ASTM E2696-09
(Practice)Standard Practice for Life and Reliability Testing Based on the Exponential Distribution
Standard Practice for Life and Reliability Testing Based on the Exponential Distribution
SIGNIFICANCE AND USE
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.
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.
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.
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.
Whenever the methodology or choice of procedures in the practice requires clarification, the user is advised to consult a qualif...
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 and health practices and determine the applicability of regulatory limitations prior to use.
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Designation: E2696 − 09 AnAmerican 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 3.1.1 See Terminology E456 for a more extensive listing of
terms in ASTM Committee E11 standards.
1.1 Thispracticepresentsstandardsamplingproceduresand
3.1.2 consumer’s risk, β,n—probability that a lot having
tablesforlifeandreliabilitytestinginprocurement,supply,and
specified rejectable quality level will be accepted under a
maintenance quality control operations as well as in research
defined sampling plan. E2555
and development activities.
3.1.2.1 Discussion—In this practice, the consumer’s risk is
1.2 This practice describes general procedures and defini-
the probability of accepting lots with mean time to failure θ .
tions of terms used in life test sampling and describes specific
3.1.2.2 Discussion—For the procedures of 9.7 and 9.8, the
procedures and applications of the life test sampling plans for
consumer’s risk may also be defined as the probability of
determining conformance to established reliability require-
accepting lots with unacceptable proportion of lot failing
ments.
before specified time, p .
1.3 This practice is an adaptation of the Quality Control and
3.1.3 life test, n—process of placing one or more units of
Reliability Handbook H-108, “Sampling Procedures and
product under a specified set of test conditions and measuring
Tables for Life and Reliability Testing (Based on Exponential
the time until failure for each unit.
Distribution),” U.S. Government Printing Office, April 29,
3.1.4 mean time to failure, θ, n— in life testing, the average
1960.
length of life of items in a lot.
1.4 A system of units is not specified in this practice.
3.1.4.1 Discussion—Also known as mean life.
1.5 This standard does not purport to address all of the
3.1.5 number of failures, n—number of failures that have
safety concerns, if any, associated with its use. It is the
occurred at the time the decision as to lot acceptability is
responsibility of the user of this standard to establish appro-
reached.
priate safety and health practices and determine the applica-
3.1.5.1 Discussion—The expected number of failures re-
bility of regulatory limitations prior to use.
quired for decision is the average of the number of failures
required for decision when life tests are conducted on a large
2. Referenced Documents
number of samples drawn at random from the same exponen-
2.1 ASTM Standards:
tial distribution.
E456 Terminology Relating to Quality and Statistics
3.1.6 producer’s risk, α,n—probability that a lot having
E2234 Practice for Sampling a Stream of Product by Attri-
specified acceptable quality level will be rejected under a
butes Indexed by AQL
defined sampling plan.
E2555 Practice for Factors and Procedures for Applying the
3.1.6.1 Discussion—In this practice, the producer’s risk is
MIL-STD-105 Plans in Life and Reliability Inspection
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
3. Terminology
producer’s risk may also be defined as the probability of
3.1 Definitions:
rejecting lots with acceptable proportion of lot failing before
specified time, p .
This practice is under the jurisdiction ofASTM Committee E11 on Quality and 3.1.7 sequential life test, n—life test sampling plan whereby
Statistics and is the direct responsibility of Subcommittee E11.30 on Statistical
neither the number of failures nor the time required to reach a
Quality Control.
decision are fixed in advance but instead decisions depend on
Current edition approved May 1, 2009. Published June 2009. DOI: 10.1520/
the accumulated results of the life test.
E2696-09.
For referenced ASTM standards, visit the ASTM website, www.astm.org, or
3.1.8 unit of product, n—that which is inspected to deter-
contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
mine its classification as defective or nondefective or to count
Standards volume information, refer to the standard’s Document Summary page on
the ASTM website. the number of defects. E2234
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E2696 − 09
3.1.9 waiting time, n—in life testing, the time elapsed from and May 1960 issues of Technometrics. Part I of the paper
the start of testing until a decision is reached as to lot contains descriptions of the mathematical and graphical pro-
acceptability. cedures as well as an extensive bibliography for reference
3.1.9.1 Discussion—The expected waiting time required for purposes. Numerical examples illustrating the statistical pro-
decision is the average of the waiting times required for cedures are included in Part II of the paper.
decision when life tests are conducted on a large number of
5.2 It is important to note that the life test sampling plans of
samples drawn at random from the same exponential distribu-
thispracticearenottobeusedindiscriminatelysimplybecause
tion.
it is possible to obtain life test data. Only after the exponential
assumption is deemed reasonable should the sampling plans be
4. Significance and Use
used.
4.1 This practice was prepared to meet a growing need for
5.3 Sections 6 and 7 describe general procedures and
the use of standard sampling procedures and tables for life and
description of life test sampling plans. Section 8 describes
reliability testing in government procurement, supply, and
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
(1) acceptance procedures; (2) expected duration of life tests
i
the ith tube fails, the data occur in such a way that t ≤ t ≤ .
andcostconsiderationsinselectionofsamplesizes;and(3)life
1 2
≤ t . The same kind of ordered observations will occur
testplansforcertainspecifiedvaluesofα,β,andθ /θ .Section
n
1 0
whether the problem under consideration deals with the life of
9 covers: (1) acceptance procedures; (2) life test plans for
electric bulbs, the life of electronic components, the life of ball
certainspecifiedvaluesofα,β,θ /θ ,and T/θ ;and (3)lifetest
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
current needed to blow a fuse, the voltage needed to break
5.4 Operating characteristic (OC) curves for the life test
down a condenser, or the force needed to rupture a physical
samplingplansof8.1-8.5,9.1-9.5,andSection10areshownin
material,thetestcanoftenbearrangedinsuchawaythatevery
Fig. A1.1 for the corresponding sampling plans in these
item in the sample is subjected to precisely the same stimulus
sectionswerematchedwithrespecttotheirOCcurves.TheOC
(current, voltage, or stress). If this is done, then clearly the
curves in Fig. A1.1 have been computed for the life test
weakest item will be observed to fail first, the second weakest
sampling plans of 8.1-8.5 but are equally applicable for the
next, and so forth. While the random variable considered
sampling plans of 9.1-9.5 and Section 10.
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,
and 10 describe specific procedures and applications of the life 6.1 Discussion of Terms and Procedures:
testsamplingplansfordeterminingconformancetoestablished 6.1.1 Purpose—This section provides definitions of terms
reliability requirements. required for the life test sampling plans and procedures of
Sections 7 through 10.
4.5 Whenever the methodology or choice of procedures in
6.1.2 Life Test—Life test is the process of placing the “unit
the practice requires clarification, the user is advised to consult
of product” under a specified set of test conditions and
a qualified mathematical statistician, and reference should be
measuring the time it takes until failure.
made to appropriate technical reports and other publications in
6.1.3 Unit of Product—The unit of product is the entity of
the field.
product that may be placed on life test.
6.1.4 Specifying Failure—The state that constitutes a failure
5. Introduction
shall be specified in advance of the life test.
5.1 The theory underlying the development of the life test
6.1.5 Life Test Sampling Plan—A life test sampling plan is
sampling plans of this section, including the operating charac-
a procedure that specifies the number of units of product from
teristic curves, assumes that the measurements of the length of
a lot that are to be tested and the criterion for determining
life are drawn from an exponential distribution. Statistical test
acceptability of the lot.
procedures for determining the validity of the exponential
distribution assumption have appeared in the technical statis-
Epstein, B., “Tests for the Validity of the Assumption that the Underlying
tical journals. Professor Benjamin Epstein published a com-
Distribution of Life is Exponential,” Technometrics , Vol 2, Feb 1960 and May
prehensive paper (in two parts) on this subject in the February 1960, pp. 83-101 and 167-183.
E2696 − 09
6.1.6 Life Test Terminated upon Occurrence of Preassigned 6.2.3 Acceptable Mean Life—The acceptable mean life, θ ,
Number of Failures—Life test sampling plans whereby testing is the minimum mean time to failure that is considered
is terminated when a preassigned termination number of satisfactory.
failures, r, occur are given in Section 8 of this practice.
6.2.4 Unacceptable Mean Life—The unacceptable mean
6.1.7 Life Test Terminated at Preassigned Time—Life test
life, θ (θ <θ ), is the mean time to failure such that lots
1 1 0
sampling plans whereby testing is terminated when a preas- having a mean life less than or equal to θ are considered
signed termination time, T, is reached are given in Section 9 of
unsatisfactory. The interval between θ and θ is a zone of
0 1
this practice.
indifference in which there is a progressively greater degree of
6.1.8 Sequential Life Test—Sequential life test is a life test dissatisfaction as the mean life decreases from θ to θ .
0 1
sampling plan whereby neither the number of failures nor the
6.3 Failure Rate:
time required to reach a decision are fixed in advance but,
6.3.1 Proportion of Lot Failing Before Specified Time—The
instead, decisions depend on the accumulated results of the life
term “proportion of lot failing before specified time,” p,
test. Information on the observed time to failure are accumu-
denotes the fraction of the lot that fails before some specified
lated over time and the results at any time determine the choice
time, T, that is:
of one among three possible decisions: (1) the lot meets the
p 5 1 2 exp 2T/θ (1)
acceptability criterion, (2) the lot does not meet the acceptabil- ~ !
ity criterion, or (3) the evidence is insufficient for either
6.3.2 Failure Rate during Period of Time—The “failure rate
decision (1) or (2) and the test must continue. Sequential life
during period of time T,” G, is given by:
test sampling plans are given in Section 10 of this practice and
have the advantage over the life test sampling plans mentioned
G 5 1 2 exp T/θ 5 p/T (2)
$ ~ !%
T
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
6.3.3 Instantaneous Failure Rate—The “instantaneous fail-
reach a decision as to lot acceptability are less for the
ure rate” or “hazard rate” is given by:
sequential life tests.
Z 51/θ (3)
6.1.9 Expected Number of Failures—Thenumberoffailures
required for decision is the number of failures that have
6.3.4 Acceptable Proportion of Lot Failing Before Specified
occurred at the time the decision as to lot acceptability is
Time—The “acceptable proportion of lot failing before speci-
reached. For the life test sampling plans mentioned in 6.1.6,
fied time,” p , is the maximum fraction of the lot that may fail
thisnumberoffailuresisknowninadvanceofthelifetest;but,
before time, T, and still result in the lot being considered
for the sampling plans mentioned in 6.1.7 and 6.1.8, this
satisfactory.
number cannot be predetermined. The expected number of
6.3.5 Unacceptable Proportion of Lot Failing Before Speci-
failures required for decision is the average of the number of
fied Time—The “unacceptable proportion of lot failing before
failures required for decision when life tests are conducted on
specifiedtime,” p ,(p > p ),istheminimumfractionofthelot
1 1 0
a large number of samples drawn at random from the same
that may fail before time, T, and results in the lot being
exponential distribution. The expected number of failures can
considered unsatisfactory. The interval between p and p is a
0 1
be predetermined for the sampling plans mentioned in
zone of indifference in which there is a progressively greater
6.1.6-6.1.8.
degree of dissatisfaction as the fraction of the lot failing before
6.1.10 Expected Waiting Time—The waiting time required
time, T, increases from p to p .
0 1
for decision is the time elapsed from the start of the life test to
6.3.6 Acceptable Failure Ra
...
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