ASTM E2696-21

Standard Practice for Life and Reliability Testing Based on the Exponential Distribution

ABSTRACT
This practice presents standard sampling procedures and tables for life and reliability testing in procurement, supply, and maintenance quality control operations as well as in research and development activities. This practice describes general procedures and definitions of terms used in life test sampling and describes specific procedures and applications of the life test sampling plans for determining conformance to established reliability requirements.
SIGNIFICANCE AND USE
4.1 This practice was prepared to meet a growing need for the use of standard sampling procedures and tables for life and reliability testing in government procurement, supply, and maintenance quality control (QC) operations as well as in research and development activities where applicable.  
4.2 A characteristic feature of most life tests is that the observations are ordered in time to failure. If, for example, 20 radio tubes are placed on life test, and ti denotes the time when the ith tube fails, the data occur in such a way that t1 ≤ t2 ≤ ... ≤ tn. The same kind of ordered observations will occur whether the problem under consideration deals with the life of electric bulbs, the life of electronic components, the life of ball bearings, or the length of life of human beings after they are treated for a disease. The examples just given all involve ordering in time.  
4.3 In destructive testing involving such situations as the current needed to blow a fuse, the voltage needed to break down a condenser, or the force needed to rupture a physical material, the test can often be arranged in such a way that every item in the sample is subjected to precisely the same stimulus (current, voltage, or stress). If this is done, then clearly the weakest item will be observed to fail first, the second weakest next, and so forth. While the random variable considered mostly in this guide is time to failure, it should be emphasized, however, that the methodology provided herein can be adapted to the testing situations mentioned above when the random variable is current, voltage, stress, and so forth.  
4.4 Sections 6 and 7 describe general procedures and definitions of terms used in life test sampling. Sections 8, 9, and 10 describe specific procedures and applications of the life test sampling plans for determining conformance to established reliability requirements.  
4.5 Whenever the methodology or choice of procedures in the practice requires clarification, the user is adv...
SCOPE
1.1 This practice presents standard sampling procedures and tables for life and reliability testing in procurement, supply, and maintenance quality control operations as well as in research and development activities.  
1.2 This practice describes general procedures and definitions of terms used in life test sampling and describes specific procedures and applications of the life test sampling plans for determining conformance to established reliability requirements.  
1.3 This practice is an adaptation of the Quality Control and Reliability Handbook H-108, “Sampling Procedures and Tables for Life and Reliability Testing (Based on Exponential Distribution),” U.S. Government Printing Office, April 29, 1960.  
1.4 A system of units is not specified in this practice.  
1.5 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.  
1.6 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.

General Information

• Status: Published
• Publication Date: 30-Apr-2021
• ICS: 03.120.10 - Quality management and quality assurance
• ICS: 21.020 - Characteristics and design of machines, apparatus, equipment
• 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 E2555-07(2018) - Standard Practice for Factors and Procedures for Applying the MIL-STD-105 Plans in Life and Reliability Inspection
• Effective Date: 01-Apr-2018
• 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 E456-13ae2 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 15-Nov-2013
• Refers: ASTM E456-13ae3 - 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 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 E2555-07(2012) - Standard Practice for Factors and Procedures for Applying the MIL-STD-105 Plans in Life and Reliability Inspection
• Effective Date: 01-May-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
• Refers: ASTM E2234-09 - Standard Practice for Sampling a Stream of Product by Attributes Indexedby AQL
• Effective Date: 01-Nov-2009
• Refers: ASTM E2234-08 - Standard Practice for Sampling a Stream of Product by Attributes Indexedby AQL
• Effective Date: 01-Oct-2008

Overview

ASTM E2696-21 is the Standard Practice for Life and Reliability Testing Based on the Exponential Distribution, developed by ASTM International. This standard provides comprehensive sampling procedures and tables for conducting life and reliability testing, primarily using the exponential distribution model. Its guidance applies to procurement, supply chain quality control, maintenance operations, and research & development across a wide range of industries. The document outlines both general and specific methodologies to assess product life, reliability, and conformance to established reliability requirements.

Key concepts include time-to-failure analysis, ordered failure observations, and the use of statistical sampling plans to ensure the reliability of components or products. The standard is particularly valuable for scenarios where testing involves measuring lifespans under controlled conditions or exposing products to consistent stress until failure.

Key Topics

• Sampling Procedures
  Defines standard plans for drawing samples to estimate product life or reliability, including both fixed-number and time-based failure terminations.

• Exponential Distribution Model
  Life test sampling plans in this standard are primarily based on the exponential distribution, a commonly used statistical model for time-to-failure data. The standard also addresses validation of this assumption.

• Acceptance Criteria and Risks
  Includes definitions and guidance for setting acceptable and unacceptable mean life (Mean Time to Failure, MTTF), acceptance proportions, producer’s risk (α), and consumer’s risk (β).

• Testing Methods
  Describes practical testing with and without replacement, sequential testing, and procedures for calculating average waiting time and the number of failures required for decision-making.

• Cost and Efficiency Considerations
  Offers insights into optimizing sample size based on factors such as test duration, cost, and desired level of assurance.

• Definitions and Terminology
  Aligns key terms with ASTM E456 and related standards, ensuring consistency in quality and reliability discussions.

Applications

ASTM E2696-21 is utilized in a broad variety of reliability engineering, quality assurance, and lifecycle management applications, especially where exponential time-to-failure is a reasonable model. Key application areas include:

• Government Procurement and Supply Chain
  Ensures supplied components meet contractual reliability requirements through standard, auditable testing.

• Maintenance and Quality Control
  Supports ongoing QC initiatives in industries such as electronics, automotive, aerospace, and defense by providing objective criteria for accepting or rejecting product lots.

• Product Development and R&D
  Assists research teams in comparing the longevity and failure characteristics of new materials, designs, or technologies.

• Destructive and Non-Destructive Testing
  Adaptable to situations where test items are exposed to stress until failure (e.g., current, voltage, mechanical force), not just time-based failures.

Related Standards

• ASTM E456: Terminology Relating to Quality and Statistics - For definitions and consistency in statistical terminology.
• ASTM E2234: Practice for Sampling a Stream of Product by Attributes Indexed by AQL - For sampling by attributes.
• ASTM E2555: Practice for Applying MIL-STD-105 Plans in Life and Reliability Inspection - For implementing military standard sampling plans.
• Quality Control and Reliability Handbook H-108: The U.S. government resource on which ASTM E2696-21 is adapted.

Practical Value

• Objective Decision-Making: Facilitates consistent, data-driven decisions about lot acceptance or rejection based on standardized sampling plans.
• Efficiency and Cost Savings: Enables organizations to balance test duration, risk, and resource allocation while ensuring product reliability.
• Regulatory and Contractual Compliance: Supports fulfillment of technical requirements in government and commercial contracts.
• Versatile Test Planning: Suitable for both periodic and continuous monitoring scenarios in a diversity of industries.

ASTM E2696-21 is essential for anyone involved in product reliability management, offering practical methods and definitions to enhance quality, minimize risk, and optimize product lifecycle performance.

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.
Designation: E2696 − 21 An American National Standard
Standard Practice for
Life and Reliability Testing Based on the Exponential
Distribution
This standard is issued under the fixed designation E2696; 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 butes Indexed by AQL
E2555 Practice for Factors and Procedures for Applying the
1.1 Thispracticepresentsstandardsamplingproceduresand
MIL-STD-105 Plans in Life and Reliability Inspection
tablesforlifeandreliabilitytestinginprocurement,supply,and
maintenance quality control operations as well as in research
3. Terminology
and development activities.
3.1 Definitions:
1.2 This practice describes general procedures and defini-
3.1.1 See Terminology E456 for a more extensive listing of
tions of terms used in life test sampling and describes specific
terms in ASTM Committee E11 standards.
procedures and applications of the life test sampling plans for
3.1.2 consumer’s risk, β,n—probability that a lot having
determining conformance to established reliability require-
ments. specified rejectable quality level will be accepted under a
defined sampling plan. E2555
1.3 This practice is an adaptation of the Quality Control and
3.1.2.1 Discussion—In this practice, the consumer’s risk is
Reliability Handbook H-108, “Sampling Procedures and
the probability of accepting lots with mean time to failure θ .
Tables for Life and Reliability Testing (Based on Exponential
3.1.2.2 Discussion—For the procedures of 9.7 and 9.8, the
Distribution),” U.S. Government Printing Office, April 29,
consumer’s risk may also be defined as the probability of
1960.
accepting lots with unacceptable proportion of lot failing
1.4 A system of units is not specified in this practice.
before specified time, p .
1.5 This standard does not purport to address all of the
3.1.3 life test, n—process of placing one or more units of
safety concerns, if any, associated with its use. It is the
product under a specified set of test conditions and measuring
responsibility of the user of this standard to establish appro-
the time until failure for each unit.
priate safety, health, and environmental practices and deter-
mine the applicability of regulatory limitations prior to use.
3.1.4 mean time to failure (MTTF), θ, n—in life testing, the
1.6 This international standard was developed in accor- average length of life of items in a lot.
dance with internationally recognized principles on standard-
3.1.4.1 Discussion—Also referred to as mean life.
ization established in the Decision on Principles for the
3.1.5 number of failures, n—number of failures that have
Development of International Standards, Guides and Recom-
occurred at the time the decision as to lot acceptability is
mendations issued by the World Trade Organization Technical
reached.
Barriers to Trade (TBT) Committee.
3.1.5.1 Discussion—The expected number of failures re-
quired for decision is the average of the number of failures
2. Referenced Documents
required for decision when life tests are conducted on a large
2.1 ASTM Standards:
number of samples drawn at random from the same exponen-
E456 Terminology Relating to Quality and Statistics
tial distribution.
E2234 Practice for Sampling a Stream of Product by Attri-
3.1.6 producer’s risk, α,n—probability that a lot having
specified acceptable quality level will be rejected under a
This practice is under the jurisdiction ofASTM Committee E11 on Quality and defined sampling plan.
Statistics and is the direct responsibility of Subcommittee E11.40 on Reliability.
3.1.6.1 Discussion—In this practice, the producer’s risk is
^Current edition approved May 1, 2021. Published June 2021. Originally
the probability of rejecting lots with mean time to failure θ .
approved in 2009. Last previous edition approved in 2018 as E2696 – 09 (2018). 0
DOI: 10.1520/E2696-21.
3.1.6.2 Discussion—For the procedures of 9.7 and 9.8, the
For referenced ASTM standards, visit the ASTM website, www.astm.org, or
producer’s risk may also be defined as the probability of
contact ASTM Customer Service at [e-mail]. For Annual Book of ASTM
rejecting lots with acceptable proportion of lot failing before
Standards volume information, refer to the standard’s Document Summary page on
the ASTM website. specified time, p .
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E2696 − 21
3.1.7 sequential life test, n—life test sampling plan whereby 5. Introduction
neither the number of failures nor the time required to reach a
5.1 The theory underlying the development of the life test
decision are fixed in advance but instead decisions depend on
sampling plans of this section, including the operating charac-
the accumulated results of the life test.
teristic curves, assumes that the measurements of the length of
life are drawn from an exponential distribution. Statistical test
3.1.8 unit of product, n—that which is inspected to deter-
procedures for determining the validity of the exponential
mine its classification as defective or nondefective or to count
distribution assumption have appeared in the technical statis-
the number of defects. E2234
tical journals. Professor Benjamin Epstein published a com-
3.1.9 waiting time, n—in life testing, the time elapsed from
prehensive paper (in two parts) on this subject in the February
the start of testing until a decision is reached as to lot
and May 1960 issues of Technometrics. Part I of the paper
acceptability.
contains descriptions of the mathematical and graphical pro-
3.1.9.1 Discussion—The expected waiting time required for
cedures as well as an extensive bibliography for reference
decision is the average of the waiting times required for purposes. Numerical examples illustrating the statistical pro-
decision when life tests are conducted on a large number of cedures are included in Part II of the paper.
samples drawn at random from the same exponential distribu-
5.2 It is important to note that the life test sampling plans of
tion.
this practice are not to be used indiscriminately simply because
it is possible to obtain life test data. Only after the exponential
4. Significance and Use
assumption is deemed reasonable should the sampling plans be
used.
4.1 This practice was prepared to meet a growing need for
the use of standard sampling procedures and tables for life and 5.3 Sections 6 and 7 describe general procedures and
reliability testing in government procurement, supply, and description of life test sampling plans. Section 8 describes
specific procedures and applications of sampling plans when
maintenance quality control (QC) operations as well as in
life tests are terminated upon the occurrence of a preassigned
research and development activities where applicable.
number of failures, and Section 9 provides sampling plans
4.2 A characteristic feature of most life tests is that the
when life tests are terminated at a preassigned time. Section 10
observations are ordered in time to failure. If, for example, 20
describes sequential life test sampling plans. Section 8 covers:
radio tubes are placed on life test, and t denotes the time when
i
(1) acceptance procedures; (2) expected duration of life tests
the ith tube fails, the data occur in such a way that t ≤ t ≤ .
1 2
andcostconsiderationsinselectionofsamplesizes;and(3)life
≤ t .The same kind of ordered observations will occur whether
n
test plans for certain specified values of α, β, and θ /θ . Section
1 0
the problem under consideration deals with the life of electric
9 covers: (1) acceptance procedures; (2) life test plans for
bulbs, the life of electronic components, the life of ball
certain specified values of α, β, θ /θ , and T/θ ; and (3) life test
1 0 0
bearings, or the length of life of human beings after they are
plans based on proportion of lot failing before specified time.
treated for a disease. The examples just given all involve
Section 10 covers: (1) acceptance procedures; (2) graphical
ordering in time.
acceptance procedures; and (3) expected number and waiting
time required for decision.
4.3 In destructive testing involving such situations as the
5.4 Operating characteristic (OC) curves for the life test
current needed to blow a fuse, the voltage needed to break
sampling plans of 8.1 – 8.5, 9.1 – 9.5, and Section 10 are
down a condenser, or the force needed to rupture a physical
shown in Fig. A1.1 for the corresponding sampling plans in
material,thetestcanoftenbearrangedinsuchawaythatevery
these sections were matched with respect to their OC curves.
item in the sample is subjected to precisely the same stimulus
The OC curves in Fig. A1.1 have been computed for the life
(current, voltage, or stress). If this is done, then clearly the
test sampling plans of 8.1 – 8.5 but are equally applicable for
weakest item will be observed to fail first, the second weakest
the sampling plans of 9.1 – 9.5 and Section 10.
next, and so forth. While the random variable considered
mostly in this guide is time to failure, it should be emphasized,
5.5 The procedures of this section are based on the premise
however, that the methodology provided herein can be adapted
that the life tests are monitored continuously. If the tests are
to the testing situations mentioned above when the random
monitored only periodically, the values obtained from the
variable is current, voltage, stress, and so forth.
tables and curves are only approximations.
4.4 Sections 6 and 7 describe general procedures and
6. General Definitions of Life and Reliability Test Terms
definitions of terms used in life test sampling. Sections 8, 9,
6.1 Discussion of Terms and Procedures:
and 10 describe specific procedures and applications of the life
6.1.1 Purpose—This section provides definitions of terms
testsamplingplansfordeterminingconformancetoestablished
required for the life test sampling plans and procedures of
reliability requirements.
Sections 7 through 10.
4.5 Whenever the methodology or choice of procedures in
the practice requires clarification, the user is advised to consult
a qualified mathematical statistician, and reference should be
Epstein, B., “Tests for the Validity of the Assumption that the Underlying
made to appropriate technical reports and other publications in
Distribution of Life is Exponential,” Technometrics, Vol 2, February and May 1960,
the field. pp. 83–101 and 167–183.
E2696 − 21
6.1.2 Life Test—Life test is the process of placing the “unit exponential distribution. The expected waiting time can be
of product” under a specified set of test conditions and predetermined for the sampling plans mentioned in 6.1.6 –
6.1.8.
measuring the time it takes until failure.
6.1.3 Unit of Product—The unit of product is the entity of
6.2 Length of Life:
product that may be placed on life test.
6.2.1 LengthofLife—Theterms“lengthoflife”and“timeto
failure” may be used interchangeably and shall denote the
6.1.4 Specifying Failure—The state that constitutes a failure
length of time it takes for a unit of product to fail after being
shall be specified in advance of the life test.
placed on life test. The length of time may be expressed in any
6.1.5 Life Test Sampling Plan—A life test sampling plan is
convenient time scale such as seconds, hours, days, and so
a procedure that specifies the number of units of product from
forth.
a lot that are to be tested and the criterion for determining
6.2.2 Mean Time to Failure—The terms “mean time to
acceptability of the lot.
failure”and“meanlife”maybeusedinterchangeablyandshall
6.1.6 Life Test Terminated upon Occurrence of Preassigned
denote the mean (or equivalently, the average) length of life of
Number of Failures—Life test sampling plans whereby testing
items in the lot. Mean life is denoted by θ.
is terminated when a preassigned termination number of
6.2.3 Acceptable Mean Life—The acceptable mean life, θ ,
failures, r, occur are given in Section 8 of this practice.
is the minimum mean time to failure that is considered
6.1.7 Life Test Terminated at Preassigned Time—Life test
satisfactory.
sampling plans whereby testing is terminated when a preas-
6.2.4 Unacceptable Mean Life—The unacceptable mean
signed termination time, T, is reached are given in Section 9 of
life, θ (θ < θ ), is the mean time to failure such that lots
1 1 0
this practice.
having a mean life less than or equal to θ are considered
6.1.8 Sequential Life Test—Sequential life test is a life test
unsatisfactory. The interval between θ and θ is a zone of
0 1
sampling plan whereby neither the number of failures nor the indifference in which there is a progressively greater degree of
time required to reach a decision are fixed in advance but, dissatisfaction as the mean life decreases from θ to θ .
0 1
instead, decisions depend on the accumulated results of the life
6.3 Failure Rate:
test. Information on the observed time to failure are accumu-
6.3.1 Proportion of Lot Failing Before Specified Time—The
lated over time and the results at any time determine the choice
term “proportion of lot failing before specified time,” p,
of one among three possible decisions: (1) the lot meets the
denotes the fraction of the lot that fails before some specified
acceptability criterion, (2) the lot does not meet the acceptabil-
time, T, that is:
ity criterion, or (3) the evidence is insufficient for either
p 5 1 2 exp 2T/θ (1)
~ !
decision (1) or (2) and the test must continue. Sequential life
test sampling plans are given in Section 10 of this practice and
6.3.2 Failure Rate during Period of Time—The “failure rate
have the advantage over the life test sampling plans mentioned
during period of time T,” G, is given by:
in 6.1.6 and 6.1.7 in that, for the same OC curve, the expected
G 5 1 2 exp T/θ 5 p/T (2)
waiting time and the expected number of failures required to $ ~ !%
T
reach a decision as to lot acceptability are less for the
6.3.3 Instantaneous Failure Rate—The “instantaneous fail-
sequential life tests.
ure rate” or “hazard rate” is given by:
6.1.9 Expected Number of Failures—Thenumberoffailures
required for decision is the number of failures that have Z 51/θ (3)
occurred at the time the decision as to lot acceptability is
6.3.4 Acceptable Proportion of Lot Failing Before Specified
reached. For the life test sampling plans mentioned in 6.1.6,
Time—The “acceptable proportion of lot failing before speci-
this number of failures is known in advance of the life test; but,
fied time,” p , is the maximum fraction of the lot that may fail
for the sampling plans mentioned in 6.1.7 and 6.1.8, this
before time, T, and still result in the lot being considered
number cannot be predetermined. The expected number of
satisfactory.
failures required for decision is the average of the number of
6.3.5 Unacceptable Proportion of Lot Failing Before Speci-
failures required for decision when life tests are conducted on
fied Time—The “unacceptable proportion of lot failing before
a large number of samples drawn at random from the same
specifiedtime,” p ,(p > p ),istheminimumfractionofthelot
1 1 0
exponential distribution. The expected number of failures can
that may fail before time, T, and results in the lot being
be predetermined for the sampling plans mentioned in 6.1.6 –
considered unsatisfactory. The interval between p and p is a
0 1
6.1.8.
zone of indifference in which there is a progressively greater
6.1.10 Expected Waiting Time—The waiting time required degree of dissatisfaction as the fraction of the lot failing before
for decision is the time elapsed from the start of the life test to
time, T, increases from p to p .
0 1
the time decision is reached as to lot acceptability. The waiting
6.3.6 Acceptable Failure Rate During Period of Time—The
time required for decision cannot be predetermined for any of “acceptable failure rate during period of time,” G,isthe
the sampling plans mentioned in 6.1.6 – 6.1.8. The expected maximum failure rate during the period of time that can be
considered satisfactory.
waiting time required for decision is the average of the waiting
times required for decision when life tests are conducted on a
6.3.7 Unacceptable Failure Rate During Period of Time—
large number of samples drawn at random from the same The “unacceptable failure rate during period of time,” G ,
E2696 − 21
(G > G ), is the minimum failure rate during the period of average time required to determine acceptability but, on the
1 0
time that results in the lot being considered unsatisfactory. The otherhand,willincreasethecostbecauseofplacingmoreunits
interval between G and G is a zone of indifference in which of product on test.
0 1
there is a progressively greater degree of dissatisfaction as the
6.7 Exponential Distribution:
failure rate increases from G to G .
0 1
6.7.1 Exponential Distribution with One Parameter—The
6.3.8 Life Test Sampling Plans Based on Failure Rates—
density function for the exponential distribution with one
Lifetestsamplingplansthatarebasedonfailureratesaregiven
parameter is given by:
in 9.7 and 9.8.
f~t;θ! 51/θexp~2t/θ! t$0, θ.0 (4)
6.4 OC Curves and Sampling Risks:
50 t,0
6.4.1 OC Curve—The OC curve of a life test sampling plan
is the curve that shows the probability that a submitted lot with
6.7.1.1 The function has the following general graphical
given mean life would meet the acceptability criterion on the
form:
basis of that sampling plan.
6.4.2 Producer’s Risk—The producer’s risk, α, is the prob-
ability of rejecting lots with mean life, θ . For the procedures
of 9.7 and 9.8, the producer’s risk may also be defined as the
probability of rejecting lots with acceptable proportion of lot
failing before specified time, p .
6.4.3 Consumer’s Risk—The consumer’s risk, β,isthe
probability of accepting lots with mean life, θ . For the
procedures of 9.7 and 9.8, the consumer’s risk may also be
defined as the probability of accepting lots with p as the
6.7.2 Exponential Distribution with Two Parameters—The
unacceptable proportion of lot failing before specified time.
density function for the exponential distribution with two
parameters is given by:
6.5 Submittal of Product:
6.5.1 Lot—The term “lot” shall mean either an “inspection
f t;θ,A 51/θexp@2 t 2 A /θ# t$ A$0 (5)
~ ! ~ !
lot,”thatis,acollectionofunitsofproductmanufacturedunder
5 0 elsewhere
essentially the same conditions from which a sample is drawn
and tested to determine compliance with the acceptability
6.7.2.1 The function has the following general graphical
criterion or, a “preproduction lot,” that is, one or more units of
form:
product submitted before the initiation of production for test to
determine compliance with the acceptability criterion.
6.6 Sample Selection:
6.6.1 Drawing of Samples—A sample is one or more units
of product drawn at random from a lot.
6.6.2 Testing without Replacement—Life test sampling
without replacement is a life test procedure whereby failed
units are not replaced.
6.6.3 Testing with Replacement—Life test sampling with
6.7.2.2 The quantity, A, is called “guarantee time” and the
replacement is a life test procedure whereby the life test is
one parameter case is a special case of the two-parameter
continued with each failed unit of product replaced by a new
distribution with a guarantee time of zero.
one, drawn at random from the same lot, as soon as the failure
6.7.3 Exponential Distribution when Number of Parameters
occurred. In the case of complex unit of product, this may be
Is Unspecified—In this practice, whenever the term “exponen-
interpreted to mean replacement of the component that caused
tial distribution” is mentioned without specific mention of the
the failure by a new component drawn at random from the
number of parameters, it shall be assumed to mean the
same lot of components.When the “sample sizes” are the same
exponential distribution with one parameter.
in both instances, the expected waiting time required for
decision when testing with replacement is less than when
7. General Description of Life Test Sampling Plans
testing without replacement.
6.6.4 Sample Size—The sample size, n, for a life test is the 7.1 Scope:
7.1.1 Purpose—Sections 7 through 10 of this practice es-
number of units of product placed on test at the start of a life
test. When testing with replacement, the total number of units tablish life test sampling plans for determining acceptability of
a product when samples are drawn at random from an
of product placed on test will, in general, be greater than the
original sample size. The sample sizes for the life test plans of exponential distribution.
Sections 8 to 10 depend on the relative cost of placing large 7.1.2 Specifying Acceptable Mean Life—Before the start of
numbers of units of product on test and the expected length of the life test, the particular value of the acceptable mean life, θ ,
time the life tests must continue to determine acceptability of shall be specified except when using the procedures of 9.7 and
the lots. Increasing the sample size will, on one hand, cut the 9.8.
E2696 − 21
7.1.3 Specifying Unacceptable Mean Life—The particular given mean life would meet the acceptability criterion on the
value of the unacceptable mean life, θ , shall be specified in basis of that sampling plan. The OC curves given in Fig. A1.1
advance of the life test when using the life test procedures of are equally applicable for the sampling plans of 8.1 – 8.5, 9.1
8.6 and 9.6. – 9.5, and Section 10. Moreover, the OC curves are also
7.1.4 Specifying Acceptable Proportion of Lot Failing be- equally applicable for both the sampling with and without
fore Specified Time—The particular value, p , of the acceptable replacement procedures. The abscissas of the OC curves are
proportion of lot failing before specified time to be used in the expressed as the ratio θ/θ in Fig. A1.1 so that the same set of
life test shall be specified in advance for the procedures of 9.7 OC curves is applicable regardless of the value of the specified
and 9.8. acceptable mean life θ .
7.1.5 Specifying Unacceptable Proportion of Lot Failing 7.3.2 Sampling Plan Code Designation—The life test sam-
before Specified Time—The particular value, p , of the unac- pling plans of 8.1 – 8.5, 9.1 – 9.5, and Section 10, along with
ceptable proportion of lot failing before specified time shall be their associated OC curves, are designated by code letters and
specified in advance of the life test when using the procedures numbers. The sample code is given in Table A1.1 and is
of 9.7 and 9.8. determined by the values of α, β, and θ /θ . The OC curves of
1 0
all sampling plans designated by the same code pass through
7.2 Sampling Risks:
the two points (1, 1-α) and (θ /θ , β = 0.10).Thus, all sampling
1 0
7.2.1 Producer’s Risk—The producer’s risk, α, is the prob-
plans that are designated by the same code offer essentially the
ability of rejecting lots with mean life, θ . For the procedures
same protection.
of 9.7 and 9.8, the producer’s risk may also be defined as the
7.3.3 Ratio θ /θ as Measure of Protection Offered by
1 0
probability of rejecting lots with p as the acceptable propor-
Sampling Plan—The consumer’s risk β has been defined in
tion of lot failing before specified time. Summarized in the
7.2.3 as the risk of accepting lots with mean life, θ . Because
following are the various numerical values of α and the master
the OC curves are drawn with abscissa, θ /θ , the ratio, θ /θ ,
1 0 1 0
sampling tables in which they are given.
is also a measure of mean life that is accepted with probability,
Procedures for Producer’s Risk Table
β.Theratio, θ /θ ,shallbegreaterthanzerobutlessthanunity.
1 0
8.1–8.5 0.01, 0.05, 0.10, 0.25, 0.50 Table A1.2
If α, β, and θ are kept constant, as θ /θ increases, the
8.6 0.01, 0.05, 0.10, 0.25 Table A1.7
0 1 0
9.1–9.5 0.01, 0.05, 0.10, 0.25, 0.50 Tables A1.8-A1.12
protection offered by the sampling plan against accepting lots
and
with low mean life also increases. Thus, Table A1.1 allows
Tables A1.13-A1.17
comparisons in the amount of protection offered by the various
9.6 0.01, 0.05, 0.10, 0.25 Table A1.18 and
Table A1.19
sampling plans, for in any column, the protection increases as
9.7 and 9.8 0.01, 0.05, 0.10 Table A1.20
θ /θ increases.
1 0
10 0.01, 0.05, 0.10, 0.25, 0.50 Tables A1.21-A1.25
7.4 Specifying Acceptance Procedures—To identify com-
7.2.2 Specifying Producer’s Risk—The particular value of α
pletely the sampling plan to be used, the following shall be
to be used in the life test shall be selected from among those
specified for the sampling plans of:
given in 7.2.1 and specified in advance of the life test.
7.2.3 Consumer’s Risk—The consumer’s risk, β,isthe 8.1–8.5 α, r, θ or sample plan code, θ
0 0
8.6 α, β, θ , θ
0 1
probability of accepting lots with mean life, θ . For the
9.1–9.5 θ , r, α, n or sample plan code, n, θ
0 0
procedures of 9.7 and 9.8, the consumer’s risk may also be
9.6 α, β, θ , θ , T
0 1
defined as the probability of accepting lots with p as the 9.7 and 9.8 α, β, p , p , T or α, β, G , G , T
0 1 0 1
10 Sample plan code, θ
unacceptable proportion of lot failing before specified time.
7.4.1 In addition, the use of life testing with or without
Summarized in the following are the various numerical values
replacement may be specified, except when using the sampling
of β and the master sampling tables in which they are given.
plans of 9.7 and 9.8.
Procedures for Consumer’s Risk Table
8.1–8.5 0.10 Table A1.2
8. Life Tests Terminated upon Occurrence of Preassigned
8.6 0.01, 0.05, 0.10, 0.25 Table A1.7
9.1–9.5 0.10 Tables A1.8-A1.12 and Number of Failures
Tables A1.13-A1.17
8.1 Life Test Sampling Plans—This part of the practice
9.6 0.01, 0.05, 0.10, 0.25 Table A1.18 and
Table A1.19
describes the procedures for use with life tests that are
9.7 and 9.8 0.01, 0.05, 0.10 Table A1.20
terminated upon the occurrence of a preassigned number of
10 0.10 Tables A1.21-A1.25
failures. Two procedures are given: (1) a procedure when
7.2.3.1 The smaller the value of β, the greater is the
testing without replacement and (2) another procedure when
protectionagainstacceptanceoflotswithlowmeanlifeorhigh
testing with replacement.
failure rate.
8.1.1 Use of Life Test Sampling Plans—To determine
7.2.4 SpecifyingConsumer’sRisk—Theparticularvalueof β
whether the lot meets the acceptability criterion with respect to
to be used in the life test shall be selected from among those
average length of life, the applicable sampling plan shall be
given in 7.2.3 and specified in advance of the life test.
used in accordance with the provisions of Section 7 and those
7.3 OC Curves: in this part of the practice.
7.3.1 OC Curve—The OC curve of a life test sampling plan 8.1.2 Drawing of Samples—All samples shall be drawn in
is the curve that shows the probability that a submitted lot with accordance with 6.6.
E2696 − 21
8.2 Selecting the Life Test Sampling Plan: where:
ˆ
8.2.1 Master Sampling Table—The master sampling table θ = estimate of the lot mean life,
r,n
for the life test sampling plans of this part of the practice is r = termination number,
n = original sample size, and
Table A1.2.
x = time when the rth failure occurs.
r,n
8.2.2 Obtaining the Sampling Plan—The life test sampling
plan consists of a sample size, a termination number, and an ˆ
8.4.3 Acceptability Criterion—Compare the quantity θ
r,n
associated acceptability constant. The sampling plan is ob- ˆ
with the acceptability constant, C, mentioned in 8.2.2.3.If θ
r,n
tained from Master Table A1.2.
is equal to or greater than C, the lot meets the acceptability
ˆ
8.2.2.1 Sample Sizes—For the procedures of 8.1 – 8.5, the
criterion; if θ is less than C, then the lot does not meet the
r,n
acceptabilityconstantsandtheOCcurvesdonotdependonthe
acceptability criterion.
number of units of product placed on test. The sample size, as
8.4.3.1 Example 1: Use of TableA1.2—Find a life test plan
mentioned in 6.6.4, depends on the relative cost of placing
that is to be stopped on the occurrence of the fifth failure and
large numbers of units of product on test and the expected
will accept a lot having an acceptable mean life of 1000 h with
length of time the life test shall continue. The sample size may
probability 0.90.
be selected by using the procedures of 8.5.
8.4.3.2 Solution—In the notation of this section, θ = 1000,
8.2.2.2 Termination Number—The termination number, r,
α = 0.10, and r = 5. In the testing without replacement case:
may be selected from among those given in Table A1.2 and
specified before the initiation of the life test. The choice of this
ˆ
θ 5 @x 1x 1x 1x 1x 1 n 2 5 x # (8)
~ !
r,n 1,n 2,n 3,n 4,n 5,n 5,n
number shall be dependent on the degree of protection desired
ˆ
(1) In the replacement case, θ =nx /5. The acceptabil-
against acceptance of material with unacceptable mean life. r,n 5,n
ity criterion is, accept the lot if:
The larger the termination number, the larger is the ratio, θ /θ ,
1 0
and, as mentioned in 7.3.3, the greater is the assurance against
ˆ
θ $ C (9)
5,n
accepting material with an unacceptable mean life.
8.2.2.3 Acceptability Constant—The acceptability constant, $ θ ~C/θ ! 5 ~1000!~0.487! 5 487
0 0
(2) The quantity C/θ = 0.487 is obtained from TableA1.2.
C, corresponding to the applicable termination number, r, and
In words, place n items on test. Wait until the first five failures
producer’s risk, α, is obtained from the master table by
ˆ ˆ
multiplying the tabled entry by the acceptable mean life, θ . occur. Compute θ . Accept the lot if θ ≥ 487; reject the lot
0 5,n 5,n
otherwise.
8.3 Lot Acceptability Procedures when Testing without Re-
(3) The code designation for the above life test sampling
placement:
plan is obtained from Table A1.2 as C-5. From Fig. A1.1, the
8.3.1 Estimate of Mean Life—The acceptability of a lot,
probability of accepting a lot with mean life of, say, 500 h may
when using a life test from this part of the practice, shall be
beobtainedbyfindingtheordinateoftheOCcurvelabeledC-5
ˆ
judged by the quantity, θ .
r,n
at the point where the abscissa θ/θ = 500⁄1000 = 0.5. The
8.3.2 Computation—The following quantity shall be com-
probability is seen to be equal to 0.47.
puted from the test results:
(4) In this example, if the termination number had been
r
1 selectedas6insteadof5,theprobabilityofacceptingalotwith
ˆ
θ 5 x 1 n 2 r x (6)
F ~ ! G
r,n ( i,n r,n
r
mean life of 500 h is obtained from the OC curve labeled C-6.
i51
The probability is seen to equal 0.41. This illustrates the
where:
remark made in 8.2.2.2 that the larger the termination number,
ˆ
θ = estimate of the lot mean life,
r,n
the higher the probability of rejecting lots with unacceptable
r = termination number,
mean life.
n = sample size, and
x = time when the ith failure occurs. i=1,2, …, r.
i,n
8.4.3.3 Example 2: Calculations for Testing Without
ˆ
Replacement—Suppose that in the life test of Example 1, 10
8.3.3 Acceptability Criterion—Compare the quantity θ
r,n
ˆ
with the acceptability constant C, mentioned in 8.2.2.3.If θ unitsofproducthadbeenplacedontest.Ifthefailedunitswere
r,n
is equal to or greater than C, the lot meets the acceptability not replaced and the first 5 failure times were 50, 75, 125, 250,
ˆ
criterion; if θ is less than C, then the lot does not meet the and 300, determine whether the lot met the acceptability
r,n
acceptability criterion. criterion.
8.4.3.4 Solution—In this case:
8.4 Lot Acceptability Procedures when Testing With Re-
placement:
5017511251250130015~300!
ˆ
θ 5 5460 (10)
5,10
8.4.1 Estimate of Mean Life—The acceptability of a lot,
(1) Since 460 < 487, the lot did not meet the acceptability
when using a life test from this part of the practice, shall be
ˆ
judged by the quantity θ . criterion.
r,n
8.4.2 Computation—The following quantity shall be com-
8.4.3.5 Example 3: Calculations for Testing With
puted from the test results:
Replacement—Suppose that in the life test of Example 1, 10
ˆ
θ 5nx /r (7) unitsofproducthadbeenplacedontest.Ifthefailedunitswere
r,n r,n
E2696 − 21
replaced immediately and the first 5 failure times were 56, 128, is a measure of the relative expected saving in time as a
176, 276, and 442, determine whether the lot met the accept- result of using larger sample sizes.
ability criterion. 8.5.4 Relative Saving in Time by Testing with Replacement
8.4.3.6 Solution—In this case: as Compared to Testing Without Replacement—When testing
with replacement, the expected waiting time required to
10 442
~ !
ˆ
θ 5 5884 (11)
observe the rth failure in a sample of size n (n ≥ r) is equal to
5,10
the quantity rθ/n. When testing without replacement, this
(1) Since 884 > 487, the lot met the acceptability criterion.
expected waiting time may be obtained from Table A1.3 or
8.5 Expected Waiting Time of Life Tests and Cost Consid-
TableA1.4 by multiplying the tabled entry by the mean life of
erations in Selection of Sample Sizes—The operating charac-
the lot θ. By dividing these two expected waiting times, the
teristics of the life test sampling plans of 8.1 – 8.4 are
mean life of the lot cancels out and the ratio
independent of the number of units of product placed on test.
Expected Waiting Time for r Failures in Sample of n When Testing With Replacement
(15)
Expected Waiting Time for r Failures in Sample of n When Testing Without Replacement
Thus, all tests based on common values of the termination
number, r, and producer’s risk, α, are equally good, and the
is a measure of the relative expected saving in time as a
choice of the sample size, n, depends only on the relative cost
result of sampling with replacement. A brief table of these
of placing a large number of units of product on test and the
ratios is given in Table A1.6.
expected waiting time required for decision. For fixed α and r,
8.5.4.1 Example 4: Saving in Time by Increasing Sample
increasing n will, on one hand, cut the expected waiting time;
Size When Testing Without Replacement—Compare the aver-
but will, on the other hand, increase the cost because of placing
age length of time needed to observe the failure of the first two
more units of product on test.This part of the practice provides
out of five units of product under test with the average length
procedures for determining the optimum sample size based on
of time required to observe the failure of two out of two units
considerations of cost.
when testing without replacement.
8.5.1 Expected Waiting Time—The mean life of the lot and, 8.5.4.2 Solution—From Table A1.3, it is seen that for r=2
as noted in 8.5, the size of the sample drawn from the lot affect
and n = 2, the expected waiting time is 1.5000θ and that for
the expected waiting time required to observe the rth failure in r = 2 and n = 5, the expected waiting time is 0.4500θ.Thus, the
a sample of size n. The rth failure is expected to occur more
relative saving in time by placing five units on test is
quickly in samples drawn from lots with low values of mean 0.4500θ/1.500θ = 0.300. This figure may also be obtained
life. The values of the expected waiting time divided by the
directly from Table A1.5. Hence, the average time required
mean life of the lot, when testing without replacement, are when five units are placed on test is 30 % of the average time
given in TableA1.3 and TableA1.4. Corresponding values for
required when only two units are used.
thetestingwithreplacementsituationarenottabledbutmaybe 8.5.4.3 Example 5: Saving in Time by Increasing Sample
calculatedbydividingtheterminationnumber,r,bythesample
Size When Testing With Replacement—Make the same com-
size, n, that is: parison as in Example 4 (8.5.4.1) if the testing had been with
replacement.
Expected Waiting Time r
5 (12)
8.5.4.4 Solution—For r = 2 and n = 2, the expected waiting
Mean Life of a Lot n
timeis θandthatforr=2andn=5isrθ/n=2θ/5=0.4θ.Thus,
8.5.2 Relative Saving in Time by Increasing Sample Size
the relative saving in time by placing five units on test is
When Testing Without Replacement—When testing without
0.4θ/θ = 0.4. Hence, the average time required when five units
replacement, the expected waiting time required to observe the
are placed on test is 40 % of the average time required when
rth failure in a sample of size n,(n ≥ r), may be obtained from
only two units are used.
TableA1.3orTableA1.4bymultiplyingthetabledentrybythe
8.5.4.5 Example 6: Saving in Time by Testing With
mean life of the lot. By dividing the expected waiting time
Replacement—Compare the average length of time needed to
when n units of product are placed on test by that when only r
observe the failure of the first five out of five units of product
units are placed on test, the mean life of the lot cancels out and
under test when testing with replacement with the average
the ratio
length of time needed when testing without replacement.
8.5.4.6 Solution—When testing with replacement, for r=5
Expected Waiting Time for r Failures in Sample of n
(13)
and n = 5, the expected waiting time is θ.When testing without
Expected Waiting Time for r Failures in Sample of r
replacement, TableA1.3 or TableA1.4 shows that the expected
is a measure of the relative expected saving in time as a
waiting time is 2.2833θ. Thus, the relative saving in time by
result of placing more units of product on test.Abrief table of
testing with replacement is θ/2.2833θ = 0.438; or the average
these ratios is given in Table A1.5.
timerequiredforadecision,byreplacingfailedunits,is43.8 %
8.5.3 Relative Saving in Time by Increasing Sample Size
of the average time required when failed units are not replaced.
When Testing With Replacement—When testing with
This figure may also be obtained directly from Table A1.6.
replacement, the expected waiting time required to observe the
8.5.5 Cost Considerations in Choice of Sample Size—
rth failure in a sample of size n is equal to the quantity rθ/ n.
Methods for finding the optimum sample size based on
By dividing the expected waiting time when n units of product
considerations of cost are given in this section.
are placed on test by that when only r units are placed on test,
8.5.5.1 Cost When Testing Without Replacement—The total
the mean life of the lot cancels out and the ratio
expected cost of any of the life test plans of 8.2 when testing
Relative Saving 5 rθ/nθ 5 r/n (14) without replacement is given by:
E2696 − 21
1 1 1
θ = acceptable mean life,
c θ 5 1 1…1 1c n (16)
S D
1 0 2
n n 2 1 n 2 r 2 1
r = termination number, and
n = sample size.
where:
8.5.5.4 Optimum Sample Size When Testing With
c = cost of waiting per unit time,
Replacement—The value of n, which minimizes the total cost,
c = cost of placing a unit of product on test,
as determined by the method of 8.5.5.3, is the optimum sample
θ = acceptable mean life,
r = termination number, and size. In general, the optimum n for the case of testing with
n = sample size.
replacement is the integer nearest to:
8.5.5.2 Optimum Sample Size When Testing Without
c θ r 1
1 0
Π1 (18)
Replacement—The value of n, which minimizes the total cost,
c 4
as determined by the method of 8.5.5.1, is the optimum sample
(1) Example 9: Calculation of Cost—Consider the problem
size. A general method of obtaining the optimum n is to use
of Example 7 (8.5.5.2(1)), that is, r = 10, θ = 1000, c = $1,
0 1
Table A1.3 or Table A1.4. The smallest n is chosen such that
and c = $100. Using the total cost formula, determine the
the difference between the expected waiting time for the rth
optimum sample size if failed units were replaced.
failure when that number of units of product are placed on test
(2) Solution—Using the formula of 8.5.5.3, the costs for
and that when n + 1 units are placed on test is less than the
various values of n, are:
quantity c /c θ .
2 1 0
Expected Cost Cost of Units
(1) Example 7: Calculation of Costs—Consider the case in
n Because of Waiting Tested Total Cost
9 1111 1800 2911
which r = 10, θ = 1000 h, c = $1 per hour, and c = $100 per
0 1 2
10 1000 1900 2900
unit of product tested. Using the total cost formula, determine
11 909 2000 2909
the optimum sample size if failed units are not replaced.
(a) The optimum sample size is thus n = 10.
(2) Solution—Using the formula of 8.5.5.1, the costs for
(3) Example 10: Obtaining Optimum Sample Size by
various values of n are:
Formula—Usethemethodof8.5.5.4todeterminetheoptimum
Expected Cost Cost of Units
sample size for the problem of Example 9.
n Because of Waiting Tested Total Cost
(4) Solution—The integer nearest to
10 2929 1000 3929
11 2020 1100 3120
12 1603 1200 2803 1~1000!~10! 1
1 5 10.012 (19)
Œ
13 1346 1300 2646
100 4
14 1168 1400 2568
15 1035 1500 2535
is 10. This is the optimum sample size as was seen in
16 931 1600 2531
Example 9 (8.5.5.4(1)).
17 847 1700 2547
8.6 Life Test Sampling Plans for Certain Specified Values of
(a) The optimum sample size is thus n = 16.
α, β, and θ /θ —A life test sampling plan may be designed so
(3) Example 8: Obtaining Optimum Sample Size With
1 0
that its OC curve meets the following prescribed conditions: if
Expected Waiting Time—Use Table A1.3 to determine the
optimum sample size for the problem of Example 7, 8.5.5.2(1). θ = θ , then the probability of the lot meeting the acceptability
criterion is less than or equal to β. This part of the practice,
(4) Solution—The quantity c /c θ is equal to 0.1 and, from
2 1 0
Table A1.3, the expected waiting times are: which may be considered an extension of 8.1 – 8.5, provides
procedures for obtaining values of the termination number, r,
Expected Waiting Time to Observe 10th Failure
n in n in n + 1 Difference
and the acceptability constant, C, when certain selected values
10 2.9290 2.0199 0.9091
of α, β, and θ /θ are specified. When other values of α, β, and
1 0
11 2.0199 1.6032 0.4167
θ /θ thanthoseprovidedinthispartofthepracticeareneeded,
12 1.6032 1.3468 0.2564 1 0
13 1.3468 1.1682 0.1786 refer to 8.1 – 8.5 to determine whether one of the life test
14 1.1682 1.0349 0.1333
sampling plans given in those paragraphs are applicable.
15 1.0349 0.9307 0.1042
A 8.6.1 Life Test Sampling Plans—From TableA1.7, values of
16 0.9307 0.8467 0.0840
17 0.8467 0.7773 0.0694
the termination number, r, and the acceptability constant, C,
may be obtained for values of α = 0.01, 0.05, 0.10, and 0.25;
A
The optimum sample size isn = 16, as was seen in Example 7 (8.5.5.2(1)), since
β=0.01,0.05,0.10,and0.25;and θ /θ =2/3,1/2,1/3,1/5,and
1 0
that is the smallest sample size for which the difference in expected waiting times
1/10. The value of r is obtained directly from Table A1.7, but
is less than c /c θ or 0.1.
2 1 0
the acceptability constant, C, is obtained by multiplying the
8.5.5.3 Cost When Testing With Replacement—The total
tabled entry by the acceptable mean life, θ .
expected cost of any of the life test plans of 8.2, when testing
8.6.1.1 Example 11—Find a life test sampling plan that
with replacement, is given by:
possesses the following OC curve: If the mean life is θ = 900
r
h, the lot is accepted with probability 0.95; if the mean life is
c θ 1c ~n1r 2 1! (17)
1 0 2
n θ = 300 h, it is accepted with probability approximately equal
to 0.10.
where:
8.6.1.2 Solution—In this example, θ /θ = 1/3, α = 0.05, and
1 0
c = cost of waiting per unit time,
β = 0.10. Looking in TableA1.7, the termination number r=8
c = cost of placing a unit of product on test,
and the acceptability constant C = θ (C/θ ) = 900(0.498) = 448
0 0
E2696 − 21
are obtained. In word form, place eight or more units of 9.2.2 Obtaining the Sampling Plan—The life test sampling
product on test. Stop life testing after eight failures have plan consists of a termination number, r, a sample size, n, and
ˆ
occurred. If the estimate of lot mean life θ is greater than or an associated termination time, T.
8n
equal to 448, the lot is acceptable; otherwise, the lot is not
9.2.2.1 Termination Number—The termination number, r,
acceptable.
shallbeselectedfromamongthosegiveninTablesA1.8-A1.12
8.6.2 Expansion of Table A1.7 for Values of θ /θ Greater
...

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: E2696 − 09 (Reapproved 2018) E2696 − 21 An American National Standard
Standard Practice for
Life and Reliability Testing Based on the Exponential
Distribution
This standard is issued under the fixed designation E2696; 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 standard sampling procedures and tables for life and reliability testing in procurement, supply, and
maintenance quality control operations as well as in research and development activities.
1.2 This practice describes general procedures and definitions of terms used in life test sampling and describes specific procedures
and applications of the life test sampling plans for determining conformance to established reliability requirements.
1.3 This practice is an adaptation of the Quality Control and Reliability Handbook H-108, “Sampling Procedures and Tables for
Life and Reliability Testing (Based on Exponential Distribution),” U.S. Government Printing Office, April 29, 1960.
1.4 A system of units is not specified in this practice.
1.5 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.
1.6 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
E2555 Practice for Factors and Procedures for Applying the MIL-STD-105 Plans in Life and Reliability Inspection
3. Terminology
3.1 Definitions:
3.1.1 See Terminology E456 for a more extensive listing of terms in ASTM Committee E11 standards.
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 Dec. 1, 2018May 1, 2021. Published December 2018June 2021. Originally approved in 2009. Last previous edition approved in 20132018 as
E2696 – 09 (2013).(2018). DOI: 10.1520/E2696-09R18.10.1520/E2696-21.
For referenced ASTM standards, visit the ASTM website, www.astm.org, or contact ASTM Customer Service at [e-mail]. For Annual Book of ASTM Standards
volume information, refer to the standard’s Document Summary page on the ASTM website.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E2696 − 21
3.1.2 consumer’s risk, β, n—probability that a lot having specified rejectable quality level will be accepted under a defined
sampling plan. E2555
3.1.2.1 Discussion—
In this practice, the consumer’s risk is the probability of accepting lots with mean time to failure θ .
3.1.2.2 Discussion—
For the procedures of 9.7 and 9.8, the consumer’s risk may also be defined as the probability of accepting lots with unacceptable
proportion of lot failing before specified time, p .
3.1.3 life test, n—process of placing one or more units of product under a specified set of test conditions and measuring the time
until failure for each unit.
3.1.4 mean time to failure, failure (MTTF), θ, n—in life testing, the average length of life of items in a lot.
3.1.4.1 Discussion—
Also known referred to as mean life.
3.1.5 number of failures, n—number of failures that have occurred at the time the decision as to lot acceptability is reached.
3.1.5.1 Discussion—
The expected number of failures required for decision is the average of the number of failures required for decision when life tests
are conducted on a large number of samples drawn at random from the same exponential distribution.
3.1.6 producer’s risk, α, n—probability that a lot having specified acceptable quality level will be rejected under a defined
sampling plan.
3.1.6.1 Discussion—
In this practice, the producer’s risk is the probability of rejecting lots with mean time to failure θ .
3.1.6.2 Discussion—
For the procedures of 9.7 and 9.8, the producer’s risk may also be defined as the probability of rejecting lots with acceptable
proportion of lot failing before specified time, p .
3.1.7 sequential life test, n—life test sampling plan whereby neither the number of failures nor the time required to reach a decision
are fixed in advance but instead decisions depend on the accumulated results of the life test.
3.1.8 unit of product, n—that which is inspected to determine its classification as defective or nondefective or to count the number
of defects. E2234
3.1.9 waiting time, n—in life testing, the time elapsed from the start of testing until a decision is reached as to lot acceptability.
3.1.9.1 Discussion—
The expected waiting time required for decision is the average of the waiting times required for decision when life tests are
conducted on a large number of samples drawn at random from the same exponential distribution.
4. Significance and Use
4.1 This practice was prepared to meet a growing need for the use of standard sampling procedures and tables for life and
reliability testing in government procurement, supply, and maintenance quality control (QC) operations as well as in research and
development activities where applicable.
4.2 A characteristic feature of most life tests is that the observations are ordered in time to failure. If, for example, 20 radio tubes
are placed on life test, and t denotes the time when the ith tube fails, the data occur in such a way that t ≤ t ≤ . ≤ t . The same
i 1 2 n
kind of ordered observations will occur whether the problem under consideration deals with the life of electric bulbs, the life of
electronic components, the life of ball bearings, or the length of life of human beings after they are treated for a disease. The
examples just given all involve ordering in time.
4.3 In destructive testing involving such situations as the current needed to blow a fuse, the voltage needed to break down a
condenser, or the force needed to rupture a physical material, the test can often be arranged in such a way that every item in the
sample is subjected to precisely the same stimulus (current, voltage, or stress). If this is done, then clearly the weakest item will
be observed to fail first, the second weakest next, and so forth. While the random variable considered mostly in this guide is time
E2696 − 21
to failure, it should be emphasized, however, that the methodology provided herein can be adapted to the testing situations
mentioned above when the random variable is current, voltage, stress, and so forth.
4.4 Sections 6 and 7 describe general procedures and definitions of terms used in life test sampling. Sections 8, 9, and 10 describe
specific procedures and applications of the life test sampling plans for determining conformance to established reliability
requirements.
4.5 Whenever the methodology or choice of procedures in the practice requires clarification, the user is advised to consult a
qualified mathematical statistician, and reference should be made to appropriate technical reports and other publications in the
field.
5. Introduction
5.1 The theory underlying the development of the life test sampling plans of this section, including the operating characteristic
curves, assumes that the measurements of the length of life are drawn from an exponential distribution. Statistical test procedures
for determining the validity of the exponential distribution assumption have appeared in the technical statistical journals. Professor
Benjamin Epstein published a comprehensive paper (in two parts) on this subject in the February and May 1960 issues of
Technometrics. Part I of the paper contains descriptions of the mathematical and graphical procedures as well as an extensive
bibliography for reference purposes. Numerical examples illustrating the statistical procedures are included in Part II of the paper.
5.2 It is important to note that the life test sampling plans of this practice are not to be used indiscriminately simply because it
is possible to obtain life test data. Only after the exponential assumption is deemed reasonable should the sampling plans be used.
5.3 Sections 6 and 7 describe general procedures and description of life test sampling plans. Section 8 describes specific
procedures and applications of sampling plans when life tests are terminated upon the occurrence of a preassigned number of
failures, and Section 9 provides sampling plans when life tests are terminated at a preassigned time. Section 10 describes sequential
life test sampling plans. Section 8 covers: (1) acceptance procedures; (2) expected duration of life tests and cost considerations in
selection of sample sizes; and (3) life test plans for certain specified values of α, β, and θ /θ . Section 9 covers: (1) acceptance
1 0
procedures; (2) life test plans for certain specified values of α, β, θ /θ , and T/θ ; and (3) life test plans based on proportion of lot
1 0 0
failing before specified time. Section 10 covers: (1) acceptance procedures; (2) graphical acceptance procedures; and (3) expected
number and waiting time required for decision.
5.4 Operating characteristic (OC) curves for the life test sampling plans of 8.1 – 8.5, 9.1 – 9.5, and Section 10 are shown in Fig.
A1.1 for the corresponding sampling plans in these sections were matched with respect to their OC curves. The OC curves in Fig.
A1.1 have been computed for the life test sampling plans of 8.1 – 8.5 but are equally applicable for the sampling plans of 9.1 –
9.5 and Section 10.
5.5 The procedures of this section are based on the premise that the life tests are monitored continuously. If the tests are monitored
only periodically, the values obtained from the tables and curves are only approximations.
6. General Definitions of Life and Reliability Test Terms
6.1 Discussion of Terms and Procedures:
6.1.1 Purpose—This section provides definitions of terms required for the life test sampling plans and procedures of Sections 7
through 10.
6.1.2 Life Test—Life test is the process of placing the “unit of product” under a specified set of test conditions and measuring the
time it takes until failure.
6.1.3 Unit of Product—The unit of product is the entity of product that may be placed on life test.
6.1.4 Specifying Failure—The state that constitutes a failure shall be specified in advance of the life test.
Epstein, B., “Tests for the Validity of the Assumption that the Underlying Distribution of Life is Exponential,” Technometrics, Vol 2, February and May 1960, pp. 83–101
and 167–183.
E2696 − 21
6.1.5 Life Test Sampling Plan—A life test sampling plan is a procedure that specifies the number of units of product from a lot
that are to be tested and the criterion for determining acceptability of the lot.
6.1.6 Life Test Terminated upon Occurrence of Preassigned Number of Failures—Life test sampling plans whereby testing is
terminated when a preassigned termination number of failures, r, occur are given in Section 8 of this practice.
6.1.7 Life Test Terminated at Preassigned Time—Life test sampling plans whereby testing is terminated when a preassigned
termination time, T, is reached are given in Section 9 of this practice.
6.1.8 Sequential Life Test—Sequential life test is a life test sampling plan whereby neither the number of failures nor the time
required to reach a decision are fixed in advance but, instead, decisions depend on the accumulated results of the life test.
Information on the observed time to failure are accumulated over time and the results at any time determine the choice of one
among three possible decisions: (1) the lot meets the acceptability criterion, (2) the lot does not meet the acceptability criterion,
or (3) the evidence is insufficient for either decision (1) or (2) and the test must continue. Sequential life test sampling plans are
given in Section 10 of this practice and have the advantage over the life test sampling plans mentioned in 6.1.6 and 6.1.7 in that,
for the same OC curve, the expected waiting time and the expected number of failures required to reach a decision as to lot
acceptability are less for the sequential life tests.
6.1.9 Expected Number of Failures—The number of failures required for decision is the number of failures that have occurred at
the time the decision as to lot acceptability is reached. For the life test sampling plans mentioned in 6.1.6, this number of failures
is known in advance of the life test; but, for the sampling plans mentioned in 6.1.7 and 6.1.8, this number cannot be predetermined.
The expected number of failures required for decision is the average of the number of failures required for decision when life tests
are conducted on a large number of samples drawn at random from the same exponential distribution. The expected number of
failures can be predetermined for the sampling plans mentioned in 6.1.6 – 6.1.8.
6.1.10 Expected Waiting Time—The waiting time required for decision is the time elapsed from the start of the life test to the time
decision is reached as to lot acceptability. The waiting time required for decision cannot be predetermined for any of the sampling
plans mentioned in 6.1.6 – 6.1.8. The expected waiting time required for decision is the average of the waiting times required for
decision when life tests are conducted on a large number of samples drawn at random from the same exponential distribution. The
expected waiting time can be predetermined for the sampling plans mentioned in 6.1.6 – 6.1.8.
6.2 Length of Life:
6.2.1 Length of Life—The terms “length of life” and “time to failure” may be used interchangeably and shall denote the length
of time it takes for a unit of product to fail after being placed on life test. The length of time may be expressed in any convenient
time scale such as seconds, hours, days, and so forth.
6.2.2 Mean Time to Failure—The terms “mean time to failure” and “mean life” may be used interchangeably and shall denote the
mean (or equivalently, the average) length of life of items in the lot. Mean life is denoted by θ.
6.2.3 Acceptable Mean Life—The acceptable mean life, θ , is the minimum mean time to failure that is considered satisfactory.
6.2.4 Unacceptable Mean Life—The unacceptable mean life, θ (θ < θ ), is the mean time to failure such that lots having a mean
1 1 0
life less than or equal to θ are considered unsatisfactory. The interval between θ and θ is a zone of indifference in which there
1 0 1
is a progressively greater degree of dissatisfaction as the mean life decreases from θ to θ .
0 1
6.3 Failure Rate:
6.3.1 Proportion of Lot Failing Before Specified Time—The term “proportion of lot failing before specified time,” p, denotes the
fraction of the lot that fails before some specified time, T, that is:
p 5 12 exp 2T/θ (1)
~ !
6.3.2 Failure Rate during Period of Time—The “failure rate during period of time T,” G, is given by:
G 5 12 exp T/θ 5 p/T (2)
$ ~ !%
T
E2696 − 21
6.3.3 Instantaneous Failure Rate—The “instantaneous failure rate” or “hazard rate” is given by:
Z 51/θ (3)
6.3.4 Acceptable Proportion of Lot Failing Before Specified Time—The “acceptable proportion of lot failing before specified
time,” p , is the maximum fraction of the lot that may fail before time, T, and still result in the lot being considered satisfactory.
6.3.5 Unacceptable Proportion of Lot Failing Before Specified Time—The “unacceptable proportion of lot failing before specified
time,” p , (p > p ), is the minimum fraction of the lot that may fail before time, T, and results in the lot being considered
1 1 0
unsatisfactory. The interval between p and p is a zone of indifference in which there is a progressively greater degree of
0 1
dissatisfaction as the fraction of the lot failing before time, T, increases from p to p .
0 1
6.3.6 Acceptable Failure Rate During Period of Time—The “acceptable failure rate during period of time,” G , is the maximum
failure rate during the period of time that can be considered satisfactory.
6.3.7 Unacceptable Failure Rate During Period of Time—The “unacceptable failure rate during period of time,” G , (G > G ),
1 1 0
is the minimum failure rate during the period of time that results in the lot being considered unsatisfactory. The interval between
G and G is a zone of indifference in which there is a progressively greater degree of dissatisfaction as the failure rate increases
0 1
from G to G .
0 1
6.3.8 Life Test Sampling Plans Based on Failure Rates—Life test sampling plans that are based on failure rates are given in 9.7
and 9.8.
6.4 OC Curves and Sampling Risks:
6.4.1 OC Curve—The OC curve of a life test sampling plan is the curve that shows the probability that a submitted lot with given
mean life would meet the acceptability criterion on the basis of that sampling plan.
6.4.2 Producer’s Risk—The producer’s risk, α, is the probability of rejecting lots with mean life, θ . For the procedures of 9.7 and
9.8, the producer’s risk may also be defined as the probability of rejecting lots with acceptable proportion of lot failing before
specified time, p .
6.4.3 Consumer’s Risk—The consumer’s risk, β, is the probability of accepting lots with mean life, θ . For the procedures of 9.7
and 9.8, the consumer’s risk may also be defined as the probability of accepting lots with p as the unacceptable proportion of lot
failing before specified time.
6.5 Submittal of Product:
6.5.1 Lot—The term “lot” shall mean either an “inspection lot,” that is, a collection of units of product manufactured under
essentially the same conditions from which a sample is drawn and tested to determine compliance with the acceptability criterion
or, a “preproduction lot,” that is, one or more units of product submitted before the initiation of production for test to determine
compliance with the acceptability criterion.
6.6 Sample Selection:
6.6.1 Drawing of Samples—A sample is one or more units of product drawn at random from a lot.
6.6.2 Testing without Replacement—Life test sampling without replacement is a life test procedure whereby failed units are not
replaced.
6.6.3 Testing with Replacement—Life test sampling with replacement is a life test procedure whereby the life test is continued with
each failed unit of product replaced by a new one, drawn at random from the same lot, as soon as the failure occurred. In the case
of complex unit of product, this may be interpreted to mean replacement of the component that caused the failure by a new
component drawn at random from the same lot of components. When the “sample sizes” are the same in both instances, the
expected waiting time required for decision when testing with replacement is less than when testing without replacement.
E2696 − 21
6.6.4 Sample Size—The sample size, n, for a life test is the number of units of product placed on test at the start of a life test. When
testing with replacement, the total number of units of product placed on test will, in general, be greater than the original sample
size. The sample sizes for the life test plans of Sections 8 to 10 depend on the relative cost of placing large numbers of units of
product on test and the expected length of time the life tests must continue to determine acceptability of the lots. Increasing the
sample size will, on one hand, cut the average time required to determine acceptability but, on the other hand, will increase the
cost because of placing more units of product on test.
6.7 Exponential Distribution:
6.7.1 Exponential Distribution with One Parameter—The density function for the exponential distribution with one parameter is
given by:
f t;θ 51/θexp 2t/θ t $ 0, θ.0 (4)
~ ! ~ !
50 t,0
6.7.1.1 The function has the following general graphical form:
6.7.2 Exponential Distribution with Two Parameters—The density function for the exponential distribution with two parameters
is given by:
f~t;θ,A! 51/θexp@2~t 2 A!/θ# t $ A $ 0 (5)
5 0 elsewhere
6.7.2.1 The function has the following general graphical form:
6.7.2.2 The quantity, A, is called “guarantee time” and the one parameter case is a special case of the two-parameter distribution
with a guarantee time of zero.
6.7.3 Exponential Distribution when Number of Parameters Is Unspecified—In this practice, whenever the term “exponential
distribution” is mentioned without specific mention of the number of parameters, it shall be assumed to mean the exponential
distribution with one parameter.
7. General Description of Life Test Sampling Plans
7.1 Scope:
E2696 − 21
7.1.1 Purpose—Sections 7 through 10 of this practice establish life test sampling plans for determining acceptability of a product
when samples are drawn at random from an exponential distribution.
7.1.2 Specifying Acceptable Mean Life—Before the start of the life test, the particular value of the acceptable mean life, θ , shall
be specified except when using the procedures of 9.7 and 9.8.
7.1.3 Specifying Unacceptable Mean Life—The particular value of the unacceptable mean life, θ , shall be specified in advance
of the life test when using the life test procedures of 8.6 and 9.6.
7.1.4 Specifying Acceptable Proportion of Lot Failing before Specified Time—The particular value, p , of the acceptable
proportion of lot failing before specified time to be used in the life test shall be specified in advance for the procedures of 9.7 and
9.8.
7.1.5 Specifying Unacceptable Proportion of Lot Failing before Specified Time—The particular value, p , of the unacceptable
proportion of lot failing before specified time shall be specified in advance of the life test when using the procedures of 9.7 and
9.8.
7.2 Sampling Risks:
7.2.1 Producer’s Risk—The producer’s risk, α, is the probability of rejecting lots with mean life, θ . For the procedures of 9.7 and
9.8, the producer’s risk may also be defined as the probability of rejecting lots with p as the acceptable proportion of lot failing
before specified time. Summarized in the following are the various numerical values of α and the master sampling tables in which
they are given.
Procedures for Producer’s Risk Table
8.1 – 8.5 0.01, 0.05, 0.10, 0.25, 0.50 Table A1.2
8.6 0.01, 0.05, 0.10, 0.25 Table A1.7
9.1 – 9.5 0.01, 0.05, 0.10, 0.25, 0.50 Tables A1.8-A1.12 and
Tables A1.13-A1.17
9.6 0.01, 0.05, 0.10, 0.25 Table A1.18 and
Table A1.19
9.7 and 9.8 0.01, 0.05, 0.10 Table A1.20
10 0.01, 0.05, 0.10, 0.25, 0.50 Tables A1.21-A1.25
7.2.2 Specifying Producer’s Risk—The particular value of α to be used in the life test shall be selected from among those given
in 7.2.1 and specified in advance of the life test.
7.2.3 Consumer’s Risk—The consumer’s risk, β, is the probability of accepting lots with mean life, θ . For the procedures of 9.7
and 9.8, the consumer’s risk may also be defined as the probability of accepting lots with p as the unacceptable proportion of lot
failing before specified time. Summarized in the following are the various numerical values of β and the master sampling tables
in which they are given.
Procedures for Consumer’s Risk Table
8.1 – 8.5 0.10 Table A1.2
8.6 0.01, 0.05, 0.10, 0.25 Table A1.7
9.1 – 9.5 0.10 Tables A1.8-A1.12 and
Tables A1.13-A1.17
9.6 0.01, 0.05, 0.10, 0.25 Table A1.18 and
Table A1.19
9.7 and 9.8 0.01, 0.05, 0.10 Table A1.20
10 0.10 Tables A1.21-A1.25
7.2.3.1 The smaller the value of β, the greater is the protection against acceptance of lots with low mean life or high failure rate.
7.2.4 Specifying Consumer’s Risk—The particular value of β to be used in the life test shall be selected from among those given
in 7.2.3 and specified in advance of the life test.
7.3 OC Curves:
E2696 − 21
7.3.1 OC Curve—The OC curve of a life test sampling plan is the curve that shows the probability that a submitted lot with given
mean life would meet the acceptability criterion on the basis of that sampling plan. The OC curves given in Fig. A1.1 are equally
applicable for the sampling plans of 8.1 – 8.5, 9.1 – 9.5, and Section 10. Moreover, the OC curves are also equally applicable for
both the sampling with and without replacement procedures. The abscissas of the OC curves are expressed as the ratio θ/θ in Fig.
A1.1 so that the same set of OC curves is applicable regardless of the value of the specified acceptable mean life θ .
7.3.2 Sampling Plan Code Designation—The life test sampling plans of 8.1 – 8.5, 9.1 – 9.5, and Section 10, along with their
associated OC curves, are designated by code letters and numbers. The sample code is given in Table A1.1 and is determined by
the values of α, β, and θ /θ . The OC curves of all sampling plans designated by the same code pass through the two points (1,
1 0
1-α) and (θ /θ , β = 0.10). Thus, all sampling plans that are designated by the same code offer essentially the same protection.
1 0
7.3.3 Ratio θ /θ as Measure of Protection Offered by Sampling Plan—The consumer’s risk β has been defined in 7.2.3 as the risk
1 0
of accepting lots with mean life, θ . Because the OC curves are drawn with abscissa, θ /θ , the ratio, θ /θ , is also a measure of
1 1 0 1 0
mean life that is accepted with probability, β. The ratio, θ /θ , shall be greater than zero but less than unity. If α, β, and θ are kept
1 0 0
constant, as θ /θ increases, the protection offered by the sampling plan against accepting lots with low mean life also increases.
1 0
Thus, Table A1.1 allows comparisons in the amount of protection offered by the various sampling plans, for in any column, the
protection increases as θ /θ increases.
1 0
7.4 Specifying Acceptance Procedures—To identify completely the sampling plan to be used, the following shall be specified for
the sampling plans of:
8.1 – 8.5 α, r, θ or sample plan code, θ
0 0
8.6 α, β, θ , θ
0 1
9.1 – 9.5 θ , r, α, n or sample plan code, n, θ
0 0
9.6 α, β, θ , θ , T
0 1
9.7 and 9.8 α, β, p , p , T or α, β, G , G , T
0 1 0 1
10 Sample plan code, θ
7.4.1 In addition, the use of life testing with or without replacement may be specified, except when using the sampling plans of
9.7 and 9.8.
8. Life Tests Terminated upon Occurrence of Preassigned Number of Failures
8.1 Life Test Sampling Plans—This part of the practice describes the procedures for use with life tests that are terminated upon
the occurrence of a preassigned number of failures. Two procedures are given: (1) a procedure when testing without replacement
and (2) another procedure when testing with replacement.
8.1.1 Use of Life Test Sampling Plans—To determine whether the lot meets the acceptability criterion with respect to average
length of life, the applicable sampling plan shall be used in accordance with the provisions of Section 7 and those in this part of
the practice.
8.1.2 Drawing of Samples—All samples shall be drawn in accordance with 6.6.
E2696 − 21
8.2 Selecting the Life Test Sampling Plan:
8.2.1 Master Sampling Table—The master sampling table for the life test sampling plans of this part of the practice is Table A1.2.
8.2.2 Obtaining the Sampling Plan—The life test sampling plan consists of a sample size, a termination number, and an associated
acceptability constant. The sampling plan is obtained from Master Table A1.2.
8.2.2.1 Sample Sizes—For the procedures of 8.1 – 8.5, the acceptability constants and the OC curves do not depend on the number
of units of product placed on test. The sample size, as mentioned in 6.6.4, depends on the relative cost of placing large numbers
of units of product on test and the expected length of time the life test shall continue. The sample size may be selected by using
the procedures of 8.5.
8.2.2.2 Termination Number—The termination number, r, may be selected from among those given in Table A1.2 and specified
before the initiation of the life test. The choice of this number shall be dependent on the degree of protection desired against
acceptance of material with unacceptable mean life. The larger the termination number, the larger is the ratio, θ /θ , and, as
1 0
mentioned in 7.3.3, the greater is the assurance against accepting material with an unacceptable mean life.
8.2.2.3 Acceptability Constant—The acceptability constant, C, corresponding to the applicable termination number, r, and
producer’s risk, α, is obtained from the master table by multiplying the tabled entry by the acceptable mean life, θ .
8.3 Lot Acceptability Procedures when Testing without Replacement:
8.3.1 Estimate of Mean Life—The acceptability of a lot, when using a life test from this part of the practice, shall be judged by
the quantity, θˆ .
r,n
8.3.2 Computation—The following quantity shall be computed from the test results:
r
ˆ
θ 5 x 1 n 2 r x (6)
F ~ ! G
r,n i,n r,n
(
r
i51
where:
θˆ = estimate of the lot mean life,
r,n
r = termination number,
n = sample size, and
x = time when the ith failure occurs. i = 1, 2, …, r.
i,n
8.3.3 Acceptability Criterion—Compare the quantity θˆ with the acceptability constant C, mentioned in 8.2.2.3. If θˆ is equal
r,n r,n
to or greater than C, the lot meets the acceptability criterion; if θˆ is less than C, then the lot does not meet the acceptability
r,n
criterion.
8.4 Lot Acceptability Procedures when Testing With Replacement:
8.4.1 Estimate of Mean Life—The acceptability of a lot, when using a life test from this part of the practice, shall be judged by
the quantity θˆ .
r,n
8.4.2 Computation—The following quantity shall be computed from the test results:
ˆ
θ 5 n x /r (7)
r,n r,n
where:
θˆ = estimate of the lot mean life,
r,n
r = termination number,
n = original sample size, and
x = time when the rth failure occurs.
r,n
E2696 − 21
8.4.3 Acceptability Criterion—Compare the quantity θˆ with the acceptability constant, C, mentioned in 8.2.2.3. If θˆ is equal
r,n r,n
to or greater than C, the lot meets the acceptability criterion; if θˆ is less than C, then the lot does not meet the acceptability
r,n
criterion.
8.4.3.1 Example 1: Use of Table A1.2—Find a life test plan that is to be stopped on the occurrence of the fifth failure and will
accept a lot having an acceptable mean life of 1000 h with probability 0.90.
8.4.3.2 Solution—In the notation of this section, θ = 1000, α = 0.10, and r = 5. In the testing without replacement case:
ˆ
θ 5 @x 1x 1x 1x 1x 1~n 2 5! x # (8)
r,n 1,n 2,n 3,n 4,n 5,n 5,n
(1) In the replacement case, θˆ = nx /5. The acceptability criterion is, accept the lot if:
r,n 5,n
ˆ
θ $ C (9)
5,n
$ θ ~C/θ !5 ~1000!~0.487!5 487
0 0
(2) The quantity C/θ = 0.487 is obtained from Table A1.2. In words, place n items on test. Wait until the first five failures
occur. Compute θˆ . Accept the lot if θˆ ≥ 487; reject the lot otherwise.
5,n 5,n
(3) The code designation for the above life test sampling plan is obtained from Table A1.2 as C-5. From Fig. A1.1, the
probability of accepting a lot with mean life of, say, 500 h may be obtained by finding the ordinate of the OC curve labeled C-5
at the point where the abscissa θ/θ = 500 ⁄1000 = 0.5. The probability is seen to be equal to 0.47.
(4) In this example, if the termination number had been selected as 6 instead of 5, the probability of accepting a lot with mean
life of 500 h is obtained from the OC curve labeled C-6. The probability is seen to equal 0.41. This illustrates the remark made
in 8.2.2.2 that the larger the termination number, the higher the probability of rejecting lots with unacceptable mean life.
8.4.3.3 Example 2: Calculations for Testing Without Replacement—Suppose that in the life test of Example 1, 10 units of product
had been placed on test. If the failed units were not replaced and the first 5 failure times were 50, 75, 125, 250, and 300, determine
whether the lot met the acceptability criterion.
8.4.3.4 Solution—In this case:
5017511251250130015~300!
ˆ
θ 5 5 460 (10)
5,10
(1) Since 460 < 487, the lot did not meet the acceptability criterion.
8.4.3.5 Example 3: Calculations for Testing With Replacement—Suppose that in the life test of Example 1, 10 units of product had
been placed on test. If the failed units were replaced immediately and the first 5 failure times were 56, 128, 176, 276, and 442,
determine whether the lot met the acceptability criterion.
8.4.3.6 Solution—In this case:
10 442
~ !
ˆ
θ 5 5 884 (11)
5,10
(1) Since 884 > 487, the lot met the acceptability criterion.
8.5 Expected Waiting Time of Life Tests and Cost Considerations in Selection of Sample Sizes—The operating characteristics of
the life test sampling plans of 8.1 – 8.4 are independent of the number of units of product placed on test. Thus, all tests based on
common values of the termination number, r, and producer’s risk, α, are equally good, and the choice of the sample size, n, depends
only on the relative cost of placing a large number of units of product on test and the expected waiting time required for decision.
For fixed α and r, increasing n will, on one hand, cut the expected waiting time; but will, on the other hand, increase the cost
because of placing more units of product on test. This part of the practice provides procedures for determining the optimum sample
size based on considerations of cost.
8.5.1 Expected Waiting Time—The mean life of the lot and, as noted in 8.5, the size of the sample drawn from the lot affect the
expected waiting time required to observe the rth failure in a sample of size n. The rth failure is expected to occur more quickly
in samples drawn from lots with low values of mean life. The values of the expected waiting time divided by the mean life of the
lot, when testing without replacement, are given in Table A1.3 and Table A1.4. Corresponding values for the testing with
replacement situation are not tabled but may be calculated by dividing the termination number, r, by the sample size, n, that is:
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Expected Waiting Time r
5 (12)
Mean Life of a Lot n
8.5.2 Relative Saving in Time by Increasing Sample Size When Testing Without Replacement—When testing without replacement,
the expected waiting time required to observe the rth failure in a sample of size n, (n ≥ r), may be obtained from Table A1.3 or
Table A1.4 by multiplying the tabled entry by the mean life of the lot. By dividing the expected waiting time when n units of
product are placed on test by that when only r units are placed on test, the mean life of the lot cancels out and the ratio
Expected Waiting Time for r Failures in Sample of n
(13)
Expected Waiting Time for r Failures in Sample of r
is a measure of the relative expected saving in time as a result of placing more units of product on test. A brief table of these
ratios is given in Table A1.5.
8.5.3 Relative Saving in Time by Increasing Sample Size When Testing With Replacement—When testing with replacement, the
expected waiting time required to observe the rth failure in a sample of size n is equal to the quantity rθ/ n. By dividing the
expected waiting time when n units of product are placed on test by that when only r units are placed on test, the mean life of the
lot cancels out and the ratio
Relative Saving 5 rθ/nθ5 r/n (14)
is a measure of the relative expected saving in time as a result of using larger sample sizes.
8.5.4 Relative Saving in Time by Testing with Replacement as Compared to Testing Without Replacement—When testing with
replacement, the expected waiting time required to observe the rth failure in a sample of size n (n ≥ r) is equal to the quantity rθ/n.
When testing without replacement, this expected waiting time may be obtained from Table A1.3 or Table A1.4 by multiplying the
tabled entry by the mean life of the lot θ. By dividing these two expected waiting times, the mean life of the lot cancels out and
the ratio
Expected Waiting Time for r Failures in Sample of n When Testing With Replacement (15)
Expected Waiting Time for r Failures in Sample of n When Testing Without Replacement
is a measure of the relative expected saving in time as a result of sampling with replacement. A brief table of these ratios is given
in Table A1.6.
8.5.4.1 Example 4: Saving in Time by Increasing Sample Size When Testing Without Replacement—Compare the average length
of time needed to observe the failure of the first two out of five units of product under test with the average length of time required
to observe the failure of two out of two units when testing without replacement.
8.5.4.2 Solution—From Table A1.3, it is seen that for r = 2 and n = 2, the expected waiting time is 1.5000θ and that for r = 2 and
n = 5, the expected waiting time is 0.4500θ. Thus, the relative saving in time by placing five units on test is 0.4500θ/1.500θ = 0.300.
This figure may also be obtained directly from Table A1.5. Hence, the average time required when five units are placed on test
is 30 % of the average time required when only two units are used.
8.5.4.3 Example 5: Saving in Time by Increasing Sample Size When Testing With Replacement—Make the same comparison as in
Example 4 (8.5.4.1) if the testing had been with replacement.
8.5.4.4 Solution—For r = 2 and n = 2, the expected waiting time is θ and that for r = 2 and n = 5 is rθ/n = 2θ/5 = 0.4θ. Thus,
the relative saving in time by placing five units on test is 0.4θ/θ = 0.4. Hence, the average time required when five units are placed
on test is 40 % of the average time required when only two units are used.
8.5.4.5 Example 6: Saving in Time by Testing With Replacement—Compare the average length of time needed to observe the
failure of the first five out of five units of product under test when testing with replacement with the average length of time needed
when testing without replacement.
8.5.4.6 Solution—When testing with replacement, for r = 5 and n = 5, the expected waiting time is θ. When testing without
replacement, Table A1.3 or Table A1.4 shows that the expected waiting time is 2.2833θ. Thus, the relative saving in time by testing
with replacement is θ/2.2833θ = 0.438; or the average time required for a decision, by replacing failed units, is 43.8 % of the
average time required when failed units are not replaced. This figure may also be obtained directly from Table A1.6.
8.5.5 Cost Considerations in Choice of Sample Size—Methods for finding the optimum sample size based on considerations of
cost are given in this section.
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8.5.5.1 Cost When Testing Without Replacement—The total expected cost of any of the life test plans of 8.2 when testing without
replacement is given by:
1 1 1
c θ 5 1 1…1 1c n (16)
S D
1 0 2
n n 2 1 n 2 r 2 1
where:
c = cost of waiting per unit time,
c = cost of placing a unit of product on test,
θ = acceptable mean life,
r = termination number, and
n = sample size.
8.5.5.2 Optimum Sample Size When Testing Without Replacement—The value of n, which minimizes the total cost, as determined
by the method of 8.5.5.1, is the optimum sample size. A general method of obtaining the optimum n is to use Table A1.3 or Table
A1.4. The smallest n is chosen such that the difference between the expected waiting time for the rth failure when that number
of units of product are placed on test and that when n + 1 units are placed on test is less than the quantity c /c θ .
2 1 0
(1) Example 7: Calculation of Costs—Consider the case in which r = 10, θ = 1000 h, c = $1 per hour, and c = $100 per unit
0 1 2
of product tested. Using the total cost formula, determine the optimum sample size if failed units are not replaced.
(2) Solution—Using the formula of 8.5.5.1, the costs for various values of n are:
Expected Cost Cost of Units
n Because of Waiting Tested Total Cost
10 2929 1000 3929
11 2020 1100 3120
12 1603 1200 2803
13 1346 1300 2646
14 1168 1400 2568
15 1035 1500 2535
16 931 1600 2531
17 847 1700 2547
(a) The optimum sample size is thus n = 16.
(3) Example 8: Obtaining Optimum Sample Size With Expected Waiting Time—Use Table A1.3 to determine the optimum
sample size for the problem of Example 7, 8.5.5.2(1).
(4) Solution—The quantity c /c θ is equal to 0.1 and, from Table A1.3, the expected waiting times are:
2 1 0
Expected Waiting Time to Observe 10th Failure
n in n in n + 1 Difference
10 2.9290 2.0199 0.9091
11 2.0199 1.6032 0.4167
12 1.6032 1.3468 0.2564
13 1.3468 1.1682 0.1786
14 1.1682 1.0349 0.1333
15 1.0349 0.9307 0.1042
A
16 0.9307 0.8467 0.0840
17 0.8467 0.7773 0.0694
A
The optimum sample size is n = 16, as was seen in Example 7 (8.5.5.2(1)), since that is the smallest sample size for which the difference in expected waiting times is
less than c /c θ or 0.1.
2 1 0
8.5.5.3 Cost When Testing With Replacement—The total expected cost of any of the life test plans of 8.2, when testing with
replacement, is given by:
r
c θ 1c n1r 2 1 (17)
~ !
1 0 2
n
where:
c = cost of waiting per unit time,
c = cost of placing a unit of product on test,
θ = acceptable mean life,
r = termination number, and
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n = sample size.
8.5.5.4 Optimum Sample Size When Testing With Replacement—The value of n, which minimizes the total cost, as determined by
the method of 8.5.5.3, is the optimum sample size. In general, the optimum n for the case of testing with replacement is the integer
nearest to:
c θ r 1
1 0
Π1 (18)
c 4
(1) Example 9: Calculation of Cost—Consider the problem of Example 7 (8.5.5.2(1)), that is, r = 10, θ = 1000, c = $1, and
0 1
c = $100. Using the total cost formula, determine the optimum sample size if failed units were replaced.
(2) Solution—Using the formula of 8.5.5.3, the costs for various values of n, are:
Expected Cost Cost of Units
n Because of Waiting Tested Total Cost
9 1111 1800 2911
10 1000 1900 2900
11 909 2000 2909
(a) The optimum sample size is thus n = 10.
(3) Example 10: Obtaining Optimum Sample Size by Formula—Use the method of 8.5.5.4 to determine the optimum sample
size for the problem of Example 9.
(4) Solution—The integer nearest to
1~1000!~10! 1
1 5 10.012 (19)
Œ
100 4
is 10. This is the optimum sample size as was seen in Example 9 (8.5.5.4(1)).
8.6 Life Test Sampling Plans for Certain Specified Values of α, β, and θ /θ —A life test sampling plan may be designed so that
1 0
its OC curve meets the following prescribed conditions: if θ = θ , then the probability of the lot meeting the acceptability criterion
is less than or equal to β. This part of the practice, which may be considered an extension of 8.1 – 8.5, provides procedures for
obtaining values of the termination number, r, and the acceptability constant, C, when certain selected values of α, β, and θ /θ
1 0
are specified. When other values of α, β, and θ /θ than those provided in this part of the practice are needed, refer to 8.1 – 8.5
1 0
to determine whether one of the life test sampling plans given in those paragraphs are applicable.
8.6.1 Life Test Sampling Plans—From Table A1.7, values of the termination number, r, and the acceptability constant, C, may be
obtained for values of α = 0.01, 0.05, 0.10, and 0.25; β = 0.01, 0.05, 0.10, and 0.25; and θ /θ = 2/3, 1/2, 1/3, 1/5, and 1/10. The
1 0
value of r is obtained directly from Table A1.7, but the acceptability constant, C, is obtained by multiplying the tabled entry by
the acceptable mean life, θ .
8.6.1.1 Example 11—Find a life test sampling plan that possesses the following OC curve: If the mean life is θ = 900 h, the lot
is accepted with probability 0.95; if the mean life is θ = 300 h, it is accepted with probability approximately equal to 0.10.
8.6.1.2 Solution—In this
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