1、Designation: G166 00 (Reapproved 2011)Standard Guide forStatistical Analysis of Service Life Data1This standard is issued under the fixed designation G166; the number immediately following the designation indicates the year oforiginal adoption or, in the case of revision, the year of last revision.
2、A number in parentheses indicates the year of last reapproval. Asuperscript epsilon () indicates an editorial change since the last revision or reapproval.1. Scope1.1 This guide presents briefly some generally acceptedmethods of statistical analyses which are useful in the inter-pretation of service
3、 life data. It is intended to produce acommon terminology as well as developing a common meth-odology and quantitative expressions relating to service lifeestimation.1.2 This guide does not cover detailed derivations, orspecial cases, but rather covers a range of approaches whichhave found applicati
4、on in service life data analyses.1.3 Only those statistical methods that have found wideacceptance in service life data analyses have been consideredin this guide.1.4 TheWeibull life distribution model is emphasized in thisguide and example calculations of situations commonly en-countered in analysi
5、s of service life data are covered in detail.1.5 The choice and use of a particular life distribution modelshould be based primarily on how well it fits the data andwhether it leads to reasonable projections when extrapolatingbeyond the range of data. Further justification for selecting amodel shoul
6、d be based on theoretical considerations.2. Referenced Documents2.1 ASTM Standards:2G169 Guide for Application of Basic Statistical Methods toWeathering Tests3. Terminology3.1 Definitions:3.1.1 material propertycustomarily, service life is consid-ered to be the period of time during which a system m
7、eetscritical specifications. Correct measurements are essential toproducing meaningful and accurate service life estimates.3.1.1.1 DiscussionThere exists many ASTM recognizedand standardized measurement procedures for determiningmaterial properties. As these practices have been developedwithin commi
8、ttees with appropriate expertise, no further elabo-ration will be provided.3.1.2 beginning of lifethis is usually determined to be thetime of manufacture. Exceptions may include time of deliveryto the end user or installation into field service.3.1.3 end of lifeOccasionally this is simple and obviou
9、ssuch as the breaking of a chain or burning out of a light bulbfilament. In other instances, the end of life may not be socatastrophic and free from argument. Examples may includefading, yellowing, cracking, crazing, etc. Such cases needquantitative measurements and agreement between evaluatorand us
10、er as to the precise definition of failure. It is also possibleto model more than one failure mode for the same specimen.(for example,The time to produce a given amount of yellowingmay be measured on the same specimen that is also tested forcracking.)3.1.4 F(t)The probability that a random unit draw
11、n fromthe population will fail by time (t). Also F(t) = the decimalfraction of units in the population that will fail by time (t). Thedecimal fraction multiplied by 100 is numerically equal to thepercent failure by time (t).3.1.5 R(t)The probability that a random unit drawn fromthe population will s
12、urvive at least until time (t). Also R(t) =the fraction of units in the population that will survive at leastuntil time (t)Rt! 5 1 2 Ft! (1)3.1.6 pdfthe probability density function (pdf), denotedby f(t), equals the probability of failure between any two pointsof time t(1) and t(2). Mathematically f
13、(t) =dF t!dt. For thenormal distribution, the pdf is the “bell shape” curve.3.1.7 cdfthe cumulative distribution function (cdf), de-noted by F(t), represents the probability of failure (or thepopulation fraction failing) by time = (t). See section 3.1.4.3.1.8 weibull distributionFor the purposes of
14、this guide,the Weibull distribution is represented by the equation:Ft! 5 1 2 e2StcDb(2)where:1This guide is under the jurisdiction of ASTM Committee G03 on Weatheringand Durability and is the direct responsibility of Subcommittee G03.08 on ServiceLife Prediction.Current edition approved July 1, 2011
15、. Published August 2011. Originallyapproved in 2000. Last previous edition approved in 2005 as G166 00(2005).DOI: 10.1520/G0166-00R11.2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at serviceastm.org. For Annual Book of ASTMStandards volume info
16、rmation, refer to the standards Document Summary page onthe ASTM website.1Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.F(t) = defined in paragraph 3.1.4t = units of time used for service lifec = scale parameterb = shape parameter3.
17、1.8.1 The shape parameter (b), section 3.1.6, is so calledbecause this parameter determines the overall shape of thecurve. Examples of the effect of this parameter on the distri-bution curve are shown in Fig. 1, section 5.3.3.1.8.2 The scale parameter (c), section 3.1.6, is so calledbecause it posit
18、ions the distribution along the scale of the timeaxis. It is equal to the time for 63.2 % failure.NOTE 1This is arrived at by allowing t to equal c in the aboveexpression.This then reduces to Failure Probability = 1e1, which furtherreduces to equal 10.368 or .632.3.1.9 complete dataA complete data s
19、et is one where allof the specimens placed on test fail by the end of the allocatedtest time.3.1.10 Incomplete dataAn incomplete data set is onewhere (a) there are some specimens that are still surviving atthe expiration of the allowed test time, (b) where one or morespecimens is removed from the te
20、st prior to expiration of theallowed test time. The shape and scale parameters of the abovedistributions may be estimated even if some of the testspecimens did not fail. There are three distinct cases where thismight occur.3.1.10.1 Time censoredSpecimens that were still surviv-ing when the test was
21、terminated after elapse of a set time areconsidered to be time censored. This is also referred to as rightcensored or type I censoring. Graphical solutions can still beused for parameter estimation. At least ten observed failuresshould be used for estimating parameters (for example slopeand intercep
22、t).3.1.10.2 specimen censoredSpecimens that were still sur-viving when the test was terminated after a set number offailures are considered to be specimen censored. This isanother case of right censored or type I censoring. See 3.1.10.13.1.10.3 Multiply CensoredSpecimens that were removedprior to th
23、e end of the test without failing are referred to as leftcensored or type II censored. Examples would include speci-mens that were lost, dropped, mishandled, damaged or brokendue to stresses not part of the test.Adjustments of failure ordercan be made for those specimens actually failed.4. Significa
24、nce and Use4.1 Service life test data often show different distributionshapes than many other types of data. This is due to the effectsof measurement error (typically normally distributed), com-bined with those unique effects which skew service life datatowards early failure (infant mortality failur
25、es) or late failuretimes (aging or wear-out failures) Applications of the prin-ciples in this guide can be helpful in allowing investigators tointerpret such data.NOTE 2Service life or reliability data analysis packages are becomingmore readily available in standard or common computer software pack-
26、ages. This puts data reduction and analyses more readily into the hands ofa growing number of investigators.5. Data Analysis5.1 In the determinations of service life, a variety of factorsact to produce deviations from the expected values. Thesefactors may be of a purely random nature and act to eith
27、erincrease or decrease service life depending on the magnitude ofthe factor. The purity of a lubricant is an example of one suchfactor. An oil clean and free of abrasives and corrosivematerials would be expected to prolong the service life of amoving part subject to wear. A fouled contaminated oil m
28、ightprove to be harmful and thereby shorten service life. Purelyrandom variation in an aging factor that can either help or harma service life might lead a normal, or gaussian, distribution.Such distributions are symmetrical about a central tendency,usually the mean.5.1.1 Some non-random factors act
29、 to skew service lifedistributions. Defects are generally thought of as factors thatFIG. 1 Effect of the Shape Parameter (b) on the Weibull Probability DensityG166 00 (2011)2can only decrease service life. Thin spots in protective coat-ings, nicks in extruded wires, chemical contamination in thinmet
30、allic films are examples of such defects that can cause anoverall failure even through the bulk of the material is far fromfailure. These factors skew the service life distribution towardsearly failure times.5.1.2 Factors that skew service life towards the high sidealso exist. Preventive maintenance
31、, high quality raw materials,reduced impurities, and inhibitors or other additives are suchfactors. These factors produce life time distributions shiftedtowards the long term and are those typically found in productshaving been produced a relatively long period of time.5.1.3 Establishing a descripti
32、on of the distribution of fre-quency (or probability) of failure versus time in service is theobjective of this guide. Determination of the shape of thisdistribution as well as its position along the time scale axis arethe principle criteria for estimating service life.5.2 Normal (Gaussian) Distribu
33、tionThe characteristic ofthe normal, or Gaussian distribution is a symmetrical bellshaped curve centered on the mean of this distribution. Themean represents the time for 50 % failure. This may be definedas either the time when one can expect 50 % of the entirepopulation to fail or the probability o
34、f an individual item tofail. The “scale” of the normal curve is the mean value (x), andthe shape of this curve is established by the standard deviationvalue (s).5.2.1 The normal distribution has found widespread use indescribing many naturally occurring distributions. Its firstknown description by C
35、arl Gauss showed its applicability tomeasurement error. Its applications are widely known andnumerous texts produce exhaustive tables and descriptions ofthis function.5.2.2 Widespread use should not be confused with justifi-cation for its application to service life data. Use of analysistechniques d
36、eveloped for normal distribution on data distrib-uted in a non-normal manner can lead to grossly erroneousconclusions. As described in Section 5, many service lifedistributions are skewed towards either early life or late life.The confinement to a symmetrical shape is the principalshortcoming of the
37、 normal distribution for service life appli-cations. This may lead to situations where even negativelifetimes are predicted.5.3 Lognormal DistributionThis distribution has shownapplication when the specimen fails due to a multiplicativeprocess that degrades performance over time. Metal fatigue isone
38、 example. Degradation is a function of the amount offlexing, cracks, crack angle, number of flexes, etc. Performanceeventually degrades to the defined end of life.35.3.1 There are several convenient features of the lognormaldistribution. First, there is essentially no new mathematics tointroduce int
39、o the analysis of this distribution beyond those ofthe normal distribution.Asimple logarithmic transformation ofdata converts lognormal distributed data into a normal distri-bution.All of the tables, graphs, analysis routines etc. may thenbe used to describe the transformed function. One note ofcaut
40、ion is that the shape parameter s is symmetrical in itslogarithmic form and non-symmetrical in its natural form. (forexample, x =16 .2s in logarithmic form translates to 10 +5.8and 3.7 in natural form)5.3.2 As there is no symmetrical restriction, the shape ofthis function may be a better fit than th
41、e normal distribution forthe service life distributions of the material being investigated.5.4 Weibull DistributionWhile the Swedish ProfessorWaloddi Weibull was not the first to use this expression,4hispaper, A Statistical Distribution of Wide Applicability pub-lished in 1951 did much to draw atten
42、tion to this exponentialfunction. The simplicity of formula given in (1), hides itsextreme flexibility to model service life distributions.5.4.1 The Weibull distribution owes its flexibility to the“shape” parameter. The shape of this distribution is dependenton the value of b. If b is less than 1, t
43、he Weibull distributionmodels failure times having a decreasing failure rate.The timesbetween failures increase with exposure time. Ifb=1,then theWeibull models failure times having constant failure rate. If b 1 it models failure times having an increasing failure rate, ifb = 2, then Weibull exactly
44、 duplicates the Rayleigh distribu-tion, as b approaches 2.5 it very closely approximates thelognormal distribution, as b approaches 3. the Weibull expres-sion models the normal distribution and as b grows beyond 4,the Weibull expression models distributions skewed towardslong failure times. See Fig.
45、 1 for examples of distributions withdifferent shape parameters.5.4.2 The Weibull distribution is most appropriate whenthere are many possible sites where failure might occur and thesystem fails upon the occurrence of the first site failure. Anexample commonly used for this type of situation is a ch
46、ainfailing when only one link separates. All of the sites, or links,are equally at risk, yet one is all that is required for total failure.5.5 Exponential DistributionThis distribution is a specialcase of the Weibull. It is useful to simplify calculationsinvolving periods of service life that are su
47、bject to randomfailures. These would include random defects but not includewear-out or burn-in periods.6. Parameter Estimation6.1 Weibull data analysis functions are not uncommon butnot yet found on all data analysis packages. Fortunately, theexpression is simple enough so that parameter estimation
48、maybe made easily. What follows is a step-by-step example forestimating the Weibull distribution parameters from experi-mental data.6.1.1 The Weibull distribution, (Eq 2) may be rearranged asshown below: (Eq 3)1 2 Ft! 5 e2StcDb(3)and, by taking the natural logarithm of both sides twice, thisexpressi
49、on becomeslnFln11 2 Ft!G5 blnt! 2 blnc (4)3Mann, N.R. et al, Methods for Statistical Analysis of Reliability and Life Data,Wiley, New York 1974 and Gnedenko, B.V. et al, Mathematical Methods ofReliability Theory, Academic Press, New York 1969.4Weibull, W., “A statistical distribution of wide applicability ,” J. Appl. Mech.,18, 1951, pp 293297.G166 00 (2011)3Eq 4 is in the form of an equation describing a straight line(y=mx+y0) withlnFln11 2 Ft!G(5)corresponding to Y, ln(t) corresponding to x and the slope ofthe line m equals the Weibull shape par
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