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. 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 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 application 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 analysis 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 should 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 property—customarily, service life is consid-ered to be the period of time during which a system meetscritical specifications. Correct measurements are essential toproducing meaningful and accurate service life estimates.3.1.1.1 Discussion—There exists many ASTM recognizedand standardized measurement procedures for determiningmaterial properties. As these practices have been developedwithin committees with appropriate expertise, no further elabo-ration will be provided.3.1.2 beginning of life—this 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 life—Occasionally this is simple and obvioussuch 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 user 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 drawn 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 survive at least until time (t). Also R(t) =the fraction of units in the population that will survive at leastuntil time (t)R~t! 5 1 2 F~t! (1)3.1.6 pdf—the probability density function (pdf), denoted byf(t), equals the probability of failure between any two points oftime t(1) and t(2). Mathematically f~t!5dF ~t!dt. For the normaldistribution, the pdf is the “bell shape” curve.3.1.7 cdf—the 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 distribution—For the purposes of this guide,the Weibull distribution is represented by the equation:F~t! 5 1 2 e2StcDb(2)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. 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

[email protected] For Annual Book of ASTMStandards volume information, refer to the standard’s Document Summary page onthe ASTM website.Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States1where:F(t) = defined in paragraph 3.1.4t = units of time used for service lifec = scale parameterb = shape parameter3.1.8.1 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 positions the distribution along the scale of the timeaxis. It is equal to the time for 63.2 % failure.NOTE 1—This is arrived at by allowing t to equal c in the aboveexpression.This then reduces to Failure Probability = 1−e−1, which furtherreduces to equal 1−0.368 or .632.3.1.9 complete data—Acomplete data set is one where all ofthe specimens placed on test fail by the end of the allocated testtime.3.1.10 Incomplete data—An 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 test 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 censored—Specimens that were still surviv-ing when the test was 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 intercept).3.1.10.2 specimen censored—Specimens 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 Censored—Specimens that were removedprior to the 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. Significance 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 failures) 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 2—Service life or reliability data analysis packages are becomingmore readily available in standard or common computer software pack-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 eitherincrease 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 mightprove to be harmful and thereby shorten service life. Purelyrandom variation in an aging factor that can either help or harmFIG. 1 Effect of the Shape Parameter (b) on the Weibull Probability DensityG166 − 00 (2011)2a 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 to skew service lifedistributions. Defects are generally thought of as factors thatcan only decrease service life. Thin spots in protectivecoatings, nicks in extruded wires, chemical contamination inthin metallic films are examples of such defects that can causean overall failure even through the bulk of the material is farfrom failure. These factors skew the service life distributiontowards early failure times.5.1.2 Factors that skew service life towards the high sidealso exist. Preventive maintenance, 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 description 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) Distribution—The 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 of 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 (σ).5.2.1 The normal distribution has found widespread use indescribing many naturally occurring distributions. Its firstknown description by Carl 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 developed 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 normal distribution for service life appli-cations. This may lead to situations where even negativelifetimes are predicted.5.3 Lognormal Distribution—This distribution has shownapplication when the specimen fails due to a multiplicativeprocess that degrades performance over time. Metal fatigue isone 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 into 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 ofcaution is that the shape parameter σ is symmetrical in itslogarithmic form and non-symmetrical in its natural form. (forexample, x¯ =16 .2σ 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 the normal distribution forthe service life distributions of the material being investigated.5.4 Weibull Distribution—While 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 attention 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, the 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 duplicates the Rayleighdistribution, as b approaches 2.5 it very closely approximatesthe lognormal distribution, as b approaches 3. the Weibullexpression models the normal distribution and as b growsbeyond 4, the Weibull expression models distributions skewedtowards long failure times. See Fig. 1 for examples ofdistributions with different 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 chainfailing 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 Distribution—This distribution is a specialcase of the Weibull. It is useful to simplify calculationsinvolving periods of service life that are subject 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 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)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 293–297.G166 − 00 (2011)31 2 F~t! 5 e2StcDb(3)and, by taking the natural logarithm of both sides twice, thisexpression becomeslnFln11 2 F~t!G5 bln~t! 2 blnc (4)Eq 4 is in the form of an equation describing a straight line(y=mx+y0) withlnFln11 2 F~t!G(5)corresponding to Y, ln(t) corresponding to x and the slope ofthe line m equals the Weibull shape parameter b.