1、Designation: E2586 10Standard 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 last revision. A number in pare
2、ntheses 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 formulas for comput-ing and presenting basic descriptive statistics using a set ofsample data containing a single va
3、riable. 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 practice is not specified.Dimen
4、sional 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 theresponsibility of the user o
5、f 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 Quality and StatisticsE2282 Guide
6、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: Unless otherwise noted, terms relating toq
7、uality 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 nonnegativecharacteristic, the ratio
8、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 estimate, nsample statistic used to approximate apopulation para
9、meter.3.1.4 histogram, ngraphical representation of the fre-quency distribution of a characteristic consisting of a set ofrectangles with area proportional to the frequency.ISO 3534-13.1.4.1 DiscussionWhile not required, equal bar or classwidths are recommended for histograms.3.1.5 interquartile ran
10、ge, IQR, nthe 75thpercentile (0.75quantile) minus the 25thpercentile (0.25 quantile), for a dataset.3.1.6 kurtosis, g2,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 s
11、ample, theoretical if a population applies) to thestandard deviation (sample, s, or population, s) raised to thefourth power, minus 3 (also referred to as excess kurtosis).3.1.7 mean, nof a population, , average or expectedvalue of a characteristic in a population of a sample, x, sumof the observed
12、values in the sample divided by the samplesize.3.1.8 median, X , nthe 50thpercentile in a population orsample.3.1.8.1 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.3.1.9 midrange, naverag
13、e of the minimum and maximumvalues in a sample.3.1.10 order statistic, x(k), nvalue of the kthobservedvalue in a sample after sorting by order of magnitude.3.1.10.1 DiscussionFor a sample of size n, the first orderstatistic x(1)is the minimum value, x(n)is the maximum value.3.1.11 parameter, nsee po
14、pulation parameter.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 May 15, 2010. Published August 2010. Originallyapproved in 2007. Last previous edition a
15、pproved in 2007 as E2586 07. DOI:10.1520/E2586-10.2For 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.3Available
16、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 C700, West Conshohocken, PA 19428-2959, United States.3.1.12 percentile, nquantile of a sample or a population,for which th
17、e fraction less than or equal to the value isexpressed as a percentage.3.1.13 population, nthe totality of items or units ofmaterial under consideration.3.1.14 population parameter, nsummary measure of thevalues of some characteristic of a population. ISO 3534-23.1.15 statistic, nsee sample statisti
18、c.3.1.16 quantile, nvalue such that a fraction f of the sampleor population is less than or equal to that value.3.1.17 range, R, nmaximum value minus the minimumvalue in a sample.3.1.18 sample, na group of observations or test results,taken from a larger collection of observations or test results,wh
19、ich serves to provide information that may be used as a basisfor making a decision concerning the larger collection.3.1.19 sample size, n, nnumber of observed values in thesample3.1.20 sample statistic, nsummary measure of the ob-served values of a sample.3.1.21 skewness, g1,g1, nfor population or s
20、ample, 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, s) raised to the third power.3.1.22 standard deviationof a population, s, the squarero
21、ot of the average or expected value of the squared deviationof a variable from its mean of a sample, s, the square root ofthe sum of the squared deviations of the observed values in thesample divided by the sample size minus 1.3.1.23 variance, s2, s2, nsquare of the standard deviationof the populati
22、on or sample.3.1.23.1 DiscussionFor a finite population, 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 square
23、d deviations ofobserved values from their average divided by one less than thesample size.3.1.24 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 arriv
24、e 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 amounts of
25、 empirical data.4.1.1 Adata 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 observed. We m
26、ay 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 of item
27、s, 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 particular study
28、.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 0s and 1
29、s 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 count of events over some interval,
30、 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 and data collectio
31、nmust 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 corresponding paramete
32、r 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 data. The method
33、sconsidered 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, variance, standard
34、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 frequency distribu-tion, cumulative frequency distribution, and cumulative rela-tive frequency distribution.4.7 Graphical m
35、ethods 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 standpoint ofinterp
36、retation 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 test condition
37、s, 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.E2586 1024.8.1 If a given collection of data consists of two or moresamples collected unde
38、r 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 such sub-groups, representing
39、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 metho
40、ds 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 da
41、ta 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
42、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
43、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 va
44、riable 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 varie
45、ty of distribution shapes are theoreticallypossible. When the curve is symmetric, we say that thedistribution is symmetric; otherwise, it is asymmetric. Adistribution having a longer tail on the right side is called rightskewed; a distribution having a longer tail on the left is calledleft skewed.5.
46、1.3 For a given density function, f (x), the relationship tocumulative area under the curve may be graphically shown inthe form of a cumulative distribution function, F (x). Thefunction F (x) plots the cumulative area under f (x)asx movesto the right. Fig. 2 shows a symmetric distribution with itsde
47、nsity function, f (x), plotted on the left-hand axis anddistribution function, F (x), plotted on the right-hand axis.5.1.4 Referring to the F (x) axis in Fig. 2, observe that F(30) = 0.5. The point x = 30 divides the distribution into twoequal halves with respect to probability (50 % on each side of
48、x). In general, where F (x) = 0.5, we call the point x the medianor 50thpercentile of the distribution. In like manner, we maydefine any percentile, for example, the 25thor the 90thpercentiles. In general, for 0 0 as in Fig. 4. If the distribution has a longertail on the left, we say that it is left
49、 skewed and g10FIG. 5 Curve with Negative Skewness, g10TABLE 2 Skewness and Kurtosis for Selected Distribution FormsDistribution Form Skewness KurtosisNormal 0 0Exponential 2 6Uniform 0 1.2PoissonA1/=l 1/lStudents tB0 6/(v4)WeibullC, b = 3.6 0 0.28Weibull, b = 0.5 6.62 84.72Weibull, b = 50.0 1 1.9AFor the Poisson distribution, l is the mean.BFor the Students t distribution, v is the degrees of freedom. When v # 4,kurtosis is infinite.CFor the Weibull distribution, b is the shape parameter.E2586 105size, n, we take the average
