ASTM E1402-13(2018)

NOTICE: This standard has either been superseded and replaced by a new version or withdrawn.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
Designation: E1402 − 13 (Reapproved 2018) An American National Standard
Standard Guide for
Sampling Design
This standard is issued under the fixed designation E1402; 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 3. Terminology
1.1 This guide defines terms and introduces basic methods 3.1 Definitions—For a more extensive list of statistical
for probability sampling of discrete populations, areas, and terms, refer to Terminology E456.
bulk materials. It provides an overview of common probability 3.1.1 area sampling, n—probability sampling in which a
sampling methods employed by users of ASTM standards. map, rather than a tabulation of sampling units, serves as the
sampling frame.
1.2 Sampling may be done for the purpose of estimation, of
3.1.1.1 Discussion—Area sampling units are segments of
comparison between parts of a sampled population, or for
land area and are listed by addresses on the frame prior to their
acceptance of lots. Sampling is also used for the purpose of
actual delineation on the ground so that only the randomly
auditing information obtained from complete enumeration of
selected ones need to be exactly identified.
the population.
3.1.2 bulk sampling, n—sampling to prepare a portion of a
1.3 No system of units is specified in this standard.
mass of material that is representative of the whole.
1.4 This standard does not purport to address all of the
3.1.3 cluster sampling, n—sampling in which the sampling
safety concerns, if any, associated with its use. It is the
unit consists of a group of subunits, all of which are measured
responsibility of the user of this standard to establish appro-
for sampled clusters.
priate safety, health, and environmental practices and deter-
3.1.4 frame, n—a list, compiled for sampling purposes,
mine the applicability of regulatory limitations prior to use.
which designates all of the sampling units (items or groups) of
1.5 This international standard was developed in accor-
a population or universe to be considered in a specific study.
dance with internationally recognized principles on standard-
ization established in the Decision on Principles for the
3.1.5 multi-stage sampling, n—sampling in which the
Development of International Standards, Guides and Recom-
sample is selected by stages, the sampling units at each stage
mendations issued by the World Trade Organization Technical
being selected from subunits of the larger sampling units
Barriers to Trade (TBT) Committee.
chosen at the previous stage.
3.1.5.1 Discussion—The sampling unit for the first stage is
2. Referenced Documents
the primary sampling unit. In multi-stage sampling, this unit is
further subdivided. The second stage unit is called the second-
2.1 ASTM Standards:
ary sampling unit. A third stage unit is called a tertiary
D7430 Practice for Mechanical Sampling of Coal
sampling unit. The final sample is the set of all last stage
E105 Practice for Probability Sampling of Materials
sampling units that are obtained. As an example of sampling a
E122 Practice for Calculating Sample Size to Estimate, With
lot of packaged product, the cartons of a lot could be the
Specified Precision, the Average for a Characteristic of a
primary units, packages within the carton could be secondary
Lot or Process
units, and items within the packages could be the third-stage
E141 Practice for Acceptance of Evidence Based on the
units.
Results of Probability Sampling
E456 Terminology Relating to Quality and Statistics
3.1.6 nested sampling, n—same as multi-stage sampling.
3.1.7 primary sampling unit, PSU, n—the item, element,
increment, segment or cluster selected at the first stage of the
This guide is under the jurisdiction of ASTM Committee E11 on Quality and
selection procedure from a population or universe.
Statistics and is the direct responsibility of Subcommittee E11.10 on Sampling /
Statistics.
3.1.8 probability proportional to size sampling, PPS,
Current edition approved Nov. 1, 2018. Published November 2018. Originally
n—probability sampling in which the probabilities of selection
approved in 2008. Last previous edition approved in 2013 as E1402 – 13. DOI:
of sampling units are proportional, or nearly proportional, to a
10.1520/E1402-13R18.
For referenced ASTM standards, visit the ASTM website, www.astm.org, or
quantity (the “size”) that is known for all sampling units.
contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
3.1.9 probability sample, n—a sample in which the sam-
Standards volume information, refer to the standard’s Document Summary page on
the ASTM website. pling units are selected by a chance process such that a
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E1402 − 13 (2018)
specified probability of selection can be attached to each 3.2.4 increment, n—(bulk sampling) individual portion of
possible sample that can be selected. material collected by a single operation of a sampling device.
3.1.10 proportional sampling, n—a method of selection in 3.3 Symbols:
stratified sampling such that the proportions of the sampling
N = number of units in the population to be sampled.
units (usually, PSUs) selected for the sample from each stratum
n = number of units in the sample.
are equal.
Y = quantity value for the i-th unit in the population.
i
3.1.11 quota sampling, n—a method of selection similar to y = quantity observed for i-th sampling unit.
i
¯
stratified sampling in which the numbers of units to be selected = average quantity for the population.
Y
from each stratum is specified and the selection is done by y¯ = average of the observations in the sample.
trained enumerators but is not a probability sample. X = value of an auxiliary variable for the i-th unit in the
i
population.
3.1.12 sampling fraction, f, n—the ratio of the number of
x = value of an auxiliary variable for the i-th sampling
i
sampling units selected for the sample to the number of
unit.
sampling units available.
P = population proportion of units having an attribute of
3.1.13 sampling unit, n—an item, group of items, or seg- interest.
p = sample proportion.
ment of material that can be selected as part of a probability
f = sampling fraction.
sampling plan.
s = sample standard deviation of the observations in the
3.1.13.1 Discussion—The full collection of sampling units
sample.
listed on a frame serves to describe the sampled population of
s = sample variance of the observations in the sample.
a probability sampling plan.
SE~y¯! = standard error of an estimated mean y¯.
3.1.14 sampling with replacement, n—probability sampling
4. Significance and Use
in which a selected unit is replaced after any step in selection
so that this sampling unit is available for selection again at the
4.1 This guide describes the principal types of sampling
next step of selection, or at any other succeeding step of the
designs and provides formulas for estimating population means
sample selection procedure.
and standard errors of the estimates. Practice E105 provides
principles for designing probability sampling plans in relation
3.1.15 sampling without replacement, n—probability sam-
to the objectives of study, costs, and practical constraints.
pling in which a selected sampling unit is set aside and cannot
Practice E122 aids in specifying the required sample size.
be selected at a later step of selection.
Practice E141 describes conditions to ensure validity of the
3.1.15.1 Discussion—Most samplings, including simple
results of sampling. Further description of the designs and
random sampling and stratified random sampling, are con-
formulas in this guide, and beyond it, can be found in textbooks
ducted by sampling without replacement.
(1-10).
3.1.16 simple random sample, n—(without replacement)
4.2 Sampling, both discrete and bulk, is a clerical and
probability sample of n sampling units from a population of N
physical operation. It generally involves training enumerators
N!
units selected in such a way that each of the subsets
and technicians to use maps, directories and stop watches so as
n! N2n !
~ !
of n units is equally probable – (with replacement) a probabil- to locate designated sampling units. Once a sampling unit is
ity sample of n sampling units from a population of N units located at its address, discrete sampling and area sampling
n
selected in such a way that, in order of selection, each of the N enumeration proceeds to a measurement. For bulk sampling,
ordered sequences of units from the population is equally material is extracted into a composite.
probable.
4.3 A sampling plan consists of instructions telling how to
list addresses and how to select the addresses to be measured
3.1.17 stratified sampling, n—sampling in which the popu-
lation to be sampled is first divided into mutually exclusive or extracted. A frame is a listing of addresses each of which is
indexed by a single integer or by an n-tuple (several integer)
subsets or strata, and independent samples taken within each
stratum. number. The sampled population consists of all addresses in
the frame that can actually be selected and measured. It is
3.1.18 systematic sampling, n—a sampling procedure in
sometimes different from a targeted population that the user
which evenly spaced sampling units are selected.
would have preferred to be covered.
3.2 Definitions of Terms Specific to This Standard:
4.4 A selection scheme designates which indexes constitute
3.2.1 address, n—(sampling) a unique label or instructions
the sample. If certified random numbers completely control the
attached to a sampling unit by which it can be located and
selection scheme the sample is called a probability sample.
measured.
Certified random numbers are those generated either from a
3.2.2 area segment, n—(area sampling) final sampling unit table (for example, Ref (11)) that has been tested for equal digit
for area sampling, the delimited area from which a character- frequencies and for serial independence, from a computer
istic can be measured.
3.2.3 composite sample, n—(bulk sampling) sample pre- 3
The boldface numbers in parentheses refer to a list of references at the end of
pared by aggregating increments of sampled material. this standard.
E1402 − 13 (2018)
program that was checked to have a long cycle length, or from The finite population correction factor depends on (a) the
a random physical method such as tossing of a coin or a population of interest being finite, (b) sampling being without
casino-quality spinner. errors and measurements for any sampled item being assumed
completely well defined for that item. When the purpose of
4.5 The objective of sampling is often to estimate the mean
sampling is to understand differences between parts of a
of the population for some variable of interest by the corre-
population (analytic as opposed to enumerative, as described
sponding sample mean. By adopting probability sampling,
by Deming (4)), actual population values are viewed as
selection bias can be essentially eliminated, so the primary goal
themselves sampled from a parent random process and the
of sample design in discrete sampling becomes reducing
finite population correction should not be used in making such
sampling variance.
comparisons.
5.4 Sample Size—The sample size required for a sampling
5. Simple Random Sampling (SRS) of a Finite
Population study depends on the variability of the population and the
required precision of the estimate. Refer to Practice E122 for
5.1 Sampling is without replacement. The selection scheme
further detail on determining sample size. Eq 2 can be
must allocate equal chance to every combination of n indexes
developed to find required sample size. First, the user must
from the N on the frame.
have a reasonable prior estimate s of the population standard
5.1.1 Make successive equal-probability draws from the
deviation, either from previous experience or a pilot study.
integers 1 to N and discard duplicates until n distinct indexes
Solving for n in Eq 2, where now SE y¯ is the required standard
~ !
have been selected.
error, gives:
5.1.2 If the N indexed addresses or labels are in a computer
file, generate a random number for each index and sort the file
n
o
2 2
n 5 where:n 5 s /SE ~y¯! (6)
o o
by those numbers. The first n items in the sorted file constitute
11n /N
o
a simple random sample (SRS) of size n from the N.
5.5 Estimating a Proportion—Formulas 1 through 5 serve
5.1.3 A method that requires only one pass through the
for proportions as well as means. For an indicator variable Y
i
population is used, for example, to sample a production
which equals 1 if the i-th unit has the attribute and 0 if not, the
process. For each item, generate a random number in the range
¯
population proportion P5Y can be recognized as the average of
0 to 1 and select the i-th item when the random number is less
ones and zeros. The sample estimate is the sample proportion
than n 2 a ⁄ N 2 i 1 1 , where a is the number of selections
~ ! ~ !
i i 2
p5y¯ and the sample variance is s 5np~1 2 p!⁄~n 2 1!.
already made up to the i-th item. For example, the first item (i
5.6 Ratio Estimates—An auxiliary variable may be used to
= 1 and a = 0) is selected with probability n/N.
improve the estimate from an SRS. Values of this variable for
5.2 The quantities observed on the variable of interest at the
each item on the frame will be denoted X . Specific knowledge
i
selected sampling units will be denoted y , y , …, y . The
1 2 n
of each and every X is not necessary for ratio estimation but
i
estimate of the mean of the sampled population is
¯
knowing the population average X is. The observed values x
i
y¯ 5 y /n (1) are needed along with the y , where the index i goes from i =
( i i
ˆ
1 to i = n, the sample size. The estimated ratio is R5y¯/x¯ and the
The standard error of the mean of a finite population using
¯ ¯
improved ratio estimate of Y is Xy¯⁄x¯. The estimated standard
simple random sampling without replacement is:
¯
error of the ratio estimate of Y is:
SE y¯ 5 s = 1 2 f /n (2)
~ ! ~ !
1 2 f
2 2
¯ ˆ ˆ
where f = n/N is the sampling fraction and s is the sample ~ ! ~ !
SE XR 5Πy 2 Rx /~n 2 1! (7)
( i i
n
variance (s, its square root, is sample standard deviation).
5.6.1 The ratio estimator works best when the relation of
2 2
s 5 ~y 2 y¯! /~n 2 1! (3)
( i
X-values to Y-values is approximately linear through the origin
The population mean that y¯ estimate
...

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.
Designation: E1402 − 13 E1402 − 13 (Reapproved 2018) An American National Standard
Standard Guide for
Sampling Design
This standard is issued under the fixed designation E1402; 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 This guide defines terms and introduces basic methods for probability sampling of discrete populations, areas, and bulk
materials. It provides an overview of common probability sampling methods employed by users of ASTM standards.
1.2 Sampling may be done for the purpose of estimation, of comparison between parts of a sampled population, or for
acceptance of lots. Sampling is also used for the purpose of auditing information obtained from complete enumeration of the
population.
1.3 No system of units is specified in this standard.
1.4 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.
1.5 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
2.1 ASTM Standards:
D7430 Practice for Mechanical Sampling of Coal
E105 Practice for Probability Sampling of Materials
E122 Practice for Calculating Sample Size to Estimate, With Specified Precision, the Average for a Characteristic of a Lot or
Process
E141 Practice for Acceptance of Evidence Based on the Results of Probability Sampling
E456 Terminology Relating to Quality and Statistics
3. Terminology
3.1 Definitions—For a more extensive list of statistical terms, refer to Terminology E456.
3.1.1 area sampling, n—probability sampling in which a map, rather than a tabulation of sampling units, serves as the sampling
frame.
3.1.1.1 Discussion—
Area sampling units are segments of land area and are listed by addresses on the frame prior to their actual delineation on the
ground so that only the randomly selected ones need to be exactly identified.
3.1.2 bulk sampling, n—sampling to prepare a portion of a mass of material that is representative of the whole.
3.1.3 cluster sampling, n—sampling in which the sampling unit consists of a group of subunits, all of which are measured for
sampled clusters.
3.1.4 frame, n—a list, compiled for sampling purposes, which designates all of the sampling units (items or groups) of a
population or universe to be considered in a specific study.
This guide 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, 2013Nov. 1, 2018. Published August 2013November 2018. Originally approved in 2008. Last previous edition approved in 20082013
ε1
as E1402 – 0813. . DOI: 10.1520/E1402-13.10.1520/E1402-13R18.
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
E1402 − 13 (2018)
3.1.5 multi-stage sampling, n—sampling in which the sample is selected by stages, the sampling units at each stage being
selected from subunits of the larger sampling units chosen at the previous stage.
3.1.5.1 Discussion—
The sampling unit for the first stage is the primary sampling unit. In multi-stage sampling, this unit is further subdivided. The
second stage unit is called the secondary sampling unit. A third stage unit is called a tertiary sampling unit. The final sample is
the set of all last stage sampling units that are obtained. As an example of sampling a lot of packaged product, the cartons of a
lot could be the primary units, packages within the carton could be secondary units, and items within the packages could be the
third-stage units.
3.1.6 nested sampling, n—same as multi-stage sampling.
3.1.7 primary sampling unit, PSU, n—the item, element, increment, segment or cluster selected at the first stage of the selection
procedure from a population or universe.
3.1.8 probability proportional to size sampling, PPS, n—probability sampling in which the probabilities of selection of sampling
units are proportional, or nearly proportional, to a quantity (the “size”) that is known for all sampling units.
3.1.9 probability sample, n—a sample in which the sampling units are selected by a chance process such that a specified
probability of selection can be attached to each possible sample that can be selected.
3.1.10 proportional sampling, n—a method of selection in stratified sampling such that the proportions of the sampling units
(usually, PSUs) selected for the sample from each stratum are equal.
3.1.11 quota sampling, n—a method of selection similar to stratified sampling in which the numbers of units to be selected from
each stratum is specified and the selection is done by trained enumerators but is not a probability sample.
3.1.12 sampling fraction, f, n—the ratio of the number of sampling units selected for the sample to the number of sampling units
available.
3.1.13 sampling unit, n—an item, group of items, or segment of material that can be selected as part of a probability sampling
plan.
3.1.13.1 Discussion—
The full collection of sampling units listed on a frame serves to describe the sampled population of a probability sampling plan.
3.1.14 sampling with replacement, n—probability sampling in which a selected unit is replaced after any step in selection so that
this sampling unit is available for selection again at the next step of selection, or at any other succeeding step of the sample
selection procedure.
3.1.15 sampling without replacement, n—probability sampling in which a selected sampling unit is set aside and cannot be
selected at a later step of selection.
3.1.15.1 Discussion—
Most samplings, including simple random sampling and stratified random sampling, are conducted by sampling without
replacement.
3.1.16 simple random sample, n—(without replacement) probability sample of n sampling units from a population of N units
N!
selected in such a way that each of the subsets of n units is equally probable – (with replacement) a probability sample
n!~N2n!!
n
of n sampling units from a population of N units selected in such a way that, in order of selection, each of the N ordered sequences
of units from the population is equally probable.
3.1.17 stratified sampling, n—sampling in which the population to be sampled is first divided into mutually exclusive subsets
or strata, and independent samples taken within each stratum.
3.1.18 systematic sampling, n—a sampling procedure in which evenly spaced sampling units are selected.
3.2 Definitions of Terms Specific to This Standard:
3.2.1 address, n—(sampling) a unique label or instructions attached to a sampling unit by which it can be located and measured.
3.2.2 area segment, n—(area sampling) final sampling unit for area sampling, the delimited area from which a characteristic can
be measured.
3.2.3 composite sample, n—(bulk sampling) sample prepared by aggregating increments of sampled material.
3.2.4 increment, n—(bulk sampling) individual portion of material collected by a single operation of a sampling device.
E1402 − 13 (2018)
3.3 Symbols:
N = number of units in the population to be sampled.
n = number of units in the sample.
Y = quantity value for the i-th unit in the population.
i
y = quantity observed for i-th sampling unit.
i
¯
= average quantity for the population.
Y
y¯ = average of the observations in the sample.
X = value of an auxiliary variable for the i-th unit in the population.
i
x = value of an auxiliary variable for the i-th sampling unit.
i
P = population proportion of units having an attribute of interest.
p = sample proportion.
f = sampling fraction.
s = sample standard deviation of the observations in the sample.
s = sample variance of the observations in the sample.
SE y¯ = standard error of an estimated mean y¯.
~ !
4. Significance and Use
4.1 This guide describes the principal types of sampling designs and provides formulas for estimating population means and
standard errors of the estimates. Practice E105 provides principles for designing probability sampling plans in relation to the
objectives of study, costs, and practical constraints. Practice E122 aids in specifying the required sample size. Practice E141
describes conditions to ensure validity of the results of sampling. Further description of the designs and formulas in this guide,
and beyond it, can be found in textbooks (1-10).
4.2 Sampling, both discrete and bulk, is a clerical and physical operation. It generally involves training enumerators and
technicians to use maps, directories and stop watches so as to locate designated sampling units. Once a sampling unit is located
at its address, discrete sampling and area sampling enumeration proceeds to a measurement. For bulk sampling, material is
extracted into a composite.
4.3 A sampling plan consists of instructions telling how to list addresses and how to select the addresses to be measured or
extracted. A frame is a listing of addresses each of which is indexed by a single integer or by an n-tuple (several integer) number.
The sampled population consists of all addresses in the frame that can actually be selected and measured. It is sometimes different
from a targeted population that the user would have preferred to be covered.
4.4 A selection scheme designates which indexes constitute the sample. If certified random numbers completely control the
selection scheme the sample is called a probability sample. Certified random numbers are those generated either from a table (for
example, Ref (11)) that has been tested for equal digit frequencies and for serial independence, from a computer program that was
checked to have a long cycle length, or from a random physical method such as tossing of a coin or a casino-quality spinner.
4.5 The objective of sampling is often to estimate the mean of the population for some variable of interest by the corresponding
sample mean. By adopting probability sampling, selection bias can be essentially eliminated, so the primary goal of sample design
in discrete sampling becomes reducing sampling variance.
5. Simple Random Sampling (SRS) of a Finite Population
5.1 Sampling is without replacement. The selection scheme must allocate equal chance to every combination of n indexes from
the N on the frame.
5.1.1 Make successive equal-probability draws from the integers 1 to N and discard duplicates until n distinct indexes have been
selected.
5.1.2 If the N indexed addresses or labels are in a computer file, generate a random number for each index and sort the file by
those numbers. The first n items in the sorted file constitute a simple random sample (SRS) of size n from the N.
5.1.3 A method that requires only one pass through the population is used, for example, to sample a production process. For
each item, generate a random number in the range 0 to 1 and select the ithi-th item when the random number is less than n
~
2 a !⁄~N 2 i 1 1!(n-a, )/(N-i+1), where a is the number of selections already made up to the i-th item. For example, the first item
i i i
(i=1(i = 1 and a =0) = 0) is selected with probability n/N.
5.2 The quantities observed on the variable of interest at the selected sampling units will be denoted y , y ,…,y, …, y . The
1 2 n
estimate of the mean of the sampled population is
y¯ 5 y /n (1)
( i
The standard error of the mean of a finite population using simple random sampling without replacement is:
The boldface numbers in parentheses refer to a list of references at the end of this standard.
E1402 − 13 (2018)
SE y¯ 5 s = 12 f /n (2)
~ ! ~ !
where f =n/N= n/N is the sampling fraction and s is the sample variance (s,(s, its square root, is sample standard deviation).
2 2
s 5 y 2 y¯ / n 2 1 (3)
~ ! ~ !
( i
The population mean that y¯ estimates is:
N
¯
Y 5 Y /N (4)
( i
i51
The expected value of s is the finite population variance defined as:
N
¯
S 5 ~Y 2 Y! / N 2 1 (5)
~ !
( i
i51
5.3 Finite Population Correction—The factor (1- f)(1 – f) in Eq 2 is the finite population correction. In conventional statistical
theory, the standard error of the average of independent, identically distributed random variables does not include this factor.
Conventional statistical theory applies for random sampling with replacement. In sampling without replacement from a finite
population, the observations are not independent. The finite population correction factor depends on (a) the population of interest
being finite, (b) sampling being without errors and measurements for any sampled item being assumed completely well defined
for that item. When the purpose of sampling is to understand differences between parts of a population (analytic as opposed to
enumerative, as described by Deming (4)), actual population values are viewed as themselves sampled from a parent random
process and the finite population correction should not be used in making such comparisons.
5.4 Sample Size—The sample size required for a sampling study depends on the variability of the population and the required
precision of the estimate. Refer to Practice E122 for further detail on determining sample size. Eq 2 can be developed to find
required sample size. First, the user must have a reasonable prior estimate s of the population standard deviation, either from
previous experience or a pilot study. Solving for n in Eq 2, where now SE y¯ is the required standard error, gives:
~ !
n
o
2 2
n 5 where:n 5 s /SE ~y¯! (6)
o o
11n /N
o
n
o
2 2
n 5 where:n 5 s /SE y¯ (6)
~ !
o o
11n /N
o
5.5 Estimating a Proportion—Formulas 1 through 5 serve for proportions as
...