1、Designation: E1402 081E1402 13 An American National StandardStandard Guide forSampling Design1This standard is issued under the fixed designation E1402; the number immediately following the designation indicates the year oforiginal adoption or, in the case of revision, the year of last revision. A n
2、umber in parentheses indicates the year of last reapproval. Asuperscript epsilon () indicates an editorial change since the last revision or reapproval.1 NOTESection 1.4 was editorially corrected in July 2010.1. Scope1.1 This guide defines terms and introduces basic methods for probability sampling
3、of discrete populations, areas, and bulkmaterials. It provides an overview of common probability sampling methods employed by users of ASTM standards.1.2 Sampling may be done for the purpose of estimation, of comparison between parts of a sampled population, or foracceptance of lots. Sampling is als
4、o used for the purpose of auditing information obtained from complete enumeration of thepopulation.1.3 No system of units is specified in this standard.1.4 This standard does not purport to address all of the safety concerns, if any, associated with its use.2. Referenced Documents2.1 ASTM Standards:
5、2D7430 Practice for Mechanical Sampling of CoalE105 Practice for Probability Sampling of MaterialsE122 Practice for Calculating Sample Size to Estimate, With Specified Precision, the Average for a Characteristic of a Lot orProcessE141 Practice for Acceptance of Evidence Based on the Results of Proba
6、bility SamplingE456 Terminology Relating to Quality and Statistics3. Terminology3.1 Definitions:Definitions3.1.1 Terminology E456 contains a more extensive list of statistical terms.For a more extensive list of statistical terms, referto Terminology E456.3.1.1 area sampling, nprobability sampling in
7、 which a map, rather than a tabulation of sampling units, serves as the samplingframe.3.1.1.1 DiscussionArea sampling units are segments of land area and are listed by addresses on the frame prior to their actual delineation on theground so that only the randomly selected ones need to be exactly ide
8、ntified.3.1.2 bulk sampling, nsampling to prepare a portion of a mass of material that is representative of the whole.3.1.3 cluster sampling, nsampling in which the sampling unit consists of a group of subunits, all of which are measured forsampled clusters.3.1.4 frame, na list, compiled for samplin
9、g purposes, which designates all of the sampling units (items or groups) of apopulation or universe to be considered in a specific study.3.1.5 multi-stage sampling, nsampling in which the sample is selected by stages, the sampling units at each stage beingselected from subunits of the larger samplin
10、g units chosen at the previous stage.1 This guide is under the jurisdiction of ASTM Committee E11 on Quality and Statistics and is the direct responsibility of Subcommittee E11.10 on Sampling / Statistics.Current edition approved Oct. 1, 2008Aug. 1, 2013. Published January 2009August 2013. Originall
11、y approved in 2008. Last previous edition approved in 2008 as E1402 081. DOI: 10.1520/E1402-08.10.1520/E1402-13.2 For referencedASTM standards, visit theASTM website, www.astm.org, or contactASTM Customer Service at serviceastm.org. For Annual Book of ASTM Standardsvolume information, refer to the s
12、tandards Document Summary page on the ASTM website.This document is not an ASTM standard and is intended only to provide the user of an ASTM standard an indication of what changes have been made to the previous version. Becauseit may not be technically possible to adequately depict all changes accur
13、ately, ASTM recommends that users consult prior editions as appropriate. In all cases only the current versionof the standard as published by ASTM is to be considered the official document.Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United Stat
14、es13.1.5.1 DiscussionThe sampling unit for the first stage is the primary sampling unit. In multi-stage sampling, this unit is further subdivided. Thesecond stage unit is called the secondary sampling unit. A third stage unit is called a tertiary sampling unit. The final sample isthe set of all last
15、 stage sampling units that are obtained. As an example of sampling a lot of packaged product, the cartons of alot could be the primary units, packages within the carton could be secondary units, and items within the packages could be thethird-stage units.3.1.6 nested sampling, nsame as multi-stage s
16、ampling.3.1.7 primary sampling unit, PSU, nthe item, element, increment, segment or cluster selected at the first stage of the selectionprocedure from a population or universe.3.1.8 probability proportional to size sampling, PPS, nprobability sampling in which the probabilities of selection of sampl
17、ingunits are proportional, or nearly proportional, to a quantity (the “size”) that is known for all sampling units.3.1.9 probability sample, na sample in which the sampling units are selected by a chance process such that a specifiedprobability of selection can be attached to each possible sample th
18、at can be selected.3.1.10 proportional sampling, na method of selection in stratified sampling such that the proportions of the sampling units(usually, PSUs) selected for the sample from each stratum are equal.3.1.11 quota sampling, na method of selection similar to stratified sampling in which the
19、numbers of units to be selected fromeach stratum is specified and the selection is done by trained enumerators but is not a probability sample.3.1.12 sampling fraction, f, nthe ratio of the number of sampling units selected for the sample to the number of sampling unitsavailable.3.1.13 sampling unit
20、, nan item, group of items, or segment of material that can be selected as part of a probability samplingplan.3.1.13.1 DiscussionThe full collection of sampling units listed on a frame serves to describe the sampled population of a probability sampling plan.3.1.14 sampling with replacement, nprobabi
21、lity sampling in which a selected unit is replaced after any step in selection so thatthis sampling unit is available for selection again at the next step of selection, or at any other succeeding step of the sampleselection procedure.3.1.15 sampling without replacement, nprobability sampling in whic
22、h a selected sampling unit is set aside and cannot beselected at a later step of selection.3.1.15.1 DiscussionMost samplings, including simple random sampling and stratified random sampling, are conducted by sampling withoutreplacement.3.1.16 simple random sample, n(without replacement) probability
23、sample of n sampling units from a population of N unitsselected in such a way that each of the N!n!N2n! subsets of n units is equally probable (with replacement) a probability sampleof n sampling units from a population of N units selected in such a way that, in order of selection, each of the Nn or
24、dered sequencesof units from the population is equally probable.3.1.17 stratified sampling, nsampling in which the population to be sampled is first divided into mutually exclusive subsetsor strata, and independent samples taken within each stratum.3.1.18 systematic sampling, na sampling procedure i
25、n which evenly spaced sampling units are selected.3.2 Definitions of Terms Specific to This Standard:3.2.1 address, n(sampling) a unique label or instructions attached to a sampling unit by which it can be located and measured.3.2.2 area segment, n(area sampling) final sampling unit for area samplin
26、g, the delimited area from which a characteristic canbe measured.3.2.3 composite sample, n(bulk sampling) sample prepared by aggregating increments of sampled material.3.2.4 increment, n(bulk sampling) individual portion of material collected by a single operation of a sampling device.E1402 1323.3 S
27、ymbols:N = number of units in the population to be sampled.n = number of units in the sample.Yi = quantity value for the i-th unit in the population.yi = quantity observed for i-th sampling unit.Y = average quantity for the population.y = average of the observations in the sample.Xi = value of an au
28、xiliary variable for the i-th unit in the population.xi = value of an auxiliary variable for the i-th sampling unit.P = population proportion of units having an attribute of interest.p = sample proportion.f = sampling fraction.s = sample standard deviation of the observations in the sample.s2 = samp
29、le variance of the observations in the sample.SEy!= standard error of an estimated mean y .4. Significance and Use4.1 This guide describes the principal types of sampling designs and provides formulas for estimating population means andstandard errors of the estimates. Practice E105 provides princip
30、les for designing probability sampling plans in relation to theobjectives of study, costs, and practical constraints. Practice E122 aids in specifying the required sample size. Practice E141describes conditions to ensure validity of the results of sampling. Further description of the designs and for
31、mulas in this guide,and beyond it, can be found in textbooks (1-10).34.2 Sampling, both discrete and bulk, is a clerical and physical operation. It generally involves training enumerators andtechnicians to use maps, directories and stop watches so as to locate designated sampling units. Once a sampl
32、ing unit is locatedat its address, discrete sampling and area sampling enumeration proceeds to a measurement. For bulk sampling, material isextracted into a composite.4.3 A sampling plan consists of instructions telling how to list addresses and how to select the addresses to be measured orextracted
33、. A frame is a listing of addresses each of which is indexed by a single integer or by an n-tuple (several integer) number.The sampled population consists of all addresses in the frame that can actually be selected and measured. It is sometimes differentfrom a targeted population that the user would
34、 have preferred to be covered.4.4 A selection scheme designates which indexes constitute the sample. If certified random numbers completely control theselection scheme the sample is called a probability sample. Certified random numbers are those generated either from a table (forexample, Ref (11) th
35、at has been tested for equal digit frequencies and for serial independence, from a computer program that waschecked to have a long cycle length, or from a random physical method such as tossing of a coin or a casino-quality spinner.4.5 The objective of sampling is often to estimate the mean of the p
36、opulation for some variable of interest by the correspondingsample mean. By adopting probability sampling, selection bias can be essentially eliminated, so the primary goal of sample designin discrete sampling becomes reducing sampling variance.5. Simple Random Sampling (SRS) of a Finite Population5
37、.1 Sampling is without replacement. The selection scheme must allocate equal chance to every combination of n indexes fromthe N on the frame.5.1.1 Make successive equal-probability draws from the integers 1 to N and discard duplicates until n distinct indexes have beenselected.5.1.2 If the N indexed
38、 addresses or labels are in a computer file, generate a random number for each index and sort the file bythose numbers. The first n items in the sorted file constitute a simple random sample (SRS) of size n from the N.5.1.3 A method that requires only one pass through the population is used, for exa
39、mple, to sample a production process. Foreach item, generate a random number in the range 0 to 1 and select the ith item when the random number is less than (n-ai)/(N-i+1),where ai is the number of selections already made up to the i-th item. For example, the first item (i=1 and a1=0) is selected wi
40、thprobability n/N.5.2 The quantities observed on the variable of interest at the selected sampling units will be denoted y1, y2,yn. The estimateof the mean of the sampled population isy 5(yi/n (1)The standard error of the mean of a finite population using simple random sampling without replacement i
41、s:3 The boldface numbers in parentheses refer to a list of references at the end of this standard.E1402 133SEy! 5s =12f!/n (2)where f =n/N is the sampling fraction and s2 is the sample variance (s, its square root, is sample standard deviation).s25(yi 2y!2/n 21! (3)The population mean that y estimat
42、es is:Y 5(i51NYi/N (4)The expected value of s2 is the finite population variance defined as:S25(i51N Yi 2Y!2/N 21! (5)5.3 Finite population correctionPopulation CorrectionThe factor (1- f) in Eq 32 is the finite population correction. Inconventional statistical theory, the standard error of the aver
43、age of independent, identically distributed random variables does notinclude this factor. Conventional statistical theory applies for random sampling with replacement. In sampling without replacementfrom a finite population, the observations are not independent.The finite population correction facto
44、r depends on (a) the populationof interest being finite, (b) sampling being without errors and measurements for any sampled item being assumed completely welldefined for that item. When the purpose of sampling is to understand differences between parts of a population (analytic as opposedto enumerat
45、ive, as described by Deming (4), actual population values are viewed as themselves sampled from a parent randomprocess and the finite population correction should not be used in making such comparisons.5.4 Sample SizeThe sample size required for a sampling study depends on the variability of the pop
46、ulation and the requiredprecision of the estimate. Refer to Practice E122 for further detail on determining sample size. Eq 32 can be developed to findrequired sample size. First, the user must have a reasonable prior estimate s0 of the population standard deviation, either fromprevious experience o
47、r a pilot study. Solving for n in Eq 32, where now SEy! is the required standard error, gives:n 5 no11no/Nwhere:no 5so2/SE y!2 (6)5.5 Estimating a proportionProportionFormulas 1 through 5 serve for proportions as well as means. For an indicatorvariable Yi which equals 1 if the i-th unit has the attr
48、ibute and 0 if not, the population proportion P 5Y can be recognized as theaverage of ones and zeros. The sample estimate is the sample proportion p 5y and the sample variance is s2 = np(1-p)/(n-1).5.6 Ratio estimatesEstimatesAn auxiliary variable may be used to improve the estimate from an SRS. Val
49、ues of thisvariable for each item on the frame will be denoted Xi. Specific knowledge of each and every Xi is not necessary for ratioestimation but knowing the population average X is. The observed values xi are needed along with the yi, where the index i goesfrom i=1 to i=n, the sample size. The estimated ratio is R5y/R5y/x and the improved ratio estimate of Y is Xy/x . The estimatedstandard error of the ratio estimate of Y is:SEXR!512fn (yi 2R xi!2/n 21! (7)5.6.1 The ratio estimator works bes
