1、Designation: E2586 16 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 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 DefinitionsUnless otherw
7、ise noted, terms relating toquality and statistics are as defined in Terminology E456.3.1.1 characteristic, na property of items in a sample orpopulation which, when measured, counted, or otherwiseobserved, helps to distinguish among the items. E22823.1.2 coeffcient of variation, CV, nfor a nonnegat
8、ivecharacteristic, the ratio of the standard deviation to the meanfor a population or sample3.1.2.1 DiscussionThe coefficient of variation is oftenexpressed as a percentage.3.1.2.2 DiscussionThis statistic is also known as therelative standard deviation, RSD.3.1.3 confidence bound, nsee confidence l
9、imit.3.1.4 confidence coeffcient, nsee confidence level.3.1.5 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) 1.3.1.5.1 DiscussionThe confidence level, 1 , reflects theproportion of cases that the confi
10、dence 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 interval but
11、 only to the long run proportion of caseswhen repeating the procedure many times.3.1.6 confidence level, nthe value, 1 , of the probabilityassociated with a confidence interval, often expressed as apercentage.3.1.6.1 Discussion is generally a small number. Confi-dence level is often 95 % or 99 %.3.1
12、.7 confidence limit, neach of the limits, L and U, of aconfidence interval, or the limit of a one-sided confidenceinterval.3.1.8 degrees of freedom, nthe number of independentdata points minus the number of parameters that have to beestimated before calculating the variance.3.1.9 estimate, nsample s
13、tatistic used to approximate apopulation parameter.3.1.10 histogram, ngraphical representation of the fre-quency distribution of a characteristic consisting of a set ofrectangles with area proportional to the frequency. ISO 3534-13.1.10.1 DiscussionWhile not required, equal bar or classwidths are re
14、commended for histograms.1This practice is under the jurisdiction of ASTM Committee E11 on Quality andStatistics and is the direct responsibility of Subcommittee E11.10 on Sampling /Statistics.Current edition approved Nov. 1, 2016. Published November 2016. Originallyapproved in 2007. Last previous e
15、dition approved in 2014 as E2586 14. DOI:10.1520/E2586-16.2For referenced ASTM standards, 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.3Av
16、ailable from American National Standards Institute (ANSI), 25 W. 43rd St.,4th Floor, 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 interquartile range, IQR, nthe 75thpercentile (0.75quan
17、tile) minus the 25thpercentile (0.25 quantile), for a dataset.3.1.12 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 population appl
18、ies) to thestandard deviation (sample, s, or population, ) raised to thefourth power, minus 3 (also referred to as excess kurtosis).3.1.13 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 sam
19、plesize.3.1.14 median, X,nthe 50thpercentile in a population orsample.3.1.14.1 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.3.1.15 midrange, naverage of the minimum and maxi-mum values i
20、n a sample.3.1.16 order statistic, x(k),nvalue of the kthobserved valuein a sample after sorting by order of magnitude.3.1.16.1 DiscussionFor a sample of size n, the first orderstatistic x(1)is the minimum value, x(n)is the maximum value.3.1.17 parameter, nsee population parameter.3.1.18 percentile,
21、 nquantile of a sample or a population,for which the fraction less than or equal to the value isexpressed as a percentage.3.1.19 population, nthe totality of items or units ofmaterial under consideration.3.1.20 population parameter, nsummary measure of thevalues of some characteristic of a populatio
22、n. ISO 3534-23.1.21 prediction interval, nan interval for a future valueor set of values, constructed from a current set of data, in a waythat has a specified probability for the inclusion of the futurevalue.3.1.22 residual, nobserved value minus fitted value, whena model is used.3.1.23 statistic, n
23、see sample statistic.3.1.24 quantile, nvalue such that a fraction f of the sampleor population is less than or equal to that value.3.1.25 range, R, nmaximum value minus the minimumvalue in a sample.3.1.26 sample, na group of observations or test results,taken from a larger collection of observations
24、 or test results,which serves to provide information that may be used as a basisfor making a decision concerning the larger collection.3.1.27 sample size, n, nnumber of observed values in thesample.3.1.28 sample statistic, nsummary measure of the ob-served values of a sample.3.1.29 skewness, 1,g1,nf
25、or 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.3.1.30 standard errorstandard deviation
26、of the populationof values of a sample statistic in repeated sampling, or anestimate of it.3.1.30.1 DiscussionIf the standard error of a statistic isestimated, it will itself be a statistic with some variance thatdepends on the sample size.3.1.31 standard deviationof a population, , the squareroot o
27、f 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 from their mean divided by the sample sizeminus 1.3.1.32 variance, 2,s2,nsquare of the standard deviationof the
28、 population or sample.3.1.32.1 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 calculated as the sum of the
29、 squared deviations ofobserved values from their average divided by one less than thesample size.3.1.33 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 tha
30、t 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 generate considerable amo
31、unts 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 thevariable being studied or observ
32、ed. 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 essentially unlimited set
33、 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 ofprimary interest in any particul
34、ar 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 consist of a sequence of
35、0s and 1s inwhich a “1” indicates that the inspected item exhibited theattribute being studied and a “0” indicates the item did notE2586 162exhibit 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 ov
36、er some interval, thenumber of times the event is observed on the 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 a
37、nd 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 samplestatistic usually has a corre
38、sponding 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 methods for summarizing a set of
39、 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, quantile, interquartilerange, va
40、riance, 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 cumulativerelative frequency distribution.4.
41、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 little value from the standp
42、oint 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 under essentially the same t
43、est 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 two or moresamples collected
44、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 suchsubgroups, representin
45、g 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 statistical control.4.9 The met
46、hods 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 containing n observa-tions. The
47、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. Populations may be finite
48、 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). The density function visuall
49、y 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 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 a
