ASTM D4460-22a

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Designation: D4460 − 22 D4460 − 22a
Standard Practice for
Calculating Precision Limits Where Values Are Calculated
from Other Test Methods
This standard is issued under the fixed designation D4460; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. Scope
1.1 Material and mixture properties such as air voids and voids in mineral aggregates (VMA) are calculated from two or three test
results, combined in simple mathematical relationships. The standard deviation equations for these calculated values can be
developed using a mathematical process called “propagation of errors” (also called “propagation of uncertainty”). This practice
includes uncertainty equations for four forms or material and mixture equations: when two test results are (1) added or subtracted,
(2) multiplied together, (3) one divided by the other, and (4) two test results divided by a third.
1.2 This approach to calculating standard deviation equations is only valid when the distributions of the test results from the two
standards are independent (that is, not correlated).
1.3 The accuracy of a calculated standard deviation is dependent on the accuracy of the standard deviations used for the individual
test result methods.
1.4 Values for the mean and standard deviation for each test method are needed to determine the standard deviation for a calculated
value.
1.5 Examples of how to use these equations are shown in Appendix X1.
1.6 A brief explanation of how standard deviation equations are derived for more complicated material and mixture equations is
also included.
1.7 The text of this standard references notes and footnotes which provide explanatory material. These notes and footnotes
(excluding those in tables and figures) shall not be considered as requirements of the standard.
1.8 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility
of the user of this standard to establish appropriate safety, health, and environmental practices and determine the applicability of
regulatory limitations prior to use.
1.9 This international standard was developed in accordance with internationally recognized principles on standardization
established in the Decision on Principles for the Development of International Standards, Guides and Recommendations issued
by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
This practice is under the jurisdiction of ASTM Committee D04 on Road and Paving Materials and is the direct responsibility of Subcommittee D04.94 on Statistical
Procedures and Evaluation of Data.
Current edition approved June 15, 2022Nov. 1, 2022. Published June 2022November 2022. Originally approved in 1985. Last previous edition approved in 20152022 as
D4460 – 97 (2015).D4460 – 22. DOI: 10.1520/D4460-22.10.1520/D4460-22A.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
D4460 − 22a
2. Referenced Documents
2.1 ASTM Standards:
C127 Test Method for Relative Density (Specific Gravity) and Absorption of Coarse Aggregate
C128 Test Method for Relative Density (Specific Gravity) and Absorption of Fine Aggregate
D1188/D1188M Test Method for Bulk Specific Gravity and Density of Compacted Asphalt Mixtures Using Coated Samples
D2172/D2172M Test Methods for Quantitative Extraction of Asphalt Binder from Asphalt Mixtures
D2726/D2726M Test Method for Bulk Specific Gravity and Density of Non-Absorptive Compacted Asphalt Mixtures
D4125/D4125M Test Methods for Asphalt Content of Asphalt Mixtures by the Nuclear Method
D6307 Test Method for Asphalt Content of Asphalt Mixture by Ignition Method
D6752/D6752M Test Method for Bulk Specific Gravity and Density of Compacted Asphalt Mixtures Using Automatic Vacuum
Sealing Method
E177 Practice for Use of the Terms Precision and Bias in ASTM Test Methods
3. Terminology
3,4,5
3.1 For definitions of terms used in this document, consult Practice E177, or a standard dictionary, or a statistical text.
4. Significance and Use
4.1 Precision statements for calculated values can be developed using this approach. Users can also evaluate how an individual
test method’s precision influences the variability of calculated values.
4.2 The standard deviation of a calculated value that is the sum, difference, product, or quotient of two or more test method results,
each with their own precision statement, can be calculated so long as the individual variables (that is, test results) are independent
and the standard deviations are small relative to their mean values. These restrictions are usually met in ASTM methods. In those
cases where these restrictions are not met, other methods can be used. Only cases complying with the restrictions are covered in
this standard.
5. Procedure
5.1 Standard deviations that can be used for precision limits for a calculated value can be calculated from the equations in this
section. The appropriate equation format is selected based on how independent individual test results are combined to calculate
the property. Examples in Appendix X1 illustrate how the equations are used.
5.2 Addition or Subtraction of Two Test Results:
5.2.1 This subsection applies when the equation for the calculated values uses the general formula:
value 5 x1y (1)
where:
x = result from first test method, and
y = result from second test method.
5.2.2 The standard deviation of the calculated value can be estimated with the following equation:
2 2
σ 5=σ 1σ (2)
x6y x y
where:
σ = standard deviation for calculated values based on either an addition or subtraction of test results,
x6y
For referenced ASTM standards, visit the ASTM website, www.astm.org, or contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM Standards
volume information, refer to the standard’s Document Summary page on the ASTM website.
Geary, R. C., “The Frequency Distribution of a Quotient,” Journal of the Royal Statistical Society, Vol 93, 1930, pp. 442–446.
Fieller, E. C., “The Distribution of the Index in a Normal Bivariate Population,” Biometrika, Vol 24, 1932, pp. 428–440.
Ku, H. H., “Notes on the Use of Propagation of Error Formulas,” Journal of Research of the National Bureau of Standards, Vol 70C, No. 4, 1966, pp. 331–341.
D4460 − 22a
σ = standard deviation for first test method, and
x
σ = standard deviation for second test method.
y
5.3 Product of Two Test Results:
5.3.1 This subsection applies when the equation for the calculated values uses the general formula:
value 5 x 3y (3)
5.3.2 The standard deviation of the calculated value can be estimated with the following equation:
2 2 2 2
σ 5= y¯ σ 1x¯ σ (4)
xy x y
where:
σ = standard deviation for calculated values based on the product of two other test results,
xy
σ = standard deviation for first test method,
x
x¯ = mean or av
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