ASTM E2555-21e1

Standard Practice for Factors and Procedures for Applying the MIL-STD-105 Plans in Life and Reliability Inspection

ABSTRACT
This practice presents a procedure and related tables of factors for adapting Practice E2234 (equivalent to MIL-STD105) 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. The procedure and tables presented in this practice are based on the use of the Weibull distribution in acceptance sampling 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 issue...

General Information

• Status: Published
• Publication Date: 30-Apr-2021
• ICS: 07.020 - Mathematics
• Technical Committee: E11 - Quality and Statistics
• Drafting Committee: E11.40 - Reliability

Relations

• Refers: ASTM E456-13a(2022)e1 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 01-Apr-2022
• Refers: ASTM E2586-19e1 - Standard Practice for Calculating and Using Basic Statistics
• Effective Date: 01-Apr-2019
• Refers: ASTM E456-13A(2017)e1 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 01-Oct-2017
• Refers: ASTM E456-13A(2017)e3 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 01-Oct-2017
• Refers: ASTM E2586-14 - Standard Practice for Calculating and Using Basic Statistics
• Effective Date: 01-Jun-2014
• Refers: ASTM E456-13ae3 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 15-Nov-2013
• Refers: ASTM E456-13ae2 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 15-Nov-2013
• Refers: ASTM E456-13ae1 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 15-Nov-2013
• Refers: ASTM E456-13a - Standard Terminology Relating to Quality and Statistics
• Effective Date: 15-Nov-2013
• Refers: ASTM E2586-13 - Standard Practice for Calculating and Using Basic Statistics
• Effective Date: 01-Oct-2013
• Refers: ASTM E456-13 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 15-Aug-2013
• Refers: ASTM E2234-09(2013) - Standard Practice for Sampling a Stream of Product by Attributes Indexed by AQL
• Effective Date: 01-Apr-2013
• Refers: ASTM E2586-12b - Standard Practice for Calculating and Using Basic Statistics
• Effective Date: 01-Oct-2012
• Refers: ASTM E456-12 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 01-May-2012
• Refers: ASTM E456-12e1 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 01-May-2012

Overview

ASTM E2555-21e1 defines standard practices for applying MIL-STD-105 sampling plans to life and reliability inspection, specifically when assessing product quality based on life length or reliability metrics. This ASTM standard extends the use of attribute-based acceptance sampling (as described in Practice E2234, equivalent to MIL-STD-105) to situations where the key quality characteristic is time-to-failure rather than discrete defects. The statistical foundation relies on the Weibull distribution, including the exponential distribution as a special case, enabling robust modeling of product life characteristics.

Key Topics

• Life and Reliability Acceptance Sampling: Procedures for selecting and applying sampling plans for items where life or survival time is the quality measure, instead of nonconforming units.
• Adapting MIL-STD-105 Plans: Tables of conversion factors allow MIL-STD-105/E2234 attribute sampling plans to be used for evaluating mean life, hazard rate, and reliable life.
• Weibull and Exponential Models: The underlying statistical model for calculations is the Weibull distribution, widely used for life data analysis, with the exponential distribution as a special, simpler case.
• Truncated Life Testing: Inspection is performed by testing a sample of items for a predetermined time or until failure, then accepting or rejecting the lot based on the observed number of failures.
• Assessment Criteria: The standard provides conversion factors for three key reliability criteria:
  • Mean Life (average expected time to failure)
  • Hazard Rate (instantaneous failure rate at a given time)
  • Reliable Life (time by which a specified proportion of items survive)
• Sample Sizes and Acceptance Numbers: Sample sizes and acceptance numbers are determined by E2234 (MIL-STD-105) plans, with adjustments made using provided factors.

Applications

• Quality Assurance in Manufacturing: Adaptable for industries where product longevity or reliability is a primary concern, such as electronics, automotive components, aerospace, and medical devices.
• Supplier and Lot Qualification: Utilized by quality professionals to decide on lot acceptance based on life test outcomes rather than observing only visible defects.
• Product Evaluation: Suitable for both single, double, and multiple sampling scenarios, enabling flexible application to products with varying expected life and failure modes.
• Regulatory Compliance: Supports compliance with internationally recognized sampling procedures, facilitating transparent and repeatable qualification practices.
• Risk Management: Incorporates consumer’s risk and acceptance quality limits (AQLs) for decision-making, allowing stakeholders to balance stringency and practicality in life test acceptance.

Related Standards

• ASTM E2234 – Practice for Sampling a Stream of Product by Attributes Indexed by AQL. The foundational document for MIL-STD-105 based sampling plans.
• MIL-STD-105 – Military Standard for Sampling Procedures and Tables for Inspection by Attributes. Provides the structure for acceptance sampling plans.
• ASTM E456 – Terminology relating to quality and statistics. Offers definitions for statistical and quality control concepts used in E2555.
• ASTM E2586 – Practice for Calculating and Using Basic Statistics. Useful for practitioners applying the statistical models in E2555.
• International Guidelines – Developed according to WTO Technical Barriers to Trade Committee principles, ensuring alignment with global standardization practices.

Practical Value

• Provides a methodical approach for adapting widely recognized attribute sampling plans to life and reliability inspection.
• Enables organizations to maintain product reliability standards while using familiar MIL-STD-105/E2234 sampling structures.
• Offers conversion tables and stepwise procedures, facilitating easy implementation without the need for complex calculations.
• Supports a wide range of products and industries where reliability and time-to-failure are central quality characteristics.
• Reinforces international acceptance and good practice in quality control systems focused on life performance.

Keywords: ASTM E2555, MIL-STD-105, acceptance sampling, reliability inspection, Weibull distribution, mean life, hazard rate, reliable life, truncated life test, quality control, E2234, exponential distribution, product life testing, attribute sampling.

Standards Content (Sample)

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.
´1
Designation: E2555 − 21 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.
ε NOTE—Editorial corrections were made to the Terminology section in January 2022.
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
are provided for three alternative criteria for lot evaluation: 3.1 Definitions:
mean life, hazard rate, and reliable life. Inspection of the
3.1.1 TheterminologydefinedinTerminologyE456applies
sampleisbyattributeswithtestingtruncatedattheendofsome
to this practice unless modified herein.
prearranged period of time. The Weibull distribution, together
3.1.2 acceptance quality limit (AQL), n—qualitylimitthatis
with the exponential distribution as a special case, is used as
theworsttolerableprocessaveragewhenacontinuingseriesof
the underlying statistical model.
lots is submitted for acceptance sampling. E2234
1.2 A system of units is not specified by this practice. 3.1.2.1 Discussion—Thisdefinitionsupersedesthatgivenin
MIL-STD-105E.
1.3 This standard does not purport to address all of the
3.1.2.2 Discussion—A sampling plan and an AQL are cho-
safety concerns, if any, associated with its use. It is the
sen in accordance with the risk assumed. Use of a value of
responsibility of the user of this standard to establish appro-
AQL for a certain defect or group of defects indicates that the
priate safety, health, and environmental practices and deter-
sampling plan will accept the great majority of the lots or
mine the applicability of regulatory limitations prior to use.
batchesprovidedtheprocessaveragelevelofpercentdefective
1.4 This international standard was developed in accor-
(or defects per hundred units) in these lots or batches are no
dance with internationally recognized principles on standard-
greater than the designated value ofAQL. Thus, theAQL is a
ization established in the Decision on Principles for the
designated value of percent defective (or defects per hundred
Development of International Standards, Guides and Recom-
units) for which lots will be accepted most of the time by the
mendations issued by the World Trade Organization Technical
sampling procedure being used. The sampling plans provided
Barriers to Trade (TBT) Committee.
herein are so arranged that the probability of acceptance at the
designated AQL value depends upon the sample size, being
2. Referenced Documents
generally higher for large samples than for small ones, for a
2.1 ASTM Standards:
given AQL. The AQL alone does not identify the chances of
E456Terminology Relating to Quality and Statistics
accepting or rejecting individual lots or batches but more
E2234Practice for Sampling a Stream of Product by Attri-
directly relates to what might be expected from a series of lots
butes Indexed by AQL
or batches, provided the steps indicated in this refer to the
E2586Practice for Calculating and Using Basic Statistics
operating characteristic curve of the plan to determine the
relative risks.
3.1.3 consumer’s risk, n—probability that a lot having
ThispracticeisunderthejurisdictionofASTMCommitteeE11onQualityand
specified rejectable quality level will be accepted under a
Statistics and is the direct responsibility of Subcommittee E11.40 on Reliability.
defined sampling plan.
Current edition approved May 1, 2021. Published June 2021. Originally
ɛ1
approved in 2007. Last previous version approved in 2018 as E2555–07 (2018) .
DOI: 10.1520/E2555-21E01.
For referenced ASTM standards, visit the ASTM website, www.astm.org, or
contact ASTM Customer Service at [e-mail]. 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
´1
E2555 − 21
3.1.4 double sampling plan, n—a multiple sampling plan in provide information that may be used as a basis for making a
which up to two samplings can be taken and evaluated to decision concerning the larger collection. E2586
accept or reject a lot. E2234
3.2 Definitions of Terms Specific to This Standard:
3.1.5 hazard rate, n—differential fraction of items failing at
3.2.1 acceptance number, n—the maximum number of
time t among those surviving up to time t, symbolized by h(t).
failed items allowed in the sample for the lot to be accepted
3.1.5.1 Discussion—h(t) is also referred to as the instanta-
using a single or multiple sampling plan.
neous failure rate at time t. It is related to the probability
3.2.2 mean life, n—average time that items in the lot or
density and cumulative distribution functions by h(t)= f(t)
population are expected to operate before failure.
/(l– F(t)).
3.2.2.1 Discussion—Thismetricisoftenreferredtoasmean
3.1.6 limiting quality level (LQL), n—quality level having a
time to failure (MTTF).
specified consumer’s risk for a given sampling plan.
3.2.3 rejection number, n—the minimum number of failed
3.1.7 lot, n—a definite quantity of a product or material
items in the sample that will cause the lot to be rejected under
accumulated under conditions that are considered uniform for
a given sampling plan.
sampling purposes.
3.1.7.1 Discussion—The lot for sampling may differ from a
3.2.4 reliable life (ρ ), n—life beyond which some specified
r
collectionofunitsdesignatedasabatchforotherpurposes,for
proportion, r, of the items in the lot or population will survive.
example, production, shipment, and so forth.
3.2.5 test truncation time (t), n—amount of time sampled
3.1.8 multiple sampling plan, n—a sampling plan in which
items are allowed to be tested.
successive samples from a lot are drawn and after each sample
3.2.6 Weibull distribution, n—probability distribution hav-
is inspected a decision is made to accept the lot, reject the lot,
ing cumulative distribution:
or to take another sample, based on quality level of the
β
combined samples. E2234
t 2 γ
function F~t! 51 2exp 2 , t.γ andprobabilitydensity
S S D D
3.1.8.1 Discussion—When the quality is much less or much
η
β21 β
more than the AQL, the decision can be made on the first
β t 2 γ t 2 γ
function f t 5 exp 2
~ ! S D S S D D
sample, which is smaller than that of a single sampling plan
η η η
with equivalent acceptance quality level. For samples that are
3.2.6.1 Discussion—TheWeibulldistributioniswidelyused
closetotheAQLinquality,additionalsamplesarerequiredand
for modeling product life. It can take a wide variety of shapes
the total sample size will be larger than the corresponding
andalsothecharacteristicsofothertypesofdistributionsbased
single sampling plan.
on the value of its parameters. γ is called the location,
3.1.9 sample, n—group of items, observations, test results, minimum life, or threshold parameter and defines the lower
or portions of material taken from a large collection of items, limit of the distribution (Fig. 1). η is called the scale or
observations,testresults,orquantitiesofmaterialthatservesto characteristic life parameter and is equal to the 63.2 percentile
FIG. 1 Effect of the Parameter γ on the Weibull Probability Den-
sity Function, f(t)
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E2555 − 21
of the distribution, minus γ (Fig. 2). β is the shape parameter 4.2.5 Compare the number of items that failed with the
(Fig. 3). The exponential distribution is the special case where number allowed under the selected Practice E2234 plan.
γ = 0 and β=l.
4.2.6 If the number that failed is equal to or less than the
acceptable number, accept the lot; if the number failing
4. Significance and Use
exceeds the acceptable number, reject the lot.
4.1 The procedure and tables presented in this practice are
4.3 Both the sample sizes and the acceptance numbers used
based on the use of the Weibull distribution in acceptance
arethosespecifiedbyPracticeE2234plans.Itwillbeassumed
sampling inspection. Details of this work, together with tables
in the section on examples that single sampling plans will be
of sampling plans of other forms, have been published previ-
4 used. However, the matching double sampling and multiple
ously. See Refs (1-3). Since the basic computations required
sampling plans provided in MIL-STD-105 can be used if
havealreadybeenmade,ithasbeenquiteeasytoprovidethese
desired. The corresponding sample sizes and acceptance and
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
has been briefly written. Readers interested in further details
4.4 The probability of acceptance for a lot under this
are referred to these previous publications. Other sources of
procedure depends only on the probability of a sample item
material on the underlying theory and approach are also
failing before the end of the test truncation time, t. For this
available (4-7).
reason, the actual life at failure need not be determined; only
4.2 The procedure to be used is essentially the same as the thenumberofitemsfailingisofinterest.Liferequirementsand
test time specifications need not necessarily be measured in
one normally used for attribute sampling inspection. The only
difference is that sample items are tested for life or survival chronologicaltermssuchasminutesorhours.Forexample,the
instead of for some other property. For single sampling, the life measure may be cycles of operation, revolutions, or miles
following are the required steps: of travel.
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
4 procedure is available. The third parameter, and the one of
Theboldfacenumbersinparenthesesrefertothelistofreferencesattheendof
this standard.
FIG. 2 Effect of the Parameter η on the Weibull Probability Den-
sity Function, f(t)
´1
E2555 − 21
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 or life-length applications, these
2000 h has been specified, and under the plan, used lots with a
factors are used in place of the corresponding p’ values
mean life of this value or greater shall have a high probability
normally used in the use of Practice E2234 plans for attribute
of acceptance.Atesting truncation time of t = 250 h has been
inspection of other item qualities.The use of these factors will
specified.Frompastexperienceithasbeendeterminedthatthe
bedemonstratedbyseveralexamples(seeSections5,7,and9).
Weibull distribution can be used as a life-length model and a
shape parameter value of 2.5 and a location or threshold
parameter value of 0 can be assumed. Single sampling is to be
used.Asample of as many as 300 items or so can be tested at
Insomedisciplines,theWeibullshapeβparameterisreferredtoasthe“Weibull
slope.” one time. An appropriate sampling inspection plan shall be
´1
E2555 − 21
selected. Also, the consumer’s risk under use of the selected parameter value, γ, of 3000 h. A Practice E2234 acceptance-
plan shall be determined. inspectionplanshallbeselectedunderwhichtheprobabilityof
5.1.1 Computation of the 100t/µ ratio at the AQL gives acceptance will be low (0.05 or less) if mean item life is 8000
100t/µ = 100 × 250/2000 = 12.5. Examination of the ratios in h or less. The sample size will be kept large to reduce the
the column for a shape parameter of 2.5 in Annex Table1A
testing period time but it cannot exceed 250 items. To reduce
discloses a value of 12.4 for anAQLof 0.40 in p’(%) terms.A further testing time, an acceptance number of 0 will be used.
plan with this AQL is accordingly to be used. Reference now
The required test truncation time must be determined; also, the
toPracticeE2234indicatesforSampleSizeCodeLetterMthe AQL.
sample size is 315; this value will accordingly be used.
5.3.1 ReferencetoPracticeE2234indicatestheCodeLetter
ExaminationoftheMasterTableforNormalInspection(Single
Lwithasamplesizeof200itemsshallbeused.Withthiscode
Sampling) in Practice E2234 shows for Sample Size Code
letter and an acceptance number of 0, the AQL in Practice
Letter M and anAQLof 0.40, the acceptance number must be
E2234 terms must be 0.065. Subtraction of the threshold
3 and the rejection number 4.
parameter value, γ, of 3000 h from the required mean value, µ,
5.1.2 The acceptance procedure will thus be to draw at
of 8000 h gives as a converted value for the mean µ = 8000 –
random a sample of 315 items and submit them to life test for
3000 = 5000 h. This converted value must now be used in
250 h.At the end of that time, the number that has failed will
working with the tables of factors. Use ofAnnex Table1C for
be determined. If three items or less have failed, the lot will be
β=3 ⁄3 Code Letter L, and an AQL of 0.065 gives a 100t/µ
accepted; if four or more have failed, it will be rejected.
value of 31 at the LQL (for P(A) = 0.05). With µ = 5000,
5.1.3 The consumer’s risk at a probability level of 0.10 can
100t /µ = 100 t /5000 = 31 or t = 1550 h. Conversion of this
0 0 0 0
be determined by use of Annex Table1B which gives 100t/µ
to absolute terms gives t = t + γ = 1550 + 3000 = 4550 h as
ratiosattheLQLforthe0.10riskvalue.Forashapeparameter
the required test truncation time.
value of 2.5, a Sample Size Code Letter M, and an AQL of
5.3.2 From Annex Table1A, the corresponding ratio at the
0.40, the 100t/µ ratio value is found to be 24. With t = 250,
AQLmaybefound.ForanAQLof0.065and b=3 ⁄3itis12.3.
100t/µ = 24 or 100 × 250/µ = 24 which gives a value for µ of
Thus, 100 t /µ = 12.3 or 100 × 1550/µ = 12.3 or µ = 12600.
0 0 0 0
1040. Thus, if the mean life for the items in the lot is 1040 h
Converting this to absolute terms givesµ=µ + γ = 12600 +
orless,theprobabilityofacceptancewillbe0.10orless.Ifthe
3000 = 15600. Thus, the mean item life for a lot shall be
lot quality for which the consumer’s risk was 0.05 was desired
15600 h or more for its probability of acceptance to be high.
instead, Annex Table1C might be used which gives ratios at
the LQL for this risk value.
6. Hazard Rate Conversion Factors
5.2 APractice E2234 plan with Sample Size Code Letter F
6.1 Another measure of lot quality is the hazard rate or
and anAQL of 4.0 has been specified for a product for which
instantaneous failure rate, h(t), at some specified period of
life length in terms of cycles of operation is the quality of
time, t. Hazard rate conversion factors or values for the
interest.Acceptanceistobeintermsofameanlifeevaluation.
dimensionless product 100t{h(t)} have been determined for all
TheWeibull distribution can be assumed to apply with a shape
of the p’ values that characterize the collection of Practice
parametervalueandalocationparametervalueof0.Testingof
E2234plans.Asforthemeanlifeplans,theseproductsmaybe
sample items is to be truncated at 5000 cycles. The operating
used in place of the corresponding p’ values when using the
characteristics in terms of mean life for this plan are required.
Practice E2234 plans for life-length and reliability applica-
5.2.1 AnnexTable1Alistsratiosof100t/µatselectedAQLs
tions.
andgivesa100t/µvalueof0.62foranAQLof4.0andashape
parameter value of ⁄3. With t = 5000, 100t/µ = 0.62 or 100 ×
6.2 Annex Table2A lists for each selected value for the
5000/µ=0.62whichgivesµ=810000.Therefore,ifthemean
shape parameter 100t{h(t)} products for each Practice E2234
item life for the lot is 810000 or more, the probability of
AQL value. Annex Table2B lists corresponding 100t{h(t)}
acceptance will be high.AnnexTable1C gives ratios 100t/µ at
products at the LQL for a consumer’s risk of 0.10. Annex
the LQL for a consumer’s risk of 0.05 and provides a 100t/µ
Table2C lists products at the LQL for a consumer’s risk of
value of 14 for Code Letter F, an AQL of 4.0, and a shape
0.05. Use of these tables of factors is similar to the method of
parameter value of ⁄3. Thus, 100 × 5000/µ = 14 or µ = 36000.
use for the mean life ratios including the variation in method
If the mean item life for the lot is 36000 cycles or less, the
requiredwhensomenonzerovalueforthelocationorthreshold
probability of acceptance will be 0.05 or less.
parameter shall be assumed.
5.2.2 The sample size and acceptance number will be those
6.2.1 Note one point of difference. The products are for
specified by Practice E2234 for Code Letter F and anAQL of
directapplicationonlyincasesinwhichthetime tatwhichthe
4.0. For single sampling, the sample size will be 20 items and
hazard rate is specified or is to be evaluated is the same as the
the acceptance number 2. For this example, as in all cases, the
time t at which the life testing of sample items is to be
matched Practice E2234 double sampling and multiple sam-
truncated. However, a table of hazard rate ratios has been
pling plans may be used instead. No additional changes in
prepared, Annex Table2D, to use in a simple modification of
procedure are required. The specified test time, which in this
method that allows the test truncation time to differ from the
case is 5000 cycles, shall be used for all samples.
timeatwhichthehazardrateisspecified.Allthatshallbedone
5.3 Assume the Weibull distribution applies with a shape istodeterminethehazardrateatthetesttruncationtimewhich
parameter value of β = 3.33 and a location or threshold corresponds to the hazard rate at the specification time.Annex
´1
E2555 − 21
Table2D provides ratios for making this conversion. It gives column for β = 0.67 shows that a Practice E2234 plan with an
forvariousvaluesoft /t thecorrespondingvaluesfortheratio AQL of 4.0% precisely meets this need.
2 1
h t /h t for all the shape parameter values for which conver-
~ ! ~ !
2 1
8. Reliable Life Conversion Factors
sion values have been provided. If the test truncation time is
shorter than the time for hazard rate specification, t is used to 8.1 A third possible reliability and life-length measure for
the items in a lot or population is reliable life (ρ). Reliable life
represent the test truncation time and h~t ! the corresponding
hazard rate at that time. In this case, t represents the time of can be defined as the life beyond which some specified
proportion of the items in the lot or population will survive.
hazard rate specification and h~t ! the specified hazard rate. If
the test truncation is longer instead, the meanings given The letter r represents this specified proportion.
8.1.1 Tables of conversion factors have been prepared for
Subscripts 1 and 2 are simply reversed.
two different proportions, r = 0.90 and r = 0.99. As for the
mean life case, these reliable life conversion factors have been
7. Examples, Hazard Rate
prepared in the form of values for the dimensionless ratio
7.1 An acceptance-inspection plan shall be selected from
100t/ρ. Ratio values have been determined for all the p’(%)
the Practice E2234 collection for an application for which the
values associated with Practice E2234 plans. Annex Table3A
Weibull distribution applies and for which it may be assumed
gives 100t/ρ values at each of the AQLs for r = 0.90; Annex
the shape parameter value is 1.67 and the location parameter
Table4A gives corresponding values for r = 0.99. Annex
value is 0.Ahazard rate of no more than 0.0005/h at 1000 h of
Table3BgivesratiovaluesattheLQLforaconsumer’sriskof
life can be tolerated so a plan under which the probability of
0.10 for r = 0.90;Annex Table4B gives corresponding values
acceptance will be low (0.10) if this rate will be exceeded at
for a consumer’s risk of 0.10 and r = 0.99. Annex Table3C
this life is required. The test truncation time is likewise to be
givesratiovaluesattheLQLforaconsumer’sriskof0.05and
1000 h.
r = 0.90; Annex Table4C gives similar ratio values at a
7.1.1 Computation of the 100t{h(t)} product gives 100 ×
consumer’s risk of 0.05 and r = 0.99. These conversion ratios
1000 × 0.0005 = 50. Thus, apian shall be used for which this
areusedinthesamemannerinwhichmeanliferatiosareused,
product is found at the LQL for which the consumer’s risk is
including the manner for application when the location param-
0.10. Examination of the column for β = 1.67 in Annex
eter is not zero. See Section 9 for an example.
Table2B discloses several close possibilities. One is for a plan
with Code Letter D and an AQL of 1.5 for which the product
9. Examples, Reliable Life
is48;anotherisCodeLetterFandanAQLof4.0forwhichthe
9.1 A sampling inspection plan shall be selected for a
product is likewise 48; still another is Code Letter G and an
product for which item life in terms of feet of travel is the
AQL of 6.5 for which the product is 53. Any of these will
quality of interest. Experience indicates the Weibull distribu-
provide fairly closely the required consumer’s protection.
tion will serve well as a statistical model with a shape
7.1.2 The last plan mentioned with its relatively large
parametervalueofapproximately1 ⁄3andalocationparameter
sample size and acceptance number will discriminate most
of 0.Alot will be considered “acceptable” if the reliable life is
sharply between good and bad lots and hence provide the most
40000 ft and the probability of acceptance for such lots shall
reasonable AQL. This will be achieved at the expense of a
be high. For lots in which reliable life is 10000 ft or less, the
relatively large number of item hours of inspection, of course.
probability of acceptance shall be low, namely 0.05 or less.
With this choice (Code Letter G and anAQL of 6.5) theAQL
Reliable life is defined as the life beyond which 90% of the
can be easily determined. Reference to Annex Table2A gives
items will survive; that is, r is to be 0.90. Testing of sample
a value for 100t{h(t)} of 11.2 for anAQL of 6.5. Thus, 100 ×
items is to be truncated at 5000 ft.
1000 h(t) = 11.2 or h(t) = 0.000 112 at t = 1000; the
9.1.1 At theAQL, the 100t/ρ factor is 100 × 5000/40000 =
“acceptable” hazard rate is therefore 0.000112 (per hour). If,
12.5. Examination of Annex Table3A shows that for β=1 ⁄3
alternatively, Code Letter D and anAQLof 1.5 had been used,
the100t/ρratioforanAQLof0.65is12.4whichisquiteclose
the “acceptable” hazard rate would be 0.0000252 (per hour)
to the desired ratio.Accordingly, a plan with thisAQLis to be
instead.
adopted.At the unacceptable or LQL, the 100t/ρ factor is 100
r
7.2 Suppose the selected sampling plan must have an
×5000/10000=50.ReferencingAnnexTable3C,whichgives
acceptable hazard rate (a rate for which the probability of
ratios at the LQL for P(A) = 0.05, shows that, for Code Letter
acceptance is high) of 0.0001 per hour at 500 h of life.
L, an AQL of 0.65 (which is required for this application, as
However, the testing of sample items shall be truncated at
indicated above) and β=1 ⁄3 the corresponding ratio is 48,
200h.Avalueof β=0.67andalocationparameterof0canbe
which is close to the desired value of 50. Thus, a Practice
assumed. A Practice E2234 plan shall be selected.
E2234 plan with Code Letter L and anAQL of 0.65 will meet
7.2.1 In this case, useAnnexTable2D. Letting t = 500 and the specified operating requirements. For single sampling,
t = 200, t /t = 500/200 = 2.5. Referencing Annex Table2D
Practice E2234 shows the sample size to be 200 items and the
1 2 1
with this ratio using the value β = 0.67 column shows acceptance number 3.
h(t )/h(t )tobe0.734.With h t 50.0001, 0.0001/h t 50.734,or
~ ! ~ !
2 1 2 1
10. Summary
h~t !50.000136. This failure rate number shall be used in
selectingtheplan.Thus,100t{h(t)}=100×200×0.000136= 10.1 This practice preserves the structure ofTR-7 for use in
2.72 (note that the testing truncation time of 200h is used as t applications in which that standard is prescribed or its use is
at this point). Referencing Annex Table2A examining the desirable.
´1
E2555 − 21
10.2 Thispracticeprovidestablesandproceduresforapply- 10.4.2 Compute the dimensionless ratio 100t/µ from the
ing three different measures of reliability in which testing is specified µ and t and enter Annex Table1A under β. Locate
performed without replacement. the nearest value of 100t/µ to that calculated and read the
10.2.1 Mean Life, µ—The expected life of the product. corresponding AQL.
10.4.3 Compute the dimensionless ratio 100t/µ from the
10.2.2 Hazard Rate, h(t)—The instantaneous failure rate at
some specified time, t. specified µ and t and enter Annex Table1B under β. Locate
thenearestvalueof100t/µ correspondingtotheAQLobtained
10.2.3 Reliable Life, ρ —The life ρ beyond which some
r
in 10.4.2 and read the sample size code letter (use Annex
specified proportion r of the items in the population will
Table1C if a limiting quality with 5% probability of accep-
survive.
tance is desired).
10.3 Procedure for Application:
10.4.4 Obtain the sample size and acceptance number for
10.3.1 Using the tables of factors provided in Annex A1,
the test from the Practice E2234 normal inspection plan.
select a suitable sampling inspection plan from those tabulated
10.4.5 Mean Life Example:
in Practice E2234 for normal inspection.
10.4.5.1 Suppose µ = 50, µ = 10, t =5, β = 1, then
0 1
10.3.2 Draw at random a sample of items of the size
100t/µ = 10 giving an AQL of 10 from Annex Table1A and
specified by the selected Practice E2234 plan.
100t/µ = 50 giving Code F from Table1B.
10.3.3 Placethesampleofitemsonlifetestforthespecified
10.4.5.2 Practice E2234 gives sample size 20. Accept on 5
period of time, t.
for Code F, AQL = 10.
10.3.4 Determine the number of sample items that failed
10.5 Selection—Hazard Rate or Reliable Life:
during the test period.
10.5.1 The selection of plans for a specified hazard rate or
10.3.5 Compare the number of items that failed with the
reliable life follows the procedure for mean life described in
number allowed under the selected Practice E2234 plan.
10.4 using appropriate dimensionless ratios and the associated
10.3.6 If the number that failed is equal to or less than the
tables from Annex A1.
acceptance number, accept the lot; if the number failing
10.5.2 Hazard rate uses the product 100t{h(t) } with the
exceeds the acceptance number, reject the lot.
Annex A1 tables of Section B.
10.4 Selection—Mean Life:
10.5.3 Reliablelifeusesthedimensionlessratio100t/ρwith
10.4.1 Specify:
the Annex A1 tables of Section C.
10.4.1.1 Acceptable mean life, µ .
11. Keywords
10.4.1.2 Unacceptable mean life, µ .
10.4.1.3 Test truncation time, t.
11.1 exponential distribution; hazard rate; mean life; MIL-
10.4.1.4 Weibull shape parameter, β. STD-105; reliability; reliable life; Weibull distribution
ANNEX
(Mandatory Information)
A1. TABLES OF CONVERSION FACTORS
TABLE 1A
100t/µ Ratios at the Acceptable Quality Level (normal inspection)
for the ASTM E2234 Plans
NOTE—These plans assume the characteristic being measured has a Weibull distribution.
NOTE—Where scientific notation is used (that is, E-x), the decimal point is moved to the left x places (for example, if the number in scientific notation is 8.03E-04, then
the decimal is moved to the left four places. The number in decimal notation is 0.000803).
Shape Parameter, β
AQL
p’(%)
0.333 0.500 0.667 1.000 1.333 1.500 1.667 2.000 2.500 3.000 3.333 3.500 4.000 5.000 10.000
0.010 1.67E-11 5.00E-07 7.52E-05 1.00E-02 0.109 0.239 0.446 1.128 2.831 5.198 7.031 7.999 11.033 17.262 41.847
0.015 5.63E-11 1.13E-06 1.38E-04 1.50E-02 0.147 0.313 0.568 1.382 3.330 5.950 7.940 8.981 12.210 18.720 43.578
0.025 2.61E-10 3.13E-06 2.97E-04 2.50E-02 0.216 0.440 0.772 1.784 4.085 7.055 9.255 10.393 13.873 20.734 45.863
0.040 1.07E-09 8.00E-06 6.02E-04 4.00E-02 0.308 0.601 1.024 2.257 4.930 8.252 10.657 11.887 15.603 22.778 48.070
0.065 4.58E-09 2.11E-05 1.25E-03 6.50E-02 0.443 0.831 1.370 2.877 5.986 9.702 12.328 13.656 17.617 25.101 50.462
0.100 1.67E-08 5.01E-05 2.38E-03 0.100 0.612 1.108 1.774 3.569 7.113 11.200 14.030 15.445 19.622 27.360 52.684
0.150 5.64E-08 1.13E-04 4.38E-03 0.150 0.830 1.452 2.263 4.372 8.366 12.822 15.845 17.344 21.716 29.673 54.866
0.250 2.61E-07 3.13E-04 9.42E-03 0.250 1.218 2.042 3.076 5.645 10.265 15.205 18.472 20.072 24.677 32.868 57.744
0.400 1.07E-06 8.03E-04 1.91E-02 0.401 1.733 2.795 4.080 7.144 12.391 17.788 21.274 22.962 27.759 36.113 60.527
´1
E2555 − 21
TABLE 1A
100t/µ Ratios at the Acceptable Quality Level (normal inspection)
for the ASTM E2234 Plans
NOTE—These plans assume the characteristic being measured has a Weibull distribution.
NOTE—Where scientific notation is used (that is, E-x), the decimal point is moved to the left x places (for example, if the number in scientific notation is 8.03E-04, then
the decimal is moved to the left four places. The number in decimal notation is 0.000803).
Shape Parameter, β
AQL
p’(%)
0.333 0.500 0.667 1.000 1.333 1.500 1.667 2.000 2.500 3.000 3.333 3.500 4.000 5.000 10.000
0.650 4.62E-06 2.13E-03 3.96E-02 0.652 2.497 3.867 5.464 9.112 15.055 20.922 24.619 26.388 31.352 39.806 63.547
1.000 1.69E-05 5.05E-03 7.58E-02 1.005 3.454 5.159 7.083 11.312 17.899 24.167 28.031 29.859 34.932 43.402 66.356
1.500 5.75E-05 1.14E-02 0.140 1.511 4.690 6.771 9.047 13.872 21.071 27.687 31.680 33.551 38.683 47.092 69.119
2.500 2.70E-04 3.20E-02 0.303 2.532 6.906 9.551 12.330 17.954 25.901 32.883 36.983 38.879 44.008 52.211 72.778
4.000 1.13E-03 8.33E-02 0.620 4.082 9.882 13.133 16.422 22.798 31.355 38.559 42.682 44.565 49.591 57.446 76.339
6.500 5.06E-03 0.226 1.311 6.721 14.362 18.311 22.149 29.253 38.275 45.530 49.569 51.388 56.174 63.469 80.242
10.000 1.95E-02 0.555 2.573 10.536 20.122 24.711 29.007 36.626 45.816 52.891 56.726 58.431 62.856 69.441 83.932
TABLE 1B
100t/µ Ratios at the Limiting Quality Level
for the ASTM E2234 Plans, Consumer’s Risk = 0.10
NOTE—These plans assume the characteristic being measured has a Weibull distribution.
NOTE—Where scientific notation is used (that is, E-x), the decimal point is moved to the left x places (for example, if the number in scientific notation is 8.03E-04, then
the decimal is moved to the left four places. The number in decimal notation is 0.000803).
Shape Parameter, β
Code AQL
Letter (p%)
0.333 0.50 0.667 1.000 1.333 1.500 1.667 2.000 2.500 3.000 3.333 3.500 4.000 5.000 10.000
A 6.500 25.433 66.274 92.927 115.129 120.933 121.682 121.789 121.073 119.240 117.369 116.235 115.707 114.281 112.025 106.605
B 4.000 7.536 29.455 50.583 76.753 89.223 92.861 95.489 98.856 101.388 102.531 102.922 103.050 103.265 103.299 102.369
C 2.500 1.628 10.604 23.509 46.052 60.826 66.059 70.282 76.573 82.650 86.478 88.298 89.056 90.885 93.267 97.271
C 10.000 11.235 38.440 61.762 87.681 98.590 101.478 103.429 105.659 106.932 107.183 107.115 107.045 106.759 106.086 103.741
D 1.500 0.397 4.142 11.616 28.782 42.756 48.289 53.012 60.537 68.485 73.938 76.686 77.865 80.809 84.899 92.805
D 6.500 2.361 13.587 28.313 52.129 66.752 71.750 75.709 81.469 86.852 90.126 91.644 92.267 93.745 95.608 98.484
D 10.000 7.688 29.850 51.091 77.265 89.669 93.274 95.871 99.185 101.658 102.759 103.128 103.247 103.437 103.437 102.437
E 1.000 9.26E-02 1.569 5.608 17.712 29.707 34.937 39.615 47.489 56.397 62.890 66.292 67.780 71.573 77.043 88.407
E 4.000 0.505 4.859 13.094 31.175 45.395 50.930 55.614 63.003 70.708 75.933 78.546 79.662 82.439 86.266 93.549
E 6.500 1.478 9.943 22.401 44.594 59.376 64.657 68.939 75.352 81.594 85.556 87.450 88.241 90.157 92.669 96.959
E 10.000 3.379 17.255 33.871 58.746 73.011 77.700 81.336 86.486 91.104 93.789 94.989 95.471 96.588 97.920 99.668
F 0.650 2.54E-02 0.663 2.939 11.513 21.505 26.215 30.592 38.287 47.470 54.478 58.255 59.930 64.265 70.683 84.679
F 2.500 0.133 1.992 6.709 19.962 32.495 37.836 42.562 50.415 59.160 65.448 68.714 70.136 73.745 78.908 89.471
F 4.000 0.369 3.940 11.189 28.073 41.963 47.492 52.224 59.786 67.805 73.325 76.114 77.312 80.306 84.476 92.574
F 6.500 0.795 6.577 16.430 36.267 50.850 56.335 60.899 67.954 75.120 79.860 82.193 83.182 85.617 88.916 94.975
F 10.000 2.566 14.362 29.516 53.596 68.156 73.089 76.980 82.608 87.821 90.964 92.410 93.001 94.398 96.140 98.758
G 0.400 6.21E-03 0.259 1.452 7.196 15.117 19.164 23.075 30.268 39.334 46.578 50.594 52.399 57.141 64.341 80.792
G 1.500 3.14E-02 0.763 3.266 12.352 22.670 27.474 31.911 39.657 48.825 55.770 59.498 61.147 65.405 71.684 85.277
G 2.500 8.46E-02 1.476 5.358 17.183 29.038 34.237 38.900 46.773 55.716 62.257 65.691 67.194 71.032 76.576 88.139
G 4.000 0.176 2.407 7.730 21.939 34.879 40.294 45.043 52.852 61.437 67.540 70.688 72.053 75.506 80.412 90.319
G 6.500 0.524 4.981 13.339 31.563 45.818 51.351 56.028 63.393 71.059 76.246 78.837 79.944 82.694 86.479 93.665
G 10.000 1.201 8.658 20.194 41.613 56.374 61.743 66.136 72.790 79.367 83.606 85.654 86.514 88.611 91.395 96.290
H 0.250 1.63E-03 0.106 0.743 4.605 10.817 14.232 17.654 24.215 32.904 40.140 44.254 46.126 51.108 58.847 77.265
H 1.000 8.09E-03 0.309 1.657 7.859 16.150 20.324 24.329 31.633 40.747 47.967 51.950 53.737 58.414 65.486 81.507
H 1.500 2.14E-02 0.590 2.694 10.865 20.592 25.223 29.548 37.194 46.384 53.437 57.252 58.947 63.342 69.869 84.191
H 2.500 4.36E-02 0.950 3.849 13.783 24.613 29.558 34.081 41.892 51.014 57.846 61.487 63.093 67.223 73.274 86.217
H 4.000 0.125 1.912 6.505 19.555 31.996 37.320 42.039 49.898 58.675 65.000 68.290 69.724 73.366 78.583 89.286
H 6.500 0.273 3.222 9.621 25.385 38.912 44.411 49.164 56.852 65.130 70.906 73.850 75.121 78.311 82.793 91.647
H 10.000 0.680 5.928 15.199 34.432 48.908 54.418 59.030 66.212 73.575 78.490 80.922 81.956 84.512 87.997 94.483
J 0.150 3.97E-044.14E-02 0.367 2.878 7.603 10.404 13.316 19.143 27.264 34.319 38.434 40.330 45.442 53.568 73.718
J 0.650 1.95E-03 0.120 0.814 4.893 11.320 14.819 18.308 24.960 33.711 40.959 45.066 46.932 51.889 59.565 77.735
J 1.000 5.10E-03 0.227 1.316 6.738 14.390 18.342 22.183 29.290 38.314 45.569 49.606 51.425 56.210 63.501 80.262
J 1.500 1.03E-02 0.362 1.868 8.513 17.147 21.436 25.523 32.922 42.070 49.262 53.211 54.977 59.593 66.541 82.161
J 2.500 2.86E-02 0.717 3.117 11.974 22.148 26.911 31.322 39.046 48.222 55.196 58.946 60.607 64.899 71.240 85.013
J 4.000 6.09E-02 1.186 4.547 15.403 26.752 31.830 36.430 44.285 53.332 60.029 63.571 65.128 69.116 74.920 87.180
J 6.500 0.145 2.119 7.025 20.584 33.251 38.618 43.353 51.194 59.891 66.120 69.349 70.753 74.312 79.393 89.745
J 10.000 0.354 3.832 10.956 27.682 41.525 47.051 51.787 59.369 67.426 72.984 75.795 77.003 80.026 84.240 92.444
K 0.100 1.04E-041.70E-02 0.188 1.842 5.440 7.726 10.188 15.315 22.807 29.575 33.618 35.502 40.645 48.994 70.500
K 0.400 5.08E-044.88E-02 0.415 3.124 8.086 10.989 13.988 19.945 28.174 35.270 39.392 41.287 46.384 54.454 74.325
K 0.650 1.32E-039.21E-02 0.669 4.292 10.261 13.580 16.925 23.378 31.991 39.210 43.330 45.209 50.217 58.025 76.724
K 1.000 2.64E-03 0.146 0.947 5.410 12.206 15.845 19.445 26.246 35.094 42.354 46.445 48.299 53.208 60.774 78.520
K 1.500 7.24E-03 0.287 1.568 7.573 15.707 19.828 23.793 31.052 40.147 47.378 51.376 53.170 57.875 65.002 81.205
K 2.500 1.52E-02 0.470 2.270 9.692 18.900 23.373 27.590 35.129 44.311 51.439 55.323 57.054 61.558 68.290 83.234
K 4.000 3.54E-02 0.825 3.465 12.849 23.351 28.206 32.675 40.447 49.602 56.509 60.206 61.840 66.054 72.252 85.614
K 6.500 8.31E-02 1.459 5.312 17.084 28.913 34.105 38.765 46.638 55.588 62.137 65.577 67.084 70.929 76.488 88.088
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TABLE 1B
100t/µ Ratios at the Limiting Quality Level
for the ASTM E2234 Plans, Consumer’s Risk = 0.10
NOTE—These plans assume the characteristic being measured has a Weibull distribution.
NOTE—Where scientific notation is used (that is, E-x), the decimal point is moved to the left x places (for example, if the number in scientific notation is 8.03E-04, then
the decimal is moved to the left four places. The number in decimal notation is 0.000803).
Shape Parameter, β
Code AQL
Letter (p%)
0.333 0.50 0.667 1.000 1.333 1.500 1.667 2.000 2.500 3.000 3.333 3.500 4.000 5.000 10.000
K 10.000 2.51E-01 3.050 9.234 24.700 38.122 43.608 48.363 56.079 64.420 70.262 73.247 74.535 11.111 82.341 91.396
L 0.065 2.54E-056.63E-039.29E-02 1.151 3.824 5.648 7.684 12.107 18.898 25.286 29.197 31.041 36.139 44.598 67.263
L 0.250 1.24E-041.90E-02 0.205 1.950 5.677 8.025 10.541 15.756 23.331 30.141 34.196 36.083 41.226 49.553 70.902
L 0.400 3.19E-043.58E-02 0.329 2.675 7.196 9.907 12.742 18.454 26.476 33.490 37.597 39.493 44.616 52.787 73.179
L 0.650 6.35E-045.66E-02 0.465 3.366 8.550 11.548 14.627 20.701 29.026 36.156 40.281 42.174 47.255 55.271 74.880
L 1.000 1.73E-03 0.110 0.766 4.696 10.977 14.419 17.863 24.453 33.163 40.403 44.515 46.386 51.360 59.079 77.417
L 1.500 3.58E-03 0.179 1.103 5.991 13.176 16.961 20.673 27.619 36.556 43.819 47.889 49.728 54.583 62.027 79.325
L 2.500 8.23E-03 0.312 1.671 7.903 16.218 20.400 24.411 31.722 40.838 48.057 52.038 53.823 58.497 65.560 81.553
L 4.000 1.89E-02 0.544 2.536 10.435 19.977 24.552 28.840 36.450 45.640 52.722 56.562 58.270 62.705 69.307 83.851
L 6.500 5.50E-02 1.109 4.323 14.892 26.083 31.122 35.699 43.544 52.617 59.357 62.931 64.503 68.535 74.416 86.887
M 0.040 6.51E-062.67E-034.70E-02 0.731 2.720 4.172 5.851 9.647 15.758 21.733 25.477 27.263 32.259 40.725 64.276
M 0.150 3.15E-057.65E-03 0.103 1.237 4.035 5.924 8.022 12.549 19.448 25.897 29.831 31.683 36.792 45.242 67.747
M 0.250 8.12E-051.44E-02 0.166 1.695 5.111 7.309 9.692 14.691 22.061 28.766 32.789 34.668 39.808 48.185 69.916
M 0.400 1.61E-042.27E-02 0.234 2.131 6.069 8.515 11.119 16.472 24.176 31.047 35.120 37.011 42.153 50.443 71.535
M 0.650 4.36E-044.40E-02 0.385 2.968 7.780 10.619 13.564 19.440 27.601 34.672 38.790 40.686 45.793 53.898 73.944
M 1.000 8.99E-047.14E-02 0.553 3.779 9.326 12.474 15.679 21.935 30.402 37.580 41.706 43.593 48.643 56.566 75.753
M 1.500 2.05E-03 0.124 0.834 4.970 11.454 14.975 18.481 25.157 33.924 41.174 45.279 47.143 52.093 59.753 77.857
M 2.500 4.65E-03 0.214 1.257 6.537 14.066 17.975 21.782 28.849 37.852 45.110 49.157 50.981 55.785 63.117 80.019
M 4.000 1.32E-02 0.429 2.120 9.260 18.265 22.674 26.846 34.338 43.511 50.664 54.572 56.316 60.861 67.671 82.856
N 0.025 1.63E-061.06E-032.35E-02 0.461 1.923 3.066 4.435 7.657 13.099 18.631 22.180 23.891 28.740 37.130 61.374
N 0.100 7.87E-063.03E-035.17E-02 0.779 2.852 4.352 6.078 9.957 16.162 22.197 25.965 27.760 32.774 41.243 64.684
N 0.150 2.02E-055.69E-038.29E-02 1.067 3.611 5.367 7.340 11.654 18.329 24.651 28.535 30.370 35.455 43.922 66.751
N 0.250 4.01E-058.98E-03 0.117 1.340 4.286 6.250 8.418 13.063 20.082 26.600 30.558 32.418 37.538 45.974 68.293
N 0.400 1.08E-041.74E-02 0.191 1.864 5.490 7.788 10.261 15.407 22.917 29.694 33.739 35.624 40.767 49.111 70.585
N 0.650 2.22E-042.81E-02 0.275 2.371 6.574 9.142 11.853 17.374 25.229 32.171 36.262 38.156 43.292 51.530 72.302
N 1.000 5.03E-044.84E-02 0.413 3.113 8.063 10.961 13.956 19.908 28.132 35.226 39.347 41.242 46.340 54.413 74.297
N 1.500 1.13E-038.34E-02 0.621 4.083 9.883 13.135 16.425 22.801 31.358 38.562 42.685 44.568 49.594 57.448 76.341
N 2.500 3.18E-03 0.166 1.040 5.759 12.791 16.519 20.188 27.078 35.982 43.245 47.324 49.169 54.046 61.538 79.012
P 0.015 3.97E-074.14E-041.16E-02 0.288 1.352 2.241 3.345 6.054 10.854 15.929 19.263 20.889 25.554 33.799 58.556
P 0.065 1.92E-061.18E-032.55E-02 0.487 2.004 3.181 4.583 7.871 13.390 18.975 22.548 24.269 29.138 37.540 61.712
P 0.100 4.93E-062.22E-034.09E-02 0.666 2.537 3.922 5.534 9.209 15.183 21.071 24.777 26.548 31.519 39.975 63.682
P 0.150 9.76E-063.50E-035.76E-02 0.837 3.010 4.565 6.345 10.321 16.633 22.734 26.530 28.335 33.367 41.840 65.150
P 0.250 2.62E-056.76E-039.43E-02 1.163 3.853 5.686 7.731 12.169 18.975 25.372 29.285 31.130 36.230 44.688 67.331
P 0.400 5.38E-051.09E-02 0.135 1.478 4.612 6.671 8.926 13.717 20.883 27.481 31.468 33.336 38.467 46.8
...