1、Designation: E 122 00An American National StandardStandard Practice forCalculating Sample Size to Estimate, With a SpecifiedTolerable Error, the Average for a Characteristic of a Lot orProcess1This standard is issued under the fixed designation E 122; the number immediately following the designation
2、 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 (e) indicates an editorial change since the last revision or reapproval.This standard has been approved for use by agenc
3、ies of the 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 prescribed precision, a measure of quality for all theunits of a lot of material, or produced by a process. Thispractice will clearl
4、y indicate the sample size required toestimate the average value of some property or the fraction ofnonconforming items produced by a production process duringthe time interval covered by the random sample. If the processis not in a state of statistical control, the result will not havepredictive va
5、lue for immediate (future) production. The prac-tice treats the common situation where the sampling units canbe considered to exhibit a single (overall) source of variability;it does not treat multi-level sources of variability.2. Referenced Documents2.1 ASTM Standards:E 456 Definitions of Terms Rel
6、ating to Statistical Methods23. Terminology3.1 DefinitionsUnless otherwise noted, all statisticalterms are defined in Definitions E 456.3.2 Symbols:E = maximum tolerable error for the sample average, thatis, the maximum acceptable difference between trueaverage and the sample average.e = E/, maximum
7、 allowable sampling error expressed asa fraction of .k = the total number of samples available from 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
8、taken from a lot or process.nj= size of sample j.nL= size of the sample from a finite lot (7.4).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
9、 values. The largest minusthe smallest observation.Rj= range of sample j.R=(j 5 1kRj/k, average 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 1
10、n(Xi X)2/(n1)1/2, an estimate of the standarddeviation s from n observation, Xi, i = 1 to n.s =(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 derivation of sample j.t = a factor (the 99.865t
11、h percentile of the Studentsdistribution) corresponding to the degrees of freedomfoof an advance estimate so(5.1).V = s/, the coefficient of variation of the lot or process.Vo= an advance estimate of V (8.3.1).v = s/ X, the coefficient of variation estimated from thesample.vj= coefficient of variati
12、on from sample j.X = numerical value of the characteristic of an individualunit being measured.1This practice is under the jurisdiction ofASTM Committee E11 on Quality andStatistics and is the direct responsibility of Subcommittee E11.10 on Sampling.Current edition approved Oct. 10, 2000. Published
13、January 2001. Originallypublished as E12289. Last previous edition E12299.2Annual Book of ASTM Standards, Vol 14.02.1Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.X=(i 5 1nXi/niaverage of n observations, Xi,i= 1 to n.4. Significance
14、 and Use4.1 This practice is intended for use in determining thesample size required to estimate, with prescribed 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
15、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 measure of
16、 quality with respect to thatcharacteristic. This practice is intended for use in determiningthe sample size required to estimate, with prescribed 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 Kn
17、owledge 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 the
18、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 oneen
19、d 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 p8, the fractionnonconforming to the lot or process. S
20、ome rough idea con-cerning the size of p8 is therefore needed, which may bederived from preliminary sampling or from previous experi-ence.5.3 Sketchy knowledge is sufficient to start with, althoughmore knowledge permits a smaller sample size. Seldom willthere be difficulty in acquiring enough inform
21、ation to computethe required size of sample. A sample that is larger than theequations indicate is used in actual practice when the empiricalknowledge is sketchy to start with and when the desiredprecision is critical.5.4 In any case, even with sketchy knowledge, the precisionof the estimate made fr
22、om a random sample may itself beestimated from the sample. This estimation of the precisionfrom one sample makes it possible to fix more economicallythe sample size for the next sample of a similar material. Inother words, information concerning the process, and thematerial produced thereby, accumul
23、ates and should be used.6. Precision Desired6.1 The approximate precision desired for the estimate 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
24、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 may wish to make an estimate of the sulfurcontent of coal with maximum allowable error of 1 %, or e= 0.01.7. Equations for Calculating Sample Size7.1 Bas
25、ed on a normal distribution for the characteristic, theequation for the size, n, of the sample is as follows:n 5 3so/E!2(1)where:so= the advance estimate of the standard deviation of thelot or process,E = the maximum allowable error between the estimate tobe made from the sample and the result of me
26、asuring(by the same methods) all the units in the lot orprocess, and3 = a factor corresponding to a low probability that thedifference between the sample estimate and the resultof measuring (by the same methods) all the units inthe lot or process is greater than E. The choice of thefactor 3 is recom
27、mended for general use. With thefactor 3, and with a lot or process standard deviationequal to the advance estimate, it is practically certainthat the sampling error will not exceed E. Where alesser degree of certainty is desired a smaller factormay be used (Note 1).NOTE 1For example, the factor 2 i
28、n place of 3 gives a probability ofabout 45 parts in 1000 that the sampling error will exceed E. Althoughdistributions met in practice may not be normal (Note 2), the followingtext table (based on the normal distribution) indicates approximateprobabilities:Factor Approximate Probability of Exceeding
29、 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 in 1000 (1 in 20)1.64 0.100 or 100 in 1000 (1 in 10)NOTE 2If a lot of material has a highly asymmetric distribution in thecharacteristic measured, the factor 3 will give a different probability,
30、possibly much greater than 3 parts in 1000 if the sample size is small.There are two things to do when asymmetry is suspected.7.1.1 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
31、 ofthe asymmetry for use with statistical theory and adjustment ofthe sample size if necessary.7.1.2 Search the lot for abnormal material and segregate itfor separate treatment.7.2 There are some materials for which s varies approxi-mately with , in which case V (=s/) remains approximatelyconstant f
32、rom 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)E122002where:Vo= (coefficient of variation) = so/othe advance estimateof the coefficient of variation, expressed as a fraction(or as a percentage),e = E/, the allowable samp
33、ling error expressed as afraction (or as a percentage) of , and = the expected value of the characteristic being mea-sured.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
34、 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 poso that Eq 1 becomes:n 5 3/E!2po1 2 po! (3)where:po= the advance estimate of the lot or process fractionnonconforming p81and E # po7.4 When the average for t
35、he 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 estimating the average of thefinite lot will be:nL5 n/1 1 n/N!# (4)where:n = the value computed from Eq 1, Eq 2, or Eq
36、 3, andN = the lot size.This reduction in sample size is usually of little importanceunless n is 10 % or more of N.8. 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 f
37、or previous samples.8.2 For Eq 1An estimate of socan be obtained fromprevious 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 ofso. A better, more stablevalue for somay be computed by pooling the s values o
38、btainedfrom several samples from similar lots. The pooled s value spfor k 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)where:sj= the standard deviation for sample j,nj= the sample size for sample j.8.2.1 If each of the previous
39、 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 the s values obtained from the several previous samples. Thecalculated s value will in general be a slightly biased estimateof so. An unbiased esti
40、mate 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).8.2.2 An even simpler, and slightly less efficient estimateforsomay be computed by using the average range ( R) takenfrom the several previous data se
41、ts 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 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 desired value of
42、E is 50 psi, and practical certainty isdesired.8.2.3.2 SolutionFrom the data of three previous lots, thevalues of the estimated standard 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 essentiall
43、y unity when Eq 1 gives the followingequation:n 3 3 203!/50!#2525 149 bricks (9)for the required size of sample to give a maximum samplingerror of 50 psi, and practical certainty is desired.8.3 For Eq 2If s varies approximately proportionatelywith for the characteristic of the material to be measure
44、d,compute both the average, X, and the standard deviation, s, forseveral samples that have the same size. An average of theseveral values of v=s/Xmay be used for Vo.8.3.1 For cases where the sample sizes are not the same, aweighted average should be used as an approximate estimatefor VoVo5 (j 5 1knj
45、2 1!vj/(j 5 1knj2 1!#1/2(10)where:vj= the coefficient of variation for sample j, andnj= the sample size for sample j.8.3.2 Example 2Use of V, the estimated coefficient ofvariation:3ASTM Manual on Presentation of Data and Control Chart Analysis, ASTM STP15D, 1976, Part 3, Table 27.TABLE 1 Values of t
46、he 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.E1220038.3.2.1 ProblemTo compute the sample size needed toestimate the average abrasion resistance of a material when thedesi
47、red value of e is 0.10 or 10 %, and practical certainty isdesired.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 estimated standard deviation are approximately proportio
48、nalto 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 360 116.9 126 10 3550 2090 678.6 19Avg 15.2AValues of s
49、tandard 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 Eq 2 gives the following:n 5 3 3 15.2!/102525 21.222 specimens (11)for the required size of sample to give a maximum samplingerror of 10 % of the expected value, and practical certainty isdesired.8.3.2.3 If a maximum allowable error of 5 % were needed,the required sample size
