ASTM G166-00(2011)
(Guide)Standard Guide for Statistical Analysis of Service Life Data
Standard Guide for Statistical Analysis of Service Life Data
SIGNIFICANCE AND USE
Service life test data often show different distribution shapes than many other types of data. This is due to the effects of measurement error (typically normally distributed), combined with those unique effects which skew service life data towards early failure (infant mortality failures) or late failure times (aging or wear-out failures) Applications of the principles in this guide can be helpful in allowing investigators to interpret such data.
Note 2—Service life or reliability data analysis packages are becoming more readily available in standard or common computer software packages. This puts data reduction and analyses more readily into the hands of a growing number of investigators.
SCOPE
1.1 This guide presents briefly some generally accepted methods of statistical analyses which are useful in the interpretation of service life data. It is intended to produce a common terminology as well as developing a common methodology and quantitative expressions relating to service life estimation.
1.2 This guide does not cover detailed derivations, or special cases, but rather covers a range of approaches which have found application in service life data analyses.
1.3 Only those statistical methods that have found wide acceptance in service life data analyses have been considered in this guide.
1.4 The Weibull life distribution model is emphasized in this guide and example calculations of situations commonly encountered in analysis of service life data are covered in detail.
1.5 The choice and use of a particular life distribution model should be based primarily on how well it fits the data and whether it leads to reasonable projections when extrapolating beyond the range of data. Further justification for selecting a model should be based on theoretical considerations.
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Standards Content (Sample)
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Designation: G166 − 00 (Reapproved 2011)
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 (´) indicates an editorial change since the last revision or reapproval.
1. Scope 3.1.1.1 Discussion—There exists many ASTM recognized
and standardized measurement procedures for determining
1.1 This guide presents briefly some generally accepted
material properties. As these practices have been developed
methods of statistical analyses which are useful in the inter-
withincommitteeswithappropriateexpertise,nofurtherelabo-
pretation of service life data. It is intended to produce a
ration will be provided.
common terminology as well as developing a common meth-
odology and quantitative expressions relating to service life
3.1.2 beginning of life—this is usually determined to be the
estimation.
time of manufacture. Exceptions may include time of delivery
to the end user or installation into field service.
1.2 This guide does not cover detailed derivations, or
special cases, but rather covers a range of approaches which
3.1.3 end of life—Occasionally this is simple and obvious
have found application in service life data analyses.
such as the breaking of a chain or burning out of a light bulb
1.3 Only those statistical methods that have found wide filament. In other instances, the end of life may not be so
acceptance in service life data analyses have been considered catastrophic and free from argument. Examples may include
in this guide. fading, yellowing, cracking, crazing, etc. Such cases need
quantitative measurements and agreement between evaluator
1.4 TheWeibulllifedistributionmodelisemphasizedinthis
anduserastotheprecisedefinitionoffailure.Itisalsopossible
guide and example calculations of situations commonly en-
to model more than one failure mode for the same specimen.
countered in analysis of service life data are covered in detail.
(forexample,Thetimetoproduceagivenamountofyellowing
1.5 Thechoiceanduseofaparticularlifedistributionmodel
may be measured on the same specimen that is also tested for
should be based primarily on how well it fits the data and
cracking.)
whether it leads to reasonable projections when extrapolating
3.1.4 F(t)—The probability that a random unit drawn from
beyond the range of data. Further justification for selecting a
the population will fail by time (t). Also F(t) = the decimal
model should be based on theoretical considerations.
fractionofunitsinthepopulationthatwillfailbytime (t).The
2. Referenced Documents
decimal fraction multiplied by 100 is numerically equal to the
percent failure by time (t).
2.1 ASTM Standards:
G169Guide for Application of Basic Statistical Methods to
3.1.5 R(t)—The probability that a random unit drawn from
Weathering Tests
the population will survive at least until time (t). Also R(t) =
the fraction of units in the population that will survive at least
3. Terminology
until time (t)
3.1 Definitions:
R~t! 51 2 F~t! (1)
3.1.1 material property—customarily, service life is consid-
ered to be the period of time during which a system meets 3.1.6 pdf—theprobabilitydensityfunction(pdf),denotedby
critical specifications. Correct measurements are essential to
f(t),equalstheprobabilityoffailurebetweenanytwopointsof
dF t
~ !
producing meaningful and accurate service life estimates.
time t(1) and t(2). Mathematically f t 5 . For the normal
~ !
dt
distribution, the pdf is the “bell shape” curve.
This guide is under the jurisdiction of ASTM Committee G03 on Weathering
and Durability and is the direct responsibility of Subcommittee G03.08 on Service 3.1.7 cdf—the cumulative distribution function (cdf), de-
Life Prediction.
noted by F(t), represents the probability of failure (or the
Current edition approved July 1, 2011. Published August 2011. Originally
population fraction failing) by time = (t). See section 3.1.4.
approved in 2000. Last previous edition approved in 2005 as G166–00(2005).
DOI: 10.1520/G0166-00R11.
3.1.8 weibull distribution—For the purposes of this guide,
For referenced ASTM standards, visit the ASTM website, www.astm.org, or
the Weibull distribution is represented by the equation:
contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
Standards volume information, refer to the standard’s Document Summary page on
t b
S D
the ASTM website. F t 51 2 e c (2)
~ !
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
G166 − 00 (2011)
where: failures are considered to be specimen censored. This is
anothercaseofrightcensoredortypeIcensoring.See3.1.10.1
F(t) = defined in paragraph 3.1.4
t = units of time used for service life
3.1.10.3 Multiply Censored—Specimens that were removed
c = scale parameter
priortotheendofthetestwithoutfailingarereferredtoasleft
b = shape parameter
censored or type II censored. Examples would include speci-
3.1.8.1 The shape parameter (b), section 3.1.6, is so called
mens that were lost, dropped, mishandled, damaged or broken
because this parameter determines the overall shape of the
duetostressesnotpartofthetest.Adjustmentsoffailureorder
curve. Examples of the effect of this parameter on the distri-
can be made for those specimens actually failed.
bution curve are shown in Fig. 1, section 5.3.
3.1.8.2 The scale parameter (c), section 3.1.6, is so called
4. Significance and Use
because it positions the distribution along the scale of the time
4.1 Service life test data often show different distribution
axis. It is equal to the time for 63.2% failure.
shapes than many other types of data.This is due to the effects
NOTE 1—This is arrived at by allowing t to equal c in the above
−1 of measurement error (typically normally distributed), com-
expression.ThisthenreducestoFailureProbability=1−e ,whichfurther
bined with those unique effects which skew service life data
reduces to equal 1−0.368 or .632.
towards early failure (infant mortality failures) or late failure
3.1.9 completedata—Acompletedatasetisonewhereallof
times (aging or wear-out failures) Applications of the prin-
thespecimensplacedontestfailbytheendoftheallocatedtest
ciples in this guide can be helpful in allowing investigators to
time.
interpret such data.
3.1.10 Incomplete data—An incomplete data set is one
where (a) there are some specimens that are still surviving at
NOTE2—Servicelifeorreliabilitydataanalysispackagesarebecoming
the expiration of the allowed test time, (b) where one or more
more readily available in standard or common computer software pack-
ages.Thisputsdatareductionandanalysesmorereadilyintothehandsof
specimens is removed from the test prior to expiration of the
a growing number of investigators.
allowedtesttime.Theshapeandscaleparametersoftheabove
distributions may be estimated even if some of the test
5. Data Analysis
specimensdidnotfail.Therearethreedistinctcaseswherethis
might occur.
5.1 In the determinations of service life, a variety of factors
3.1.10.1 Time censored—Specimens that were still surviv- act to produce deviations from the expected values. These
ing when the test was terminated after elapse of a set time are
factors may be of a purely random nature and act to either
consideredtobetimecensored.Thisisalsoreferredtoasright
increaseordecreaseservicelifedependingonthemagnitudeof
censored or type I censoring. Graphical solutions can still be
the factor. The purity of a lubricant is an example of one such
used for parameter estimation. At least ten observed failures
factor. An oil clean and free of abrasives and corrosive
should be used for estimating parameters (for example slope
materials would be expected to prolong the service life of a
and intercept).
moving part subject to wear. A fouled contaminated oil might
3.1.10.2 specimen censored—Specimens that were still sur- prove to be harmful and thereby shorten service life. Purely
viving when the test was terminated after a set number of randomvariationinanagingfactorthatcaneitherhelporharm
FIG. 1 Effect of the Shape Parameter (b) on the Weibull Probability Density
G166 − 00 (2011)
a service life might lead a normal, or gaussian, distribution. 5.3.1 Thereareseveralconvenientfeaturesofthelognormal
Such distributions are symmetrical about a central tendency, distribution. First, there is essentially no new mathematics to
usually the mean. introduce into the analysis of this distribution beyond those of
5.1.1 Some non-random factors act to skew service life thenormaldistribution.Asimplelogarithmictransformationof
data converts lognormal distributed data into a normal distri-
distributions. Defects are generally thought of as factors that
can only decrease service life. Thin spots in protective bution.Allofthetables,graphs,analysisroutinesetc.maythen
coatings, nicks in extruded wires, chemical contamination in be used to describe the transformed function. One note of
thin metallic films are examples of such defects that can cause caution is that the shape parameter σ is symmetrical in its
an overall failure even through the bulk of the material is far logarithmic form and non-symmetrical in its natural form. (for
from failure. These factors skew the service life distribution example, x¯=1 6 .2σ in logarithmic form translates to 10 +5.8
towards early failure times. and −3.7 in natural form)
5.1.2 Factors that skew service life towards the high side 5.3.2 As there is no symmetrical restriction, the shape of
thisfunctionmaybeabetterfitthanthenormaldistributionfor
also exist. Preventive maintenance, high quality raw materials,
reduced impurities, and inhibitors or other additives are such the service life distributions of the material being investigated.
factors. These factors produce life time distributions shifted
5.4 Weibull Distribution—While the Swedish Professor
towardsthelongtermandarethosetypicallyfoundinproducts
Waloddi Weibull was not the first to use this expression, his
having been produced a relatively long period of time.
paper, A Statistical Distribution of Wide Applicability pub-
5.1.3 Establishing a description of the distribution of fre-
lished in 1951 did much to draw attention to this exponential
quency (or probability) of failure versus time in service is the
function. The simplicity of formula given in (1), hides its
objective of this guide. Determination of the shape of this
extreme flexibility to model service life distributions.
distribution as well as its position along the time scale axis are
5.4.1 The Weibull distribution owes its flexibility to the
the principle criteria for estimating service life.
“shape” parameter. The shape of this distribution is dependent
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=1,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 = 2, then Weibull exactly duplicates the Rayleigh
population to fail or the probability of an individual item to
distribution, as b approaches 2.5 it very closely approximates
fail.The“scale”ofthenormalcurveisthemeanvalue(x¯),and
the lognormal distribution, as b approaches 3. the Weibull
the shape of this curve is established by the standard deviation
expression models the normal distribution and as b grows
value (σ).
beyond 4, the Weibull expression models distributions skewed
5.2.1 The normal distribution has found widespread use in
towards long failure times. See Fig. 1 for examples of
describing many naturally occurring distributions. Its first
distributions with 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-
uted in a non-normal manner can lead to grossly erroneous
5.5 Exponential Distribution—This distribution is a special
conclusions. As described in Section 5, many service life
case of the Weibull. It is useful to simplify calculations
distributions are skewed towards either early life or late life.
involving periods of service life that are subject to random
The confinement to a symmetrical shape is the principal
failures. These would include random defects but not include
shortcoming of the normal distribution for service life appli-
wear-out or burn-in periods.
cations. This may lead to situations where even negative
6. Parameter Estimation
lifetimes are predicted.
6.1 Weibull data analysis functions are not uncommon but
5.3 Lognormal Distribution—This distribution has shown
not yet found on all data analysis packages. Fortunately, the
application when the specimen fails due to a multiplicative
expression is simple enough so that parameter estimation may
process that degrades performance over time. Metal fatigue is
be made easily. What follows is a step-by-step example for
one example. Degradation is a function of the amount of
estimating the Weibull distribution parameters from experi-
flexing,cracks,crackangle,numberofflexes,etc.Performance
mental data.
eventually degrades to the defined end of life.
6.1.1 TheWeibull distribution, (Eq 2) may be rearranged as
shown below: (Eq 3)
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 − 00 (2011)
t b
were still burning. A data sh
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