1、Designation: E2586 12b An 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
2、last 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 ofsa
3、mple 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 Qu
6、ality 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 Unles
7、s 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
8、nonnegativecharacteristic, 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 conf
9、idence 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 andwith confidence level 1 , where Pr(L U) 1.3.1.6.1 DiscussionThe confidence level, 1 , reflects theproportion of cases that t
10、he confidence interval L, U wouldcontain or cover the true parameter value in a series of repeatedrandom samples under identical conditions. Once L and U aregiven values, the resulting confidence interval either does ordoes not contain it. In this sense “confidence“ applies not to theparticular inte
11、rval but only to the long run proportion of caseswhen repeating the procedure many times.3.1.7 confidence level, nthe value, 1 , of the probabilityassociated with a confidence interval, often expressed as apercentage.3.1.7.1 Discussion is generally a small number. Confi-dence level is often 95 % or
12、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.3.1.10 estimate,
13、nsample statistic used to approximate apopulation parameter.3.1.11 histogram, ngraphical representation 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 E1
14、1 on Quality andStatistics and is the direct responsibility of Subcommittee E11.10 on Sampling /Statistics.Current edition approved Oct. 1, 2012. Published November 2012. Originallyapproved in 2007. Last previous edition approved in 2012 as E2586 12a. DOI:10.1520/E2586-12B.2For referenced ASTM stand
15、ards, 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.3Available from American National Standards Institute (ANSI), 25 W. 43rd St.,4th Floor,
16、 New York, NY 10036, http:/www.ansi.org.Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States13.1.11.1 DiscussionWhile not required, equal bar or classwidths are recommended for histograms.3.1.12 interquartile range, IQR, nthe 75thpercentil
17、e (0.75quantile) minus the 25thpercentile (0.25 quantile), for a dataset.3.1.13 kurtosis, 2,g2,nfor 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 popu
18、lation applies) to thestandard deviation (sample, s, or population, ) raised to thefourth power, 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
19、 by the samplesize.3.1.15 median,X,nthe 50thpercentile in a population orsample.3.1.15.1 DiscussionThe 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-mu
20、m values in a sample.3.1.17 order statistic, x(k),nvalue of the kthobserved valuein a sample 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 p
21、ercentile, nquantile of a sample or a population,for which the fraction less than or equal to 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
22、 population. ISO 3534-23.1.22 statistic, nsee sample statistic.3.1.23 quantile, nvalue such 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,take
23、n from a larger collection of observations or test results,which serves to provide information 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 va
24、lues of a sample.3.1.28 skewness, 1,g1,nfor population or sample, ameasure of symmetry of 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, ) raised to the third power
25、.3.1.29 standard errorstandard deviation of the populationof values of a sample statistic 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 dev
26、iationof a population, , the squareroot of the average or expected value of the squared 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, 2,s2,nsquare of th
27、e standard deviationof the population or sample.3.1.31.1 DiscussionFor a finite population, 2is calcu-lated as the sum of squared deviations of values from the mean,divided by n. For a continuous population, 2is calculated byintegrating (x )2with respect to the density function. For asample, s2is ca
28、lculated as the sum of the squared deviations ofobserved values from their average 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 asa
29、mple of n observations that arrive in the form of a data set.Large data sets from organizations, 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
30、 generate considerable amounts of empirical data.4.1.1 A data set containing a single 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 thevariab
31、le being studied or observed. We may refer to eachobservation of a variable as an item 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
32、 essentially unlimited set of items, or a process. In aprocess, the items originate 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 ofprimar
33、y interest in any particular study.4.2 The data (measurements and observations) may 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
34、 consist of a sequence of 0s and 1s inwhich a “1” indicates that the inspected item 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 c
35、ount of events over some interval, thenumber of times the event is observed on the inspectionE2586 12b2interval 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
36、, the process of sampling and data collectionmust be considered, at least potentially, repeatable. Descriptivestatistics are calculated using real sample data that will vary inrepeating the sampling process. As such, a statistic is a randomvariable subject to variation in its own right. The samplest
37、atistic usually has a corresponding parameter in the popula-tion that is unknown (see 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 metho
38、ds for summarizing a set of data. The methodsconsidered in this practice are used for 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, quant
39、ile, interquartilerange, variance, standard deviation, Z-score, coefficient ofvariation, skewness and kurtosis, and standard error.4.6 Tabular methods described in this practice are:4.6.1 Frequency distribution, relative frequencydistribution, cumulative frequency distribution, and cumulativerelativ
40、e frequency distribution.4.7 Graphical methods described in this practice are:4.7.1 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 l
41、ittle value from the standpoint ofinterpretation unless the data quality is acceptable 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 u
42、nder essentially the same test conditions, on amaterial or product, all of which 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 tw
43、o or moresamples collected under different test conditions or represent-ing material 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
44、 suchsubgroups, representing significantly different conditions, maylead to a 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 stat
45、istical control.4.9 The methods developed in Sections 6, 7, and 8 apply tothe 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 conta
46、ining n observa-tions. The data set may be denoted as:x1, x2, x3 xn(1)4.9.1 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
47、. Populations may be finite or unlimited in sizeand may be existing or continuing 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). T
48、he density function visually describes the shape of thedistribution as for 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 equivale
49、nt 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 # t! 5 *stfx!dx (2)5.1.2 A great variety of distribution shapes are theoreticallypossible. When the curve is symmetric, we say that thedistribution is symmetric; otherwise, it is asymmetric. Adistribution hav
