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Table of Contents

THE USE OF WEIBULL IN DEFECT DATAANALYSIS

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Information Sources . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Application to Sampled Defect Data . . . . . . . . . . . . . . . . . 1

DATA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Quality of Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Quantity of Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

THE MECHANICS OF WEIBULL ANALYSIS . . . . . . . . . . . . . . 4

The Value of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 4

Evaluating the Weibull Parameters . . . . . . . . . . . . . . . . . . 5

INTERPRETATION OF WEIBULL OUTPUT . . . . . . . . . . . . . . 7

Concept of Hazard . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Scale Parameter or Characteristic LIfe . . . . . . . . . . . . . . . . 10

Location Parameter or Minimum Life . . . . . . . . . . . . . . . . 11

PRACTICAL DIFFICULTIES WITH WEIBULL PLOTTING. . . . . . . 13

Scatter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Extrapolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Multi-Modal Failures . . . . . . . . . . . . . . . . . . . . . . . . . 13

Confidence Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Censoring of Sample Data . . . . . . . . . . . . . . . . . . . . . . . 15

COMPARISON WITH HAZARD PLOTTING . . . . . . . . . . . . . . 16

CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

TWO CYCLE WEIBULL PAPER . . . . . . . . . . . . . . . . . . 18

PROGRESSIVE EXAMPLE OF WEIBULL PLOTTING . . . . . . 20

ESTIMATION OF WEIBULL LOCATION PARAMETER. . . . 31

EXAMPLE OF A 3-PARAMETER WEIBULL PLOT. . . . . . . . . . 32

THE EFFECT OF SCATTER . . . . . . . . . . . . . . . . . . . . . . 37

95% CONFIDENCE LIMITS FOR WEIBULL . . . . . . . . . . . . . 39

WEIBULL PLOT OF MULTIPLY-CENSORED DATA . . . . . . . . 42

THE USE OF WEIBULL IN DEFECT DATAANALYSIS

1 INTRODUCTION

These notes give a brief introduction to Weibull analysis and its potential contribution toequipment maintenance and lifing policies. Statistical terminology has been avoided wher-ever possible and those terms which are used are explained, albeit briefly. Weibull analysisoriginated from a paper, Reference 1, published in 1951 by a Swedish mechanical engineer, Pro-fessor Waloddi Weibull. His original paper did little more than propose a multi-parameter dis-tribution, but it became widely appreciated and was shown by Pratt and Whitney in 1967 tohave some application to the analysis of defect data.

1.1 Information Sources

The definitive statistical text on Weibull is cited at Reference 2, and publications closer to theworking level are given at Reference 3 and 4. A set of British Standards, BS 5760 Parts 1 to 3 cov-ering a broad spectrum of reliability activities are being issued. Part 1 on Reliability Pro-gramme Management was issued in 1979 but is of little value here except for its comments onthe difficulties of obtaining adequate data. Part 2, Reference 5, contains valuable guidance forthe application of Weibull analysis although this may be difficult to extract. The third of theStandard contains authentic practical examples illustrating the principles established in Parts1 and 2. One further source of information is an I Mech E paper by Sherwin and Lees at Refer-ence 6. Part 1 of this paper is a good review of current Weibull theory and Part 2 provides someinsight into the practical problems inherent in its use.

1.2 Application to Sampled Defect Data

It is important to define the context in which the following Weibull analysis may be used. Allthat is stated subsequently is applicable to sampled defect data. This is a very different situa-tion to that which exists on, say, the RB-211 for which Rolls Royce has a complete data base.They know at any time the life distribution of all the in-service engines and their components,and their analysis can be done from a knowledge of the utilizations at failure and the currentutilisation for all the non-failed components. Their form of Weibull analysis is unique to thissituation of total visibility. It is assumed here, however, that most organisations are not inthis fortunate position; their data will at best be of some representative sample of the failureswhich have occurred, and of utilization of unfailed units. It is stressed too highly, though, that

Warwick Manufacturing Group 1Revision date: 6 December, 2004

THE USE OF WEIBULL IN DEFECT DATA ANALYSIS

life of unfailed units must be known if a realistic estimate of lifetimes to failure is to bemade, and, therefore, data must be collected on unfailed units in the sample.life of unfailed units must be known if a realistic estimate of lifetimes to failure is to bemade, and, therefore, data must be collected on unfailed units in the sample.

2 DATA

The basic elements in defect data analysis comprise:

• a population, from which some sample is taken in the form of times to failure (heretime is taken to mean any appropriate measure of utilisation),

• an analytical technique such as Weibull which is then applied to the sample offailure data to derive a mathematical model for the behaviour of the sample, andhopefully of the population also, and finally

• some deductions which are generated by an examination of the model. Thesedeductions will influence the decisions to be made about the maintenance strategyfor the population.

The most difficult part of this process is the acquisition of trustworthy data. No amount of ele-gance in the statistical treatment of the data will enable sound judgements to be made frominvalid data.

Weibull analysis requires times to failure. This is higher quality data than a knowledge of thenumber of failures in an interval. A failure must be a defined event and preferably objectiverather than some subjectively assessed degradation in performance. A typical sample, there-fore, might at its most superficial level comprise a collection of individual times to failure forthe equipment under investigation.

2.1 Quality of Data

The quality of data is a most difficult feature to assess and yet its importance cannot be over-stated. When there is a choice between a relatively large amount of dubious data and a rela-tively small amount of sound data, the latter is always preferred. The quality problem hasseveral facets:

• The data should be a statistically random sample of the population. Exactly whatthis means in terms of the hardware will differ in each case. Clearly the modifica-tion state of equipments may be relevant to the failures being experienced andfailure data which cannot be allocated to one or other modification is likely to bemisleading. By an examination of the source of the data the user must satisfyhimself that it contains no bias, or else recognise such a bias and confine the deduc-tions accordingly. For example, data obtained from one user unit for an item expe-riencing failures of a nature which may be influenced by the quality ofmaintenance, local operating conditions/practices or any other idiosyncrasy of thatunit may be used providing the conclusions drawn are suitably confined to the unitconcerned.



• 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 mustalso be appropriate for the major failure modes. As will be seen later, an analysis atequipment level can be totally misleading if there are several significant failuremodes each exhibiting their own type of behaviour. The view of the problem atequipment level may give a misleading indication of the counter-strategies to beemployed. The more meaningful deeper examination will not be possible unlessthe data contains mode information at the right depth and degree of integrity.

• It is necessary to know any other details which may have a bearing on the failuresensitivity of the equipment; for example the installed position of the failureswhich comprise the sample. There are many factors which may render elements ofa sample unrepresentative including such things as misuse or incorrect diagnosis.

2.2 Quantity of Data

Whereas the effects of poor quality are insidious, the effects of inadequate quantity of data aremore apparent and can, in part, be countered. To see how this may be done it is necessary toexamine one of the statistical characteristics used in Weibull analysis. An equipment undergo-ing in-service failures will exhibit a cumulative distribution function (F(t)), which is the distri-bution in time of the cumulative failure pattern or cumulative percent failed as a function oftime, as indicated by the sample.

Consider a sample of 5 failures (sample size n = 5). The symbol i is used to indicate the failurenumber once the failure times are ranked in ascending order; so here i will take the integervalues 1 to 5 inclusive. Suppose the 5 failure times are 2, 7, 13, 19 and 27 cycles. Now the firstfailure 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 = 1/5 or 0.2 or 20%

Similarly for the second failure time of 7 cycles, the corresponding F(t) is 40% and so on. Onthis basis, this data is suggesting that the fifth failure at 27 cycles corresponds to a cumulativepercent failed of 100%. In other words, on the basis of this sample, 100% of the population willfail by 27 cycles. Clearly this is unrealistic. A further sample of 10 items may contain one ormore which exceed a 27 cycle life. A much larger sample of 1000 items may well indicate thatrather than correspond to a 100% cumulative failure, 27 cycles corresponds to some lessercumulative 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 ben calculated which givesa 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 consid-ered, the F(t) values corresponding to the 5 ascending failure times quoted are not20%, 40%, 60%, 80% and 100%, but are 12.9%, 31.4%, 50%, 68.6% and 87.1%. It isthis latter set of F(t) use values which should be plotted against the correspondingranked failure times on a Weibull plot. Median rank values give the best estimatefor the primary Weibull parameter and are best suited to some later work on confi-dence limits.



• Sample Size Less Than 100. For sample sizes less than 100, in the absence ofMedian Rank tables the true median rank values can be adequately approximatedusing Bernard’s Approximation:

F(t) = (i - 0.3)/(n + 0.4)

• Sample Sizes Greater Than 100. Above a sample size of about 100 the problem ofsmall sample bias is insignificant and the F(t) values may be calculated from theexpression for the Mean Ranks:

i/(n + 1)

3 THE MECHANICS OF WEIBULL ANALYSIS

3.1 The Value of Analysis

On occasions, an analysis of the data reveals little that was not apparent from engineeringjudgement 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 themost experienced assessor. It is always necessary to keep a balance between deductions drawnfrom data analysis and those which arise from an examination of the mechanics of failure.Ideally, these should be suggesting complementary rather than conflicting counter-strategiesto unreliability.

There are many reliability characteristics of an item which may be of interest and significantlymore reliability measures or parameters which can be used to describe those characteristics.Weibull will provide meaningful information on 2 such characteristics. First, it will give somemeasure of how failures are distributed with time. Second, it will indicate the hazard regimefor the failures under consideration. The significance of these 2 measures of reliability isdescribed later.

Weibull is a 3-parameter distribution which has the great strength of being sufficiently flexibleto encompass almost all the failure distributions found in practice, and hence provide informa-tion on the 3 failure regimes normally encountered. Weibull analysis is primarily a graphicaltechnique although it can be done analytically. The danger in the analytical approach is that ittakes away the picture and replaces it with apparent precision in terms of the evaluatedparameters. However, this is generally considered to be a poor practice since it eliminates thejudgement and experience of the plotter. Weibull plots are often used to provide a broad feelfor the nature of the failures; this is why, to some extent, it is a nonsense to worry about errorsof about 1% when using Bernard’s approximation, when the process of plotting the points andfitting the best straight line will probably involve significantly larger “errors”. However, theaim is to appreciate in broad terms how the equipment is behaving. Weibull can make suchprofound statements about an equipment’s behaviour that ±5% may be relatively trivial.



3.2 Evaluating the Weibull Parameters

The first stage of Weibull analysis once the data has been obtained is the estimation of the 3Weibull parameters:

� : Shape parameter.

� : Scale parameter or characteristic life.

� : Location parameter or minimum life.

The general expression for the Weibull F(t) is:

F t e

t

1

�

�

�

This can be transformed into:

log log log log1

1 F tt� � � �

It follows that if F(t) can be plotted against t (corresponding failure times) on paper which hasa reciprocal double log scale on one axis and a log scale on the other, and that data forms astraight line, then the data can be modelled by Weibull and the parameters extracted from theplot. A piece of 2 cycle Weibull paper (Chartwell Graph Data Ref C6572) is shown at Annex Aand this is simply a piece of graph paper constructed such that its vertical scale is a double logreciprocal and its horizontal scale is a conventional log.

The mechanics of the plot are described progressively using the following example and theassociated illustrations in plots 1 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.

• 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 tabula-tion is shown at table 1 (Annex B).

• 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 axisagainst the corresponding F(t) value on the vertical axis (plot 3).

• If the points constitute a reasonable straight line then construct that line. Notethat real data frequently snakes about the straight line due to scatter in the data;this is not a problem providing the snaking motion is clearly to either side of theline. When determining the position of the line give more weight to the laterpoints rather than the early ones; this is necessary both because of the effects ofcumulation and because the Weibull paper tends to give a disproportionate empha-sis to the early points which should be countered where these are at variance withthe subsequent points. Do not attempt to draw more than one straight linethrough the data and do not construct a straight line where there is manifestly a



curve. In this example the fitting of the line presents no problem (plot 4). Notealso that on the matter of how much data is required for a Weibull plot that any 4or so of the pieces of data used here would give an adequate straight line. In suchcircumstances 4 points may well be enough. Generally, 7 or so points would be areasonable minimum, depending on their shape once plotted.

• The fact that the data produced a straight line when initially plotted enables 2statements to be made:

• The data can apparently be modelled by the Weibull distribution.

• The location parameter or minimum life (�) is approximately zero. Thisparameter is discussed later.

• The next step is to construct a perpendicular from the Estimation Point in the topleft hand corner of the paper to the plotted line (plot 5).

• Once the plotted line is obtained, information based on the sample can beextracted. For example, plot 6 illustrates that this data is indicating that a 400 hourlife would result in about 15% of in-service failures for these equipments. Con-versely, an acceptable level of in-service failure may be converted into a life; forexample it can be seen from plot 6 that an acceptable level of in-service failure ofsay, 30% would correspond to a life of about 550 hours, and so on.

• At plot 7 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 arepossible.

• 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 8).

• At plot 9 a dotted horizontal line is highlighted corresponding to an F(t) value of63.2%. Now the scale parameter or characteristic life estimate �� is the life whichcorresponds to a cumulative mortality of 63.2% of the population. Hence to deter-

mine �� it is necessary only to follow the � Estimator line horizontally until it inter-

sects the plotted line and then read off the corresponding time on the lower scale.Plot 10 shows that, based on this sample, these components have an �� of about 830hours. By this time 63.2% of them will have failed.

• At plot 11 the evaluation of the proportion failed corresponding to the mean of thedistribution of the times to failure (Pµ) is shown to be 52.5% using the point ofintersection of the perpendicular and the Pµ scale. This value is inserted in the F(t)scale and its intersection with the plotted line determines the estimated mean ofthe distribution of the times to failure (�). In this case this is about 740 hours.

• One additional piece of information which can be easily extracted also is themedian life; that is to say the life corresponding to 50% mortality. This is shown atplot 12 to be about 720 hours, based on this sample.



4 INTERPRETATION OF WEIBULL OUTPUT

4.1 Concept of Hazard

Before examining the significance of the Weibull shape parameter � it is necessary to knowsomething of the concept of hazard and the 3 so-called failure regimes. The parameter of inter-est here is the hazard rate, h(t). This is the conditional probability that an equipment will failin a given interval of unit time given that it has survived until that interval of time. It is, there-fore, the instantaneous failure rate and can in general be thought of as a measure of the prob-ability of failure, where this probability varies with the time the item has been in service. The3 failure regimes are defined in terms of hazard rate and not, as is a common misconception, interms of failure rate.

The 3 regimes are often thought of in the form of the so-called ‘bath-tub’ curve; this is a validconcept for the behaviour of a system over its whole life but is a misleading model for the vastmajority of components and, more importantly, their individual failure modes (see Reference5 and 7). An individual mode is unlikely to exhibit more than one of the 3 characteristics ofdecreasing, constant or increasing hazard.

Shape Parameter Less Than Unity.

A � value of less than unity indicates that the item or failure mode may be characterised by thefirst regime of decreasing hazard. This is sometimes termed the early failure of infant mortal-ity period and it is a common fallacy that such failures are unavoidable. The distribution oftimes to failure will follow a hyper-exponential distribution in which the instantaneous prob-ability of failure is decreasing with time in service. This hyper-exponential distributionmodels a concentration of failure times at each end of the time scale; many items fail early orelse go on to a substantial life, whilst relatively few fail between the extremes. The extent towhich � is below 1 is a measure of the severity of the early failures; 0.9 for example would be arelatively 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 �� , then clearly componentlifing is inappropriate since the replacement is more likely to fail than the replaced item. Justas importantly, a �� gives a powerful indication of the causes of these failures, which are clas-sically attributed to two deficiencies. First such failures result from poor quality control in themanufacturing process or some other mechanism which permits the installation of lowquality components. It is for this reason that burn-in programmes are the common counter-strategy to poor quality control for electronic components which would otherwise generatean unacceptably high initial in-service level of failure. The second primary cause of infant mor-tality is an inadequate standard of maintenance activity, and here the analysis is pointing to alack of quality rather than quantity in the work undertaken. The circumstance classically asso-ciated with infant mortality problems is the introduction of a new equipment, possibly ofnew design, which is unfamiliar to its operators and its maintainers. Clearly in such situa-tions, the high initial level of unreliability should decrease with the dissemination of experi-ence and the replacement of weakling components with those of normal standard. Theproblem of infant mortality has been shown to be much more prevalent than might have beenanticipated. In one particular study (Part 2 of Reference 6) it was found to be the dominantfailure regime on a variety of mechanical components of traditional design.



Shape Parameter Equal to Unity.

When the shape parameter has a value of approximately one, the Weibull analysis is indicatingthat constant hazard conditions apply. This is the special case where the degree of hazard isnot changing with time in service and such terms as failure rate, MTBF and MTTF may beused meaningfully. This is the most frequently assumed distribution because to do so simpli-fies the mathematical manipulation significantly and opens up the possibility of using manyother reliability techniques which are based on, but rarely state, the precondition that con-stant hazard conditions apply. To assume constant hazard, with its associated negative expo-nential distribution of times to failure, over some or all of an equipment’s life must frequentlyproduce misleading conclusions. The term ‘random failures’ is often used to describe constanthazard and refers to the necessary conditions that failures be independent of each other and oftime. Equipments which predominantly suffer constant hazard over their working livesshould not be lifed since, by definition, the replacement has the same hazard or instantaneousprobability of failure as the replaced item.

Individual failure modes with � = 1 tend to be the exception. Frequently, an equipment willappear to exhibit constant hazard because it has several failure modes of a variety of types,none of which is dominant. This summation effect is a particular characteristic of complexmaintained systems comprising multiple series elements whether they be electronic, electri-cal, mechanical or some combination, particularly when their lives have been randomized byearlier failure replacements. The difficulty here is that the counter-strategy for the individualfailure modes may well be significantly different to those suggested by constant hazard condi-tions for the system or equipment as a whole. There may well be, therefore, a need for a deeperanalysis at mode level. Typical counter-strategies to known constant hazard conditionsinclude de-rating, redundancy or modification.

Shape Parameter Greater Than Unity.

If the Weibull shape parameter is greater than one the analysis is indicating that increasinghazard conditions apply. The instantaneous probability of failure is therefore increasing withtime; the higher the � value, the greater is the rate of increase. This is often called the ‘wear-

-out’ phase, although again this term can be misleading. The time dependence of failures nowpermits sensible consideration of planned replacement providing the total cost of a failurereplacement is greater than the total cost of a planned replacement. The interval for suchreplacements should be optimised and there is at least one general technique (Reference 8)which will do this directly from the Weibull parameters, providing the total costs are known.

Various values of � can be associated with certain distributions of times to failure and the com-monest causes of such distributions. A � value of about 2 arises from a times to failure distribu-tion which is roughly log-normal - see Figure 1:

Such distributions may be attributable to a wear-out phenomenon but are classically gener-ated by situations where failure is due to the nucleation effect of imperfections or weaknesses,such as in crack propagation. A shape parameter of about 2 is an indication, therefore, offatigue failure. As the � value increases above 2, the shape of the pdf approaches the symmetri-cal normal distribution until at � = 3.4 the pdf is fully normal (Figure 2).

A � value of this order indicates at least one dominant failure mode which is being caused bywear or attrition. As the � value rises still further so does the rate of wear-out. Such situationsneed not necessarily be viewed with alarm; if the combined analysis for the 3 Weibull



parameters indicates a pdf of the form shown below, of which a very high �, say about 6 or 7, isjust one element, then clearly a strategy to replace at t0 might be highly satisfactory, particu-larly if it is a critical component, since the evidence suggests there will be no in-service failuresonce that life is introduced (Figure 3).



pdf for = 2�

time

Figure 1 Probability Density Function for a Shape Parameter of 2

pdf for = 3.4�

time

Figure 2 Probability Density Function for a Shape Parameter of 3.4

pdf for = 6 or 7�

t0time

Figure 3 Probability Density Function for a Shape Parameter of 6 to 7

The initiation of increasing hazard conditions and their rate of increase may be a function ofthe maintenance policy adopted and the operating conditions imposed on the equipment.

Some General Comments on �

The Weibull shape parameter provides a clear indication of which failure regime is the appro-priate one for the mode under investigation and quantifies the degree of decreasing or increas-ing hazard. It can be used therefore, to indicate which counter-strategies are most likely tosucceed and aids interpretation of the physics of failure. It can also be used to quantify theeffects of any modifications or maintenance policy changes. Although the use of median ranksprovides the best estimate of � by un-biasing the sample data, it is important to remember thatthe confidence which can be placed on the � estimate for any given failure mode is primarily afunction of the sample size and quality of the data for that mode.

4.2 Scale Parameter or Characteristic LIfe

As stated earlier, � is the value in time by which 63.2% of all failures will have occurred. In thissense, � is just one point on the time scale, providing some standard measure of the distribu-tion of times to failure.

Looking back at the example of 10 items, it was found that �� = 2.4 and �� = 830 hours. Thisinformation helps the construction of a picture of the appropriate pdf.

To say here that the characteristic life is 830 hours is to say simply that roughly two thirds ofall failures will occur by that time, according to this sample. As Sherwin showed in his studyat Reference 6, this is a very useful means of quantifying the effects of some change in mainte-nance strategy. There are, however, others some of which were evaluated in the example. Themean of this log-normalish distribution for these items was found to be about 740 hours andcorresponded to a percent failed of 52.5%. Figure 5 can be sketched using these estimates:

Alternatively the median or 50% life was found to be about 720 hours:

Here the 3 measures of time are all doing roughly the same thing. The characteristic life,however, is taken as the standard measure of position. Its significance is strengthened by thefact that when constant hazard conditions apply, ie � = 1, then the � value becomes the mean



time

f(t)

� = 2.4

� = 830

63.2%

Figure 4 Probability Density Function and Characteristic Life

time between failures (MTBF) for a repairable equipment or a mean time to failure (MTTF)for a non-repairable equipment, and is therefore the inverse of the constant hazard failure rate.This is the only circumstance in which � may be termed an MTBF/MTTF.

4.3 Location Parameter or Minimum Life

It was briefly stated during the example that if a reasonable straight line could be fitted to theinitial plot, then the value of the location parameter is approximately zero. Sometimes,however, the first plot may appear concave when viewed from the bottom right hand cornerof the sheet (Figure 7):

When this occurs it is necessary to subtract some quantity of time (��) from every failure timeused to plot the curve. This is best done by a method attributed to General Motors and shownin Annex C. Using this or any other suitable method, an estimate of �, termed ��, can beobtained. The estimate is enhanced by subtracting its value from every failure time and replot-ting the data: if �� is too small the curve will remain concave but to a lesser degree than before:if �� is too large the plot will become convex; and the best estimate of � is that value which whensubtracted from all the failure times gives the best straight line.



time

f(t)

� = 2.4

= 740

52.7%

Figure 5 Probability Density Function and Mean Life

time

f(t)

� = 2.4

life = 720

50%

Figure 6 Probability Density Function and Median Life

The significance of � is that it is some value of time by which the complete distribution oftimes to failure is shifted, normally to the right, hence the term ‘location’. In the earlierexample the distribution with �� = 0 is shown at Figure 4. If, however, �� had taken some posi-tive value, say 425 hours, then this value must be added to all the times to failure extractedfrom the subsequent analysis of the straight line, and Figure 4 would have changed to that illus-trated at Figure 8.

Here two thirds of the population do not fail until 1245 hours and most importantly the �

value or minimum life value has shifted the time origin such that no failures are anticipated inthe first 425 hours of service. The existence of a positive location parameter is therefore ahighly desirable feature in any equipment and the initial plot should always be examined for apotential concave form.

A further example of a 3-parameter Weibull plot is given at Annex D.



x

x

x

x

x

x

x

F(t)

time

Figure 7 Representing Points on a Curve using Weibull Paper

time

f(t)

� = 2.4

63.2%

� = 425 830

new = 830 + 425 = 1255�

Figure 8 Effect of Location Parameter

5 PRACTICAL DIFFICULTIES WITH WEIBULL PLOTTING

5.1 Scatter

The problem of scatter in the original data and the resultant snaking effect this can producehas been briefly mentioned. At Annex E, however, is a plot using 11 pieces of real data whichillustrates a severe case of snaking. It is possible to plot a line and an attempt has been made inthis case which gives the necessary added weight to later points. The difficulty is obvious; it isnecessary to satisfy yourself that you are seeing true snaking about a straight line caused byscatter of the points about the line and not some other phenomenon.

5.2 Extrapolation

Successful Weibull plotting relies on having historical failure data. Inaccuracies will arise if thespan in time of that data is not significantly greater than the mean of the distribution of timesto failure. If data obtained over an inadequate range is used as a basis for extrapolation (i.e.)extending the plotted line significantly, estimates of the 3 parameters are likely to be inaccu-rate and may well fail to reveal characteristics of later life such as a bi-modal wear-out phe-nomenon. The solution is comprehensive data at the right level.

5.3 Multi-Modal Failures

The difficulty of multi-modal failures has been mentioned previously. In the same way thatthe distribution of times to failures for a single mode will be a characteristic of that mode, sothe more modes there are contributing to the failure data, the more the individual characteris-tics of number of failure modes often tends to look like constant hazard (� = 1.0). In somecases this has been found to be so even when the modes themselves have all had a high wear-out characteristic (� � 3 or 4). This tendency is strongest when there are many modes none ofwhich is dominant. Hence a knowledge of the failure regimes of the individual failure modesof an equipment is more useful in formulating a maintenance policy than that of the failureregime of the equipment itself. The solution once again is data precise enough to identify the



F(t)

time

or

Figure 9 Representation of Multi-Modal Behaviour on Weibull Paper

characteristics of all the significant failure modes. A Weibull plot using data gathered at equip-ment level may or may not indicate multi-modal behaviour. The most frequent manifestationof such behaviour is a convex or cranked plot as shown in Figure 9.

The cranked plot shown above should not normally be drawn since it implies the existence of2 failure regimes, one following the other in time. This is rarely the case; in general the bi- ormulti-modal plots will be found to be mixed along both lines, because the distributions oftimes to failure themselves overlap. This is illustrated in Figure 10.

One example of this bi-modal behaviour is quoted in Reference 6. There a vacuum pump wasfound to have one mode of severe infant mortality (� = 0.42) combined with another of wear-out (� = 3.2). It is most unlikely that an analysis of their combined times to failure would havesuggested an adequate maintenance strategy for the item as a whole. The convex curve alsoshown in Figure 9 indicates the presence of corrupt or multi-modal data. One form of corrup-tion stems from the concept of a negative location parameter; if life is consumed in storage butthe failure data under analysis is using an in-service life measured once the items are issuedfrom store, then clearly the data is corrupt in that only a part life is being used in the analysis.

Once adequate multi-modal data has been obtained it is possible to separate the data for eachmode and replot all the data in such a way as to make maximum use of every piece of life infor-mation. This approach provides more confidence than simply plotting failure data for the indi-vidual mode and is best done using an adaptation of the technique for dealing withmultiply-censored data; this topic is covered later.

5.4 Confidence Limits

As was pointed out earlier, most forms of analysis will give a false impression of accuracy andWeibull is no exception, particularly when the same size is less than 50. The limitations of thedata are best recognised by the construction of suitable confidence limits on the original plot.The confidence limits normally employed are the 95% lower confidence limit (LCL) or 5%Ranks, and the 5% upper confidence limit (UCL) or 95% Ranks, although other levels of confi-dence can be used. With these notes are tables of LCL and UCL ranks which can be seen to be afunction solely of sample size. The technique for using these ranks consists of entering the



mode 1 — < 1, hence infant mortality�

mode 2 — > 1, showingtime dependent failures

�

f(t)

time

Figure 10 Multiple Probability Density Functions

vertical axis of the Weibull plot at the ith F(t) value quoted in the tables for the appropriatesample size. A straight horizontal line should be drawn from the point of entry to intersect theline constructed from the data. From the point of intersection, move vertically up (for a lowerlimit) or down (for an upper limit) until horizontal with the corresponding ith plotted point.The technique is shown at Plot 1 of Annex F for the lower bound using the same example as inAnnex B. The first value obtained from the table for a sample size of 10 is 0.5; this cannot beused since it does not intersect the plotted line. The next value is 3.6 and this is shown in Plot 1to generate point (1) on the lower bound. The third point of entry is at 8.7 and this is shown toproduce point (2) which is level with the third plotted point for the straight line, and so on.

The primary use of this lower bound curve constructed through the final set of points is that itis a visual statement of how bad this equipment might be and still give rise to the raw dataobserved, with 95% confidence. Hence it can be said here that although the best estimate for ��

is 830 hours, we can be only 95% confident, based on the data used, that the true � is greaterthan or equal to 615 hours. Similarly at Plot 2, which shows the construction of a 95% upperbound, we can be 95% confident that the true � is less than or equal to 1040 hours. These 2statements can be combined to give symmetrical 90% confidence limits of between 615 and1040 hours. This range can only be reduced by either diminishing the confidence level (andtherefore increasing the risks of erroneous deduction) or by increasing the quantity of data.

5.5 Censoring of Sample Data

Often samples contain information on incomplete times to failure in addition to the moreobviously useful consumed lives at failure. This incomplete data may arise because an itemhas to be withdrawn for some reason other than the failure which is being studied. If the equip-ment suffers multi-modal failures then in an analysis of a particular mode, failure times attrib-utable to all other modes become censorings. Alternatively the data collection period may endwithout some equipments failing, ie unknown finish times. The outcome of such situations isgenerally a series of complete failure times and a series of incomplete failure times or censor-ings for the mode under investigation. This latter information, this collection of times whenthe equipment did not fail for the particular reason cannot be ignored since to do so would biasthe analysis, and diminish the confidence level associated with subsequent statements drawnfrom the plot. The assumption is generally made that the non-failures would have failed withequal probability at any time between the known failures or censored lives or after all of them.Therefore an item removed during inspection because it is nearing unacceptable limits is closerto a failure and is not a censoring.

The mechanics of dealing with censored data require the determination of a mean ordernumber for each failure; this may be considered as an alternative to the failure number i usedpreviously, the primary difference being that the mean order number becomes a non-integeronce the first censoring is reached. The technique is outlined using the example at Annex G.As a first step a table is constructed with columns a and b listing in ascending order the failureand censoring times respectively. Column (c) is calculated as the survivors prior to each eventin either of columns a or b; where the event is a censoring the corresponding surviving numberis shown in parenthesis by convention. Clearly the data in the sample is multiply-censored inthat it is a mixture of failure and censored times; a total of 7 failures and 9 censorings gives asample size n of 16.



Column (d) is obtained using the formula:

m mn m

ki i

i

i

1

11

1

Where mi = current mean order number

mi-1= previous mean order number

n = total sample size for failure and censorings

ki = number of survivors prior to the failure or censoring underconsideration

Mean order number values are determined only for failures. Once the first censoring occurs at65, all subsequent mi values are non-integers. The median rank values at column (e) are takenfrom the median rank tables using linear interpolation when necessary. For purposes of com-parison only, the equivalent median ranks obtained from Bernard’s Approximation, (i-0.3)/(n+ 0.4) are included at column (f). These are obtained by substituting mi for i in the standardexpression. These can be seen to be largely in agreement with the purer figures in column (e).Finally 5% LCL AND 95% UCL figures are included at columns (g) and (h). These are obtainedfrom the tables using linear interpolation where necessary.

The median rank figures in column (e) are plotted on Weibull paper against the correspondingfailure times at column a in the normal way. The plot is illustrated at Plot 1 of Annex G, andproduces �, � and � estimates without difficulty. For completeness, Plot 2 shows the 5% LCLAND 95% UCL curves; a 90% confidence range for �� of between 90 and 148 units of time isobtained.

6 COMPARISON WITH HAZARD PLOTTING

It is often thought that Weibull plots are no better than plotting techniques based on thecumulative hazard function calculated from sample data. Such methods will give estimates ofthe 3 Weibull parameters and the mechanics of obtaining them are often slightly simpler thanfor the equivalent application of Weibull. However, cumulative hazard plots give little feel forthe behaviour of the equipment in terms of the levels of risk of in-service failures for a proposedlife. More importantly, such methods contain no correction for small sample bias and aretherefore less suitable for use with samples smaller than 50. This limitation is is compoundedby the difficulty of attempting the evaluation of confidence limits on a cumulative hazardplot. Finally, cases have occurred where cumulative hazard plots have failed to indicate multi-modal behaviour which was readily apparent from a conventional Weibull plot from the samedata.



7 CONCLUSIONS

The ability of the Weibull distribution to model failure situations of many types, includingthose where non-constant hazard conditions apply, make it one of the most generally usefuldistributions for analyzing failure data. The information it provides, both in terms of the mod-elled distribution of times to failure and the prevailing failure regime is fundamental to theselection of a successful maintenance strategy, whether or not component lifing is an elementin that strategy.

Weibull’s use of median ranks helps overcome the problems inherent in small samples. Thedegree of risk associated with small samples can be quantified using confidence limits and thiscan be done for complete or multiply-censored data. Weibull plots can quantify the risks asso-ciated with a proposed lifing policy and can indicate the likely distribution of failure arisings.In addition, they may well indicate the presence of more than one failure mode. However,Weibull is not an autonomous process for providing instant solutions; it must be used in con-junction with a knowledge of the mechanics of the failures under study. The final point to bemade is that Weibull, like all such techniques, relies upon data of adequate quantity andquality; this is particularly true of multi-modal failure patterns.

REFERENCES

1. Weibull W. A statistical distribution function of wide application. ASME paper 51-A-6,Nov 1951.

2. Mann R N, Schafer R E and Singpurwalla N D. Methods for statistical analysis of reli-ability and life data. Wiley 1974.

3. Bompas-Smith J H. Mechanical survival - the use of reliability data. McGraw-Hill 1973.4. Carter A D S. Mechanical reliability. Macmillan 1972.5. British Standard 5760: Part 2: 1981. Reliability of systems, equipments and components;

guide to the assessment of reliability.6. Sherwin D J and Lees F P. An investigation of the application of failure data analysis to

decision making in the maintenance of process plant. Proc Instn Mech Engrs, Vol 194, No29, 1980.

7. Carter ADS. The bathtub curve for mechanical components - fact or fiction. Conference onImprovement of Reliability in Engineering, Instn Mech Engrs, Loughborough 1973.

8. Glasser G J. Planned replacement: some theory and its application. Journal of QualityTechnology, Vol 1,No 2. April 1969.



ANNEX A

TWO CYCLE WEIBULL PAPER



ANNEX B

PROGRESSIVE EXAMPLE OF WEIBULL PLOTTING

Arranging the Raw Data

Failure Number

(i)

Ranked Hours

at Failure

(ti)

Median Rank

Cumulative % Failed

F(t)

1 300 6.7

2 410 16.2

3 500 25.9

4 600 35.5

5 660 45.2

6 750 54.8

7 825 64.5

8 900 74.1

9 1050 83.8

10 1200 93.3

The following plots illustrate Weibull plotting.



ANNEX C

ESTIMATION OF WEIBULL LOCATION PARAMETER

Steps:

1. Plot the data initially, observing a concave curve when viewed from the bottom righthand corner.

2. Select 2 extreme points on the vertical scale (say a and b), and determine the corre-sponding failure times (t1 and t3).

3. Divide the physical distance between points a and b in half without regard for thescale of the vertical axis, and so obtain point c.

4. Determine the failure time corresponding to point c (ie t2).5. he estimate of the location parameter is given by:

�� tt t t t

t t t t2

3 2 2 1

3 2 2 1



Weibull Plot

� = t - (t - t )(t - t )2 3 2 2 1

(t - t ) - (t - t )3 2 2 1

time

a

b

t1 t2 t3

Figure 11 Estimation of Location Parameter

ANNEX D

EXAMPLE OF A 3-PARAMETER WEIBULL PLOT

Problem: to determine the Weibull parameters for the following (ordered) sample times tofailure:

1000, 1300, 1550, 1850, 2100, 2450 and 3000 hours.

Steps:

1. Plot initially (Plot 1).2. Having identified a concave form apply the technique at Annex C (Plot 2).3. Determine � and evaluate modified times to failure.4. Plot modified points and confirm a straight line (Plot 3).5. Extract � and � in the normal way remembering to add � to the straight line value for

� (Plot 4).6. Sketch the probability density function (Plot 5).

Plotting the raw data:

Failure Number

(i)

Ranked Hours

at Failure

(ti)

Median Rank

Cumulative % Failed

F(t)

1 1000 9.4

2 1300 22.8

3 1550 36.4

4 1850 50.0

5 2100 63.6

6 2450 77.2

7 3000 90.6



From Plot 2:

t1 = 810 hours

t2 = 1500 hours

t3 = 4000 hours

General expression from Annex D:

�

�

�

�

tt t t t

t t t t2

3 2 2 1

3 2 2 1

15004000 1500 1500 810

4000 1500 1500 810

1500 953

547 hours

Replot using:

1000 - 547 = 453

1300 - 547 = 753

1550 - 547 = 1003

1850 - 547 = 1303

2100 - 547 = 1553

2450 - 547 = 1903

3000 - 547 = 2453



b = 1.9

f(t)

� = 547 1560 time

� = 547 + 1560 = 2107

63.2%

Figure 12 Probability Density Function

ANNEX E

THE EFFECT OF SCATTER



ANNEX F

95% CONFIDENCE LIMITS FOR WEIBULL



ANNEX G

WEIBULL PLOT OF MULTIPLY-CENSORED DATA





Fai

lure

Tim

est i

(a)

Cen

sori

ng

Tim

esc i (b

)

Su

rviv

ors

ki

(c)

Mea

nO

rder

Nu

mb

erM

i

(d)

Med

ian

Ran

ks

%

(e)

Ber

nar

ds

Ap

pro

x% (f

)

5%

Ran

kL

ow

erB

ou

nd

(g)

95%

Ran

kU

pp

erB

ou

nd

(h)

31.7

16

14.2

4.2

70.3

17

39.2

15

210.2

10.3

72.2

26

57.5

14

316.3

16.4

65.3

34

65.0

13)

——

——

—

65.8

12

316

13

112

408

.22.8

923.0

59.3

242.4

8

70.0

11

408

16

14

08

111

516

..

.29.4

929.6

313.8

49.1

2

75.0

75.0

87.5

88.3

84.2

101.7

(10)

(9)

(8)

(7)

(6)

(5)

——

——

—

105.8

47.5

344.0

344.0

925.6

564.1

8

109.2

(3)

——

——

—

110.0

210.6

963.3

163.3

543.1

480.4

5

130.0

(1)

——

——

—

Mu

ltip

lyC

en

sore

dD

ata

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