1、Designation: E 2655 08Standard Guide forReporting Uncertainty of Test Results and Use of the TermMeasurement Uncertainty in ASTM Test Methods1This standard is issued under the fixed designation E 2655; the number immediately following the designation indicates the year oforiginal adoption or, in the
2、 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 guide provides concepts necessary for understand-ing the term “uncertainty” when a
3、pplied to a quantitative testresult. Several measures of uncertainty can be applied to agiven measurement result; the interpretation of some of thecommon forms is described.1.2 This guide describes methods for expressing test resultuncertainty and relates these to standard statistical methodol-ogy.
4、Relationships between uncertainty and concepts of preci-sion and bias are described.1.3 This guide also presents concepts needed for a labora-tory to identify and characterize components of method per-formance. Elements that an ASTM method can include toprovide guidance to the user on estimating unc
5、ertainty for themethod are described.1.4 The system of units for this guide is not specified.Dimensional quantities in the guide are presented only asillustrations of calculation methods and are not binding onproducts or test methods treated.1.5 This standard does not purport to address all of thesa
6、fety 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:2E29 Practice for Using Signifi
7、cant Digits in Test Data toDetermine Conformance with SpecificationsE 122 Practice for Calculating Sample Size to Estimate,With Specified Precision, the Average for a Characteristicof a Lot or ProcessE 141 Practice for Acceptance of Evidence Based on theResults of Probability SamplingE 177 Practice
8、for Use of the Terms Precision and Bias inASTM Test MethodsE 456 Terminology Relating to Quality and StatisticsE 691 Practice for Conducting an Interlaboratory Study toDetermine the Precision of a Test MethodE 2554 Practice for Estimating and Monitoring the Uncer-tainty of Test Results of a Test Met
9、hod in a SingleLaboratory Using a Control Sample Program2.2 Other Standard:ISO 17025 General Requirements for the Competence ofTesting and Calibration Laboratories33. Terminology3.1 DefinitionsAdditional statistical terms are defined inTerminology E 456.3.1.1 accepted reference value, na value that
10、serves as anagreed-upon reference for comparison, and which is derivedas: (1) a theoretical or established value, based on scientificprinciples, (2) an assigned or certified value, based on experi-mental work of some national or international organization, or(3) a consensus or certified value, based
11、 on collaborativeexperimental work under the auspices of a scientific orengineering group. E 1773.1.2 error of result, na test result minus the acceptedreference value of the characteristic.3.1.3 expanded uncertainty, U, nuncertainty reported as amultiple of the standard uncertainty.3.1.4 random err
12、or of result, na component of the errorthat, in the course of a number of test results for the samecharacteristic, varies in an unpredictable way.3.1.4.1 DiscussionUncertainty due to random error can bereduced by averaging multiple test results.3.1.5 sensitivity coeffcient, ndifferential effect of t
13、hechange in a factor on the test result.1This guide is under the jurisdiction of ASTM Committee E11 on Quality andStatistics and is the direct responsibility of Subcommittee E11.20 on Test MethodEvaluation and Quality Control.Current edition approved Oct. 1, 2008. Published November 2008.2For refere
14、nced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at serviceastm.org. For Annual Book of ASTMStandards volume information, refer to the standards Document Summary page onthe ASTM website.3Available from American National Standards Institute (ANSI), 25 W. 43rd
15、 St.,4th Floor, New York, NY 10036, http:/www.ansi.org.1Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.3.1.6 standard uncertainty, u, nuncertainty reported as thestandard deviation of the estimated value of the quantitysubject to mea
16、surement.3.1.7 systematic error of result, na component of the errorthat, in the course of a number of test results for the samecharacteristic, remains constant or varies in a predictable way.3.1.7.1 DiscussionSystematic errors and their causes maybe known or unknown. When causes are known, systemat
17、icerror can sometimes be reduced by incorporating correctionsinto the calculation of the test result.3.1.8 uncertainty, nan indication of the magnitude oferror associated with a value that takes into account bothsystematic errors and random errors associated with the mea-surement or test process.3.1
18、.9 uncertainty budget, na tabular listing of uncertaintycomponents for a given measurement process giving themagnitudes of contributions to uncertainty of the result fromthose sources.3.1.10 uncertainty component, na source of error in a testresult to which is attached a standard uncertainty.4. Sign
19、ificance and Use4.1 Part A of the “Blue Book,” Form and Style for ASTMStandards, introduces the statement of measurement uncer-tainty as an optional part of the report given for the result ofapplying a particular test method to a particular material.4.2 Preparation of uncertainty estimates is a requ
20、irement forlaboratory accreditation under ISO 17025. This guide describessome of the types of data that the laboratory can use as thebasis for reporting uncertainty.5. Concepts for Reporting Uncertainty of Test Results5.1 Uncertainty is part of the relationship of a test result tothe property of int
21、erest for the material tested. When a testprocedure is applied to a material, the test result is a value fora characteristic of the material. The test result obtained willusually differ from the actual value for that material. Multiplecauses can contribute to the error of result. Errors of samplinga
22、nd effects of sample handling make the portion actually testednot identical to the material as a whole. Imperfections in thetest apparatus and its calibration, environmental, and humanfactors also affect the result of testing. Nonetheless, aftertesting has been completed, the result obtained will be
23、 used forfurther purposes as if it were the actual value. Reportingmeasurement uncertainty for a test result is an attempt toestimate the approximate magnitude of all these sources oferror. In common cases the measurement will be reported in theform x 6 u, in which x represents the test result and u
24、represents the uncertainty associated with x.5.2 Practice E 177 describes precision and bias. Uncertaintyis a closely related but not identical concept. The primarydifference between concepts of precision and of uncertainty isthe object that they address. Precision (repeatability andreproducibility)
25、 and bias are attributes of the test method. Theyare estimates of statistical variability of test results for a testmethod applied to a given material. Repeatability and interme-diate precision measure variation within a laboratory. Repro-ducibility refers to interlaboratory variation. Uncertainty i
26、s anattribute of the particular test result for a test material. It is anestimate of the quality of that particular test result.5.3 In the case of a quantity with a definition that does notdepend on the measurement or test method (e.g., concentration,pH, modulus, heat content), uncertainty measures
27、how close itis believed the measured value comes to the quantity. Forresults of test methods where the target is only definablerelative to the test method (e.g., flash points, extractablecomponents, sieve analysis), uncertainty of a test result must beinterpreted as a measure of how closely an indep
28、endent,equally competent test result would agree with that beingreported.5.4 In the simplest cases, uncertainty of a test result isnumerically equivalent to test method precision. That is, if anunknown sample is tested, and the test precision is known to besigma, then uncertainty of the result of te
29、st is sigma. The termuncertainty, however, is correct to apply where variation ofrepeated test results is not relevant, as in the followingexamples.5.4.1 ExampleThe Newtonian constant of gravitation, G,is 6.6742 3 10-116 0.0010 3 10-11m3kg-1s-2based on 2002CODATA recommended values (1).40.0010 3 10-
30、11m3kg-1s-2is the standard uncertainty. The value and the uncertaintytogether represent the state of knowledge of this fundamentalphysical constant. It is not naturally thought of in terms ofvariation of repeated measurements. Both G and its uncertaintyare derived from the analysis and comparison of
31、 a variety ofmeasurement data using methods that are an elaboration ofthose presented in this guide.5.4.2 ExampleA length is measured but the result onlyreported to the nearest inch (for example, a measuring rodgraduated in inches was used to obtain the measurement).Precision of the reported value,
32、in the sense of variation ofrepeated measurements, is zero when all reported lengths arethe same. In this case it is not possible to detect randomvariation in the series of repeated measurements. Uncertaintyof the length is primarily composed of the systematic error of60.5 inch due to the resolution
33、 of the measurement apparatus.5.5 The goal in reporting uncertainty is to take account of allpotential causes of error in the test result. In many cases,uncertainty can be related to components of variability due tosampling and to testing. Both of these should be taken intoaccount for the uncertaint
34、y of the measurement when thepurpose of the result is to estimate the property for the entire lotof material from which the sample was taken. Uncertainty ofthe lot property value based on a single determination is then=s121 s221 u32where s1is an estimate of the samplingstandard deviation, s2is an es
35、timate of the standard deviationof the test method, and u3is standard uncertainty due to factorsthat affect all measurements under consideration.5.6 A commonly cited definition (2, 3) defines uncertaintyas “a parameter, associated with the measurement result, or testresult, that characterizes the di
36、spersion of values that couldreasonably be attributed to the quantity subject to measurementor characteristic subject to test.” This definition emphasizes4The boldface numbers in parentheses refer to the list of references at the end ofthis standard.E2655082uncertainty as an attribute of the particu
37、lar result, as opposed tostatistical variation of test results. The uncertainty parameter isa measure of spread (for example, the standard deviation) of aprobability distribution used to represent the likelihood ofvalues of the property.55.7 The methodology for uncertainty estimates has beenclassifi
38、ed as Type A and Type B as discussed in (4). Type Aestimates of uncertainty include standard error estimates basedon knowledge of the statistical character of observations, andbased on statistical analysis of replicate measurements. Type Bestimates of uncertainty include approximate values derivedfr
39、om experience with measurement processes similar to the onebeing considered, and estimates of standard uncertainty de-rived from the range of possible measurement values for agiven material and an assumed distribution of values withinthat range. See Practice E 122 for examples (e.g., rectangular,tri
40、angular, normal) where a standard deviation is derived froma range without data from samples being available. Complexestimates of test result uncertainty are calculated by combiningType A and Type B component standard uncertainties forfactors contributing to error (see Section 8).5.8 Forms of Uncert
41、ainty Expression:5.8.1 Standard UncertaintyThe uncertainty is reported asthe standard deviation of the reported value. The report x 6 uimplies that the value should be between x u and x + u withapproximate probability two-thirds, where x is the test result.5.8.2 Relative Standard UncertaintyThe unce
42、rtainty isreported as a fraction of the reported value. For a measuredvalue and a standard uncertainty, x 6 u, the relative standarduncertainty is u/x. This method of expressing uncertainty maybe useful when standard uncertainty is proportional to the valueover a wide range. However, for a particula
43、r result, reportingthe value and standard uncertainty is preferred.5.8.3 Expanded UncertaintyThe uncertainty is reportedas x 6 U, where the value of U is a multiple of the standarduncertainty u. The most common multiple used is 2, which isapproximately equal to the 1.96 factor for a 95 % two-sidedco
44、nfidence interval for the mean of a normal distribution (see5.8.4).5.8.4 Confidence IntervalsA confidence interval for aparameter (the actual value of the material property subject tomeasurement) consists of upper and lower limits generatedfrom sample data by a method that ensures the limits bracket
45、the parameter value with a stated probability 1-a, referred to asthe confidence coefficient.5.8.4.1 From statistical theory, a 95 % confidence intervalfor the mean of a normal distribution, given n independentobservations x1, x2, xndrawn from the distribution, isx6 ts /=n where xis the sample mean,
46、s is the standarddeviation of the observations, and t is the 0.975 percentile ofthe Students t distribution with n-1 degrees of freedom.Because Students t distribution approaches the Normal as nincreases, the value of t approaches 1.96 as n increases. This isthe basis for using the factor 2 for expa
47、nded uncertainty.5.8.5 Measurement UncertaintyMeasurement uncertaintyis uncertainty reported for a test result without taking intoaccount sampling variation or heterogeneity of the material ofinterest. The report of measurement uncertainty then refersspecifically to the particular sample presented f
48、or analysis.5.8.6 Reporting Uncertainty with a Bias ComponentGoodmeasurement practice requires that biases due to environmentaland other factors should be corrected in the reported resultwhen there is a sound basis for correction and the error in thecorrection terms themselves is not greater than th
49、e bias. Suchcorrections are part of the calculation of the result within thetest method. The symmetrical form of reporting a measurementwith standard uncertainty, x 6 u, is adequate for measurementswhere bias is absent or corrected. If the measurement processhas a bias for which there is an estimate of magnitude and it isnot corrected in the reported value x, a form of reporting shouldbe used making clear both bias and random components. Atypical form to highlight the asymmetry caused by bias isx ul/+uh, where ul= bias standard uncertainty and uh= bias
