1、Designation: C 1215 92 (Reapproved 2006)Standard Guide forPreparing and Interpreting Precision and Bias Statements inTest Method Standards Used in the Nuclear Industry1This standard is issued under the fixed designation C 1215; the number immediately following the designation indicates the year ofor
2、iginal adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon (e) indicates an editorial change since the last revision or reapproval.INTRODUCTIONTest method standards are required to contain precision and
3、bias statements. This guide contains aglossary that explains various terms that often appear in these statements as well as an exampleillustrating such statements for a specific set of data. Precision and bias statements are shown to varyaccording to the conditions under which the data were collecte
4、d. This guide emphasizes that the errormodel (an algebraic expression that describes how the various sources of variation affect themeasurement) is an important consideration in the formation of precision and bias statements.1. Scope1.1 This guide covers terminology useful for the preparationand int
5、erpretation of precision and bias statements.1.2 In formulating precision and bias statements, it isimportant to understand the statistical concepts involved and toidentify the major sources of variation that affect results.Appendix X1 provides a brief summary of these concepts.1.3 To illustrate the
6、 statistical concepts and to demonstratesome sources of variation, a hypothetical data set has beenanalyzed in Appendix X2. Reference to this example is madethroughout this guide.1.4 It is difficult and at times impossible to ship nuclearmaterials for interlaboratory testing. Thus, precision stateme
7、ntsfor test methods relating to nuclear materials will ordinarilyreflect only within-laboratory variation.2. Referenced Documents2.1 ASTM Standards:2E 177 Practice for Use of the Terms Precision and Bias inASTM Test MethodsE 691 Practice for Conducting an Interlaboratory Study toDetermine the Precis
8、ion of a Test Method2.2 ANSI Standard:ANSI N15.5 Statistical Terminology and Notation forNuclear Materials Management33. Terminology for Precision and Bias Statements3.1 Definitions:3.1.1 accuracy (see bias)(1) bias. (2) the closeness of ameasured value to the true value. (3) the closeness of ameasu
9、red value to an accepted reference or standard value.3.1.1.1 DiscussionFor many investigators, accuracy isattained only if a procedure is both precise and unbiased (seebias). Because this blending of precision into accuracy canresult occasionally in incorrect analyses and unclear statementsof result
10、s, ASTM requires statement on bias instead of accu-racy.43.1.2 analysis of variance (ANOVA)the body of statisticaltheory, methods, and practices in which the variation in a set ofdata is partitioned into identifiable sources of variation.Sources of variation may include analysts, instruments,samples
11、, and laboratories. To use the analysis of variance, thedata collection method must be carefully designed based on amodel that includes all the sources of variation of interest. (SeeExample, X2.1.1)3.1.3 bias (see accuracy)a constant positive or negativedeviation of the method average from the corre
12、ct value oraccepted reference value.3.1.3.1 DiscussionBias represents a constant error asopposed to a random error.1This guide is under the jurisdiction of ASTM Committee C26 on Nuclear FuelCycle and is the direct responsibility of Subcommittee C26.08 on Quality Assur-ance, Statistical Applications,
13、 and Reference Materials.Current edition approved Jan. 1, 2006. Published February 2006. Originallyapproved in 1992. Last previous edition approved in 1997 as C 121592(1997).2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at serviceastm.org. For
14、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 St.,4th Floor, New York, NY 10036, http:/www.ansi.org.4Refer to Form and Style for ASTM Standards, 8th Ed., 1989,
15、ASTM.1Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.(a) A method bias can be estimated by the difference (orrelative difference) between a measured average and an ac-cepted standard or reference value. The data from which theestimat
16、e is obtained should be statistically analyzed to establishbias in the presence of random error. A thorough bias investi-gation of a measurement procedure requires a statisticallydesigned experiment to repeatedly measure, under essentiallythe same conditions, a set of standards or reference material
17、s ofknown value that cover the range of application. Bias oftenvaries with the range of application and should be reportedaccordingly.(b) In statistical terminology, an estimator is said to beunbiased if its expected value is equal to the true value of theparameter being estimated. (See Appendix X1.
18、)(c) The bias of a test method is also commonly indicated byanalytical chemists as percent recovery. A number of repeti-tions of the test method on a reference material are performed,and an average percent recovery is calculated. This averageprovides an estimate of the test method bias, which is mul
19、ti-plicative in nature, not additive. (See Appendix X2.)(d) Use of a single test result to estimate bias is stronglydiscouraged because, even if there were no bias, random erroralone would produce a nonzero bias estimate.3.1.4 coeffcient of variationsee relative standard devia-tion.3.1.5 confidence
20、intervalan interval used to bound thevalue of a population parameter with a specified degree ofconfidence (this is an interval that has different values fordifferent random samples).3.1.5.1 DiscussionWhen providing a confidence interval,analysts should give the number of observations on which theint
21、erval is based. The specified degree of confidence is usually90, 95, or 99 %. The form of a confidence interval depends onunderlying assumptions and intentions. Usually, confidenceintervals are taken to be symmetric, but that is not necessarilyso, as in the case of confidence intervals for variances
22、.Construction of a symmetric confidence interval for a popula-tion mean is discussed in Appendix X3.It is important to realize that a given confidence-interval estimateeither does or does not contain the population parameter. The degree ofconfidence is actually in the procedure. For example, if the
23、interval (9,13) is a 90 % confidence interval for the mean, we are confident that theprocedure (take a sample, construct an interval) by which the interval(9, 13) was constructed will 90 % of the time produce an interval thatdoes indeed contain the mean. Likewise, we are confident that 10 % ofthe ti
24、me the interval estimate obtained will not contain the mean. Notethat the absence of sample size information detracts from the usefulnessof the confidence interval. If the interval were based on five observa-tions, a second set of five might produce a very different interval. Thiswould not be the ca
25、se if 50 observations were taken.3.1.6 confidence levelthe probability, usually expressed asa percent, that a confidence interval will contain the parameterof interest. (See discussion of confidence interval in AppendixX3.)3.1.7 error modelan algebraic expression that describeshow a measurement is a
26、ffected by error and other sources ofvariation. The model may or may not include a sampling errorterm.3.1.7.1 DiscussionA measurement error is an error attrib-utable to the measurement process. The error may affect themeasurement in many ways and it is important to correctlymodel the effect of the e
27、rror on the measurement.(a) Two common models are the additive and the multiplicativeerror models. In the additive model, the errors are independent of thevalue of the item being measured. Thus, for example, for repeatedmeasurements under identical conditions, the additive error modelmight beXi5 1 b
28、 1ei(1)where:Xi= the result of the ithmeasurement, = the true value of the item,b = a bias, andei= a random error usually assumed to have a normaldistribution with mean zero and variance s2.In the multiplicative model, the error is proportional to the truevalue. A multiplicative error model for perc
29、ent recovery (see bias)might be:Xi5 bei(2)and a multiplicative model for a neutron counter measurement mightbe:Xi5 1 b 1 ei5 1 1 b 1ei! (3)( b) Clearly, there are many ways in which errors may affect a finalmeasurement. The additive model is frequently assumed and is thebasis for many common statist
30、ical procedures. The form of the modelinfluences how the error components will be estimated and is veryimportant, for example, in the determination of measurement uncer-tainties. Further discussion of models is given in the Example ofAppendix X2 and in Appendix X4.3.1.8 precisiona generic concept us
31、ed to describe thedispersion of a set of measured values.3.1.8.1 DiscussionIt is important that some quantitativemeasure be used to specify precision. A statement such as,“The precision is 1.54 g” is useless. Measures frequently usedto express precision are standard deviation, relative standarddevia
32、tion, variance, repeatability, reproducibility, confidenceinterval, and range. In addition to specifying the measure andthe precision, it is important that the number of repeatedmeasurements upon which the precision estimated is based alsobe given. (See Example, Appendix X2.)(a) It is strongly recom
33、mended that a statement on precision of ameasurement procedure include the following:(1) A description of the procedure used to obtain the data,(2) The number of repetitions, n, of the measurementprocedure,(3) The sample mean and standard deviation of themeasurements,(4) The measure of precision bei
34、ng reported,(5) The computed value of that measure, and(6) The applicable range or concentration.The importance of items (3) and (4) lies in the fact that with these areader may calculate a confidence interval or relative standard deviationas desired.(b) Precision is sometimes measured by repeatabil
35、ity and reproduc-ibility (see Practice E 177, and Mandel and Laskof (3). The ANSI andC 1215 92 (2006)2ASTM documents differ slightly in their usages of these terms. Thefollowing is quoted from Kendall and Buckland (2):“In some situations, especially interlaboratory comparisons, preci-sion is defined
36、 by employing two additional concepts: repeatability andreproducibility. The general situation giving rise to these distinctionscomes from the interest in assessing the variability within severalgroups of measurements and between those groups of measurements.Repeatability, then, refers to the within
37、-group dispersion of themeasurements, while reproducibility refers to the between-group dis-persion. In interlaboratory comparison studies, for example, the inves-tigation seeks to determine how well each laboratory can repeat itsmeasurements (repeatability) and how well the laboratories agree withe
38、ach other (reproducibility). Similar discussions can apply to thecomparison of laboratory technicians skills, the study of competingtypes of equipment, and the use of particular procedures within alaboratory. An essential feature usually required, however, is thatrepeatability and reproducibility be
39、 measured as variances (or standarddeviations in certain instances), so that both within- and between-groupdispersions are modeled as a random variable.The statistical tool usefulfor the analysis of such comparisons is the analysis of variance.”( c) In Practice E 177 it is recommended that the term
40、repeatabilitybe reserved for the intrinsic variation due solely to the measurementprocedure, excluding all variation from factors such as analyst, timeand laboratory and reserving reproducibility for the variation due to allfactors including laboratory. Repeatability can be measured by thestandard d
41、eviation, sr,ofn consecutive measurements by the sameoperator on the same instrument. Reproducibility can be measured bythe standard deviation, sR,ofm measurements, one obtained from eachof m independent laboratories. When interlaboratory testing is notpractical, the reproducibility conditions shoul
42、d be described.(d) Two additional terms are recommended in Practice E 177. Theseare repeatability limit and reproducibility limit. These are intended togive estimates of how different two measurements can be. Therepeatability limit is defined as 1.96 =2 sr, and the reproducibilitylimit is defined as
43、 1.96 =2 sR, where sris the estimated standarddeviation associated with repeatability, and sRis the estimated standarddeviation associated with reproducibility. Thus, if normality can beassumed, these limits represent 95 % limits for the difference betweentwo measurements taken under the respective
44、conditions. In thereproducibility case, this means that “approximately 95 % of all pairs oftest results from laboratories similar to those in the study can beexpected to differ in absolute value by less than 1.96 =2 sR.” It isimportant to realize that if a particular sRis a poor estimate of sR, the9
45、5 % figure may be substantially in error. For this reason, estimatesshould be based on adequate sample sizes.3.1.9 propagation of variancea procedure by which themean and variance of a function of one or more randomvariables can be expressed in terms of the mean, variance, andcovariances of the indi
46、vidual random variables themselves(Syn. variance propagation, propagation of error).3.1.9.1 DiscussionThere are a number of simple exactformulas and Taylor series approximations which are usefulhere (4, 5).3.1.10 random error(1) the chance variation encounteredin all measurement work, characterized
47、by the random occur-rence of deviations from the mean value. (2) an error thataffects each member of a set of data (measurements) in adifferent manner.3.1.11 random sample (measurements)a set of measure-ments taken on a single item or on similar items in such a waythat the measurements are independe
48、nt and have the sameprobability distribution.3.1.11.1 DiscussionSome authors refer to this as a simplerandom sample. One must then be careful to distinguishbetween a simple random sample from a finite population of Nitems and a simple random sample from an infinite population.In the former case, a s
49、imple random sample is a sample chosenin such a way that all samples of the same size have the samechance of being selected. An example of the latter case occurswhen taking measurements. Any value in an interval isconsidered possible and thus the population is conceptuallyinfinite. The definition given in 3.1.11 is then the appropriatedefinition. (See representative sample and Appendix X5.)3.1.12 rangethe largest minus the smallest of a set ofnumbers.3.1.13 relative standard deviation (percent) the samplestandard deviation expressed as a percent of
