1、Designation: E 122 07An 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 E 122; the number immediately following the designation indicat
2、es 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 (e) indicates an editorial change since the last revision or reapproval.1. Scope1.1 This practice covers simple methods for calc
3、ulating 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 indicate the sample size required to estimate theaverage value of some property or the f
4、raction 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 for immediate (future) production. The practice treats thecommon situation where the
5、 sampling units can be consideredto exhibit a single (overall) source of variability; it does nottreat multi-level sources of variability.2. Referenced Documents2.1 ASTM Standards:2E 456 Terminology Relating to Quality and Statistics3. Terminology3.1 Definitions: Unless otherwise noted, all statisti
6、cal termsare defined in Terminology E 456.3.2 Symbols: Symbols used in all 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 .k = the total number of samples available f
7、rom the sameor similar 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).
8、p8 = fraction of a lot or process whose units have thenonconforming characteristic under investigation.p0= an advance estimate of p8.p = fraction nonconforming in the sample.R = range of a set of sampling values. The largest minusthe smallest observation.Rj= range of sample j.R=(j 5 1kRj/k, average
9、of the range of k samples, all of thesame size (8.2.2).s = lot or process standard deviation of X, the result ofmeasuring all of the units of a finite lot or process.s0= an advance estimate of s.s =(i 5 1n(Xi X )2/(n1)1/2, an estimate of thestandard deviation s from n observation, Xi, i = 1 to n.s =
10、(j 5 1kSj/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.t = a factor (the 99.865thpercentile of the Studentsdistribution) corresponding to the degrees of freedomfoof an advance
11、estimate so(5.1).Vo= an advance estimate of V, equal to do/o.v = s/ X, the coefficient of variation estimated from thesample.vj= coefficient of variation from sample j.X = numerical value of the characteristic of an individualunit being measured.1This practice is under the jurisdiction ofASTM Commit
12、tee E11 on Quality andStatistics and is the direct responsibility of Subcommittee E11.10 on Sampling.Current edition approved Oct. 1, 2007. Published November 2007 . Originallypublished as E 12289. Last previous edition approved in 2000 as E 12200.2For referenced ASTM standards, visit the ASTM websi
13、te, 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.1Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United State
14、s.Copyright by ASTM Intl (all rights reserved); Fri Sep 5 02:59:46 EDT 2008Downloaded/printed byGuo Dehua (CNIS) pursuant to License Agreement. No further reproductions authorized.X=(i 5 1nXi/niaverage of n observations, Xi,i= 1 to n.4. Significance and Use4.1 This practice is intended for use in de
15、termining 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 conforming to prescribedstandards.The level of a characteristic ma
16、y 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 measure of quality with respect to thatcharacteristic. This pr
17、actice 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. Empirical Knowledge Needed5.1 Some empirical knowledge of the pro
18、blem 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 the range or spread of the characteristic from itslowest
19、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 oneend to the other (Section 9).5.2 If the aim is to estim
20、ate 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 p8, the fractionnonconforming in the lot or process. Some rough idea con-cerning the size of p8 is therefor
21、e needed, which may bederived from preliminary sampling or from previous experi-ence.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 that is largerthan the equations indicate is used in a
22、ctual 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 from one sample makes it possible to fixmore economicall
23、y 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 estimate mustbe prescribed. That is, it must be decided wh
24、at 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 proportion, e, or a percentage, 100 e. Forexample, one m
25、ay 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 3so/E!2(1)The multiplier 3 is a factor corresponding to a l
26、ow 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 or processstandard deviation equal to the advance es
27、timate, 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 a probabil-ity of about 45 parts in 1000 that the sampling error will exceed E.A
28、lthough 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 or 10 in 10002 0.045 or 45 in 10001.96 0.050 or 50
29、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 is suspected.7.1.1.1 Probe the material with a vie
30、w 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 the lot for abnormal material and segregate itfor s
31、eparate treatment.7.2 There are some materials for which s varies approxi-mately with , in which case V (=s/) remains approximatelyconstant from large to small values of .7.2.1 For the situation of 7.2, the equation for the samplesize, n, is as follows:n 5 3 Vo/e!2(2)If the relative error, e, is to
32、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 fraction noncon-forming, then so2is replaced by po(1po) so tha
33、t Eq 1becomes:n 5 3/E!2po1 2 po! (3)E122072Copyright by ASTM Intl (all rights reserved); Fri Sep 5 02:59:46 EDT 2008Downloaded/printed byGuo Dehua (CNIS) pursuant to License Agreement. No further reproductions authorized.7.4 When the average for the production process is notneeded, but rather the av
34、erage of a particular lot is needed, thenthe required sample size is less than Eq 1, Eq 2, and Eq 3indicate. The sample size for estimating the average of thefinite lot will be:nL5 n/1 1 n/N!# (4)where n is the value computed from Eq 1, Eq 2, or Eq 3. Thisreduction in sample size is usually of littl
35、e importance unless nis 10 % or more of N8. Reduction of Empirical Knowledge to a NumericalValue of so(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 Eq 1An estimate of socan be obtained fromprevi
36、ous sets of data. The standard deviation, s, from any givensample is computed as:s 5 (i 5 1nXi2 X!2/n 2 1!#1/2(5)The s value is a sample estimate of so. A better, more stablevalue for somay be computed by pooling the s values obtainedfrom several samples from similar lots. The pooled s value spfor k
37、 samples is obtained by a weighted averaging of the kresults from use of Eq 5.sp5 (j 5 1knj2 1!sj2/(j 5 1knj2 1!#1/2(6)8.2.1 If each of the previous data sets contains the samenumber of measurements, nj, then a simpler, but slightly lessefficient estimate for somay be made by using an average ( s)of
38、 the s values obtained from the several previous samples. Thecalculated s value will in general be a slightly biased estimateof so. An unbiased estimate of sois computed as follows:so5sc4(7)where the value of the correction factor, c4, depends on thesize of the individual data sets (nj)(Table 13).8.
39、2.2 An even simpler, and slightly less efficient estimate forsomay be computed by using the average range ( R) takenfrom the several previous data sets that have the same groupsize.so5Rd2(8)The factor, d2, from Table 1 is needed to convert the averagerange into an unbiased estimate of so.8.2.3 Examp
40、le 1 Use 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 standard deviation were found
41、 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 3 203!/5025 12.225 149 bricks (
42、9)8.3 If s varies approximately proportionately with for thecharacteristic of the material to be measured, compute both theaverage, X, and the standard deviation, s, for several samplesthat have the same size. An average of the several values ofv=s/Xmay be used for Vo. Then use Eq 2.8.3.1 For cases
43、where the sample sizes are not the same, aweighted average should be used as an approximate estimatefor VoVo5 (j 5 1knj2 1!vj/(j 5 1knj2 1!#1/2(10)8.3.2 Example 2 Use of V, the estimated coefficient ofvariation:8.3.2.1 ProblemTo compute the sample size needed toestimate the average abrasion resistan
44、ce (i.e., average numberof cycles) of a material when the value of e is 0.10 or 10 %, andpractical certainty is desired.8.3.2.2 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 es
45、timated standard deviation are approximately proportionalto the observed averages, as shown in the following text table:Lot No.SampleSizesAvgCyclesObservedrange, REstimate ofso=R/3.08ACoefficientof Varia-tion, %1 10 90 40 13.0 142 10 190 100 32.5 173 10 350 140 45.5 134 10 450 220 71.4 165 10 1000 3
46、60 116.9 126 10 3550 2090 678.6 19Avg 15.2AValues of standard deviation, s, may be used instead of the estimates madefrom the range, if they are preferred or already available. The use of s would bemore efficient.The use of the average of the observed values of thecoefficient of variation for Voin E
47、q 2 gives the following forthe required size of sample to give a maximum sampling errorof 10 % of the expected value:n 5 3 3 15.2!/1025 4.625 21.222 test specimens (11)8.3.2.3 If a maximum allowable error of 5 % were needed,the required sample size would be 85 specimens. The datasupplied by the pres
48、cribed sample will be useful for the studyin hand and also for the next investigation of similar material.3ASTM Manual on Presentation of Data and Control Chart Analysis, ASTM STP15D, 1976, Part 3, Table 27.TABLE 1 Values of the Correction Factor C4and d2for SelectedSample Sizes njASample Size3,(nj)
49、 C4d22 .798 1.134 .921 2.065 .940 2.338 .965 2.8510 .973 3.08AAs njbecomes large, C4approaches 1.000.E122073Copyright by ASTM Intl (all rights reserved); Fri Sep 5 02:59:46 EDT 2008Downloaded/printed byGuo Dehua (CNIS) pursuant to License Agreement. No further reproductions authorized.8.4 For Eq 3Compute the estimated fraction nonconform-ing, p, for each sample. Then for the weighted average use thefollowing equation:p 5total number nonconforming in all samplestotal number of units in all samples(12)8.4.1 Example 3U
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