ASTM E2555-07(2018)
(Practice)Standard Practice for Factors and Procedures for Applying the MIL-STD-105 Plans in Life and Reliability Inspection
Standard Practice for Factors and Procedures for Applying the MIL-STD-105 Plans in Life and Reliability Inspection
SIGNIFICANCE AND USE
4.1 The procedure and tables presented in this practice are based on the use of the Weibull distribution in acceptance sampling inspection. Details of this work, together with tables of sampling plans of other forms, have been published previously. See Refs (1-3).4 Since the basic computations required have already been made, it has been quite easy to provide these new factors. No changes in method or details of application have been made over those described in the publications referenced above. For this reason, the text portion of this report has been briefly written. Readers interested in further details are referred to these previous publications. Other sources of material on the underlying theory and approach are also available (4-7).
4.2 The procedure to be used is essentially the same as the one normally used for attribute sampling inspection. The only difference is that sample items are tested for life or survival instead of for some other property. For single sampling, the following are the required steps:
4.2.1 Using the tables of factors provided in Annex A1, select a suitable sampling inspection plan from those tabulated in Practice E2234.
4.2.2 Draw at random a sample of items of the size specified by the selected Practice E2234 plan.
4.2.3 Place the sample of items on life test for the specified period of time, t.
4.2.4 Determine the number of sample items that failed during the test period.
4.2.5 Compare the number of items that failed with the number allowed under the selected Practice E2234 plan.
4.2.6 If the number that failed is equal to or less than the acceptable number, accept the lot; if the number failing exceeds the acceptable number, reject the lot.
4.3 Both the sample sizes and the acceptance numbers used are those specified by Practice E2234 plans. It will be assumed in the section on examples that single sampling plans will be used. However, the matching double sampling and multiple sampling plans provided in MIL-...
SCOPE
1.1 This practice presents a procedure and related tables of factors for adapting Practice E2234 (equivalent to MIL-STD-105) sampling plans to acceptance sampling inspection when the item quality of interest is life length or reliability. Factors are provided for three alternative criteria for lot evaluation: mean life, hazard rate, and reliable life. Inspection of the sample is by attributes with testing truncated at the end of some prearranged period of time. The Weibull distribution, together with the exponential distribution as a special case, is used as the underlying statistical model.
1.2 A system of units is not specified by this practice.
1.3 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility of the user of this standard to establish appropriate safety, health, and environmental practices and determine the applicability of regulatory limitations prior to use.
1.4 This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
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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
...
This document is not an ASTM standard and is intended only to provide the user of an ASTM standard an indication of what changes have been made to the previous version. Because
it may not be technically possible to adequately depict all changes accurately, ASTM recommends that users consult prior editions as appropriate. In all cases only the current version
of the standard as published by ASTM is to be considered the official document.
Designation: E2555 − 07 (Reapproved 2012) 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. A number in parentheses indicates the year of last reapproval. A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. Scope
1.1 This practice presents a procedure and related tables of factors for adapting Practice E2234 (equivalent to MIL-STD-105)
sampling plans to acceptance sampling inspection when the item quality of interest is life length or reliability. Factors are provided
for three alternative criteria for lot evaluation: mean life, hazard rate, and reliable life. Inspection of the sample is by attributes with
testing truncated at the end of some prearranged period of time. The Weibull distribution, together with the exponential distribution
as a special case, is used as the underlying statistical model.
1.2 A system of units is not specified by this practice.
1.3 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility
of the user of this standard to establish appropriate safety safety, health, and healthenvironmental practices and determine the
applicability of regulatory limitations prior to use.
1.4 This international standard was developed in accordance with internationally recognized principles on standardization
established in the Decision on Principles for the Development of International Standards, Guides and Recommendations issued
by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
2. Referenced Documents
2.1 ASTM Standards:
E456 Terminology Relating to Quality and Statistics
E2234 Practice for Sampling a Stream of Product by Attributes Indexed by AQL
2.2 Other Documents:
MIL-STD-105E Sampling Procedures and Tables for Inspection by Attributes
3. Terminology
3.1 Definitions:
3.1.1 The terminology defined in Terminology E456 applies to this practice unless modified herein.
3.1.2 acceptance quality level (AQL), n—quality limit that is the worst tolerable process average when a continuing series of
lots is submitted for acceptance sampling. E2234
3.1.2.1 Discussion—
This term is often referred to as the “acceptance quality limit.”
3.1.2.2 Discussion—
This definition supersedes that given in MIL-STD-105E.
3.1.2.3 Discussion—
This practice is under the jurisdiction of ASTM Committee E11 on Quality and Statistics and is the direct responsibility of Subcommittee E11.40 on Reliability.
Current edition approved May 1, 2012April 1, 2018. Published May 2012May 2018. Originally approved in 2007. Last previous version approved in 20072012 as
E2555 – 07.E2555 – 07 (2012). DOI: 10.1520/E2555-07R12.10.1520/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 Standards
volume information, refer to the standard’s Document Summary page on the ASTM website.
MIL-STD-105E is also commonly referred to as “MIL-STD-105.” It is virtually identical in content to its predecessor, MIL-STD-105D. These documents are out of print.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E2555 − 07 (2018)
A sampling plan and an AQL are chosen in accordance with the risk assumed. Use of a value of AQL for a certain defect or group
of defects indicates that the sampling plan will accept the great majority of the lots or batches provided the process average level
of percent defective (or defects per hundred units) in these lots or batches are no greater than the designated value of AQL. Thus,
the AQL is a designated value of percent defective (or defects per hundred units) for which lots will be accepted most of the time
by the sampling procedure being used. The sampling plans provided herein are so arranged that the probability of acceptance at
the designated AQL value depends upon the sample size, being generally higher for large samples than for small ones, for a given
AQL. The AQL alone does not identify the chances of 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 operating characteristic
curve of the plan to determine the relative risks.
3.1.3 consumer’s risk, n—probability that a lot having specified rejectable quality level will be accepted under a defined
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 accept
or reject a lot.
3.1.5 limiting quality level (LQL), n—quality level having a specified consumer’s risk for a given sampling plan.
3.1.6 lot, n—a definite quantity of a product or material accumulated under conditions that are considered uniform for sampling
purposes.
3.1.6.1 Discussion—
The lot for sampling may differ from a collection of units designated as a batch for other purposes, for example, production,
shipment, and so forth.
3.1.7 multiple sampling plan, n—a sampling plan in which successive samples from a lot are drawn and after each sample is
inspected a decision is made to accept the lot, reject the lot, or to take another sample, based on quality level of the combined
samples.
3.1.7.1 Discussion—
When the quality is much less or much more than the AQL, the decision can be made on the first sample, which is smaller than
that of a single sampling plan with equivalent acceptance quality level. For samples that are close to the AQL in quality, additional
samples are required and the total sample size will be larger than the corresponding single sampling plan.
3.1.8 sample, n—group of items, observations, test results, or portions of material taken from a large collection of items,
observations, test results, or quantities of material that serves to provide information that may be used as a basis for making a
decision concerning the larger collection. E2234
E2555 − 07 (2018)
3.2 Definitions of Terms Specific to This Standard:
3.2.1 acceptance number, n—the maximum number of failed items allowed in the sample for the lot to be accepted using a single
or multiple sampling plan.
3.2.2 hazard rate, n—differential fraction of items failing at 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 instantaneous failure rate at time t. It is related to the probability density and cumulative distribution
functions by h(t) = f(t) /(l – F(t)).
3.2.3 mean life, n—average time that items in the lot or population are expected to operate before failure.
3.2.3.1 Discussion—
This metric is often referred to as mean time to failure (MTTF) or mean time before failure (MTBF).
3.2.4 rejection number, n—the minimum number of failed items in the sample that will cause the lot to be rejected under a given
sampling plan.
3.2.5 reliable life (ρ ) , ), n—life beyond which some specified proportion, r, of the items in the lot or population will survive.
r
3.2.6 test truncation time (t), n—amount of time sampled items are allowed to be tested.
3.2.7 Weibull distribution, n—probability distribution having cumulative distribution:
β
t 2 γ
function F~t! 512exp 2 , t.γ and probability density
S S DD
η
β21 β
β t 2 γ t 2 γ
function f t 5 exp 2
~ ! S D S S DD
η η η
3.2.7.1 Discussion—
The Weibull distribution is widely used for modeling product life. It can take a wide variety of shapes and also the characteristics
of other types of distributions based 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 characteristic life parameter and is equal to the 63.2
percentile of the distribution, minus γ (Fig. 2). β is the shape parameter (Fig. 3). The exponential distribution is the special case
where γ = 0 and β = l.
FIG. 1 Effect of the Parameter γ on the Weibull Probability Den-
sity Function, f(t)
E2555 − 07 (2018)
FIG. 2 Effect of the Parameter η on the Weibull Probability Den-
sity Function, f(t)
FIG. 3 Effect of the Parameter β on the Weibull Probability Den-
sity Function, f(t)
4. Significance and Use
4.1 The procedure and tables presented in this practice are based on the use of the Weibull distribution in acceptance sampling
inspection. Details of this work, together with tables of sampling plans of other forms, have been published previously. See Refs
(1-3). Since the basic computations required have already been made, it has been quite easy to provide these new factors. No
changes in method or details of application have been made over those described in the publications referenced above. For this
reason, the text portion of this report has been briefly written. Readers interested in further details are referred to these previous
publications. Other sources of material on the underlying theory and approach are also available (4-7).
4.2 The procedure to be used is essentially the same as the one normally used for attribute sampling inspection. The only
difference is that sample items are tested for life or survival instead of for some other property. For single sampling, the following
are the required steps:
4.2.1 Using the tables of factors provided in Annex A1, select a suitable sampling inspection plan from those tabulated in
Practice E2234.
4.2.2 Draw at random a sample of items of the size specified by the selected Practice E2234 plan.
The boldface numbers in parentheses refer to the list of references at the end of this standard.
E2555 − 07 (2018)
4.2.3 Place the sample of items on life test for the specified period of time, t.
4.2.4 Determine the number of sample items that failed during the test period.
4.2.5 Compare the number of items that failed with the number allowed under the selected Practice E2234 plan.
4.2.6 If the number that failed is equal to or less than the acceptable number, accept the lot; if the number failing exceeds the
acceptable number, reject the lot.
4.3 Both the sample sizes and the acceptance numbers used are those specified by Practice E2234 plans. It will be assumed in
the section on examples that single sampling plans will be used. However, the matching double sampling and multiple sampling
plans provided in MIL-STD-105 can be used if desired. The corresponding sample sizes and acceptance and rejection numbers are
used in the usual way. The specified test truncation time, t, must be used for all samples.
4.4 The probability of acceptance for a lot under this procedure depends only on the probability of a sample item failing before
the end of the test truncation time, t. For this reason, the actual life at failure need not be determined; only the number of items
failing is of interest. Life requirements and test time specifications need not necessarily be measured in chronological terms such
as minutes or hours. For example, the life measure may be cycles of operation, revolutions, or miles of travel.
4.5 The underlying life distribution assumed in this standard is the Weibull distribution (note that the exponential distribution
is a special case of the Weibull). The Weibull model has three parameters. One parameter is a scale or characteristic life parameter.
For these plans and procedures, the value for this parameter need not be known; the techniques used are independent of its
magnitude. A second parameter is a location or “guaranteed life” parameter. In these plans and procedures, it is 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 procedure is available. The third parameter, and the one of importance, is the shape
parameter, β. The magnitude of the conversion factors used in the procedures described in this report depends directly on the value
for this parameter. For this reason, the magnitude of the parameter shall be known through experience with the product or shall
be estimated from past research, engineering, or inspection data. Estimation procedures are available and are outlined in Ref (1).
4.6 For the common case of random chance failures with the failure rate constant over time, rather than failures as a result of
“infant mortality” or wearout, a value of 1 for the shape parameter shall be assumed. With this parameter value, the Weibull
distribution reduces to the exponential. Tables of conversion factors are provided in Annex A1 for 15 selected shape parameter
values ranging from ⁄2 to 10, the range commonly encountered in industrial and technical practice. The value 1, used for the
exponential case, is included. Factors for other required shape parameter values within this range may be obtained approximately
by interpolation. A more complete discussion of the relationship between failure patterns and the Weibull parameters can be found
in Refs (1-3).
4.7 One possible acceptance criterion is the mean life for items making up the lot (μ). Mean life conversion factors or values
for the dimensionless ratio 100t/μ have been determined to correspond to or replace all the p’ or percent defective values associated
with Practice E2234 plans. In this factor, t represents the specified test truncation time and μ the mean item life for the lot. For
reliability or life-length applications, these factors are used in place of the corresponding p’ values normally used in the use of
Practice E2234 plans for attribute inspection of other item qualities. The use of these factors will be demonstrated by several
examples (see Sections 5, 7, and 9).
4.8 Annex Table 1A lists, for each selected shape parameter value, 100t/μ ratios for each of the Practice E2234 AQL [p’(%)]
values. With acceptance inspection plans selected in terms of these ratios, the probability of acceptance will be high for lots whose
mean life meets the specified requirement. The actual probability of acceptance will var
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