

6.2.2 When examining the difference between testresults obtained by the same measurement method,the bias of the measurement method may have noinfluence and can be ignored, unless it is a functionof the level of the test. When comparing test resultswith a value specified in a contract, or a standardvalue where the contract or specification refers to thetrue value p and not to the level of the test m, or whencomparing test results obtained using differentmeasurement methods, the bias of the measurementmethod must be taken into account.

6.3 Term B

6.3.1 B is a term representing the deviation of alaboratory, for one or more reasons, from m, irre-spective of the random error e occurring in every testresult. Under repeatability conditions in one labora-tory, B is considered constant and is called the “lab-oratory component of bias”.

6.3.2 However, when using a measurement methodroutinely, it is apparent that embodied within anoverall value for B are a large number of effects whichare due, for example, to changes in the operator, theequipment used, th-e calibration of the equipment, andthe environment (temperature, humidity, air pollution,etc.). The statistical model [equation (1)] can then berewritten in the form:

y=m+Bo+B U) +B(2) + . . . + e . . . (3)

or

y=p+f5+ Bo+B(l)+B(2) +...+ e . ..(4)

where B is composed of contributions from variates

Bo, B(l), B(2) ..:. ,and can account for a number of inter-mediate precmon factors.

In practice, the objectives of a study and consider-ations of the sensitivity of the measurement methodwill govern the extent to which this model is used. Inmany cases, abbreviated forms will suffice.

6.4 Terms Bo,B(l), B(2),etc.

6.4.1 Under repeatability conditions, these terms allremain constant and add to the bias of the test re-sults. Under intermediate precision conditions, B. isthe fixed effect of the factor(s) that remained thesame (state 1 of table 1), while B(l), B(2), etc. are therandom effects of the factor(s) which vary (state 2 oftable 1). These no longer contribute to the bias, butincrease the intermediate precision standard deviationso that it becomes larger than the repeatabilitydard deviation.

6.4.2 The effects due to differences between

stan-

oper-ators include personal habits in operating measure-ment methods (e.g. in reading graduations on scales,etc.). Some of these differences should be removableby standardization of the measurement method, par-ticularly in having a clear and accurate description ofthe techniques involved. Even though there is a biasin the test results obtained by an individual operator,that bias is not always constant (e.g. the magnitudeof the bias will change according to hislher mentaland/or physical conditions on that day) and the biascannot be corrected or calibrated exactly. The magni-tude of such a bias should be reduced by use of aclear operation manual and training. Under such cir-cumstances, the effect of changing operators can beconsidered to be of a random nature.

6.4.3 The effects due to differences betweenequipment include the effects due to different placesof installation, particularly in fluctuations of the indi-cator, etc. Some of the effects due to differencesbetween equipment can be corrected by exact cali-bration. Differences due to systematic causes be-tween equipment should be corrected by calibration,and such a procedure should be included in the stan-dard method. For example, a change in the batch ofa reagent could be treated that way. An acceptedreference value is needed for this, for whichISO Guide 33 and ISO Guide 35 shall be consulted.The remaining effect due to equipment which hasbeen calibrated using a reference material is con-sidered a random effect.

4

IS 15393 (Pati 3):2003ISO 5725-3:1994

6.4.4 The effects due to time may be caused byenvironmental differences, such as changes in roomtemperature, humidity, etc. Standardization of en-vironmental conditicsns should be attempted to mini-mize these effects.

6.4.5 The effect of skill or fatigue of an operator maybe considered to be the interaction of operator andtime. The performance of a set of equipment may bedifferent at the time of the start of its use and afterusing it for many hours: this is an example of inter-action of equipment and time. .When the populationof operators is small in number and the population ofsets of equipment even smaller, effects caused bythese factors may be evaluated as fixed (not random)effects

6.4.6 The proceduresveloped assuming thatcomponents of bias is

given in ISO 5725-2 are de-the distribution of laboratoryapproximately normal, but in

practice they work for most distributions provided “thatthese distributions are unimodal. The variance of B iscalled the “between-laboratory variance”, expressedas

Var(B) = at . . . (5)

However, it will also include effects of changes ofoperator, equipment, time and environment. From aprecision experiment using different operators,measurement times, environments, etc., in a nesteddesign, intermediate precision variances can be cal-culated. Var(13) is considered to be composed of in-dependent contributions of laborato~, operator, dayof experiment, environment, etc.

Var(B) = Var(BO) + Var(B(l)) + Var(B(2)) + ...

. . . (6)

The variances are denoted by

Var(BO) = ti~o)

Var(B(l)) = u~l)

(7)va@(2)) = 42p etc. . . .

Var(B) is estimated in practical terms as s: and similarintermediate precision estimates may be obtainedfrom suitably designed experiments.

6.5 Errorterm,e

6.5.1 This term represents a random error occurringin every test result and the procedures giventhroughout this part of ISO 5725 were developed as-suming that the distribution of this error variable isapproximately normal, but in practice they work formost distributions provided that they are unimodal.

6.5.2 Wkhin a single laboratory, its variance is calledthe within-laboratory variance and is expressed as

Var(e) = & . . . (8)

6.5.3 It may be expected that u~ will have differentvalues in different laboratories due to differences suchas in the skills of the operators, but in this part ofISO 5725 it is assumed that, for a properly standard-ized measurement method, such differences be-tween laboratories should be small and that it isjustifiable to establish a common value of within-Iaboratory variance for all the laboratories using themeasurement method. This common value, which isestimated by the mean of the within-laboratory vari-ances, is called the “repeatability variance” and isdesignated by

a:= Var(e) . . . (9)

This mean value is taken over all the laboratories tak-ing part in the accuracy experiment which remain af-ter outliers have been excluded.

7 Choice of measurement conditions

7.1 In applying a measurement method, manymeasurement conditions are conceivable within aIaboratov, as follows:

a)

b)

c)

d)

e)

repeatability conditions (four factors constant);

several intermediate precision conditions with onefactor different;

several intermediate precision conditions with twofactors different;

several intermediate precision conditions withthree factors different;

intermediate precision conditions with four factorsdifferent.

In the standard for a measurement method, it is notnecessary (or even feasible) to state ail possible pre-

5

IS 15393 (Part 3):2003ISO 5725-3:1994

cisioi-r measures, although the repeatability standarddeviation should always be specified. As regards in-termediate precision measures, common commercialpractice should indicate the conditions normally en-countered, and it should be sufficient to specify onlythe one suitable intermediate precision measure, to-gether with the detailed stipulation of the specificmeasurement conditions associated with it. Themeasurement condition factor(s) to be changedshould be carefully defined; in particular, for time-different intermediate precision, a practical mean timeinterval between successive measurements shouldbe specified.

7.2 It is assumed that a standardized measurementmethod will be biased as little as possible, and thatthe bias inherent in the method itself should havebeen dealt with by technical means. This part of

of factor(s) between each measurement. It is rec-ommended that n should bq at Ieast 15. This may notbe satisfactory for the laboratory, and this method ofestimating intermediate precision measures within alabor&ory cannot be regarded as efficient whencompared with other procedures. The analysis issimple, however, and it can be useful for studyingtime-different intermediate precision by making suc-cessive measurements on the same sample on suc-cessive days, or for study”mg the effects of calibrationbetween measurements.

A graph of (yk– Y) versus the measurement numberk, where yk is the kth test result of n replicate test re-sults and Y is the mean of the n replicate test results,is recommended to identify potential outliers. A moreformal test of outliers consists of the application ofGrubbs’ test as given in subclause 7.3.4 ofISO 5725-2:1994.

ISO 5725, therefore, deals only with the bias coming The estimate of the intermediate precision standardfrom the measurement conditions. deviation with M factor(s) different is given by

/zIn7.3 A change in the factors of the measurement Slo = (yk - ~)’ . . . (lo)n–1conditions Kline, calibration, operator and equipment) k=l

from repeatability conditions (i.e. from state 1 to 2 oftable 1) will increase the variability of test results.

where symbols denoting the intermediate precision

However, the expectation of the mean of a numberconditions should appear inside the parentheses.

of test results will be less biased than under repeat-ability conditions. The increase in the standard devi-ation for the intermediate precision conditions may be 8.2 An alternative methodovercome by not relying on a single test result but byusing the mean of several test results as the finalquoted result.

7.4 Practical considerations in most laboratories,such as the desired precision (standard deviation) ofthe final quoted result and the cost of performing themeasurements, will govern the number of factors andthe choice of the factor(s) whose changes can bestudied in the standardization of the measurementmethod.

8 Within-laboratory study and analysisof intermediate precision measures

8.2.1 An alternative method considers r groups ofmeasurements, each comprising n replicate test re-sults. For example, within one laboratory, a set of rmaterials could each be measured, then the inter-mediate precision factor(s) could be altered and the rmaterials remeasured, the procedure being repeateduntil there are n test results on each of the r materials.Each group of n test results shall be obtained on oneidentical sample (or set of presumed identical samplesin the case of destructive testing), but it is not es-sential that the materials be identical, It is only re-quired that the t materials all belong to the interval oftest levels within which one value of the intermediate,precision standard deviation with M factor(s) differentcan be considered to apply. It is recommended thatthe value of t(n – 1) should be at least 15.

8.1 Simplest approach EXAMPLE

The simplest method of estimating an intermediate One operator performs a single measurement onprecision standard deviation within one laboratory each of the t materials, then this is repeated by aconsists of taking one sample (or, for destructive second operator, and possibly by a third operator, andtesting, one set of presumably identical samples) and so on, allowing an estimate of s,(o) to be calculated.performing a series of n measurements with a change

6

IS 15393 (Part 3):2003ISO 5725-3:1994

8.2.2 A graph of (~ – jj) versus the material numberj, where yjkis the k test result on the jth material and~ is the average of the n results on the jth material, isrecommended to identify potential outliers. A moreformal test of outliers consists of the application ofGrubbs’ test as given in subclause 7.3.4 ofISO 5725-2:1994 either for each group separately orfor all m test results combined.

The estimate of the intermediate precision standarddeviation with M factor(s) different, S,(), is then givenby

I

For n = 2 (i.e. two test results on each material), theformula simplifies to

(12)

8.3 Effectof the measurement conditions onthe final quoted result

8.3.1 The expectation of Y is different between onecombination and another of time, calibration, operatorand equipment, even when only one of the four fac-tors changes. This is a limitation on the usefulness ofmean values. In chemical analysis or physical testing,Y is reported as the final quoted result. In trading rawmaterials, this final quoted result is often used forquality evaluation of the raw materials and affects theprice of the product to a considerable extent.

EXAMPLE

In the international trading of coal, the size of theconsignment can often exceed 70000 t, and the ashcontent is determined finally on a test portion of only1 g. In a contract stipulating that each difference of1 ?4. in ash content corresponds to USD 1,5 per tonneof coal, a difference of 1 mg in the weighing of ashby a chemical balance corresponds to 0,1 YO in ashcontent, or -USD 0,15 per tonne, which for such aconsignment amounts to a difference in proceeds ofUSD 10500 (from 0,1 x 1,5 x 70 000).

8.3.2 Consequently, the final quoted result ofchemical analysis or physical testing should be suf-ficiently precise, highly reliable and, especially, uni-versal and reproducible. A final quoted result which

can be guaranteed only under conditions of a specificoperator, equipment or time may not be good enoughfor commercial considerations.

9 Interlabcwatory study and analysis ofintermediate precision measures

9.1 Underlying assumptions

Estimation of intermediate measures of precisionfrom interlaboratory studies relies on the assumptionthat the effect of a particular factor is the same acrossall laboratories, so that, for example, changing oper-ators in one laboratory has the same effect as chang-ing operators in another laboratory, or that variationdue to time is the same across all laboratories. If thisassumption is violated, then the concept of inter-mediate measures of precision does not make sense,nor do the techniques proposed in the subsequentsections to estimate these intermediate measures ofprecision. Careful attention to outliers (not necessarilydeletion of outliers) must be paid as this will help indetecting departure from the assumptions necessaryto pool information from all laboratories. One powerfultechnique to detect potential outliers is to depict themeasurements graphically as a function of the variouslevels of the factors or the various laboratories in-eluded in the study.

9.2 Simplest approach

If material at q levels is sent to p laboratories whoeach perform measurements on each of the q levelswith a change of intermediate precision factor(s) be-tween each of the n measurements, then the analysisis by the same method of calculation as explained inISO 5725-2, except that an intermediate precisionstandard deviation is estimated instead of the repeat-ability standard deviation.

9.3 Nested experiments

A further way of estimating intermediate precisionmeasufes is to conduct more sophisticated exper-iments. These can be fully-nested or staggered-nested experiments (for definitions of these terms,see ISO 3534-3). The advantage of employing anested experimental design is that it is possible, atone time and in one interlaborato~ experiment, toestimate not only repeatability and reproducibilitystandard deviations but also one or more intermediateprecision standard deviations. There are, however,

7

IS 15393 (Part 3) :2003ISO 5725-3:1994

certain caveats which must be considered, as will beexplained in 9.8.

9.4 Fully-nested experiment

A schematic layout of the fully-nested experiment ata particular level of the test is given in figure 1.

By carrying out the three-factor fully-nested exper-iment collaboratively in several laboratories, one in-termediate precision measure can be obtained at thesame time as the repeatability and reproducibilitystandard deviations, i.e. U(O, a(l) and a, can be esti-

1mated. Likewise the four- actor fully-nested exper-iment can be used to obtain two intermediateprecision measures, i.e. U(0), u(l), a(z) and a, can beestimated.

The subscripts i, j and k suffixed to the data y infigure 1 a) for the three-factor fully-nested experimentrepresent, “for example, a laboratory, a day of exper-iment and a replication under repeatability conditions,respectively.

The subscripts i, j, k and f suffixed to the data y infigure 1 b) for the four-factor fully-nested experimentrepresent, for example, a laborato~, a day of exper-iment, an operator and a replication under repeatabilityconditions, respectively.

Analysis of the results of -an n-factor fully-nested ex-periment is carried out by the statistical technique“analysis of variance” (ANOVA) separately for eachlevel of the test, and is described in detail inannex B.

FACTOR

O (laboratory)

1

2 (residual) k ---

i ---

j---

Y/j/t Yin Yi12 Yi21 Yi22

a) Three-factor ful~y-nested experiment

FACTOR

O (laboratory) i ---

1 J----1

2 k ---I

3 (residual} ( ---I 1

Y ijkl Y /111 Yil12 Yi121 Yi122 Yi211 Yi212 Yi221 Yi222

b) Four-factor futly-nested experiment

Figure 1 — Schematic layouts for three-factor and four-factor fully-nested experiments

8

tS 15393 (Part 3):20031s0 5725-3:1994

9.5 Staggered-nested

A schematic layout of theiment at a particular levelfigure 2.

FACTOR

O (Laboratory)

1

2

experiment

staggered-nested exper-of the test is given in

i --- I

should be in the lowest ranks, the lowest factor beingconsidered as a residual variation. For example, in afour-factor design such as illustrated in figure 1 b andfigure 2, factor O could be the Iaboratoty, factor 1 theoperator, factor 2 the day on which the measurementis carried out, and factor 3 the replication. This maynot seem important in the case of the fully-nestedexperiment due to its symmetty.

9.7 Comparison of the nested design withthe procedure given in ISO 5725-2

1 1

I 1 I I3 (residual) j ---

I I I I

Y!, Y/1 Yi2 Yi3 Yi4

Figure 2 — Schematic layout of a four-factorstaggered-nested experiment

The three-factor staggered-nested experiment re-quires each laboratory i to obtain three test results.Test results yil and yiz shall be obtained under re-peatability conditions, and yi~ under intermediate pre-cision conditions with M factor(s) different(M= 1,2 or 3), for example under time-different in-termediate precision conditions (by obtaining yi~ on adifferent day from that on which yil and Yiz were ob-tained).

In a four-factor staggered-nested experiment, Yio shallbe obtained under intermediate precision conditionswith one more factor different, for example, under[time + operator]-different intermediate precisionconditions by changing the day and the operator.

Again, analysis of the results of an n-factorstaggered-nested experiment is carried out by thestatistical technique “analysis of variance” (ANOVA)separately for each level of the test, and is describedin detail in annex C.

9.6 Allocation of factors in a nestedexperimental design

The allocation of the factors in a nested experimentaldesign is arranged so that the factors affected mostby systematic effects should be in the highest ranks

(O, 1, ...). and those affected most by random effects

The procedure given in ISO 5725-2, because theanalysis is carried out separately for each level of thetest (material), is, in fact, a two-factor fully-nested ex-perimental design and produces two standard devi-ations, the repeatability and reproducibility standarddeviations. Factor O is the laboratory and factor 1 thereplication. If this design were increased by one fac-tor, by having two operatocs in each laborato~ eachobtaining two test results under repeatability con-ditions, then, in addition to the repeatability andreproducibility standard deviations, one could deter-mine the operator-different intermediate precisionstandard deviation. Alternatively, if each laboratoryused only one operator but repeated the experimenton another day, the time-different intermediate pre-cision standard deviation would be determined by thisthree-factor fully-nested experiment. The addition ofa further factor to the experiment, by each laboratoryhaving two operators each carrying out twomeasurements and the whole experiment being re-peated the next day, would allow determination of therepeatability, reproducibility, operator-different, time-different, and [time + operator]-different standarddeviations.

9.8 Comparison of fully-nested andstaggered-nested experimental designs

An n-factor fully-nested experiment requires 2“ -‘ test~esults from each laboratory, which can be an ex-cessive requirement on the%laboratories. This is themain argument for the staggered-nested experimentaldesign. This design requires less test results to pro-duce the same number of standard deviations, al-though the analysis is slightly more complex and thereis a larger uncertainty in the estimates of the standarddeviations due to the smaller number of test results.

9

IS 15393 (Part 3) :2003ISO 5725-3: 1994

Annex A(normative)

u

A

b

B

B.

ql), ~(2)1 etc.

c

c, c’, c“

Symbols and abbreviations

Intercept in the relationship

s=ai-bm

Factor used to calculate the uncer-tainty of an estimate

Slope in the relationship

s=a+bm

Component in a test result rep-resenting the deviation of a labora-tory from the general average

(laboratory component of bias)

Component of B representing allfactors that do not change in inter-mediate precision conditions

Components of B representing fac-tors that vary in intermediate pre-cision conditions

Intercept in the relationship

Igs=c+dlgm

Test statistics

cc’crm, ,,lt, C“~~i~ Critical values for statistical tests

CDP

CRP

d

e

f

FP(v1, V~)

G

h

10

Critical difference for probability P

Critical range for probability P

Slope in the relationship

Igs=c+dlgm

Component in a test result rep-resenting the random error occur-ring in every test result

Critical range factor

p-quantile of the F-distribution withVI and V2degrees of freedom

Grubbs’ test statistic

Mandel’s between-laboratory con-sistency test statistic

k

LCL

m

M

N

n

P

P

q

r

R

RM

s

:

T

t

UCL

w

w

~

Y

used in ISO 5725

Mandel’s within-laboratory consistency test sta-tistic

Lower control limit (either action limit or warninglimit)

General mean of the test property; level

Number of factors considered in intermediateprecision conditions

Number of iterations

Number of test results obtained in one labora-tory at one level (i.e. per cell)

Number of laboratories participating in the inter-Iaboratory experiment

Probability

Number of levels of the test property in theinterlaboratory experiment

Repeatability limit

Reproducibility limit

Reference material

Estimate of a standard deviation

Predicted standard deviation

Total or sum of some expression

Number of test objects or groups

Upper control limit (either action limit or warninglimit)

Weighting factor used in calculating a weightedregression

Range of a set of test results

Datum used for Grubbs’ test

Test result

Arithmetic mean of test results

Grand mean of test results

Significance level

Type II error probability

Ratio of the reproducibility standard deviation tothe repeatability standard deviation (u~/a,)

Laboratory bias

Estimate of A

Bias of the measurement method

Estimate of 6

Detectable difference between two laboratorybiases or the biases of two measurementmethods

True value or accepted reference value of a testproperty

Number of degrees of freedom

Detectable ratio between the repeatability stan-

dard deviations of method B and method A

True value of a standard deviation

Component in a test result representing thevariation due to time since last calibration

Detectable ratio between the square roots ofthe between-laboratory mean squares ofmethod B and method A

p-quantile of the ~2-distribution with v degreesof freedom

IS 15393 (Part 3) :2003ISO 5725-3:1994

Symbols used as subscripts

c

E

i

1( )

j

k

L

m

M

o

P

r

R

T

w

1, 2, 3...

(1), (2), (3)

Calibration-different

Equipment-different

Identifier for a particular laboratory

Identifier for intermediate measures ofprecision; in brackets, identification ofthe type of intermediate situation

Identifier for a particular level

(ISO 5725-2).Identifier for a group of tests or for afactor (ISO 5725-3)

Identifier for a particular test result in aIaboratoy i at level j

Between-laboratory (interlaboratory)

Identifier for detectable bias

Between-test-sample

Operator-different

Probability

Repeatability

Reproducibility

Time-different

Within-1 aboratory (intralaboratory)

For test results, numberingof obtaining them

For test results, numberingof increasing magnitude

in the order

in the order

11

IS 15393 (Part 3) :2003ISO 5725-3:1994

Annex B(normative)

Analysis of variance for fully-nested experiments

The analysis of variance described in this annex has to be carried out separately for each level of the test includedin the interlaboratory experiment. For simplicity, a subscript indicating the level of the test has not been suffixedto the data. It should be noted that the subscript j k used in this part of ISO 5725 for factor 1 (factor O being thelaboratory), while in the other parts of ISO 5725 it is used for the level of the test.

The methods described in subclause 7.3 of ISO 5725-2:1994 should be applied to check the data for consistencyand outliers. With the designs described in this annex, the exact analysis of the data is very complicated whensome of the test results from a laboratory are missing. If it is decided that some of the test results from a lab-oratory are stragglers or outliers and should be excluded from the analysis, then it is recommended that all thedata from that laboratory (at the level affected) should be excluded from the analysis.

B.1 Three-factor fully-nested experiment

The data obtained in the experiment are denoted by ytik, and the mean values and ranges are as follows:

jij = + (yijl + Yij2) ~j = ~ (Jil + fi*)

‘ij( 1)= lYijl – Yij21 ‘i(2) = l~il – 121

where p is the number of laboratories which have

The total sum of squares, SST, can be subdivided

participated in the interlaboratory experiment.

as

SST = ~ ~ ~(yijk – ~)z = SS0 + SS1 + SSei jk

where

Since the degrees of freedom for the sums of squares SS0, SS1 and SSe are p – 1, p and 2P, respectively, theANOVA table is composed as shown in table B.1.

12

IS 15393 (Part 3) :2003ISO 5725-3:1994

Table 6.1 — ANOVA table for a three-factor fully-nested experiment

SourceSum of Degreesof

freedomMean square

squaresExpected mean square

o Sso p–1 MSO=SSO/(p -1) u; + 2af1)4 4&

1 Ssl P MS1 = SS1/p u; + 2U:1)

Residual SSe 2p MSe = SSe/(2p) a:

Total SST 4p–1

The unbiased estimates s;), s~l) and s: of o~o), a~l) and o?, respectively, can be obtained from the mean squaresMSO, MSI and MSe as ‘ ‘ ~‘

,,. ,

1 (MSO – MS1)St, = ~

S~I) = ~ (MS1 – MSe)

s: = MSe

The estimates of the repeatability variance, one-factor-differentvariance are, respectively, as follows:

s:

S;l) = s: i- s~l)

s; = s: + $, + St)

B.2 Four-factor fully-nested experiment

intermediate precision variance and reproducibility

The data obtained in the experiment are denoted by ytiu, and mean values and ranges are as follows:

~jk = ~ b’ijkl + yijk2) ‘ijk(l) = lYijkl – Yijk21

Yi = ~ (Xl+ %2) ‘j(3) = Iyil — yjzl

where p is the number of laboratories which have participated in the interlaboratory experiment.

The total sum of squares, SST, can be subdivided as follows:

SST = ~ ‘~ ~, ~,~ijkl - ~)2 = SS0 + SS1 + SS2 + SSei Jkl

where

13

Since the degrees of freedom for the sums of squares SS0, SS1, SS2 and SSe are p – 1, p, 2p and 4p, respectively,the ANOVA table is composed as shown in table B.2.

Table B.2 — ANOVA table for a four-factor fullv-nested experiment.

SourceSum of Degrees of

freedomMean square

squares Expected mean square

o Sso p–1 MSO = SSO/(jJ– 1) u: + 2uf2) + 4U~1) + 80;o)

1 Ssl P MS1 = SS1/p a; + 2U:2)+ 4U:1)

2 SS2 2p MS2 = Ss2/(2p) a: + 2uf2)

Residual SSe 4p 2MSe = SSe/(4p) Clr

Total SST 8p–1

The unbiased estimates $), s~l), s;, and s: of a~o),a~l), a~2)and a:, respectively, can be obtained from the meansquares MSO, MSI, MS2 and MSe as follows:

S~O)= ~ (MSO –’MS1)

~ (MS2 –“MS1)s;,, = 4

S~Z)= ~ (MS I – MSe)

s? = MSe

The estimates of the repeatability variance, one-factor-different intermediate precision variance, two-factors dif-ferent intermediate precision variance and reproducibility variance are, respectively, as follows:

s:

S;l) = s: + S72)

S;z) = s; + S72)+ S:l)

14

IS 15393 (Part 3) :2003ISO 5725-3:1994

Annex C(normative)

Analysis of variance for staggered-nested experiments

The analysis of variance described in this annex has to be carried out separately for each level of the test includedin the interlaboratory experiment. For simplicity, a subscript indicating the level of the test has not been suffixedto the data. It should be noted that the subscript j is used in this part of ISO 5725 for the replications within thelaboratory, while in the other parts of ISO 5725 it is used for the level of the test.

The methods described in subclause 7.3 of ISO 5725-2:1994 should be applied to check the data for consistencyand outliers. With the designs described in this annex, the exact analysis of the data is very complicated whensome of the test results from a Iaboratow are missing. If it is decided that some of the test results from a lab-oratory are stragglers or outliers and should be excluded from the analysis, then it is recommended that all thedata from that laborat~ (at the level affected) should .be excluded from the analysis.

C.1 Three-factor staggered-nested experiment

The data obtained in the experiment within laborato~ i are denoted by yij (j= 1, 2, 3), and the mean values andranges are as follows:

Yi(l) = + (Yil + Yi2) ‘i(l) = lYil - Yi21

Yi(2) = ~ (Yil + Yi2 + Y13) ‘i(2) = lYi(l)- Yi31

where p is the number of laboratories which have participated in the interlaboratory experiment

The total sum of squares, SST, can be subdivided as follows:

SST = ~, ~4(yo –j)* = SS0 +-SS1 + SSei j

where

Sso = 3X (7i(*))2 – 3P(;)2i

Since the degrees of freedom for the sums of squares SS0, SS1 and SSe are p – 1,p and p, respectively, theANOVA table is composed as shown in table C.1.

15

IS 15393 (Part 3) :2003ISO 5725-3:1994

Table C.1 — ANOVA table for a three-factor staggered-nested experiment

SourceSum ofsquares

o Sso

1 Ssl

Residual SSe

Total I SST

Degreesoffreedom I Meansquare I Expected mean square I

p–1 sso/(p -1) u; + : u~,, + 30;0)

P Ssl /p u; + + Uf,,

P SSe/p 0:

3p–1 I I I

The unbiased estimates s~o),s~l) and s; of o~o)t CJ~I) and d, respectively, can be obtained from the mean squaresMSO, MS1 and MSe as follows:

5 MS1 + ~ MSes~o)= ~ MSO – ~

3 MSes~l) = : MS1 –~

S: = MSe

The estimates of the repeatability variance, one-factor-differentvariance are, respectively, as follows:

s:

+;l) = s; + Sfl)

s; = s; + s~l) + s~o)

C.2 Four-factor staggered-nested experiment

intermediate precision variance and reproducibility

The data obtained in the experiment within laboratory i are denoted by yu (j= 1,2,3, 4), and the mean values andranges are as follows:

X(l) = ~ (Yil + Yi2) ‘i(1) = IYil – Yi21

1(2) = ~ b’il +X2+ Y~) ‘i(2) = E“(l)– Y131

Y(3) = + (Yil + YIZ? +%3 + Yi4) ‘i(3) = 1.Ii(2) – Yi41

where p is the number of taboratories which have participated in the interlaboratory experiment.

The ANOVA table is composed as shown in table C.2.

16

IS .15393 (Part 3) :2003ISO 5725-3:1994

Table C.2 — ANOVA table for a four-factor staggered-nested experiment

F

Degrees offreedom

Mean square Expectedmean square1

ro12ResidualIu!-_

Sum of squares

p-1 sso/(p -1) Eizd: + ~ u~z)+ ~ a(l)+ 4ufoJ2

P Ssl /p Tzszu; +—U(2)+T u(,)6

P ss2/p LIZa; + — U(2)3

P SSe/p u;

4p–1 I I

C.3 Five-factor staggered-nested experiment

The data obtained in the experiment within laboratory i are denoted by ytiand ranges are as follows:

(i= 1,2,3,4, 5), and the mean values

Ji(l)= + (Yil + Yi2) ‘i(l) = lYil – Yi21

X(2) = ~ (Yil + Yi2 + Y13) ‘i(2) = lYi(l) – Yi31

~i(3) = ~ (Yil + Yi2 + Yi3 + Yi4) ‘i(3) = IX(2) – Yi41

3(4) = ~ (Yil + Yi2 + Yi3 + Yi4 + Yi5) ‘i(4) = IX(3) – Yi!jl

where p is the number of laboratories which have participated in the interlaboratory experiment

The ANOVA table is composed as shown in table C.3.

Table C.3 — ANOVA table for a five-factor stacmered-nestedex~eriment

SourGe

D

1

2

3

Residual

Total

Sum of squaresDegrees offreedom

5UX(4))2 - 5P&2 p-1i

$ ;W;4) P

+ ;W;3) P1

* ;42) P

4 D& P

Mean square

sso/(p -1)

Ssl /p

ss2/p

ss3/p

SSe/p

Expected mean square

-zXYij- ;)2 5p-1;, I I

17

IS 15393 (Part 3) :2003ISO 5725-3:1994

C.4 Six-faetor staggered-nested experiment

The data obtained in the experiment within Iaboratow i are denoted by yij (j= 1, 2, 3, 4, 5, 6), and the mean valuesand ranges are as follows:

1(1) = ~ (Yil + Yi2) ‘i(l)= b’il ‘Yi21

7i(2) = ~ (Yil + Yi2 + YKJ ‘i(2) = IYi(l) – Yi31

Yi(3) = ~ (Yil + Yi2 + Yi3 + Yi4) ‘i(3) = Fi(2) ‘.Yi41

E(4) = ; (Yil + Yi2 + Yi3 + Yi4 + Yi5) ‘i(4) = 17i(3) – Yi51

X(5) = ~ (yil + yi2 + Yi3 + yi4 + Yis + yi6) ‘i(s) = Iyi (4) – yi61

;=+z Ji(5)

i

where p is the number of laboratories which have participated in the interiaboratory experiment.

The ANOVA table is composed as shown in table C.4.

Table C.4 — ANOVA table for a six-factor staggered-nested ex~eriment

Source

o

1

2

3

4

Residual

Total

--

Sum of squares Degrees offreedom Mean square

Expected mean square

18

IS 15393 (Part 3):2003ISO 5725-3:1994

Annex D(informative)

Examples of the statistical analysis of intermediate precision experiments

D.1 Example 1 — Obtaining the [time +operator]-different intermediate precisionstandard deviation, q (To)#within a Specific

laboratory at a pmticular level of the test

D.1.1 Background

a)

b)

c)

Measurement method:carbon content in steel

Determination of theby vacuum emission

spectrometry with test results expressed as per-centages by mass.

Source: Routine report of a steel works in Nov-ember 1984.

Experimental design: In a specific Iaboratoty, arandomly selected sample out of the materialsanalysed on one day was analysed again on thenext day by a different analyst. In a month 29 pairsof such data were obtained (see table D.1 ).

SampleNo.

j

123456789101112

13

14

15

First day

Yil

0,1300,1400,0780,1100,1260,0360,0500,1430,0910,0400,1100,1420,1430,1690,169

D.1..2 Analysis

The data yjl, y,z and Wj= ]yjl – y,zl are shown intable D.1. The analysis follows the procedure given in8.2.

A plot of the data [deviations from the mean of themeasurements on both days (yjk– T) versus thesample number~~ is shown in figure D.1. This plot andthe application of Cochran’s test indicate that theranges for samples with numbers 20 and 24 areoutliers. There is a large discrepancy between th’edaily measurements on these samples which is mostlikely due to errors in recording the data. The valuesfor these two samples were removed from the com-putation of the [time + operator]-different intermedi-ate precision standard deviation, ~\(T~\, which iscalculated according to equation (12) as “

r 27Sl(-ro)= 12x27 x WJ2 = 2,87 X 10-3j=lTable D.1 — Original data — Carbon content, ‘Yo (rdm)

Next day I RangeYr2 I Wj

0,1270,1320,0800,1130,1280,0320,0470,1400,0890,0300,1130,1450,1500,1650,173

0,0030,0080,0020,0030,0020,0040,0030,0030,0020,0100,0030,0030,0070,0040,004

SampleNo.

first day Next day Range

j Yjl Y,2 Wj

16 0,149 0,144 0,00517 0,044 0,044 0,00018 0,127 0,122 0,00519 0,050 0,048 0,00220 0,042 0,146 0,10421 0,150 0,145 0,00522 0,135 0,133 0,00223 0,044 0,045 0,00124 0,100 0,161 0,06125 0,132 0,131 0,00126 0,047 0,045 0,00227 0,168 0,165 0,00328 0,092 0,088 0,00429 0,041 0,043 0,002

I

19

IS 15393 (Part 3) :2003ISO 5725-3:1994

I0,06

0,02

0

-0,02

-0,04

•1c1 ++-+ ❑ a[

+

.nnfJ~1++++

L❑ First day-1- Next day-0,06 I I I I

+

❑ ❑ln IJnr In IJn+u

+ ++ ++~ +++0

D

r1

I I I I I I I I I I I I

1 5 10 15 20 25 30

Sample number, j _

Figure D.1 — Carbon content in steel — Deviations from the mean of the measurements on both daysversus the sample number

D.2 Example 2 — Obtaining thetime-different intermediate precisionstandard deviation by interlaboratoryexperiment

D.2.I Background

a) Measurement method: Determination of thevanadium content in steel by the atomic absorp-tion spectrometric method descriied in the in-structions for the experiment. Test results areexpressed as percentages by mass.

b) Source lSO/TC 17, Stee//SC 1, Methods of de-termination of chemica/ composition. Experimentcarried out in May 1985.

c) Experimental design: A three-factor staggered-nested experiment was carried out with 20 lab-oratories each reporting two test results obtainedunder repeatability conditions on one day, fol-lowed by one further test result on the next dayat each of the six levels included in the exper-iment. All measurements in any laboratory werecarried out by one operator, using the samemeasurement equipment.

20

IS 15393 (Part 3) :2003ISO 5725-3:1994

D.2.2 Analysis oratory. There is a large discrepancy between the testresult for day 2 and the mean value for day 1, which

The data for all six levels are given in table D.2. is ve~ large compared with the test results for the

The analysis of variance is presented for only one ofother laboratories. This laboratory was removed fromthe calculations of the precision measures.

the levels, i.e. level 1.

A plot of the data (test results for day 1 and day 2 In accordance with C.1 of annex C, Wi(l), Wi(z) and

versus laboratory number i) is shown in figure D.2. ](p) were computed and the results are shown in ta-

This plot indicates that laboratory 20 is an outlier lab- bieD.3.

I150

-Y0

x Iko

~

.5

aQ

E2 130c0

:

:

c

~ 120

110

100

90

80

70

•l Day 1, first test result-1- Day 1, second test result* clay 2

I I

I I

+

1 1,+

❑

❑

✎✌✎ 1—

*f

+*m’

-r

•1 -m

*

I I 1 I I I I I I

*

*

n++CI*

c1+

I I 1

1 4 8 ?2 16 20

Laboratory number, i —

Figure D.2 — Vanadium content in steel — Test results for day 1 and 2 at level 1 versus laboratory number

21

Level 1 (0,01%) Level 2 (0,04 %) Level 3 (O,1%)Laboratory

Level 4 (0,2 %) Level 5 (0,5 %) Level 6 (O,75 ‘7.)

No. Day 1 Day 2 Day 1 Day 2 Day 1 Day 2 Day 1 Day 2 Day 1 Day 2 Day 1 Day 2

i Yil Y(2 Y(3 Yil Y,2 Yi3 Yil Y,2 Y,3 Y,l Y,2 Ya Yil Y,3 Yi2 Y,l Yi2 Yi3

1 0,0091 0,0102 0,0098 0,0382 0,b38 8 0,0385 0,101 0,103 0,102 0,214 0,211 0,210 0,514 0,510 0,513 0,755 0,753 0

IS 15393 (Part 3) :2003ISO 5725-3:1994

Table D.3 — Values of w,(,), w,~,,and Y,,,,.,!, .,-,

- ,(-)

LaboratoryNo. Wi(l) ‘i(2) X(2)

i

1 0,!)01 1 0,00015 0,0097002 0,0000 0,00100 0,0096673 0,0005 0,00015 0,0093004 0,0003 0,00045 0,0080005 0,0000 0,00000 0,0100006 0,0005 0,00025 0,0092337 0,0001 0,00025 0,0099338 0,0002 0,00040 0,0096339 0,0010 0,00010 0,00993310 0,001 1 0,001 55 0,01073311 0,0000 0,00100 0,00966712 0,0006 0,001 50 0,01070013 0,0005 0,00025 0,00966714 0,0000 0,00040 0,00973315 0,0008 0,00130 0,00906716 0,0002 0,00040 0,00976717 0,0003 0,00085 0,01063318 0,0002 0,00140 0,01086719 0,000 “2 0,00070 0,009933

The sums of squares of Wi(l), Wi(z) and Y(2) and themean value ~ are computed as

z – 5,52x10-6Wfi,, –i

E– 12,44x 10-6w,~2)–

i

X(X,2,)2= 1 832,16X ,0-6i

1Y=19–~ X(2)

= 0,00979825

From these values, the sums of squares SS0, SS1 andSSe are obtained, and the ANOVA table is preparedas shown in table D.4.

The unbiased estimates of variances between labora-tories S$), between days within a laboratory s~l), andthe estimated repeatability variance s: are obtainedas

S$, = 0,278 X 10-s

$, = 0,218 X 10-G

S: =0,14.5X 10-6

The reproducibility standard deviation SR, time-different intermediate precision standard deviationS(F), and repeatability standard deviation s, are ob-tained as

SR = dt + fI) + $) = 0,801 X 10-3

Sr = J_S2r = 0,381 X 10-3The values of these standard deviations for the sixlevels of vanadium content are summarized intable D.5 and shown in figure D.3.

Table D,4 — ANOVA table — Vanadium content

SourceSum of Degreas of Meansquares freedom Expected mean squaresquare

O (laboratory) 24,16 x10-” 18 1,342 x10-G a; + : u;,, + 3fJ&,

1 (day) 8,29 X 10-6 19 0,436 X 10-6 42u; + ~u(l)

Residual 2,76 X 10-6 19 0,145 x 10-6 a;t ,

Total I 35,21 X10-6 I 56 I I I

23

IS 15393 (Part 3) :2003ISO 5725-3:1994

Table D.5 — -Values of s,,Sin,ands~ for six levels of vanadium content in steel.,. ,

OutlierLevel laboratories Average (%) s, (%)

No.

1 20 0,0098 0,381 X 10-3

2 2 0,0378 0,820 X 10-3

3 — 0,1059 1,739 X1 O-3

4 6 and 8 0,2138 3,524 X 10-3

5 20 0,5164 6,237 X 10-3

6 20 0,7484 9,545 x 10-3

12

8

‘$

o

—

—

—

—

0,001

—.------

0.01

—

—

—

...-

1sl(T) @) s~ (%)

0,603X 10-3

0,902X 10-3

2,305X 10-3

4,7 IO X1 O-3

6,436 X 10-3

9,545 x 10-3

0,1

0,801 X 10-3

0,954 x 10-3

2,650 X 10-3

4,826 X 10-3

9,412 x10-3

15,962 X 10-3

Concentration level, % —

Figure D.3 — Vanadium content in steel— Repeatability standard deviation s,, time-different intermh@ateprecision standard deviation Slfl)and reproducibility standard deviations~ as functions of the concentra~n

level

24

IS 15393 (Part 3) :2003ISO 5725-3:1994

[1]

[2]

[3]

[4]

Annex E(informative)

Bibliography

ISO 3534-2:1993, Statistics — Vocabulary andsymbols — Part 2: Statistical quality control.

ISO 3534-3:1985, Statistics — Vocabulary andsymbols — Part 3: Design of experiments. [5]

ISO 5725-4:1994, Accuracy (trueness and pre-cision) of measurement methods and resu/ts —Part 4: Basic methods for the determination of [6]the trueness of a standard measurementmethod,

[7]ISO 5725-5:—11, Accuracy (trueness and pre-cision) of measurement methods and results —

Part 5: Alternative methods for the determi-nation of the precision of a standard measure-ment method.

ISO 5725-6:1994, Accuracy (trueness and pre-cision) of measurement methods and results —Part 6: Use in practice of accuracy values.

WINER, B.J. Statistical principles in experimentaldesign. McGraw-Hill, 1962.

SNEDECOR,G.W. and COCHRAN,W.G. Statisticalmethods. Iowa University Press, 1967.

1) To be published.

25

( Continuedfrom secomfcover)

IS No. Title

IS 15393 ( Part 4 ) : 2003/ Accuracy ( trueness and precision ) of measurement methods andISO 5725-4:1994 results : Part 4 Basic methods for the determination of the trueness of

a standard measurement method

IS 15393 ( Part 5 ) : 2003/ Accuracy ( trueness and precision ) of measurement methods andISO 5725-5:1994 results: Part 5 Alternative methods for the determination of the precision

of a standard measurement method

IS 15393 ( Part 6 ) : 2003/ Accuracy ( trueness and precision ) of measurement methods andISO 5725-6:1994 results : Part 6 Use in practice of accuracy values

The concerned Technical Committee has reviewed the provisions of following International Standardsreferred in this adopted standard and decided that it is acceptable for use with this standard:

1s0 3534-1:1993 Statistics — Vocabulary and symbols — Part 1: Probability and generalstatistical terms

ISO Guide 33:1989 Uses of certified reference materials

ISO Guide 35:1989 Certification of reference materials — General and statistical principles

This standard also gives Bibliography in Annex E, -which is informative.

Bureau of Indian Standards

BIS is a statutory institution established under the Bureau of Indian Standards Act, 1986 to promoteharmonious development of the activities of standardization, marking and quality certification of goods andattending to connected matters in the country,

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This Indian Standard has been developed from Doc : No. BP 25 ( 0190 ).

Amendments Issued Since Pu-blication

Amend No. Date of Issue Text Affected

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