ASTM E122-17
(Practice)Standard Practice for Calculating Sample Size to Estimate, With Specified Precision, the Average for a Characteristic of a Lot or Process
Standard Practice for Calculating Sample Size to Estimate, With Specified Precision, the Average for a Characteristic of a Lot or Process
ABSTRACT
This practice covers simple methods for calculating how many units to include in a random sample in order to estimate with a specified precision, a measure of quality for all the units of a lot of material or produced by a process. It also treats the common situation where the sampling units can be considered to exhibit a single source of variability; it does not treat multi-level sources of variability.
This practice is intended for use in determining the sample size required to estimate, with specified precision, a measure of quality of a lot or process. It applies when quality is expressed as either the lot average for a given property, or as the lot fraction not conforming to prescribed standards. The level of a characteristic may often be taken as an indication of the quality of a material.
SIGNIFICANCE AND USE
4.1 This practice is intended for use in determining the sample size required to estimate, with specified precision, a measure of quality of a lot or process. The practice applies when quality is expressed as either the lot average for a given property, or as the lot fraction not conforming to prescribed standards. The level of a characteristic may often be taken as an indication of the quality of a material. If so, an estimate of the average value of that characteristic or of the fraction of the observed values that do not conform to a specification for that characteristic becomes a measure of quality with respect to that characteristic. This practice is intended for use in determining the sample size required to estimate, with specified precision, such a measure of the quality of a lot or process either as an average value or as a fraction not conforming to a specified value.
SCOPE
1.1 This practice covers simple methods for calculating how many units to include in a random sample in order to estimate with a specified precision, a measure of quality for all the units of a lot of material, or produced by a process. This practice will clearly indicate the sample size required to estimate the average value of some property or the fraction of nonconforming items produced by a production process during the time interval covered by the random sample. If the process is not in a state of statistical control, the result will not have predictive value for immediate (future) production. The practice treats the common situation where the sampling units can be considered to exhibit a single (overall) source of variability; it does not treat multi-level sources of variability.
1.2 The system of units for this standard is not specified. Dimensional quantities in the standard are presented only as illustrations of calculation methods. The examples are not binding on products or test methods treated.
1.3 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.
General Information
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Designation:E122 −17 An American National Standard
Standard Practice for
Calculating Sample Size to Estimate, With Specified
Precision, the Average for a Characteristic of a Lot or
1
Process
This standard is issued under the fixed designation E122; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision.Anumber in parentheses indicates the year of last reapproval.A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
This standard has been approved for use by agencies of the U.S. Department of Defense.
1. Scope 3. Terminology
3.1 Definitions—Unlessotherwisenoted,allstatisticalterms
1.1 Thispracticecoverssimplemethodsforcalculatinghow
are defined in Terminology E456.
many units to include in a random sample in order to estimate
3.1.1 pooled standard deviation, s,n—the estimate of a
with a specified precision, a measure of quality for all the units
p
standard deviation derived by combining sample standard
ofalotofmaterial,orproducedbyaprocess.Thispracticewill
deviations of several samples, weighting squared standard
clearly indicate the sample size required to estimate the
deviations by their degrees of freedom.
average value of some property or the fraction of nonconform-
ing items produced by a production process during the time
3.2 Symbols—Symbols used in all equations are defined as
interval covered by the random sample. If the process is not in
follows:
a state of statistical control, the result will not have predictive
E = the maximum acceptable difference between the true
valueforimmediate(future)production.Thepracticetreatsthe
average and the sample average.
common situation where the sampling units can be considered
e = E/µ, maximum acceptable difference expressed as a
to exhibit a single (overall) source of variability; it does not
fraction of µ.
treat multi-level sources of variability.
f = degrees of freedom for a standard deviation estimate
1.2 The system of units for this standard is not specified.
(7.5).
Dimensional quantities in the standard are presented only as
k = thetotalnumberofsamplesavailablefromthesameor
illustrations of calculation methods. The examples are not
similar lots.
binding on products or test methods treated.
µ = lot or process mean or expected value of X, the result
of measuring all the units in the lot or process.
1.3 This international standard was developed in accor-
µ = an advance estimate of µ.
0
dance with internationally recognized principles on standard-
N = size of the lot.
ization established in the Decision on Principles for the
n = size of the sample taken from a lot or process.
Development of International Standards, Guides and Recom-
n = size of sample j.
j
mendations issued by the World Trade Organization Technical
n = size of the sample from a finite lot (7.4).
L
Barriers to Trade (TBT) Committee.
p' = fraction of a lot or process whose units have the
nonconforming characteristic under investigation.
2. Referenced Documents
p = an advance estimate of p'.
0
2
p = fraction nonconforming in the sample.
2.1 ASTM Standards:
R = rangeofasetofsamplingvalues.Thelargestminusthe
E456Terminology Relating to Quality and Statistics
smallest observation.
R = range of sample j.
j
k
¯
R =
R /k , average of the range of k samples, all of the
(
1 j
This practice is under the jurisdiction ofASTM Committee E11 on Quality and
j51
Statistics and is the direct responsibility of Subcommittee E11.10 on Sampling /
same size (8.2.2).
Statistics.
σ = lot or process standard deviation of X, the result of
Current edition approved April 1, 2017. Published April 2017. Originally
ɛ1
measuring all of the units of a finite lot or process.
approved in 1958. Last previous edition approved in 2009 as E122–09 . DOI:
10.1520/E0122-17. σ = an advance estimate of σ.
0
n 1⁄2
2
For referenced ASTM standards, visit the ASTM website, www.astm.org, or 2
s =
H
~X 2X! / n 2 1 , an estimate of the standard
F ~ !G
( i
contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
i51
Standards volume information, refer to the standard’s Document Summary page on
deviation σ from n observation, X, i=1 to n.
i
the ASTM website.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
1
---------------------- Page: 1 ----------------------
E122−17
k
5.4 The precision of the estimate made from a random
s¯ =
S /k , average s from k samples all of the same size
( j
sample may itself be estimated from the sample. This estima-
j51
(8.2.1). tion of the precision from one sample makes it possible to fix
s = pooled (weighted average) s from k samples, not all of
more economically the sample size for the next sample of a
p
the same size (8.2).
similar material. In other w
...
This document is not an ASTM standard and is intended only to provide the user of an ASTM standard an indication of what changes have been made to the previous version. Because
it may not be technically possible to adequately depict all changes accurately, ASTM recommends that users consult prior editions as appropriate. In all cases only the current version
of the standard as published by ASTM is to be considered the official document.
´1
Designation: E122 − 09 E122 − 17 An American National Standard
Standard Practice for
Calculating Sample Size to Estimate, With Specified
Precision, the Average for a Characteristic of a Lot or
1
Process
This standard is issued under the fixed designation E122; 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.
This standard has been approved for use by agencies of the U.S. Department of Defense.
1
ε NOTE—Editorial corrections were made to 8.4.1.2 in November 2011.
1. Scope
1.1 This practice covers simple methods for calculating how many units to include in a random sample in order to estimate with
a specified precision, a measure of quality for all the units of a lot of material, or produced by a process. This practice will clearly
indicate the sample size required to estimate the average value of some property or the fraction of nonconforming items produced
by a production process during the time interval covered by the random sample. If the process is not in a state of statistical control,
the result will not have predictive value for immediate (future) production. The practice treats the common situation where the
sampling units can be considered to exhibit a single (overall) source of variability; it does not treat multi-level sources of
variability.
1.2 The system of units for this standard is not specified. Dimensional quantities in the standard are presented only as
illustrations of calculation methods. The examples are not binding on products or test methods treated.
1.3 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.
2. Referenced Documents
2
2.1 ASTM Standards:
E456 Terminology Relating to Quality and Statistics
3. Terminology
3.1 Definitions—Unless otherwise noted, all statistical terms are defined in Terminology E456.
3.1.1 pooled standard deviation, s , n—the estimate of a standard deviation derived by combining sample standard deviations
p
of several samples, weighting squared standard deviations by their degrees of freedom.
3.2 Symbols—Symbols used in all equations are defined as follows:
E = the maximum acceptable difference between the true average and the sample average.
e = E/μ, maximum acceptable difference expressed as a fraction of μ.
f = degrees of freedom for a standard deviation estimate (7.5).
k = the total number of samples available from the same or similar lots.
μ = lot or process mean or expected value of X, the result of measuring all the units in the lot or process.
μ = an advance estimate of μ.
0
N = size of the lot.
n = size of the sample taken from a lot or process.
n = size of sample j.
j
1
This practice is under the jurisdiction of ASTM Committee E11 on Quality and Statistics and is the direct responsibility of Subcommittee E11.10 on Sampling / Statistics.
Current edition approved Aug. 1, 2009April 1, 2017. Published September 2009April 2017. Originally approved in 1958. Last previous edition approved in 20072009 as
ɛ1
E122 – 07.E122 – 09 . DOI: 10.1520/E0122-09E01.10.1520/E0122-17.
2
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.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
1
---------------------- Page: 1 ----------------------
E122 − 17
n = size of the sample from a finite lot (7.4).
L
p' = fraction of a lot or process whose units have the nonconforming characteristic under investigation.
p = an advance estimate of p'.
0
p = fraction nonconforming in the sample.
R = range of a set of sampling values. The largest minus the smallest observation.
R = range of sample j.
j
k
R¯ =
R /k , average of the range of k samples, all of the same size (8.2.2).
(
j
j51
σ = lot or process standard deviation of X, the result of measuring all of the units of a finite lot or process.
σ = an advance estimate of σ.
0
n
2
s =
H 1⁄2
~ !
F X 2X /~n21! G , an estimate of the standard de
...
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