ASTM E122-09
(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
SIGNIFICANCE AND USE
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.
General Information
Relations
Buy Standard
Standards Content (Sample)
NOTICE: This standard has either been superseded and replaced by a new version or withdrawn.
Contact ASTM International (www.astm.org) for the latest information
An American National Standard
Designation: E122 – 09
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
k = the total number of samples available from the same
or similar lots.
1.1 Thispracticecoverssimplemethodsforcalculatinghow
µ = lot or process mean or expected value of X, the result
many units to include in a random sample in order to estimate
of measuring all the units in the lot or process.
withaspecifiedprecision,ameasureofqualityforalltheunits
µ = an advance estimate of µ.
ofalotofmaterial,orproducedbyaprocess.Thispracticewill 0
N = size of the lot.
clearly indicate the sample size required to estimate the
n = size of the sample taken from a lot or process.
average value of some property or the fraction of nonconform-
n = size of sample j.
j
ing items produced by a production process during the time
n = size of the sample from a finite lot (7.4).
L
interval covered by the random sample. If the process is not in
p8 = fraction of a lot or process whose units have the
a state of statistical control, the result will not have predictive
nonconforming characteristic under investigation.
valueforimmediate(future)production.Thepracticetreatsthe
p = an advance estimate of p8.
0
common situation where the sampling units can be considered
p = fraction nonconforming in the sample.
to exhibit a single (overall) source of variability; it does not
R = range of a set of sampling values. The largest minus
treat multi-level sources of variability.
the smallest observation.
R = range of sample j.
j
2. Referenced Documents
k
¯
R =
2
R/k, average of the range of k samples, all of the
2.1 ASTM Standards: ( j
j 51
E456 Terminology Relating to Quality and Statistics
same size (8.2.2).
s = lot or process standard deviation of X, the result of
3. Terminology
measuring all of the units of a finite lot or process.
3.1 Definitions—Unless otherwise noted, all statistical
s = an advance estimate of s.
0
n
terms are defined in Terminology E456.
s =
2 1/2
[ (X − X ) /(n−1)] , an estimate of the
( i
3.2 Symbols—Symbols used in all equations are defined as
i 51
follows:
standarddeviation sfromnobservation,X,i=1ton.
i
k
s¯ =
S/k,average sfrom ksamplesallofthesamesize
( j
j 51
E = the maximum acceptable difference between the true
(8.2.1).
average and the sample average.
s = pooled(weightedaverage)sfromksamples,notallof
p
e = E/µ, maximum acceptable difference expressed as a
the same size (8.2).
fraction of µ.
s = standard deviation of sample j.
j
f = degrees of freedom for a standard deviation estimate
V = an advance estimate of V, equal to d /µ .
o o o
(7.5).
¯
v = s/X, the coefficient of variation estimated from the
sample.
v = pooled (weighted average) of v from k samples (8.3).
p
v = coefficient of variation from sample j.
1 j
ThispracticeisunderthejurisdictionofASTMCommitteeE11onQualityand
X = numerical value of the characteristic of an individual
Statistics and is the direct responsibility of Subcommittee E11.10 on Sampling /
unit being measured.
Statistics.
n
¯
Current edition approved Aug. 1, 2009. Published September 2009. Originally
X =
approved in 1958. Last previous edition approved in 2007 as E122–07. DOI: X/n average of n observations, X,i=1 to n.
(
i i i
i 51
10.1520/E0122-09.
2
For referenced ASTM standards, visit the ASTM website, www.astm.org, or
4. Significance and Use
contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
4.1 This practice is intended for use in determining the
Standards volume information, refer to the standard’s Document Summary page on
the ASTM website. sample size required to estimate, with specified precision, a
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.
1
---------------------- Page: 1 ----------------------
E122 – 09
measure of quality of a lot or process. The practice applies 7. Equations for Calculating Sample Size
when quality is expressed as either the lot average for a given
7.1 Basedonanormaldistributionforthecharacteristic,the
property, or as the lot fraction not conforming to prescribed
equation for the size, n, of the sample is as follows:
standards.Thelevelofacharacteristicmayoftenbetakenasan
2
n 5 3s /E (1)
~ !
o
indication of the quality of a material. If so, an estimate of the
average value of that characteristic or of the fraction of the
The multiplier 3 is a factor corresponding to
...
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:E122–07 Designation: E 122 – 09
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
1.1 Thispracticecoverssimplemethodsforcalculatinghowmanyunitstoincludeinarandomsampleinordertoestimatewith
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
byaproductionprocessduringthetimeintervalcoveredbytherandomsample.Iftheprocessisnotinastateofstatisticalcontrol,
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:
E456 Terminology Relating to Quality and Statistics
3. Terminology
3.1 Definitions: Unless otherwise noted, all statistical terms are defined in Terminology E456.
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
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). 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. 1, 2007. Published November 2007 . Originally published as E122–89. Last previous edition approved in 2000 as E122–00. on
Sampling/Statistics.
Current edition approved Aug. 1, 2009. Published September 2009. Originally approved in 1958. Last previous edition approved in 2007 as E122–07.
2
ForreferencedASTMstandards,visittheASTMwebsite,www.astm.org,orcontactASTMCustomerServiceatservice@astm.org.For Annual Book ofASTM 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 ----------------------
E122–09
n
s =
2 1/2
[ (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). 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 deviation 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 = an advance estimate of V, equal to d /µ .
o o o
¯
v = s/ X, the coefficient of variation estimated from the sample.
v = pooled (weighted average) of v from k samples (8.3).
p
v = coefficient of variation from sample j.
j
X = numerical value of the characteristic of an individual unit being measured.
n
¯
X =
X/n average of n observations, X,i=1 to n.
(
i i i
i 51
4. Significance and Use
4.1 This practice is intended for us
...
Questions, Comments and Discussion
Ask us and Technical Secretary will try to provide an answer. You can facilitate discussion about the standard in here.