1、Designation: E122 17 An American National StandardStandard Practice forCalculating Sample Size to Estimate, With SpecifiedPrecision, the Average for a Characteristic of a Lot orProcess1This standard is issued under the fixed designation E122; the number immediately following the designation indicate
2、s 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.This standard has been approved for use by agencies of the
3、 U.S. Department of Defense.1. Scope1.1 This practice covers simple methods for calculating howmany units to include in a random sample in order to estimatewith a specified precision, a measure of quality for all the unitsof a lot of material, or produced by a process. This practice willclearly indi
4、cate the sample size required to estimate theaverage value of some property or the fraction of nonconform-ing items produced by a production process during the timeinterval covered by the random sample. If the process is not ina state of statistical control, the result will not have predictivevalue
5、for immediate (future) production. The practice treats thecommon situation where the sampling units can be consideredto exhibit a single (overall) source of variability; it does nottreat multi-level sources of variability.1.2 The system of units for this standard is not specified.Dimensional quantit
6、ies in the standard are presented only asillustrations of calculation methods. The examples are notbinding on products or test methods treated.1.3 This international standard was developed in accor-dance with internationally recognized principles on standard-ization established in the Decision on Pr
7、inciples for theDevelopment of International Standards, Guides and Recom-mendations issued by the World Trade Organization TechnicalBarriers to Trade (TBT) Committee.2. Referenced Documents2.1 ASTM Standards:2E456 Terminology Relating to Quality and Statistics3. Terminology3.1 DefinitionsUnless othe
8、rwise noted, all statistical termsare defined in Terminology E456.3.1.1 pooled standard deviation, sp,nthe estimate of astandard deviation derived by combining sample standarddeviations of several samples, weighting squared standarddeviations by their degrees of freedom.3.2 SymbolsSymbols used in al
9、l equations are defined asfollows:E = the maximum acceptable difference between the trueaverage and the sample average.e = E/, maximum acceptable difference expressed as afraction of .f = degrees of freedom for a standard deviation estimate(7.5).k = the total number of samples available from the sam
10、e orsimilar lots. = lot or process mean or expected value of X, the resultof measuring all the units in the lot or process.0= an advance estimate of .N = size of the lot.n = size of the sample taken from a lot or process.nj= size of sample j.nL= size of the sample from a finite lot (7.4).p = fractio
11、n of a lot or process whose units have thenonconforming characteristic under investigation.p0= an advance estimate of p.p = fraction nonconforming in the sample.R = range of a set of sampling values. The largest minus thesmallest observation.Rj= range of sample j.R=(j51kRj/k , average of the range o
12、f k samples, all of thesame size (8.2.2). = lot or process standard deviation of X, the result ofmeasuring all of the units of a finite lot or process.0= an advance estimate of .s =F(i51nXi2XH!2/n 2 1!G12, an estimate of the standarddeviation from n observation, Xi, i = 1 to n.1This practice is unde
13、r the jurisdiction ofASTM Committee E11 on Quality andStatistics and is the direct responsibility of Subcommittee E11.10 on Sampling /Statistics.Current edition approved April 1, 2017. Published April 2017. Originallyapproved in 1958. Last previous edition approved in 2009 as E122 091. DOI:10.1520/E
14、0122-17.2For referenced 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.Copyright ASTM International, 100 Barr Harbor Drive,
15、PO Box C700, West Conshohocken, PA 19428-2959. United StatesThis international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for theDevelopment of International Standards, Guides and Recommendations issued
16、 by the World Trade Organization Technical Barriers to Trade (TBT) Committee.1s =(j51kSj/k , average s from k samples all of the same size(8.2.1).sp= pooled (weighted average) s from k samples, not all ofthe same size (8.2).sj= standard deviation of sample j.Vo= an advance estimate of V, equal to o/
17、o.v = s/X, the coefficient of variation estimated from thesample.vp= pooled (weighted average) of v from k samples (8.3).vj= coefficient of variation from sample j.X = numerical value of the characteristic of an individualunit being measured.X=(i51nXi/niaverage of n observations, Xi,i=1ton.4. Signif
18、icance and Use4.1 This practice is intended for use in determining thesample size required to estimate, with specified precision, ameasure of quality of a lot or process. The practice applieswhen quality is expressed as either the lot average for a givenproperty, or as the lot fraction not conformin
19、g to prescribedstandards.The level of a characteristic may often be taken as anindication of the quality of a material. If so, an estimate of theaverage value of that characteristic or of the fraction of theobserved values that do not conform to a specification for thatcharacteristic becomes a measu
20、re of quality with respect to thatcharacteristic. This practice is intended for use in determiningthe sample size required to estimate, with specified precision,such a measure of the quality of a lot or process either as anaverage value or as a fraction not conforming to a specifiedvalue.5. Empirica
21、l Knowledge Needed5.1 Some empirical knowledge of the problem is desirablein advance.5.1.1 We may have some idea about the standard deviationof the characteristic.5.1.2 If we have not had enough experience to give a preciseestimate for the standard deviation, we may be able to state ourbelief about
22、the range or spread of the characteristic from itslowest to its highest value and possibly about the shape of thedistribution of the characteristic; for instance, we might be ableto say whether most of the values lie at one end of the range,or are mostly in the middle, or run rather uniformly from o
23、neend to the other (Section 9).5.2 If the aim is to estimate the fraction nonconforming,then each unit can be assigned a value of 0 or 1 (conforming ornonconforming), and the standard deviation as well as theshape of the distribution depends only on p, the fractionnonconforming in the lot or process
24、. Some rough idea con-cerning the size of p is therefore needed, which may be derivedfrom preliminary sampling or from previous experience.5.3 More knowledge permits a smaller sample size. Seldomwill there be difficulty in acquiring enough information tocompute the required size of sample. A sample
25、that is largerthan the equations indicate is used in actual practice when theempirical knowledge is sketchy to start with and when thedesired precision is critical.5.4 The precision of the estimate made from a randomsample may itself be estimated from the sample. This estima-tion of the precision fr
26、om one sample makes it possible to fixmore economically the sample size for the next sample of asimilar material. In other words, information concerning theprocess, and the material produced thereby, accumulates andshould be used.6. Precision Desired6.1 The approximate precision desired for the esti
27、mate mustbe prescribed. That is, it must be decided what maximumdeviation, E, can be tolerated between the estimate to be madefrom the sample and the result that would be obtained bymeasuring every unit in the lot or process.6.2 In some cases, the maximum allowable sampling error isexpressed as a pr
28、oportion, e, or a percentage, 100 e. Forexample, one may wish to make an estimate of the sulfurcontent of coal within 1 %, or e = 0.01.7. Equations for Calculating Sample Size7.1 Based on a normal distribution for the characteristic, theequation for the size, n, of the sample is as follows:n 5 3o/E!
29、2(1)The multiplier 3 is a factor corresponding to a low probabil-ity that the difference between the sample estimate and theresult of measuring (by the same methods) all the units in thelot or process is greater than E. The value 3 is recommendedfor general use. With the multiplier 3, and with a lot
30、 or processstandard deviation equal to the advance estimate, it is practi-cally certain that the sampling error will not exceed E. Wherea lesser degree of certainty is desired a smaller multiplier maybe used (Note 1).NOTE 1For example, multiplying by 2 in place of 3 gives aprobability of about 45 pa
31、rts in 1000 that the sampling error will exceedE.Although distributions met in practice may not be normal, the followingtext table (based on the normal distribution) indicates approximateprobabilities:Factor Approximate Probability of Exceeding E3 0.003 or 3 in 1000 (practical certainty)2.56 0.010 o
32、r 10 in 10002 0.045 or 45 in 10001.96 0.050 or 50 in 1000 (1 in 20)1.64 0.100 or 100 in 1000 (1 in 10)7.1.1 If a lot of material has a highly asymmetric distribu-tion in the characteristic measured, the sample size as calcu-lated in Eq 1 may not be adequate. There are two things to dowhen asymmetry
33、is suspected.7.1.1.1 Probe the material with a view to discovering, forexample, extra-high values, or possibly spotty runs of abnor-mal character, in order to approximate roughly the amount ofthe asymmetry for use with statistical theory and adjustment ofthe sample size if necessary.7.1.1.2 Search t
34、he lot for abnormal material and segregate itfor separate treatment.7.2 There are some materials for which varies approxi-mately with , in which case V(=) remains approximatelyconstant from large to small values of .E122 1727.2.1 For the situation of 7.2, the equation for the samplesize, n, is as fo
35、llows:n 5 3 Vo/e!2(2)If the relative error, e, is to be the same for all values of ,then everything on the right-hand side of Eq 2 is a constant;hence n is also a constant, which means that the same samplesize n would be required for all values of .7.3 If the problem is to estimate the lot fractionn
36、onconforming, then o2is replaced by po(1po) so that Eq1 becomes:n 5 3/E!2po1 2 po! (3)7.4 When the average for the production process is notneeded, but rather the average of a particular lot is needed, thenthe required sample size is less than Eq 1, Eq 2, and Eq 3indicate. The sample size for estima
37、ting the average of thefinite lot will be:nL5 n/11n/N!# (4)where n is the value computed from Eq 1, Eq 2,orEq 3.This reduction in sample size is usually of little importanceunless n is 10 % or more of N.7.5 When the information on the standard deviation islimited, a sample size larger than indicated
38、 in Eq 1, Eq 2, andEq 3 may be appropriate. When the advance estimate 0isbased on f degrees of freedom, the sample size in Eq 1 may bereplaced by:n 5 30/E!211=2/f! (5)NOTE 2The standard error of a sample variance with f degrees offreedom, based on the normal distribution, is =24/f. The factor11=2/f!
39、 has the effect of increasing the preliminary estimate 02byone times its standard error.8. Reduction of Empirical Knowledge to a NumericalValue of o(Data for Previous Samples Available)8.1 This section illustrates the use of the equations inSection 7 when there are data for previous samples.8.2 For
40、Eq 1An estimate of ocan be obtained fromprevious sets of data. The standard deviation, s, from any givensample is computed as:s 5F(i51nXi2 X!2/n 2 1!G1/2(6)The s value is a sample estimate of o. A better, more stablevalue for omay be computed by pooling the s values obtainedfrom several samples from
41、 similar lots. The pooled s value spfor k samples is obtained by a weighted averaging of the kresults from use of Eq 6.sp5F(j51knj2 1!sj2/(j51knj2 1!G1/2(7)8.2.1 If each of the previous data sets contains the samenumber of measurements, nj, then a simpler, but slightly lessefficient estimate for oma
42、y be made by using an average (s)of the s values obtained from the several previous samples. Thecalculated s value will in general be a slightly biased estimateof o. An unbiased estimate of ois computed as follows:o5sc4(8)where the value of the correction factor, c4, depends on thesize of the indivi
43、dual data sets (nj)(Table 13).8.2.2 An even simpler, and slightly less efficient estimate foromay be computed by using the average range (R) taken fromthe several previous data sets that have the same group size.o5Rd2(9)The factor, d2, from Table 1 is needed to convert the averagerange into an unbia
44、sed estimate of o.8.2.3 Example 1Use of s.8.2.3.1 ProblemTo compute the sample size needed toestimate the average transverse strength of a lot of bricks whenthe value of E is 50 psi, and practical certainty is desired.8.2.3.2 SolutionFrom the data of three previous lots, thevalues of the estimated s
45、tandard deviation were found to be215, 192, and 202 psi based on samples of 100 bricks. Theaverage of these three standard deviations is 203 psi. The c4value is essentially unity when Eq 1 gives the followingequation for the required size of sample to give a maximumsampling error of 50 psi:n 5 3 320
46、3!/50#25 12.225 149 bricks (10)8.3 For Eq 2If varies approximately proportionatelywith for the characteristic of the material to be measured,compute the average, X, the standard deviation, s, and thecoefficient of variation v for each sample. The pooled V valuevpfor k samples, not necessarily of the
47、 same size, is obtainedby a weighted average of the k results. Then use Eq 2.vp5F(j51knj2 1!vj2/(j51knj2 1!G1/2(11)8.3.1 Example 2Use of V, the estimated coefficient ofvariation:8.3.1.1 ProblemTo compute the sample size needed toestimate the average abrasion resistance (that is, averagenumber of cyc
48、les) of a material when the value of e is 0.10 or10 %, and practical certainty is desired.8.3.1.2 SolutionThere are no data from previous samplesof this same material, but data for six samples of similarmaterials show a wide range of resistance. However, the valuesof estimated standard deviation are
49、 approximately proportionalto the observed averages, as shown in the following text table:3ASTM Manual on Presentation of Data and Control Chart Analysis, ASTMMNL 7A, 2002, Part 3.TABLE 1 Values of the Correction Factor C4and d2for SelectedSample Sizes njASample Size3,(nj) C4d22 .798 1.134 .921 2.065 .940 2.338 .965 2.8510 .973 3.08AAs njbecomes large, C4approaches 1.000.E122 173Lot No.SampleSizeAvgCyclesStandardDeviationCoefficientof Varia-tion, %110 9013142 10 190 32 173 10 350 45 134 10
