ASTM G166-00

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Designation:G166–00
Standard Guide for
Statistical Analysis of Service Life Data
This standard is issued under the fixed designation G166; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision.Anumber in parentheses indicates the year of last reapproval.A
superscript epsilon (e) indicates an editorial change since the last revision or reapproval.
1. Scope withincommitteeswithappropriateexpertise,nofurtherelabo-
ration will be provided.
1.1 This guide presents briefly some generally accepted
3.1.2 beginning of life—this is usually determined to be the
methods of statistical analyses which are useful in the inter-
time of manufacture. Exceptions may include time of delivery
pretation of service life data. It is intended to produce a
to the end user or installation into field service.
common terminology as well as developing a common meth-
3.1.3 end of life—Occasionally this is simple and obvious
odology and quantitative expressions relating to service life
such as the breaking of a chain or burning out of a light bulb
estimation.
filament. In other instances, the end of life may not be so
1.2 This guide does not cover detailed derivations, or
catastrophic and free from argument. Examples may include
special cases, but rather covers a range of approaches which
fading, yellowing, cracking, crazing, etc. Such cases need
have found application in service life data analyses.
quantitative measurements and agreement between evaluator
1.3 Only those statistical methods that have found wide
anduserastotheprecisedefinitionoffailure.Itisalsopossible
acceptance in service life data analyses have been considered
to model more than one failure mode for the same specimen.
in this guide.
(forexample,Thetimetoproduceagivenamountofyellowing
1.4 TheWeibulllifedistributionmodelisemphasizedinthis
may be measured on the same specimen that is also tested for
guide and example calculations of situations commonly en-
cracking.)
countered in analysis of service life data are covered in detail.
3.1.4 F(t)—The probability that a random unit drawn from
1.5 Thechoiceanduseofaparticularlifedistributionmodel
the population will fail by time (t). Also F(t) 5 the decimal
should be based primarily on how well it fits the data and
fraction of units in the population that will fail by time (t).The
whether it leads to reasonable projections when extrapolating
decimal fraction multiplied by 100 is numerically equal to the
beyond the range of data. Further justification for selecting a
percent failure by time (t).
model should be based on theoretical considerations.
3.1.5 R(t)—The probability that a random unit drawn from
2. Referenced Documents the population will survive at least until time (t). Also R(t) 5
the fraction of units in the population that will survive at least
2.1 ASTM Standards:
until time (t)
ASTM Standard Guide for the Application of Basic Statis-
tical Methods to Weathering Tests
R~t! 51 2 F~t! (1)
3.1.6 pdf—the probability density function (pdf), denoted
3. Terminology
byf(t),equalstheprobabilityoffailurebetweenanytwopoints
3.1 Definitions:
dF ~t!
3.1.1 material property—customarily,servicelifeisconsid- of time t(1) and t(2). Mathematically f(t) 5 . For the
dt
ered to be the period of time during which a system meets
normal distribution, the pdf is the “bell shape” curve.
critical specifications. Correct measurements are essential to
3.1.7 cdf—the cumulative distribution function (cdf), de-
producing meaningful and accurate service life estimates.
noted by F(t), represents the probability of failure (or the
3.1.1.1 Discussion—There exists many ASTM recognized
population fraction failing) by time 5 (t). See section 3.1.4.
and standardized measurement procedures for determining
3.1.8 weibull distribution—For the purposes of this guide,
material properties. As these practices have been developed
the Weibull distribution is represented by the equation:
t b
S D
c
F~t! 51 2 e (2)
This guide is under the jurisdiction of ASTM Committee G-3 on Weathering
and Durability and is the direct responsibility of Subcommittee G3.08 on Service
F(t) 5 defined in paragraph 3.1.4
Life Prediction.
t 5 units of time used for service life
Current edition approved Jan. 00, 2000. Published March 2000.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.
G166
censored or type II censored. Examples would include speci-
c 5 scale parameter
mens that were lost, dropped, mishandled, damaged or broken
b 5 shape parameter
duetostressesnotpartofthetest.Adjustmentsoffailureorder
3.1.8.1 The shape parameter (b), section 3.1.6, is so called
can be made for those specimens actually failed.
because this parameter determines the overall shape of the
curve. Examples of the effect of this parameter on the distri-
4. Significance and Use
bution curve are shown in Fig. 1, section 5.3.
4.1 Service life test data often show different distribution
3.1.8.2 The scale parameter (c), section 3.1.6, is so called
shapes than many other types of data.This is due to the effects
because it positions the distribution along the scale of the time
of measurement error (typically normally distributed), com-
axis. It is equal to the time for 63.2% failure.
bined with those unique effects which skew service life data
NOTE 1—This is arrived at by allowing t to equal c in the above
towards early failure (infant mortality failures) or late failure
−1
expression. This then reduces to Failure Probability 5 1−e , which
times (aging or wear-out failures) Applications of the prin-
further reduces to equal 1−0.368 or .632.
ciples in this guide can be helpful in allowing investigators to
3.1.9 complete data—A complete data set is one where all
interpret such data.
of the specimens placed on test fail by the end of the allocated
NOTE 2—Servicelifeorreliabilitydataanalysispackagesarebecoming
test time.
more readily available in standard or common computer software pack-
3.1.10 Incomplete data—An incomplete data set is one
ages.This puts data reduction and analyses more readily into the hands of
where (a) there are some specimens that are still surviving at
a growing number of investigators.
the expiration of the allowed test time, (b) where one or more
5. Data Analysis
specimens is removed from the test prior to expiration of the
allowedtesttime.Theshapeandscaleparametersoftheabove 5.1 In the determinations of service life, a variety of factors
distributions may be estimated even if some of the test act to produce deviations from the expected values. These
specimensdidnotfail.Therearethreedistinctcaseswherethis factors may be of a purely random nature and act to either
might occur. increaseordecreaseservicelifedependingonthemagnitudeof
3.1.10.1 Time censored—Specimens that were still surviv- the factor. The purity of a lubricant is an example of one such
ing when the test was terminated after elapse of a set time are factor. An oil clean and free of abrasives and corrosive
consideredtobetime censored.This is also referredtoasright materials would be expected to prolong the service life of a
censored or type I censoring. Graphical solutions can still be moving part subject to wear. A fouled contaminated oil might
used for parameter estimation. At least ten observed failures prove to be harmful and thereby shorten service life. Purely
should be used for estimating parameters (for example slope randomvariationinanagingfactorthatcaneitherhelporharm
and intercept). a service life might lead a normal, or gaussian, distribution.
3.1.10.2 specimen censored—Specimens that were still sur- Such distributions are symmetrical about a central tendency,
viving when the test was terminated after a set number of usually the mean.
failures are considered to be specimen censored. This is 5.1.1 Some non-random factors act to skew service life
anothercaseofrightcensoredortypeIcensoring.See3.1.10.1 distributions. Defects are generally thought of as factors that
3.1.10.3 Multiply Censored—Specimens that wereremoved can only decrease service life. Thin spots in protective coat-
priortotheendofthetestwithoutfailingarereferredtoasleft ings, nicks in extruded wires, chemical contamination in thin
FIG. 1 Effect of the Shape Parameter (b) on the Weibull Probability Density
G166
metallic films are examples of such defects that can cause an logarithmic form and non-symmetrical in its natural form. (for
overallfailureeventhroughthebulkofthematerialisfarfrom example, x¯ 51 6.2sinlogarithmicformtranslatesto10+5.8
failure.Thesefactorsskewtheservicelifedistributiontowards and −3.7 in natural form)
early failure times. 5.3.2 As there is no symmetrical restriction, the shape of
5.1.2 Factors that skew service life towards the high side thisfunctionmaybeabetterfitthanthenormaldistributionfor
also exist. Preventive maintenance, high quality raw materials, the service life distributions of the material being investigated.
reduced impurities, and inhibitors or other additives are such 5.4 Weibull Distribution—While the Swedish Professor
factors. These factors produce life time distributions shifted Waloddi Weibull was not the first to use this expression, his
towardsthelongtermandarethosetypicallyfoundinproducts paper, A Statistical Distribution of Wide Applicability pub-
having been produced a relatively long period of time. lished in 1951 did much to draw attention to this exponential
5.1.3 Establishing a description of the distribution of fre- function. The simplicity of formula given in (1), hides its
quency (or probability) of failure versus time in service is the extreme flexibility to model service life distributions.
objective of this guide. Determination of the shape of this 5.4.1 The Weibull distribution owes its flexibility to the
distribution as well as its position along the time scale axis are “shape” parameter. The shape of this distribution is dependent
the principle criteria for estimating service life. on the value of b. If b is less than 1, the Weibull distribution
5.2 Normal (Gaussian) Distribution—The characteristic of modelsfailuretimeshavingadecreasingfailurerate.Thetimes
the normal, or Gaussian distribution is a symmetrical bell betweenfailuresincreasewithexposuretime.Ifb 51,thenthe
shaped curve centered on the mean of this distribution. The Weibull models failure times having constant failure rate. If b
meanrepresentsthetimefor50%failure.Thismaybedefined > 1 it models failure times having an increasing failure rate, if
as either the time when one can expect 50% of the entire b 5 2, then Weibull exactly duplicates the Rayleigh distribu-
population to fail or the probability of an individual item to tion, as b approaches 2.5 it very closely approximates the
fail.The“scale”ofthenormalcurveisthemeanvalue(x¯),and lognormal distribution, as b approaches 3. the Weibull expres-
the shape of this curve is established by the standard deviation sion models the normal distribution and as b grows beyond 4,
value (s). the Weibull expression models distributions skewed towards
5.2.1 The normal distribution has found widespread use in longfailuretimes.SeeFig.1forexamplesofdistributionswith
describing many naturally occurring distributions. Its first different shape parameters.
known description by Carl Gauss showed its applicability to 5.4.2 The Weibull distribution is most appropriate when
measurement error. Its applications are widely known and therearemanypossiblesiteswherefailuremightoccurandthe
numerous texts produce exhaustive tables and descriptions of system fails upon the occurrence of the first site failure. An
this function. example commonly used for this type of situation is a chain
5.2.2 Widespread use should not be confused with justifi- failing when only one link separates.All of the sites, or links,
cation for its application to service life data. Use of analysis areequallyatrisk,yetoneisallthatisrequiredfortotalfailure.
techniques developed for normal distribution on data distrib- 5.5 Exponential Distribution—This distribution is a special
uted in a non-normal manner can lead to grossly erroneous case of the Weibull. It is useful to simplify calculations
conclusions. As described in Section 5, many service life involving periods of service life that are subject to random
distributions are skewed towards either early life or late life. failures. These would include random defects but not include
The confinement to a symmetrical shape is the principal wear-out or burn-in periods.
shortcoming of the normal distribution for service life appli-
6. Parameter Estimation
cations. This may lead to situations where even negative
6.1 Weibull data analysis functions are not uncommon but
lifetimes are predicted.
not yet found on all data analysis packages. Fortunately, the
5.3 Lognormal Distribution—This distribution has shown
expression is simple enough so that parameter estimation may
application when the specimen fails due to a multiplicative
be made easily. What follows is a step-by-step example for
process that degrades performance over time. Metal fatigue is
estimating the Weibull distribution parameters from experi-
one example. Degradation is a function of the amount of
mental data.
flexing,cracks,crackangle,numberofflexes,etc.Performance
6.1.1 TheWeibull distribution, (Eq 2) may be rearranged as
eventually degrades to the defined end of life.
shown below: (Eq 3)
5.3.1 Thereareseveralconvenientfeaturesofthelognormal
t b
distribution. First, there is essentially no new mathematics to
S D
c
1 2 F~t! 5 e (3)
introduce into the analysis of this distribution beyond those of
thenormaldistribution.Asimplelogarithmictransformationof and, by taking the natural logarithm of both sides twice, this
data converts lognormal distributed data into a normal distri- expression becomes
bution.Allofthetables,graphs,analysisroutinesetc.maythen
ln ln 5bln~t! 2blnc (4)
be used to describe the transformed function. One note of F G
1 2 F~t!
caution is that the shape parameter s is symmetrical in its
Eq 4 is in the form of an equation describing a straight line
(y 5mx+y ) with
[Ref. Mann, 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 of Weibull, W., “A statistical distribution of wide applicability”, J. Appl. Mech.,
Reliability Theory, Academic Press, New York 1969). 18, 1951, pp 293–297
G166
1 TABLE 1 Time to Failure (days of operation) for Incandescent
ln ln (5)
F G
Lamps
1 2 F t!
~
Lamp ID Days of Operation Lamp ID Days of Operation
corresponding toY, ln(t) corresponding
...