ASTM_D_4460_-_97_2015.pdf

Designation:D446097(Reapproved 2015)Standard Practice forCalculating Precision Limits Where Values are Calculatedfrom Other Test Methods1This standard is issued under the fixed designation D4460;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()indicates an editorial change since the last revision or reapproval.1.Scope1.1 This practice covers techniques for calculating precisionlimits when values are calculated from two other methodshaving precision limits.1.2 This standard does not purport to address all of thesafety concerns,if any,associated with its use.It is 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:2D1188 Test Method for Bulk Specific Gravity and Density ofCompacted Bituminous Mixtures Using Coated SamplesD2041 Test Method for Theoretical Maximum SpecificGravity and Density of Bituminous Paving MixturesD3203 Test Method for Percent Air Voids in CompactedDense and Open Bituminous Paving MixturesE177 Practice for Use of the Terms Precision and Bias inASTM Test Methods3.Definitions3.1 For definitions of terms used in this document,consultPractice E177,or a standard dictionary,or a statistical text.3,4,54.Significance and Use4.1 Precision limits for a test result that is calculated byaddition,subtraction,multiplication,or division of two othertest results that have valid precision limits can be calculateddirectly.This saves the cost and delay of conducting aninterlaboratory study.4.2 At the heart of statistical theory is the concept of afrequency distribution of a random variable.The precisionlimit of the random variable is determined by the standarddeviation of the variable.The standard deviation of a randomvariable that is the sum,difference,product,or quotient of twoother random variables can be calculated simply so long as theindividual variables are independent and the standard devia-tions are small relative to their mean values.These restrictionsare usually met in ASTM methods.In those cases where theserestrictions are not met,other methods can be used.Only casescomplying with the restrictions are covered in this standard.5.Procedure5.1 The standard deviation on which precision limits for atest result are based can be calculated from the followingequations:x6y5=x21y2(1)where:x6y=standard deviation for determining precision limits ofa test result for a new standard based on either anaddition or subtraction of test results from two otherstandards,x=standard deviation from precision statement of one ofthe standards on which new standard is based,andy=standard deviation from precision statement of otherstandard on which new standard is based.The distributions of the test results from the two standardsshould be independent.xy5=y2x21x2y2(2)where:xy=standard deviation for determining precision limits oftest results for a new standard based on the products oftwo other test results from two other standards,x=standard deviation from precision statement of one ofthe standards on which new standard is based,x=mean or average value of X variable,1This practice is under the jurisdiction of ASTM Committee D04 on Road andPaving Materials and is the direct responsibility of Subcommittee D04.94 onStatistical Procedures and Evaluation of Data.Current edition approved Jan.1,2015.Published February 2015.Originallyapproved in 1985.Last previous edition approved in 2009 as D4460 97(2009).DOI:10.1520/D4460-97R15.2For 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.3Geary,R.C.,“The Frequency Distribution of a Quotient,”Journal of the RoyalStatistical Society,Vol 93,1930,pp.442446.4Fieller,E.C.,“The Distribution of the Index in a Normal Bivariate Population,”Biometrika,Vol 24,1932,pp.428440.5Ku,H.H.,“Notes on the Use of Propagation of Error Formulas,”Journal ofResearch of the National Bureau of Standards,Vol 70C,No.4,1966,pp.331341.Copyright ASTM International,100 Barr Harbor Drive,PO Box C700,West Conshohocken,PA 19428-2959.United States1 y=standard deviation from precision statement of otherstandard on which new standard is based,andy=mean or average value of Y variable.xy5y 2x21x 2y2y 4(3)wherexy=standard deviation for determining precision limits oftest results for a new standard based on the quotient oftwo test results from two other standards.x,y,x,y 5 definitions given above.(4)An example in Appendix X1.illustrates how the equationsare applied.6.Keywords6.1 precision limits;standard deviationAPPENDIX(Nonmandatory Information)X1.EXAMPLE OF CALCULATED PRECISION LIMITSX1.1 Test Method D3203 calculates t