1、Designation: E2586 12aAn American National StandardStandard Practice forCalculating and Using Basic Statistics1This standard is issued under the fixed designation E2586; the number immediately following the designation indicates the year oforiginal adoption or, in the case of revision, the year of l
2、ast revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon () indicates an editorial change since the last revision or reapproval.1. Scope1.1 This practice covers methods and equations for comput-ing and presenting basic descriptive statistics using a set ofsam
3、ple data containing a single variable. This practice includessimple descriptive statistics for variable data, tabular andgraphical methods for variable data, and methods for summa-rizing simple attribute data. Some interpretation and guidancefor use is also included.1.2 The system of units for this
4、practice is not specified.Dimensional quantities in the practice are presented only asillustrations of calculation methods. The examples are notbinding on products or test methods treated.1.3 This standard does not purport to address all of thesafety concerns, if any, associated with its use. It is
5、theresponsibility of the user of this standard to establish appro-priate safety and health practices and determine the applica-bility of regulatory limitations prior to use.2. Referenced Documents2.1 ASTM Standards:2E178 Practice for Dealing With Outlying ObservationsE456 Terminology Relating to Qua
6、lity and StatisticsE2282 Guide for Defining the Test Result of a Test Method2.2 ISO Standards:3ISO 3534-1 StatisticsVocabulary and Symbols, part 1:Probability and General Statistical TermsISO 3534-2 StatisticsVocabulary and Symbols, part 2:Applied Statistics3. Terminology3.1 Definitions:3.1.1 Unless
7、 otherwise noted, terms relating to quality andstatistics are as defined in Terminology E456.3.1.2 characteristic, na property of items in a sample orpopulation which, when measured, counted, or otherwiseobserved, helps to distinguish among the items. E22823.1.3 coeffcient of variation, CV, nfor a n
8、onnegativecharacteristic, the ratio of the standard deviation to the meanfor a population or sample3.1.3.1 DiscussionThe coefficient of variation is oftenexpressed as a percentage.3.1.3.2 DiscussionThis statistic is also known as therelative standard deviation, RSD.3.1.4 confidence bound, nsee confi
9、dence limit.3.1.5 confidence coeffcient, nsee confidence level.3.1.6 confidence interval, nan interval estimate L, Uwith the statistics L and U as limits for the parameter u andwith confidence level 1 a, where Pr(L # u # U) $ 1a.3.1.6.1 DiscussionThe confidence level, 1 a, reflectsthe proportion of
10、cases that the confidence interval L, Uwould contain or cover the true parameter value in a series ofrepeated random samples under identical conditions. Once Land U are given values, the resulting confidence interval eitherdoes or does not contain it. In this sense 9confidence9 appliesnot to the par
11、ticular interval but only to the long run proportionof cases when repeating the procedure many times.3.1.7 confidence level, nthe value, 1 a, of the probabil-ity associated with a confidence interval, often expressed as apercentage.3.1.7.1 Discussiona is generally a small number. Confi-dence level i
12、s often 95 % or 99 %.3.1.8 confidence limit, neach of the limits, L and U, of aconfidence interval, or the limit of a one-sided confidenceinterval.3.1.9 degrees of freedom, nthe number of independentdata points minus the number of parameters that have to beestimated before calculating the variance.3
13、.1.9.1 DiscussionThe term degrees of freedom is bestdefined in the specific context of its use. For a generaldiscussion, the following comments were reprinted from Box,Hunter, and Hunter,3.1.10 estimate, nsample statistic used to approximate apopulation parameter.3.1.11 histogram, ngraphical represe
14、ntation of the fre-quency distribution of a characteristic consisting of a set ofrectangles with area proportional to the frequency.ISO 3534-11This practice is under the jurisdiction of ASTM Committee E11 on Quality andStatistics and is the direct responsibility of Subcommittee E11.10 on Sampling /S
15、tatistics.Current edition approved Feb. 15, 2012. Published March 2012. Originallyapproved in 2007. Last previous edition approved in 2012 as E2586 12. DOI:10.1520/E2586-12A.2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at serviceastm.org. For
16、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.1Copyright ASTM International, 100 Barr Harbor Drive, PO Box
17、 C700, West Conshohocken, PA 19428-2959, United States.3.1.11.1 DiscussionWhile not required, equal bar or classwidths are recommended for histograms.3.1.12 interquartile range, IQR, nthe 75thpercentile (0.75quantile) minus the 25thpercentile (0.25 quantile), for a dataset.3.1.13 kurtosis, g2,g2, nf
18、or a population or a sample, ameasure of the weight of the tails of a distribution relative tothe center, calculated as the ratio of the fourth central moment(empirical if a sample, theoretical if a population applies) to thestandard deviation (sample, s, or population, s) raised to thefourth power,
19、 minus 3 (also referred to as excess kurtosis).3.1.14 mean, nof a population, , average or expectedvalue of a characteristic in a population of a sample, x, sumof the observed values in the sample divided by the samplesize.3.1.15 median, X , nthe 50thpercentile in a population orsample.3.1.15.1 Disc
20、ussionThe sample median is the (n + 1)/2order statistic if the sample size n is odd and is the average ofthe n/2 and n/2 + 1 order statistics if n is even.3.1.16 midrange, naverage of the minimum and maxi-mum values in a sample.3.1.17 order statistic, x(k), nvalue of the kthobservedvalue in a sample
21、 after sorting by order of magnitude.3.1.17.1 DiscussionFor a sample of size n, the first orderstatistic x(1)is the minimum value, x(n)is the maximum value.3.1.18 parameter, nsee population parameter.3.1.19 percentile, nquantile of a sample or a population,for which the fraction less than or equal t
22、o the value isexpressed as a percentage.3.1.20 population, nthe totality of items or units ofmaterial under consideration.3.1.21 population parameter, nsummary measure of thevalues of some characteristic of a population. ISO 3534-23.1.22 statistic, nsee sample statistic.3.1.23 quantile, nvalue such
23、that a fraction f of the sampleor population is less than or equal to that value.3.1.24 range, R, nmaximum value minus the minimumvalue in a sample.3.1.25 sample, na group of observations or test results,taken from a larger collection of observations or test results,which serves to provide informati
24、on that may be used as a basisfor making a decision concerning the larger collection.3.1.26 sample size, n, nnumber of observed values in thesample3.1.27 sample statistic, nsummary measure of the ob-served values of a sample.3.1.28 skewness, g1,g1, nfor population or sample, ameasure of symmetry of
25、a distribution, calculated as the ratioof the third central moment (empirical if a sample, andtheoretical if a population applies) to the standard deviation(sample, s, or population, s) raised to the third power.3.1.29 standard errorstandard deviation of the populationof values of a sample statistic
26、 in repeated sampling, or anestimate of it.3.1.29.1 DiscussionIf the standard error of a statistic isestimated, it will itself be a statistic with some variance thatdepends on the sample size.3.1.30 standard deviationof a population, s, the squareroot of the average or expected value of the squared
27、deviationof a variable from its mean; of a sample, s, the square rootof the sum of the squared deviations of the observed values inthe sample divided by the sample size minus 1.3.1.31 variance, s2, s2, nsquare of the standard deviationof the population or sample.3.1.31.1 DiscussionFor a finite popul
28、ation, s2is calcu-lated as the sum of squared deviations of values from the mean,divided by n. For a continuous population, s2is calculated byintegrating (x )2with respect to the density function. For asample, s2is calculated as the sum of the squared deviations ofobserved values from their average
29、divided by one less than thesample size.3.1.32 Z-score, nobserved value minus the sample meandivided by the sample standard deviation.4. Significance and Use4.1 This practice provides approaches for characterizing asample of n observations that arrive in the form of a data set.Large data sets from o
30、rganizations, businesses, and govern-mental agencies exist in the form of records and otherempirical observations. Research institutions and laboratoriesat universities, government agencies, and the private sectoralso generate considerable amounts of empirical data.4.1.1 A data set containing a sing
31、le variable usually consistsof a column of numbers. Each row is a separate observation orinstance of measurement of the variable. The numbers them-selves are the result of applying the measurement process to thevariable being studied or observed. We may refer to eachobservation of a variable as an i
32、tem in the data set. In manysituations, there may be several variables defined for study.4.1.2 The sample is selected from a larger set called thepopulation. The population can be a finite set of items, a verylarge or essentially unlimited set of items, or a process. In aprocess, the items originate
33、 over time and the population isdynamic, continuing to emerge and possibly change over time.Sample data serve as representatives of the population fromwhich the sample originates. It is the population that is ofprimary interest in any particular study.4.2 The data (measurements and observations) may
34、 be ofthe variable type or the simple attribute type. In the case ofattributes, the data may be either binary trials or a count of adefined event over some interval (time, space, volume, weight,or area). Binary trials consist of a sequence of 0s and 1s inwhich a “1” indicates that the inspected item
35、 exhibited theattribute being studied and a “0” indicates the item did notexhibit the attribute. Each inspection item is assigned either a“0” or a “1.” Such data are often governed by the binomialdistribution. For a count of events over some interval, thenumber of times the event is observed on the
36、inspectioninterval is recorded for each of n inspection intervals. ThePoisson distribution often governs counting events over aninterval.4.3 For sample data to be used to draw conclusions aboutthe population, the process of sampling and data collectionmust be considered, at least potentially, repeat
37、able. Descriptivestatistics are calculated using real sample data that will vary inrepeating the sampling process. As such, a statistic is a randomE2586 12a2variable subject to variation in its own right. The samplestatistic usually has a corresponding parameter in the popula-tion that is unknown (s
38、ee Section 5). The point of using astatistic is to summarize the data set and estimate a correspond-ing population characteristic or parameter.4.4 Descriptive statistics consider numerical, tabular, andgraphical methods for summarizing a set of data. The methodsconsidered in this practice are used f
39、or summarizing theobservations from a single variable.4.5 The descriptive statistics described in this practice are:4.5.1 Mean, median, min, max, range, mid range, orderstatistic, quartile, empirical percentile, quantile, interquartilerange, variance, standard deviation, Z-score, coefficient ofvaria
40、tion, skewness and kurtosis, and standard error.4.6 Tabular methods described in this practice are:4.6.1 Frequency distribution, relative frequency distribu-tion, cumulative frequency distribution, and cumulative rela-tive frequency distribution.4.7 Graphical methods described in this practice are:4
41、.7.1 Histogram, ogive, boxplot, dotplot, normal probabilityplot, and q-q plot.4.8 While the methods described in this practice may beused to summarize any set of observations, the results obtainedby using them may be of little value from the standpoint ofinterpretation unless the data quality is acc
42、eptable and satisfiescertain requirements. To be useful for inductive generalization,any sample of observations that is treated as a single group forpresentation purposes must represent a series of measurements,all made under essentially the same test conditions, on amaterial or product, all of whic
43、h have been produced underessentially the same conditions. When these criteria are met,we are minimizing the danger of mixing two or more distinctlydifferent sets of data.4.8.1 If a given collection of data consists of two or moresamples collected under different test conditions or represent-ing mat
44、erial produced under different conditions (that is,different populations), it should be considered as two or moreseparate subgroups of observations, each to be treated inde-pendently in a data analysis program. Merging of such sub-groups, representing significantly different conditions, maylead to a
45、 presentation that will be of little practical value.Briefly, any sample of observations to which these methods areapplied should be homogeneous or, in the case of a process,have originated from a process in a state of statistical control.4.9 The methods developed in Sections 6, 7, and 8 apply tothe
46、 sample data. There will be no misunderstanding when, forexample, the term “mean” is indicated, that the meaning issample mean, not population mean, unless indicated otherwise.It is understood that there is a data set containing n observa-tions. The data set may be denoted as:x1, x2, x3. xn(1)4.9.1
47、There is no order of magnitude implied by thesubscript notation unless subscripts are contained in parenthe-sis (see 6.7).5. Characteristics of Populations5.1 A population is the totality of a set of items underconsideration. Populations may be finite or unlimited in sizeand may be existing or conti
48、nuing to emerge as, for example,in a process. For continuous variables, X, representing anessentially unlimited population or a process, the population ismathematically characterized by a probability density function,f (x). The density function visually describes the shape of thedistribution as for
49、example in Fig. 1. Mathematically, the onlyrequirements of a density function are that its ordinates be allpositive and that the total area under the curve be equal to 1.5.1.1 Area under the density function curve is equivalent toprobability for the variable X. The probability that X shall occurbetween any two values, say s and t, is given by the area underthe curve bounded by the two given values of s and t. This isexpressed mathematically as a definite integral over the densityfunction between s and t:P s , X#
