1、Designation: D4460 97 (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 ca
2、se 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
3、 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 regu
4、latory 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
5、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. Signific
6、ance 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 stat
7、istical 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 c
8、an 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
9、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 base
10、d 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 t
11、he 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
12、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 ap
13、proved 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 AS
14、TMStandards 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 Populatio
15、n,”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
16、 States1y= standard deviation from precision statement of otherstandard on which new standard is based, andy = mean or average value of Y variable. xy5y2x21x2y2y4(3)wherexy = standard deviation for determining precision limits oftest results for a new standard based on the quotient oftwo test result
17、s 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 the
18、 percent air voids inpaving mixtures by using values obtained from Test MethodsD1188 and D2041. Both of these test methods have precisionstatements so a precision statement can be calculated for TestMethod D3203.X1.2 Calculating Air Voids:X1.2.1 The equation for calculating air voids is:Percent air
19、voids 5 100 1 2 bulk sp gr/theoretical maximum sp gr!(X1.1)The bulk specific gravity of Test Method D1188 is dividedby the theoretical maximum specific gravity of Test MethodD2041. Therefore, Eq 3 involving a quotient is the proper oneto use in determining a precision statement for air voids.X1.2.2
20、Test Method D1188 has one of the older precisionstatements which does not give the standard deviation but onlygives the D2S limit for multilaboratory precision. The testingoperations in Test Methods D1188 and D2041 are very similarso it is not surprising that the D2S limits for multilaboratoryprecis
21、ion are almost identical being 0.02 for Test MethodD1188 and 0.019 for Test Method D2041. Therefore, the tablefor nonporous aggregate from Test Method D2041 is used asthe basis for both test methods as follows:Standard DeviationAcceptable Range ofTwo ResultsTest and type index (1S) (D2S)Single-opera
22、tor precision 0.0040 0.011Multilaboratory precision 0.0067 0.019During a testing program, the bulk specific gravity (TestMethod D1188) of an asphalt mixture was determined to be2.423 and the theoretical maximum specific gravity (TestMethod D2041) was determined to be 2.523.X1.3 Eq 3 was used to calc
23、ulate the single-operator preci-sion standard deviation for these specific gravity values.Standard deviation for Test Method D3203 is as follows:Single-operator precision: xy52.523!20.004!212.423!20.004!22.523!4(X1.2)= 0.0022Multilaboratory precision: xy52.523!20.0064!212.423!20.0064!22.523!4(X1.3)=
24、 0.0035These standard deviations are in terms of specific gravity.Since Test Method D3203 is reported in percent they must bemultiplied by 100 to get the required percentages yielding thefollowing values:StandardDeviationRange ofTwo ResultsTest and Type Index (1S %) (D2S %)Single-operator precision
25、0.22 0.62Multilaboratory precision 0.35 0.99This is the complete precision statement for Test MethodD3203 where nonporous aggregates are used.D4460 97 (2015)2ASTM International takes no position respecting the validity of any patent rights asserted in connection with any item mentionedin this standa
26、rd. Users of this standard are expressly advised that determination of the validity of any such patent rights, and the riskof infringement of such rights, are entirely their own responsibility.This standard is subject to revision at any time by the responsible technical committee and must be reviewe
27、d every five years andif not revised, either reapproved or withdrawn. Your comments are invited either for revision of this standard or for additional standardsand should be addressed to ASTM International Headquarters. Your comments will receive careful consideration at a meeting of theresponsible
28、technical committee, which you may attend. If you feel that your comments have not received a fair hearing you shouldmake your views known to the ASTM Committee on Standards, at the address shown below.This standard is copyrighted by ASTM International, 100 Barr Harbor Drive, PO Box C700, West Consh
29、ohocken, PA 19428-2959,United States. Individual reprints (single or multiple copies) of this standard may be obtained by contacting ASTM at the aboveaddress or at 610-832-9585 (phone), 610-832-9555 (fax), or serviceastm.org (e-mail); or through the ASTM website(www.astm.org). Permission rights to photocopy the standard may also be secured from the Copyright Clearance Center, 222Rosewood Drive, Danvers, MA 01923, Tel: (978) 646-2600; http:/ 97 (2015)3
