ASTM E2555-07(2018)

NOTICE: This standard has either been superseded and replaced by a new version or withdrawn.
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Designation: E2555 − 07 (Reapproved 2018) An American National Standard
Standard Practice for
Factors and Procedures for Applying the MIL-STD-105 Plans
in Life and Reliability Inspection
This standard is issued under the fixed designation E2555; 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 2.2 Other Documents:
MIL-STD-105ESampling Procedures and Tables for In-
1.1 This practice presents a procedure and related tables of
spection by Attributes
factors for adapting Practice E2234 (equivalent to MIL-STD-
105) sampling plans to acceptance sampling inspection when
3. Terminology
the item quality of interest is life length or reliability. Factors
3.1 Definitions:
are provided for three alternative criteria for lot evaluation:
3.1.1 TheterminologydefinedinTerminologyE456applies
mean life, hazard rate, and reliable life. Inspection of the
to this practice unless modified herein.
sampleisbyattributeswithtestingtruncatedattheendofsome
3.1.2 acceptancequalitylevel(AQL),n—qualitylimitthatis
prearranged period of time. The Weibull distribution, together
theworsttolerableprocessaveragewhenacontinuingseriesof
with the exponential distribution as a special case, is used as
lots is submitted for acceptance sampling. E2234
the underlying statistical model.
3.1.2.1 Discussion—This term is often referred to as the
1.2 A system of units is not specified by this practice.
“acceptance quality limit.”
1.3 This standard does not purport to address all of the
3.1.2.2 Discussion—Thisdefinitionsupersedesthatgivenin
safety concerns, if any, associated with its use. It is the
MIL-STD-105E.
responsibility of the user of this standard to establish appro-
3.1.2.3 Discussion—A sampling plan and an AQL are cho-
priate safety, health, and environmental practices and deter-
sen in accordance with the risk assumed. Use of a value of
mine the applicability of regulatory limitations prior to use.
AQL for a certain defect or group of defects indicates that the
1.4 This international standard was developed in accor-
sampling plan will accept the great majority of the lots or
dance with internationally recognized principles on standard-
batchesprovidedtheprocessaveragelevelofpercentdefective
ization established in the Decision on Principles for the
(or defects per hundred units) in these lots or batches are no
Development of International Standards, Guides and Recom-
greater than the designated value ofAQL. Thus, theAQL is a
mendations issued by the World Trade Organization Technical
designated value of percent defective (or defects per hundred
Barriers to Trade (TBT) Committee.
units) for which lots will be accepted most of the time by the
sampling procedure being used. The sampling plans provided
2. Referenced Documents
herein are so arranged that the probability of acceptance at the
2.1 ASTM Standards:
designated AQL value depends upon the sample size, being
E456Terminology Relating to Quality and Statistics
generally higher for large samples than for small ones, for a
E2234Practice for Sampling a Stream of Product by Attri-
given AQL. The AQL alone does not identify the chances of
butes Indexed by AQL
accepting or rejecting individual lots or batches but more
directly relates to what might be expected from a series of lots
or batches, provided the steps indicated in this refer to the
ThispracticeisunderthejurisdictionofASTMCommitteeE11onQualityand
Statistics and is the direct responsibility of Subcommittee E11.40 on Reliability.
operating characteristic curve of the plan to determine the
Current edition approved April 1, 2018. Published May 2018. Originally
relative risks.
approved in 2007. Last previous version approved in 2012 as E2555–07 (2012).
DOI: 10.1520/E2555-07R18.
For referenced ASTM standards, visit the ASTM website, www.astm.org, or
contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM MIL-STD-105Eisalsocommonlyreferredtoas“MIL-STD-105.”Itisvirtually
Standards volume information, refer to the standard’s Document Summary page on identical in content to its predecessor, MIL-STD-105D.These documents are out of
the ASTM website. print.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E2555 − 07 (2018)
3.1.3 consumer’s risk, n—probability that a lot having 3.2 Definitions of Terms Specific to This Standard:
specified rejectable quality level will be accepted under a 3.2.1 acceptance number, n—the maximum number of
defined sampling plan. failed items allowed in the sample for the lot to be accepted
using a single or multiple sampling plan.
3.1.4 double sampling plan, n—a multiple sampling plan in
which up to two samplings can be taken and evaluated to 3.2.2 hazard rate, n—differential fraction of items failing at
accept or reject a lot. time t among those surviving up to time t, symbolized by h(t).
3.2.2.1 Discussion—h(t) is also referred to as the instanta-
3.1.5 limiting quality level (LQL), n—quality level having a
neous failure rate at time t. It is related to the probability
specified consumer’s risk for a given sampling plan.
density and cumulative distribution functions by h(t)= f(t)
3.1.6 lot, n—a definite quantity of a product or material
/(l– F(t)).
accumulated under conditions that are considered uniform for
3.2.3 mean life, n—average time that items in the lot or
sampling purposes.
population are expected to operate before failure.
3.1.6.1 Discussion—The lot for sampling may differ from a
3.2.3.1 Discussion—Thismetricisoftenreferredtoasmean
collectionofunitsdesignatedasabatchforotherpurposes,for
time to failure (MTTF) or mean time before failure (MTBF).
example, production, shipment, and so forth.
3.2.4 rejection number, n—the minimum number of failed
3.1.7 multiple sampling plan, n—a sampling plan in which
items in the sample that will cause the lot to be rejected under
successive samples from a lot are drawn and after each sample
a given sampling plan.
is inspected a decision is made to accept the lot, reject the lot,
or to take another sample, based on quality level of the
3.2.5 reliable life (ρ ), n—life beyond which some specified
r
combined samples.
proportion, r, of the items in the lot or population will survive.
3.1.7.1 Discussion—When the quality is much less or much
3.2.6 test truncation time (t), n—amount of time sampled
more than the AQL, the decision can be made on the first
items are allowed to be tested.
sample, which is smaller than that of a single sampling plan
3.2.7 Weibull distribution, n—probability distribution hav-
with equivalent acceptance quality level. For samples that are
ing cumulative distribution:
closetotheAQLinquality,additionalsamplesarerequiredand
β
t 2 γ
the total sample size will be larger than the corresponding
function F t 51 2exp 2 , t.γ andprobabilitydensity
~ ! S S D D
single sampling plan. η
β21 β
β t 2 γ t 2 γ
3.1.8 sample, n—group of items, observations, test results,
function f t 5 exp 2
~ ! S D S S D D
η η η
or portions of material taken from a large collection of items,
3.2.7.1 Discussion—TheWeibulldistributioniswidelyused
observations,testresults,orquantitiesofmaterialthatservesto
for modeling product life. It can take a wide variety of shapes
provide information that may be used as a basis for making a
andalsothecharacteristicsofothertypesofdistributionsbased
decision concerning the larger collection. E2234
on the value of its parameters. γ is called the location,
minimum life, or threshold parameter and defines the lower
limit of the distribution (Fig. 1). η is called the scale or
FIG. 1 Effect of the Parameter γ on the Weibull Probability Den-
sity Function, f(t)
E2555 − 07 (2018)
characteristic life parameter and is equal to the 63.2 percentile 4.2.5 Compare the number of items that failed with the
of the distribution, minus γ (Fig. 2). β is the shape parameter number allowed under the selected Practice E2234 plan.
(Fig. 3). The exponential distribution is the special case where
4.2.6 If the number that failed is equal to or less than the
γ = 0 and β=l.
acceptable number, accept the lot; if the number failing
exceeds the acceptable number, reject the lot.
4. Significance and Use
4.3 Both the sample sizes and the acceptance numbers used
4.1 The procedure and tables presented in this practice are
arethosespecifiedbyPracticeE2234plans.Itwillbeassumed
based on the use of the Weibull distribution in acceptance
in the section on examples that single sampling plans will be
sampling inspection. Details of this work, together with tables
used. However, the matching double sampling and multiple
of sampling plans of other forms, have been published previ-
sampling plans provided in MIL-STD-105 can be used if
ously. See Refs (1-3). Since the basic computations required
desired. The corresponding sample sizes and acceptance and
havealreadybeenmade,ithasbeenquiteeasytoprovidethese
new factors. No changes in method or details of application rejection numbers are used in the usual way.The specified test
have been made over those described in the publications truncation time, t, must be used for all samples.
referencedabove.Forthisreason,thetextportionofthisreport
4.4 The probability of acceptance for a lot under this
has been briefly written. Readers interested in further details
procedure depends only on the probability of a sample item
are referred to these previous publications. Other sources of
failing before the end of the test truncation time, t. For this
material on the underlying theory and approach are also
reason, the actual life at failure need not be determined; only
available (4-7).
thenumberofitemsfailingisofinterest.Liferequirementsand
4.2 The procedure to be used is essentially the same as the
test time specifications need not necessarily be measured in
one normally used for attribute sampling inspection. The only
chronologicaltermssuchasminutesorhours.Forexample,the
difference is that sample items are tested for life or survival
life measure may be cycles of operation, revolutions, or miles
instead of for some other property. For single sampling, the
of travel.
following are the required steps:
4.2.1 Using the tables of factors provided in Annex A1, 4.5 Theunderlyinglifedistributionassumedinthisstandard
select a suitable sampling inspection plan from those tabulated
is the Weibull distribution (note that the exponential distribu-
in Practice E2234.
tion is a special case of the Weibull). The Weibull model has
4.2.2 Drawatrandomasampleofitemsofthesizespecified
threeparameters.Oneparameterisascaleorcharacteristiclife
by the selected Practice E2234 plan.
parameter. For these plans and procedures, the value for this
4.2.3 Place the sample of items on life test for the specified
parameter need not be known; the techniques used are inde-
period of time, t.
pendent of its magnitude. A second parameter is a location or
4.2.4 Determine the number of sample items that failed
“guaranteedlife”parameter.Intheseplansandprocedures,itis
during the test period.
assumed that this parameter has a value of zero and that there
is some risk of item failure right from the start of life. If this is
not the case for some applications, a simple modification in
Theboldfacenumbersinparenthesesrefertothelistofreferencesattheendof
procedure is available. The third parameter, and the one of
this standard.
FIG. 2 Effect of the Parameter η on the Weibull Probability Den-
sity Function, f(t)
E2555 − 07 (2018)
FIG. 3 Effect of the Parameter β on the Weibull Probability Den-
sity Function, f(t)
importance, is the shape parameter, β. The magnitude of the 4.8 AnnexTable1Alists, for each selected shape parameter
conversion factors used in the procedures described in this value, 100t/µ ratios for each of the Practice E2234 AQL
reportdependsdirectlyonthevalueforthisparameter.Forthis [p’(%)] values. With acceptance inspection plans selected in
reason,themagnitudeoftheparametershallbeknownthrough termsoftheseratios,theprobabilityofacceptancewillbehigh
experience with the product or shall be estimated from past for lots whose mean life meets the specified requirement. The
research, engineering, or inspection data. Estimation proce-
actualprobabilityofacceptancewillvaryfromplantoplanand
dures are available and are outlined in Ref (1). maybereadfromtheassociatedoperatingcharacteristiccurves
suppliedinMIL-STD-105.Thecurvesareenteredbyusingthe
4.6 Forthecommoncaseofrandomchancefailureswiththe
corresponding p’(%) value.Annex Table1B lists 100t/µ ratios
failurerateconstantovertime,ratherthanfailuresasaresultof
attheLQLforthequalitylevelatwhichtheconsumer’sriskis
“infant mortality” or wearout, a value of 1 for the shape
0.10. Annex Table1C lists corresponding 100t/µ ratios for a
parameter shall be assumed. With this parameter value, the
consumer’s risk of 0.05.
Weibull distribution reduces to the exponential. Tables of
4.8.1 These ratios are to be used directly for the usual case
conversion factors are provided in Annex A1 for 15 selected
for which the value for the Weibull location or threshold
shape parameter values ranging from ⁄2 to 10, the range
parameter (γ) can be assumed as zero. If γ is not zero but has
commonly encountered in industrial and technical practice.
someotherknownvalue,allthatshallbedoneistosubtractthe
Thevalue1,usedfortheexponentialcase,isincluded.Factors
value for γ from t to get t and from m to get m . These
for other required shape parameter values within this range
0 0
transformedvalues, t and m ,arethenemployedintheuseof
may be obtained approximately by interpolation. A more
0 0
thetablesandforallothercomputations.Asolutionintermsof
complete discussion of the relationship between failure pat-
m and t can then be converted back to actual or absolute
terns and the Weibull parameters can be found in Refs (1-3).
0 0
values by adding the value for γ to each.
4.7 One possible acceptance criterion is the mean life for
items making up the lot (µ). Mean life conversion factors or
5. Examples, Mean Life Ratio
valuesforthedimensionlessratio100t/µhavebeendetermined
tocorrespondtoorreplaceallthep’orpercentdefectivevalues
5.1 A Practice E2234 acceptance sampling inspection plan
associated with Practice E2234 plans. In this factor, t repre-
istobeappliedtoincominglotsofproductforwhichthemean
sentsthespecifiedtesttruncationtimeandµthemeanitemlife
item life is the property of interest.An acceptable mean life of
for the lot. For reliability o
...