WARWICK MANUFACTURING GROUP Product Excellence using 6 Sigma (PEUSS) WWeeiibbuullll aannaallyyssiiss Se Sec c t t i i o on n 88 Warwick Manufacturing Group THE USE OF WEIBULL IN DEFECT DATA ANALYSIS Contents 1Introduction1 2Data1 3The mechanics of Weibull analysis5 4Interpretation of Weibull output8 5Practical difficulties with Weibull plotting15 6Comparison with hazard plotting20 7Conclusions20 8References21 9ANNEX A two cycle Weibull paper22 10ANNEX B Progressive example of Weibull plotting23 11ANNEX C Estimation of Weibull location parameter35 12ANNEX D Example of a 3-parameter Weibull plot36 13ANNEX E the effect of scatter40 13ANNEX E the effect of scatter40 14ANNEX F 95% confidence limits for Weibull42 15ANNEX G Weibull plot of multiply censored data44 15ANNEX G Weibull plot of multiply censored data44 Warwick Manufacturing Group Copyright 2007 University of Warwick Warwick Manufacturing Group The use of Weibull in defect data analysis Page 1 THE USE OF WEIBULL IN DEFECT DATA ANALYSIS 1Introduction ThesenotesgiveabriefintroductiontoWeibullanalysisanditspotentialcontributionto equipment maintenance and lifing policies. Statistical terminology has been avoided wherever possibleandthosetermswhichareusedareexplained,albeitbriefly.Weibullanalysis originated from a paper [1] published in 1951 by a Swedish mechanical engineer, Professor WaloddiWeibull.Hisoriginalpaperdidlittlemorethanproposeamulti-parameter distribution, but it became widely appreciated and was shown by Pratt and Whitney in 1967 to have some application to the analysis of defect data. 1.1Information sourcesThe definitive statistical text on Weibull is cited at [2], and publications closer to the working level are given at [3] and [4]. A set of British Standards, BS 5760 Parts 1 to 3 cover a broad spectrum of reliability activities. Part 1 on Reliability Programme Management was issued in 1979butisoflittlevaluehereexceptforitscommentsonthedifficultiesofobtaining adequate data. Part 2 [5] contains valuable guidance for the application of Weibull analysis although this may be difficult to extract. The third of the Standard contains authentic practical examplesillustratingtheprinciplesestablishedinParts1and2.Onefurthersourceof information is an I Mech E paper by Sherwin and Lees [6]. Part 1 of this paper is a good review of current Weibull theory and Part 2 provides some insight into the practical problems inherent in its use. 1.2Application to sampled defect data It is important to define the context in which the following Weibull analysis may be used. All thatisstatedsubsequentlyisapplicabletosampleddefectdata.Thisisaverydifferent situation to that which exists on, say, the RB-211 for which Rolls Royce has a complete data base.Theyknowatanytimethelifedistributionofallthein-serviceenginesandtheir components, and their analysis can be done from knowledge of the utilizations at failure and the current utilisation for all the non-failed components. Their form of Weibull analysis is unique to this situation of total visibility. It is assumed here, however, that most organisations are not in this fortunate position; their data will at best be of some representative sample of the failures which have occurred, and of utilization of unfailed units. It cannot be stressed too highly, though, that life of unfailed units must be known if a realistic estimate of lifetimes to failure is to be made, and, therefore, data must be collected on unfailed units in the sample. 2Data The basic elements in defect data analysis comprise: a population, from which some sample is taken in the form of times to failure (here time is taken to mean any appropriate measure of utilisation),Warwick Manufacturing Group The use of Weibull in defect data analysis Page 2 an analytical technique such as Weibull which is then applied to the sample of failure data to derive a mathematical model for the behaviour of the sample, and hopefully of the population also, and finallysomedeductionswhicharegeneratedbyanexaminationofthemodel.These deductions will influence the decisions to be made about the maintenance strategy for the population.The most difficult part of this process is the acquisition of trustworthy data. No amount of elegance in the statistical treatment of the data will enable sound judgements to be made from invalid data. Weibull analysis requires times to failure. This is higher quality data than knowledge of the number of failures in an interval. A failure must be a defined event and preferably objective ratherthansomesubjectivelyassesseddegradationinperformance.Atypicalsample, therefore,mightatitsmostsuperficiallevelcompriseacollectionofindividualtimesto failure for the equipment under investigation. 2.1Quality of dataThe quality of data is a most difficult feature to assess and yet its importance cannot be over-stated.Whenthereisachoicebetweenarelativelylargeamountofdubiousdataanda relatively small amount of sound data, the latter is always preferred. The quality problem has several facets: The data should be a statistically random sample of the population. Exactly what this means in terms of the hardware will differ in each case. Clearly the modification state of equipments may be relevant to the failures being experienced and failure data which cannotbeallocatedtooneorothermodificationislikelytobemisleading.Byan examination of the source of the data the user must satisfy himself that it contains no bias,orelserecognisesuchabiasandconfinethedeductionsaccordingly.For example, data obtained from one user unit for an item experiencing failures of a nature whichmaybeinfluencedbythequalityofmaintenance,localoperating conditions/practices or any other idiosyncrasy of that unit may be used providing the conclusions drawn are suitably confined to the unit concerned. A less obvious data quality problem concerns the measure of utilisation to be used; it must not only be the appropriate one for the equipment as a whole, but it must also be appropriate for the major failure modes. As will be seen later, an analysis at equipment levelcanbetotallymisleadingifthereareseveralsignificantfailuremodeseach exhibiting their own type of behaviour. The view of the problem at equipment level may give a misleading indication of the counter-strategies to be employed. The more meaningfuldeeperexaminationwillnotbepossibleunlessthedatacontainsmode information at the right depth and degree of integrity. Itisnecessarytoknowanyotherdetailswhichmayhaveabearingonthefailure sensitivity of the equipment; for example the installed position of the failures which Warwick Manufacturing Group The use of Weibull in defect data analysis Page 3 comprise the sample. There are many factors which may render elements of a sample unrepresentative including such things as misuse or incorrect diagnosis. 2.2Quantity of dataWhereas the effects of poor quality are insidious, the effects of inadequate quantity of data are more apparent and can, in part, be countered. To see how this may be done it is necessary to examineoneofthestatisticalcharacteristicsusedinWeibullanalysis.Anequipment undergoing in-service failures will exhibit a cumulative distribution function (F(t)), which is the distribution in time of the cumulative failure pattern or cumulative percent failed as a function of time, as indicated by the sample. Consider a sample of 5 failures (sample size n =5). The symbol i is used to indicate the failure number once the failure times are ranked in ascending order; so here i will take the integer values 1 to 5 inclusive. Suppose the 5 failure times are 2, 7, 13, 19 and 27 cycles. Now the first failure at 2 cycles may be thought to correspond to an F(t) value of i/n, where i =1 and n =5. ie F(t) @ 2 cycles =1/5or 0.2 or 20% Similarly for the second failure time of 7 cycles, the corresponding F(t) is 40% and so on. On this basis, this data is suggesting that the fifth failure at 27 cycles corresponds to a cumulative percent failed of 100%. In other words, on the basis of this sample, 100% of the population will fail by 27 cycles. Clearly this is unrealistic. A further sample of 10 items may contain one or more which exceed a 27 cycle life. A much larger sample of 1000 items may well indicate thatratherthancorrespondtoa100%cumulativefailure,27cyclescorrespondstosome lesser cumulative failure of any 85 or 90%.This problem of small sample bias is best overcome as follows: Sample Size Less Than 50. A table of Median Ranks has been calculated which gives a best estimate of the F(t) value corresponding to each failure time in the sample. This table is issued with these notes. It indicates that in the example just considered, the F(t) values corresponding to the 5 ascending failure times quoted are not 20%, 40%, 60%, 80% and 100%, but are 12.9%, 31.4%, 50%, 68.6% and 87.1%. It is this latter set of F(t) use values which should be plotted against the corresponding ranked failure times on a Weibull plot. Median rank values give the best estimate for the primary Weibull parameter and are best suited to some later work on confidence limits. Sample Size Less Than 100. For sample sizes less than 100, in the absence of Median Ranktablesthetruemedianrankvaluescanbeadequatelyapproximatedusing Bernards Approximation: ( 0.3)( )( 0.4)iF tn=+ Sample Sizes Greater Than 100. Above a sample size of about 100 the problem of smallsamplebiasisinsignificantandtheF(t)valuesmaybecalculatedfromthe expression for the Mean Ranks: Warwick Manufacturing Group The use of Weibull in defect data analysis Page 4 ( )( 1)iF tn=+ 2.3Trends in data A trend may be in relation to any other time-base than the one being used to assess reliability.For instance, if the reliability of vehicle engines is being assessed against vehicle miles, and a trend is observed in respect to the date of engine manufacture, or in respect of calendar time, then this is evidence that there is a trend.Before attempting to use a mathematical model such as weibul analysis is important to check thehomogeneityofthedata.Ifthedataindicatesthepresenceofatimeseries,itis inappropriate to apply a model that requires homogeneity.A simple approach to this would betochartthefailuredataintheformofcumulativefailuresagainstcumulativetime. Deviation from a straight line would indicate a trend. A mathematical approach to this would be to employ the Laplace Trend Test. 2.3.1Laplace trend test The Laplace Trend test assesses the distribution of failure events with respect to the ordering indicated by the criteria being tested.The Trend ordering may simply be that the failures occurred in a specific order, rather than randomly. For instance, if the date of engine manufacture is being considered as an ordering criteria, then the failure times should be put on a time-line as follows, as ordered by date of engine manufacture. Common sense will indicate that the later manufactured engines are not lasting as long to failure as the first.Therefore, there is clear evidence of a trend. However, to put this on a statistical basis, the Laplace Trend Test is used: ( )( )Test St at i st i c U N ttN t t, . = 12 05where U =standardised normal deviateif the calculated value is less than that taken from tables appropriate to the confidence required (e.g 95%), then trend is not proven.If it is greater than the tabled value, then trend is demonstrated. t =sum of failure times (on cumulative scale: see diagram above) Warwick Manufacturing Group The use of Weibull in defect data analysis Page 5 N(t) =total number of failures t =total test time Crudely, ( )tN t t 0.5 indicates increasing failure rate 2.3.2Amendment for Failure Terminated Data Wheredatahasbeencollected,terminatingcollectionatthepointofafailure,thenthe Laplace Trend Test is amended as follows: ( )( )Test St at i st i c U N ttiN t tninn n, . ==12105111 where the last failure is not counted, whereas the final time interval is counted. 3The mechanics of Weibull analysis 3.1The value of analysisOn occasions, an analysis of the data reveals little that was not apparent from engineering judgement applied to the physics of the failures and an examination of the raw data. However, on other occasions, the true behaviour of equipments can be obscured when viewed by the mostexperiencedassessor.Itisalwaysnecessarytokeepabalancebetweendeductions drawn from data analysis and those which arise from an examination of the mechanics of failure. Ideally, these should be suggesting complementary rather than conflicting counter-strategies to unreliability. There are many reliability characteristics of an item which may be of interest and significantly more reliability measures or parameters which can be used to describe those characteristics. Weibull will provide meaningful information on two such characteristics. First, it will give some measure of how failures are distributed with time. Second, it will indicate the hazard regimeforthefailuresunderconsideration.Thesignificanceofthesetwomeasuresof reliability is described later. Weibullisa3-parameterdistributionwhichhasthegreatstrengthofbeingsufficiently flexible to encompass almost all the failure distributions found in practice, and hence provides Warwick Manufacturing Group The use of Weibull in defect data analysis Page 6 information on the 3 failure regimes normally encountered. Weibull analysis is primarily a graphicaltechniquealthoughitcanbedoneanalytically.Thedangerintheanalytical approach is that it takes away the picture and replaces it with apparent precision in terms of the evaluated parameters. However, this is generally considered to be a poor practice since it eliminatesthejudgementandexperienceoftheplotter.Weibullplotsareoftenusedto provide a broad feel for the nature of the failures; this is why, to some extent, it is a nonsense to worry about errors of about 1% when using Bernards approximation, when the process of plotting the points and fitting the best straight line will probably involve significantly larger errors. However, the aim is to appreciate in broad terms how the equipment is behaving. Weibull can make such profound statements about an equipments behaviour that 5% may be relatively trivial. 3.2Evaluating the Weibull parametersThe first stage of Weibull analysis once the data has been obtained is the estimation of the 3 Weibull parameters: :Shape parameter. : Scale parameter or characteristic life. : Location parameter or minimum life. The general expression for the Weibull F(t) is: =) (1 ) (te t F This can be transformed into: log ) log()) ( 1 (1log log =tt F It follows that if F(t) can be plotted against t (corresponding failure times) on paper which has a reciprocal double log scale on one axis and a log scale on the other, and that data forms a straight line, then the data can be modelled by Weibull and the parameters extracted from the plot. A piece of 2 cycle Weibull paper (Chartwell Graph Data Ref C6572) is shown at Annex A and this is simply a piece of graph paper constructed such that its vertical scale is a double log reciprocal and its horizontal scale is a conventional log. The mechanics of the plot are described progressively using the following example and the associated illustrations in plots 2 to 12 of Annex B. Consider the following times to failure for a sample of 10 items: 410, 1050, 825, 300, 660, 900, 500, 1200, 750 and 600 hours. Warwick Manufacturing Group The use of Weibull in defect data analysis Page 7 Assemble the data in ascending order and tabulate it against the corresponding F(t) values for a sample size of 10, obtained from the Median Rank tables. The tabulation is shown at Section 16. Mark the appropriate time scale on the horizontal axis on a piece of Weibull paper (plot 2). Plot on the Weibull paper the ranked hours at failure (ti) on the horizontal axis against the corresponding F(t) value on the vertical axis (plot 3). If the points constitute a reasonable straight line then construct that line. Note that real data frequently snakes about the straight line due to scatter in the data; this is not a problemprovidingthesnakingmotionisclearlytoeithersideoftheline.When determining the position of the line give more weight to the later points rather than the early ones; this is necessary both because of the effects of cumulation and because the Weibullpapertendstogiveadisproportionateemphasistotheearlypointswhich should be countered where these are at variance with the subsequent points. Do not attempt to draw more than one straight line through the data and do not construct a straight line where there is manifestly a curve. In this example the fitting of the line presentsnoproblem(plot4).Notealsothatonthematterofhowmuchdatais required for a Weibull plot that any 4 or so of the pieces of data used here would give anadequatestraightline.Insuchcircumstances4pointsmaywellbeenough. Generally, 7 or so points would be a reasonable minimum, depending on their shape once plotted. Thefactthatthedataproducedastraightlinewheninitiallyplottedenables2 statements to be made: oThe data can apparently be modelled by the Weibull distribution. oThelocationparameterorminimumlife()isapproximatelyzero.This parameter is discussed later. At plot 5 a scale for the estimate of the Shape Parameter , is highlighted. This scale can be seen to range from 0.5 to 5, although values outside this range are possible. The next step is to construct a perpendicular from the Estimation Point in the top left hand corner of the paper to the plotted line (plot 6). The estimated value of , termed, is given by the intersection of the constructed perpendicular and the scale. In this example, is about 2.4 (plot 7). Atplot8adottedhorizontallineishighlightedcorrespondingtoanF(t)valueof 63.2%.Nowthescaleparameterorcharacteristiclifeestimateisthelifewhich corresponds to a cumulative mortality of 63.2% of the population. Hence to determine itsvalueitisnecessaryonlytofollowtheEstimatorlinehorizontallyuntilit intersects the plotted line and then read off the corresponding time on the lower scale. Warwick Manufacturing Group The use of Weibull in defect data analysis Page 8 Plot 9 shows that, based on this sample, these components have a characteristic life of about 830 hours. By this time 63.2% of them will have failed. At plot 10 the evaluation of the proportion failed corresponding to the mean of the distributionofthetimestofailure(P)isshowntobe52.7%usingthepointof intersection of the perpendicular and the P scale. This value is inserted in the F(t) scale and its intersection with the plotted line determines the estimated mean of the distribution of the times to failure ( ). In this case this is about 740 hours. The median life can also be easily extracted; that is to say the life corresponding to 50% mortality. This is shown at plot 11 to be about 720 hours, based on this sample. Finally, plot 12 illustrates that this data is indicating that a 400 hour life would result in about 15% of in-service failures for these equipments. Conversely, an acceptable level of in-service failure may be converted into a life; for example it can be seen from plot 12 that an acceptable level of in-service failure of say, 30% would correspond to a life of about 550 hours, and so on. 4Interpretation of Weibull output 4.1Concept of hazardBefore examining the significance of the Weibull shape parameter it is necessary to know somethingoftheconceptofhazardandthe3 so-calledfailureregimes.Theparameterof interest here is the hazard rate, h(t). This is the conditional probability that an equipment will fail in a given interval of unit time given that it has survived until that interval of time. It is, therefore, the instantaneous failure rate and can in general be thought of as a measure of the probability of failure, where this probability varies with the time the item has been in service. The3failureregimesaredefinedintermsofhazardrateandnot,asisacommon misconception, in terms of failure rate. The 3 regimes are often thought of in the form of the so-called bath-tub curve; this is a valid concept for the behaviour of a system over its whole life but is a misleading model for the vast majority of components and, more importantly, their individual failure modes (see [5] and [7]).Anindividualmodeisunlikelytoexhibitmorethanoneofthe3characteristicsof decreasing, constant or increasing hazard. 4.1.1Shape parameter less than unity.A value of less than unity indicates that the item or failure mode may be characterised by the first regime of decreasing hazard. This is sometimes termed the early failure or infant mortalityperiodanditisacommonfallacythatsuchfailuresareunavoidable.The distributionoftimestofailurewillfollowahyper-exponentialdistributioninwhichthe instantaneous probability of failure is decreasing with time in service. This hyper-exponential distribution models a concentration of failure times at each end of the time scale; many items fail early or else go on to a substantial life, whilst relatively few fail between the extremes. The extent to which is below 1 is a measure of the severity of the early failures; 0.9 for Warwick Manufacturing Group The use of Weibull in defect data analysis Page 9 example would be a relatively weak early failure effect, particularly if the sample size and therefore the confidence, was low. If there is a single or a predominant failure mode with a