ASTM E122-07
(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
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.
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An American National Standard
Designation: E 122 – 07
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.
1. Scope
e = E/µ, maximum acceptable difference expressed as a
fraction of µ.
1.1 Thispracticecoverssimplemethodsforcalculatinghow
k = the total number of samples available from the same
many units to include in a random sample in order to estimate
or similar lots.
withaspecifiedprecision,ameasureofqualityforalltheunits
µ = lot or process mean or expected value of X, the result
ofalotofmaterial,orproducedbyaprocess.Thispracticewill
of measuring all the units in the lot or process.
clearly indicate the sample size required to estimate the
µ = an advance estimate of µ.
0
average value of some property or the fraction of nonconform-
N = size of the lot.
ing items produced by a production process during the time
n = size of the sample taken from a lot or process.
interval covered by the random sample. If the process is not in
n = size of sample j.
j
a state of statistical control, the result will not have predictive
n = size of the sample from a finite lot (7.4).
L
valueforimmediate(future)production.Thepracticetreatsthe
p8 = fraction of a lot or process whose units have the
common situation where the sampling units can be considered
nonconforming characteristic under investigation.
to exhibit a single (overall) source of variability; it does not
p = an advance estimate of p8.
0
treat multi-level sources of variability.
p = fraction nonconforming in the sample.
R = range of a set of sampling values. The largest minus
2. Referenced Documents
the smallest observation.
2
2.1 ASTM Standards:
R = range of sample j.
j
k
¯
E456 Terminology Relating to Quality and Statistics
R =
R/k, average of the range of k samples, all of the
(
j
j 51
3. Terminology
same size (8.2.2).
3.1 Definitions: Unless otherwise noted, all statistical terms
s = lot or process standard deviation of X, the result of
are defined in Terminology E456.
measuring all of the units of a finite lot or process.
3.2 Symbols: Symbols used in all equations are defined as
s = an advance estimate of s.
0
n
follows: s =
2 1/2
[ (X − X ) /(n−1)] , an estimate of the
( i
i 51
standarddeviation sfromnobservation,X,i=1ton.
i
E = the maximum acceptable difference between the true k
s¯ =
S/k,average sfrom ksamplesallofthesamesize
average and the sample average.
( j
j 51
(8.2.1).
s = pooled(weightedaverage)sfromksamples,notallof
p
1
ThispracticeisunderthejurisdictionofASTMCommitteeE11onQualityand
the same size (8.2).
Statistics and is the direct responsibility of Subcommittee E11.10 on Sampling.
s = standard deviation of sample j.
j
Current edition approved Oct. 1, 2007. Published November 2007 . Originally
th
t = a factor (the 99.865 percentile of the Student’s
published as E122–89. Last previous edition approved in 2000 as E122–00.
2
For referenced ASTM standards, visit the ASTM website, www.astm.org, or distribution) corresponding to the degrees of freedom
contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
f of an advance estimate s (5.1).
o o
Standards volume information, refer to the standard’s Document Summary page on
V = an advance estimate of V, equal to d /µ .
o o o
the ASTM website.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.
1
---------------------- Page: 1 ----------------------
E122–07
¯ 6. Precision Desired
v = s/ X, the coefficient of variation estimated from the
sample.
6.1 Theapproximateprecisiondesiredfortheestimatemust
v = coefficient of variation from sample j.
j
be prescribed. That is, it must be decided what maximum
X = numerical value of the characteristic of an individual
deviation, E, can be tolerated between the estimate to be made
unit being measured.
from the sample and the result that would be obtained by
n
¯
X =
measuring every unit in the lot or process.
X/n average of n observations, X,i=1 to n.
( i i i
i 51
6.2 Insomecases,themaximumallowablesamplingerroris
4. Significance and Use
expressed as a proportion, e, or a percentage, 100 e. For
example, one may wish to make an estimate of the sulfur
4.1 This practice is intended for use in determining the
content of coal within 1%, or e =0.01.
sample size required to estimate, with specified precision, a
measure of quality of a lot or process. The practice applies
7. Equations for Calculating Sample S
...
This document is not anASTM standard and is intended only to provide the user of anASTM 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.
An American National Standard
Designation:E122–00 Designation: E 122 – 07
Standard Practice for
Calculating Sample Size to Estimate, With a Specified
Tolerable Error,Precision, the Average for a Characteristic of
1
a Lot or 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 (e) indicates an editorial change since the last revision or reapproval.
This standard has been approved for use by agencies of the Department of Defense.
1. Scope
1.1 Thispracticecoverssimplemethodsforcalculatinghowmanyunitstoincludeinarandomsampleinordertoestimatewith
a prescribedspecified precision, a measure of quality for all the units of a lot of material, or produced by a process. This practice
willclearlyindicatethesamplesizerequiredtoestimatetheaveragevalueofsomepropertyorthefractionofnonconformingitems
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.
2. Referenced Documents
2
2.1 ASTM Standards:
E456Definitions of Terms Relating to Statistical Methods Terminology Relating to Quality and Statistics
3. Terminology
3.1 Definitions—Unless : Unless otherwise noted, all statistical terms are defined in DefinitionsTerminology E456.
3.2 Symbols: Symbols: Symbols used in all equations are defined as follows:
E = maximum tolerable error for the sample average, that is, the maximum acceptable difference between true average and
the sample average. the maximum acceptable difference between the true average and the sample average.
e = E/µ, maximum allowable sampling erroracceptable difference expressed as a fraction of µ.
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
n = size of the sample from a finite lot (7.4).
L
p8 = fraction of a lot or process whose units have the nonconforming characteristic under investigation.
p = an advance estimate of p8.
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
j 51
s = lot or process standard deviation of X, the result of measuring all of the units of a finite lot or process.
s = an advance estimate of s.
0
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.
Current edition approved Oct. 10, 2000. Published January 2001. Originally published as E122–89. Last previous edition E122–99.
Current edition approved Oct. 1, 2007. Published November 2007 . Originally published as E122–89. Last previous edition approved in 2000 as E122–00.
2
ForreferencedASTMstandards,visittheASTMwebsite,www.astm.org,orcontactASTMCustomerServiceatservice@astm.org.For Annual Book ofASTM Standards
, Vol 14.02.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 ----------------------
E122–07
n
s =
2 1/2
[ ( X −¯X − X ) /(n−1)] , an estimate of the standard deviation s from n observation, X, i=1 to n.
(
i i
ki 51
s¯ =
S/k, average s from k samples all of the same size (8.2.1).
( j
j 51
s = pooled (weighted average) s from k samples, not all of the same size (8.2).
p
s = standard derivationdeviation of sample j.
j
th
t = a factor (the 99.865 percentile of the Student’s distribution) corresponding to the degrees of freedom f of an advance
o
estimate s (5.1).
o
V = s/µ, the coefficient of variation of the lot or process.
V = an advance estimate of V(8.3.1). , equal to d /µ .
o o o
¯
v = s/ X, the coefficient of variation estimated from the sample.
v = coefficient of variation from sample j
...
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