Disclosure to Promote the Right To Information

Whereas the Parliament of India has set out to provide a practical regime of right to information for citizens to secure access to information under the control of public authorities, in order to promote transparency and accountability in the working of every public authority, and whereas the attached publication of the Bureau of Indian Standards is of particular interest to the public, particularly disadvantaged communities and those engaged in the pursuit of education and knowledge, the attached public safety standard is made available to promote the timely dissemination of this information in an accurate manner to the public.

इंटरनेट मानक

“!ान $ एक न' भारत का +नम-ण”Satyanarayan Gangaram Pitroda

“Invent a New India Using Knowledge”

“प0रा1 को छोड न' 5 तरफ”Jawaharlal Nehru

“Step Out From the Old to the New”

“जान1 का अ+धकार, जी1 का अ+धकार”Mazdoor Kisan Shakti Sangathan

“The Right to Information, The Right to Live”

“!ान एक ऐसा खजाना > जो कभी च0राया नहB जा सकता है”Bhartṛhari—Nītiśatakam

“Knowledge is such a treasure which cannot be stolen”

“Invent a New India Using Knowledge”

है”ह”ह

IS 15393-2 (2003): Accuracy ( Trueness and Precision ) ofMeasurement Methods and Results, Part 2: Basic Method forthe Determination of Repeatability and Reproducibility of aStandard Measurement Method [PGD 25: Engineering Metrology]

IS 15393 (Part 2): 2003ISO 5725-2:1994

w-l-m ~*dbn m“llw$r

hdian Standard

ACCURACY ( TRUENESS AND PRECISION ) OFMEASUREMENT METHODS AND RESULTSPART 2 BASIC METHOD FOR THE DETERMINATION OF REPEATABILITY

AND REPRODUCIBILITY OF A STANDARD MEASUREMENT METHOD

ICS 17.020; 03.120.30

0 BIS 2003

BUREAU OF INDIAN STANDARDSMANAK BHAVAN, 9 BAHADUR SHAH ZAFAR MARG

NEW DELHI 110002

October 2003 Price Group 13

Engineering Metrology Sectional Committee, BP 25

NATIONAL FOREWORD

This Indian Standard ( Part 2 ) which is identical with ISO 5725-2:1994 ‘Accuracy ( trueness andprecision ) of measurement methods and results — Part 2: Basic method for the determination ofrepeatability and reproducibility of a standard measurement method’ issued by the International Organizationfor Standardization ( ISO ) was adopted by the Bureau of Indian Standards on the recommendations ofEngineering Metrology Sectional Committee and approval of the Basic and Production EngineeringDivision Council.

This standard amplifies the general principles to be obsewed in designing, experiments for the numericalestimation of the precision of measurement methods by means of a collaborative inter-laboratoryexperiment. It-also provides a detail practical description of the basic method for routine use in estimatingthe precision of measurement methods and guidance to ail personal concerned with designing, performingor analyzing the results of the tests for estimating precision.

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 S-tandard’ appear referrrng 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.

Technical Corrigendum 1 to the above International Standard has been given at the end of this publication.

In the adopted standard, reference appears to the following International Standards for which IndianStandards also exists. The corresponding Indian Standard which is to be substituted in its place islisted below along with its degree of equivalence for the edition 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 principles-and definitions and definitions

This standard ( Part 2 ) covers the basic method for the determination of repeatability and reproducibilityof a standard measurement method. The other parts of this standard are:

IS No.

IS 15393 ( Part 1 ) : 2003/ISO 5725-1:1994

1S 15393 ( Part 3 ) : 2003/ISO 5725-3:1994

IS 15393 ( Part 4 ) : 2003/ISO 5725-4:1994

Title

Accuracy ( trueness and precision ) of measurement methods andresults : Part 1 General principles and definitions

Accuracy ( trueness and precision ) of measurement methods andresults : Part 3 Intermediate measures of the precision of a standardmeasurement method

Accuracy ( trueness and precision ) of measurement methods andresults : Part 4 Basic methods for the determination of the trueness ofa standard measurement method

.

( Continued on third cover)

IS 15393 (Part 2):2003ISO 5725-2:1994

Introduction

0.1 1S0 5725 uses two terms “trueness” and “precision” to describethe accuracy of a measurement method. “Trueness” refers to the close-ness of agreement between the arithmetic mean 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 consider~tion of these quantities is given in ISO 5725-1 andso is not repeated in this part of ISO 5725. ISO 5725-1 should be read inconjunction with all other parts of ISO 5725, including this part, becauseit gives the underlying definitions and general principles.

0.3 This part of ISO 5725 is concerned solely with estimating by meansof the repeatability standard deviation and reproducibility standard devi-ation. Although other types of experiment (such as the split-level exper-iment) are used in certain circumstances for the estimation of precision,they are not dealt with in this part of ISO 5725 -but rather are the subjectof ISO 5725-5. Nor does this part of ISO 5725 consider any other meas-ures of precision intermediate between the two principal measures; thoseare the subject of ISO 5725-3.

0.4 In certain circumstances, the data obtained from an experimentcarried out to estimate precision are used also to estimate trueness. Theestimation of trueness is not considered in this part of ISO 5725; all as-pects of the estimation of trueness are the subject of 1S0 5725-4.

(i)

IS 15393 (Part 2):2003ISO 5725-2:1994

Indian Standard

ACCURACY ( TRUENESS AND PRECISION ) OFMEASUREMENT METHODS AND RESULTSPART 2 BASIC METHOD FOR THE DETERMINATION OF REPEATABILITY

AND REPRODUCIBILITY OF A STANDARD MEASUREMENT METHOD

1 Scope

1.1 This part of ISO 5725

— amplifies the general principles ‘to be observed indesigning experiments for the numerical esti-mation of the precision of measurement methodsby means of a collaborative interlaboratory exper-iment;

— provides a detailed practical description of thebasic method for routine use in estimating theprecision of measurement methods;

— provides guidance to all personnel concerned withdesigning, performing or analysing the results ofthe tests for estimating precision.

NOTE 1 Modifications to this basic method for particularpurposes are given in other parts of ISO 5725.

Annex B provides practical examples of estimatingthe precision of measurement methods by exper-iment.

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 this single value may bethe outcome of a calculation from a set of observa-tions.

1.3 It assumes that in the design and performanceof the precision experiment, all the principles as laiddown in ISO 5725-1 have been observed.. The basicmethod uses the same number of test results in eachlaboratory, with each laborato~ analysing the samelevels of test sample; i.e. a balanced uniform-levelexperiment. The basic method applies to proceduresthat have been standardized and are in regular use ina number of laboratories.

NOTE 2 Worked examples are given to demonstrate bal-anced uniform sets of test results, although in one examplea variable number of replicates per cell were reported (un-balanced design) and in another some data were missing.This is because’an experiment designed to be balanced canturn out to be unbalanced. Stragglers and outliers are alsoconsidered.

1.4 The statistical model of clause 5 ofISO 5725-1:1994 is accepted as a suitable basis forthe interpretation and analysis of the test results, thedistribution of which is approximately normal.

1.5 The basic method, as described in this part ofISO 5725, will (usually) estimate the precision .of ameasurement method:

a) when it is required to determine the repeatabilityand reproducibility standard deviations as definedin ISO 5725-1;

b) when the materials to be used are homogeneous,or when the effects of heterogeneity can be in-cluded in the precision values; and

1

IS 15393 (Part 2):2003ISO 5725-2:1994

c) when the use of a balanced uniform-level layoutis acceptable.

1.6 The same approach can be used to make apreliminary estimate of precision for measurementmethods which have not reached standardization orare not in routine use.

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 symb-ols — Part 1: Probability and general statisticalterms.

ISO 5725-1:1994, Accuracy (trueness and precision)of measurement methods and results — Part 7.“General principles and definitions.

3 Definitions

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

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

4 Estimates of the parameters in thebasic model

4.1 The procedures given in this part of ISO 5725are based on the statistical model given in clause 5of ISO 5725-1:1994 and elaborated upon in subclause1.2 of ISO 5725-1:1994. In particular, these pro-cedures are based onof ISO 5725-1:1994.

The model is

y=m+B+e

equations (2) to (6) of clause 5

where, for the particular material tested,

m is the general mean (expectation);

2

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

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

4.2 Equations (2) to (6) of ISO 5725-1:1994,clause 5 are expressed in terms of the true standarddeviations of the populations considered. In practice,the exact values of these standard deviations are notknown, and estimates of precision values must bemade from a relatively small sample of all the possiblelaboratories, and within those laboratories from asmall sample of all the possible test results.

4.3 In statistical practice, where the true value of astandard deviation, a, is not known and is replaced by :an estimate based upon a sample, then the symbol ais replaced by s to denote that it is an estimate. Thishas to be done in each of the equations (2) to (6) ofISO 5725-1:1994, giving:

is the estimate of the between-laboratoryvariance;

is the estimate of the within-laboratory vari-ance;

is the arithmetic mean of & and is the esti-mate of the repeatability variance; this arith-metic mean is taken over all thoselaboratories taking part in the accuracy ex-periment which remain after outliers havebeen excluded;

is the estimate of the reproducibility vari-ance:

s; =s; +s: . . . (1)

5 Requirements for a precisionexperiment

5.1 Layout of the experiment

5.1.1 In the layout used in the basic method, sam-ples from q batches of materials, representing q dif-ferent levels of the test, are sent to p laboratorieswhich each obtain exactly n replicate test results un-der repeatability conditions at each of the g levels.This type of experiment is called a balanced uniform-Ievel experiment.

IS 15393 (Part2) :2003ISO 5725-2:1994

5.1.2 The performance of these measurements shallbe organized and instructions issued as follows.

a) Any prelimina~ checking of equipment shall beas specified in the standard method.

b) Each group of n measurements belonging to onelevel shall be carried out under repeatability con-ditions, i.e. within a short interval of time and bythe same operator, and without any intermediaterecalibration of the apparatus unless this is an in-tegral part of performing a measurement.

c)

d)

e)

f)

g)

It is essential that a group of n tests under re-peatability conditions be performed independentlyas if they were n tests on different materials. Asa rule, however, the operator will know thathe/she is testing identical material, but the pointshould be stressed in the instructions that thewhole purpose of the experiment is to determinewhat differences in results can occur in actualtesting. If it is feared that, despite this warning,previous results may influence subsequent testresults and thus the repeatability variance, itshould be considered whether to use n separatesamples at each of the q levels, coded in such away that the operator will not know which are thereplicates for a given level. However, such a pro-cedure could cause problems in ensuring that re-peatability conditions will apply betweenreplicates. This would only be possible if themeasurements were of such a nature that all theqn measurements could be performed within ashort interval of time.

It is not essential that all the q groups of nmeasurements each be performed strictly withina short interval; different groups of measurementsmay be carried out on different days.

Measurements of all q levels shall be performedby one and the same operator and, in addition, then measurements at a given level shall be per-formed using the same equipment throughout.

If in the course of the measurements an operatorshould become unavailable, another operator maycomplete the measurements, provided that thechange does not occur within a group of nmeasurements at one level but only occurs be-tween two of the q groups. Any such change shallbe reported with the results.

A time limit shall be given within which allmeasurements shall be completed. This may benecessary to limit the time allowed to elapse be-

tween the day the samples are received and theday the measurements are performed.

h) All samples shall be clearly Iabelled with the nameof the experiment and a sample identification.

5.1.2) In 5.1.2 and elsewhere in this part ofISO 5725, reference is made to the operator. Forsome measurements, there may in fact be a team ofoperators, each of whom performs some specific partof the procedure. In such a case, the team shall beregarded as “the operator” and any change in theteam shall be regarded as providing a different “op-erator”.

5.1.4 In commercial practice, the test results maybe rounded rather crudely, but in a precision exper-iment test results shall be reported to at least onemore digit than specified in the standard method. Ifthe method does not specify the number of digits, therounding shall not be coarser than half the repeatabil-ity standard deviation estimate. When precision maydepend on the level m, different degrees of roundingmay be needed for different levels.

5.2 Recruitment of the laboratories

5.2.1 The general principles regarding recruitmentof the laboratories to participate in an interlaboratoryexperiment are given in 6.3 of ISO 5725-1:1994. Inenlisting the cooperation of the requisite number oflaboratories, their responsibilities shall be clearlystated. An example of a suitable enlistment question-naire is given in “figure 1.

5.2.2 For the purposes of this part of ISO 5725, a“laborato~” is considered to be a combination of theoperator, the equipment and the test site. One testsite (or Iaboratow in the conventional sense) may thusproduce several “laboratories” if it can provide severaloperators each with independent sets of equipmentand situations in which to perform the work.

5.3 Preparation of the materials

5.3.1 A discussion of the points that need to beconsidered when selecting materials for use in a pre-cision experiment is given in 6.4 of ISO 5725-1:1994.

5.3.2 When deciding on the quantities of material tobe provided, allowance shall be made for accidentalspillage or errors in obtaining some test results whichmay necessitate using extra material. The amount ofmaterial prepared shall be sufficient to cover the ex-periment bnd allow an adequate stock in reserve.

3

IS 15393 (Part 2) :2003

ISO 5725-2:1994

Questionnaire for interlaboratory study

‘itle of measurement method (copy attached) .........................................................

1. Our laboratory is willing to participate in the precision experiment for this stan-dard measurement method.

‘fEs ❑ NO ❑ (tickappropriate box)

L As a participant, we understand that:

a) all essential apparatus, chemicals and other requirements specified in themethod must be available in our laboratory when the programme begins;

b) specified “timing” requirements such as starting date, order of testingspecimens and finishing date of the programme must be rigidly met;

c) the method must be strictly adhered to;

d) samples must be handled in accordance with instructions;

e) a qualified operator must perform the measurements.

-faving studied the method and having made a fair appraisal of our capabilities and‘acilities, we feel that we will be adequately prepared for cooperative testing of thisnethod.

1. Comments(Signed) .......................................................

(Company orlaboratory) ....................................................

Figure 1 — Enlistment questionnaire for rnterlaboratory study

5.3.5 For the samples at each level, n se~arate con-5.3.3 Itshould be considered whether it is desirablefor some laboratories to obtain some preliminary testresults for familiarization with the measurementmethod before obtaining the official test result and, ifso, whether additional material (not precision exper-iment samples) should be provided for this purpose.

5.3.4 When a material has to be homogenized, thisshall be done in the manner most appropriate for thatmaterial. When the material to be tested is nothomogeneous, it is important to prepare the samplesin the manner specified in the method, preferablystarting with one batch of commercial material foreach level. In the case of unstable materials, specialinstructions on storage’ and treatment shall be speci-fied.

tainers shall be used for each laboratory if ~here is anydanger of the materials deteriorating once the con-tainer has been opened (e.g. by oxidation, by losingvolatile components, or with hydroscopic material). Inthe case of unstable materials, special instructions onstorage and treatment shall be specified. Precautionsmay be needed to ensure that samples remain iden-tical up to the time the measurements are made. Ifthe material to be measured consists of a mixture ofpowders of different relative density or of differentgrain size, some care is needed because segregationmay re-suit from shaking, for example during transport.When reaction with the atmosphere may be ex-pected, the specimens may be sealed into ampoules,either evacuated or filled with an inert gas. For per-ishable materials such as food or blood samples, it

4

may be necessary to send them in a deep-frozen stateto the participating laboratories with detailedstructions for the procedure for thawing.

6 Personnel involved in a precisionexperiment

NOTE 3 The methods of operation within differentoratories are not expected to be identical. Therefore

in-

lab-the

contents of this clause are only intended as a guide to bemodified as appropriate to cater for a particular situation.

6.1 Panel

6.1.1 The panel should consist of experts familiarwith the measurement method and its application.

6.1.2 The tasks of the panel are:

a) to plan and coordinate the experiment;

b) to decide on the number of laboratories, levelsand measurements to be made, and the numberof significant figures to be required;

c) to appoint someone for the statistical functions(see 6.2);

d) to appoint someone for the executive functions

(see 6.3);

e) to consider the instructions to be issued to thelaboratory supervisors in addition to the standardmeasurement method;

f) to decide whether some operators may be al-lowed to carry out a few unofficial measurementsin order to regain experience of the method aftera long interval (such measurements shall neverbe carried out on the official collaborative sam-ples);

g) to discuss the report of the statistical analysis oncompletion of the analysis of the test results;

h) to establish final values for the repeatability stan-dard deviation and the reproducibility standarddeviation;

i) to decide if further actions are required to improvethe standard for the measurement method or withregard to laboratories whose test results havebeen rejected as outliers.

6.2 Statistical functions

At least one member of theperience in statistical design}ments. His/her tasks are:

panel should have ex-and analysis of exper-

a)

b)

c)

IS 15393 -(Part 2) :2003

ISO 5725-2:1994

to contribute his/her specialized knowledge in de-signing the experiment;

to analyse the data;

to write “a report for submission to the panel fol-lowing the instructions contained in 7.7.

6.3 Executivefunctions

6.3.1 The actual organization of the experimentshould be entrusted to a single laborato~. A memberof the staff of that laborato~ should take full respon-sibility; he/she is called the executive officer and isappointed by the panel.

6.3.2 The tasks of the executive officer are:

a)

b)

c)

d)

e)

f)

g)

to enlist the cooperation of the requisite numberof laboratories and to ensure that supervisors areappointed;

to organize and supervise the preparation of thematerials and samples and the dispatch of thesamples; for each level, an adequate quantity ofmaterial should be set aside as a reserve stock;

to draft instructions covering all the points in 5.1.2a) to h), and circulate them to the supervisorsearly enough in advance for them to raise anycomments or queries and to ensure that operatorsselected are those who would normally carry outsuch measurements in routine operations;

to design suitable forms for the operator to useas a working record and for the supervisor to re-port the test results to the requisite number ofsignificant figures (such forms may include thename of the operator, the dates on which sam-ples were received and measured, the equipmentused and any other relevant information);

to deal with any queries from laboratories regard-ing the performance of the measurements;

to see that an overall time schedule is maintained;

to collect the data forms and present them to thestatistical expert.

6.4 Supervisors

6.4.1 A staff member in each of the participatinglaboratories should be made responsible for organiz-ing the actual performance of the measurements, inkeeping with instructions received from the executiveofficer, and for reporting the tesl results.

5

IS 15393 (Part2) :2003

ISO 5725-2:1994

6.4.2 The tasks of the supervisor are:

a)

b)

c)

d)

e)

f)

ensure that the operators selected are those whowould normally carry out such measurements inroutine operations;to hand out the samples to the operator(s) inkeeping with the instructions of the executive of-ficer (and to provide material for familiarizationexperiments, if necessary);

to supervise the execution of the measurements(the supervisor shall not take part in performingthe measurements);

to ensure that the operators carry out the requirednumber of measurements;

to ensure adherence to the set timetable for per-forming the measurements;

to collect the test results recorded to the agreednumber of decimal places, including any anom--alies and difficulties experienced, and commentsmade by the operators.

6.4.3 The supervisor of each laboratory should writea full report which should contain the following infor-mation:

a)

b)

c)

d)

e)

f)

g)

h)

the test results, entered legibly by their originatoron the forms provided, not transcribed or typed

(computer or testing machine printout may be ac-ceptable as an alternative);

the original observed values or readings (if any)from which the test results were derived, enteredlegibly by the operator on the forms provided, nottranscribed or typed;

comments by the operators on the standard forthe measurement method;

information about irregularities or disturbancesthat may have occurred during the measure-ments, including any change of operator that mayhave occurred, together with a statement as towhich measurements were performed by whichoperator, and the reasons for any missing results;

the date(s) on which the samples were received;

the date(s) on which each sample was measured;

information about the equipment used, if relevant;

any other relevant information.

6.5 Operators

6.5.1 In each laborato~ the measurements shall becarried out by one operator selected as being repre-sentative of those likely to perform the measure-ments in normal operations.

6

6.5.2 Because the object of the experiment is todetermine the precision obtainable by the generalpopulation of operators working from the standardmeasurement method, in general the operatorsshould not be given amplifications to the standard forthe measurement method. However, it should bepointed out to the operators that the purpose of theexercise is to discover the extent to which results canvary in practice, so that there will be less temptationfor them to discard or rework results that they feel areinconsistent.

6.5.3 Although normally the operators should re-ceive no supplementary amplifications to the standardmeasurement method, they should be encouraged tocomment on the standard and, in particular, to statewhether the instructions contained in it are sufficientlyunambiguous and clear.

6.5.4 The tasks of the operators are:

a)

b)

c)

to perform the measurements according to thestandard measurement method;

to report any anomalies or difficulties experi-enced; it is better to report a mistake than to ad-just the test results because one or two missingtest results will not spoil the experiment andmany indicate a deficiency in the standard;

to comment on the adequacy of the instructionsin the standard; operators should report any oc-casions when they are unable to follow their in-structions as this may also indicate a deficiency inthe standard.

Statistical analysis of a precision7experiment

7.1 Preliminary considerations

7.1.1 The analysis of the data, which should beconsidered as a statistical problem to be solved by astatistical expert, involves three successive stages:

a) critical examination of the data in order to identifyand treat outliers or other irregularities and to testthe suitability of the model;

b) computation of preliminary values of precision andmeans for each level separately;

c) establishment of final values of precision andmeans, including the establishment of a relation-ship between precision and the level m when theanalysis indicates that such a relationship mayexist.

IS 15393 (Part 2) :2003ISO 5725-2:1994

7.1.2 The analysis first computes, for each levelseparately, estimates of

— the repeatability variance s:

— the between-laboratory variance s:

— the reproducibility variances; = S; + S:

— the mean m.

7.1.3 The analysis includes a systematic applicationof statistical tests for outliers, a great variety of whichare available from the literature and which could beused for the purposes of this part of ISO 5725. Forpractical reasons, only a limited number of thesetests, as explained in 7.3, have been incorporated.

7.2 Tabulation of the results and notationused

7.2.1 Cells

Each combination of a laboratory and a level is calleda cell of the precision experiment. In the ideal case,the results of an experiment with p laboratories andq levels consist of a table with pq cells, each contain-ing n replicate test results that can all be used forcomputing the repeatability .standacd deviation and thereproducibility standard deviation. This ideal situationis not, however, always attained in practice. Depar-tures occur owing to redundant data, missing data andoutliers.

7.2.2 Redundant data

Sometimes a laboratory may carry out and reportmore than the n test results officially specified. In thatcase, the supervisor shall report why this was doneand which are the correct test results. If the answeris that they are all equally valid, then a random se-lection should be made from those available test re-sults to choose the planned number of test results foranalysis.

7.2.3 Missing data

In other cases, some of the test results may bemissing, for example because of loss of a sample ora mistake in performing the measurement. Theanalysis recommended in 7.1 is such that completelyempty cells can simply be ignored, while partly emptycells can be taken into account by the standard com-putational procedure.

7.2.4 Outliers

These are entries among the original test results, orin the tables derived from them, that deviate so much

from the comparable entries in the same table thatthey are considered irreconcilable with the other data.Experience has taught that outliers cannot always beavoided and they have to be taken into considerationin a similar way to the treatment of missing data.

7.2.5 Outlying laboratories

When several unexplained abnormal test results occurat different levels within the same laboratory, thenthat laboratory may be considered to be an outlier,having too high a within-laboratory variance and/or toolarge a systematic error in the level of its test results.It may then be reasonable to discard some or all ofthe data from such an outlying Iaboratow.

This part of ISO ‘5725 does not -provide a statisticaltest by which suspected laboratories may be judged.The prima~ decision should be the responsibility ofthe statistical expert, but all rejected laboratories shallbe reported to the panel for further action.

7.2.6 Erroneous data

Obviously erroneous data should be investigated andcorrected or discarded.

7.2.7 Balanced uniform-level test results

The ideal case is p laboratories called i(i= 1, 2, .... p), each testing q levels called j(j=l, 2, .... q) with n replicates at each level (eachij combination), giving a total of pqn test results. Be-cause of missing (7.2.3) or deviating (7.2.4) test re-sults, or outlying laboratories (7.2.5) or erroneous data

(7.2.6), this ideal situation is not always attained. Un-der these conditions the notations given in 7.2.8 to7.2.10 and the procedures of 7.4 allow for differingnumbers of test results. Specimens of recommendedforms for the statistical analysis are given in figure 2.For convenience, they will be referred to simply asforms A, B and C (of figure 2).

7.2.8 Original test results

See form A of figure 2, where

nij is the number of test results in the cell forlaborato~ i at level j;

)’ijk is w one of these test results(k= 1, 2, .... nu);

Pj is the number of laboratories reporting atleast one test result for level j (after elim-inating any test results designated asoutliers or as erroneous).

7

IS 15393 (Part 2) :2003

ISO 5725-2:1994

Form A — Recommended form for the collation of the original data

LevelLaboratory

1 2 J . q–1 q

1

2

.

. . . .i Yijk

. . . .

. .

P

Form B — Recommended form for the collation of the means

LevelLaboratory

1 2 . J . . . q–1 q

1

2

. .

i jij

.

P

FormC — Recommendedform for the collationof the measures of spread within cells

LevelLaborato~

1 2 . . J . . . q–1 q

1

2

. .

i %. .

P

Figure 2 — Recommended forms for the collation of results for analysis

7.2.9 Cell means (form B of figure 2)

These are derived from form A as follows:

The cell means should be recorded to one more sig-nificant figure than the test result in form A.

7.2.10 Measures of cell spread (form C of figure 2)

(2) These are derived from form A (see 7.2.8) and formB (see 7.2.9) as follows.

IS 15393 (Part 2):2003ISO 5725-2:1994

For the general case, use the intracell standard devi- oratories or values that appear to be inconsistent withation all other laboratories or values may change the esti-

‘V=w (3)

mates, and decisions have to be made with respectto these values. Two approaches are introduced:

a) graphical consistency technique;

or, equivalently b) numerical outlier tests.

r

L1 %s;,=

[ 11Lnti-l ~b’ijk)’ ‘+ ~Xjk 2

k=l k=l

. . . (4)

In using these equations, care shall be taken to retaina sufficient number of digits in the calculations; i.e.every intermediate value shall be calculated to at leasttwice as many digits as in the original data.

NOTE 4 If a ceil v contains two test results, the intracellstandard deviation is

rTj=l Y@- Y~21/ “2 . . . (5)

Therefore, for simplicity, absolute differences can be usedinstead of standard deviations if all cells contain two testresults.

7.3.1 Graphical consistency technique

Two measures called Mandel’s h and k statistics areused. It may be noted that, as well as describing thevariability of the measurement method, these help inlaborato~ evaluation.

7.3.1.1 Calculate the between-laboratory consist-ency statistic, h, for each laboratory by dividing thecell deviation (cell mean minus the grand mean forthat level) by the standard deviation among cell means(for that level):

.

hv =Rj – Yj

i . . . (6)

The standard deviation should be expressed to one in which, for ~j see 7.2.9, and for ~. see 7.4.4.

more significant figure than the results in form A.Plot the ho values for each cell in order of laboratory,

For values of nti less than 2, a dash should be inserted in groups for each level (and separately grouped for

in form C. the several levels examined by each laboratory) (seefigure B.7).

7.2.11 Corrected or rejected data

As some of the data may be corrected or rejected onthe basis of the tests mentioned in 7.1.3, 7.3.3 and7.3,4, the values of ytik, nij and pj used for the finaldeterminations of precision and mean may be differ-ent from the values referring to the original test re-sults as recorded in forms A, B and C of figure 2.Hence in reporting the final values for precision andtrueness, it shall always be stated what data, if any,have been corrected or discarded.

7.3 Scrutiny of results for consistency andoutliers

See reference [3].

From data collected on a number of specific levels,repeatability and reproducibility standard deviationsare to be estimated. The presence of individual lab-

.7.3.1.2 Calculate the within-laboratory consistencystatistic, k, by first calculating the pooled within-cellstandard deviation

for each level, and then calculate

rSij p,kij =

r zs;

for each laboratory within each level.

. . . (7)

Plot the kti values for each cell in order of laboratory,in groups for each level (and separately grouped forthe several levels examined by each laborato~) (seefigure B.8).

9

IS 15393 (Part 2) :2003ISO 5725-2:1994

7.3.1.3 Examination of the h and k plots may indicatethat specific laboratories exhibit patterns of resultsthat are markedly different from the others in thestudy. This is indicated by consistently high or lowwithin-cell variation and/or extreme cell means acrossmany levels. If this occurs, the specific Iaboratowshould be contacted to t~ to ascertain the cause ofthe discrepant behaviour. On the basis of the findings,the statistical expert could:

a) retain the laboratory’s data for the moment;

b) ask the laboratory to redo the measurementfeasible);

c) remove the Iabocatory’s data from the study.

(if

7.3.1.4 Various patterns can appear in the h plots.All laboratories can have both positive and negative hvalues at different levels of the experiment. Individuallaboratories may tend to give either all positive or allnegative h values, and the number of laboratoriesgiving negative values is approximately equal to thosegiving positive values. Neither of these patterns isunusual or requires investigation, although the secondof these patterns may suggest that a common sourceof laborato~ bias exists. On the other hand, if all theh values for one Iaboratoty are of one sign and the hvalues for the other laboratories are all of the othersign, then the reason should be sought. Likewise, ifthe h values for a laboratory are extreme and appearto .depend on the experimental level in some sys-tematic way, then the reason should be sought. Linesare drawn on the h plots corresponding to the indica-tors given in 8.3 (tables 6 and 7). These indicator linesserve as guides when examining patterns in the data.

7.3.1.5 If one laboratory stands out on the k plot ashaving many large values, then the reason should besought: this indicates that it has a poorer repeatabilitythan the other laboratories. A laboratory could giverise to consistently small k values because of suchfactors as excessive rounding of its data or an insen-sitive measurement scale. Lines are drawn on the kplots corresponding to the indicators given in 8.3 (ta-bles 6 and 7). These indicator lines serve as guideswhen examining patterns in the data.

7.3.1.6 When an h or k plot grouped by laboratorysuggests that one laborato~ has several h or k valuesnear the critical value line, the corresponding plotgrouped by level should be studied. Often a value thatappears large in a plot grouped by laborato~ will turnout to be reasonably consistent with other labora-tories for the same level. If it is revealed as stronglydifferent from values for the other laboratories, thenthe reason should be sought.

7.3.1.7 In addition to these h and k graphs,histograms of cell means and cell ranges can revealthe presence of, for example, two distinct popu-lations. Such a case wo-uld require special treatmentas the general underlying principle behind the meth-ods described here assumes a single unimodal popu-lation.

.

7.3.2 Numerical outlier technique

7.3.2.1 The following practice is recommended fordealing with outliers.

a)

b)

c)

The tests recommended in 7.3.3 and 7.3.4 areapplied to identify stragglers or outliers:

—

—

.

It

if the test statistic is less than or equal to its5 % critical value, the item tested is acceptedas correct

if the test statistic is greater than its 5 YO crit-ical value and less than or equal to its 1 ‘Yo

critical value, the item tested is called astraggler and is indicated by a single asterisk;

if the test statistic is greater than its 1 ‘%0 crit-ical value, the item is called a statistical outlierand is indicated by a double asterisk.

is next investigated whether the stragglersand/or statistical outliers can be explained bysome technical error, for example

— a slip in performing the measurement,

— an error in computation,

— a simple clerical error in transcribing a test re-sult, or

— analysis of the wrong sample.

Where the error was one of the computation ortranscription type, the suspect result should bereplaced by the correct value; where the error-was from analysing a wrong sample, the resultshould be placed in its correct cell. After suchcorrection has been made, the examination forstragglers or cdiers should be -repeated. If theexplanation of the technical error is such that itproves impossible to replace the suspect test re-sult, then it should be discarded as a “genuine”outlier that does not belong to the experimentproper.

When any stragglers and/or statistical outliers re-main that have not been explained or rejected as

10

IS 15393 (Part 2):2003ISO 5725-2:1994

belonging to an outlying laborato~, the stragglers

are retained as correct items and the statisticaloutliers are discarded unless the statistician forgood reason decides to retain them.

d) When the data for a cell have been rejected forform B of figure 2 under the above procedure,then the corresponding data shall be rejected forform C of figure 2, and vice versa.

7.3.2;2 The tests given in 7.3.3 and 7.3.4 are of twotypes. Cochran’s test is a test of the within-laboratoryvariabilities and should be applied first, then anynecessary action should be taken, with repeated testsif necessary. The other test (Grubbs’) is primarily atest of between-laborato~ variability, -and can also beused (if n > 2) where Cochran’s test has raised sus-picions as to whether the high within-laboratory vari-ation is attributable to only one of the test results in‘the cell.

7.3.3 Cochran’s test

‘7.3.3.1 This part of ISO 5725 assumes that betweenlaboratories only small differences exist in the within-laborato~ variances. Experience, however, showsthat this is not always the case, so that a test hasbeen included here to test the validity of this as-sumption. Several tests could be used for this pur-pose, but Cochran’s test has been chosen.

7.3.3:2 Given a set of p standard deviations Si, allcomputed from the same number (n) of replicate re-sults, Cochran’s test statistic, C, is

C=* . . . (8)

where Smaxis the highest standard deviation in the set.

a) If the test statistic is less than or equal to its 5 ‘%0

critical value, the item tested is accepted as cor-rect.

b) If the test statistic is greater than its 5 YO criticalvalue and less then or equal to its 1 YO criticalvalue, the item tested is called a straggler and isindicated by a single asterisk.

c) If the test statistic is greater than its 1 ‘Yo criticalvalue, the item is called a statistical outlier and isindicated by a double asterisk.

Critical values for Cochran’s test are given in 8.1

(table 4).

Cochran’s test has to be applied to form C of figure 2at each level separately.

7.3.3.3 Cochran’s criterion applies strictly only whenall the standard deviations are derived from the samenumber (n) of test results obtained under repeatabilityconditions. In actual cases, this number may varyowing to missing or discarded data. This part ofISO 5725 assumes, however, that in a properly or-ganized experiment such variations in the number oftest results per cell will be limited and can be ignored,and therefore Cochran’s criterion is applied using forn the number of test results occurring in the majorityof cells.

7.3.3.4 Cochran’s criterion tests only the highestvalue in a set of standard deviations and is thereforea one-sided outlier test. Variance heterogeneity mayalso, of course, manifest itself in some of the stan-dard deviations being comparatively too low. How-ever, small values of standard deviation may be verystrongly influenced by the degree of rounding of theoriginal data and are for that reason not very reliable.In addition, it seems unreasonable to reject the datafrom a laborato~ because it has accomplished ahigher precision in its test results than the other lab-oratories. Hence Cochran’s criterion is considered ad-equate.

7.3.3.5 A critical examination 01 form C of figure 2may sometimes reveal that the standard deviationsfor a particular laboratory are at all or at most levelslower than those for other laboratories. This may in-dicate that the laborato~ works with a lower repeat-ability standard deviation than the other laboratories,which in turn may be caused either by better tech-nique and equipment or by a modified or incorrectapplication of the standard measurement method. Ifthis occurs it should be reported to the panel, whichshould then decide whether the point is worthy of amore detailed investigation. (An example of this islaboratory 2 in the experiment detailed in B.1.)

7.3.3.6 If the highest standard deviation is classedas an outlier, then the value should be omitted andCochran’s test repeated on the remaining values. Thisprocess can be repeated but it may lead to excessiverejections when, as is sometimes the case, theunderlying assumption of normality is not sufficientlywell approximated to. The repeated application ofCochran’s test is here proposed only as a helpful toolin view of the lack of a statistical test designed fortesting several outliers together. Cochran’s test is notdesigned for this purpose and great caution should beexercised in drawing conclusions. When two or thFee

11

IS 15393 (Part 2):2003ISO 5725-2:1994

laboratories give results having high standard devi-ations, particularly if this is within only one of the lev-els, conclusions from Cochran’s test should beexamined carefully. On the other hand, if severalstragglers and/or statistical outliers are found at dif-ferent levels within one laborato~, this may be astrong indication that the laboratory’s within-Iaboratory variance is exceptionally high, and thewhole of the data from that Iaboratoy should be re-jected.

7.3.4 Grubbs’ test

7.3.4.1 One outlying observation

Given a set of data xi for i = 1, 2, .... p, arranged inascending order, then to determine whether the larg-est observation is an outlier using Grubbs’ test, com-pute the &ubb’s statistic, GP.

GP= (JP– 2) /s . . . (9)

where

~=

and

_.$=

To test

1 ‘x,x-F’.

j=l

(lo)

/

*$(xi-z)’ (II),=1

the significance of the smallest observation,compute the test statistic

a)

b)

c)

G, = (~–X1)/S

If the test statistic is less than or equal to its 5 YO

critical value, the item tested is accepted as cor-rect.

If the test statistic is greater than its 5 YO criticalvalue and less than or equal to its 1 YO criticalvalue, the item tested is called a straggler and isindicated by a single asterisk.

if the test statistic is greater than its 1 ‘Yo criticalvalue, the item is called a statistical outlier and isindicated by a double asterisk.

7.3.4.2 Two outlying observations

To test whether the two largest observations may beoutliers, compute the Grubbs’ test statistic G:

G=s2 P–l,pld ... (12)

where

(13)

and

p–2

s’ x(2

p–l,p= xi —Ip_l,p)

i=l

(14)

and

p–2

1’Fp_lp=—

zP–2 i=lxi. . . (15)

Alternatively, to test the two smallest observations,compute the Grubbs’ test statistic G:

G = s:,2/s: (16)

where

P2

s,,’ = x( Xi – 11,2)2

i=s

(17)

and

-P— 1 x (18)‘1,2= ~_z i=3xi . . .

Critical values for Grubbs’ test are given in 8.2

(table 5).

7.3.4.3 Application of Grubbs’ test

When analysing a precision experiment, Grubbs’ testcan be applied to the following.

a) The cell averages (formlevel j, in which case

~i = yti

and

p=pj

where j is fixed.

Taking the data at one

B of figure 2) for a given

level, apply the Grubbs’test for one outlying observation to “cell means asdescribed in 7.3.4.1. If a cell mean is shown to bean outlier by this test, exclude it, and repeat thetest at the other extreme cell mean (e.g. if thehighest is an outlier then look at the lowest withthe highest excluded), but do not apply the

12

IS 15393 (Part 2) :2003ISO 5725-2:1994

Grubbs’ test for two outlying observations de-scribed in 7.3.4.2. If the Grubbs’ test does notshow a cell mean to be an outlier, then apply thedouble-Grubbs’ test described in 7.3.4.2.

b) A single result within a cell, where Cochran’s testhas shown the cell standard deviation to be sus-pect.

7.4.4 Calculation of the general mean ~

For level j, the general mean is

D

7.4 Calculation of the general mean andvariances

7.4.5 Calculationof variances

7.4.1 Meithod of analysis

The method of analysis adopted in this part ofISO 5725 involves carrying out the estimation of mand the precision for each level separately. The resultsof the computation are expressed in a table for eachvalue of j.

7.4.2 Basic data

The basic data needed for the computations are pre-sented in the three tables given in figure 2:

—

—

—

table A containing the original test results;

table B containing the cell means;

table C containing the measures of within-cellspread.

7.4.3 Non-empty cells

AS a consequence of the rule stated in 7.3.2.1 d), thenumber of non-empty cells to be used in the compu-tation will, for a specific level, always be the same intables B and C. An exception might occur if, owing tomissing data, a cell in table A contains only a singletest result, which will entail an empty cell in table Cbut not in table B. In that case it is possible

a)

b)

to discard the solitary test result, which will leadto empty cells in both tables B and C, or

if this is considered an undesirable loss of infor-mation, to insert a dash in table C.

The number of non-empty cells may be different fordifferent levels, hence the index j in pj.

. . . (19)

Three variances are calculated for each level. They arethe repeatability variance, the between-laborato~variance and the reproducibility variance.

7.4.5.1 The repeatability variance is

7.4.5.2 The between-laboratory variance is

2.dj – s;

s;= *

‘j

where

. . . (20)

. . . (21)

‘+$~jbij-~)’s:.-

i=l

[ 1‘* ~nij(Yij)2 - 6j2~nij...(22)1=1 1=1

and

ij=—

P:l. . . (23)

These calculations are illustrated in the examples inB.1 and B.3 in annex B.

13

IS 15393 (Part 2) :2003ISO 5725-2:1994

7.4.5.3 For the particular case where all nu = n = 2,the simpler formulae may be used, giving

1Px(s:’=~i=l Yijl – Yij2)2

and

These are illustrated by the example given in 6.2.

7.4.5.4 Where, owing to random effects, a negativevalue for s; is obtained from these calculations, thevalue should be assumed to be zero.

7.4.5.5 The reproducibility variance is

S;j = s; + S:j . . . (24)

7.4.6 Dependence of the variances upon m

Subsequently, it should be investigated whether theprecision depends upon m and, if so, the functionalrelationship should be determined.

7.5 Establishing a functional relationshipbetween precision values and the meanlevel m

7.5.1 It cannot always be taken for granted thatthere exists a regular functional relationship betweenprecision and m. in particular, where matefialheterogeneity forms an inseparable pan of the vari-ability of the test results, there will be a functionalrelationship only if this heterogeneity is a regularfunction of the level m. With solid materials of differ-ent composition and coming from different productionprocesses, a regular functional relationship is in noway certain. This point should be decided before thefollowing procedure is applied. Alternatively, separatevalues of precision would have to be established foreach material investigated.

7.5.2 The reasoning and computation procedurespresented in 7.5.3 to 7.5.9 apply both to repeatabilityand reproducibility standard deviations, but are pre-sented here for repeatability only in the interests ofbrevity. Only three types of relationship will be con-sidered:

1: S,= bm (a straight line through the origin)

11: s,= a + bm (a straight line with a positive inter-cept)

Ill: Ig Sr=c + d Ig m (or s,= Cm9; d <1 (an expo-nential relationship)

It is to be expected that in the majority of cases atleast one of these formulae will give a satisfacto~ fit.If not, the statistical expert carrying out the analysisshould seek an alternative solution. To avoid con-fusion, the constants u, b, c, C and d occurring inthese equations may be distinguished by subscripts,ar, br, . . . for repeatability and aR, bR, . . . when consid-ering reproducibility, but these have been omitted inthis clause again to simplify the notat’hms. Alsos, hasbeen abbreviated simply to s to allow a suffix for thelevel j.

7.5.3 In general d >0 so that relationships I and Illwill lead to s = O for m = O, which may seem un-acceptable from an experimental point of view. How-ever, when repotting the precision data, it should bemade clear that they apply only within the levels cov-ered by the interlaboratory precision experiment.

7.5.4 For a = O and d = 1, all three relationships areidentical, so when a lies near zero and/or d lies nearunity, two or all three of these relationships will yieldpractically equivalent fits, and in such a case relation-ship I should be preferred because it permits the fol-lowing simple statement.

“Two test results are considered as suspect whenthey differ by more than (100 b) ‘%o. ”

In statistical terminology, this is a statement that thecoefficient of variation (100 s/m) is a constant for alllevels.

7.5.5 If in a plot of Sj against AJ or a plot of Ig Sj

against lg fi} the set of points are found to lie rea-sonably close to a straight line, a line drawn by handmay provide a satisfacto~ solution; but if for somereason a numerical method of fitting is preferred, theprocedure of 7.5.6 is recommended for relationshipsI and 11,and that of 7.5.8 for relationship Ill.

7.5.6 From a statistical viewpoint, the fitting of astraight line is complicated by the fact that both $ andsj are estimates and thus subject to error. But as theslope b is usually small (of the order of 0,1 or less),then errors in ~ have little influence and the errors inestimatings predominate.

7.5.6.1 A good estimate of the parameters ofregression line requires a weighted regressioncause the standard error of s is proportional topredicted value of sj (~).

thebe-the

14

IS 15393 (Part 2):2003ISO 5725-2:1994

The weighting factors have to be proportional to

1/($)2, where $ is the predicted repeatability standarddeviation for level j. However $ depends on par-ameters that have yet to be calculated.

A mathematically correct procedure for finding esti-mates corresponding to the weighted least-squaresof residuals may be complicated. The following pro-cedure, which has proved to be satisfacto~ in prac-tice, is recommended.

7.5.6.2 With weighting factor wj equal to 1/(~’j)2,where N=O, 1, 2 ... for successive iterations, thenthe calculated formulae are:

T, = ~Wj

Then for relationshipby TrJT3.

I (s= h), the value of b is given

For relationship II (s= a + bin):

a=[’~::~$land

b=[Ti7i?l

. . . (25)

~(sjlaj)

b=Jq

7.5.6.4 For relationship 11,the initial values $ojare theoriginal values of s as obtained “by the -proceduresgiven in 7.4. These are used to calculate

W~j= l/(i~j)2 ~= 1, 2, .... ~)

and to calculate al and b, as in 7.5.6.2.

This leads to

The computations are then repeated with

Wlj = 1 /(~1j)2 to produce

;*j = * + b.#2j

The same procedure could now be repeated onceagain with weighting factors W2j = 1/(;zj)2 derivedfrom these equations, but this will only lead to unim-portant changes. The step from WOjto W,J is effectivein eliminating gross errors in the weights, and theequations for ;2j should be considered as the final re-sult .

7.5.7 The standard error of Igs is independent of sand so an unweighed regression of Ig s on Ig ~ isappropriate.

7.5.8 For relationship Ill, the computational formulaeare:

T, =z

19 *j

i

. . . (26)

‘4= ~ (19 ‘j) (19 .Jj

and no iteration is necessary.

7.5.6.3 For relationship 1, algebraic substitution forthe weighting factors Wj = 1/($2 with $ = b~j leadsto the simplified expression:

. . . (27)

and thence

T2 TB– T, T4c .

qT2 – Tf

and

~=qTd– T1T3

qT2 - lf

. . .(28)

. . . (29)

15

IS 15393 (Part 2) :2003

ISO 5725-2:1994

7.5.9 Examples of fitting relationships 1, II and Ill of 7.5.9.1 An example of fitting relationship I is given7.5.2 to the same set of data are now given in 7.5.9.1 in table 1.

to 7.5.9.3. The data are taken from the case study ofB.3 and have been used here only to illustrate the 7.5.9.2 An example of fitting relationship II is givennumerical procedure. They will be further discussed in table2 (& Sj are as in 7.5.9.1).in B.3,

7.5.9.3 An example of fitting relationship Ill is givenin table 3.

Table 1 — Relationship 1:s = bm

ii 3,94 8,28 14,18 15,59 20,41

Sj 0,092 0,179 0,127 0,337 0,393

sj/l$’Ij 0,0234 0,0216 0,0089 0;021 6 W193

~ (sj/Aj) 0,0948b=’q 5

= 0,019 1..

s=bm 0,075 0,157 0,269 0,286 0,388

Table 2 — Relationship 11:s = u + bm

w~j 118 31 62 8,8 6,5

s, = 0,058+ 0,0090 m

AS,j 0,093 0,132 0,185 0,197 0,240

W,j 116 57 29 26 17

Sz= 0,030 + 0,0156 m

&j 0,092 0,159 0,251 0,273 0,348

w*j 118 40 16 13 8

S3= 0,032 + 0,0154 m

A 1)s3j 0,093 0,160 0,251 0,273 0,348

NOTE — The values of the weighting factors are not critical; two significant figures suffice.

1) The difference from Szis negligible.

Table 3 — Relationship Ill: Ig s = c + d Ig m

Ig &j + 0,595 + 0,918 +1,152 + 1,193 + 1,310

Ig Soi -1,036 -0,747 – 0,896 – 0,472 – 0,406

lgs=–l,5065 +0,7721gm

ors = 0,031 mO’7T

s 0,089 0,158 0,239 0,257 0,316

IS 15393 (Part 2) :2003ISO 5725-2:1994

7.6 Statistical analysis as a step-by-stepprocedure

NOTE5 Figure 3indicates inastepwise fashion the pro-cedure given in 7.6.

7.6.1 Collect all available test results in one form,form A of figure 2 (see 7.2). It is recommended thatthis form be arranged into p rows, indexedi=l, 2 ,. ... p (representing the p laboratories thathave contributed data) and q columns, indexedj=l, 2, .... q (representing the q levels in increasing

order).

In a uniform-level experiment the test results withina cell of form A need not be distinguished and maybe put in any desired order.

7.6.2 Inspect form A for any obvious irregularities,investigate and, if necessary, discard any obviouslyerroneous data (for example, data outside the rangeof the measuring instrument or data which are im-possible for technical reasons) and report to the panel.It is sometimes immediately evident that the test re-sults of a particular laboratory or in a particular cell lieat a level inconsistent with the other data. Such obvi-ously discordant data shall be discarded immediately,but the fact shall be repo~ed to the panel for furtherconsideration (see 7.7.1).

7.6.3 From form A, corrected according to 7.6.2when needed, compute form B containing cell meansand form C containing measures of within-cell spread.

When a cell in form A contains only a single test re-sult, one of the options of 7.4.3 should be adopted.

7.6.4 Prepare the Mandel h and k plots as describedin 7.3.1 and examine them for consistency of thedata. These plots may indicate the suitability of thedata for further analysis, the presence of any possibleoutlying values or outlying laboratories. However, nodefinite decisions are taken at this stage, but are de-layed until completion of 7.6.5 to 7.6.9.

7.6.5 Inspect forms B and C (see figure 2) level bylevel for possible stragglers and/or statistical outliers[see 7.3.2.1 a)]. Apply the statistical tests given in 7.3to all suspect items, marking the stragglers with asingle asterisk and the statistical outliers with a dou-ble asterisk. If there are no stragglers or statisticaloutliers, ignore steps 7.6.6 to 7.6.10 and proceed di-rectly with 7.6.11.

7,6.6 Investigate whether there is or may be sometechnical explanation for the stragglers and/or stat-istical outliers and, if possible, verify such an expla-nation. Correct or discard, as required, thosestragglers and/or statistical outliers that have beensatisfactorily explained, and apply corresponding cor-rections to the forms. If there are no stragglers orstatistical outliers left that have not been explained,ignore steps 7.6.7 to 7.6.10 and proceed directly with7.6.11.

NOTE 6 A large number of stragglers and/or statisticaloutliers may indicate a pronounced variance inhomogeneityor pronounced differences between laboratories and maythereby cast doubt on the suitability of the measurementmethod. This should be reported to the panel.

7.6.7 If the distribution of the unexplained stragglersor statistical outliers in form B or C does not suggestany outlying laboratories (see 7.2.5), ignore step 7.6.8and proceed directly with 7.6.9.

7.6.8 If the evidence against some suspectedoutlying laboratories is considered strong enough tojustify the rejection of some or all the data from thoselaboratories, then discard the requisite data and reportto the panel.

The decision to reject some or all data from a partic-ular laboratory is the responsibility of the statisticalexpert carrying out the analysis, but shall be reportedto the panel for further consideration (see 7.7.1).

7.6.9 If any stragglers a~d/or statistical outliers re-main that have not been explained or attributed to anoutlying laboratory, discard the statistical outliers butretain the stragglers.

7.6.10 If in the previous steps any entry in form Bhas been rejected, then the corresponding entry inform C has to be rejected also, and vice versa.

7.6.11 From the entries tt’rat have been retained ascorrect in forms B and C, compute, by the proceduresgiven in 7.4, for each level separately, the mean level&j and the repeatability and reproducibility standarddeviations.

7.6.12 If the experiment only used a single level, orif it has been decided that the repeatability and re-producibility standard deviations should be given sep-arately for each level (see 7.5.1 ) and not as functionsof the level, ignore steps 7.6.13 to 7.6.18 and proceeddirectly with 7.6.19.

NOTE 7 The following steps 7.6.13 to 7.6.17 are appliedto s, and SRseparately, but for brevity they are written outonly in terms ofs~

17

IS 15393 (Part 2) :2003

ISO 5725-2:1994

Drawup form A.I

any obvious irregularities ? Discard discordant data. INo

IComputeforms B and-C.Prepare Mandel’s h and k fAots.

any stragglers/outliers in technical explanation for the Discardor correct thoseform Bor C? Testsare itemsexplained.

No

L?distribution of unexplained Yes Discardsomeor all datastragglers/outliers indicate

No

Oiscardout~iers.Retainstragglers.

Noentries in form Bor Cbeen

I

DiscardcorrespondingYes entry in form B or C.

I

t

No~Figure 3 — Flow diagram of the principal steps in the statistical analysis (continued cm page 19)

18

IS 15393 (Patt 2):2003ISO 5725-2:1994

1“

Compute,for each level separately,using the procedures given in 7.4:- meanm;

- repeatability standard deviation Sr;- reproducibility standard deviation s,Q.

Yes

Level been used or has it Yes Calculate the values of sp andbeen decided to quote s, and sRfor apparently independent of m? s# to apply to all values of m.

ye5 Obtain the Linear relationshiprelationship between sr or s,Qand m be considered linear 7 by applying the computational

procedure given in 7.5.

relationship between lg Sr or ye5 Obtain the linear relationship

(gs# and lg m be considered by app(ying the computational

other relationship between s,EstabUsh that relationship.

or s,Qand m be established 7

Report results to panel (7.7).

Figure 3 — Flow diagram of the principal steps in the statistical analysis

19

IS 15393 (Part 2):2003ISO 5725-2:1994

7.6.13 Plot Sj against &j and judge from this plotwhether s depends on m or not. If s is considered todepend on m, ignore step 7.6.14 and proceed with7.6.15. If s is judged to be independent of m, proceedwith step 7.6.14. If there should be doubt, it is bestto work out both cases and let the panel decide.There exists no useful statistical test appropriate forthis problem, but the technical experts familiar withthe measurement method should have sufficient ex-perience to take a decision.

7.6.14 Use ~zsj =s, as the final value of the re-peatability standard deviation. Ignore steps 7.6.15 to7.6.18 and proceed directly with 7.6.1”9.

7.6.15 Judge from the plot of 7.6.13 whether therelationship between s and m can be represented bya straight line and, if so, whether relationship I(s= bm) or relationship II (s= a + bm) is appropriate

(see 7.5.2). Determine the parameter b, or the twoparameters a and b, by the procedure of 7.5.6. If thelinear relationship is considered satisfactory, ignorestep 7.6.16 and proceed directly with 7.6.17. If not,proceed with step 7.6.16.

7.6.16 Plot Ig Sj against Ig fij and judge from thiswhether the relationship between Ig s and Ig m canreasonably be represented by a straight line. If this isconsidered satisfacto~, fit the relationship III

(!g s = c + d Ig m) using the procedure given in 7.5.8.

7.6.17 If a satisfactory relation has been establishedin step 7.6.15 m 7.6.16, then the final values of S, (orSR)are the smoothed values obtained from this re-lationship for given values of m. Ignore step 7.6.18and proceed with 7.6.19.

7.6.18 If no satisfacto~ relation has been estab-lished in step 7.6.15 or 7.6.16, the statistical expertshould decide whether some other relation betweens and m can be established, or alternatively whetherthe data are so irregular that the establishment of afunctional relationship is considered to be impossible.

7.6.19 Prepare a report showing the basic data andthe results and conclusions from the statistical analy-sis, and present this to the panel. The graphical pres-entations of 7.3.1 may be .usef.ul in presenting theconsistency or variability of the results.

7.7 The report to, and the decisions to betaken by, the panel

7.7.1 Report by the statistical expert

Having completed the statistical analysis, the statisti-cal expert should write a“report to be submitted to the

panel. In this report the following in~ormation shouldbe given:

a)

b)

c)

d)

e)

a full account of the observations received fromthe operators and/or supervisors concerning thestandard for the measurement method;

a full account of the laboratories that have beenrejected as outlying laboratories in steps 7,6.2 and7.6.8, together with the reasons for their re-jection;

a full account of any stragglers and/or statisticaloutliers that were discovered, and whether thesewere explained and corrected, or discarded;

a form of the final results & s, and SR and an ac-count of the -conclusions reached in steps 7.6.13,7.6.15 or 7.6.16, illustrated by one of the plotsrecommended in these steps;

forms A, B and C (figure 2) used in the statisticalanalysis, possibly as an annex.

7.7.2 Decisions to be taken by the panel

The panel should then discuss this report and takedecisions concerning the following questions.

a)

b)

c)

d)

Are the discordant results, stragglers or outliers,if any, due to defects in the description of thestandard for the measurement method?

What action should be taken with respect to re-jected outlying laboratories?

Do the results of the outlying laboratories and/orthe comments received from the operators andsupervisors indicate the need to improve thestandard for the measurement method? if so,what are the improvements required?

Do the results of the mecision ex~eriment justifythe establishment of ‘values of the repeatabilitystandard deviation and reproducibility standarddeviation? If so, what are those values, in whatform should they be published, and what is theregion in which the precision data apply?

7.7.3 Full report

A report setting out the reasons for the work and howit was organized, including the report by the statisti-cian and setting out agreed conclusions, should beprepared by the executive officer for approval by thepanel. Some graphical ~resentation of consistency orvariability is often useful. The report -should be circu-lated to those responsible for authorizing the workand to other interested parties.

20

IS 15393 (Part 2):2003ISO 5725-2:1994

8 Statistical tables

8.1 Critical values for Cochran’s test (see 7.3.3) aregiven in table4.

Table 4 — Critical waluesfor Cochran’s test

n=z n=s n=d n =.5 n=6P

1 ‘%0 570 1 ‘%0 5 % 1 y. 5 % 1 70 5 Y. 1 % 5 !40

2 — — 0,995 0,975 0,979 0,939 0,959 0,906 0,937 0,8773 0,993 0,967 0,942 0,871 0,883 0,798 0,834 0,746 0,793 0,7074 0,968 0,906 0,864 0,768 0,781 0,684 0,721 0,629 0,676 05905 0,928 0,841 0,788 0,684 0,696 0,598 0,633 0,544 0,588 0,5066 0,883 0,781 0,722 0,616 0,626 0,532 0,564 0,480 0,520 0,4457 0,838 0,727 0,664 0,561 0,568 0,480 0,508 0,431 0,466 0,3978 0,794 0,680 0,615 0,516 0,521 0,438 0,463 0,391 0,423 0,3609 0,754 0,638 0,573 0,478 0,481 0,403 0,425 0,358 0,387 0,32910 0,718 0,602 0,536 0,445 0,447 0,373 0,393 0,331 0,357 0,30311 0,6S4 0,570 0,504 0,417 0,418 0,348 0,366 0,308 0,332 0,28112 0,653 0,541 0,475 0,392 0,392 0,326 0,343 0,288 0,310 0,26213 0,624 0,515 0,450 0,371 0,369 0,307 0,322 0,271 0,291 0,24314 0,599 0,492 0,427 0,352 0,349 0,291 0,304 0,255 0,274 0,23215 0,575 0,471 0,407 0,335 0,332 0,276 0,288 0,242 0,259 0,22016 0,553 0,452 0,388 0,319 0,316 0,262 0,274 0,230 0,246 0,20817 0,532 0,434 0,372 0,305 0,301 0,250 0,261 0,219 0,234 0,19818 0,514 0,418 0,356 “0,293 0,288 0,240 0,249 0,209 0,223 0,18919 0,496 0,403 0,343 0Z81 0,276 0,230 0,238 0,200 0,214 0.18120 0,480 0,388 0,330 0,270 0,265 0,220 0,229 0,192 0,205 0,17421 0,465 0,377 0,318 0,261 0,255 0,212 0,220 0,185 0,197 0,16722 0,450 0,365 0,307 0,252 0,246 0,204 0,212 0,178 0,189 0,16023 0,437 0,354 0,297 0,243 0,238 0,197 0,204 0,172 0,182 0,15524 0,425 0,343 0,287 0,235 0,230 0,191 0,197 0,166 0,176 0,14925 0,413 0,334 0,278 0,228 0,222 0,185 0,190 0,160 0,170 0,14426 0,402 0,325 0,270 0,221 0,215 0,179 0,184 0,155 0,164 0,14027 0,391 0,316 0,262 0,215 0,209 0,173 0,179 0,150 0,159 0,13528 0,382 0,308 0,255 0,209 0,202 0,168 0,173 0,146 0,154 0,13129 0,372 0,300 0,248 0,203 0,196 0,164 0,168 0,142 0,150 0,12730 0,363 0,293 0,241 0,198 0,191 0,159 0,164 0,138 0,145 0,12431 0,355 0,286 0,235 0,193 0,186 0.155 0,159 0,134 0,141 0,12032 0,347 0,280 0,229 0,188 0,181 0,151 0,155 0,131 0,138 0,11733 0,339 0,273 0,224 0,184 0,177 0,147 0,151 0,127 0,134 0,11434 0,332 0,267 0,218 0,179 0,172 0,144 -0,147 0,124 0,131 0,11135 0,325 0,262 0,213 0,175 0,168 0,140 0,144 0,121 0,127 0,10836 0,318 0,256 0,208 0,172 0,165 0,137 0,140 0,118 0,124 0,10637 0,312 0,251 0,204 0,168 0,161 0,134 0,137 0,116 0,121 0,10338 0,306 0,246 0,200 0,164 0,157 0,131 0,134 0,113 0,119 0,10139 0,300 0,242 0,196 0,161 0,154 0,129 0,131 0,111 0,116 0,09940 0,294 0,237 0,192 0,158 0,151 0,126 0,128 0,108 0,114 0,097

p = number-of laboratories at a given level

n = number of test results per cell (see 7.3.3.3)

21

IS 15393 (Part 2) :2003ISO 5725-2: 1994

8.2 Critical values for Grubbs’ test (see 7.3.4) are For the Grubbs’ test for two outlying observations,

given In table 5. outliers and stragglers give rise to values which aresmaller than the tabulated 1 YO and 5 YO critical values

For the Grubbs’ test for one outlying observation, respectively.

outliers and stragglers give rise to values which arelarger than the tabulated 1 ‘%0 and 5 ‘%. critical values 8.3 Indicators for Mandel’s h and k statistics (seerespectively. 7.3.1) are given in tables 6 and 7.

Table 5 — Critical values for Grubbs’ test

One largest or one smallest Two largest or two smallestP

Upper 1 ‘%0 Upper 5 % Lower 1 Y. Lower 5 ?!.

3 1,155 1,155 — —

4 1,496 1,481 0,0000 0,00025 1,764 1,715 0,0018 0,00906 1,973 1,887 0,0116 0,03497 2,139 2,020 0,D308 0,07088 2,274 2,126 0,0563 0,11019 2,387 2,215 0,0851 0,149210 2,482 2,290 0,1150 0,186411 2,564 2,355 0,1448 0,221312 2,636 2,412 0,1738 0,253713 2,699 2,462 0,2016 0,283614 2,755 2,507 0,2280 0,311215 2,806 2,549 0,2530 0,336716 2,852 2,585 0,2767 0,360317 2,894 2,620 0,2990 0,382218 2,932 2,651 0,3200 0,402519 2,968 2,681 0,3398 0,421-420 3,001 2,709 0,3585 0,439121 3,031 2,733 0,3761 0,455622 3,060 2,758 0,3927 0,471 123 3,087 2,781 0,4085 0,485724 3,112 2,802 0,4234 0,499425 3,135 2,822 0,4376 0,512326 3,157 2,841 0,4510 0,524527 3,178 2,859 0,4638 0,536028 3,199 2,876 0.4759 0,547029 3,218 2,893 0,4875 0,557430 3,236 2,908 0,4985 0,567231 3,253 2,924 0,5091’ 0,576632 3,270 2,938 0,5192 0,585633 3,286 2,952 0,5288 0,594134 3,301 2,965 0,5381 f),602 335 3,316 2,979 0,5469 o,kyo 136 3,330 2,991 0,5554 0,617537 3,343 3,003 0,5636 0,624738 3,356 3,014 0,5714 0,631639 3,369 3,025 0,5789 0,638240 3,381 3,036 0,5862 0,6445

Reproduced, with the permission of the American Statistical Association, from reference[4] in annex C.

-.. —L---.,- L-------- –. , ,P = rlumOer or laDoralorles a~a given level

22

IS 15393 (Part 2) :2003

ISO 5725-2:1994

Table 6 — Indicators for Mandel’s h and k statisticsat the 1 ‘Yo significance level

k

p h n

2 3 4 5 6 7 8 9 10

3 1,15 1,71 1,64 1,58 1,53 1,49 1,46 1,43 1,41 1,394 1,49 1,91 1,77 1,67 1,60 1#55 1,51 1,48 1,45 1,435 1,72 2,05 1,85 1,73 1,65 159 1,55 1,51 1,48 1,466 1,87 2,14 1,90 1,77 1,68 1,62 1,57 1,53 1,50 1,477 1,98 2,20 1,94 1,79 1,70 1,63 1,58 1,54 1,51 1,488 2,06 2,25 1#97 1,81 1,71 1,65 1,59 1,55 1,52 1,499 2,13 2,29 1,99 1,82 1,73 1,66 1,60 1,56 1,53 1,5010 2,18 2,32 2,00 1,84 1,74 1,66 1,61 1,57 1,53 1,5011 2,22 2,34 2,01 1,85 1,74 1,67 1,62 1,57 1,54 1,5112 2,25 2,36 2,02 1,85 1,75 1,68 1,62 1,58 1,54 1,5113 2,27 2,38 2,03 1,86 1,76 1,68 1,63 1,58 1,55 1,5214 2,30 2,39 2,04 1,87 1,76 1,69 1,63 1,58 1,55 1,5215 2,32 2,41 2,05 1,87 1,76 1,69 1,63 1,59 1,55 1,5216 2,33 2,42 2,05 1,88 1,77 1,69 1,63 1,59 1,55 1,5217 2,35 2,44 2,06 1,88 1,77 1,69 1,64 1,59 1,55 1,5218 2,36 2,44 2,06 1,88 1,77 1,70 1,64 1,59 1,56 1,5219 2,37 2,44 2,07 1,89 1,78 1,70 1,64 1,59 1,56 1,5320 2,39 2,45 2,07 1,89 1,78 1,70 1,64 1,60 1,56 1,5321 2,39 2,46 2,07 1,89 1,78 1,70 1,64 1,60 1,56 1,5322 2,40 2,46 2,08 1,90 1,78 1,70 1,65 1,60 1,56 1,5323 2,41 2,47 2,08 1,90 1,78 1,71 1,65 1,60 1,56 1,5324 2,42 2,47 2,08 1,90 1,79 1,71 1,65 1,60 1,56 1,5325 2,42 2,47 2,08 1,90 1,79 1,71 1,65 1,60 1,56 1,5326 2,43 2,48 2,09 1,90 1,79 1,71 1,65 1,60 1,56 1,5327 2,44 2,48 2,09 1,90 1,79 1,71 1,65 1,60 1,56 1,5328 2,44 2,49 2,09 1,91 1,79 1,71 1,65 1,60 1,57 1,5329 2,45 2,49 2,09 1,91 1,79 1,71 1,65 1,60 1,57 1,5330 2,45 2,49 2,10 1,91 1,79 1,71 1,65 1,61 1,57 1,53

p = number of laboratories at a given level

n = number of replicates within each Ipboratory at that level

NOTE — Supplied by Dr. J. Mandel and published with his permission.

23

IS 15393 (Part 2):2003

ISO 5725-2:1994

Table 7 — Indicators for Mandel’s h and k statisticsat the 5 YO significance level

k

P h n

2 3 4 5 6 7 8 9 10

3 1,15 1,65 1.53 1,45 1,40 1,37 1,34 1,32 1,30 1,294 1,42 1,76 1,59 1,50 1,44 1,40 1,37 1,35 1,33 1,315 1,57 1,81 1,62 1,53 1,46 1,42 1,39 1,36 1,34 1,326 1,66 1,85 1,64 1,54 1,48 1,43 1,40 1,37 1,35 1,337 1,71 1,87 1,66 1,55 1,49 1,44 1,41 1,38 1,36 1,348 1,75 1,88 1,67 1,56 1,50 1,45 1,41 1,38 1,36 1,349 1,78 1,90 1,68 1,57 1,50 1,45 1,42 1,39 1,36 1,3510 1,80 1,90 1,68 1,57 1,50 1,46 1,42 1,39 1,37 1,3511 1,82 1,91 1,69 1,58 1,51 1,46 1,42 1,39 1,37 1,3512 1,83 1,92 1,69 1,58 1,51 1,46 1,42 1,40 1,37 1,3513 1,84 1,92 1,69 1,58 1,51 1,46 1,43 1,40 1,37 1,3514 1,85 1,92 1,70 1,59 1,52 1,47 1,43 1,40 1,37 1,3515 1,86 1,93 1,70 1,59 1,52 1,47 1,43 1,40 1,38 1,3616 1,86 1,93 1,70 1,59 1;52 1,47 1,43 1,40 1,38 1,3617 1,87 1,93 1,70 1,59 1,52 1,47 1,43 1,40 1,38 1,3618 1,88 1,93 1,71 1,59 1,52 1,47 1,43 1,40 1,38 1,3619 1,88 1,93 1,71 1,59 1,52 1,47 1,43 1.40 1,38 1,3620 1,89 1,94 1,71 1,59 1,52 1,47 1,43 1,40 1,38 1,3621 1,89 1,94 1,71 1,60 1,52 1,47 1,44 1,41 1,38 1,3622 1,89 1,94 1,71 1,60 1,52 1,47 1,44 1,41 1,38 1,3623 1,90 1,94 1,71 1,60 1,53 1,47 1,44 1,41 1,38 1,3624 1,90 1,94 1,71 1,60 1,53 1,48 1,44 1,41 1,38 1,3825 1;90 1,94 1,71 1,60 1,53 1,48 1,44 1,41 1,38 1,3626 1,90 1,94 1,71 1,60 1,53 1,48 1,44 1,41 1,38 1,3627 1,91 1,94 1,71 1,60 1,53 1,48 1,44 1,41 1,38 1,3628 1,91 1,94 1,71 1,60 1,53 1,48 1,44 1,41 1,38 1,3629 1,91 1,94 1,72 1,60 1,53 1,48 1,44 1,41 1,38 1,3630 1,91 1,94 1,72 1,60 1,53 1,48 1,44 1,41 1,38 1;36

p = number Of laboratories at a given level

n = number Of replicates within each laborato~ at that level

NOTE — Supplied by Dr. J. Mandel and published with his permission.

24

IS 15393 (Part 2):2003ISO 5725-2:1994

Annex A4(normative)

B.

B(l)! f72)f etc.

c

c, c’, c“

c~fi~, ~~~i~, ~’~,1~

CDP

CRP

d

e

f

~p(vl,V2)

G

h

Symbols and abbreviations

Intercept in the relationship

s=a+bm

Factor used to calculatetainty of an estimate

Slope in the relationship

s=a+bm

the uncer-

Component in a test result repre-senting the deviation of a laborato~from the general average (laboratorycomponent 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

Critical values for statistical tests

Critical difference for probability P

Critical range for probability P

Slope in the relationship

Igs=c+dlgm

Component in a test result repre-senting the random error occurringin every test result

Critical range factor

p-quantile of the F-distribution withVTand V2 degrees of freedom

Grubbs’ test statistic

Mandel’s between-laboratory con-sistency test statistic

k

LCL

m

M

N

n

P

P

4

r

R

RM

s

$

T

t

UCL

w

w

x

Y

used in ISO 5725

Mandel’s within-laboratory consistency teststatistic

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-laborato~ 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 limitlimit)

or warning

Weighting factor used in calculating a weightedregression

Range of a set of test results

Datum used for Grubbs’ test

Test result

25

IS 15393 (Part 2):2003ISO 5725-2:1994

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 (aR/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 testpropefiy

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

Comporrent in a test result representing thevariation due 10 time since last calibration

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

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

Symbols used as subscripts

c

E

i

1()

J

k

L

m

M

o

P

r

R

T

w

Calibration-different

Equipment-different

Identifier for a particular laborato~

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 alaboratory i at level j

Between-laboratory (interlaboratory)

Identifier for detectable bias

Between-test-sample

Operatordifferent

Probability

Repeatability

Reproducibility

Time-different

Within-laboratory (intralaboratory)

1, 2, 3... For test results, numbering in the orderof obtaining them

(1), (2), (3)... For test results, numbering in the orderof increasing magnitude

26

. . ..___-—_— ——

IS 15393 (Part 2) :2003ISO 5725-2:1994

Annex B(informative)

Examples of the statistical analysis of precision experiments

B.1 Example 1: Determination of thesulfur content of coal (Several levels withno missing or outlying data)

6.1.1 Background

a)

b)

c)

d)

Measurement method

Determination of the sulfur content in coal withtest results expressed as a percentage by mass.

Source

Tomkins, S. S. Industrial and Engineering Chem-istry. (See reference [6] in annex C.)

Description

Eight laboratories participated in the experiment,carrying out the analysis according to a standard-ized measurement method described in thesource cited. Laboratory 1 reported four test re-sults and laborato~ 5 reported four or five; theother laboratories all carried out three measure-ments.

Graphical presentation

Mandel’s h and k statistics should be plotted, butbecause in this example they showed little of notethey have been omitted in order to allow space fora different example of the graphical presentationof data. Mandel’s plots are fully illustrated anddiscussed in the example given in B.3.

6.1.2 Originaldata

These are given, as percentage by mass [% (m/m)],in table B. 1 in the format of form A of figure 2 (see7.2.8) and do not invite any specific remarks.

Graphical presentations of these data are given in fig-ures B.1 to B.4.

Table B.1 — Original data: Sulfur content of coal

Laboratoryi1

0,71

10,710,700,71

0,692 0,67

0,68

0,663 0,65

0,69

0,674 0,65

0.66

0,700,69

5 0,660,710,69

0,736 0,74

0,73

t+

0,717 0,71

0,69

0,708 0,65

0,68

2

Levelj

314

1,201,181,231,21

1,221,211,22

1,281,311,30

1,231,181,20

1,311,221,221,24—

1,391,361,37

1,201,261,26

1,241,221,30

1,68 3,261,70 3,261,68 3,201,69 3,24

1,64 3,201,64 3,201,65 3,20

1,61 3,371,61 3,361,62 3,38

+

1,68 3,161,66 3,221,65 3,23

1,64 3,201,67 3,191,60 3,181,66 3,271,68 3,24

+

1,70 3,271,73 3,311,73 3,29

1,69 3,271,70 3,241,68 3,23

1,67 3,251,68 3,261,67 3,26

NOTE 8 For the experiment quoted in table B.1. the lab-oratories were not instructed as to how many measure-ments were to be made, only a minimtim number. By therecommended procedures given in this part of ISO 5725, forlaboratories 1 and 5 a random selection should be madefrom -the values given in order to reduce all cells to exactlythree test results. However, in order to illustrate the com-putational procedures for variable numbers of test results,all test results have been retained in this example. Thereader may make random selections to reduce the numberof test results to three in each cell if he/she wishes to verifythat such a procedure has relatively Iittki ‘effect on the vakues of & s, and Sk.

27

IS 15393 (Part 2):2003ISO 5725-2:1994

B.1.3 Computation ofcellmeans(}j)

The cell means are given, as a percentage by mass

[% (tire)], in table 6.2 in the format of form B of fig-

ure2 (see 7.2.9).

B.1.4 Computation of standard deviations

(%j)

The standard deviations are given, as a percentage bymass [’Yo (tire)], in table B.3 in the format of form Cof figure2 (see 7.2.10).

B. 1.5 Scrutiny for consistency and outliers

Cochran’s test with n = 3 for p = 8 laboratories givescritical values of 0,516 for 5 ‘XO and 0,615 for 1 ‘Yo.

For level 1, largest value of s is in laboratory 8:

ZS2 = 0,001 82; test value = 0,347

For level 2, largest value of s k in laboratory 5:

ZS2 = 0,006 36; test value = 0,287

For level 3, largest value of s is in laboratory 5:

ZS2 = 0,001 72; test value = 0,598

For level 4, largest value of s is in laborato~ 4:

ZS2 = 0,004 63; test value = 0,310

Table B.2 — Cell maans Sulfur content of coal

I Levelj

Laboratoryi I 1

I E,

12345678

0,7080,6800,6670,6600,6900,733047030,677

12

T‘ij $j

4 1,2053 1,2173 1,2973 1,2035 1,2483 1,3733 1,240“3 1,253 1

‘U

43334333

3

Al&1,688 41,643 31,613 31,667 31,650 51,720 31,690 31,673 3

4

3,240 43,200 33,370 33,203 33,216 53,290 33,247 33,257 3

Table B.3 — Standard deviations: Sulfur content of coalh

LaboratoryiL

1

1

2345678

0,0050,0100,021010100,0190,0060,0120,025

%

43335333

2

%

0,0210,0060,0150,0250,0430,0150,0350,042

Levelj

%

43334333

31

%

0,01-00,0060,0060,0120,0320,0170,0100,006T%43335333

4

0,0280,0000,0100,0380,0380,0200,0210.006

ni,

43335333

28

1{

2

3

4

5{

6<

7{

8

xx

xx

xxx

x

x x

I

0,65

<;x

<x

-cxxx

<

1 I I

70 0,75 0,80 m.%Ii%

Figure B.1 — Sulfur content of coal, aampla

1

2{

3

4

5{

6

7{

o

Xxxx

xc

xx x

Ix

x

xx

I I

1,1 1.2

xxx

x

xx

x

1,3 1,6 1,5 m,%

IS 15393 (Part 2) :2003ISO 5725-2:1994

1’

Figure B.2 — Sulfur content of coal, sample 2

29

IS 15393 (Part 2) :2003ISO 5725-2:1994

1{

2{

3{

4{

5{

6{

7

8{

xx>

xxx

xx

1,50 1,55 1,60 1,1

<<xx

x

:x

xxx

1,70 1,75 1,80 1,8S m,%

33

Figure B,3 — Sulfur content ofcoal, sample3

1<

2{

3

.4

5

6

7

8{

xxx

x xx

xxx x

x:

I 1 13,0 3.1 3,2

xxx

Kxx

x

xx

1 I 1

3,3 3,4 3,5 6 m,%

figure B.4 — Sulfur content of coal, sample 4

30

ts 15393 (Part 2):2003ISO 5725-2:1994

This indicates that one cell in level 3 maybe regardedas a straggler, and there are no outliers. The straggleris retained in subsequent calculations.

Grubbs’ tests were applied to the cell means, givingthe values shown in table B.4. There are no singlestragglers or outliers. At levels 2 and 4, the high re-sults for laboratories 3 and 6 are stragglers accordingto the double-high test; these were retained in theanalysis.

B.1.6 Computation of &j, srjand s~j

The variances defined in 7.4.4 and 7.4.5 are calculatedas toilows, using level 1 as an example.

Number of laboratories, p = 8

TI = Zni ~i = 18,642

T2 = Zni (X)2 = 12,8837

TB= Xni = 27

Td = Xnf = 95

T~ = Z(ni – l)s~= 0,004411

s:=~=TB-p

0,0002322

= 0,0004603

s: = s; + S: = 0,0006925

~ = 0,69044‘=T~

S,= 0,01524

S// = 0,02632

The calculations for levels 2, 3 and 4 may be carriedout similarly to give the results shown in table B.5.

B.1.7 Dependenceof precisionon m

An examination of the data in table B.5 does not indi-cate any dependence and average values can beused.

B.1.8 Conclusions

The precision of the measurement method should bequoted, as a percentage by mass, as

repeatability standard deviation, Sr= 0,022

reproducibility standard deviation, SR= 0,045

Table 6.4 — Application of Grubbs’ test to cell means. .

LevelSingle Single Double Double

low high low highTypeof teat

1 1,24 1,80 0,539 0,2982 0,91 2,09 0,699 0,108 Grubbs’ test3 1,67 1,58 0,378 0,459 statistics4 0,94 2,09 0,679 0,132

Stragglers 2,126 2,126 0,1101 0,1101 Grubbs’ critical

Outliers 2,274 2,274 0,0563 0,0563 values

Table B.5 — Computed values of ~j, Srjand s~jfor sulfur content of coal

Levelj Pj &j srj ‘R]

1 8 0,690 0,015 0,0262 8 1,252 0,029 0,0613 8 1,667 0,017 “0,0354 8 3,250 0,026 0,058

31

IS 15393 (Part 2) :2003ISO 5725-2:1994

These values may be applied within a range0,69 ‘Y.(m/m) to 3,25 ‘%.(m/m). They were deter-mined from a uniform-level experiment involving 8laboratories covering that range of values, in whichfour stragglers were detected and retained.

B.2 Example 2: Softening point of pitch(Several levels with missing data)

B.2.1 Background

a)

b)

c)

d)

e)

Measurement method

The determination of the softening pornt of pitchby ring and ball.

Source

Standard methods for testing tar and its products;Pitch section; Method Serial No. PT3 using neutral

glycerine (reference [5] in annex C).

Material

This was selected from commercial batches ofpitch collected and prepared as specified in the“Samples” chapter of the pitch section of refer-ence [5].

Description

This was the determination of a prope~y involvingtemperature measurement in degrees Celsius.

Sixteen laboratories cooperated. It was intendedto measure four specimens at about 87,5 “C,92,5 “-C, 97,5 ‘C and 102,5 “C to cover the normal

commercial range of products, but wrong materialwas chosen for level 2 with a mean temperatureof about 96 “C which was similar to level 3. Lab-oratory 5 applied the method incorrectly at firston the sample for level 2 (the first one theymeasured) and there was then insufficient ma-terial remaining for more than one determination.Laboratory 8 found that they did not have a sam-ple for level 1 (they had two specimens for level4).

Graphical presentations

Mandel’s h and k statistics should be plotted, butagain in this example they have been omitted in

order to provide for another type of graphicalpresentation of data. Mandel’s plots are fullyillustrated and discussed in the example given inB.3.

B.2.2 Original data

These are presented in table B.6, in degrees Celsius,in the format of form A of figure 2 (see 7.2.8).

Table B.6 — Originaldata:Softeningpointofpitch (“C).,

LaboratoryiLevelj

1 2 3 4

191,0 97,0 96,5 104,089,6 97,2 97,0 104,0

289,7 98,5 97,2 102,689,8 972 97,0 103,6

388,0 97,8 94,2 103,087,5 94,5 95,8 99,5

489,2 96,8 96,0 102,588,5 97,5 98,0 103,5

589,0 97,2 98,2 101,090,0 — 98,5 100,2

688,5 97,8 99,5 102,290,5 97,2 103,2 102,0

788,9 96,6 98,2 102,888,2 97,5 99,0 102,2

8— 96,0 98,4 102,6— 97,5 97,4 103,9

990,1 95,5 98,2 102,888,4 96,8 96,7 102,0

1086,0 95,2 94,8 99,885,8 95,0 93,0 100,8

1187,6 93,2 93,6 98,284,4 93,4 93,9 97,8

1288,2 95,8 95,8 101,787,4 95,4 95,4 101,2

1391,0 98,2 98,0 104,590,4 99,5 97,0 105,6

1487,5 97,0 97,1 105,287,8 95,5 96,6 101,8

1587,5 95,0 97,8 101,587,6 95,2 99,2 100,9

1688,8 95.0 97,2 99,585,0 93,2 97,8 99,8

NOTE — There are no obvious stragglers or statisticaloutliers.

32

IS 15393 (Part 21:2003ISO 5725-2:1994

6.2.3 Cell means 6.2.4 Absolute differenceswithin cells

These are given in table B.7, in degrees Celsius, in the In this example there are two test results per cell andformat of form B of figure2 (see 7.2.9). the absolute difference can be used to represent the

variability. The absolute differences within cells, inA graphical presentation of these data is given in fig- degrees Celsius, are given in table 6.8, in the formature B.5. of form C of figure 2 (see 7.2.10).

A graphical presentation of these data is given in fig-ure B.6.

Table B.7 — Cell means Softening point of pitch (“C)

LeveljLaboratoryi

1 2 3 4

1 90,30 97,10 96,75 104,002 89,75 97,85 97,10 103,103 87,75 96,15 95,00 101,254 88,85 97,15 97,00 103,005 89,50 — 98,35 100,606 89,50 97,50 101,35 102,107 88,55 97,05 98,60 102,508 — 96,75 97,90 103,259 89,25 96,15 97,45 102,4010 85,90 95,10 93,90 100,3011 86,00 93,30 93,75 98,0012 87,80 95,60 95,60 101,4513 90,70 98,85 97,50 105,0514 87,65 96,25 96,85 103,5015 87,55 95,10 98,50 101,2016 86,90 94,10 97,50 !39,65

NOTE — The entry for i = 5, j = 2 has been dropped (see 7.4.3).

Table B.8 — Absolute differences within cells: Softening point of pitch (“C)

LeveljLaboratoryi

1 2 3 4

1 1#4 0,2 0,5 0,02 0,1 1,3 0,2 1,03 0,5 3,3 1,6 3,54 0,7 0,7 2,0 1,05 1,0 — 0,3 0,86 2,0 0,6 3,7 0,27 0,7 0,9 0,8 0,68 . 1,5 1,0 1,39 1,7 1,3 1,5 0,810 0,2 0,2 1,8 1,011 3,2 0,2 0,3 0,412 0,8 0,4 0,4 0,513 0,6 1,3 1,0 1,114 0,3 1,5 0,5 3,415 0,1 0,2 1,4 0,616 3,8 1,8 0,6 0,3

33

IS 15393 (Part 2):2003ISO 5725-2:1994

8.2.5 Scrutiny for consistencyand outliers

Application of Cochran’s test leads to the values ofthe test statistic C given in table B.9.

The critical values (see 8.1 ) at the 5 Y. significancelevel are 0,471 for p = 15 and 0,452 for p = 16 wheren = 2. No stragglers are indicated.

Grubbs’ tests were applied to the cell means. Nosingle or double stragglers or outliers were found.

B.2.6 Computation of ~j, S,jand ~~j >

These are calculated as in 7.4.4 and 7.4.5.

Using level 1 for example, the calculations are as fol-lows. To ease the arithmetic, 80,00 has been sub-tracted from all the data. The method for n = 2replicates per cell is used.

Number of laboratories, p = 15

Number of replicates, n = 2

T, = Zy = 125,9500

Tz = Z(x)z = 1087,9775

T~ = (yil – yi~)z = 36,9100

s:=+= 1,2303

[1

pTz – Tf’s; = –$=1,5575

P(P-1)

S: = S; + S: = 2,787 8

* = ~ (add 80,00) = 88,3966

s, =1,1092

S// = 1,6697

The values for all four levels are given in table 6.11.

Table 6.9 — Values of Cochran’s test statistic,C

I Levelj 1 2 3 4 I, 1 1 1

c 0,391 (15) 0,424 (15) 0,434 (16) 0,380 (16)

NOTE — Number of laboratories is given in parentheses.

Table B.1O — Application of Grubbs’ test to cell means

Level;n Single Single Double Doublelow high low high Typeof test

1;15 1,69 1,56 0,546 0,6622; 15 2,04 1,77 0,478 0,646 Grubbs’ test3; 16 1,76 2,27 0,548 0,5664; 16

statistics2,22 1,74 0,500 0,672

Stragglersn=15 2,549 2,549 0,3367 0,3367n=16 2,585 2,585 0,3603 0,3603 Grubbs’ critical

Outliers values

n=15 2,806 2,806 0,2530 0,2530n=16 2,852 2,852 0,2767 0.2767

34

IS 15393 (Part 2):2003ISO 5725-2:1994

Table B.11 — Computed values of ~j, Srjand s~jfor softening point of pitch

Levelj Pj &j (“c) % s~j

1 15 88,40 1,109 1,6702 15 96,27 0,925 1,5973 16 97,07 0,993 2,0104 16 101,96 1,004 1,915

,Fd!d!iL,eveL,

[

16139

—

J3diL7

14 615941282

1116103113Level 2

&15 914 6

1612751310113421

Level 1I I

80 90 100 110Temperature,“C

Figure B.5 — Softening point of pitch: Cell means

35

IS 15393 (Part 2) :2003ISO 5725-2:1994

w.uca!3val M913

kLevel 3

15Y

11

10 9

1Leve( 2

Level 1I I 1 I I I

o 1 2 3 4 5Temperature, “C

Figure 6.6 — Softening point of pitch: Absolute differences within cells

B.2.7 Dependence of precision on m

A cursory examination of table 6.11 does not revealany marked dependence, except perhaps in repro-ducibility. The changes over the range of values of m,if any at all, are too small to be considered significant.Moreover, in view of the small range of values of mand the nature of the measurement, a dependenceon m is hardly to be expected. It seems safe to con-clude that precision does not depend on m in thisrange, which was stated as covering normal com-mercial material, so that the means may be taken asthe final values for repeatability and reproducibilitystandard deviations.

6.2.8 Conclusions

For practical applications, the precision values for themeasurement method can be considered as inde-pendent of the level of material, and are

repeatability standard deviation, S,= 1,0 “C

reproducibility standard deviation, SR = 1,8 “C

B.3 Example 3: Thermometric titrationof creosote oil (Several levels withoutlying data)

6.3.1 Background

a)

b)

c)

Source

Standard methods for testing tar and its products;Creosote oil section; Method Serial No. Co. 18(reference [5] in annex C).

Material

This was selected from commercial batches ofcreosote oil collected and prepared as specified inthe “.Samples” chapter of the creosote oil sectionof reference [5].

Description

This was a standard measurement method forchemical analysis involving a thermometrictitration, with results expressed as a percentageby mass. Nine laboratories participated by meas-uring five specimens in duplicate, the specimens

36

IS 15393 (Part 2) :2003ISO 5725-2:1994

measured having been selected so as to cover thenormal range expected to be encountered in gen-eral commercial application. These were chosento lie at the approximate levels of 4, 8, 12, 16 and20 [Y. (tire)]. The usual practice would be to re-cord test results to only one decimal place, but forthis experiment operators were instructed to workto two decimal places.

B.3.2 Original data

These are presented in tableB.12, as a percentage bymass, in the format of form A of figure 2 (see 7.2.8).

The test results for Iaboratoy 1 were always higher,and at some levels considerably higher, than those ofthe other laboratories.

The second test result for laboratory 6 at level 5 issuspect; the value recorded would fit much better atlevel 4.

These points are discussed further in 6.3.5.

6.3.3 Cell means

These are given in table B. 13, as a percentage bymass, in the format of form B of figure 2 (see 7.2.9).

Table B.12 — Original data: Thermometric titration of creosote oil

I LaboratoryiLevelj

1 1213141s II I I I 11 4,44 4,39 9,34 9,34 17,40 16,90 19,23 19,23 24,28 “24,00

2 4,03 4,23 8,42 8,33 14,42 14,50 16,06 16,22 20,40 19,91

3 3,70 3,70 7,60 7,40 13,60 13,60 14,50 15,10 19,30 19,70

4 4,10 4,10 8,93 8,80 14,60 14,20 15,60 15,50 20,30 20,30

5 3,97 4,04 7,89 8,12 13,73 13,92 15,54 15,78 20,53 20,88

6 3,75 4,03 8,76 9,24 13,90 14,06 16,42 16,58 18,56 16,58

7 3,70 3;80 8,00 8,30 14,10 14,20 14,90 16,00 19,70 20,50.

8 3,91 3,90 8,04 8,07 14,84 14,84 15,41 15,22 21,10 20,78

9 4,02 4,07 8,44 8,17 14,24 14,10 15,14 15,44 20,71 21,66

Table B.13 — Cell means: Thermometric titration of creosote oil-.

LaboratoryiLevelj

1 2 3 4 5

1 4,415 9,340 17,150** 19,230”” 24,140”2 4,130 8,375 14,460 16,140 20,1553 3,700 7,500 13,600 14,800 19,5004 4,100 8,865 14,400 15,550 20,3005 4,005 8,005 13,825 15,660 20,7056 3,890 9,000 13,980 16,500 17,5707 3,750 8,150 14,150 15,450 20,1008 3,905 8,055 14,840 15,315 20,9409 4,045 8,305 14,170 15,290 21,185

● Regarded as a straggler.

● ● Regarded as a statistical outlier.

37

IS 15393 (Part2) :2003

ISO 5725-2:1994

6.3.4 Absolute differences within cells

These are given in table B.14, as WYby mass, in the format of form C7.2,10).

6.3.5 Scrutiny for consistency

as a percentage

of figure 2 (see

and outliers

Calculation of Mandel’s h and k consistency statistics(see 7.3.1) gave the values shown in figures B.7 andB.8, Horizontal lines are shown corresponding to thevalue of Mandel’s indicators taken from 8.3.

The h graph (figure B.7) shows clearly that laboratory1 obtained much higher test results than all otherlaboratories at all levels. Such results require attentionon the part of the committee running the interlabora-tory study. If no explanations can be found for thesetest results, the members of the committee shoulduse their judgement, based on additional and perhapsnon-statistical considerations, in deciding whether toinclude or exclude this laboratory in the calculation ofthe precision values.

The k graph (figure B.8) exhibits rather large variabilitybetween replicate test results for laboratories 6and 7. However, these test results do not seem sosevere as to’ require any special action beyond asearch for possible explanations and, if necessary,remedial action for these test results.

Application of Cochran’s test yields the following re-sults.

At level 4, the absolute difference 1,10 gave a teststatistic value of 1,102/1 ,8149 = 0,667.

At level 5, the absolute difference 1,98 gave a teststatistic value of 1,982/6,166 3 = 0,636.

For p = 9, the critical values for Cochran’s test are0,638 for 5 ‘X.,and 0,754 for 1 ‘X..

The value 1,10 at level 4 is clearly a straggler, and thevalue 1,98 at level 5 is so near the 5 Y. level as to bealso a possible straggler. As these two values are sodifferent from all the others, and as their presence hasinflated the divisor used in Cochran’s test statistic,they have both been regarded as stragglers andmarked with an asterisk. The evidence against themso far, however, cannot be regarded as sufficient forrejection, although Mandel’s k plot (figure B.8) alsogives rise to suspicion of these values.

Application of Grubbs’ tests to the cell means givesthe results shown in tableB.15.

For levels 3 and 4, because the single Grubbs testindicates an outlier, the double Grubbs test is not ap-plied (see 7.3.4).

The cell means for laboratory 1 in levels 3 and 4 arefound to be outliers. The cell mean for this laboratoryfor level 5 is also high. This is also clearly indicatedon Mandel’s h plot (figure B.7).

On further enqui~, it was learned that at least one ofthe samples for laborato~ 6, level 5, might by mistakehave come from level 4. As the absolute differencefor this cell was also suspect, it was decided that thispair of test results may dSO have to be Fejected.Without the “help” of this pair of values, the test re-sult for laboratory 1 at level 5 is now definitely suspi-cious.

Table B.14 — Cell ranges Thermometric titration .of creosote oil

Laboratoryi

1

23456789

1 12

0,05 0,000,20 0,090,00 0,200,00 0,130,07 0,230,28 0,480,10 0,300,01 0,030,05 0,27

Levelj

3 1415

0,50 0,00 0,280,08 0,16 0,490,00 0,60 0,400,40 0,10 0,000,19 0,24 0,350,16 0,16 1,98*0,10 1,1O* 0,800,00 0,19 0,320,14 0,30 0,95

● Regarded as a straggler. I

38

IS 15393 (Part 2) :2003ISO 5725-2:1994

Because of these test results, it was-decided to reject Without these test results, the Cochran’s test statistic

the pair of test results from laborato~ 6 for level 5 at level 4 was then compared with the critical value

because it was uncertain what material had been for 8 laboratories (0,680 at 5 %) and this no longer

measured and to reject all the test results from lab- appeared as a straggler and was retained.

oratory 1 as coming from an outlying laboratory.

Table B.15 — Application of Grubbs’ test to cell means. .

Level Single Single Double Doublelow high low high Typeof test

1 1,36 1,95 0,502 0,3562 1,57 1,64 0,540 0,3953 0,86 2,50

Grubbs’ test— —

4 0,91 2,47statistics— —

5 1,70 2,10 0,501 0,318

Stragglers 2,215 2,215 0,1492 0,1492 Grubbs’ critical

Outliers 2,387 2,387 0,0851 0,0851 values

6

5 -

4 -

.!? -----------------------------------------------------------------~ ---------2 ;2 -------- ,---------------------------------------------------------------%m‘4almcz 0‘ 1..1 I I II I I

111 II’ II 111’qI-1 -

---------------------------------------------------2 =

------------------------.---------------------------- ----------------------------------- ------------.

-3 . 1 2 3 4 5 6 7 a 9

Laboratory i

Figure B.7 — Titration of creosote oil: Mandel’s between-laboratory consistency statistic, h, grouped bylaboratories

39

IS 15393 (Part 2) :2003

ISO 5725-2:1994

.’+ 3 -z.-% ------------------------- ----------------------% --- .------ .-----------------U12 :---------. ------------------------------------- J-- ------- ------------------‘4%c:1

0 1 I 1111 I I I 1 I Ill II I 1 n II I IL

-1 -1 2 3 4 5 6 7 8 9

Laboratory i

Figure B.8 — Tfiration of creosote oil: Mandel’s within-laboratory consistency statistic, k, grouped bylaboratories

B.3.6 Computation of~j, s,jands~j

The values of& s,jands~jcomputed withoutthe testresults oflaborato~ 1 and the pair of test results fromlaboratory, level 5, are given intable B.16, asa per-centage by mass, calculated asin7.4.4and 7.4.5.

B.3.7 Dependence of precision on m

From table B.16, it seems clear that the standard de-viations tend to increase with higher values of m, soit is likely that it might be permissible to establishsome form of functional relationship. This view wassupported by a chemist familiar with the measure-ment method, who was of the view that the precisionwas likely to be dependent on the level.

The actual calculations for fitting a functional relation-ship are not given here as they have already been setout in detail for S, in 7.5.9. The values of S,j and s~j areplotted against fij in figure B.9.

From figure B.9 it is evident that the value for level 3is strongly divergent and could not be improved by

any alternative procedures (see 7.5.2).

For repeatability, a straight line through the originseems adequate.

For reproducibility, all three lines show adequate fitwith the data, relationship Ill showing the best fit.

Someone familiar with the requirements for a stan-dard measurement method for creosote oil may beable to select the most suitable relationship.

B.3.8 Final values of precision

The final values, duly rounded, should be

repeatability standard deviation, Sr= 0,019m

reproducibility standard deviation,s~ = 0,086 + 0,030m or

S// = 0,078m0’72

B.3.9 Conclusions

There are no statistical reasons for preferring eitherone of the two equations for SR in B.3.8. The panelshould decide which one to use.

The reason for the outlying test results of laboratory1 should be investigated.

This seems to have been a rather unsatisfactory pre-cision experiment. One of the 9 laboratories had tobe rejected as an outlier, and another laborato~ hadtested a wrong specimen. The material for level 3seems to have been wrongly selected, having almostthe same value as level 4 instead of lying midway

40

IS 15393 (Part 2):2003ISO 5725-2:1994

between levels 2 and 4. Moreover, the material for other material. It might be worthwhile to repeat this

level 3 seems to have been somewhat different in experiment, taking more care over the selection of the

nature, perhaps being more homogeneous than the materials for the different levels.

Table 6.16 — Computed values of *Y Srjand $Rj for thermometric titration ofcreosote oil

Levelj Pj hi % s//j

1 8 3,94 0,092 0,1712 8 8,28 0,179 0,498

3 8 14,18 0,127 0,4004 8 15,59 0,337 0,5795 7 20,41 0,393 0,637

SR= 0,086 + OJ130m*I \

Figure 6.9 — Plot

w:I SR= 0,04mw \ \A

“ o— 5 10 15 20 m

and .$Rj against*j of the datafromtable6.16,showingthefreed in 7.5from thesedata

functionalrelationships

41

IS 15393 (Part2):2003ISO 5725-2:1994

Annex C(informative)

Bibliography

[1] ISO Guide 33:1989, Uses of certified referencematerials.

[2] ISO Guide 35:1989, Certification of referencematerials — General and statistical principles.

[3] ASTM .E691 -87, Standard Practice for Conduct-ing an Irtterlaboratory Study to Determine thePrecision of a Test Method. American Societyfor Testing and Materials, Philadelphia, PA,USA.

[4] GRUBBS, F.E. and BECK, G. Extension of sample

sizes and percentage points for significance

tests of outlying observations. Technometrics,14, 1972, pp. 847-854.

[5] “Standard Methods for Testing Tar and itsProducts”. 7th Ed. Standardisation of Tar Prod-ucts Tests Committee, 1979.

[6] TOMKINS, S.S. /ndustria/ and Engineering Chem-istry (Analytical edition), 14, 1942, pp. 141-145.

[7] GRUBBS, F.E. Procedures for detecting outlying

observations in samples. Technornetrics, 11,

1969, pp. 1-21.

[8] ISO 3534-2:1993, Statistics — Vocabulary andsymbo/s — Part 2: Statistical quality control.

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

[1 O] ISO 5725-3:1994, Accuracy (trueness and pre-cision) of measurement methods and results —Part 3: Intermediate measures of the precision ,of a standard measurement method.

[11] ISO 5725-4:1994, Accuracy (trueness and pre-cision) of measurement methods and results —Part 4: Basic methods for the determination ofthe trueness of a standard measurementmethod.

[12] ISO 5725-5:– ‘1, 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.

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

1) To be published.

42

IS 15393 (Part 2):2003ISO 572.5-2:1994

Accuracy (trueness and precision) of measurement methods andresults —

Part 2:Basic method for the determination of repeatability andreproducibility of a standard measurement method

TECHNICAL CORRIGENDUM 1

Technical Corrigendum 1 to International Standard ISO 5725-2:1994 was prepared by Technical CommitteelSO/TC 69, Applications of statistical methods, Subcommittee SC 6, Measurement methods and resu/ts.

Page 12, subclause 7.3.4.3

Add the following note at the end of the subclause 7.3.4.3:

“NOTE According to 7.3.2.1, an item is to be called a statistical outlier if the test statisticis greater than its 4 ‘%.criticalvalue. When the Grubbs’ test is first applied to a group of cell means, a critical value from Table 5 is used to test thehighest cell mean using a testat the 0,5% level, an~to test the lowest cell mean at the 0,5% level. This amounts to a restof the most extreme cell mean at the 1 % level in ac~ordance with 7.3.2.1. If the most extreme cell mean is found to be astatistical outlier, the Grubbs’ test is then applied to the other extreme cell mean. It can be argued that a one-sided testshould now be used. However, the procedure recommended in this part of 1S0 5725 is to use only the critical valuestabulated in Table 5 (the critical values for two-sided tests at the 1 ‘Y.signifimnce level) in order that all cell means betreated consistently.A similarargument maybe used to justifjrthe use of the two-sided5 % criticalvalues in Table 5 for alltests for statisticalstragglers.”

Page 22, Table 5

Add the following note at the end of Table 5:

“NOTE The criticalvalues given in this table are appropriatewhen two-sidedtests are required. They are the criticalvalues required by the procedure for applyingthe Grubbs’outlier tests described in 7.3.4 of this part of ISO 5725. Theyhave been derived from the criticalvalues for the correspondingone-sided testa as given in reference [4] of annex C.”

43

( Continued from secordcover)

IS No. Title

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 ISO 3534-1 :1993 ‘Statistics —Vocabulary and symbols : Part 1 Probability and general statistical terms’ referred in this adoptedstandard and decided that it is acceptable for use with this standard.

This standard also gives Bibliography in Annex C, 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.

Copyright

BIS has the copyright of all its publications. No part ofthese publications maybe reproduced in any form withoutthe prior permission in writing of BIS. This does not preclude the free use, in the course of implementing thestandard, of necessary details, such as symbols and sizes, type or grade designations. Enquiries relating to fcopyright be addressed to the Director (Publications), BIS.

Review of Indian Standards i

Amendments are issued to standards as the need arises on the basis of comments. Standards are also reviewedperiodically; a standard along with amendments is reaffirmed when such review indicates that no changes areneeded; if the review indicates that changes are needed, it is taken up for revision. Users of Indian Standardsshould ascertain that they are in possession of the latest amendments or edition by referring to the latest issueof ‘BIS Catalogue’ and ‘Standards : Monthly Additions’.

This Indian Standard has been developed from Doc : No. BP 25 ( 0189 ).

Amendments Issued Since Publication

Amend No. Date of Issue Text Affected

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{

2323761723233841

{23378499,2337856123378626,23379120

{

603843609285

{

22541216,2254144222542519,22542315

{

28329295,2832785828327891,28327892

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