ASTM E3264-21

Standard Guide for Homogeneity of Samples and Reference Materials Used for Inter- and Intra-Laboratory Studies

SIGNIFICANCE AND USE
5.1 This guide presents techniques and guidance for evaluating and assuring homogeneity of individual samples or bulk materials and can be used for either interlaboratory or intra-laboratory studies. The types of studies include, but are not limited to, studies to determine precision estimates for test methods, proficiency testing programs, and studies related to quality control of testing within a single laboratory.  
5.2 Because the test results of any laboratory study are affected by the quality of the samples tested, producing homogeneous samples and determining the degree of homogeneity is important for interpreting the results of the study.  
5.3 Five techniques are presented in this guide to evaluate sample homogeneity for a range of circumstances and degrees of rigor. The circumstances under which the studies are conducted and the degree of rigor required may differ. The user should consider the circumstances listed in each technique to determine which is appropriate for the study at hand.  
5.4 Each of the Techniques 1, 2, and 3 provides a procedure for testing and evaluating sample homogeneity when replicate testing of the samples is possible. Technique 4 provides a plan to evaluate sample homogeneity when replicate testing is not possible. Technique 5 recommends practices for producing homogeneous samples for circumstances when homogeneity testing is not possible.  
5.5 When the conditions of adequate within-sample homogeneity and between-sample homogeneity are satisfied, any differences in test results on multiple samples can reasonably be attributed to testing variation and not due to sample variation.  
5.6 When differences within or between samples are discovered and the samples are deemed insufficiently homogeneous, the sample preparation process can be improved or corrected and a new set of samples can be prepared. Or, in cases where the sample homogeneity cannot be improved or for other reasons when the samples must be used, the method of eval...
SCOPE
1.1 This guide presents techniques and guidance for evaluating and assuring homogeneity of individual samples and bulk materials used for interlaboratory and intra-laboratory studies.  
1.2 This guide is applicable to samples and reference materials used for proficiency testing programs and for interlaboratory studies to determine precision estimates for test methods. It may also be useful for activities related to quality control of testing within a single laboratory.  
1.3 Five techniques are presented for assessing sample homogeneity. The five techniques are not an exhaustive list of available techniques for assessing homogeneity of samples, but the techniques were chosen to cover a range of circumstances (and various degrees of rigor required) for laboratory studies of various types and purposes.  
1.4 Each of the first four techniques provides a scheme for testing for homogeneity and a statistical procedure for evaluating the results of the homogeneity testing. The circumstances are described for which each of the techniques is suited.  
1.5 For circumstances when homogeneity testing is not possible, the fifth technique provides guidance for producing homogeneous samples.  
1.6 The appendixes of this guide provide example spreadsheets for Techniques 1, 2, 3, and 4.  
1.7 This guide is not intended for evaluation of certified reference materials (CRMs) or materials used for calibration.  
1.8 Units—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.  
1.9 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 ...

General Information

• Status: Published
• Publication Date: 31-Mar-2021
• ICS: 19.020 - Test conditions and procedures in general
• Technical Committee: E11 - Quality and Statistics
• Drafting Committee: E11.20 - Test Method Evaluation and Quality Control

Relations

• Refers: ASTM D6299-23a - Standard Practice for Applying Statistical Quality Assurance and Control Charting Techniques to Evaluate Analytical Measurement System Performance
• Effective Date: 01-Dec-2023
• Refers: ASTM E2282-23 - Standard Guide for Defining the Test Result of a Test Method
• Effective Date: 01-Nov-2023
• Refers: ASTM E1402-13(2023) - Standard Guide for Sampling Design
• 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 E1402-13(2018) - Standard Guide for Sampling Design
• Effective Date: 01-Nov-2018
• Refers: ASTM D6299-17b - Standard Practice for Applying Statistical Quality Assurance and Control Charting Techniques to Evaluate Analytical Measurement System Performance
• Effective Date: 15-Dec-2017
• Refers: ASTM D6299-17a - Standard Practice for Applying Statistical Quality Assurance and Control Charting Techniques to Evaluate Analytical Measurement System Performance
• Effective Date: 15-Nov-2017
• Refers: ASTM E456-13A(2017)e3 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 01-Oct-2017
• Refers: ASTM E456-13A(2017)e1 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 01-Oct-2017
• Refers: ASTM D6299-17 - Standard Practice for Applying Statistical Quality Assurance and Control Charting Techniques to Evaluate Analytical Measurement System Performance
• Effective Date: 01-Jan-2017
• Refers: ASTM E178-16 - Standard Practice for Dealing With Outlying Observations
• Effective Date: 01-Jun-2016
• Refers: ASTM C33/C33M-16e1 - Standard Specification for Concrete Aggregates
• Effective Date: 01-Feb-2016
• Refers: ASTM C1128-15 - Standard Guide for Preparation of Working Reference Materials for Use in Analysis of Nuclear Fuel Cycle Materials
• Effective Date: 01-Feb-2015
• Refers: ASTM E2282-14 - Standard Guide for Defining the Test Result of a Test Method
• Effective Date: 01-Oct-2014
• Refers: ASTM D7915-14 - Standard Practice for Application of Generalized Extreme Studentized Deviate (GESD) Technique to Simultaneously Identify Multiple Outliers in a Data Set
• Effective Date: 01-May-2014

Overview

ASTM E3264-21: Standard Guide for Homogeneity of Samples and Reference Materials Used for Inter- and Intra-Laboratory Studies provides comprehensive guidance on evaluating and ensuring the homogeneity of samples and reference materials. Developed by ASTM International, this standard is vital for laboratories involved in interlaboratory studies, proficiency testing programs, and internal quality control procedures.

Homogeneity of samples is a crucial factor that impacts the accuracy and comparability of laboratory results. By following the recommendations in ASTM E3264-21, laboratories can reliably produce or select adequately homogeneous materials, ensuring that variation in test outcomes is due to the testing process rather than inconsistencies in the sample itself.

Key Topics

• Definition and Importance of Homogeneity: The guide clearly distinguishes between within-sample homogeneity (uniformity within a single sample) and between-sample homogeneity (consistency across a set of samples).

• Techniques for Evaluating Homogeneity: ASTM E3264-21 describes five techniques for assessing homogeneity across various circumstances:

  • Techniques 1, 2, and 3: Procedures for testing and evaluating homogeneity when replicate testing is possible.
  • Technique 4: A plan for situations where replicate testing is not possible.
  • Technique 5: Best practices for producing homogeneous samples when testing cannot be performed.

• Statistical Evaluation: The standard outlines statistical tools and procedures for evaluating the results, including screening for outliers and calculating variances.

• Practical Procedures: Step-by-step guidance on sample preparation, testing, data analysis, and decision-making regarding sample suitability.

• Scope Limitations: The guide is not intended for certified reference materials (CRMs) or calibration materials, which are subject to additional requirements.

Applications

ASTM E3264-21 is widely applicable in laboratory quality assurance and method validation, including:

• Interlaboratory Studies: Ensures that materials distributed to multiple laboratories are sufficiently homogeneous so that observed test result variation is attributable to test methods, not sample inconsistencies.
• Proficiency Testing Programs: Assures fairness in program results by confirming material consistency across all participants.
• Intra-Laboratory Quality Control: Supports consistent results within a single laboratory by assessing the uniformity of bulk or batch materials.
• Test Method Precision Studies: Aids in generating reliable data when determining the precision of analytical methods.
• Sample Preparation Optimization: Provides guidance for improving processes where homogeneity issues are identified.
• Research and Development: Useful in material development or new method validation by ensuring sample uniformity in early-stage studies.

By applying ASTM E3264-21, laboratories, quality managers, and program coordinators can minimize erroneous results and enhance confidence in testing outcomes.

Related Standards

For comprehensive laboratory practice, the following related standards should be considered alongside ASTM E3264-21:

• ASTM E691: Practice for Conducting an Interlaboratory Study to Determine the Precision of a Test Method
• ISO 13528: Statistical Methods for Use in Proficiency Testing by Interlaboratory Comparison
• ASTM D6299: Practice for Statistical Quality Assurance and Control Charting Techniques
• ASTM E178: Practice for Dealing with Outlying Observations
• ASTM E105: Guide for Probability Sampling of Materials
• ISO Guide 30: Reference Materials - Selected Terms and Definitions

These documents offer additional methodologies for analytical quality control, statistical evaluation, and sampling design, complementing the homogeneity evaluation procedures detailed in ASTM E3264-21.

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Keywords: ASTM E3264-21, sample homogeneity, reference material, interlaboratory studies, proficiency testing, laboratory quality control, statistical evaluation, sample preparation, analytical precision, laboratory standards.

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: E3264 − 21 An American National Standard
Standard Guide for
Homogeneity of Samples and Reference Materials Used for
Inter- and Intra-Laboratory Studies
This standard is issued under the fixed designation E3264; 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 priate safety, health, and environmental practices and deter-
mine the applicability of regulatory limitations prior to use.
1.1 This guide presents techniques and guidance for evalu-
1.10 This international standard was developed in accor-
ating and assuring homogeneity of individual samples and bulk
dance with internationally recognized principles on standard-
materials used for interlaboratory and intra-laboratory studies.
ization established in the Decision on Principles for the
1.2 This guide is applicable to samples and reference
Development of International Standards, Guides and Recom-
materials used for proficiency testing programs and for inter-
mendations issued by the World Trade Organization Technical
laboratory studies to determine precision estimates for test
Barriers to Trade (TBT) Committee.
methods. It may also be useful for activities related to quality
2. Referenced Documents
control of testing within a single laboratory.
2.1 ASTM Standards:
1.3 Five techniques are presented for assessing sample
C33/C33M Specification for Concrete Aggregates
homogeneity. The five techniques are not an exhaustive list of
C1128 Guide for Preparation of Working Reference Materi-
available techniques for assessing homogeneity of samples, but
als for Use in Analysis of Nuclear Fuel Cycle Materials
the techniques were chosen to cover a range of circumstances
D5956 Guide for Sampling Strategies for Heterogeneous
(and various degrees of rigor required) for laboratory studies of
Wastes
various types and purposes.
D6299 Practice for Applying Statistical Quality Assurance
1.4 Each of the first four techniques provides a scheme for
and Control Charting Techniques to Evaluate Analytical
testing for homogeneity and a statistical procedure for evalu-
Measurement System Performance
ating the results of the homogeneity testing. The circumstances
D7915 Practice for Application of Generalized Extreme
are described for which each of the techniques is suited.
Studentized Deviate (GESD) Technique to Simultane-
1.5 For circumstances when homogeneity testing is not ously Identify Multiple Outliers in a Data Set
possible, the fifth technique provides guidance for producing
E105 Guide for Probability Sampling of Materials
homogeneous samples. E178 Practice for Dealing With Outlying Observations
E456 Terminology Relating to Quality and Statistics
1.6 The appendixes of this guide provide example spread-
E691 Practice for Conducting an Interlaboratory Study to
sheets for Techniques 1, 2, 3, and 4.
Determine the Precision of a Test Method
1.7 This guide is not intended for evaluation of certified
E1402 Guide for Sampling Design
reference materials (CRMs) or materials used for calibration.
E2282 Guide for Defining the Test Result of a Test Method
2.2 ISO Standards:
1.8 Units—The system of units for this standard is not
ISO Guide 30 Reference Materials – Selected Terms and
specified. Dimensional quantities in the standard are presented
Definitions
only as illustrations of calculation methods. The examples are
ISO 13528 Statistical Methods for Use in Proficiency Test-
not binding on products or test methods treated.
ing by Interlaboratory Comparison
1.9 This standard does not purport to address all of the
ISO 3534 Statistics Vocabulary and Symbols – Part 2 Ap-
safety concerns, if any, associated with its use. It is the
plied Statistics
responsibility of the user of this standard to establish appro-
For referenced ASTM standards, visit the ASTM website, www.astm.org, or
contact ASTM Customer Service at [e-mail]. For Annual Book of ASTM
This guide is under the jurisdiction of ASTM Committee E11 on Quality and Standards volume information, refer to the standard’s Document Summary page on
Statistics and is the direct responsibility of Subcommittee E11.20 on Test Method the ASTM website.
Evaluation and Quality Control. Available from International Organization for Standardization (ISO), ISO
Current edition approved April 1, 2021. Published January 2022. DOI: 10.1520/ Central Secretariat, Chemin de Blandonnet 8, CP 401, 1214 Vernier, Geneva,
E3264-21. Switzerland, https://www.iso.org.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E3264 − 21
3. Terminology 3.2.1 between–sample homogeneity, n—the degree to which
the structure or composition with respect to one or more
3.1 Definitions—Unless otherwise noted in this guide, all
specified properties of each sample is similar to every other
terms relating to quality and statistics are defined in E456.
sample in a set of samples.
3.1.1 heterogeneity, n—the condition of a population under
3.2.1.1 Discussion—Cans of paint obtained from the same
which items of the population are not of uniform structure or
manufacturer and the same batch would be expected to have
composition with respect to the characteristic of interest.
good between-sample homogeneity and test results on multiple
3.1.1.1 Discussion—The variation in test results for a prop-
cans would be expected to be similar. The between-sample
erty of interest can be used to measure the degree of hetero-
homogeneity of cans of paint obtained from different batches
geneity of a reference material with respect to the specified
or from different manufacturers would not be expected to be as
property.
good since the ingredients and their proportions may not be the
3.1.2 homogeneity, n—condition of being of uniform struc-
same and the test results from multiple cans would exhibit
ture or composition with respect to one or more specified
greater variation.
properties. ISO Guide 30, C1128
3.2.2 between-sample variance, s , n—estimate of the vari-
s
3.1.2.1 Discussion—A reference material is said to be ho-
ance of test results due to differences between samples.
mogenous with respect to a specified property if the variation
3.2.2.1 Discussion—The between-sample variance, s , is
of test results for that property obtained by a specified test s
determined using the variance of the cell averages, s , the
¯
method is found to lie within a specified acceptable limit.
X
i
within-sample variance, s , and the number of replicates, k, as
3.1.3 representative sample, n—a sample collected in such a w
follows:
manner that it reflects one or more characteristics of interest of
the population from which it was collected. D5956 2
2 2
s 5 s ¯ 2 ~s ⁄ k! (1)
s w
X
i
3.1.3.1 Discussion—The term representative is often used
with other terms to describe when the object of discussion 3.2.3 cell, i, n—the set of results obtained from testing
(such as a sample, a group of samples, a test specimen, a performed on a single sample.
portion of a bulk material, a subset of data, or a test unit)
¯
3.2.4 cell average, X , n—the average of the results of
i
reflects characteristics of the population as a whole.
replicate tests performed on a single sample.
3.1.4 test result, n—the value of a characteristic obtained by
3.2.5 cell difference, D , n—the difference between duplicate
i
carrying out a specified test method. ISO 3534–2, E2282
results in a single cell.
3.1.4.1 Discussion—The test method specifies that one or a
3.2.6 cell standard deviation, s , n—the square root of the
number of individual observations be made, and their average
i
cell variance.
or another appropriate function, (such as the median or the
standard deviation), be reported as the test result. It can also
3.2.7 cell variance, s , n—calculated statistical variance of
i
require standard corrections to be applied, such as correction of
the replicate test results contained in a single cell.
gas volumes to standard temperature and pressure. Thus, a test
3.2.8 replicate test, n—the process of repeating a test on the
result can be a result calculated from several observed values.
same specimen or on sub-samples obtained from a single
In the simple case, the test result is the observed value itself.
sample.
3.1.5 test specimen, n—the portion of a test unit needed to
3.2.9 sub-sample, n—a quantity of material to be used as a
obtain a single test determination. E2282
test specimen, obtained from a larger sample in a manner
3.1.5.1 Discussion—When used for a physical test, this is
consistent with the recommended industry protocols for the
sometimes called “test piece.” For a chemical test, it is
material to be tested. (See test specimen.)
sometimes called test portion or test sample. For optical and
other tests, it is also sometimes called test sample. In interla- 3.2.10 variance of the cell averages, s , n—calculated
¯
X
i
boratory evaluation of test methods and other statistical
statistical variance of the cell averages for the set of samples
procedures, it is best to reserve the word sample for the whole selected for homogeneity analysis.
amount of material involved and not the individual test
3.2.11 within-sample homogeneity, n—the degree to which
specimens, pieces or portions being tested.
the structure or composition with respect to one or more
3.1.6 test unit, n—the total quantity of material (containing
specified properties of a single sample is similar throughout.
one or more test specimens) needed to obtain a test result as
3.2.11.1 Discussion—For example, if a can of paint is
specified in the test method. (See test result.)
allowed to sit for a long period of time, the components of the
3.1.6.1 Discussion—In this guide, the term sample is used
paint tend to separate into layers. A test specimen from the top
interchangeably with test unit and is defined in the context of
of the can would not be expected to have the same test
this guide as a quantity of material, an item, or an artifact
properties as a specimen taken from the bottom of the can and
provided to a laboratory to be tested as a whole or to be
the within-sample homogeneity would be poor. If the paint in
subdivided into multiple test specimens by sub-sampling in a
the can is stirred before sampling, the within-sample homoge-
manner consistent with industry protocols most suitable for
neity would be improved and test specimens from the top and
that type of material, item or artifact. E2282
bottom of the can would be expected to have similar test
3.2 Definitions of Terms Specific to This Standard: results.
E3264 − 21
3.2.12 within-sample variance, s , n—average of the cell methods, proficiency testing programs, and studies related to
w
variances for the set of samples selected for homogeneity quality control of testing within a single laboratory.
analysis.
5.2 Because the test results of any laboratory study are
affected by the quality of the samples tested, producing
3.2.12.1 Discussion—Assuming that the replicate test speci-
homogeneous samples and determining the degree of homoge-
mens obtained from each sample are identical, the within-
neity is important for interpreting the results of the study.
sample variance is an estimate of the repeatability precision of
the test method.
5.3 Five techniques are presented in this guide to evaluate
sample homogeneity for a range of circumstances and degrees
4. Summary of Guide of rigor. The circumstances under which the studies are
conducted and the degree of rigor required may differ. The user
4.1 The guide presents techniques and provides recommen-
should consider the circumstances listed in each technique to
dations for evaluating and assuring homogeneity of individual
determine which is appropriate for the study at hand.
samples or bulk materials and can be used for either interla-
5.4 Each of the Techniques 1, 2, and 3 provides a procedure
boratory or intra-laboratory studies.
for testing and evaluating sample homogeneity when replicate
4.2 Five techniques are presented covering a range of
testing of the samples is possible. Technique 4 provides a plan
circumstances encountered with laboratory studies of various
to evaluate sample homogeneity when replicate testing is not
types and purposes. Each technique describes the circum-
possible. Technique 5 recommends practices for producing
stances for which the technique is suited. See Appendix X1,
homogeneous samples for circumstances when homogeneity
Table X1.1 for an overview of the guide in tabular form.
testing is not possible.
4.3 Generally speaking, the basic steps for ensuring homo-
5.5 When the conditions of adequate within-sample homo-
geneous samples are (1) use a process that will produce
geneity and between-sample homogeneity are satisfied, any
samples that are as homogeneous as possible, (2) test a subset
differences in test results on multiple samples can reasonably
of samples (or a portion of the reference material), (3) screen
be attributed to testing variation and not due to sample
results for outliers, and (4) estimate the sample variation, and
variation.
(5) decide whether the samples are sufficiently homogeneous
5.6 When differences within or between samples are discov-
for the intended purpose.
ered and the samples are deemed insufficiently homogeneous,
4.4 To accomplish the first step, recommended practices for
the sample preparation process can be improved or corrected
preparing homogeneous samples and bulk materials for all
and a new set of samples can be prepared. Or, in cases where
types of laboratory studies are presented in Technique 5. Those
the sample homogeneity cannot be improved or for other
practices are also recommended for the preparation of samples
reasons when the samples must be used, the method of
used with Techniques 1, 2, 3, and 4.
evaluation for the laboratory study should account for the effect
of differences between samples.
4.5 For accomplishing the remaining steps, Techniques 1, 2,
3, and 4 are each designed to provide guidance for homoge-
5.7 When used in conjunction with studies to develop
neity testing and evaluation appropriate to the circumstances
precision estimates, the guidance in this standard can be used
described for that technique.
to help quantify sources of test variation (such as effects due to
sampling, test method repeatability, and the degree of inhomo-
4.6 The practices recommended in Technique 5 for produc-
geneity) and, therefore, can be useful for determining and
tion of individual samples (or units) distributed for use in a
stating the conditions under which the precision estimates are
laboratory study are also generally applicable to bulk quantities
valid.
of material. However, for simplicity, Techniques 1, 2, 3, and 4
in this guide do not specifically state how the techniques are to
5.8 For proficiency testing programs, the guidance in this
be applied to bulk materials. Since evaluation of the homoge-
standard can provide information to prevent laboratories from
neity of a bulk quantity of material requires testing of multiple
being unfairly penalized for testing variation due to inherent
representative samples obtained from the bulk quantity, evalu-
differences between samples.
ation of the homogeneity of a bulk material is accomplished by
5.9 In a single laboratory, the guidance in this standard
applying Techniques 1, 2, 3 or 4 to multiple representative
could be used to evaluate the homogeneity of samples for
samples obtained from the bulk quantity of material. The
studies to measure test variation over time or for studies to
results of homogeneity testing of the multiple representative
compare the results of tests performed by different technicians.
samples can then be used as an indication of the degree of
5.10 To minimize the resources required for homogeneity
homogeneity of the bulk material.
testing, a testing design using a minimum of ten samples with
two replicate tests performed on each sample is recommended
5. Significance and Use
in Techniques 1, 2, and 3 of this guide. This test design is used
5.1 This guide presents techniques and guidance for evalu-
in other international standards. See Ref (1) and ISO 13528.
ating and assuring homogeneity of individual samples or bulk
materials and can be used for either interlaboratory or intra-
laboratory studies. The types of studies include, but are not
The boldface numbers in parentheses refer to the list of references at the end of
limited to, studies to determine precision estimates for test this standard.
E3264 − 21
Technique 4, used when replicate testing is not possible, important to identify, quantify, and account for those variations
similarly recommends testing a minimum of ten samples. That when determining if the samples are suitable for the laboratory
does not preclude the use of more than ten samples or more study at hand.
than two replicates. 6.4.1 Sample homogeneity is evaluated for both, within-
NOTE 1—The spreadsheets provided in this guide for the examples in
sample variation and between-sample variation.
Techniques 1, 2, and 3 show the calculations when two replicate tests are
6.5 Effective homogeneity testing includes ways to detect
performed on each sample. The spreadsheets shown for Techniques 1 and
2 may be adjusted using the equations provided in the text when more than extreme test results and samples that are out of the ordinary;
two replicate tests are used. Use of Technique 3, as presented in Section
and ways to treat those instances where extreme test results or
9, is limited to duplicate testing (that is, k = 2). To use Technique 3 when
out of the ordinary samples are encountered. Techniques 1, 2,
k > 2, preliminary testing for consistency of replicate results can be
3, and 4 present different outlier routines to identify inconsis-
performed using the general form of the Cochran’s Test as presented in
tent test results or samples.
Technique 1, and the homogeneity analysis can be performed as described
in the Appendix, X4.3. Also, if desired, the homogeneity criterion in
6.5.1 The outlier routines were selected in an attempt to
Technique 3 can be used with the calculations using the spreadsheets
have calculations that match the homogeneity testing design
shown in Technique 2.
and the calculations used in the second step of the technique.
5.11 This guide is not sufficient for evaluation of certified
6.5.2 Although the outlier routines presented for identifying
reference materials (CRMs) or materials used for calibration.
inconsistent test results in Techniques 1, 2, and 3 are not
Even though homogeneity is required for CRMs, CRMs and
equivalent, they can generally be considered as alternatives.
calibration materials are typically subject to additional require-
See Note 2. Other suitable outlier routines may be substituted.
ments (such as traceability and estimates of uncertainty) that
NOTE 2—Cochran’s C-test and Mandel’s K-test are neither equivalent
are not addressed in this guide.
nor interchangeable as presented, but since C = K ⁄ n, the two tests can
be made equivalent with the right choice of critical values. For example,
6. General Considerations for Sampling and Evaluation
if there are 10 samples, Cochran’s C-test with 95 % “confidence” is
equivalent to Mandel’s K-test with 0.5 % significance, except for the
of Homogeneity
rounding of the tabulated critical values. Both critical values in this case
6.1 Effective assurance and evaluation of sample homoge-
are derived from the 99.5th percentile of the same F-distribution. In
neity requires good sample production, good sampling and
general, if the critical value for K is calculated as described in E691,
Annex A1, using the 1 − α ⁄n fractile of F, and if the K-test is applied in the
sub-sampling to obtain representative specimens for homoge-
same manner as Cochran’s C-test, the same outcome is achieved as
neity testing, and good testing of the representative specimens
Cochran’s C with significance level α. Cochran’s C-test is designed so that
used to evaluate the homogeneity.
if it were applied to every cell, it would give an expected number of false
6.1.1 In this guide, the term “homogeneity” applies in two
rejections. The K-test is designed to give an expected proportion of false
ways. The first is within-sample homogeneity. The second is
rejections, or a specified probability of false rejection for each cell.
between-sample homogeneity.
6.6 Techniques 1, 2, and 3 each analyze agreement of
replicate results for within-cell consistency. Failure to elimi-
6.2 As stated previously, the basic steps for ensuring homo-
nate extreme replicate results that truly represent either poor
geneous samples are (1) use a process that will produce
sub-sampling or a testing error will tend to inflate the apparent
samples that are as homogeneous as possible, (2) test a subset
within-sample variability and cause an overestimate of the
of samples (or a portion of the reference material), (3) screen
results for outliers, and (4) estimate the sample variation, and within-sample variance, s . Conversely, falsely eliminating
w
extreme results that are truly representative of expected testing
(5) decide whether the samples are sufficiently homogeneous
for the intended purpose. To assure a suitable outcome, care variation has the opposite effect. Both types of error can affect
the final evaluation of sufficient between-sample homogeneity
and critical consideration are needed at each step.
for the laboratory study at hand. Therefore, extreme results
6.3 Problems that affect initial sample quality can occur
should be investigated to confirm whether the data are valid.
during sample production. Examples are such things as mate-
Valid data are those data that would be reported as resulting
rials from a low-grade source, improper handling or storage of
from the normal performance of laboratory testing of repre-
materials or components, poor control of the production
sentative samples. Possible causes of invalid data include
process, improper or poorly maintained equipment, inadequate
deviations from the test method, instrument malfunctions,
blending, sloppy fabrication, use of an ineffective sampling
unexpected occurrences during testing, arithmetic errors, and
procedure, segregation or separation of components, and deg-
typographical errors. After investigation, data deemed to be
radation. To address some of these issues, the practices
invalid should be excluded from further analysis. See Ref (2).
recommended in Technique 5 provide guidance for producing
good samples. Strict compliance with the practices in Tech- 6.7 Techniques 1, 2, 3, and 4 also provide guidance for
nique 5, or established sampling practices from other sources, eliminating extreme samples (or cells) before calculating the
appropriate for the material under study are recommended. between-sample variance, s .
s
Those same practices, from Technique 5 or other sources, are
6.8 As with the analysis of replicate testing, failure to
also recommended for the preparation of samples used with
eliminate the extreme results of a sample that truly represent a
Techniques 1, 2, 3, and 4.
testing or production error will tend to inflate the apparent
6.4 Regardless of the care taken in sample production, there between-sample variability and cause an overestimate of the
will always be some degree of sample variation and some between-sample variance, s . Conversely, falsely eliminating
s
degree of testing variation when testing for homogeneity. It is the results of an extreme sample that is truly representative of
E3264 − 21
expected variation between samples has the opposite effect. 7.2 Circumstances—This technique can be used to deter-
Again, both types of errors can affect the final evaluation of mine if the homogeneity between samples is acceptable for the
intended purpose when all of the following circumstances
sufficient between-sample homogeneity for the laboratory
apply:
study at hand.
7.2.1 An expected target standard deviation (σ ) for the
ET
6.9 When evaluating extreme values (whether analyzing
laboratory study in which the samples will ultimately be used
consistency of replicate test results or consistency between
is unknown or cannot be specified prior to the study.
samples), it can be difficult to distinguish between a bad test
7.2.1.1 An example is when the precision of the test method
result and a bad sample. So, it is important to have as much
used in the laboratory study is not known. This may be the case
confidence as possible in any sampling performed to obtain
during the development of a new test method.
homogeneity test specimens and in the competence of the
7.2.1.2 Another example would be when using an existing
analysis testing performed on the homogeneity test specimens.
test method on a new material type.
Therefore, use of competent testing agencies along with
7.2.2 Homogeneity testing of the entire set of samples is
well-established sampling practices and test methods are rec-
performed under repeatability conditions.
ommended for evaluating sample homogeneity.
7.2.3 Since replicate (or duplicate) testing is necessary for
homogeneity testing using this technique, the samples to be
6.10 Generally, the test method used for homogeneity test-
used in the laboratory study must be such that replicate testing
ing is the test under study. However, special consideration
can be performed on each sample. Replicate testing may be
should be given when each sample will be subjected to a
performed by retesting the entire sample or by testing replicate
battery of different tests. It is possible that homogeneity may be
specimens obtained as representative sub-samples of the origi-
sufficient for one test property, but not sufficient for another test
nal sample. The technique would not be appropriate for a
property determined using a more discriminating test method.
destructive test method where the entire sample must be used
Ideally the homogeneity of the samples would be evaluated for
for the test specimen.
each test method to which the samples are to be subjected. If
7.2.4 During homogeneity testing, it can be assumed that a
this is not practical, a single test method may be used to
high degree of homogeneity has been achieved within each
evaluate homogeneity. In this case the test method used for
individual sample and the sampling variance associated with
homogeneity testing (it need not necessarily be one of the test
obtaining replicate specimens is small.
methods included in the battery of tests used in the study)
7.2.5 The replicate specimens (obtained by sub-sampling)
should be chosen carefully to be one that is practical (in terms
from each sample are essentially identical and the within-
of time and expense) and can most likely be able to discrimi-
sample standard deviation is a good estimate of the repeatabil-
nate differences between samples that would affect the results
ity of the test method.
of the various tests included in the battery of tests.
7.2.6 The test method used for homogeneity testing is
6.11 In some cases, it may be desirable to consider the type capable of discerning differences between the samples to the
degree necessary for purposes of the study. This can usually be
of statistical distribution that best describes the test results. The
evaluation techniques used in this guide are based on a normal assumed to be true if the test method used for homogeneity
testing is the same test method as will be used in the laboratory
distribution and assume that the normal (Gaussian) model is
study.
adequate for the evaluation of sample homogeneity. If the
7.2.6.1 In some cases, a proxy test method, of equal or
homogeneity test results are not normally distributed, the
better discrimination power for the property of interest, can be
homogeneity evaluation and the decision regarding whether the
used for the homogeneity testing. That is a test method other
samples are sufficiently homogeneous can be affected.
than the one to be used in the laboratory study can be used for
NOTE 3—An assumption of a normal distribution is usually adequate.
testing sample homogeneity. This may be done in order to
However, if there is concern regarding the type of distribution that best
describes the results, the results can be subjected to investigation for better discriminate differences between samples, or for practi-
normality. The normality of the test results can be visually evaluated using
cal reasons such as time or expense. For example, it may be
a normal probability plot. If further investigation is desired, an example of
desirable to use a test for the particle size distribution of soil
a statistical normality test is the Anderson-Darling test for normality. A
samples as a substitute to evaluate the homogeneity of samples
description of the Anderson-Darling test for normality is not included in
used in a study of a different physical property of the soil such
this guide, but the test is available in some statistical software packages.
The issue of normality is discussed in ASTM Practice D6299. The as compaction density.
construction of a normal probability plot and the Anderson-Darling test
7.3 Design—Select n ≥ 10 random samples from the group
are described in the Annex of D6299.
of samples to be distributed for testing. Perform k ≥ 2 replicate
tests on each sample. Replicate testing may be performed by
7. Technique 1 for Evaluating Homogeneity
retesting the entire sample or by testing replicate specimens
7.1 Objective—The objective of homogeneity testing using
obtained as representative sub-samples of the original sample.
this technique is to verify that the variance of test results
Perform testing of the entire set of samples under repeatability
between samples (estimated by the homogeneity testing pro-
conditions. To reduce possible effects of testing trends, the
tocol proposed in this technique) is not significantly larger than replicate tests must be performed in random order. For each
the variance of test results of replicate testing on individual
sample, list the replicate test results as X , X , . , X where
i,1 i,2 i,k
samples. See the criterion in 7.7. i corresponds to the cell (or row) number, as in Table 1.
E3264 − 21
TABLE 1 Technique 1 Example of Preliminary Test for Consistency of Replicate Results Using Cochran’s Test
1 2 3 4 5 6
Cell Average (or Mean) of A
Cell Variance
“i” refers to the Cell
A
2 2
Sample Replicate 1 Replicate 2 the Replicates
¯ ¯
sX 2 X d 1sX 2 X d
(that is, row)
i,1 i i,2 i
sX 1 X d⁄ k
i,1 i,2
k21
¯ 2
ID/No. Cell, i X X
X s
i,1 i,2 i
i
FM1 1 3.0762 3.0491 3.06265 0.0003672
FM2 2 3.0799 3.0646 3.07225 0.0001170
FM3 3 3.0588 3.0589 3.05885 0.0000000
FM4 4 3.0502 3.0621 3.05615 0.0000708
FM5 5 3.0506 3.0750 3.06280 0.0002977
FM6 6 3.0761 3.0627 3.06940 0.0000898
FM7 7 3.0797 3.0636 3.07165 0.0001296
FM8 8 3.0466 3.0745 3.06055 0.0003892
FM9 9 3.0571 3.0541 3.05560 0.0000045
FM10 10 3.0576 3.0573 3.05745 0.0000000
FM11 11 3.0520 3.1325 3.09225 0.0032401
A
The formulas in Table 1 are simplified for k = 2.
7.3.1 An example data set is shown in Columns 1, 2, 3, and s in Column 6 of Table 1, where
i
4 of Table 1 where n = 11 and k = 2. The replicate results listed
2 2 2
2 ¯ ¯ ¯
@ #
s 5 ~X 2 X ! 1 ~X 2 X ! 1 … 1 ~X 2 X ! ⁄ ~k
in Columns 3 and 4 of Table 1, and displayed in Fig. 1, are for i i,1 i i,2 i i,k i
the fineness modulus of a fine aggregate material. 2 1! (3)
n 2
7.4 Preliminary Test for Consistency of Replicate Results
7.4.4 Calculate the sum of the Cell Variances, Σ s , where
i51 i
Using the Cochran’s Test:
n 2 2 2 2
Σ s 5 s 1s 1…1s (4)
i51 i 1 2 n
7.4.1 This step is to assure that the homogeneity tests have
n 2
all been performed properly and that there are no invalid values
For the example, Σ s = 0.004706.
i51 i
in the list of replicate test results. See Cochran’s Test in Ref (3)
7.4.5 Identify the largest of the Cell Variances, s , in Table
max
and (4).
1, Column 6. In the example, s = 0.0032401.
max
7.4.2 For each sample, calculate the Cell Average of the
Replicates, listed as X in Column 5 of Table 1, where k = 2 for
7.4.6 Then calculate the Cochran’s Test statistic, C, where
i
replicate results, X and X , and
i,1 i,2
2 n 2
C 5 s ⁄ Σ s (5)
max i51 i
¯
X 5 X 1 X 1 … 1 X ⁄ k (2)
~ !
i i,1 i,2 i,k 2 n 2
For the example, C5s ⁄ Σ s = 0.0032401/0.004706 =
max i51 i
7.4.3 For each sample, calculate the Cell Variance, listed as 0.6885.
FIG. 1 All Replicate Results
E3264 − 21
7.4.7 Determine the Cochran’s Test critical value, C ,
crit
X , where n is the number of samples remaining after cells
homog h
from Table 2, where n = 11. It is recommended that the
containing invalid replicate results were eliminated.
Cochran’s Test statistic be evaluated at the 95 % or 99 %
n
5 h
confidence level. The example uses the 99 % confidence level.
¯
X 5 X (7)
homog h
(
n
For the example, C = 0.6852. h51
h
crit
7.4.8 If the value of the test statistic, C, exceeds its critical
For the example, the Cochran’s Test for variance outliers
value, C , listed in Table 2, then the data for the sample
crit
indicated that replicate results for sample FM11 were incon-
2 2
corresponding to the largest s (that is, s ) are considered
i max
sistent and, assuming an investigation determined the results to
inconsistent and should be investigated to confirm whether the
be invalid, sample FM11 was eliminated from further analysis.
data are valid.
Therefore n = 10 and the value for the Overall Average is
h
7.4.9 In the example, C = 0.6885, is greater than the critical
X = 3.062735. See Column 6 of Table 3.
homog
value of 0.6852 for i = 11. Therefore, duplicate results for
7.5.4 For each sample, calculate the Square of the Deviation
Sample FM11 are inconsistent with the duplicate results of the
of each Replicate from the Cell Average, d .
other samples and, assuming an investigation has found suffi- h,j
cient reason to believe the data to be invalid, Sample FM11 is
¯
d 5 ~X 2 X ! (8)
h,j h,j h
eliminated from further analysis. If there were not sufficient
reason to believe the data for Sample FM11 were invalid, the where h = 1, 2, ., n and j = 1, 2, ., k. In the example, d
h h,j
data would have been retained for the homogeneity analysis in
values are listed in Columns 7 and 8 of Table 3.
7.5.
7.5.5 For each sample, calculate the Total of the Squares of
7.4.10 A column chart of the cell variance for each of the
the Deviations of the Replicates for each Cell, d , where
h,total
samples is a good graphic for this application. See Fig. 2. In the
2 2 2 2
example, the value of the cell variance for Sample 11 appears d 5 d 1d 1…1d (9)
h,total h,1 h,2 h,k
to be unusually large, visually verifying the results of the
See Column 9 of Table 3.
Cochran’s Test for variance outliers.
7.5.6 Calculate the Sum of Squares Within Samples (or
7.5 Homogeneity Analysis — Estimation of Within-Sample
Cells), SS , where
w
Variation:
n 2
h
SS 5 Σ d (10)
w h51 h,total
7.5.1 This step uses the replicate test results remaining after
cells (that is, samples) containing invalid test results have been For the example, SS = 0.001466, the sum of the values in
w
eliminated. It is assumed that the replicate test results were Column 9 of Table 3.
obtained under repeatability conditions so that the estimation
7.5.7 Calculate the Within-Sample Degrees of Freedom, d ,
fw
of the within-sample variation will represent testing precision and the Mean Square Within Samples, MS , where
w
comparable to the repeatability that can be expected for the
df 5 n k 2 1 (11)
~ !
w h
material under study. The calculations use the F-test in a
and
one-way analysis of variance to determine whether there are
any statistically significant differences between the means (that
MS 5 SS ⁄ df (12)
w w w
is, within-sample means) of multiple independent groups (that
For the example, d = 10 (2 – 1) = 10 and MS = 0.001466
is, cells) and can be found in textbooks. fw w
/ 10 = 0.0001466.
7.5.2 For the n samples remaining after samples with
h
invalid replicate results have been removed, list the replicate
7.6 Homogeneity Analysis – Estimation of Between-Sample
test results as X ,X ,… ,X where h corresponds to the new
Variation:
h,1 h,2 h,k
cell (or row) number, as shown in Columns 3 and 4 of Table 3.
7.6.1 For each sample, calculate the Square of the Deviation
For each sample, calculate the Cell Average of the replicate test ¯
of the Cell Average from the Overall Average, ~X 2 X ! ,
h homog
¯
results, X . For the example, see Column 5 of Table 3.
h as listed in Column 10 of Table 3.
¯
7.6.2 Multiply the sum of the values in Column 10,
X 5 X 1 X 1 … 1 X ⁄ k (6)
~ !
h h,1 h,2 h,k
n ¯
h
Σ ~X 2 X ! , by the number of replicates, k, to determine
h51 h homog
7.5.3 For the example, the number of replicate results k = 2.
Sum of Squares Between Samples, SS , where
b
Calculate the Overall Average of All Replicate Test Results,
5 2
n ¯
h ~ !
SS 5 kΣ X 2 X (13)
b h51 h homog
5 2
A n ¯
h ~ !
TABLE 2 Cochran’s Test Critical Values for k = 2 For the example, Σ X 2 X = 0.0003565 and SS =
h51 h homog b
n 95 % Confidence 99 % Confidence 2(0.0003565) = 0.0007130.
7 0.7271 0.8376
7.6.3 Calculate the Between-Sample Degrees of Freedom,
8 0.6798 0.7945
df , and the Mean Square Between Samples, MS , where
9 0.6385 0.7544
b b
10 0.6020 0.7175
df 5 n 2 1 (14)
~ !
b h
11 0.5715 0.6852
12 0.541 0.6528
and
A
Values of C are from Table 15 of Ref (3).
crit
MS 5 SS ⁄ df (15)
b b b
E3264 − 21
FIG. 2 Column Chart of Cell Variances, s
i
TABLE 3 Technique 1 Example for Homogeneity Analysis
1 2 3 4 5 6 7 8 9 10
Square of the Square of the Total of the
Square of the
Overall Average
Squares of the
Deviation of Deviation of
Cell Average of
“h” refers to the Deviation of the
of All Replicate
Replicate 1 Replicate 2 Deviations of
A
Sample Cell (that is, Replicate 1 Replicate 2 Replicates Cell Average
Test Results
the Replicates
from the Cell from the Cell
n
row) h from the Over-
sX 1 X d⁄k 1 A
h,1 h,2 Average Average
¯ for Each Cell
X
o
h 2 2 all Average
n
h51 ¯ ¯ 2 2
h
sX 2 X d sX 2 X d
d 1d
h,1 h h,2 h h,1 h,2
5 5 2
¯ 2 2 2
¯
ID/No. Cell, h X X d d d
X s d
h,1 h,2 h X h,1 h,2 h,total X 2 X
homog h homog
FM1 1 3.0762 3.0491 3.06265 3.062735 0.00018360 0.00018360 0.000367 0.000000007
FM2 2 3.0799 3.0646 3.07225 3.062735 0.00005852 0.00005852 0.000117 0.000090535
FM3 3 3.0588 3.0589 3.05885 3.062735 0.00000000 0.00000000 0.000000 0.000015093
FM4 4 3.0502 3.0621 3.05615 3.062735 0.00003540 0.00003540 0.000071 0.000043362
FM5 5 3.0506 3.0750 3.06280 3.062735 0.00014884 0.00014884 0.000298 0.000000004
FM6 6 3.0761 3.0627 3.06940 3.062735 0.00004489 0.00004489 0.000090 0.000044422
FM7 7 3.0797 3.0636 3.07165 3.062735 0.00006480 0.00006480 0.000130 0.000079477
FM8 8 3.0466 3.0745 3.06055 3.062735 0.00019460 0.00019460 0.000389 0.000004774
FM9 9 3.0571 3.0541 3.05560 3.062735 0.00000225 0.00000225 0.000004 0.000050908
FM10 10 3.0576 3.0573 3.05745 3.062735 0.00000002 0.00000002 0.000000 0.000027931
n
h
ss 5 d 50.001466
o
w h,total
h51
n
h 5 2
¯
s d
X 2 X 50.0003565
o h homog
hj51
A
The formulas in Table 3 are simplified for k = 2.
For the example, df = (10 – 1) = 9, and MS = 0.0007130 / 7.7.2 Typically, the samples are deemed sufficiently homo-
b b
9 = 0.0000792. geneous if F ≤ F for α = 0.05. However, the value of α is
crit
somewhat arbitrary.
7.7 Homogeneity Analysis — Evaluation and Application of
7.7.3 Calculate the value of the F-Statistic where
Technique 1 Criterion:
7.7.1 The homogeneity of the samples can be evaluated
F 5 MS ⁄ MS (16)
b w
using the F-test to compare MS to MS , where df = (n – 1)
b w b h
and df = n (k – 1). For the example, F = 0.0000792 / 0.0001466 = 0.54.
w h
E3264 − 21
7.7.4 Determine the critical value of the F-Statistic, F , 8.2.3.2 For intra-laboratory testing, where repeatability (σ )
crit r
from Table 4, where α = 0.05, df = 9 and df = 10. may be the basis for the value of σ , it is obvious that the
b w ET
For the example, F = 3.02. above requirement cannot be met and another means must be
crit
7.7.5 The samples are considered sufficiently homogeneous used to assess the test method used for homogeneity testing.
if the value of the F-Statistic does not exceed F . In the
8.2.3.3 Another criterion for assessing the test method used
crit
example, F ≤ F (that is, 0.54 < 3.02) and the homogeneity of for homogeneity testing is the ratio of σ / (expected test
crit
r
the samples is deemed satisfactory.
result). If the repeatability is deemed by the user to be
sufficiently small compared to the expected test result for the
8. Technique 2 for Evaluating Homogeneity
samples, the test method may be accepted as satisfactory.
8.2.4 Sample preparation and sample homogeneity can be
8.1 Objective—The objective of homogeneity testing using
precisely controlled to a degree comparable to the precision of
this technique is to verify that the between-sample standard
the test method used for the homogeneity evaluation. Little is
deviation (s ) of the samples to be distributed for laboratory
s
gained by homogeneity testing in accordance with this tech-
testing is sufficiently small as to have little effect on the results
nique if sample differences are identified and cannot be
of the laboratory study – that is, heterogeneity between
eliminated from the sample preparation process.
samples is deemed sufficiently small in comparison to the
8.2.5 Since replicate testing is necessary for homogeneity
expected standard deviation (σ ) for the testing to be analyzed
ET
testing using this technique, the samples to be used in the
in the laboratory study. See the criterion in 8.7.
laboratory study must be such that replicate testing can be
8.2 Circumstances—This technique can be used when all of
performed on each sample. Replicate testing may be performed
the following circumstances apply (The circumstances are
by retesting the entire sample or by testing replicate specimens
similar to those of Technique 3.):
obtained as representative sub-samples of the original sample.
8.2.1 The expected target standard deviation in the labora-
8.2.6 The sub-sampling variance associated with obtaining
tory study (σ ) is known, or can be specified, in advance.
ET
replicate specimens for homogeneity testing is small relative to
8.2.1.1 For example, for interlaboratory studies of test
the within-sample (that is, repeatability) testing variance.
results from different laboratories, the value of σ may be
ET
based on the published reproducibility standard deviation (σ ) 8.3 Design—Select n ≥ 10 random samples from the group
R
for the test method used in the study. For proficiency testing of samples to be distributed for testing. Perform k ≥ 2 replicate
programs, σ may be the value of the standard deviation used tests on each sample. Replicate testing may be performed by
ET
for the evaluation of laboratory testing. For intra-laboratory retesting the entire sample or by testing replicate specimens
studies, the value of σ may be based on the repeatability obtained as representative sub-samples of the original sample.
ET
standard deviation (σ ). The value used for σ may also be Perform testing of the entire set of samples under repeatability
r ET
derived from the requirements of a specification, industry conditions. To reduce possible effects of testing trends, the
standard or other source. replicate tests must be performed in random order. For all
8.2.1.2 The value for the estimate for σ may not be replicate tests, list the replicate test results as X ,X ,… ,X
ET i,1 i,2 i,k
available until after the first round of the laboratory study has
where i corresponds to the cell (or row) number in Table 5.
been completed. In that case, if there are any available data,
8.3.1 An example data set using n = 11 and k = 2 is shown
even from an internal round robin study, that data could
in Table 5. The replicate results listed in Columns 2 and 3 of
provide a provisional estimate for σ .
ET
Table 5, and displayed in Fig. 3, are for the fineness modulus
8.2.2 Homogeneity testing of the entire set of samples is
of a fine aggregate
...