Designation: E2696 − 09 (Reapproved 2013) An American National StandardStandard Practice forLife and Reliability Testing Based on the ExponentialDistribution1This standard is issued under the fixed designation E2696; the number immediately following the designation indicates the year oforiginal adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon (´) indicates an editorial change since the last revision or reapproval.1. Scope1.1 This practice presents standard sampling procedures andtables for life and reliability testing in procurement, supply, andmaintenance quality control operations as well as in researchand development activities.1.2 This practice describes general procedures and defini-tions of terms used in life test sampling and describes specificprocedures and applications of the life test sampling plans fordetermining conformance to established reliability require-ments.1.3 This practice is an adaptation of the Quality Control andReliability Handbook H-108, “Sampling Procedures andTables for Life and Reliability Testing (Based on ExponentialDistribution),” 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 thesafety concerns, if any, associated with its use. It is theresponsibility of the user of this standard to establish appro-priate safety and health practices and determine the applica-bility of regulatory limitations prior to use.2. Referenced Documents2.1 ASTM Standards:2E456 Terminology Relating to Quality and StatisticsE2234 Practice for Sampling a Stream of Product by Attri-butes Indexed by AQLE2555 Practice for Factors and Procedures for Applying theMIL-STD-105 Plans in Life and Reliability Inspection3. Terminology3.1 Definitions:3.1.1 See Terminology E456 for a more extensive listing ofterms in ASTM Committee E11 standards.3.1.2 consumer’s risk, β,n—probability that a lot havingspecified rejectable quality level will be accepted under adefined sampling plan. E25553.1.2.1 Discussion—In this practice, the consumer’s risk isthe probability of accepting lots with mean time to failure θ1.3.1.2.2 Discussion—For the procedures of 9.7 and 9.8, theconsumer’s risk may also be defined as the probability ofaccepting lots with unacceptable proportion of lot failingbefore specified time, p1.3.1.3 life test, n—process of placing one or more units ofproduct under a specified set of test conditions and measuringthe time until failure for each unit.3.1.4 mean time to failure, θ, n— in life testing, the averagelength of life of items in a lot.3.1.4.1 Discussion—Also known as mean life.3.1.5 number of failures, n—number of failures that haveoccurred at the time the decision as to lot acceptability isreached.3.1.5.1 Discussion—The expected number of failures re-quired for decision is the average of the number of failuresrequired for decision when life tests are conducted on a largenumber of samples drawn at random from the same exponen-tial distribution.3.1.6 producer’s risk, α,n—probability that a lot havingspecified acceptable quality level will be rejected under adefined sampling plan.3.1.6.1 Discussion—In this practice, the producer’s risk isthe probability of rejecting lots with mean time to failure θ0.3.1.6.2 Discussion—For the procedures of 9.7 and 9.8, theproducer’s risk may also be defined as the probability ofrejecting lots with acceptable proportion of lot failing beforespecified time, p0.3.1.7 sequential life test, n—life test sampling plan wherebyneither the number of failures nor the time required to reach adecision are fixed in advance but instead decisions depend onthe accumulated results of the life test.3.1.8 unit of product, n—that which is inspected to deter-mine its classification as defective or nondefective or to countthe number of defects. E22341This practice is under the jurisdiction of ASTM Committee E11 on Quality andStatistics and is the direct responsibility of Subcommittee E11.30 on StatisticalQuality Control.Current edition approved April 1, 2013. Published April 2013. Originallyapproved in 2009. Last previous edition approved in 2009 as E2696 – 09ε1. DOI:10.1520/E2696-09R13.2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at

[email protected] For Annual Book of ASTMStandards volume information, refer to the standard’s Document Summary page onthe ASTM website.Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States13.1.9 waiting time, n—in life testing, the time elapsed fromthe start of testing until a decision is reached as to lotacceptability.3.1.9.1 Discussion—The expected waiting time required fordecision is the average of the waiting times required fordecision when life tests are conducted on a large number ofsamples drawn at random from the same exponential distribu-tion.4. Significance and Use4.1 This practice was prepared to meet a growing need forthe use of standard sampling procedures and tables for life andreliability testing in government procurement, supply, andmaintenance quality control (QC) operations as well as inresearch and development activities where applicable.4.2 A characteristic feature of most life tests is that theobservations are ordered in time to failure. If, for example, 20radio tubes are placed on life test, and tidenotes the time whenthe ith tube fails, the data occur in such a way that t1≤ t2≤ .≤ tn. The same kind of ordered observations will occur whetherthe problem under consideration deals with the life of electricbulbs, the life of electronic components, the life of ballbearings, or the length of life of human beings after they aretreated for a disease. The examples just given all involveordering in time.4.3 In destructive testing involving such situations as thecurrent needed to blow a fuse, the voltage needed to breakdown a condenser, or the force needed to rupture a physicalmaterial, the test can often be arranged in such a way that everyitem in the sample is subjected to precisely the same stimulus(current, voltage, or stress). If this is done, then clearly theweakest item will be observed to fail first, the second weakestnext, and so forth. While the random variable consideredmostly in this guide is time to failure, it should be emphasized,however, that the methodology provided herein can be adaptedto the testing situations mentioned above when the randomvariable is current, voltage, stress, and so forth.4.4 Sections 6 and 7 describe general procedures anddefinitions of terms used in life test sampling. Sections 8, 9,and 10 describe specific procedures and applications of the lifetest sampling plans for determining conformance to establishedreliability requirements.4.5 Whenever the methodology or choice of procedures inthe practice requires clarification, the user is advised to consulta qualified mathematical statistician, and reference should bemade to appropriate technical reports and other publications inthe field.5. Introduction5.1 The theory underlying the development of the life testsampling plans of this section, including the operating charac-teristic curves, assumes that the measurements of the length oflife are drawn from an exponential distribution. Statistical testprocedures for determining the validity of the exponentialdistribution assumption have appeared in the technical statis-tical journals. Professor Benjamin Epstein published a com-prehensive paper (in two parts) on this subject in the Februaryand May 1960 issues of Technometrics.3Part I of the papercontains descriptions of the mathematical and graphical pro-cedures as well as an extensive bibliography for referencepurposes. Numerical examples illustrating the statistical pro-cedures are included in Part II of the paper.5.2 It is important to note that the life test sampling plans ofthis practice are not to be used indiscriminately simply becauseit is possible to obtain life test data. Only after the exponentialassumption is deemed reasonable should the sampling plans beused.5.3 Sections 6 and 7 describe general procedures anddescription of life test sampling plans. Section 8 describesspecific procedures and applications of sampling plans whenlife tests are terminated upon the occurrence of a preassignednumber of failures, and Section 9 provides sampling planswhen life tests are terminated at a preassigned time. Section 10describes sequential life test sampling plans. Section 8 covers:(1) acceptance procedures; (2) expected duration of life testsand cost considerations in selection of sample sizes; and (3) lifetest plans for certain specified values of α, β, and θ1/θ0. Section9 covers: (1) acceptance procedures; (2) life test plans forcertain specified values of α, β, θ1/θ0, and T/θ0; and (3) life testplans based on proportion of lot failing before specified time.Section 10 covers: (1) acceptance procedures; (2) graphicalacceptance procedures; and (3) expected number and waitingtime required for decision.5.4 Operating characteristic (OC) curves for the life testsampling plans of 8.1 – 8.5, 9.1 – 9.5, and Section 10 areshown in Fig. A1.1 for the corresponding sampling plans inthese sections were matched with respect to their OC curves.The OC curves in Fig. A1.1 have been computed for the lifetest sampling plans of 8.1 – 8.5 but are equally applicable forthe sampling plans of 9.1 – 9.5 and Section 10.5.5 The procedures of this section are based on the premisethat the life tests are monitored continuously. If the tests aremonitored only periodically, the values obtained from thetables and curves are only approximations.6. General Definitions of Life and Reliability Test Terms6.1 Discussion of Terms and Procedures:6.1.1 Purpose—This section provides definitions of termsrequired for the life test sampling plans and procedures ofSections 7 through 10.6.1.2 Life Test—Life test is the process of placing the “unitof product” under a specified set of test conditions andmeasuring the time it takes until failure.6.1.3 Unit of Product—The unit of product is the entity ofproduct that may be placed on life test.6.1.4 Specifying Failure—The state that constitutes a failureshall be specified in advance of the life test.6.1.5 Life Test Sampling Plan—A life test sampling plan isa procedure that specifies the number of units of product froma lot that are to be tested and the criterion for determiningacceptability of the lot.3Epstein, B., “Tests for the Validity of the Assumption that the UnderlyingDistribution of Life is Exponential,” Technometrics, Vol 2, February and May 1960,pp. 83–101 and 167–183.E2696 − 09 (2013)26.1.6 Life Test Terminated upon Occurrence of PreassignedNumber of Failures—Life test sampling plans whereby testingis terminated when a preassigned termination number offailures, r, occur are given in Section 8 of this practice.6.1.7 Life Test Terminated at Preassigned Time—Life testsampling plans whereby testing is terminated when a preas-signed termination time, T, is reached are given in Section 9 ofthis practice.6.1.8 Sequential Life Test—Sequential life test is a life testsampling plan whereby neither the number of failures nor thetime required to reach a decision are fixed in advance but,instead, decisions depend on the accumulated results of the lifetest. Information on the observed time to failure are accumu-lated over time and the results at any time determine the choiceof one among three possible decisions: (1) the lot meets theacceptability criterion, (2) the lot does not meet the acceptabil-ity criterion, or (3) the evidence is insufficient for eitherdecision (1) or (2) and the test must continue. Sequential lifetest sampling plans are given in Section 10 of this practice andhave the advantage over the life test sampling plans mentionedin 6.1.6 and 6.1.7 in that, for the same OC curve, the expectedwaiting time and the expected number of failures required toreach a decision as to lot acceptability are less for thesequential life tests.6.1.9 Expected Number of Failures—The number of failuresrequired for decision is the number of failures that haveoccurred at the time the decision as to lot acceptability isreached. 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, thisnumber cannot be predetermined. The expected number offailures required for decision is the average of the number offailures required for decision when life tests are conducted ona large number of samples drawn at random from the sameexponential distribution. The expected number of failures canbe predetermined for the sampling plans mentioned in 6.1.6 –6.1.8.6.1.10 Expected Waiting Time—The waiting time requiredfor decision is the time elapsed from the start of the life test tothe time decision is reached as to lot acceptability. The waitingtime required for decision cannot be predetermined for any ofthe sampling plans mentioned in 6.1.6 – 6.1.8. The expectedwaiting time required for decision is the average of the waitingtimes required for decision when life tests are conducted on alarge number of samples drawn at random from the sameexponential distribution. The expected waiting time can bepredetermined 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 tofailure” may be used interchangeably and shall denote thelength of time it takes for a unit of product to fail after beingplaced on life test. The length of time may be expressed in anyconvenient time scale such as seconds, hours, days, and soforth.6.2.2 Mean Time to Failure—The terms “mean time tofailure” and “mean life” may be used interchangeably and shalldenote the mean (or equivalently, the average) length of life ofitems in the lot. Mean life is denoted by θ.6.2.3 Acceptable Mean Life—The acceptable mean life, θ0,is the minimum mean time to failure that is consideredsatisfactory.6.2.4 Unacceptable Mean Life—The unacceptable meanlife, θ1(θ1p0), is the minimum fraction of the lotthat may fail before time, T, and results in the lot beingconsidered unsatisfactory. The interval between p0and p1is azone of indifference in which there is a progressively greaterdegree of dissatisfaction as the fraction of the lot failing beforetime, T, increases from p0to p1.6.3.6 Acceptable Failure Rate During Period of Time—The“acceptable failure rate during period of time,” G0,isthemaximum failure rate during the period of time that can beconsidered satisfactory.6.3.7 Unacceptable Failure Rate During Period of Time—The “unacceptable failure rate during period of time,” G1,(G1 G0), is the minimum failure rate during the period oftime that results in the lot being considered unsatisfactory. Theinterval between G0and G1is a zone of indifference in whichthere is a progres