ASTM_D_4686_-_91_2003.pdf

Designation:D 4686 91(Reapproved 2003)Standard Guide forIdentification and Transformation of FrequencyDistributions1This standard is issued under the fixed designation D 4686;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 parentheses indicates the year of last reapproval.Asuperscript epsilon(e)indicates an editorial change since the last revision or reapproval.1.Scope1.1 This guide gives the rudiments of identification of someof the most common and useful frequency distributions.It doesnot give rigorous identification.To achieve exactitude,theprocedures similar to those given by Shapiro2should be used.1.2 This guide provides a key to identify frequency distri-butions.1.3 This guide gives ways to select the proper transforma-tion to use to transform a particular set of data to one which canbe modeled by the normal distribution,if such a transformationcan be found at all.1.4 This guide includes the following topics:Topic TitleSectionScope1Referenced Documents2Terminology3Significance and Use4Key to Distributions5Binomial Frequency Distribution6Poisson Frequency Distribution7Normal Frequency Distribution8Sample Average Distribution9Other Distributions10Transformations of Data112.Referenced Documents2.1 ASTM Standards:3D 123 Terminology Relating to TextilesD 4392 Terminology for Statistically Related TermsE 456 Terminology Relating to Quality and Statistics2.2 ASTM Adjuncts:TEX-PAC4NOTE1Tex-Pac is a group of PC programs on floppy disks,availablethrough ASTM Headquarters,100 Barr Harbor Drive,Conshohocken,PA19428,USA.Many of the transformations described in this Guide can bemade using a program in this adjunct.3.Terminology3.1 Definitions:3.1.1 Bernoulli distributionsee binomial distribution.3.1.2 binomial distribution,nthe frequency distributionwhich has the probability function:Pr!5n!/r!n 2 r!#!prqn2r(1)where:P(r)=the probability of obtaining exactly r“successes”inn independent trials,p=the probability,constant from trial to trial,of obtain-ing a“success”in a single trial,andq=1 p.(Syn.Bernoulli distribution)3.1.3 distributionsee frequency distribution of a sampleand frequency distribution of a population.3.1.4 frequency distribution,nof a population,a functionthat,for a specific type of distribution,gives for each value ofa random discrete variate,or each group of values of a randomcontinuous variate,the corresponding probability of occur-rence.3.1.5 frequency distribution,nof a sample,a table givingfor each value of a discrete variate,or for each group of valuesof a continuous variate,the corresponding number of observa-tions.3.1.6 normal distribution,nthe distribution that has theprobability function:fx!51/s!2p!1/2exp2x 2!2/2s2#(2)where:x=a random variate,=the mean of the distribution,ands=the standard deviation of the distribution.(Syn.Gaussian distribution,law of error)3.1.7 Poisson distribution,nthe distribution which has asits probability function:Pr!5 e2r/r!(3)1This guide is under the jurisdiction of ASTM Committee D13 on Textiles andis the direct responsibility of Subcommittee D13.93 on Statistics.Current edition approved July 15,1991.Published September 1991.Originallypublished as D 4686 87.Last previous edition D 4686 90.2Shapiro,Samuel S.,How to Test Normality and Other Distribution Assump-tions,American Society for Quality Control,Milwaukee,WI,1980.Vol.3 of theseries,The ASQC Basic References in Quality Control:Statistical Techniques,Edward J.Dudewitz,ed.3For referenced ASTM standards,visit the ASTM website,www.astm.org,orcontact ASTM Customer Service at serviceastm.org.ForAnnual Book of ASTMStandards volume information,refer to the standards Document Summary page onthe ASTM website.4PC programs on floppy disks are available through ASTM.For a 312inch diskrequest PCN:12-429040-18,for a 514inch disk request PCN:12-429041-18.1Copyright ASTM International,100 Barr Harbor Drive,PO Box C700,West Conshohocken,PA 19428-2959,United States.where:P(r)=probability of obtaining exactly r occurrences of anevent in one unit,such as a unit of time or area,=both mean and variance of distribution,ande=base of natural logarithms.3.1.8 probability function,nof a continuous variate,themathematical expression whose definite integral gives theprobability that a variate will take a value within the two limitsof integration.3.1.9 probability function,nof a discrete variate,themathematical expression which gives the probability that avariate will take a particular value.3.1.10 transformation,nthe change from one set of vari-ables,x,to another set,y,by the use of a function,y=f(x).3.1.11 For definitions of textile terms used in this guide,refer to Terminology D 123.For definitions of other statisticalterms used in this guide,refer to Terminology D 4392,Termi-nology E 456,or appropriate textbooks on statistics.4.Significance and Use4.1 In measuring,testing,and experimenting,statisticaltests are made to determine whether