ASTM E122-17(2022)

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.
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

• Status: Published
• Publication Date: 31-Mar-2022
• ICS: 03.120.30 - Application of statistical methods
• Technical Committee: E11 - Quality and Statistics
• Drafting Committee: E11.10 - Sampling / Statistics

Relations

• Refers: ASTM E456-13a(2022)e1 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 01-Apr-2022
• Refers: ASTM E456-13A(2017)e1 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 01-Oct-2017
• Refers: ASTM E456-13A(2017)e3 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 01-Oct-2017
• Refers: ASTM E456-13ae3 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 15-Nov-2013
• Refers: ASTM E456-13ae2 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 15-Nov-2013
• Refers: ASTM E456-13ae1 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 15-Nov-2013
• Refers: ASTM E456-13a - Standard Terminology Relating to Quality and Statistics
• Effective Date: 15-Nov-2013
• Refers: ASTM E456-13 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 15-Aug-2013
• Refers: ASTM E456-12e1 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 01-May-2012
• Refers: ASTM E456-12 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 01-May-2012
• Refers: ASTM E456-08e4 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 01-Apr-2008
• Refers: ASTM E456-08 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 01-Apr-2008
• Refers: ASTM E456-08e2 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 01-Apr-2008
• Refers: ASTM E456-08e3 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 01-Apr-2008
• Refers: ASTM E456-08e1 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 01-Apr-2008

Overview

ASTM E122-17(2022): Standard Practice for Calculating Sample Size to Estimate, With Specified Precision, the Average for a Characteristic of a Lot or Process is an ASTM International standard designed to help organizations determine the sample size needed for estimating the average value or nonconforming fraction of a characteristic in a lot of material or production process. The standard offers clear procedures for sample size calculation, enabling users to make informed decisions on quality assessment with specified levels of precision. This practice is widely used in quality control, sampling plans, and manufacturing process evaluation.

Key Topics

• Sample Size Determination: Provides straightforward methods to calculate how many units should be included in a random sample to achieve a specified level of statistical precision for a given characteristic.
• Single Source of Variability: The standard applies when the material or process characteristics exhibit a single, overall source of variability. It does not address multi-level or nested sources of variability.
• Quality Measurement: Guidance is given for estimating both average values (means) and fractions of nonconforming units (defective items) within a lot or process.
• Precision and Confidence: Emphasizes the importance of selecting a maximum tolerable error (E) and understanding the probability that the true value differs from the sample estimate by more than this amount.
• Empirical Knowledge: Highlights the importance of using prior knowledge or empirical data, such as standard deviation, range, and historical rates of nonconformance, to inform calculation and increase accuracy.
• Cost Considerations: Discusses balancing the cost of larger samples with the desired precision, and adapting the sample size based on information gained from previous sampling.

Applications

ASTM E122-17(2022) is valuable across a variety of industries and laboratory settings where quality assurance, product compliance, and statistical sampling are critical:

• Manufacturing and Production: Used to determine how many products or materials to randomly test to ensure production meets quality standards.
• Materials Testing: Supports laboratories in calculating the number of specimens needed to reliably estimate average properties such as strength, composition, or durability.
• Quality Assurance and Control: Enables quality professionals to establish statistically valid sampling plans for routine inspections and audits.
• Statistical Process Control: Helps process engineers and statisticians estimate process averages or defect rates with a specified confidence level.
• Regulatory Compliance: Assists organizations in fulfilling industry or governmental requirements for sampling and quality reporting.
• Resource Optimization: Applies in situations where the cost or effort of testing all units is prohibitive, allowing for the estimation of quality from a representative subset.

Related Standards

• ASTM E456: Terminology Relating to Quality and Statistics - defines statistical terms used in ASTM E122.
• ISO 2859 and ISO 3951: International standards for sampling procedures for inspection by attributes and variables.
• ASTM E2234: Practice for Sampling a Stream of Product for Lot or Batch Inspection.
• ASTM E246: Practice for Describing and Measuring Performance of Single-Item Continuous Sampling Plans.

Organizations operating under quality management systems such as ISO 9001 or those adhering to regulatory guidelines will find ASTM E122-17(2022) a crucial reference for statistical sampling and quality estimation. By following this standard, users ensure that their sampling strategies are statistically sound, cost-effective, and suitable for reliably assessing the quality of lots or production processes.

Keywords: sample size calculation, ASTM E122, quality control, lot sampling, process average, nonconformance rate, statistical precision, sampling plan, quality assessment.

Standards Content (Sample)

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.
Designation: E122 −17 (Reapproved 2022) An American National Standard
Standard Practice for
Calculating Sample Size to Estimate, With Specified
Precision, the Average for a Characteristic of 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 (´) 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.
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'.
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, 2022. Published April 2022. Originally
measuring all of the units of a finite lot or process.
approved in 1958. Last previous edition approved in 2017 as E122–17. DOI:
10.1520/E0122-17R22. σ = an advance estimate ofσ.
n 1⁄2
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 [e-mail]. For Annual Book of ASTM
i51
Standards volume information, refer to the standard’s Document Summary page on
deviationσ 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
E122 − 17 (2022)
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 words, information concerning the
s = standard deviation of sample j.
j process, and the material produced thereby, accumulates and
V = an advance estimate of V, equal toσ /µ .
o o o should be used.
¯
v = s/X, the coefficient of variation estimated from the
sample.
6. Precision Desired
v = pooled (weighted average) of v from k samples (8.3).
p
6.1 Theapproximateprecisiondesiredfortheestimatemust
v = coefficient of variation from sample j.
j
X = numerical value of the characteristic of an individual be prescribed. That is, it must be decided what maximum
deviation, E, can be tolerated between the estimate to be made
unit being measured.
n
¯
X = from the sample and the result that would be obtained by
X /n average of n observations, X,i=1to n.
( i i
i
i51
measuring every unit in the lot or process.
4. Significance and Use
6.2 Insomecases,themaximumallowablesamplingerroris
expressed as a proportion, e, or a percentage, 100 e. For
4.1 This practice is intended for use in determining the
example, one may wish to make an estimate of the sulfur
sample size required to estimate, with specified precision, a
content of coal within 1%, or e =0.01.
measure of quality of a lot or process. The practice applies
when quality is expressed as either the lot average for a given
7. Equations for Calculating Sample Size
property, or as the lot fraction not conforming to prescribed
standards.Thelevelofacharacteristicmayoftenbetakenasan
7.1 Basedonanormaldistributionforthecharacteristic,the
indication of the quality of a material. If so, an estimate of the
equation for the size, n, of the sample is as follows:
average value of that characteristic or of the fraction of the
n 5 3σ /E (1)
~ !
o
observed values that do not conform to a specification for that
characteristicbecomesameasureofqualitywithrespecttothat
The multiplier 3 is a factor corresponding to a low probabil-
characteristic. This practice is intended for use in determining
ity that the difference between the sample estimate and the
the sample size required to estimate, with specified precision,
result of measuring (by the same methods) all the units in the
such a measure of the quality of a lot or process either as an
lot or process is greater than E. The value 3 is recommended
average value or as a fraction not conforming to a specified
for general use.With the multiplier 3, and with a lot or process
value.
standard deviation equal to the advance estimate, it is practi-
cally certain that the sampling error will not exceed E. Where
5. Empirical Knowledge Needed
a lesser degree of certainty is desired a smaller multiplier may
5.1 Some empirical knowledge of the problem is desirable
be used (Note 1).
in advance.
NOTE 1—For example, multiplying by 2 in place of 3 gives a
5.1.1 We may have some idea about the standard deviation
probability of about 45 parts in 1000 that the sampling error will exceed
of the characteristic.
E.Althoughdistributionsmetinpracticemaynotbenormal,thefollowing
5.1.2 Ifwehavenothadenoughexperiencetogiveaprecise
text table (based on the normal distribution) indicates approximate
estimateforthestandarddeviation,wemaybeabletostateour probabilities:
belief about the range or spread of the characteristic from its Factor Approximate Probability of Exceeding E
3 0.003 or 3 in 1000 (practical certainty)
lowest to its highest value and possibly about the shape of the
2.56 0.010 or 10 in 1000
distributionofthecharacteristic;forinstance,wemightbeable
2 0.045 or 45 in 1000
to say whether most of the values lie at one end of the range, 1.96 0.050 or 50 in 1000 (1 in 20)
1.64 0.100 or 100 in 1000 (1 in 10)
or are mostly in the middle, or run rather uniformly from one
end to the other (Section 9).
7.1.1 If a lot of material has a highly asymmetric distribu-
tion in the characteristic measured, the sample size as calcu-
5.2 If the aim is to estimate the fraction nonconforming,
lated in Eq 1 may not be adequate. There are two things to do
theneachunitcanbeassignedavalueof0or1(conformingor
when asymmetry is suspected.
nonconforming), and the standard deviation as well as the
7.1.1.1 Probe the material with a view to discovering, for
shape of the distribution depends only on p', the fraction
example, extra-high values, or possibly spotty runs of abnor-
nonconforming in the lot or process. Some rough idea con-
mal character, in order to approximate roughly the amount of
cerningthesizeofp'isthereforeneeded,whichmaybederived
theasymmetryforusewithstatisticaltheoryandadjustmentof
from preliminary sampling or from previous experience.
the sample size if necessary.
5.3 More knowledge permits a smaller sample size. Seldom
7.1.1.2 Searchthelotforabnormalmaterialandsegregateit
will there be difficulty in acquiring enough information to
for separate treatment.
compute the required size of sample. A sample that is larger
than the equations indicate is used in actual practice when the 7.2 There are some materials for which σ varies approxi-
empirical knowledge is sketchy to start with and when the mately with µ, in which case V(=σ⁄µ) remains approximately
desired precision is critical. constant from large to small values of µ.
E122 − 17 (2022)
7.2.1 For the situation of 7.2, the equation for the sample s¯
σ 5 (8)
o
size, n, is as follows: c
n 5 3 V /e (2)
~ !
o where the value of the correction factor, c , depends on the
size of the individual data sets (n)(Table 1 ).
j
If the relative error, e, is to be the same for all values of µ,
8.2.2 Anevensimpler,andslightlylessefficientestimatefor
then everything on the right-hand side of Eq 2 is a constant;
¯
σ maybecomputedbyusingtheaveragerange(R)takenfrom
o
hence n is also a constant, which means that the same sample
the several previous data sets that have the same group size.
size n would be required for all values of µ.
¯
R
7.3 If the problem is to estimate the lot fraction
σ 5 (9)
o
2 d
nonconforming, thenσ is replaced by p (1−p ) so that Eq
o o o
1 becomes:
Thefactor, d ,fromTable1isneededtoconverttheaverage
range into an unbiased estimate ofσ .
n 5 3/E p 1 2 p (3)
~ ! ~ ! o
o o
8.2.3 Example 1—Use of s¯.
7.4 When the average for the production process is not
8.2.3.1 Problem—To compute the sample size needed to
needed,butrathertheaverageofaparticularlotisneeded,then
estimatetheaveragetransversestrengthofalotofbrickswhen
the required sample size is less than Eq 1, Eq 2, and Eq 3
the value of E is 50 psi, and practical certainty is desired.
indicate. The sample size for estimating the average of the
8.2.3.2 Solution—From the data of three previous lots, the
finite lot will be:
values of the estimated standard deviation were found to be
n 5 n/@11~n/N!# (4)
215, 192, and 20
...