ASTM E1402-13(2023)

Standard Guide for Sampling Design

ABSTRACT
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. This guide also describes the principal types of sampling designs and provides formulas for estimating population means and standard errors of the estimates.
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).3  
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 t...
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.

General Information

• Status: Published
• Publication Date: 31-Oct-2023
• ICS: 01.040.19 - Testing (Vocabularies)
• ICS: 19.020 - Test conditions and procedures in general
• Technical Committee: E11 - Quality and Statistics
• Drafting Committee: E11.10 - Sampling / Statistics

Relations

• Replaces: ASTM E1402-13(2018) - Standard Guide for Sampling Design
• Effective Date: 01-Nov-2023
• Refers: ASTM E141-10(2023) - Standard Practice for Acceptance of Evidence Based on the Results of Probability Sampling
• Effective Date: 01-Nov-2023
• Refers: ASTM E456-13a(2022)e1 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 01-Apr-2022
• Refers: ASTM E141-10(2018) - Standard Practice for Acceptance of Evidence Based on the Results of Probability Sampling
• Effective Date: 01-Apr-2018
• Referred By: ASTM E1367-03(2023) - Standard Test Method for Measuring the Toxicity of Sediment-Associated Contaminants with Estuarine and Marine Invertebrates
• Effective Date: 01-Nov-2023
• Referred By: ASTM C1285-21 - Standard Test Methods for Determining Chemical Durability of Nuclear, Hazardous, and Mixed Waste Glasses and Multiphase Glass Ceramics: The Product Consistency Test (PCT)
• Effective Date: 01-Nov-2023
• Referred By: ASTM E2655-14(2020) - Standard Guide for Reporting Uncertainty of Test Results and Use of the Term Measurement Uncertainty in ASTM Test Methods
• Effective Date: 01-Nov-2023
• Referred By: ASTM E2334-09(2023) - Standard Practice for Setting an Upper Confidence Bound for a Fraction or Number of Non-Conforming items, or a Rate of Occurrence for Non-Conformities, Using Attribute Data, When There is a Zero Response in the Sample
• Effective Date: 01-Nov-2023
• Referred By: ASTM E3264-21 - Standard Guide for Homogeneity of Samples and Reference Materials Used for Inter- and Intra-Laboratory Studies
• Effective Date: 01-Nov-2023
• Referred By: ASTM E105-21 - Standard Guide for Probability Sampling of Materials
• Effective Date: 01-Nov-2023
• Referred By: ASTM D7659-21 - Standard Guide for Strategies for Surface Sampling of Metals and Metalloids for Worker Protection
• Effective Date: 01-Nov-2023
• Referred By: ASTM F1089-18 - Standard Test Method for Corrosion of Surgical Instruments
• Effective Date: 01-Nov-2023
• Referred By: ASTM E3177-18 - Standard Guide on Sampling for Process Analytical Technology
• Effective Date: 01-Nov-2023
• Referred By: ASTM E456-13a(2022) - Standard Terminology Relating to Quality and Statistics
• Effective Date: 01-Nov-2023
• Referred By: ASTM F3207-17 - Standard Guide for <emph type="bdit">in vivo</emph> Evaluation of Rabbit Lumbar Intertransverse Process Spinal Fusion Model
• Effective Date: 01-Nov-2023

Overview

ASTM E1402-13(2023): Standard Guide for Sampling Design is an internationally recognized reference that introduces fundamental methods for probability sampling of discrete populations, geographical areas, and bulk materials. Developed by ASTM, this guide provides an essential overview of probability sampling methods, designs, and terminology to support the accurate estimation of population characteristics, critical for users of ASTM standards. By standardizing sampling terminology and methodologies, ASTM E1402 aims to help organizations ensure statistical validity and reduce selection bias in sampling activities across various sectors.

Key Topics

• Probability Sampling: The guide emphasizes probability-based techniques, including simple random, stratified, systematic, and cluster sampling, as well as probability proportional to size (PPS) methods, to ensure representative samples.
• Sampling Design Principles: Key concepts include defining the sampling frame, distinguishing between target and sampled populations, and appropriate use of certified random numbers for unbiased sample selection.
• Sample Size and Error Estimation: Provides guidance on calculating sample sizes and estimating standard errors, referencing complementary practices like ASTM E122 for precision in sample estimates.
• Discrete and Bulk Sampling: Discusses both discrete sampling (targeting individual items or addresses) and procedures for extracting representative samples from bulk materials, including use of composite samples and nested sub-sampling.
• Formation of Sampling Units: Explains area, cluster, and multi-stage sampling units, offering guidance on how to segment populations or materials appropriately for various study purposes.
• Ensuring Validity: Outlines conditions and best practices, such as adequate training of personnel and use of validated sampling frames, to safeguard result integrity.

Applications

ASTM E1402-13(2023) is widely applicable across industries and disciplines, supporting activities where sampling design is critical for accuracy and reliability, including:

• Quality Control: Assists manufacturers in creating statistically valid sampling plans for batch acceptance or quality audits.
• Environmental and Agricultural Science: Supports soil, crop, or geographical field sampling using area or cluster-based approaches.
• Resource and Mining Sectors: Guides bulk material sampling, such as mechanical sampling of coal, where correct sampling design ensures representation of large lots.
• Statistical Auditing: Facilitates unbiased audits by providing methods for randomly verifying information from census-type enumerations.
• Supply Chain and Inventory Assessment: Enables structured and representative sampling for stock evaluation and resource allocation.

By employing the standardized sampling procedures in ASTM E1402, organizations enhance their capability to draw accurate, objective conclusions about populations or lots, reduce the risk of bias, and ensure compliance with regulatory and industry best practices.

Related Standards

ASTM E1402 references and complements several other key ASTM standards and guides, including:

• ASTM E105 – Guide for Probability Sampling of Materials: Offers principles for designing probability-based sampling plans.
• ASTM E122 – Practice for Calculating Sample Size: Supports determination of adequate sample sizes for desired estimation precision.
• ASTM E141 – Practice for Acceptance of Evidence Based on Probability Sampling: Addresses the validation and acceptance of sampling results.
• ASTM E456 – Terminology Relating to Quality and Statistics: Provides comprehensive definitions for statistical and quality-related terms.
• ASTM D7430 – Practice for Mechanical Sampling of Coal: Applies sampling methods to coal, a key example of bulk material sampling.

These related standards, in conjunction with ASTM E1402, form an integrated framework for designing, executing, and evaluating probability-based sampling across various industries and applications.

Keywords: sampling design, probability sampling, ASTM sampling standard, simple random sampling, stratified sampling, cluster sampling, systematic sampling, bulk material sampling, area sampling, sampling plans, sample size calculation, statistical sampling, ASTM E1402

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: E1402 − 13 (Reapproved 2023) 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 Guide 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, 2023. Published November 2023. Originally
n—probability sampling in which the probabilities of selection
approved in 2008. Last previous edition approved in 2018 as E1402 – 13 (2018).
of sampling units are proportional, or nearly proportional, to a
DOI: 10.1520/E1402-13R23.
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 [e-mail]. 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 (2023)
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.
3.3 Symbols:
3.1.10 proportional sampling, n—a method of selection in
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.
X = value of an auxiliary variable for the i-th unit in the
trained enumerators but is not a probability sample.
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.
ment of material that can be selected as part of a probability p = sample proportion.
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
in which a selected unit is replaced after any step in selection 4. Significance and Use
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
3.1.17 stratified sampling, n—sampling in which the popu- list addresses and how to select the addresses to be measured
or extracted. A frame is a listing of addresses each of which is
lation to be sampled is first divided into mutually exclusive
subsets or strata, and independent samples taken within each indexed by a single integer or by an n-tuple (several integer)
number. The sampled population consists of all addresses in
stratum.
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-
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 (2023)
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. Simple Random Sampling (SRS) of a Finite 5.4 Sample Size—The sample size required for a sampling
study depends on the variability of the population and the
Population
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
¯ ˆ ˆ
where f = n/N is the sampling fraction and s is the sample
SE~XR! 5 ~y 2 Rx ! / n 2 1 (7)
Π~ !
( 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¯ estimates is: with the variance of Y for given X approximately proportional
to X. Other estimates using the auxiliary variable include
N
¯
regression estimators and difference estimators (2). The best
Y 5 Y /N (4)
( i
i51
form of estimate depends on the relation of X to Y values and
the relation between the variance of Y for given X.
The expected value of s is the finite population variance
defined as:
6. Systematic Selection (SYS)
N
2 ¯ 6.1 For systematic selection of a sample of n from a list of
S 5 ~Y 2 Y! / N 2 1 (5)
~ !
( i
i51
N sampling units when N/n = k is integer, a random integer
5.3 Finite Population Correction—The factor (1 – f) in Eq 2 between 1 and k should be selected for the start and every k-th
is the finite population correction. In conventional statistical unit thereafter. When N/n is not integer, then a random integer
theory, the standard error of the average of independent, between 1 and N should be selected for the start and the nearest
identically distributed random variables does not include this integer to N/n added successively, subtracting N when
factor. Conventional statistical theory applies for random exceeded, to get selected units. Multiple starts should be used
sampling with replacement. In sampling without replacement to create replicated samples (Practice E141) for estimating
from a finite population, the observations are not independent. sampling error if sample size n is large.
E1402 − 13 (2023)
n n
6.2 If an auxiliary variable, the X of 5.6, is available, it can
1 y 1 y
i i i
¯
y¯ 5 5 X (12)
S D
PPS ( (
be used to sort the units of the frame so that a systematic N π n x
i51 i i51 i
sample will contain a balanced cross section of the X values.
i
nx
i
¯
where X5C ⁄N is the population mean size. π 5 is the
N i
6.3 The sample average y¯ is an unbiased estimate of the C
N
inclusion probability for unit i.
population mean. An estimate of the standard error of y¯ based
on the first differences is:
The first formula of Eq 12 is known as the Horvitz-
Thompson estimate (13). An approximate formula for the
n
standard error of y¯ is due to Hartley and Rao (14). If
SE y¯ 5 y 2 y / n 2 1 (8) PPS
~ ! Π~ ! ~ !
( j j21
2n
j52
selection probabilities are exactly proportional to Y , then the
i
standard error of the PPS estimate y¯ is zero.
6.4 When K replicated subsamples are used, each sub-
PPS
sample mean, y¯ , estimates the population mean and the
k
SE~y¯ !5
PPS
average of all, y%, is the overall estimate. A preferred number of
n n N
1 Y Y
replicate subsamples is five to ten. The standard error is:
i j
1 2 π 1π 1 π /n 2
ΠF ~ ! GS D
2 i j k
( ( (
N n 2 1 π π
~ !
i51 j5i11 k51 i j
K
(13)
SE y% 5Œ y¯ 2 y% / K 2 1 (9)
~ ! ~ ! ~ !
( k
K
l51
7.5 An alternative to this form of unequal probability
7. Probability Proportional to Size (PPS) Sampling
sampling is to stratify the population by size, and conduct
7.1 When the frame lists an auxiliary (“size”) variable X for stratified sampling with the size categories as strata.
i
every address and the X-values are correlated with the
Y-values, then it may be efficient to select the sampling units
8. Stratified Sampling
with probability proportional to the X values.
i
8.1 The frame for stratified sampling includes division of
7.2 Cumulate sizes X to get C 5ΣX summing over j less
the sampling units into disjoint and exhaustive subsets of
i i j
than or equal to i. If the X are decimal, multiply by a power of
similar sampling units, called strata. Addresses are two-digit
i
ten to make usable integers. C is the overall sum. A random
indexes where the first number refers to the stratum while the
N
integer, say r, in the range 1 to C will lie in some interval
second identifies the sampling unit within each stratum. Strati-
N
C ,r,5C and selects unit i with probability proportional to
fied sampling requires that some item be sampled from every
i21 i
X . Generating n such integers with replacement selects a PPS
stratum on the stratified frame.
i
with replacement sample. Duplicated selections, if any, are
8.2 After listing the sampling units in each stratum on a
measured again.
frame, the selection is made of n from the N in the first
1 1
7.3 Data from a with-replacement PPS sample are converted
stratum, of N from N in the second, and so on to n from N
2 2 L L
to ratios z 5y ⁄x , which are independently and identically
in the last stratum.
i i i
distributed with mean equal to the sum of Y-values divided by
8.3 The numbers n , n , …, n are called an allocation.
1 2 L
¯
the sum of X-values. The estimate of the population mean, Y,
Common allocations are:
is:
(1) Proportional to N ,
h
(2) Neyman (15), proportional to N S (where S is stratum
¯
h h h
¯y 5 z¯X (10)
PPS
standard deviation),
with standard error:
(3) Optimum, proportional to N S /=C where C is cost
h h h h
per observation in stratum h,
n
¯
SE~y¯ ! 5 XŒ1/n ~z 2 z¯! /~n 2 1! (11) (4) Equal, all n equal, and
PPS ( j h
i51
0.5
(5) Compromise, proportional to N (exponents other
h
NOTE 1—Simple PPS sampling without replacement can be conducted
than 0.5 can also be used).
by independent draws selecting sampling unit i, if it remains unselected,
at each step with probability proportional to X . However, the resulting
i
8.4 The first three require increasing amounts of preliminary
probabilities of inclusion in the sample for each item are not exactly
information so that the second and third are seldom used.
proportional to their size. Modified PPS schemes are reviewed by Brewer
and Hanif (12). Proportional allocation has the convenient property that the
estimate of the overall population mean is the unweighted
7.4 A PPS sampling without replacement method with the
sample average. Equal allocation is appropriate if comparisons
property that inclusion probabilities are proportional to sizes
between strata or means for individual strata are of interest
can be accomplished. Form cumulative sums C following 7.2.
i
(Practice E105). The compromise allocation mediates between
If there are large units with size X > C / n then they must be
i N
goals of estimating stratum averages and estimating the overall
selected for sure, removed from the probability sampling
population mean. Values of the exponent less than 0.5 better
frame, and cumulative sums recomputed to select the remain-
estimate stratum mean differences. Exponent 0.0 gives equal
der of the sample. Systematically sample n integers from the
allocation. Values greater than 0.5 are better for estimating the
cumulative size range 1 to C in accord with 6.1 and then
N
overall mean. Exponent 1.0 gives proportional allocation.
measure the units thus selected.
7.4.1 The estimate of the population mean for this system- 8.5 The estimate of the population mean from a stratified
atic PPS without replacement sampling is: sample is:
E1402 − 13 (2023)
L
selected clusters are measured, then the estimate of the overall
y¯ 5 N /N y¯ (14)
~ !
st ( h h
average across units is:
h51
n
The estimated standard error of the mean is:
m y¯
( i i
i51
L
y% 5 (18)
n
2 2
SE y¯ 5 N /N 1 2 n /N s /n (15)
~ ! Π~ ! @ ~ !#
st ( h h h h h
m
i
h51 (
i51
8.6 Stratum divisions may be clear from the need to include
where m is the number of units in the i-th sampled cluster
i
various parts of the frame in the sample or from earlier surveys
and y¯ is their average. The estimated standard error of y% is
i
of the same type. If one has au
...