SIST EN 12603:2004
(Main)Glass in building - Procedures for goodness of fit and confidence intervals for Weibull distributed glass strength data
Glass in building - Procedures for goodness of fit and confidence intervals for Weibull distributed glass strength data
This standard describes procedures of evaluation of sample data by means of a two-parameter WEIBULL distribution function.
Glas im Bauwesen - Bestimmung der Biegefestigkeit von Glas - Schätzverfahren und Bestimmung der Vertrauensbereiche für Daten mit Weibull-Verteilung
Die vorliegende Europäische Norm legt Verfahren für die Auswertung von Stichprobenergebnissen mit der zweiparametrigen Weibull Verteilungsfunktion fest.
Verre dans la construction - Procédures de validité de l'ajustement et intervalles de confiance des données de résistance du verre au moyen de la loi de Weibull
La présente Norme européenne spécifie les procédures d'évaluation des données d'échantillon au moyen d'une fonction de la loi de Weibull a deux parametres.
Steklo v stavbah – Postopki ugotavljanja upogibne trdnosti in intervala zaupanja za podatke z Weibullovo porazdelitvijo
General Information
Standards Content (Sample)
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.Steklo v stavbah – Postopki ugotavljanja upogibne trdnosti in intervala zaupanja za podatke z Weibullovo porazdelitvijoGlas im Bauwesen - Bestimmung der Biegefestigkeit von Glas - Schätzverfahren und Bestimmung der Vertrauensbereiche für Daten mit Weibull-VerteilungVerre dans la construction - Procédures de validité de l'ajustement et intervalles de confiance des données de résistance du verre au moyen de la loi de WeibullGlass in building - Procedures for goodness of fit and confidence intervals for Weibull distributed glass strength data81.040.20Steklo v gradbeništvuGlass in buildingICS:Ta slovenski standard je istoveten z:EN 12603:2002SIST EN 12603:2004en01-september-2004SIST EN 12603:2004SLOVENSKI
STANDARD
SIST EN 12603:2004
EUROPEAN STANDARDNORME EUROPÉENNEEUROPÄISCHE NORMEN 12603November 2002ICS 81.040.20English versionGlass in building - Procedures for goodness of fit andconfidence intervals for Weibull distributed glass strength dataVerre dans la construction - Procédures de validité del'ajustement et intervalles de confiance des données derésistance du verre au moyen de la loi de WeibullGlas im Bauwesen - Bestimmung der Biegefestigkeit vonGlas - Schätzverfahren und Bestimmung derVertrauensbereiche für Daten mit Weibull-VerteilungThis European Standard was approved by CEN on 7 September 2002.CEN members are bound to comply with the CEN/CENELEC Internal Regulations which stipulate the conditions for giving this EuropeanStandard the status of a national standard without any alteration. Up-to-date lists and bibliographical references concerning such nationalstandards may be obtained on application to the Management Centre or to any CEN member.This European Standard exists in three official versions (English, French, German). A version in any other language made by translationunder the responsibility of a CEN member into its own language and notified to the Management Centre has the same status as the officialversions.CEN members are the national standards bodies of Austria, Belgium, Czech Republic, Denmark, Finland, France, Germany, Greece,Iceland, Ireland, Italy, Luxembourg, Malta, Netherlands, Norway, Portugal, Spain, Sweden, Switzerland and United Kingdom.EUROPEAN COMMITTEE FOR STANDARDIZATIONCOMITÉ EUROPÉEN DE NORMALISATIONEUROPÄISCHES KOMITEE FÜR NORMUNGManagement Centre: rue de Stassart, 36
B-1050 Brussels© 2002 CENAll rights of exploitation in any form and by any means reservedworldwide for CEN national Members.Ref. No. EN 12603:2002 ESIST EN 12603:2004
EN 12603:2002 (E)2ContentspageForeword.3Introduction.41Scope.52Normative references.53Terms and definitions.54Symbols and abbreviated terms.55Goodness of fit.66Point estimators for the parameters bb and qq of the distribution.76.1Censored sample.76.2Uncensored (complete) sample.97Assessment of data and tests.117.1The Weibull diagram.117.2Graphical representation of the estimated distribution function.117.3Plotting of sample data in the Weibull diagram.117.3.1Single values.117.3.2Classified values.127.4Assessment of sample data.128Confidence intervals.128.1Confidence interval for the shape parameter bb.128.2Confidence interval for the value of the distribution function G(x) at a given value of x, of theattribute X.158.3Confidence interval for the scale parameter qq.188.3.1Method for all samples.188.3.2Method for uncensored samples.188.4Confidence interval for the value x of the attribute X at a given value G(x) of the distributionfunction.218.4.1Method for all samples.218.4.2Method for uncensored samples.22Annex A (informative)
Examples.23A.1Uncensored sample.23A.1.1Data.23A.1.2Statistical evaluation.24A.2Censored sample.27A.2.1Data.27A.2.2Statistical evaluation.29Annex B (informative)
Weibull graph.32Bibliography.33SIST EN 12603:2004
EN 12603:2002 (E)3ForewordThis document (EN 12603:2002) has been prepared by Technical Committee CEN/TC 129 "Glass in building", thesecretariat of which is held by IBN.This European Standard shall be given the status of a national standard, either by publication of an identical text orby endorsement, at the latest by May 2003, and conflicting national standards shall be withdrawn at the latest byMay 2003.In this standard the annexes A, B and C are informative.According to the CEN/CENELEC Internal Regulations, the national standards organizations of the followingcountries are bound to implement this European Standard: Austria, Belgium, Czech Republic, Denmark, Finland,France, Germany, Greece, Iceland, Ireland, Italy, Luxembourg, Malta, Netherlands, Norway, Portugal, Spain,Sweden, Switzerland and the United Kingdom.SIST EN 12603:2004
EN 12603:2002 (E)4IntroductionThis European Standard is based on the assumption that the statistical distribution of the attribute taken intoconsideration can be represented by one single Weibull distribution function, even where in certain cases (e.g. lifetimemeasurements) mixed distributions have frequently been observed.
For this reason, the user of the standard has tocheck by a goodness of fit test whether the measured data of a sample can be represented by means of one singleWeibull function.
Only in this case can the hypothesis be accepted and the procedures described in this standard beapplied.The user decides on this question also considering all previous relevant data and the general state of knowledge in thespecial field.
Every extrapolation into ranges of fractiles not confirmed by measured values requires utmost care, themore so the farther the extrapolation exceeds the range of measurements.NOTEThe three-parameter Weibull function is:úúûùêêëé÷øöçèæ---=bq0exp1)(xxxG(1)If xo = 0 is assumed, the two-parameter Weibull function results:úúûùêêëé÷øöçèæ--=bqxxGexp1)((2)which can be written as:bq1)(11lnúûùêëé÷÷øöççèæ-=xGx(3)The calculation can be based either on an uncensored or a censored sample.
There are several methods of censoring.In this standard only the following method of censoring is considered:- given a number r < n of specimens of which attribute values xi were measured.SIST EN 12603:2004
EN 12603:2002 (E)51 ScopeThis European Standard specifies procedures for the evaluation of sample data by means of a two-parameter Weibulldistribution function.2 Normative referencesThis European Standard incorporates by dated or undated reference, provisions from other publications.
Thesenormative references are cited at the appropriate places in the text, and the publications are listed hereafter. For datedreferences, subsequent amendments to or revisions of any of these publications apply to this European Standard onlywhen incorporated in its amendment or revision.
For undated reference, the latest edition of the publications referred toapplies (including amendments).ISO 2854:1976, Statistical interpretation of data - Techniques of estimation and tests relating to means and variances.ISO 3534, Statistics - Vocabulary and symbols.3 Terms and definitionsFor the purposes of this European Standard, the terms and definitions given in ISO 3534 apply.4 Symbols and abbreviated termsXattribute taken into consideration;x, xi, xrvalues of X;G(x)distribution function of X = percentage of failure;xo, b, qparameters of the three-parameter Weibull function;^identification label for point estimators (e.g. bˆ, qˆ, Gˆ);1-aconfidence level;ivalue used in the goodness of fit test;Lvalue used in the goodness of fit test;nsample size;rnumber of specimens of which attribute values xi were measured;NOTEThe sample is ordered, i.e. x1 £ x2 £ x3 .
£ xrr
£
n;f,f1,f2degrees of freedom;kn,kr;nfactors used in estimating bˆ;SIST EN 12603:2004
EN 12603:2002 (E)6Cr;nfactor used in estimating qˆ;sint(0,84n) = largest integer < 0,84n ;h,xordinate and abcissa of the Weibull diagram;c2chi-square distribution function;y,v,gauxiliary factors used in estimating the confidence limits of G(x);A,B,Cconstants used in evaluating v ;H(f2)variable used in evaluating g ;Tn;a/2,Tn;1-a/2coefficients used in estimating the confidence limits of q ;Subscripts:unlower confidence limit;obupper confidence limit;zconfidence interval limited on two sides.5 Goodness of fitSort the r values of x into rank ascending order.Compute for each value from i = 1 to i = r - 1:()úúúúûùêêêêëé÷øöçèæ++-÷øöçèæ++---=+1434ln143)1(4lnln)ln()ln(1ninninxxiii(4)Compute the quantity:()ëûëûëûëûåå=-+=-=2/1112/2/2/1riirriirrL(5)where the symbol ëû2/r is used to denote the largest integer less than or equal to r/2.Reject the hypothesis that the data is from a Weibull distribution at the a significance level if:SIST EN 12603:2004
EN 12603:2002 (E)7()ëûëû()2/2,2/12rrFL-³a(6)The values of the fractiles of the F distribution can be found for example in Table IV of ISO 2854:1976.6 Point estimators for the parameters b and q of the distribution6.1 Censored samplex-xrn=ir=irr;nlnlnˆ1åkb(7)úûùêëébqˆ1lnexpˆC-x=nr;r(8)The factors kr;n and Cr;n are listed in Table 1 and Table 2.SIST EN 12603:2004
EN 12603:2002 (E)8Table 1 — Coefficient kkr;nnr/n0,10,20,30,40,50,60,70,80,950,22310,48130,8018100,10540,21720,33690,46670,60980,77150,96161,202200,05130,15830,27210,39440,52770,67560,84481,0481,316300,06840,17590,29040,41370,54820,69790,86971,0771,357400,07700,18480,29960,42330,55840,70900,88221,0921,378500,08210,19010,30510,42910,56460,71580,88981,1011,391600,08550,19360,30880,43300,56870,72020,89491,1081,400700,08790,19610,31140,43570,57170,72350,89851,1121,406800,08980,19800,31340,43780,57390,72590,90121,1151,410900,09120,19950,31490,43940,57560,72770,90331,1181,4141000,09240,20070,31620,44070,57700,72920,90501,1201,417kp0,102650,211290,327230,452340,589370,742740,920261,13821,4436d1-1,0271-1,0622-1,1060-1,1634-1,2415-1,3540-1,5313-1,8567-2,6929d20,0000,0300,0540,0890,1450,2420,4330,9062,796Asymptotic estimate for large n : kr,n = kp + d1/n + d2/n2SIST EN 12603:2004
EN 12603:2002 (E)9Table 2 — Coefficient Cr,nnr/n0,10,20,30,40,50,60,70,80,910-2,880-1,826-1,267-0,8681-0,5436-0,2574 0,01200,28370,584620-2,547-1,658-1,147-0,7691-0,4548-0,17270,09790,37760,702230-2,444-1,605-1,108-0,7364-0,4253-0,14430,12690,40980,744640-2,394-1,578-1,089-0,7202-0,4106-0,13010,14150,42620,766450-2,365-1,562-1,077-0,7105-0,4018-0,12160,15030,43600,779660-2,345-1,522-1,069-0,7040-0,3959-0,11590,15620,44260,788570-2,331-1,544-1,064-0,6994-0,3917-0,11180,16040,44730,794980-2,321-1,539-1,060-0,6959-0,3886-0,10880,16350,45090,799890-2,313-1,534-1,056-0,6932-0,3861-0,10640,16600,45370,8035100-2,307-1,531-1,054-0,6911-0,3841-0,10450,16790,45590,8065cp-2,2504-1,4999-1,0309-0,67173-0,36651-0,087420,185630,475890,83403a1-5,5743-3,0740-2,2859-1,9301-1,7619-1,7114-1,7727-2,0110-2,7773a2-7,201-1,886-0, 767-0,335-0,0910,1110,3690,8912,825Asymptotic estimate for large n : Cr,n = cp + a1/n + a2/n26.2 Uncensored (complete) samplex-xs-nsn=is=iin+s=inlnlnˆ11ååkb(10)úûùêëé+å=niixn=115772,0ln1expˆbq(11)The factors kn are listed in Table 3.SIST EN 12603:2004
EN 12603:2002 (E)10Table 3 — Coefficient kknnknnkn20,6931321,466530,9808331,479541,1507341,492051,2674351,504061,3545361,515671,1828371,526681,2547381,479591,3141391,4904101,3644401,5009111,4079411,5110121,4461421,5208131,3332431,5303141,3686441,4891151,4004451,4984161,4293461,5075171,4556471,5163181,4799481,5248191,3960491,5331201,4192501,5411211,4408511,5046221,4609521,5126231,4797531,5204241,4975541,5279251,5142551,5352261,4479561,5424271,4642571,5096SIST EN 12603:2004
EN 12603:2002 (E)11Table 3 (continued)281,4796581,5167291,4943591,5236301,5083601,5304311,5216¥1,56927 Assessment of data and tests7.1 The Weibull diagramThe probability diagram for the Weibull distribution is drawn up in such a way that the distribution function of a two-parameter Weibull distribution is represented by a straight line.The ordinate axis is graduated according to the function÷÷øöççèæ÷÷øöççèæG(x)-=11lnlnh(12)and the abscissa axis according to the functionx=lnx or x=logx(13)NOTESuch forms are available.
As a rule, diagrams should be used with a range of G-values from G = 1 ´ 10-3 = 0,1 % toG = 0,999 = 99,9 %.
The necessary range of x-values depends on the value b of the shape parameter.7.2 Graphical representation of the estimated distribution functionThe point estimators of the shape parameter b and the scale parameter q define a straight line in the Weibull diagram; itis appropriate to define this straight line though the following two points:qˆ=x%21,636321,0)(==xG(14)bq101005,0ˆ´=x%101,0)(==xG(15)This straight line shall be plotted into the diagram.7.3 Plotting of sample data in the Weibull diagram7.3.1 Single valuesMeasurements of a censored or uncensored sample yield r or n values xi, respectively, of the attribute X.
These valuesxi shall be ordered to make up an Ordered Sample.SIST EN 12603:2004
EN 12603:2002 (E)12Each value xi of the Ordered Sample shall be co-ordinated to an estimated value:4,03,0ˆ+n-i=)x(Gi(16)This way the points representing the measured values of the sample shall be plotted into the Weibull diagram.7.3.2 Classified valuesIn the case of a very large sample, the range of measured x-values can be subdivided into classes, usually containingthe same number of values.
The proportion of x-values summed up in any class considered shall be plotted at theupper limit of that class.7.4 Assessment of sample dataThe straight line plotted according to 7.2 and the points which represent the measured values of the sample,plotted according to 7.3 can be compared visually.Systematic deviations can examined in detail taking into consideration the general knowledge of the basic technicaland scientific facts and the results of previous relevant research.
For instance, if the distribution of the attribute valuescan be approximated by segments of straight lines with different slopes, a mixed Weibull distribution may be assumed.This can be taken as a hint that several basic mechanisms determine the attribute values xi.
Such a detailedexamination is beyond the scope of this standard.8 Confidence intervalsThe equations of the following sub-clauses are valid for the case that the confidence intervals are limited on two sides(subscript z).
Where the confidence intervals are limited only on one side, a/2 shall be replaced by a in the followingequations.The confidence level (1 - a) is to be chosen by the user of this standard.8.1 Confidence interval for the shape parameter bbThe upper limit of the confidence interval for the shape parameter b at the confidence level (1 - a)
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