EPA-600-R-03-027
Guidance for Obtaining
Representative Laboratory
Analytical Subsamples from
Particulate Laboratory
Samples
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* EPA/600/R-03/027
° November 2003
Guidance for Obtaining
Representative Laboratory
Analytical Subsamples from
Particulate Laboratory Samples
by
Robert W. Gerlach
Lockheed Martin Environmental Services
and
John M. Nocerino
U.S. Environmental Protection Agency
U.S. Environmental Protection
Region 5, Library (PL-12J)
77 West Jackson Boulevard, 12th Floor
Chicago. IL 60604-3590
223CMB03.RPT * 1/8/04
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Notice
The U.S. Environmental Protection Agency (U.S. EPA), through its Office of Research and
Development (ORD), fancied the work described here under GSA contract number GS-35F-4863G (Task
Order Number 9T1Z006TMA to Lockheed Martin Environmental Services). It has been subjected to the
Agency's peer and administrative review and has been approved for publication as an EPA document.
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Foreword
The basis for this document started in 1988. We were in a quality assurance research group dealing
with the analysis of many different kinds of samples. Historically, the focus of our work was on the
analytical method, and sampling was pretty much taken for granted. However, it soon became clear that
sampling is perhaps the major source of error in the measurement process, and, potentially, sampling is an
overwhelming source of error for heterogenous particulate materials, such as soils. It was also clear that
classical statistical sampling theory was not adequate for such samples. Simple random sampling may
work for very "homogeneous" samples, for instance, marbles of the very same size, weight, and shape
where the only difference is the color of the marble. But the color of the marble is not a factor that
contributes to the selection process of that marble! To be an effective sampling method, the factors that
contribute to the selection process must be considered.
We knew that geostatistics offered some answers, such as the sample support (mass, volume, and
orientation) and particle size (diameter) make a difference. That is only common sense. The larger the
mass of the sample, the closer it should resemble the composition of the lot that it came from. But, taking
ever larger (or more) samples was not a practical answer to getting a representative sample. Less intuitive
may be that most of the heterogeneity should be associated with the larger particles and fragments.
However, grinding an entire lot of material to dust was also not a practical alternative.
We searched for a non-conventional statistical sampling theory that actually takes into account the
nature of particulate materials and, in 1989, we hit "pay dirt." Dr. Francis Pitard offered a short course at
the Colorado School of Mines on the Pierre Gy sampling theory for particulate materials. Dr. Pitard had
taught this course many times before to mining students, but this was his first offering directed toward the
environmental community. Although this theory was developed in the mid-1950s by the French chemist,
Pierre Gy, the theory was not widely known to those outside of the mining community, and it was
seemingly only put into practice by a few mining engineers where the bottom line really counts, namely,
gold mining. Dr. Pitard had the foresight to see the importance of introducing this theory to the
environmental sciences.
Needless to say, we came back from the short course very excited that we had found our answer. But
it was a hard sell. Over the ensuing years, we were only moderately successful at transferring this
technology to the environmental community so that it might be implemented. We started by sponsoring a
couple of short courses given by Dr. Pitard and we distributed some technical transfer notes. Although
this theory has proven itself in practice many times over in the mining industry, there has been very little
published with substantiating experimental evidence for this theory (it has been virtually nonexistent in
the environmental arena). The effectiveness of the Gy theory, and the extent to which it is applicable,
was also not well-established for environmental samples. Therefore, we were compelled to start a
research program to explore the effectiveness and the application of the Gy theory for all types of
environmental samples, and, where there are limitations, to expand upon the theory. Such a research
program would not only help to provide the needed (and published) experimental verification of the Gy
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theory, but it should also give credence to the theory for those not yet convinced (and justify the
application) of this theory for the environmental sciences.
We started our experimental investigations on the various Gy sampling theory errors, using fairly
"uncomplicated" matrix-analyte combinations, as applied to obtaining a representative analytical
subsample (the material that gets physically or chemically analyzed) from a laboratory sample (the bottle
that comes to the laboratory containing the sample to be analyzed). We felt that this was the easiest place
to start, using our limited resources, while still producing an impact. The weakest link, and the potential
for the most error, could very well be from taking a non-representative grab sample "off the top" of the
material in the laboratory sample bottle! (By the way, the Gy theory defines what a representative sample
should be.) The result of our ongoing investigations is the first version of this guidance document. We
welcome any (constructive) comments.
This document provides general guidelines for obtaining representative samples for the laboratory
analysis of particulate materials using "correct" sampling practices and "correct" sampling devices.
However, this guidance is general and is not limited to environmental samples. The analysis is also not
limited to the laboratory; that is, this guidance is also applicable to samples analyzed in the field. The
information in this guidance should also be useful in making reference standards as well as taking
samples from reference standards. Similarly, this guidance should be of value in: monitoring laboratory
performance, creating performance evaluation materials (and how to sample them), certifying
laboratories, running collaborative trials, and performing method validations. For any of those
undertakings, if there seems to be a lot of unexplained variability, then sampling or sample preparation
may be the culprit, especially if one is dealing with heterogeneous particulate materials.
The material presented here: outlines the issues involved with sampling particulate materials,
identifies the principal causes of uncertainty introduced by the sampling process, provides suggested
solutions to sampling problems, and guides the user toward appropriate sample treatments. This
document is not intended to be a simple "cookbook" of approved sampling practices.
The sections of this guidance document are divided into the following order of topics: background,
theory, tools, observations, strategy, reporting, and a glossary. Many informative references are provided
and should be consulted for more details. Unless one is familiar with the Gy sampling theory, correct
sampling practices, and correct sampling devices, it is strongly recommended that one reads through mis
document at least once, especially the section on theory. The glossary can easily be consulted for
unfamiliar terms. If one is familiar with the Gy sampling theory and is just interested in developing a
sampling plan, or simply wants to answer the question, "How do I get a representative analytical
subsample?", then go ahead and jump to the section on "Proposed Strategies." This section gives a
general and somewhat extensive strategy guide for developing a sampling plan. A sampling strategy can
be general, and not all of it, necessarily, has to be followed. However, a sampling plan is necessarily
unique for each study. Any sampling endeavor should have some sort of sampling plan.
The basic strategic theme in this document is that if "correct" sampling practices are followed and
"correct" sampling devices are used, then all of the sampling errors should become negligible, except for
the minimum sampling error that is fundamental to the physical and chemical composition of the material
being sampled. Since this minimum fundamental sampling error can be estimated before any sampling
takes place, one can use this relative variance of the fundamental error to develop a sampling plan.
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At first, it may seem that following this guidance is a lot of effort just to analyze a small amount of
material. And, when one is in a hurry and has a large case load, it may seem downright overwhelming.
But, remember that the seemingly simple task of taking a small amount of material out of a laboratory
sample bottle could possibly be the largest source of error in the whole measurement process. And not
taking a representative subsample could produce meaningless results, which is at the very least a waste of
resources and, at the very most, could lead to incorrect decisions.
Remember that sampling is one of those endeavors that you "get what you pay for," at least in terms
of effort. But, with the right knowledge and a good sampling plan, the effort is not necessarily that much.
It pays to have a basic understanding of the theory. Become familiar with what causes the different
sampling errors and how to minimize them through correct sampling practices. For example, always try
to take as many random increments as you can, with a correctly designed sampling device, when
preparing your subsample; and if you can only take a few increments, then you are still better off than
taking a grab sample "off the top" from the sample bottle, and you will at least be aware of the
consequences. Be able to specify what constitutes a representative subsample. Know what your
sampling tools are capable of doing and if they can correctly select an increment. Always do a sample
characterization (at least a visual inspection) first. At a minimum, always have study objectives and a
sampling plan for each particular case. If possible, take a team approach when developing the study
objectives and the sampling plan. Historical data or previous studies should be reviewed. And be sure to
record the entire process!
An understanding of the primary sources of sampling uncertainty should prevent unwarranted claims
and guide future studies toward correct sampling practices and more representative results. Best wishes
with all of your sampling endeavors.
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Acknowledgments
The authors express their gratitude to the following individuals for their useful suggestions and their
timely review of this manuscript: Brian Schumacher (U.S. EPA), Evan Englund (U.S. EPA), Chuck
Ramsey (EnviroStat), Patricia Smith (Alpha Star), and Brad Venner (U.S. EPA). The authors also convey
their thanks to Eric Nottingham and the U.S. EPA National Enforcement Investigation Center (NEIC) for
the use of their facilities in performing many of the laboratory experiments pertinent to this guidance.
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Dedication
This sampling guidance document is dedicated to Dr. Pierre Gy to commemorate his fifty years
toward the development and practice of his sampling theory and to Dr. Francis F. Pitard for his diligence
in proliferating the Gy sampling theory and other theories for particulate sampling, for his lifetime of
dedication to correct sampling practices, and for pointing those of us in the environmental analytical
sciences in the right direction. The authors sincerely hope that this work expresses our gratitude and not
our ignorance. We also dedicate this manuscript to all of those individuals that are involved with
sampling heterogenous particulate material and we welcome any suggestions for improvement to this
work.
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Abstract
An ongoing research program has been established to experimentally verify the application of the Gy
theory to environmental samples, which serves as a supporting basis for the material presented in this
guidance. Research results from studies performed by the United States Environmental Protection
Agency (U.S. EPA) have confirmed that the application of the Gy sampling theory to environmental
heterogeneous particulate materials is the appropriate state-of-the-science approach for obtaining
representative laboratory subsamples. This document provides general guidelines for obtaining
representative subsamples for the laboratory analysis of particulate materials using the "correct" sampling
practices and the "correct" sampling devices based on Gy theory. Besides providing background and
theory, this document gives guidance on: sampling and comminution tools, sample characterization and
assessment, developing a sampling plan using a general sampling strategy, and reporting
recommendations. Considerations are given to: the constitution and the degree of heterogeneity of the
material being sampled, the methods used for sample collection (including what proper tools to use), what
it is that the sample is supposed to represent, the mass (sample support) of the sample needed to be
representative, and the bounds of what "representative" actually means. A glossary and a comprehensive
bibliography have been provided, which should be consulted for more details.
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Table of Contents
Notice iii
Foreword v
Acknowledgments ix
Dedication xi
Abstract xiii
Section 1 — Introduction 1
1.1 Overview 1
1.2 Purpose 3
1.3 Scope and Limitations 4
1.4 Intended Audience and Their Responsibilities 4
1.5 Previous Guidance 5
1.6 The Measurement and Experiment Process 6
1.7 Data Quality Objectives 8
1.8 Defining the Term, Sample, and Other Related Terms 9
1.8.1 Heterogeneity 12
1.8.2 Laboratory Subsampling: The Need for Sample Mass Reduction 14
Section 2 - Overview of Gy Sampling Theory 15
2.1 Background 15
2.2 Uncertainty Mechanisms 16
2.3 Gy Sampling Theory: Some Assumptions and Limitations 16
2.4 Gy Sampling Theory: Errors 17
2.5 Subsample Selection Issues 18
2.6 The Relationship Between the Gy Sampling Theory Errors 19
2.7 The Short-Range Heterogeneity Fluctuation Error, CE! 22
2.8 The Fundamental Error (FE) - the Heterogeneity of Particulate Constitution 23
2.9 The Grouping and Segregation Error (GE) - the Heterogeneity of Particle Distributions .... 24
2.10 The Long-Range Heterogeneity Fluctuation Error (CE2) 27
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Table of Contents, Continued
2.1 1 The Periodic Heterogeneity Fluctuation Error (CE3) ................................. 28
2.12 The Increment Materialization Error (ME), the Increment Delimitation Error (DE) and the
Increment Extraction Error (EE): Subsampling Tool Design and Execution .............. 28
2.13 The Preparation Error (PE) - Sample Integrity ......................... ............. 34
2.14 The Importance of Correctly Selected Increments ...................... ............. 35
2.15 Increment Sampling and Splitting Sampling ....................................... 37
2.16 Correct Sampling (Correct Selection) Defined ..................................... 38
2.17 Representative Sample Defined ................................................. 39
Section 3 - Fundamental Error Fundamentals ............................................. 41
3.1 Estimating the Relative Variance of the Fundamental Error, s^2 ....................... 41
3.2 Estimating the Factors for the Relative Variance of the Fundamental Error Equation ....... 43
3.2.1 Estimating the Mineralogical (Composition) Factor, c ............. , ........... 43
3.2.2 Estimating the Liberation Factor, 1 ........................................ 44
3.2.3 Estimating the Shape Factor, f ............................................ 45
3.2.4 Estimating the Granulometric Factor, g ..................................... 46
3.2.5 Estimating the Nominal Particle Size, d .................................... 47
3.2.6 Estimating the Required Sample Mass, M,. .................................. 47
3,2.7 Rearrangement of the Relative Variance for the Fundamental Error Equation to
Determine the Sample Mass (MJ ......................................... 47
3.2.8 Two-tiered Variance Comparison to Estimate IHL and the Sample Mass (MJ ....... 48
3.2.9 Visman Sampling Constants Approach for Determining the Sample Mass (Mg) ..... 50
3.3 Developing a Sample Mass or Comminution Strategy: The Sampling Nomograph ........ 50
3.3.1 Constructing a Sampling Nomograph ............................ ........... 51
3,3.2 Hypothetical Example ........................................ ........... 53
3.3.3 Some Subsampling Strategy Points to Consider .............................. 56
3.4 Low Analyte Concentration Considerations ....................................... 57
3.4.1 Low-frequency of Analyte (Contaminant) Particles ........................... 57
3.4.2 A Low Concentration Approximation of s^2 ................................ 58
3.4.3 A Simplified Low Concentration Approximation of s^2 ....................... 59
Section 4 - Subsampling Techniques [[[ 61
4.1 Subsampling Methods [[[ 61
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Table of Contents, Continued
4.1.1.2 Disadvantages 62
4.1.2 Paper Cone Sectorial Splitting 62
4.1.3 Incremental Sampling 63
4.1.4 Riffle Splitting 64
4.1.5 Alternate Shoveling 65
4.1.6 Coning and Quartering 65
4.1.7 Rolling and Quartering 66
4.1.8 Fractional Shoveling 66
4.1.9 Degenerate Fractional Shoveling 67
4.1.10 Table Sampler 68
4.1.11 V-Blender 68
4.1.12 Vibratory Spatula 69
4.1.13 Grab Sampling 69
4.2 Minimizing Particle and Mass Fragment Correlation 69
4.3 Ranking Subsampling Methods 70
Section 5 - Comminution (Particle Size Reduction) Methods 73
Section 6 - Sample Characterization and Assessment 75
6.1 Identifying Important Sample Characteristics 75
6.2 Visual Characteristics 76
6.2.1 Analyte Particles: Color, Texture, Shape, and Number 76
6.2.2 Unique or Special Features 77
6.2.3 Density 77
6.3 Moisture Content and Thermally Sensitive Materials 77
6.4 Particle Size, Classification, and Screening Decisions 78
6.5 Concentration Distribution 80
6.6 Comminution (Grinding and Crushing) 80
6.6.1 Caveats 81
Section 7 - Proposed Strategies 83
7.1 The Importance of Historical and Preliminary Information 85
7.2 A Generic Strategy to Formulate an Analytical Subsampling Plan 86
7.3 An Example Quick Estimate Protocol 91
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Table of Contents, Continued
Section 8 - Case Studies 93
8.1 Case Study: Increment Subsampling and Sectorial Splitting Subsampling 93
8.2 Case Study: The Effect of a Few Large Particles on the Uncertainty from Sampling 95
8.3 Case Study: Sampling Uncertainty Due to Contaminated Particles with Different Size
Fractions 96
8.4 Case Study: The Relative Variance of the Fundamental Error and Two Components 96
8.5 Case Study: fflL Example 97
8.6 Case Study: Selecting a Fraction Between Two Screens 98
8.7 Case Study: Subsampling Designs 100
8.7.1 Subsampling Design A 101
8.7.2 Subsampling Design B 101
8.7.3 Subsampling Design C 102
8.7.4 SubsamplingDesignD 103
8.8 Case Study: Subsampling Designs Summary 103
Section 9 - Reporting Results 105
9.1 Introduction 105
9.2 For the Analyst 106
9.3 For the Scientists and Statisticians 108
9.4 For the Managers 109
9.5 For the Decision Makers 110
Section 10 - Summary and Conclusions Ill
Recommendations 113
References 115
Bibliography 119
Glossary of Terms 125
XVlll
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List of Figures
Figure # Description Page
1 The experiment and measurement process 7
2 A depiction of the sample acquisition process 12
3 Contour plot of contaminant level across a hazardous waste site 17
4 A depiction of the fundamental error (FE) 23
5 The grouping and segregation error (GE) 25
6 The effect of sample size when there are few analyte particles 26
7 The increment delimitation error (DE) 29
8 Increments selected with incorrect and correct devices 30
9 Correct and incorrect increment delimitation when sampling from a moving belt 31
10 (a) Delimitation for the extended increment 32
(b) Ideal increment extraction 32
(c) An increment extraction error (EE) can occur when particles cross the extended
increment boundary 32
(d) Delimitation for the extended increment for a cylindrical sampling device for a two-
dimensional sample 32
(e) The particles that get extracted into the increment have their center of gravity within
the extended increment boundary of the sampling device 32
11 For correct increment extraction, the shape of the sampling device's cutting edges must
be designed with respect to the center of gravity of the particle and its chance to be part
of the sample or part of the rejects 33
12 This highly segregated lot is randomly sampled with increments of the same total area .. 37
13 An example of a sampling nomograph 52
14 A sectoral splitter with eight sectors 54
15 A paper cone sectorial splitter with eight sectors 62
16 A riffle splitter with 20 chutes and two collection pans 64
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List of Figures, Continued
17 The alternate shoveling procedure 65
18 The coning and quartering procedure 65
19 The fractional shoveling procedure 67
20 The degenerate fractional shoveling procedure 67
21 A table sampler 68
22 A V-blender 68
23 (a) Sample estimate bias and cumulative bias versus run number for the
incremental subsampling runs 94
(b) Sample estimate bias and cumulative bias versus run number for the sectorial
sampling runs 94
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List of Tables
Table # Description Page
1 The seven steps in the DQO process 8
2 Gy sampling theory error types for particulate materials 18
3 Mechanisms for increased bias and variability due to sample integrity and the PE 35
4 Liberation parameter estimates by material description 45
5 Examples of shape parameters 46
6 Granulometric factor values identified by Gy (1998) 47
7 The relationship of the particle diameter, the analyte concentration, and the desired
uncertainty level to the sample mass 60
8 Authors' relative rankings (from best to worst) for subsampling methods 72
9 Soil particle classes by particle size, and classification names for soil fragments from
clay to boulders 79
10 The minimum sample mass, M,., and the maximum particle size, d, for SpE < 15%
(density = 2.5, analyte weight proportion = 0.05) 81
11 The influence of particle size on uncertainty 96
12 Case study: IHL example parameters 98
13 Case study: Parameters for selecting a fraction between two screens 99
14 A summary of the results from the case study designs 104
XXI
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Section 1
Introduction
Please note that there is a glossary in the back of this guidance document that should help the reader
understand unfamiliar words or concepts. For more extensive explanations of sampling topics that are not
covered in this text, please refer to the bibliography.
1.1 Overview
Unless a heterogeneous population (for example: a material, a product, a lot, or a contaminated site)
can be completely and exhaustively measured or analyzed, sampling is the first physical step in any
measurement process or experimental study of that population. The characteristics of collected samples
are used to make estimates of the characteristics of the population; thus, samples are used to infer
properties about the population in order to formulate new hypotheses, deduce conclusions, and implement
decisions about the population. The assumption is that the samples both accurately and precisely
represent the population. Without special attention to that assumption, sampling could be the weakest
link leading to the largest errors in the measurement process or the experimental study.
Sampling plays an especially important role in environmental studies and decisions. In most
environmental studies, field samples (often specimens) are collected from various field locations. The
characteristics measured in those samples and any consequent subsamples (a sample of a sample),
including laboratory subsamples, are then considered, de facto, as being "representative" of the site from
which they were collected. However, just because a sample comes from the site under consideration, it
does not mean that the sample represents that site. Considerations must be given about the constitution of
the material being sampled, the degree of heterogeneity of the material being sampled, the methods used
for sample collection (including what proper tools to use), what it is that the sample is supposed to
represent, the mass (sample support) of the sample needed to be representative, and the bounds of what
"representative" actually means. If a collection of samples does not represent the population from which
they are drawn, then the statistical analyses of the generated data may lead to misinformed conclusions
and consequent (and perhaps costly) decisions.
Technical issues related to subsampling seem to fall into a gap between the concerns about the
number and location of field samples (e.g., in a field sampling plan) and the concerns about the
performance of an analytical method for individual subsamples. This is partly because subsampling is
viewed as a transitional event that often appears trivial compared to the field activity and laboratory
Everything should be made as simple as possible, but not simpler.
- Albert Einstein
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analysis steps on either end of the measurement process. Hence, subsampling is rarely evaluated to assess
its effect on subsequent analysis steps or on decisions based on the results.
But, the error introduced by subsampling should not be ignored. The uncertainty associated with this
activity can exceed analytical method uncertainties by an order of magnitude or more (Jenkins et al.,
1997; and Gerlach et al, 2002). Biased results from incorrect sample mass reduction methods can negate
the influence of the best field sampling designs for sample location, number, and type. Improper
subsampling can lead to highly variable and biased analytical results that are not amenable to control
through standard quality control measures. This can cause misleading results for decision makers relying
on measurement results to support corrective actions.
Any lot (e.g., a site, a section from a site, or a batch) of particulate material consists of particles
having diverse characteristics with respect to size, shape, density, distribution, as well as physical and
chemical constitution. This diversity in the particle properties, the lot-specific uniqueness of the
distribution of the analytes of interest, and the uncertainties due to the subsampling techniques in the field
and the laboratory, often lead to a large variability among the analytical results of the samples that are
supposed to represent the lot.
Correct subsampling requires an understanding of those particulate material characteristics for the
population under study (e.g., a lot, a site, a sample) and the technical decisions that the results are
intended to support. The sample features, as well as the reasons for sampling, guide the sampler hi
identifying the sampling activities that are helpful and avoid sampling actions that can lead to increased
bias and uncertainty.
A collected sample must be both accurate and precise, within set specifications, at the same time in
order to be representative of the lot. This is true not only for the collection of the primary (or field)
sample, but also of any sample reduction or subsampling step. Such steps include sample preparation,
comminution (crushing or grinding), "homogenization," blending, weighing, and other mass reductions or
the splitting of samples. Taking out a portion of material from the laboratory sample bottle for weighing
and analysis (the analytical subsample) is a sample mass reduction step and should be performed with
"correct" subsampling practices in order to get a representative result. Laboratory subsampling errors
(e.g., incorrectly taking an aliquot from a sample bottle for analysis) could potentially overwhelm other
errors, including other sampling steps and the analytical error, associated with the analyses of samples. It
is quite a "lot" to ask of the tiny (on the order of a few grams, and often much lower) laboratory analytical
subsample to be representative of each of the larger and larger (parent) samples in the chain from which it
was derived, up to the entire lot (which could be many tons). Therefore, it is imperative that each
subsample is as representative as possible of the parent sample from which it is derived. Any
subsampling error is only going to propagate down the chain from the largest sample to the smallest
laboratory analytical subsample.
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1.2 Purpose
This guidance is a product of the ongoing research in our chemometrics program to improve or
develop methods to reduce data uncertainty in the measurement or experiment process. Since sampling is
usually a very early stage in that process, we searched for ways to reduce sampling errors and obtain
representative samples for particulate materials. Fortunately, there is an extensive and complete sampling
theory, known as the Pierre Gy sampling theory, developed mainly for the mining industry, that addresses
the issue of obtaining representative samples from particulate materials. Although this theory has proven
itself in practice in the mining industry, very little evidence exists in the literature that verifies this theory
experimentally, and this theory has only recently received attention for environmental studies. Our goals
are to verify Gy sampling theory experimentally for environmental particulate samples, discover any
limitations hi the theory for such samples, and to develop extensions to the theory if such limitations
exist.
Since the laboratory subsample can potentially have the greatest error in representing the lot and
because of its manageable size and relative simplicity (the long-range "field" type heterogeneities can be
regarded as trivial), we focused our initial experimental studies, and this ensuing guidance, on using
"correct" sampling methods to obtain "representative" laboratory analytical subsamples of particulate
materials. The terms, "correct" and "representative," will be used as defined by Francis Pitard (Pitard,
1993) and they will be described in detail in this document.
One of the main purposes of this document is to present a general subsampling strategy. Based on
that strategy, individual sampling plans may then be developed for each unique case that should produce
representative analytical subsamples by following correct sampling practices. By following correct
sampling practices, all of the "controllable" sampling biases and relative variances defined by the Gy
sampling theory should be minimized such that a representative subsample can simply be defined by the
relative variance of just one sampling error, the fundamental error (FE). This is the minimum and
"natural" relative variance associated with the lot (for our purposes, the primary laboratory sample from
which a representative analytical subsample is to be taken), and is based on the physical and chemical
characteristics and composition of the particulate materials (and other items) that make up that lot. Those
chemical and physical differences between the different items of the lot material are due to the
constitution heterogeneity (CH). The Gy sampling theory can quantify this relative variance of the
fundamental error (sra2) through an equation based on the chemical and physical characteristics of the lot.
Hence, we can estimate what the SFE 2 should be for the analytical subsample, and, therefore, should be
able to develop a strategy to obtain a representative analytical subsample a priori - that is, before
engaging in the subsampling operation - simply based on observations about the chemical and physical
characteristics of the lot!
Thus, this guidance identifies the subsampling activities that minimize biased or highly variable
results. This guidance also suggests which practices to avoid. Provided in this document is a general
introduction to subsampling, followed by specific suggestions and proposed laboratory subsampling
procedures. This guidance focuses on a strategy to minimize uncertainty through the use of correct
sampling techniques to obtain representative samples.
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1.3 Scope and Limitations
This guidance is not intended as a guide for field sampling at a hazardous waste site. The Agency has
developed a series of documents to assist in that process (see U.S. EPA 1994,1996a-c, 1997,1998, and
2000a-d). Correct sampling practices to obtain representative field samples is the subject of ongoing
research and a future guidance document.
Instead, the primary focus of this guidance document is on identifying ways to obtain representative
laboratory analytical subsamples, the ideal subsample being one with characteristics identical to the
original laboratory sample. This document provides guidance on the laboratory sample processing and
mass reduction methods associated with laboratory analytical subsampling practices. Laboratory
subsampling takes place every time an analyst selects an analytical subsample from a laboratory sample.
(This guidance is general and is not limited to environmental samples; it also applies to selecting field
analytical subsamples.)
This guidance focuses on the issues and actions related to samples composed of particulate materials.
It is not intended for samples selected for analysis of volatile or reactive constituents, and it does not
extend to sampling biological materials, aqueous samples, or viscous materials such as grease or oil
trapped in a particulate matrix, e.g., crude oil in beach aggregate. Research into the correct sampling of
those analytes and matrices is ongoing and guidance will be prepared once research results provide a
foundation for appropriate practices.
This guidance is intended as a technical resource for individuals who select subsamples for analysis
or other purposes, such as those individuals directing others in this activity. It also contains information
of interest to anyone else that deals with the subsampling of particulate material in a secondary manner,
including anyone reviewing study results from the analysis of particulate samples. Examples and
discussions relevant to these issues can be found in a number of the references. The following references
contain extensive or particularly valuable material on the topics in this document: Mason, 1992; Myers,
1996; Pitard, 1993; and Smith, 2001; also refer to the extensive bibliography at the end of this guidance
document.
1.4 Intended Audience and Their Responsibilities
Sampling is of critical interest to each person involved in the measurement process - from designing
the sampling plan to taking the samples to making decisions from the results. Sampling issues are
important in all aspects of environmental studies, including planning, execution (sample acquisition and
analysis), interpretation, and decision making.
Decision Makers should know enough about sampling to ask or look for supporting evidence that
correct and representative sampling took place. At a minimum, they should note whether or not
sampling concerns are addressed. However, their interest can extend to evaluating whether or not
the sampling activities meet the cost and benefit goals, result hi acceptable risks, or meet legal
and policy requirements.
Managers of technical studies should include "correct" sampling as an item that must be considered
in every study. They should identify whether or not sampling issues are addressed in the
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planning stage and if related summary information is presented in the final report. Their attention
is often focused on cost and benefit issues, but they should not lose sight of the technical
requirements that the results must meet.
Scientists and Statisticians need to address sampling issues with as much concern as they apply to
other statistical and scientific design questions, such as how many samples to take, which location
and time are appropriate, and what analytical method is compatible with the type of sample and
the required accuracy and precision. A clear statement of the technical issue(s) that need to be
answered should be available. A list of the required data and a discussion of how it will be
processed should be part of the study plans. The final report should include an assessment of the
effect that sampling had on the study. The importance of sampling should be assessed in the
context of all the other factors that might affect the conclusions as part of a standard sensitivity
analysis.
Laboratory and Field Analysts need to understand sampling issues to ensure that their activities
provide results that are appropriate for each study. Their results should be reported in the context
of the technical question that is being addressed. "Correct" subsampling methods should be
selected that provide "representative" analytical values that are appropriate for decisions.
1.5 Previous Guidance
Previously, the Agency has relied on individual project leaders to address any sampling or
subsampling issues. The technical guidance in EPA SW-846 (U.S. EPA, 1986) can be summarized as
"sampling is important," and "sampling should be done correctly." Other Agency documents identify
subsampling practices as an area of concern but provide little or no direction specific to representative
subsampling. There is an excellent report (van Ee, et ai, 1990) on assessing errors when sampling soils,
but there is no discussion on how to minimize their presence. Comprehensive Agency guidelines for soil
sampling have minimal information on representative subsampling, suggesting protocols such as to dry,
sieve, mix, and prepare subsamples as a description of how to treat soil samples in a laboratory setting
(U.S. EPA, 1989). No specific sample splitting methods are mentioned in this last document.
An EPA pocket guide discusses numerous soil characteristics and how to measure them (U.S. EPA,
1991). However, it does not address whether the soil sample acquisition methods were correct or biased,
and provides only one method for mass reduction: quartering followed by incremental sampling from
each quarter. While a pocket guide is not expected to contain comprehensive instructions, there is
minimal discussion regarding the appropriate sample mass reduction strategies. Instead, the emphasis is
on how to use available sampling devices, the use of appropriate quality assurance and quality control
(QA/QC) practices, and the measurement of various soil properties. While all of the above is valuable,
the mass reduction or subsampling step is prone to large uncertainties that can result in a failure to meet
the study objectives.
When sampling particulate material, the assumption in most Agency documents is that study
designers or managers will consult a sampling expert for advice. However, there are a few exceptions.
An extensive discussion of the QA/QC concerns and recommendations for particulate sampling is
summarized by Earth et al., (1989). A comprehensive report by Mason (1992) offers excellent insight in
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areas of particulate sampling. Mason covers sampling concerns from statistical number and physical
location to subsampling practices and cost estimation.
Independent sources of sampling guidance provide a somewhat more detailed discussion of the issues
related to particulate sampling. Several American Society for Testing and Materials (ASTM) standards
include sections relevant to sampling environmental matrices, including solid waste (ASTM 1997).
ASTM Standard D 6044-96, "Representative Sampling for Management of Waste and Contaminated
Media," provides general guidance focusing on selecting the sample from a site. However, it does not
provide specific or comprehensive sampling procedures and does not attempt to give a step-by-step
account of how to develop a sampling design.
Standard D 6051-96 focuses on a limited number of issues related to composite sampling, such as
their advantages, field procedures for mixing the composite sample, and procedures to collect an unbiased
subsample from a larger sample. It does not provide information on designing a sampling plan, the
number of samples to composite, or how to determine the bias from the procedures used.
ASTM Standard D 5956-96 provides general guidance to the overall sampling issue with an emphasis
on identifying statistical design characteristics such as the location and the number of samples,
partitioning a site into strata, and implementation difficulties, such as gaining access to a sampling
location. It does not provide comprehensive sampling procedures.
The ASTM guidance documents are valuable references that should be consulted before attempting a
study involving sampling; however, the ASTM documents do not provide the details at the level of
sample processing as discussed in this document. The ASTM documents do mention the types of
problems one should be aware of, such as the composition heterogeneity or that one may have to subject
the sample to a particle size reduction step prior to subsampling. In summary, the ASTM guidance
recognizes the issues and problems that may need to be addressed, but leaves the reader to their own
resources when a specific activity is required. The ASTM standards include guidance based on theory
and expert opinion, but there are few relevant experimental studies directly demonstrating their
recommendations in environmental applications. This guidance document attempts to give the details to
the rest of the entire laboratory subsampling process where previous guidance has not had the theory or
the experimental foundation on which to formulate appropriate guidance.
1.6 The Measurement and Experiment Process
Most scientific studies that involve measurements, or experiments and measurements, proceed in a
manner similar to that depicted in Figure 1. Typically, those measurements or experiments are being
done to determine something about a lot (a batch, a population, or populations). Unless the entire lot can
be measured or used hi the experiment (which may not be practical because the lot is too large or because
of constraints on resources), samples representative of that lot must be taken in order to make estimates
about that lot. The measurement and experiment process should follow a well thought-out plan based
upon a study design. That design would not only describe all of the tools and methods needed for the
various steps for each stage of the process (note that each stage uses statistical methods), but would also
include a sampling design, an experimental design, and a decision design. Those designs use developed
strategies based on theory, the literature, earlier studies (including familiarity studies, screening studies,
and pilot studies).
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Study Design
n
Sample Acquisition
Experiment
Measurement
Analysis
Reject
QA
QC
Review
Accept
Generate Data Base
Assessment and Decision
Figure 1. The experiment and
measurement process.
The sample acquisition stage may consist of several steps: a stratification of the lot into strata (logical
subsets), a splitting of the entire lot into samples (with a subset or subsets selected for the experiments or
measurements), or through mass reduction methods (successive subsampling steps; that is, taking samples
of samples).
The measurement or experiment stage usually involves one, or multiple, analyses or measurements
(the variables) on each sample that, hopefully, represents an observation of the original lot. Other
additional samples may be needed that are representative of the measured samples for quality assurance
and quality control purposes (QA/QC); e.g., to make sure that the measurement or experiment process is
in control.
A review of the data should then be performed for data validation, completeness, making sure the
QA/QC was met and the process was in control, and to make sure that the data makes some sense. A
failure in that review may lead to new designs, experiments, or measurements before the data can be
accepted into the data base. The data base results should undergo a complete statistical and decision
analysis before being accepted for the end use. The results of those analyses may generate a new study,
again following the stages in Figure 1.
Our focus in this guidance document will be on developing a subsampling strategy for just one step in
the sample acquisition stage of the measurement and experiment process (that is, the laboratory sample to
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the analytical subsample step); however, since our developing sampling strategy is bounded by our
decision strategy, we will briefly discuss some of the decision aspects (such as the data quality objective
(DQO) process, the bounds of what makes a representative sample, and some items to include in reporting
the results). Although this focus is narrow, it behooves the analysts (unless they are involved in a "blind"
study), and it certainly behooves the statisticians, scientists, managers, and decision makers, to consider
the analytical subsample in the context of the entire measurement and experiment process.
1.7 Data Quality Objectives
The primary reason that samples are being taken is to make some determination about the lot (e.g., a
contaminated site). The study goals and objectives determine the acceptable statistical characteristics for
the study. If a decision depends on the analytical results, then the first issue is to determine what type of
measurements are needed and how accurate and precise they should be. The Agency refers to these goals
as Data Quality Objectives (DQOs). The details for the development of DQOs are discussed elsewhere
(U.S. EPA, 1994,1996a, 1996c, 1997,2000a, and 2000d) and these references should be consulted along
with this guidance when developing sampling strategies.
Table 1. The seven steps in the DQO
process.
The DQO process is summarized in Table 1 as a
seven-step procedure (U.S. EPA, 1994). DQOs include
minimum performance criteria, such as the required
quantitative range or the minimum uncertainty in the
decision statistic. For example, several samples may be
taken with the intention of making a cleanup
recommendation based on the upper 95% confidence limit
of the mean. If subsampling greatly increases the overall
measurement uncertainty, by either increased variance or
bias, then there might not be credible evidence on which
to base a decision. In terms of statistical decisions, one
may decide to clean up a site when it is unnecessary.
Alternatively, one may decide that a site has not been
shown to be contaminated because it is indistinguishable
from the background as a result of the high variability
associated with the measurement process.
The DQO step to be developed during planning that is most relevant to this document is step 6, where
the user-specified limits for a representative sample are determined. Such limits should be consistent, and
be developed in conjunction, with the sampling strategy to obtain a representative sample based on the
chemical and physical properties of the lot that is to be represented. Approaches to such strategies will be
developed in this guidance.
Step
1
2
3
4
5
6
7
Activity
State the Problem
Identify the Decision
Identify Inputs
Define Study Boundaries
Develop a Decision Rule
Specify Limits on Decision Errors
Optimize the Design
1.8 Defining the Term, Sample, and Other Related Terms
Before too much confusion sets in, it may be prudent at this point to define the term, sample, and
some other related terms that will be used frequently in this text. There is also a glossary at the end of the
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text that can be consulted for unfamiliar terms. The terminology used in this document will generally
follow that of the Gy sampling theory (Pitard, 1993).
The Notion of Sample Size: Samples seem to come in all sizes and shapes and, depending on the
context, one person's definition of a sample may not be recognized by another person. There
may be agreement that a statistical sample consists of a number of units from a target population.
However, a simple question about sample size might be answered as 68 samples by someone
concerned with the statistical aspects of a study, but as 50 g by the laboratory analyst. Due to this
difference in terminology, anyone dealing with samples and sampling needs to be careful when
summarizing the sampling process so that there is no misunderstanding.
The Notion of a Representative Sample; Strictly speaking, Pitard (1993) defines a correct sample as
"a part of the lot obtained by the reunion of several increments and (which is) meant to represent
the lot." The key word to being acceptable as a sample is "representative." As we will see later,
there are degrees for being representative that are defined by the user, and a representative sample
can only be ensured by using correct sampling practices. Thus, we should always use the
qualifiers, nonrepresentative or representative, or incorrectly or correctly selected, when we use
the terms, sample or subsample. Any material collected that is outside of the imposed limit of
being representative should be qualified as nonrepresentative and anything collected that meets
the user's definition of being representative should be qualified as representative. Thus, we can
have a nonrepresentative sample or a representative sample, and a nonrepresentative subsample or
a representative subsample. More properly, a nonrepresentative sample (or subsample) or an
"incorrectly" taken sample (or subsample) should be called a specimen and not a sample.
The Notions of a Lot and a Sampling Unit: A lot is the collective body of material under
investigation to be represented; e.g., a batch, a population, or populations. A lot may consist of
several discrete units (e.g., drums, canisters, bags, or residences), each called a sampling unit, or
it may be an entire hazardous waste site. Since it is often too difficult to analyze an entire lot, a
sample (a portion) is taken from the lot in order to make estimations about the characteristics of
that lot. For example, sample statistics, such as the sample mean and sample variance, are used to
estimate population parameters, such as the population mean and population variance. Because
sampling is never perfect and because there is always some degree of heterogeneity in the lot,
there is always a sampling error. To get accurate estimates of the lot by the sample(s) and to
minimize the total sampling error, a "representative" sample is sought by using "correct sampling
practices."
The Notion of Correct Sampling (also known as Correct Selection): Unless correct sampling
practices are used, the results from analyzing a subsample will usually be biased compared to the
true value in the original sample. Correct sampling practices give each item (particle, fragment)
of the lot an equal and constant probability of being selected from the lot to be part of the sample.
Likewise, any item that is not considered to be part of the lot (that is, should not be represented
by the sample) should have a zero probability of being selected. Any procedure that favors one
part of the sample over another is incorrect. Correct sampling practices minimize the
"controllable" errors by using correctly designed sampling devices, common sense, and by
correctly taking many random increments combined to make the sample. To be truly
representative, a correct sample must mimic (be representative of) the lot in every way, including
the distribution of the individual items or members (particles, analytes, and other fragments or
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materials) of that lot. Thus, correct sampling should produce a subsample with the same physical
and chemical constitution, and the same particle size distribution, as the parent sample. However,
depending on predefined specifications, the sample may only have to be representative of only
one (or more) characteristics of the lot, and estimated within acceptable bounds.
The Notions of a Subsample and Sampling Stages: Usually there is more than one sampling step or
stage; that is, sampling can take place successively to obtain ever smaller masses from larger
masses of material; i.e., taking samples of samples. The sampling process begins with the initial
mass of the material to be represented, called the lot (also known as the population or a batch). A
correct sample of the lot is a subset of the original mass collected using correct sampling practices
with the intent of selecting a representative sample that mimics the lot hi every way (or at least
mimics the characteristics, chosen by the user, of the lot). Subsampling is simply a repetition of
this selection process whereby the sample now becomes the new lot (since it is now the material
to be represented) and is itself sampled. A subsample is simply a sample of a sample. We will
generically use the term, subsample, as the smaller mass that is taken from the larger mass (which
is called the sample) during the sampling (or, equivalently, the subsampling) step. We will also
use the terms, parent sample and daughter sample, to describe this sample to subsample
relationship, respectively. To literally describe all of the successive sampling steps in order from
larger to smaller samples (or subsamples), the terms: lot (batch, site, or stratum), primary sample,
secondary sample (or the subsample taken from the primary sample), tertiary sample (or the
subsample taken from the secondary sample), and so on down to the end (or analytical) sample
(or subsample) will be used. Assuming that the analytical error is relatively small and in control,
and that correct sampling practices have been followed, the final analytical result can be termed a
"representative measurement" (within user specifications) of the final analytical subsample. By
extension, that measurement should be representative of each of the previous sampling stages
right up to the original lot.
The Notion of a Perfect Sample: The perfect sample of a lot is one that is selected such that every
individual object (particle, fragment, or other item) of that lot has an equal and independent
probability of being included in the sample. Ideally, each object should be examined in turn, and
selected or rejected based on a random draw with a fixed probability. In practice, the quality of
any sampling tool or method is determined by how well the sample approximates the lot. Apart
from the fundamental error due to (that is, "naturally" occurring from) the physical and chemical
constitutional heterogeneity (differences) of the objects making up the lot, all of the sampling
errors discussed in this document ultimately arise from the failure to select the lot's objects with
equal probability, or from the failure to select them independently.
The Notions of an Increment, a Composite Sample, and a Specimen: A few other terms that are
related to, or sometimes confused with, the term, "sample," should be mentioned. An
"increment" is a segment, section, or small volume of material removed hi a single operation of
the sampling device from the lot or sample (that is, the material to be represented). Many
increments taken randomly are combined to form the sample (or subsample). This process is
distinct from creating a "composite sample," which is formed by combining several distinct
samples (or subsamples). A "specimen" is a portion of the lot taken without regard to correct
sampling practices and therefore should never be used as a representative sample of the lot. A
specimen is a nonprobabilistic sample; that is, each object (item, particle, or fragment) does not
have an equal and constant probability of being selected from the lot to be part of the sample.
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Likewise, for a specimen, any object that is not considered to be part of the lot (that is, should not
be represented by the sample) does not have a zero probability of being selected. A specimen is
sometimes called a "purposive" or "judgement sample." An example of a specimen is a "grab
sample" or an "aliquot."
The Notion of Sample Support: Another term, the sample "support," affects the estimation of the lot
(or population) parameters. The support is the size (mass or volume), shape, and orientation of
the sampling unit or that portion of the lot that the sample is selected from. Factors associated
with the support are the sample mass and the lot dimensionality.
The Notion of the Dimension of a Lot: If the components of a lot are related by location or time,
then they are associated with a particular dimension. Lot dimensions can range from zero to four.
Dimensions of one, two, or three, imply the number of long dimensions compared to significantly
shorter dimensions. Bags of charcoal on a production line represent a one-dimensional lot.
Surface contamination at a used transformer storage site is a two-dimensional lot. A railroad car
fall of soil contaminated with PCBs is a three-dimensional lot. However, determining the
average level of PCBs in a train load of railroad cars deals with a zero-dimensional lot as long as
the cars are considered a set of randomly ordered objects. Zero-dimensional lots are composed of
randomly occurring objects (where the order of the units is unimportant), and this feature allows
them to be characterized with the simplest experimental design. Four dimensions include time
and the three spatial dimensions. The higher dimensional lots are more difficult to sample, but
can often be transformed to have a smaller dimension.
Figure 2 shows one possible depiction of the sampling steps in the measurement process. The
uncertainty in the estimate of the analyte concentration increases with every step in the process. A
preliminary study of the sample matrix can be used to estimate the amount of sample necessary to achieve
the study requirements. While this is not the primary goal of this document, it is directly related to the
conceptual model supporting this guidance.
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The
Lot
Sampling Unit Sampling Unit i
Incremental Incremental
Sampling Sampling
i
c
Sampling Unit
Incremental
Sampling
Field
Sample
|
T i
j
r
Sample Split Sample Split
Inorganics Pesticides
+
Sample Split
PAHs
i y, 1
/Particle Size\
\Reductlotv/
Laboratory Sample
for Testing
^ i
r
Archived Analytical
SSSfl
^
Analytical
Subsample
Replicate 2
Figure 2. A depiction of the sample acquisition process.
1.8.1 Heterogeneity
Much of this guidance deals with understanding and reducing the errors associated with
heterogeneity. The reason that samples do not exactly mimic the lot that they are supposed to represent is
because of the errors associated with heterogeneity. Heterogeneity is the condition of a population (or a
lot) when all of the individual items are not identical with respect to the characteristic of interest. For this
guidance, the focus is on the differences in the chemical and physical properties (which are responsible
for the constitution heterogeneity, CH) of the paniculate material and the distribution of the particles
(which leads to the distribution heterogeneity, DH). Conversely, homogeneity is the condition of a
population (or a lot) when all of the individual items are identical with respect to the characteristic of
interest. Homogeneity is the lower bound of heterogeneity as the difference between the individual items
of a population approach zero (which cannot be practically achieved).
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Thus, one can infer that, within predefined boundaries, heterogeneity is a matter of scale. That is, all
materials exhibit heterogeneity at some level. With a very pure liquid, one might have to go to the
molecular level before heterogeneous traits are identifiable; but, with particulate samples, heterogeneity is
usually obvious on a macroscopic scale. This lack of uniformity is the primary reason for the added
uncertainty when attempting to obtain a representative sample.
Since a sample cannot be completely identical to the lot (or parent sample), the next best goal is for it
to be as similar to the lot (or parent sample) as possible. In terms of the particulate sample structure, this
criterion is the same as requiring the physical or chemical constitution, and the distribution, of the
particles to be as similar as possible for each type of particle in the sample as in the lot. Any process that
increases heterogeneity will expand the differences, resulting in increased bias or increased variability,
between the sample and the lot.
Representative particulate samples will have a finite mass, which means that there is a lower limit of
the number of particles of any given form and type (physical or chemical characteristics). If the sample
mass is too small, compared to the amount of material to be represented (the lot), then there may not be
enough of the different types of particles in the sample to exactly mimic the lot, and the sample could
have any one subset of numerous possible particle combinations. Any measured feature of the sample
will be different depending on exactly which combination of particles ended up in the sample. The
variability associated with selecting enough particles at random is the minimum uncertainty that will be
present no matter how one takes a sample. The catch is knowing when enough particles are selected at
random to be representative of the lot. A small sample mass (below this lower limit) can be achieved
through a subsampling strategy involving comminution (for example, see the section on the sampling
nomograph). The upper limit for the sample mass is obviously the entire lot mass.
Except for this natural fundamental error (FE) inherent to the particles being chemically or physically
different, other contributions to heterogeneity can be minimized through "correct" sampling practices.
Correct sampling (or selection) will be discussed in more detail later, but it can be associated with three
practices: (1) taking many (N >30) increments to make up the subsample (to minimize the grouping and
segregation error, GE), (2) using correctly designed sampling tools (to minimize the materialization error,
ME), and (3) using common sense and vigilance (to minimize the preparation error, PE).
There are only two ways to reduce the effect of the relative variance of the fundamental heterogeneity
(SpE2) associated with the physical and chemical constitution of particulate samples. One way is by
increasing the sample mass. If the sample mass is increased, the constitution of the different particles
(and the distribution of the different particles) will more likely closely match the original particle
distribution. A larger sample size also means that the relative influence of any given particle on the
property of interest is smaller. The other way to reduce the uncertainty due to the heterogeneity of
particle types is to decrease the influence of any given particle by breaking up the larger particles into
several smaller particles (reducing the scale of heterogeneity). This crushing or grinding process is
known as "comminution." The smaller the particle size, the smaller the effect of including or excluding
any type of particle in the sample. Comminution also has the advantage of liberating more contaminant
that may be occluded in a larger particle, which could otherwise be masked from the analytical method.
The result of either increasing the sample mass or reducing the particle size is a more likely representative
estimate for the measured sample characteristic.
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For the purposes of environmental sampling, one can now deduce a quick lule-of-thumb: the sample
should be fairly representative of the lot if the largest contaminated particles of the sample are
representative of the largest contaminated particles of the lot. Remember that it is the physical and
chemical constitution of the particles that leads to the constitution heterogeneity and the fundamental
variability, and the greatest fundamental variability (s^2) associated with contamination should be the
largest contaminated particles (we will see later that this contribution to variability will show up as the
cube of the diameter of the largest contaminated particles, d3, in the equation describing the relative
variance of the fundamental error, SpE2).
18.2 Laboratory Subsampling: The Need for Sample Mass Reduction
There are several reasons for laboratory sample mass reduction. The most common reason is to select
the amount of sample required in an analytical protocol. Field samples are generally much larger than
needed for laboratory analysis. Low mass requirements for analytical methods are driven by improved
technology and by the cost savings associated with ever smaller amounts of reagents, equipment, and
waste per sample run. For example, a chemical extraction might call for 2 g of material. However, if the
original sample amount is 164 g, it is not immediately obvious how one should process the sample to
obtain a 2 g subsample that is representative of that entire 164 g sample. Another reason for subsampling
may be to generate quality control information, such as some replicate analyses using the same or an
alternate analysis method. The study design may also call for a separate determination of the
concentration of other analytes or additional physical or chemical properties of the sample, each requiring
a separate subsample. If decisions are to be made with respect to a bulk property, then the subsample
should accurately and precisely represent that property. This problem of selecting a representative sample
has been extensively studied in the mineral extraction industries, culminating with Pierre Gy's theory of
sampling particulate material (Gy, 1982,1998; Pitard, 1993; and Smith, 2001). Though there are several
alternative approaches to this problem (Visman, 1969; Ingamells and Switzer, 1973; and Ingamells,
1976), it has been shown that each type of theoretical approach is similar to Gy sampling theory
(Ingamells and Pitard, 1986). Before a representative sample is defined and a strategy to obtain a
representative sample is developed, an understanding of some of the salient points of the Gy sampling
theory would be beneficial.
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Section 2
Overview of Gy Sampling Theory
The uncertainty associated with sampling is a product of both the sample (physical and chemical
attributes) and the sampling process (involving statistical issues and sampling technique). These topics
are discussed in the context of the Gy sampling theory as applied to environmental samples. The
sampling theory of Pierre Gy has been applied very effectively in the mining industry since he introduced
it in 1953; however, very little experimental verification has been attempted and even less research has
been demonstrated for environmental samples. We are focusing on Gy theory because we believe that it
is the state-of-the-science sampling theory that identifies and minimizes the errors associated with
sampling particulate materials, and we have done experiments to verify and demonstrate the effectiveness
of Gy theory. This theory not only covers the statistical issues of sampling particulate materials, but also
blends in the effects of the physical and chemical attributes of the particulate material! Gy theory
introduces the notion of "correct" sampling to obtain a representative subsample. The relevant
components of Gy sampling theory are discussed and factors that are indicative of large errors are
identified. Specific examples are provided to highlight the conditions that may result in highly uncertain
results caused by incorrect sampling practices.
2.1 Background
Sampling theories have been developed in an effort to move toward sampling methods that reduce or
minimize the uncertainty from the sampling component of the measurement process. Gy sampling theory
is a comprehensive approach to understanding and assessing all of the sources of uncertainty affecting the
estimation of a target analyte concentration (Pitard, 1993; and Smith, 2001). While a comprehensive
discussion of Gy sampling theory is beyond the scope of this document, an abbreviated introduction to
the principal concepts and terminology is provided throughout this document.
Historically, Gy's methods for sampling heterogeneous particulate solids were developed in the mid-
1950s for the mining industry to sample crushed ore. The ore is thought of as consisting of an inert
substance (gangue) and a valuable material (gold or some other metal or metals), which are intimately
intermixed. Even though the ore is crushed to some degree of fineness, some of the 'Value" is hidden
from the assay by a covering or armor of the gangue. The smaller the diameter of the particles, the more
the "value" is released to be measured in the assay.
The Gy sampling theory is generally applicable to matrices of particulate solids, with the analytes of
interest (contaminants) presumably being no more volatile than organic semi-volatile compounds.
Nonetheless, the application of Gy theory to environmental samples, especially containing semi-volatile
and volatile compounds, needs to be further researched.
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2.2 Uncertainty Mechanisms
If a study generates data with very large errors, then the uncertainty in the results may prevent one
from making a sound scientific conclusion. There are many possible sources of uncertainty to consider
when processing or analyzing a sample, and the study designs or an analyst's expertise is relied upon to
identify and avoid as many of those sources of error as possible. Many of those error mechanisms can
occur when obtaining any types of samples, including particulate material samples. While it is important
to address all of the factors (such as analytical errors, AE) that might produce an incorrect value, the
following discussion is limited to the uncertainty mechanisms related to selecting a particulate subsample.
As will be seen, the uncertainty associated with discrete particles may be much larger than expected. In
order to identify an appropriate sampling method, one must first understand the different types of errors
that can arise from particulate samples. The rest of this section discusses the types of errors related to
sampling particulate materials.
2.3 Gy Sampling Theory: Some Assumptions and Limitations
Before trying to apply Gy sampling theory, one must first verify that the sample matrix meets certain
assumptions, otherwise some alternative sample splitting guidance should be followed. Gy sampling
theory is applicable to samples composed of particulate material, with most applications related to
extractable compounds existing as high concentration mineral grains or inclusions produced by
anthropomorphic or geochemical processes. Particle types are usually presented as if there is a discrete
set of compositions or structures rather than a continuous range. Environmental hazardous materials
often coat natural materials, are absorbed into the particle, or exist as separate particles. Those cases are
all accommodated by Gy sampling theory; however, the application of the Gy sampling theory to
environmental hazardous materials is in its infancy and examples are rare in the literature. Such cases
warrant more research.
Some sample types appear to be described by Gy sampling theory, but closer scrutiny reveals that
they fail to meet one or more assumptions. For example, PCBs in sediments occupy the interstitial spaces
between the particle and are absorbed into the particles. If the PCBs are fully integrated into the particles,
then Gy sampling theory can be applied, otherwise it may be inappropriate. Similarly, suppose the
sample was a mixture of sand and gravel from a beach. If the contaminant was crude oil, Gy sampling
theory may not fully apply, as the analyte was not a solid and is present as an interstitial fluid. Again, this
is an area for future sampling research.
Gy sampling theory assumes a mathematical model for the analyte level across the entire range of
possible samples, and that model is that the analyte concentration varies about a mean. With this model,
the primary goal is to estimate the mean value, and any difference from the mean is considered an error.
Suppose a Superfund site is contaminated very highly in the center and the contaminant levels drop as one
moves outward. If the entire site is to be represented, then Gy sampling theory would imply that there is
a high (positive) error at the center, a high (negative) error near the edges, and a low error in a circular
zone about the center (see Figure 3). Only samples near the circular zone would have analyte levels near
the average for the site. Of course, samples would not be taken from just one (or even a few) location,
and many random increments would be taken to make up each sample to represent the lot. Nonetheless,
one must be careful in the interpretation and the application of Gy sampling theory in environmental
characterizations. If it is important that the average for the entire lot (site) is known, then the application
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of Gy theory for the entire lot is appropriate. Otherwise, it may be more informative to strategically
divide (stratify) the lot into several smaller lots (strata) and find the average concentration for each
smaller lot (stratum). The strategies employed for such stratification are covered in the field of
geostatistics using screening strategies and tools like semi-variograms. Such topics will be discussed in
the (planned) future field subsampling guidance. However, the sampling theory for large lots is generally
applicable for smaller lots, such as laboratory samples.
o
Concentration
Below Average
Concentration
Above Average
East
Average concentration across the site
(No bias)
Figure 3. Contour plot of contaminant level across a hazardous waste site.
2A Gy Sampling Theory: Errors
Gy sampling theory identifies the distinct activities of the sampling process and partitions the error
between them. Of the seven basic sampling errors identified by Gy sampling theory (see Table 2), only
five of those errors will be considered and are relevant to preparing laboratory subsamples from
particulate samples. The long-range heterogeneity fluctuation error (CE2) and the periodic heterogeneity
fluctuation error (CE3) will not be considered here and are presumed to be negligible or inconsequential
for preparing laboratory subsamples from relatively small (in mass) particulate samples. However, those
errors should be considered for larger samples (e.g., drums and large field samples) or larger lots (sites or
strata from sites). Those errors will be considered and discussed in a planned guidance for obtaining
representative field samples.
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Table 2. Gy sampling theory error types for participate materials.
1.
2.
3.
4.
5.
6.
7.
Notation
FE
GE
CE2
CE3
DE
EE
PE
Error Type
Fundamental Error
Grouping and Segregation
Error
Long-Range Heterogeneity
Fluctuation Error
Periodic Heterogeneity
Fluctuation Error
Increment Delimitation Error
Increment Extraction Error
Preparation Error
Subject / Description
A result of the constitutional heterogeneity, CH (the
particles being chemically or physically different).
A result of the distributional heterogeneity, DH.
Trends across space or over time.
Periodic levels across space or over time.
Identifying the correct sample to take. Considers
the volume boundaries of a correct sampling device.
Removing the intended sample. Considers the
shape of the sampling device cutting edges.
Sample degradation, gross errors, analyte loss or
gain.
Note that these Gy errors are relative variance (squared relative deviation) errors and are with respect
to the simple model that all of the sample results are supposed to be at the mean value for the lot. Any
deviation from the model is considered an error. The five remaining Gy sampling error types play an
important role in determining the uncertainty levels for particulate samples and will be discussed in more
detail below.
2.5 Subsample Selection Issues
Little attention is usually paid to the actual selection of a subsample. However, in many
circumstances, a highly biased value will result if sampling procedures are not appropriately matched to
the sample matrix. The variability associated with subsampling depends on several physical and chemical
characteristics, including:
• particle shapes
• particle sizes
• number of particles
• number of particle types
• particle mass (or density)
• particle chemical composition
• analyte chemical composition
• other gangue (matrix) chemical and physical composition (moisture content, liquids,
amorphous solids, or other occluded or interstitial materials)
• analyte concentrations in each particle type
• particle size distribution
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These characteristics primarily affect the fundamental error, which is associated with the constitution
(or composition) heterogeneity (chemical and physical differences between the particles), and with the
grouping and segregation error, which is associated with the distribution heterogeneity (due largely to
gravitational effects across the sample). The random distribution of the particles will result in some
degree of heterogeneity of the target analyte even if the sample is free from additional heterogeneity
effects such as gravitational fractionation. The sample mass required to meet the study requirements will
depend on all of the above characteristics and will also be a function of the desired level of accuracy and
precision required by the study.
The correct selection of the subsample also depends on the sample support, the dimensionality of the
lot, and the design of the sampling tool or device - all of which affect the materialization error.
Common sense used in the selection process is the key to reducing the preparation error.
2.6 The Relationship Between the Gy Sampling Theory Errors
To correctly apply Gy sampling theory, one needs to understand the nature and source of all of the
components of sample variation. If the magnitude of each error component can be determined, then the
dominant error sources can be identified and one can avoid efforts that would have no substantial impact
on the variability.
The overall estimation error (OE) is the difference between the final analytical estimate of the
characteristic of interest (such as an estimation of the average concentration of an analyte in the
lot based on sample analysis) and the true (usually unknown) value of that characteristic of
interest (such as the true average concentration of that analyte in the lot). The overall estimation
error would be the error associated with the final value given in the measurement and experiment
process. Pitard (1993) gives this as the sum of the total sampling error (TE) and the analytical
error (AE). That is,
OE = TE +AE
The analytical error (AE) includes all of the uncertainty and errors introduced during the laboratory
phase of a study. The analytical error is the cumulative error associated with each stage of the
analytical method, such as chemical extraction, physical concentration, electronic detection,
uncertainty in the standards, fluctuations due to temperature variations, etc. Care should be taken
when reviewing claims about analytical error. Occasionally these claims refer only to one
component of the analytical procedure, such as the stability of the measurement apparatus.
Instrumental errors are typically quite small compared to the error associated with characterizing
the sample. The analytical error is not a sampling error and is the subject of fields such as
analytical chemistry and experimental design and will not be discussed here in any detail. It
should be noted that the variability added by the sampling process can easily exceed the
uncertainty associated with analytical chemistry methods (Jenkins et al, 1997). The total
sampling error is the subject for discussion in this text.
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When Gy sampling theory is presented in the context of determining the level of an analyte (or a
pollutant) in a lot, error usually refers to any deviation from the mean. The total sampling error
(TE) can be defined as
where aL is the actual (mean) content of the analyte in the lot and a,, is a measure of the (mean)
content of the analyte in the sample (it is the sample estimator of aL). The conceptual additive
linear model given by Gy for the total sampling error is:
TE = £ (SEn + PEn)
i=l
where SE is the sampling or selection error, PE is the preparation error, and n is the index for the
sampling stage. The total sampling (or selection) error, TE, for one stage is then:
The selection error (SE) is a linear combination of the continuous selection error, CE, and the
materialization error, ME, and is given as
SE = CE+ME
The continuous selection error, CE, is a linear combination of the short-range heterogeneity
fluctuation error, CEj, the long-range heterogeneity fluctuation error CE2, and the periodic
heterogeneity fluctuation error, CE3, and is given by
CE = O?, + CE2 + CE3
The short-range heterogeneity fluctuation error, CElf is given by a linear combination of the
fundamental error, FE, and the grouping and segregation error, GE
CEj=FE + GE
The increment materialization error is given by a linear combination of the delimitation error (DE)
and the extraction error (EE)
ME ^DE
The total sampling error is then
TE = FE + GE + CE2 + CE3 + DE + EE + PE
If correct sampling practices are used, then the terms, GE, DE, EE, andPE are minimized; that
is, GE + DE + EE + PE » 0. Assuming that CE2 + CE3 = 0 (that is, they are negligible for
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laboratory subsampling), and if correct sampling practices are used, then the total sampling error
becomes
TE = FE = —
This is the minimum sampling error due simply to the nature of the material being sampled (the
constitution heterogeneity) and represents a goal ofGy's correct sampling practices.
Likewise, the additive relative variances linear model given by Gy for the relative variance of
the total sampling error is
STE ~ SSE "•" SPE
where sSE2 is the relative variance of the sampling or selection error and sPE2 is the relative
variance of the preparation error. The relative variance of the selection error is given as
SSE = SCE "*" SME
Since the relative variance of the continuous selection error is given by
2 2 2, 2
and the relative variance of the short-range heterogeneity fluctuation error is given by a linear
combination of the relative variance of the fundamental error and the relative variance of the
grouping and segregation error
„ 22, „ 2
''CEl hFE ^ bGE
and the relative variance of the increment materialization error is given by a linear combination
of the relative variance of the increment delimitation error and the relative variance of the
increment extraction error
SME ~ SDE ~*~ SEE
then the relative variance of the total sampling error is
STE ~~ SFE SGE SCE2 SCE3 SDE SEE SPE •
If correct sampling practices are used, then the terms, sGE2, s^2, s^2, and sPE2 are minimized', that
is, sw2 + sDE2 + s^2 + sPE2 ~ 0. Assuming that sCE22 + Sc^2 = 0 (that is, they are negligible for
laboratory subsampling), and if correct sampling practices are used, then the relative variance of
the total sampling error becomes
STE = SFE
This is the minimum sampling relative variance due simply to the nature of the material being
sampled (the constitution heterogeneity) and is the basis for developing a representative
sampling strategy using Gy's correct sampling practices.
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The mean of the total sampling error is
m(TE) =
<>L
The mean of the fundamental error (under the above conditions of correct sampling practices and
negligible effects from CE2 and CE3) is expected to be negligible; that is,
m(FE) =
ai
The relative variance of the total sampling error is given by
S\TE) = 1^
And, under the above conditions of correct sampling practices and negligible effects from
and s^, the relative variance of the fundamental error is
SFE
<*L
Thus, the greater the variation is in the physical and chemical characteristics between each of the
particles (or other materials) in the lot, the greater the variance is in the constitution heterogeneity
and, consequently, the larger will be the relative variance of the fundamental error, s^2.
2.7 The Short-Range Heterogeneity Fluctuation Error, CE1
All particulate samples are heterogeneous; it is just a matter of scale. How closely one particle
resembles another particle is dependent upon the focus of the sampler. One common feature of
heterogenous particulate material is that it consists of a distribution of particles (and perhaps other
materials, such as oils) with diverse physical and chemical characteristics, including different: particle
sizes, particle shapes, textures, concentrations of various chemical constituents, and densities. The
differences in the chemical and physical properties of the constituents of the lot are responsible for the
constitution heterogeneity (sometimes called the composition heterogeneity), CH. The constitution
heterogeneity is the source of an expected minimum error, the fundamental error (FE). If there are groups
of items (particles, fragments, or other objects) in the lot that do not have the same average composition,
then there is a distribution heterogeneity, which is often caused by gravity. It is the distribution
heterogeneity, DH, that leads to the grouping and segregation error, GE.
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The short-range heterogeneity fluctuation error is a linear combination of those two errors and is
given by
2.8 The Fundamental Error (FE) - the Heterogeneity of Particulate
Constitution
This error is fundamental to the composition of the particles, being chemically or physically different
and is a result of the constitution heterogeneity (CH) (see Figure 4). It is the minimum sampling error and
the expected error if the sampling operation is perfect. It is also the only error that can be estimated
before the sampling operation. The fundamental error is the error expected if the individual particles for a
sample are selected at random from the particles making up the lot. The fundamental error is only a result
of the chemical and physical constitution heterogeneity of the material, and not the sampling process.
The fundamental error is usually dominated by particle size and composition properties. While sampling
activities can add additional error, the sampling process cannot reduce the fundamental error. The bias of
the fundamental error is expected to be negligible for most cases (Pitard, 1993; p. 167) and the relative
variance of the fundamental error may be reduced by decreasing the diameter of the largest particles of
the matrix to be represented, or by increasing the mass of the sample, M,, [g].
D
H
O
A
D/
A
D
D
A
D0
A
/
/
r
C
O
D
O
Figure 4. A depiction of the fundamental error (FE) due to the composition of the particles (or other items or
fractions) of the lot being chemically or physically different. It is a result of the constitution
heterogeneity (CH) of the lot; thus, this is the only sampling error that can never cancel out.
The fundamental error (FE) is the error of the measured subsample property that is expected if the
particles in the subsample were selected one at a time from the sample at random (that is, each random
increment making up the sample is a single particle. Of course, this is not a practical way to sample!).
Any subsampling method will result in a subsample that has a slightly different property than the original
sample. The variability in results that is independent of the subsampling process will depend on the
variability associated with the individual particles. One can consider the effect of including or excluding
an individual particle and asking what the relative effect will be. If all of the particles are small, then
there will be a small change associated with adding or removing one particle. If some particles are large
(or highly concentrated), then there may be a larger change associated with the inclusion or exclusion of a
single particle.
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The fundamental error is unaffected by whatever sample selection practice is used. It is always
present. Any sample selection practice that provides a less random selection of particles will result in a
larger uncertainty. One cannot stress too highly that a minimum error level is expected that depends only
on the physical and chemical constitution of the particles in the sample. The fundamental error is
unrelated to the selection method used in generating a subsample.
As mentioned above, there are only two ways to decrease the relative variance of the fundamental
error. One way is to change the sample mass, M,.. If the sample mass is doubled, then the uncertainty in
results will be lower (if sampling is done "correctly"). This makes common sense - the more mass of the
lot that is sampled, the more the sample becomes like the lot. Taking a larger sample has the same type of
effect on the magnitude of the relative variance of the fundamental error as one gets by analyzing multiple
samples to reduce the uncertainty hi estimating the mean. How large the sample needs to be in order to
achieve the study goals depends on how large the relative variance of the fundamental error is compared
to the maximum allowed error for the study. If the sample contains large particles, then one may need a
relatively large sample to meet the study uncertainty requirements. Often the size of sample needed to
achieve the DQOs can be too large for a standard laboratory to process. The other method for changing
the effect of the fundamental error is to alter the nature of the sample by crushing or grinding (the process
of comminution) it to reduce the maximum particle size. A smaller particle size will lower the relative
variance of the fundamental error for a fixed subsample mass.
There are a number of other factors that can increase the sampling variance beyond the fundamental
error. Minimizing sampling variability without particle size reduction can be accomplished through the
careful selection of all phases of the subsampling process. Whether or not this is cost-effective depends
on the relative size of the different error components (see Table 2) and the sample mass needed to achieve
success. However, minimizing those other error components has no effect on the fundamental error. If
the relative variance of the fundamental error is too large, the results will not be conclusive no matter
how carefully the rest of the sampling and analysis procedures are performed!
Because our strategy to obtain a representative sample relies on correct sampling methods to
minimize all of the "controllable" errors, we will first discuss those controllable errors and then focus in
on the fundamental error (specifically s^2) with more details in the section appropriately entitled,
"Fundamental Error Fundamentals."
2.9 The Grouping and Segregation Error (GE) - the Heterogeneity of
Particle Distributions
Another source of uncertainty is related to the particle distribution and the distribution of the analyte
throughout the sample. This error is due to the non-random short range spatial distribution of the
particles and the analyte due to grouping and segregation (usually because of gravity); i.e., incremental
samples are different (see Figure 5). This grouping and segregation error (GE) is a result of the
variability due to the heterogeneous distribution of the particles, known as the distribution heterogeneity
(DH). This error may be minimized by combining many random increments, taken correctly from the lot
to be represented, to form the sample.
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O
D
n n
n n
n D
n
°0°°°o°0°°o° °
O O A 0 U00°U00
O A Q V
3 ^p A €
A * A A A A
D A A A
3^
o
o
JA
A A
AA
a
a
H
Figure 5. The grouping and segregation error (GE) is due to the distribution heterogeneity
(DH). The non-random short range spatial grouping and segregation of the
particles and the analyte are usually because of gravity; i.e., incremental samples
are different. This error may be minimized by combining many random
increments to form the sample.
The grouping and segregation error is always present, and its effect depends on the sampling
(selection) process. This effect can be very large at low concentration levels. One segment of the sample
may have a higher density of an analyte than another segment. That situation could be purely by chance,
which is especially true when the analyte is present in only a limited fraction of the particles. Gravity
often plays an important role in causing this error by differentially segregating one type of particle from
another. Gravitational segregation can occur because of density, particle size, and even particle shape
(e.g., the angle of repose) differences. However, even in the absence of any other mechanisms, the
random distribution of the analyte particles in a sample will result in nonhomogeneous concentrations as
one considers smaller and smaller portions of the sample.
One of the most common sample characteristics related to the grouping and segregation error is where
the particles containing the analyte of interest have densities significantly different from the other
particles. Denser particles will tend to settle to the bottom of the sample if the other sample particles are
not too small. When this happens, sampling techniques, such as grab sampling, end up underestimating
the concentration, which could result in decision errors. For example, a site may be declared clean
despite the fact that the cleanup levels were not achieved.
Conversely, if the analyte-rich particles are lighter than the other particles, analysis of a grab sample
off the top might cause unneeded additional treatment and cleanup activities at a hazardous waste site,
wasting resources. Thus, it is important to use sampling techniques that produce a representative sample
by minimizing those sampling errors.
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The fewer the number of analyte particles, the higher will be the relative variance of the short-range
heterogeneity fluctuation error (that is, due to s^2 or s^2) expected from sampling. High analyte
concentration levels associated with those particles also cause an increase in variance. This is easily seen
in Figure 6 (Pitard, 1993, p. 368) which shows a two-dimensional lot with a small number of high-level
concentration analyte particles. The rate at which contaminant particles are found is expected to be very
low. Analytical results from the samples can be approximated by a normal distribution if those samples
contain more than a few analyte particles. As the sample mass decreases or the number of analyte
• y
• *
n "i * $
•
* * •
• • •
•
•. '-a •
•
• * V * •
• B
• • t
• •
•
•a- / .
• * »
• • •
^
» f •
• *
• •
.*• * '
•• B»
•
. 7 Q • •
• •* •
t« *
* • •
• • * •
*
•'• '?' '
•
• ?• '
•
• . .
• • •• *
." • * •
•'• D.V
*
• * •
• . *
• *
* •
• • Ff V '
t •
,"•'. ' •
Figure 6. The effect of sample size when there are few analyte particles (black dots) (Pitard, 1993; p. 368).
particles drops to fewer than 5 to 7 per sample, the variability of the analyte in the samples will change.
Instead of a Gaussian distribution, the data will then follow a Poisson distribution when plotted as the
probability that a discrete number of analyte particles, P(r), will appear in a sample versus that number of
analyte partilces, r, in a sample. Note that the Poisson distribution becomes more symmetrical, like a
Gaussian distribution, as the number of analyte particles per sample increases above 4 (Pitard, 1993; pp.
357 ff).
The expected effects from taking a smaller sample are reported by Starr et al. (1995), who notes, "The
smaller diameter samples gave smaller means, greater skewness, and higher variances ...." A smaller
sample mean is expected if there is less chance that a high analyte concentration level particle will be
included in the sample, reducing the reported average concentration unless exhaustive analysis is
performed. When a high-level analyte particle is present in a sample, the smaller the size of that sample,
the larger will be the concentration estimate for that sample. And that will result in a higher estimate for
the variance. A lot with a low number of analyte particles is a difficult case to deal with. Often the most
difficult part in properly sampling this type of lot is to identify whether or not the sample falls into this
category in the first place.
Many analysts rely on mixing (or blending) as a preliminary "homogenization" step before taking a
grab sample. Unfortunately, many samples cannot be made homogeneous enough for sampling by
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mixing, and such a procedure should not be relied upon to reduce GE. Segregation of particles by
gravitational effects usually occurs at the moment that the mixing has stopped. Some samples will remain
segregated even during the mixing process. Even if the mixing was effective, the subsampling step will
still involve the same minimum error contributions from the fundamental error and the grouping error due
to the random placement of analyte particles within the sample. The incorrect nature of grab sampling
exacerbates the uncertainty by maximizing the error components from grouping, segregation,
delimitation, and extraction processes. Grab sampling has been shown to be an unacceptable sampling
method and should not be used with particulate samples (Allen & Khan, 1970; and Gerlach et al, 2002).
The relative variance due to the grouping and segregation error, s^2, can be made relatively small
compared to the relative variance due to the fundamental error, Spg2, by increasing the number of random
increments, N. For most cases, one can assume (Pitard, 1993; p. 189) that sGE2 < sFE2; therefore, since
s 2 _ 2 + 2 < 2 S™2
(Note that this is not always true; for example, a sample made of only one increment containing a highly
segregated fine material may have a very small s^2 but a much larger sGE2). Then, for N increments, taken
with correct sampling practices, we can write (Pitard, 1993, p. 388)
SGE2*SFE2/N
and, under such conditions, the relative variance of the total sampling error becomes
Thus, one can see that s^2 > s^2 ~ SpE2 if N is made large enough. At least N = 30 increments are
recommended as a rule of thumb to reduce sGE2 compared to SjrE2 (Pitard, 1993; p. 187). If this is too
difficult in practice, then try to get at least 10 randomly selected increments. The goal is to reduce sGE2
relative to SpE2 and you more or less "get what you pay for" in terms of this effort.
2.10 The Long-Range Heterogeneity Fluctuation Error (CE2)
The long-range heterogeneity fluctuation error refers to non-random, non-periodic trends across one
or more dimensions of the lot and is commonly identified by variographic experiments. This error term
will not be considered for laboratory subsampling. In environmental studies, identifying this type of
heterogeneity is usually considered an objective of the sampling program, such as mapping concentration
trends across a site such as one sees in Figure 3. The error is inherent to the distribution of analyte across
a site and cannot be reduced by taking additional samples. CE2 is the regionalization term used in
geostatitics; that is, it is the region of autocorrelation between the nugget effect, V0, and the sill of the
semi-variogram. Taking additional samples helps to characterize this type of spatial heterogeneity instead
of reducing it.
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2.11 The Periodic Heterogeneity Fluctuation Error (CE3)
This error is identified by variographic experiments and may be "smoothed out" by reducing the size
of the strata or taking many increments to form the sample. The periodic heterogeneity fluctuation error
is a (typically) long-range error with a repeating intensity pattern. An example of this type of error may
be found when sampling soils over time and analyzing for nitrogen. One might find periodic fluctuations
in the nitrogen levels through several years of data related to seasonal growth and decay patterns. As with
the long-range heterogeneity fluctuation error, such information is more likely be the object of a study
rather than in determining the mean analyte level. Again, mis error is usually not a concern for laboratory
subsampling.
2.12 The Increment Materialization Error (ME), the Increment Delimitation
Error (DE) and the Increment Extraction Error (EE): Subsampling Tool
Design and Execution
The increment materialization error (ME). Another potentially overwhelming, but often overlooked,
source of uncertainty from sampling is from the variability added during the actual physical process
of selection- that is, how to correctly select, prepare, and form the increments that are combined to
make the sample (or subsample) using correctly designed sampling tools. The error associated with
the execution of the increment selection and sample preparation process is called the increment
materialization error (ME) and is technically the sum of three errors: (1) the increment delimitation
error (DE), (2) the increment extraction error (EE), and (3) the preparation error (PE). That is,
technically, ME = DE + EE + PE. However, only the increment delimitation error and the increment
extraction error are associated with the increment selection process. Therefore, we will follow
Pitard's suggestion (Pitard, 1993) and separate out the preparation error from the materialization
error; that is, we will use ME = DE + EE. Those two errors will be discussed in this section. The
preparation error results from a nonselective process and will be discussed in a separate section.
The increment delimitation error (DE). Each sampling protocol describes a process by which the
subsample is taken. If the protocol does not follow correct sampling practices, there may be a
delimitation error. The delimitation error arises when the sampling process does not give an equal
probability of selection for all parts of the sample. An error will be introduced if the sampling device
selects or includes particles from any part of the lot with unequal probability.
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This error involves the physical aspects of selecting the increment using a correctly designed
sampling device. The volume boundaries of a correct sampling device must give all of the fractions
collected an equal and constant chance of being part of the sample. For example, a "one-dimensional"
pile should be completely transected perpendicularly by a scoop with parallel sides and a flat bottom (see
Figure 7). The increment delimitation error occurs when an incorrectly designed sampling device
delimits (forming the boundary limits of the extended increment) the volume of the increment giving a
nonuniform probability for each item (fraction or particle) to be collected within the boundaries of the
sampling device.
Figure 7. The increment delimitation error (DE) involves the physical aspects of
selecting the increment using a correctly designed sampling device,
where the volume boundaries of the device correctly delimit (forming the
boundary limits of the extended increment) the volume of the increment,
giving all of the fractions collected an equal and constant chance of
being part of the sample.
The top portion of Figure 8 shows an example of an incorrect increment delimitation using a "round"
spatula or scoop. This method did not give an equal chance for selecting the particles at the top and at the
bottom of the sample, as shown by the nonrepresentative concentration gradient to the right of the figure.
The bottom portion of Figure 8 shows the same example using a correct increment delimitation with a
"square" spatula or scoop. This method gives an equal chance for selecting the particles at the top and at
the bottom of the sample, as shown by the representative concentration gradient to the right of the figure.
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Sections Taken with Semicircular Scoop
Sampled
Gradient
True
Concentration
Gradient
Sections Taken with Rectangular Scoop
True
Concentration
Gradient
Figure 8. Top: increments selected with an incorrect device.
Bottom: increments selected with a correct device.
Another example of correct or incorrect increment delimitation is shown in Figure 9 (a & b), when an
increment is taken from a moving belt. If the sample cutter uses a constant velocity as it collects the
increment across the belt (which is also moving at a constant velocity), then a correctly delimited
increment will be achieved (see Figure 9 (a)). However, if the sample cutter uses a varying velocity as it
collects the increment across the belt (still moving at a constant velocity), then it will collect more
material from one side of the belt than the other, resulting in an incorrectly delimited increment and bias
(see Figure 9 (b)).
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(a)
Total sample as a function of position across the belt.
x Y
X
-------
(a)
Oo
(d)
(b)
(c)
(e)
Figure 10. (Pitard, 1993; p. 223) (a) Delimination for the extended increment. Numerous fragments lie
across the target boundary, (b) Ideal increment extraction, (c) An increment extraction error (EE)
can occur when particles cross the extended increment boundary. The particles with the "x"
symbols, which signify their center of gravity, should have been included in the increment since
their center of gravity is within the boundaries of the sampling device, (d) Delimitation for the
extended increment for a cylindrical sampling device for a two-dimensional sample, (e) The
particles that get extracted into the increment have their center of gravity within the extended
increment boundary of the sampling device.
Thus, this error also involves the physical aspects of taking the sample and using a correctly designed
sampling device. If a core sample is taken and small rocks lie across the extended boundary of the
sampling device, some rocks will be forced completely into the core sampler and some will be forced
completely out of the sample into the rejects (see Figure lOd and lOe). This alters the composition of the
increment inside the cylindrical sampling device from the sample material that is to be represented. But,
we know that in order to have a correctly selected increment, there must be an equal chance for all of the
parts of the increment to be part of the sample or part of the rejects. To avoid this increment
materialization selection error, the shape of the sampling device's cutting edges must be designed with
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respect to the center of gravity of the particle and its chance to be part of the sample or part of the rejects
(see Figure 11). The sampling device should go completely through the pile or surface and at a slow,
even rate. A rule of thumb for correctly collecting an increment with respect to the increment extraction
error is that the inside diameter of the sampling device should be at least 3 times the diameter of the
largest particle.
Figure 11. For correct increment extraction, the shape of the sampling
device's cutting edges must be designed with respect to the
center of gravity of the particle and its chance to be part of the
sample or part of the rejects. The sampling device should go
completely through the pile or surface and at a slow, even rate
and the inside diameter of the sampling device should be at
least 3 times the diameter of the largest particle.
There is a practical limit on how small the increment can be for a correctly designed sampling device
to remain correct with respect to the increment extraction error, and it has to do with the size of the space
of the inner walls of the sampling device. Obviously, this space must be large enough to accommodate
the diameter of the largest fragments to be sampled. Not immediately obvious, however, is that the action
of the cutter moving through the sample could cause some of the fragments that should be part of the
increment (that is, those fragments that have their center of gravity within the increment extended
boundary) to move from the leading cutting edge making contact with those fragments to past the
opposite trailing cutting edge if that space is too small. Therefore, those fragments do not become part of
the sample and the quest for a fragment to have an equal chance to be part of the sample or to not be part
of the sample is lost, and a sampling bias is introduced. Pitard (1993, pp. 292 ft) gives several rules for
cutter speed and inner wall sampler width. Those rules are generally summarized giving: a maximum
cutter speed of 0.6 m/s; an inner wall sampler width of 3d for coarse materials (d >3 mm) and 3d + 10
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nun for very fine materials, thus giving a minimum of 10 mm; and a sampler depth of at least 3d. The
cutting angle should be either zero (the cutting edge is perpendicular to the extended increment) or greater
than or equal to 45 degrees. Those rules may be quite constraining for the very small increments that may
be needed for laboratory subsampling. If a width of less than 10 mm is used, we can only suggest
(without our own empirical evidence) that the cutter speed should be much slower.
For riffle splitters, Pitard (1993, p. 302) recommends a correct riffle chute width of 2d + 5 mm. No
minimum width is recommended for sectorial splitters, although an opening somewhat larger than d is
obvious; however, Pitard (1993, p. 303) does recommend that the sector slope should be at least 45° for
dry materials and 60° for slightly moist materials. Since true splitting methods select the splitting
increments at random, the extraction bias (EE) should cancel out for those methods.
2.13 The Preparation Error (PE) - Sample Integrity
Gy theory identifies error associated with sample integrity as the preparation error (PE; see Table 2).
This error involves: gross errors, such as losses, contamination, alteration (e.g., sample degradation);
uncertainty added during sample handling, shipping, storage, preservation; or any process that could alter
the analyte level between when the sample is obtained and when it is analyzed. This error can be
minimized by being careful, being honest, and using common sense. This does not include the error from
a chemical extraction step performed as part of a chemical analysis (Gy theory would identify that error
as part of the analytical error, AE). A listing of the most common PE types is shown in Table 3.
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Table 3. Mechanisms for increased bias and variability due to sample integrity and the PE.
Area of Concern
Contamination
Chemical modification
Physical alteration
Biological alteration
Unintentional mistakes
Intentional error
Examples
• Dust from other samples (Schumacher ef a/. , 1 990).
• Cross-contamination from sampling equipment; e.g., drill corer not
cleaned between borings.
• Carryover from previous sample via contaminated analysis equipment.
• Addition of material from abrasion of sampling/preparation equipment;
e.g., trace Cr analysis after using stainless steel sampling apparatus.
• Addition of material from corrosion of sampling/preparation equipment.
• Reactions adding material; e.g., oxidation of sulfur.
• Loss of chemical constituents; e.g., starting with a hydrate and ending
with an anhydrate.
• Analyte binding to sample container or processing equipment.
• Addition of a critical component; e.g., absorption of water.
• Loss of a critical component; e.g., evaporation of elemental Hg or volatile
organic compounds.
• Loss due to heating. Volatile and semi-volatiles lost while grinding the
sample.
• Dust lost preferentially, reducing or enriching a constituent.
• Loss of material in processing equipment; e.g., very fine grinding of gold
bearing rocks containing elemental gold results in gold-plated equipment.
• Unequal loss of material by fraction or type.
• Microbial consumption of organic constituent.
• Dropping the sample.
• Mixing labels.
• Equipment failure.
• Error in implementing the method.
• Transcription error.
• Fraud.
• Sabotage.
2.14 The Importance of Correctly Selected Increments
As the number of increments, N, taken "correctly" from the sample increases, the relative variance
due to the grouping and segregation error, s^2, decreases. However, there is a point of diminishing
returns. The limitation for increments is to subsample one particle at a time; that is, one increment equals
one particle.
The relative variance of the short-range fluctuation error, scm2, can be minimized to be about the
magnitude of the fundamental error if more increments are taken for each subsample. Recall that
SCEI = SFE "*" SGE •
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For most cases, one can assume (Pitard, 1993; p. 189) that
SGE ^ SFE
~ - 2 •
Then
(Note that this is not always true; for example, a sample made of only one increment containing a highly
segregated fine material may have a very small sm2 but a much larger sGE2). Then, for N increments, taken
with correct sampling practices, we can write (Pitard, 1993, p. 388)
sOE2»sPE2/N
and
Thus, one can see that s^2 z s^2 « s^2 if N is made large enough. At least N = 30 increments are
recommended as a rule of thumb to reduce s^2 compared to s^2 (Pitard, 1993; p. 187).
A grab sample may consist of one large increment. Grab sampling is an incorrect procedure because
it consists of just one increment, taken with judgement and taken incorrectly (without respect to
minimizing the GE, DE, and EE). It is incorrect because most particulate samples are impossible to mix
so that all of the particles are available to the grab sample with the same probability. However, in
addition to being incorrectly taken, the single increment grab sample will include some correlation
(because of GE) between particles in the selected and in the unselected components, which is related to
increased bias and uncertainty. In a sense, correct sampling is a requirement that produces the correct
mean value over the long term. However, there are many different ways to obtain the correct mean value,
and each is associated with a different variability. Correct sampling by itself does not guarantee low
sampling error (because of the presence of the fundamental error).
Subsampling methods can be roughly ranked with respect to the number of increments they use. The
use of many increments is one way to avoid the random effects that leave large fractions of sample with
very low or very high concentrations due to problems with fractionation or from random chance.
Figure 12 shows an example where increment sampling overcomes the uncertainty from grouping and
segregation effects, demonstrating that certain errors can be reduced by using appropriate subsampling
techniques. In this figure, the lot contains 30 "triangular" (in two-dimensions) particles and 10 "circular"
particles, giving a ratio of 3 .0 for triangular-to-circular particles. The GE in this lot is significant, and any
one of the three increments (say, acting as a "grab" sample) does not represent that ratio very accurately.
The ratio for increment 1 is 1.25, the ratio for increment 2 is 8.0, and the ratio for increment 3 is 1.33.
However, after the three increments are combined to make the sample, the ratio is "averaged out" to
2.125, which is much more "representative" of the lot ratio of 3.0.
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Figure 12. This highly segregated lot is randomly sampled with increments of the
same total area. None of the increments alone represent the lot very
well; however, when combined (thus, reducing GE), the sample is much
more representative of the lot.
The concept of many increments can be extended to include particle size. Large particles can be
thought of as large increments. Having selected one part of the particle, one automatically gets the other
part. The mass components making up a large particle can be thought of as intrinsically correlated
fragments. The correlation between parts of a large fragment is similar to the correlation between
particles taken as part of an increment or with a scoop in various subsampling procedures. The
correlation between different mass components, whether between or within particles is a key indicator for
the level of uncertainty obtained with any subsampling method. Sampling methods that minimize
correlation among all mass fragments will help to minimize uncertainty.
2.15 Increment Sampling and Splitting Sampling
The increment selection process and the splitting selection process are both sampling (mass reduction)
techniques that involve taking correctly selected increments, using correctly designed sampling tools, and
combining those increments to form the (hopefully representative) sample. But, there is a fundamental
difference between the two techniques. For the increment sampling process, the selection of where and
how to take the increment occurs before the actual materialization (the delimitation and the extraction of
taking that increment, and the combination of those increments to form the sample as they are
sequentially taken). For the splitting sampling process, the fractions are first correctly delimited;
however, the selection of which fractions (the splits) to use as the sample or to combine to make the
sample occurs after the extraction of those fractions. Thus, even if the fractions are systematically or
technically biased, if the fractions or the splits are chosen at random, the bias should average out to be
negligible. In the increment sampling process, the increments have already been sequentially selected and
combined to make the sample; that is, the selection of the increments preceded the materialization
(formation) of the sample, so that there is no "turning back." Hence, it is very important that the
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increments are taken correctly and are technically unbiased for increment sampling. The difference
between increment sampling and split sampling is clearly illustrated in the case studies section, called,
"Case Study: Increment Subsampling and Sectoral Splitting Subsampling." This difference of the
random selection of fragments can be an advantage for the splitting sampling methods. Some splitting
methods combine the best of both the increment and the splitting sampling processes. For example, each
fraction from a sectorial splitter (see section on sectorial splitters under "Subsampling Techniques") is
made up of many small increments, yet the fractions that are selected to make up the sample can be
chosen at random.
2.16 Correct Sampling (Correct Selection) Defined
The endeavor to reproduce a subsample having the same physical and chemical constitution as the
parent sample gives confidence that the subsample has reached the minimum relative variance of the
fundamental error (s^) that was present in the parent sample. The effort to reproduce the same particle
size distribution in the subsample as in the parent sample is enacted by:
• Minimizing the effect of the grouping and segregation error (GE) by correctly taking and
combining many random increments.
• Minimizing the effect of the delimitation error (DE) by using a correct sampling device that can
extend through the sample to give an increment volume selected to give each constituent of the
sample an equal chance to enter the boundaries of that sampling device. That correctly delimited
sampling device would be a scoop with parallel sides and a flat bottom for a sample in a one-
dimensional (one long dimension) pile or a cylinder with a constant cross section for a two-
dimensional (two long dimensions) sample.
• Minimizing the effect of the extraction error (EE) by using a correct sampling device that, as it cuts
through the correctly delimited increment, gives each constituent an equal chance of being selected
as part of the increment or not. Thus, a correctly designed sampling device must have cutting edges
that allow, with equal probability, the constituent to be part of the increment if its center of gravity
is within the extended bounds of correct delimitation, or the constituent not to be part of the
increment if its center of gravity is outside of the extended bounds of correct delimitation.
• Minimizing the effect of the preparation error (PE) - those "human" mistakes, such as
contamination, losing some sample, or altering the composition of the sample - by being careful,
honest, and using common sense.
Correct sampling practices, therefore, challenge us to minimize those errors that we have some control
over (that is, m^2, s^2, sDE2, s^2, and sPE2 become negligible) so that we can produce a subsample that has
the same intrinsic minimum relative variability (s^2) inherited from the original parent sample.
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2.17 Representative Sample Defined
A goal of the Gy sampling theory is to obtain a representative sample, defined as a sample that is both
accurate (within a specified level of bias) and precise (within a specified level of relative variance) at the
same time (Pitard, 1993; p. 415). The degree of representativeness, rTE2, is given by
2 2
rTE - IIITE
where TE is the total sampling error, rTE2 is the mean square of the total sampling error, m^2 is the square
of the mean of the total sampling error, and s^2 is the relative variance of the total sampling error. The
total sampling error, TE, refers to the relative difference between the expected (true or assumed) value of
the proportion of the analyte in the lot, aL, and the estimated value of the proportion of the analyte in the
sample, as , when there is preparation error, PE:
a* ~ ar
Note that TE = SE + PE = CE + ME = FE + GE + CE2 + CE3 + DE + EE + PE.
A subsample should be representative of the sample it was taken from and be an estimation of the
original lot, as defined by the study objectives (e.g., data quality objectives, DQOs) and the sampling
plan. In the case of laboratory subsampling, the analytical subsample (e.g., for chemical analysis) must
be representative of the entire contents of the laboratory sample bottle.
A sample is representative when
where roTE2 is a specified and quantitative measure of a representative sample (the smaller this number, the
more representative is the sample); that is, it is a level of representativeness regarded as acceptable.
But, without knowing aL, can we know how representative our samples are of the lot? Yes, we should
be able to using the strategy that the bias is negligible (m^2 = 0) and the "controllable" relative variances
are minimized when the sampling practices are perfectly correct (that is, when GE = DE = EE = PE = 0).
It is desirable to keep SjE2 < soTE2, where soTE2 is a level of the relative variance of the total sampling error
within user specifications. If sampling practices are perfectly correct (that is, sGE2 = SpE2 = sEE2 = sPE2 = 0),
then sTE2 = SFEZ (the relative variance of the fundamental error). Thus, if correct sampling practices are
applied and all of those "controllable" errors are minimized, then a representative sample could be
characterized by keeping the relative variance of the fundamental error below a specified level, sFE2 <
soFE2. Under such conditions,
r 2 ~ c__2 <• e 2 ~ r 2
1TE ~ ^'FE - BoFE ~ JoTE
This equation will serve as the basis of our strategy to obtain a representative subsample for chemical
analysis. This is fortuitous for our planning purposes, and for formulating our study objectives (e.g.,
DQOs), since s^2 is the only error that can be calculated, based on the physical and chemical properties
of the participate material, a priori; that is, before sampling even takes place!
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As can be seen from the above definition, like heterogeneity, and because of its relationship to
heterogeneity, representativeness is a matter of scale and depends on the focus and requirements of the
user. Thus, the user should now have a logical approach (and a sampling strategy) to specifying the
tolerable amounts of bias and variability (set in the DQOs).
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Section 3
Fundamental Error Fundamentals
The fundamental error is the minimum sampling error generated even when all of the sampling
operations are perfect. The fundamental error is a result of the constitution heterogeneity of the materials
making up the lot, those materials being chemically or physically different. When correct sampling
practices are used (that is, GE + DE + EE + PE ~ 0) and the long-range fluctuation errors are assumed to
be negligible for laboratory subsampling (that is, CE2 + CE3 = 0), the fundamental error can be
expressed as a proportion of the true mean for the lot being sampled:
a, - ar
FE = -? ^
°L
where aL is the actual content of the analyte in the lot and as is the measure content of the analyte in the
sample. The mean of the fundamental error (under the above conditions) is expected to be negligible; that
is,
m(a') - aT
= -LiL £ « o
The relative variance of the fundamental error, under the above conditions, can be expressed as:
S\FE) = LQL
Estimation of the relative variance of the fundamental error provides a baseline from which one can plan
a subsampling strategy that can meet the study DQOs. Several examples and test cases are provided to
demonstrate both the estimation methods and how to use the information once it is acquired.
3.1 Estimating the Relative Variance of the Fundamental Error, sFE2
Several formulas have been developed as approximate estimates of the relative variance of the
fundamental error. Each formula is the result of a number of approximations and assumptions. While the
assumptions may not always be exact, the results are often still useful as an aid in making sampling
decisions.
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The relative variance of the fundamental error is related to several properties of a sample Gy (1982),
(see also Pitard (1993)). The relationship between the relative variance of the fundamental sampling error
(FE) and several key sample properties is:
2 1 1 i 1 1 i 1 1
** f •*• •*• \ J3 _/T / •*• •* \ /"f J T / -*• •*- \ YVT
SFF ~ (— ~ —)cflgd = (— - —)Cd - (— - —)IH,
rr* V»JT i f J s & \ , .- , r / \ - f •* * * I*
M Mr M M' M, M,
s It S Li S Z*
where Ms is the sample mass [g], ML is the mass of the lot [g], c is the mineralogical (or composition)
factor [g cm"3], / is the dimensionless liberation factor, f is the dimensionless particle shape factor, g is the
dimensionless particle size range (or granulometric) factor, d is the nominal size of the particles [cm], C =
cflg is the sampling constant [g cm"3], and IHL is the constant factor of constitution heterogeneity (also
called the invariant heterogeneity); those factors will be covered in more detail below.
If the lot is large compared to the sample, that is, ML » M,,, then the term, 1/ML, is negligible and the
relative variance of the fundamental error is:
s2 = Cflgd3 _ Cd3 .
FE M. M.
,
3
This large lot approximation is of great importance in using information about the constant factor of
constitution heterogeneity. The sample mass associated with a relative variance of 1.0 is IHL. That is,
SFE^S ~ -fl^sJandif $j^ ~ l.thenA/y = IHL. A relative variance of 1 .0 corresponds to a relative
standard deviation of 1 .0, which is a 100% relative error. One can use IHL to estimate a sample mass
associated with a variance of 1 .0, and then scale this value to determine the sample mass required to
achieve a particular level of uncertainty .
If the lot is merely split in two (as in one pass through a riffle splitter), then the relative variance of
the fundamental error is:
2 1 1
c2 = /_!_ - \fff =
t9 U"|7 1 — — — • jj^ £ —
X 2M/ L 2M, M,
5 a 3 Li
Thus, the range of the relative variance of the fundamental error becomes larger as the mass of the sample
becomes smaller, and vice versa.
A single value (IHL) used to represent the terms, c, f, 1, and g, in the equation for the fundamental
error illustrates two important features about that equation. One is that the equation is only approximate
and should only be expected to provide gross guidance in terms of design. Each parameter involves
approximations and one should not spend a great deal of effort getting several very accurate estimates of
some of those terms when others may be largely uncertain. The other feature is that the particle diameter
is very influential in determining sampling error (since it is a cubed term).
Inspection of the equation for the relative variance of the fundamental error shows that there are two
ways to reduce this expected minimum uncertainty. One is to increase the sample mass by taking a larger
sample. However, operationally, this may not be feasible. Even if a larger sample could be taken, it may
not be possible to analyze the full sample. The other way to reduce this fundamental uncertainty is to
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reduce the particle size. The grouping and segregation error may also decrease when the particle sizes
become smaller.
If the relative variance of the fundamental error becomes greater than somewhere around 17%,
significant biases may start to occur when limited samples are available. Often, this is an indication that
the results may follow a Poisson distribution. Since the statistical analysis of skewed distributions has not
been shown to be very accurate in terms of estimating confidence limits or other statistics related to the
location of the upper tail of the distribution function, one should try to transform the sample
characteristics so that the uncertainty is adequately modeled by the normal distribution.
3.2 Estimating the Factors for the Relative Variance of the Fundamental
Error Equation
Additional guidance in estimating the factors in the equation for the relative variance of the
fundamental error is available from a number of sources (Pitard, 1989, 1993; Smith, 2001; and Myers,
1996). Some parameters in the fundamental error equation are often chosen from experience after
inspection of the sample. Other parameters require a few initial measurements or assumptions. The
following sections provide guidance on obtaining estimates for each factor.
3.2. 1 Estimating the Mineralogical (Composition) Factor, c
Pitard (1993) defines the mineralogical factor (also known as the composition factor), c, as the
maximum heterogeneity generated by the constituent (analyte or contaminant) of interest in the lot. This
maximum is reached when the constituent of interest is completely liberated (from the matrix or gangue).
The calculation of c [g/cm3] assumes that the constituent of interest is completely liberated, implying
that the sample consists of two fractions, one containing the constituent of interest and the other
containing no constituent; thus, the mineralogical factor can be estimated as:
c =
where aL is the decimal proportion [unitless] of the analyte in the sample, AM is the density of particles
containing the analyte [g cm"3], and Ag is the density of the gangue [g cm"3]. Note that a 3% contaminant
concentration is a decimal proportion of 0.03. The composition factor may also be calculated by the
following equation:
_ (i - atf yig
C- ""*" --------- «,.!_.,«-,_, ar.-n-n-.- ,„-,.,, ^^^ .
°L I
where I is the average density of the critical component and the gangue.
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For environmental applications where the decimal proportion of contaminant is low (aL < 0.1), the
estimate of the mineralogy factor is further simplified to:
°L
Likewise, when the decimal proportion of contaminant is high (% > 0.9), the estimate of the
mineralogy factor is simplified to:
c = (1 - aj)Xg
The accuracy of these estimates may need to be evaluated in situations where hazardous waste
concentrations are close to regulatory levels. However, if just used for planning, the only impact on the
study should be the need to implement a design that reflects the increased need for accuracy so that the
expected measured levels can be differentiated from the regulatory level.
3.2.2 Estimating the Liberation Factor, I
Because, for the calculation of the mineralogical factor, we needed to assume that the constituent of
interest is completely liberated from the gangue (matrix), we now need a correction factor, the liberation
factor (0 [dimensionless], for when complete liberation is not the case. Thus, 0 s / <, 1. Gy (1998, p. 67)
recommends setting I = I for environmental samples when no additional criteria are available. For most
pollutants, this should be acceptable, as the methods are for total analyte.
Pitard (1993) provides guidance on choosing / based on an assumed degree of heterogeneity of the
analyte (Table 4) and gives alternative methods of calculation that rely on the true (estimated as an
average) critical content of the constituent of interest (analyte or contaminant) in the lot, aL, and the
maximum critical content of the constituent of interest in the coarsest fragments of the lot, a^, where the
critical content is in terms of the proportional amount of the constituent of interest in the sample. The
formula based on the critical contents is:
fl_,_ - a,
r _ max L
= 1 - %
This equation assumes that one knows the maximum critical content, am!K, that all size fractions have
approximately the same critical content, aL, and that, for each fraction, the critical constituent is
segregated into individual particles.
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Table 4. Liberation parameter estimates by material description.
/
1.0
0.8
0.4
0.2
0.1
0.05
Type of Material
Analyte 100% available (recommended for most environmental applications when no criteria
are available)
Very Heterogeneous
Heterogeneous Material
Average Material
Homogeneous Material
Very Homogeneous Material
If one has a mineral-like analyte, then the liberation factor can also be estimated by (Gy, 1982):
/ =
where d} is the diameter needed to liberate the contaminant. Francois-Bongarcon and Gy (1999)
developed a generalized version of this kst equation, expressed as:
/ _
where b is an adjustable parameter that needs to be estimated for each application for the best results. For
most mining applications, where metal grains are distinct from the surrounding matrix, the value of b is
3/2.
These equations should only be used for samples where the analyte of interest exhibits mineral-like
properties. In addition, we note that these equations are simplified versions of a more complex model.
The estimates assume that the liberation factor takes on only one value for all size fractions unless one
can predict how o^ varies as a function of particle size.
3.2.3 Estimating the Shape Factor, f
The shape factor, f, also known as the coefficient of cubicity, is a dimensionless measure of how close
the particle's shape is to a perfect cube (where f = 1.0). The shape factor relates volume, V, to a fragment
size diameter, d, of unity:
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Particles with flat shapes have low shape factors, such as mica (f = 0.1) or gold flakes (f - 0.2). Most
minerals have shape factors in the mid-range (f = 0.5). A sphere has a shape factor of
= a
or about 0.5 (for d = 1), and minerals with needle shapes, such as asbestos, can have shape factors up
to 10.
If there are several types of shapes present, then the shape factor should be selected for the particle
types that contain the analyte of interest, since those particles are expected to contribute more toward the
sample uncertainty than any of the other particle types. The majority of particulate samples have shape
factors from 0.3 to 0.5. Typical hazardous waste samples particles are roughly spherical and have shape
factors close to 0. 5. Please refer to Table 5 .
Table 5. Examples of shape parameters.
f
s 1.0
1.0
0.5
0.2
0.1
Description
Needle-like materials, such as asbestos.
All of the particles are cubes (by definition).
All of the particles are spheres; most minerals; most hazardous waste samples.
Soft homogeneous solids, such as tar, or gold flakes.
Flaky materials, like mica.
3.2.4 Estimating the Granulometric Factor, g
The dimensionless granulometric factor, g, accounts for the range of particle sizes in the sample. The
granulometric factor is also known as the particle size range factor, the particle size distribution factor,
and the volume coefficient. While theoretical models could be greatly simplified if each particle had the
same size, real sets of particles have a range of sizes. The granulometric factor accounts for the range of
particle sizes by adjusting the particle sizes to a nominal value. The more uniform the particles, the
higher the value of g. For non-calibrated particles, such as particles resulting from a particle crusher and
most soils, g - 0.25. If the material was retained between two adjacent screen openings, then g = 0.55. If
the material is naturally calibrated, such as rice grains, then g = 0.75. Perfectly calibrated materials, with
all of the fragments having the same diameter, have g = 1. Also, see Table 6.
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Table 6. Granulometric factor values identified by Gy (1998).
g
0.25
0.40
0.50
0.60/0.75
0.75
1.0
Description
Undifferentiated, unsized material (most soils).
Material passing through a screen.
Material retained by a screen.
Material sized between two screens.
Naturally sized materials, e.g., cereal grains, certain sands.
Uniform size (e.g., ball bearings).
3.2.5 Estimating the Nominal Particle Size, d
The nominal particle size, d, identifies how large the largest particles are in the sample. One estimate
for the nominal particle size is the linear dimension (in cm) of a square mesh retaining no more than 5%
oversize. Another definition is that 95% of the particles must have linear dimensions less than d. The
level of 95% represents a practical compromise that avoids parameter selections based on extremes from
observed distribution functions. The value of d should be readily estimated by examination. The
influence of d in the fundamental error equation is very high.
3.2.6 Estimating the Required Sample Mass, Ms
Several procedures have been proposed to estimate the sample weight required to achieve a study's
DQOs. The two-tiered and Visman procedures may at times be more appropriate for determining field
sample requirements than for evaluating laboratory samples; however, those procedures still demonstrate
the needs and limitations related to identifying the sample mass. Understanding those requirements will
provide two benefits. One is to generate an understanding of alternate types of solutions that are available
and when they can be used. The other is to prevent wasted effort on trying to solve this issue when it is
not cost effective.
3.2.7 Rearrangement of the Relative Variance for the Fundamental Error
Equation to Determine the Sample Mass
The most common method of estimating the sample mass, MS, utilizes a rearrangement of the
equation for the relative variance for the fundamental error:
J_ - JL
M. M,
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Rearranging gives:
WLML cflgd3ML
M. = =
' Ms" + IH
j-fj- r »> jj*u* ' J.J.J.J
it "-M-I if
And, if M,.« ML, then:
The minimum variance is either known or estimated from the decision criteria that need to be met.
The mass of the lot either is known or it can be ignored if it is large compared to the subsample mass.
The other parameters are estimated based on the information from the sample, or other preliminary
samples, or from reasonable estimates from references or tables (such as Pitard, 1993). This sample mass
(MJ estimate is a minimum value, since other sources of error (besides Sp/) could contribute to the
overall variance. Thus, a larger mass may be required to meet the DQOs. However, this estimate should
prove useful in identifying if a reasonable sample mass will provide the necessary information.
3.2.8 Two-tiered Variance Comparison to Estimate IHL and the Sample Mass
Estimating IHL can be used to determine the required sample mass using a two-tiered variance
comparison and is discussed in Pitard (1993, pp. 363, 386) and Meyers (1997). It is most appropriately
applied to cases involving 2-dimensional and 3-dimensional lots, but may also provide information for
zero-dimensional and one-dimensional lots. Although this method may be limited in the application to
producing representative laboratory analytical subsamples (the subject of this guidance), it is presented
here because this method can be used to characterize the type and amount of heterogeneity and it is a way
to optimize a sampling plan.
The number of samples and analyses associated with mis procedure result in time and analysis
expenses that are justified only under certain circumstances. Any one of the following conditions might
justify the use of this method.
• A large number of similar samples are expected.
• A large economic expense is associated with the decisions based on the analysis results.
• The time and cost for an analysis is low.
Two sets of samples are required, with one sample for each set taken from the same location. Pitard
recommends the number of samples to be N = 50 (Pitard, 1993, p. 365), but reasonable estimates are
expected for sample sizes down to N = 30 (note that each sample is made up of at least 1 0 random
increments). However, if 30 samples are prohibitive, Pitard (1993, p. 385) suggests using only 10
samples per set. While 10 is too low to produce a close estimate of the uncertainty, it is large enough to
identify the order of magnitude for the sampling uncertainty.
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Assume that the mass of each sample in the first set is Msl, and the mass of each sample in the second
set of samples is M^, with Ms2 > 10Msl. The larger mass can be as large as 100 times the smaller mass
without affecting the comparison. Each sample must now be analyzed in its entirety. No subsampling is
allowed. This restriction often limits the upper sample mass level. A test sample should be run to check
the compatibility of the sample mass with the analytical method.
The variance for both sets of samples is now estimated and compared. There are four possible
outcomes:
Case 1. S} > s2 .
This is the most likely outcome. It means that effects such as grouping and segregation errors differ
over the scale change in mass. The conclusion is that mass is important. The recommendation is that the
sample weights should be based on the larger mass. One consequence is that the small mass samples
should be ignored while the larger mass sample results can be retained as part of the study.
Case 2. Sj ~ s2 , and both are small.
This outcome is rare and arises if the material is relatively homogeneous. The conclusion is that the
small mass is acceptable for characterizing the sample. However, one should realize that this outcome
can also occur when both sample sizes present estimates of the background concentration, and neither
sample mass was collected in an area with high levels of the hazardous substance. This may occur if the
contaminant occurs at only a few locations within the lot, or if both sample sizes are small with respect
the size of the lot and the contaminant is distributed like a series of discrete points rather than
continuously affecting the entire area. Assuming that the small mass does give acceptable results, then
both sets of results may be retained.
Case 3. sl ~ s2 , and both are large.
This outcome suggests that sample mass is not a primary source of variance and usually is a result of
a low concentration of pollutant or large fragments, or the effects of the heterogeneity fluctuation errors,
CE2 and CE3, in the field. The variance estimates probably represent the large-scale heterogeneity of the
matrix. The conclusion that sample mass is not important implies that the small mass level may be
adequate. Again, both sets of results can be retained as part of any further work.
Case 4. sl < s2 .
This outcome means that both of the sample masses are too small. The only conclusion is that an
acceptable result requires a sample mass greater than M2. None of the results should be retained as part of
the study. Another set of samples with a larger mass M^ should be run. Pitard (1993, p. 387)
recommends that M^ =10 M^ be selected. One can also compare the average concentrations. If the
average concentration of the smaller masses is lower than the average concentration of the larger sample
masses, then the smaller samples cannot be representative of the low occurrence of pollutant or low-
frequency pollutant clusters.
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3.2.9 Visman Sampling Constants Approach for Determining the Sample Mass
Visman (Visman, 1969; and Myers, 1996, p. 440) introduced two sampling constants, A and B, that
can be used to estimate the optimal and the minimum sample weight. Sampling constant A is related to
the heterogeneity of the sample and sampling constant B represents the variance effects from segregation,
grouping, long-range fluctuations, and periodic fluctuations. Using the summary information from the
two sets of samples with different masses described in the section, entitled "Two-tiered Variance
Comparison to Estimate IHL and the Sample Mass (M,,)," the Visman sampling constants can be
calculated as:
The optimal sample weight (Ingamells and Switzer, 1973; Ingamells, 1974; and Ingamells and Pitard,
1986) is M^ = A/B, and the minimum sample weight is M^j, = A/(a, - aB)2, where aB is the background
concentration and a, is the average concentration in sample set 1 .
3.3 Developing a Sample Mass or Comminution Strategy: The Sampling
Nomograph
Gy sampling theory provides a relatively easy way to develop a sample mass or comminution
strategy, or to optimize a sampling protocol (e.g., to stay within the predefined error limit set by the DQO
process) using a sampling nomograph (Pitard, 1993, and Myers, 1997). One of the key results from the
Gy sampling theory is the relationship between the relative variance of the fundamental error (s^2), the
particle size (d), and the mass of the sample (M,.). The sampling nomograph is a two-dimensional
summary of this relationship among the three factors in the fundamental error equation:
3 3
- -)0/
ML ML
If we assume that the mass of the lot is very large compared to the mass of the sample, ML » Ms, then:
2 a/3
SFE
M.
where C is termed the sampling constant. Because it is cubed, the particle size obviously plays a key role
when estimating the fundamental error. The development of this equation included several assumptions,
so one must remember it is only an approximate relationship. However, for most particle samples, the
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assumptions are justified. The above equation is now recast by taking the logarithm (base 10) of each
side, resulting in:
This equation shows that, for a given particle size, d, plotting the logarithm of the relative variance of the
fundamental error versus the logarithm of the sample mass gives a straight line with a slope of minus one.
This equation will be used to construct the sampling nomograph.
3.3. 1 Constructing a Sampling Nomograph
Figure 13 shows a sampling nomograph and the following example explains how to construct and use
it. To create a nomograph one has to identify at least one solution to the last equation. (Any solution to
the relative variance of the fundamental error equation will do. Typically, one knows the largest value of
d and the maximum SpE12 is usually set through the DQO process; the value of Ms is calculated from the
equation for the relative variance of the fundamental error.) This solution specifies the diameter
associated with some combination of the relative variance of the fundamental error and the sample mass.
The relative variance of the fundamental error and the sample mass determine the location of a point on
the sampling nomograph. A line with slope of -1 is then drawn through this point and labeled with the
particle diameter.
It is the placement of the slope diameter lines that makes each nomograph different. By changing d
and keeping either the relative variance of the fundamental error or the sample mass constant, additional
points can be found and a series of lines representing different particle diameters can be drawn. The
positions of these lines all have some uncertainty associated with them and the nomograph is only as good
as the model used to construct it. The farther away one goes from the original point, the more uncertainty
that there is in the predictions from the nomograph. One should try to use the nomograph in areas as
close to the initial point as possible. For more accurate predictions, additional points should be identified
from independent examples instead of extrapolating from the single point.
The assumptions made using the starting equation (the log,0 of the relative variance of the
fundamental error equation) include:
• The sample mass is small compared to lot mass
• The estimates of the parameters used to calculate the sampling constant are reasonably accurate
The assumptions made when deriving the starting equation include (Myers, 1997, Appendix C;
Pitard, 1993; and Gy, 1982):
• An average fragment mass is used
• The critical content of all of the fractions with a given density is set to the average critical content
for that density fraction
• The proportion of the mass of a size density fraction in the lot to the mass of the density fraction in
the lot is replaced by the proportion of mass of the size fraction of the lot to the mass of the lot
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10'
<0
1
8.
3456 7891
100
763
Mass [g]
Figure 13. An example of a sampling nomograph.
The sampling nomograph uses log-log coordinates (and, therefore, can be plotted on log-log paper).
The sampling nomograph has an x-axis for the sample mass that is logarithmic in scale (Iog10 grams].
Each major vertical grid line represents a 10-fold change in mass. The y-axis is the base 10 logarithm of
the fundamental error in [Iog10 relative variance units]. A bold horizontal line may be drawn in to
represent the maximum relative variance of the fundamental error that is tolerated, as set by the study
objectives (e.g., DQOs). The family of slanted lines (each with a slope of-1) represent the diameter (d)
of the largest particle size in a given sample. The distance between the slanted parallel lines is also
logarithmic (base 10). A vertical drop from a point on one slanted line to the point below on another
slanted line indicates a comminution stage (d becomes smaller). Going from one point (representing a
larger mass) on the slanted line to another point (representing a smaller mass) indicates a subsampling
stage (d and IHL remain constant; M^ becomes smaller). Using various comminution and subsampling
stages (that is, going from point-to-point along the various slanted lines) to achieve the final subsample
mass while staying within our maximum tolerated relative variance comprises our subsampling strategy.
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3.3.2 Hypothetical Example
A sample has been evaluated and it is represented in Figure 13 at point A. The sample at point A has
a mass of 763 g, with the largest particle being about 0.3 cm. A preliminary screening analysis of other
samples from the lot to be represented by this sample revealed that the average content of the contaminant
is about 100 ppm; that is aL = 0.0001 = 10"4. The contaminant, metallic lead, has a density of AM = 11.4 g
cm"3. The liberation diameter was determined to be d, = 9.5 x 10"5 cm. The factors for the relative
variance of the fundamental error for this sample are:
MS = 763 g
aL = 0.0001 = 104
AM = 11.4 g cm'3
c = A.M/aL = (11.4gcrn3/10"4) = 1.14xl05gcm-3
f = 0.5
g = 0.25
d = 0.3cm
d, = 9.5xlO-5cm
/ = (d,/d)'/2 = (9.5xlO-5cm/0.3cmf = 0.018
C = cf/g = 254 g cm"3
Assuming that the sample came from a much larger lot (i.e., ML » Mg), the relative variance of the
fundamental error for the sample is:
SpE2 = Cd3 / Ms = (254 g cm"3)(0.3 cm)3 / 763 g = 9 x 10"3
The study design objectives require an analysis using a maximum of 5 g of material with a maximum
relative variance for the fundamental error of 0.00625 or SjE2 = 6.25 x 10"3. The analytical method has an
uncertainty (CV) of less than 5% (RSD = 0.05), or a maximum relative variance of (0.05)2 = 2.5 x 10"3,
which is only somewhat smaller than the maximum s^2 tolerated by the study objectives. Note that both
the relative variance for the fundamental error (9 x 10'3) and the mass (763 g) for the sample are larger
than the requirements set by the study objectives (a maximum SpE2 of 6.25 x 10"2 and an analytical sample
mass of 5 g). This is clearly the case when looking at point A on the sampling nomograph in Figure 13.
Thus, in order to meet the study objectives, the sample needs to be altered by a subsampling and
comminution strategy, which can be developed using the sampling nomograph.
The path from point A to point E on the sampling nomograph in Figure 13 may be a route that can be
followed to get to the 5 g analytical sample. The first step is to go to point B. The only difference
between point A and point B is that the maximum particle size is smaller. Grinding (the comminution
stage) the sample until the largest particle diameters are no greater than 0.1 cm moves the sample location
from point A to point B on the sampling nomograph. Note that when d is changed, as in the comminution
stage, the liberation factor also changes because of the relationship, / = (d, / d)'/z. This also means that the
sampling constant, C, changes since C = cf/g.
A microscopic inspection of the ground material revealed that, because of the malleable nature of the
lead, a change in the liberation diameter may have occurred. A mineralogical investigation showed that
the largest particles (0.1 cm) of the ground material contained a maximum metallic lead content of 44,000
ppm, or a,^ = 0.044. Thus, the liberation factor for the ground material (at point B on the nomograph) is:
Page 53 of 134
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- a
'(0.10cm)
0.044 - 0.0001
1 - 0.0001
= 0.044
and the liberation diameter for the ground material is now:
(o.iOc«)
= (0.044)2 (0.10 cm) = L92xW~4cm
Note that, in a comminution stage where d changes and there has not been a subsampling stage (where d
does not change, and ML = N^ = constant), it would be inappropriate to use the equation,
~ ~MIHl = (~M
M,
M.
JL
M,
to calculate the relative variance of the fundamental error for the ground sample (at point B on the
sampling nomograph).
Using correct subsampling methods (e.g., a sectorial splitter; see Figure 14), a subsample of 100 g
could then be taken to move along the BC diagonal to point C, which is at the maximum tolerable relative
variance for the fundamental error as set by the study objectives (Spg2 = 6.25 x 10"3 on the sampling
nomograph). Note that any subsamples taken along the BC line will have the same maximum diameter of
0.1 cm for the largest particles.
Stream
Receiving
Container
\
o
Rotating Axis
Figure 14. A sectorial splitter with eight sectors.
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Grinding the 100 g sample to a size of no more than 0.03 cm moves that intermediate subsample from
point C to point D. Again, because of the malleable nature of the lead, another microscopic inspection of
the ground material revealed another possible change in the liberation diameter. Another mineralogical
investigation showed that the largest particles (0.03 cm) of the ground material contained a maximum
metallic lead content of 65,000 ppm, or a^ = 0.065. Thus, the liberation factor for the ground material
(at point B on the nomograph) is:
, _ amax " °L _ 0.065 - 0.0001 _
/ /n /yi ™«\ ~~ ~ - ~~ - - - - ~
(0.03 cm) \ - aL 1 - 0.0001
and the liberation diameter for the ground material is now:
4(0.03 cm) = /2^ = (0.065)2(0.03c/n) = L27jclO~4cm
Correctly subsampling from that 100 g subsample along the diagonal line DE until 5 g is reached
brings that final analytical subsample to point E on the sampling nomograph with an SpE2 = 5.0 x 10'3,
which is below the maximum set by the study objectives (s^2 = 6.25 x 10"3).
However, each subsampling event is associated with a minimum addition to the starting point
uncertainty (propagation of error). The total relative variance of the fundamental error for this
subsampling strategy is the sum of the relative variances of the fundamental error for each subsampling
stage along that strategy path; that is, from points A to E, the subsampling stages are the line from point B
to point C and the line from point D to point E.
From point B to point C:
4 (0.10 cm) = (-L - -Uqflgrf*
Ms ML
4j (0.10 cm) = (— 1— - — L- )(1.14jcl05gcm-3)(0.5)(0.044)(0.25)(0.10cro)3
5(0.10 cm) = 5.4 x
From point D to point E:
4, (0.03 cm) - (-L - -L)cflgd3
MS ML
5^(0.03 cm) = (— - — L_ )(1.14xl05£cm-3)(0.5)(0.065)(0.25)(0.03c/n)3
5g
^ (0.10 cm) = 4.75
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The total relative variance of the fundamental error for the subsampling strategy as displayed in the
sampling nomograph (Figure 13) is:
s$E(Total:pathA^E) = s^(O.lOcm) + ^(0.03 cm)
SM (Total: path A -E) = 5.4x10'* + 4.75 xl
-------
The grouping component of the error is zero when each fragment is independently and randomly
selected. This is not usually a very practical alternative. The practical action is to maximize the
number of increments (or splits) and to select those many increments from locations randomly
distributed throughout the sample (or lot). Methods with a large number of increments (N > 30)
that select particles representative of the entire sample should produce the best results.
Mixing the sample is not a guarantee that segregation effects have been eliminated. This means
that subsampling methods which minimize or eliminate segregation effects are preferred (again,
through many increments or splits taken with correct sampling practices).
One can modify the equation of the relative variance of the fundamental error used to generate the
sampling nomograph if MS is on the order of ML; e.g., for one pass through a riffle splitter, ML =
2MS and
sL = (-L - -L)cflgd3 = (— - — )Cd3 = (— - — )Cd3 = (—)Cd3
FE M M, M M/ M, 2M/ 2M/
S Jj S J-i S S )»
3.4 Low Analyte Concentration Considerations
3.4.1 Low-frequency of Analyte (Contaminant) Particles
It is important to determine if low numbers of contaminant particles are present. If a sparse analyte
distribution is not identified, then it is all too easy to misjudge the contamination level of the lot as
nonexistent. When only a rare occurrence is the cause of contamination (for instance, the occasional
presence of a lead shot), then it is probable that analyzing only a few of the possible subsamples may miss
the contaminant. When the fraction of particles containing the contaminant is very small, the distribution
of the results is usually more like a Poisson distribution than a Gaussian distribution, and the parameter
estimates, such as the mean, will have relatively large uncertainties if only a few results are available.
These difficulties may be minimized if the sample mass is increased so that at least 4 or 5 analyte particles
are expected in each subsample (Pitard, 1993, p. 370), although, at the very least, 6 analyte particles are
recommended (or the estimate of the average analyte concentration may be poor). For laboratory samples
this rule-of-thumb would have to be applied based on prior knowledge about the number and distribution
of analyte particles. Either the frequency of the analyte particles needs to be known from a prior study or
additional representative samples need to be provided to the laboratory for a few test runs.
Sparse distributions of high concentration analyte particles are one of the more difficult sampling
problems. The increased relative variability is easily envisioned for cases where only 1 or 2 analyte
particles, on average, may end up in a subsample. When only a handful of target particles is involved, it
is quite possible for samples to contain no analyte particles, just one particle, 3 particles, or even 5
particles. The relative variability in estimating the overall analyte level would be expected to be very
high. Even more important, if less than one particle is expected per subsample, most subsamples might
contain no particles and occasionally a sample with one or more particles would be analyzed, resulting in
a high analyte concentration.
Sample matrices fitting this description could be repeatedly sampled and subsampled and only rarely
would a concentrated particle be selected. A histogram of the analytical results might be highly skewed,
Page 57 of 134
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with the majority of results at very low concentrations. The occasional very high concentration would
appear, for all intents and purposes, as an outlier. If the sparse nature of analyte particles in the sample is
not understood, then decisions may be based on the assumption that the data follow a normal probability
distribution. Declaring the one subsample with relatively high analyte levels to be an outlier might easily
lead to the incorrect conclusion that a contaminated site has met cleanup standards. This is why it is
important to determine how the analyte is distributed among the sample particles. If the sample
characteristics fail to match the statistical assumptions, then the resulting decisions may not be justified or
correct.
Of course, there may be instances when a high value really is an outlier due to random chance (one
just happened to select a subsample with very rare high-level contaminant concentration) or to an
inadvertent error such as transposing numbers when recording a value. The important question is to
evaluate the results in light of any other information so that one minimizes making an incorrect decision.
If the result of multiple subsample analysis is near zero for most subsamples and very high for just one
subsample, then the question to answer is whether or not the high value is large enough to present a
problem. This may be the case if the average analyte concentration exceeds a target cleanup standard or
if the net amount of hazardous compound present in the high concentration subsample presents a health
hazard. However, whether the outlier is false or real, if its presence does not affect the study outcome,
then one should note its presence before continuing with the study.
If the outlier results change the study decision, then additional analysis should be carried out. To
increase the chance that such concentration patterns are accounted for rather than ignored, one needs to
determine when high-concentration analyte particles may be present in low numbers. Several samples,
designed to include at least one high level particle with a 90% or a 95% chance, should be taken. Those
samples can be ground to eliminate the effect of particle size in the subsampling procedure (or the entire
sample can be exhaustively analyzed). If the concentration is significantly higher from these analyses
compared to the results from individual subsamples, then one may be dealing with a matrix that not only
requires special processing techniques, but also requires special field sampling practices to ensure that the
site is correctly characterized.
For a discussion on outliers, please refer to Barnett and Lewis (1995) or to Singh and Nocerino
(1995). Software has been developed for the methods discussed in the latter reference by Singh and
Nocerino; the software is called Scout (Scout, 1999).
3.4.2 A Low Concentration Approximation ofsFE2
ML
If the analyte levels (a^) are low and Ms < - , the equation for the relative variance of the
fundamental error can be approximated (Pitard, 1993, p. 334, Equation 18.19) as:
4 - - - 2)4.
<*Lc
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where Lc is the particle size class of interest, F^. is the average fragment of the class, Lc, f is the shape
factor, A is the density of the material [g/cm3], M,, is the sample mass, dFLc is the average particle diameter
MLc
of FLC in Lc, aic = —- is the proportion of Lc in the lot, L, and is the critical content to be estimated.
ML
3.4.3 A Simplified Low Concentration Approximation ofSp/
The above low concentration approximation can be further simplified by selecting the parameters
that one might expect for typical samples. Many samples can be modeled with f = 0.5, A = 2.7, and d set
to represent the largest particle size (see below). The sample mass required to achieve a particular error
level then becomes:
_ (0.5)(2.7) ( 1 _ 3 _ 1.35. 1 _ 2_3
Ms ~ 2 ^Z)"FLc ~ ^~{ Z)"FLc
SFE aLc SFE °Lc
Results using this simplified equation are shown in Table 7 for: analyte proportions of 0.1, 0.05, and
0.01; percent relative standard deviations of 50, 25, 10, and 5; and large particle diameters of 0.01, 0.05,
0.10, 0.50, and 1.0 cm.
The results for this series of stereotypical samples are very informative. When the diameter of the
largest particles is about 1.0 cm, the amount of sample required to identify the concentration within 50%
RSD is from 40 g to 500 g. Dropping the diameter by a factor of 2 to 0.5 cm reduces the sample mass
requirements by a factor of 8. However, the minimum mass needed to estimate the result within 50% still
ranges from 5 g to 60 g. Reducing the maximum particle size to 0.1 cm results in sample sizes on the
order of 1 g to 10 g for a relative standard deviation of 10%. If the maximum particle size is reduced to
0.05 cm, then sample sizes of 0.5 g to 6 g are expected to have RSDs of about 5%. The obvious
conclusion is that many samples with particle diameters over 0.1 cm will have significant sampling
uncertainty when small sample masses (1 g to 5 g) are utilized for chemical analysis. However, as the
particle size drops below 0.1 cm, the mass of the sample associated with large uncertainty levels drops
rather steeply.
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Table 7. The relationship of the particle diameter, the analyte concentration, and the desired
uncertainty level to the sample mass for low concentration samples of average
density (A = 2.7 g cm"3). Sample sizes in columns 3 through 6 are shown to 2
significant figures in units of grams.
Maximum
Diameter
(cm)
0.01
0.05
0.10
0.5
1.0
Analyte
Proportion
0.1
0.05
0.01
0.1
0.05
0.01
0.1
0.05
0.01
0.1
0.05
0.01
0.1
0.05
0.01
% Relative Standard Deviation
M. = 5g
0.0043
0.097
0.053
0.54
1.2
6.6
4.3
9.7
53
540
1,200
6,600
4,300
9,700
53,000
M,«10g
0.0011
0.0024
0.013
0.14
0.30
1.7
1.1
2.4
13
140
300
1,700
1,100
2,400
13,000
M. = 25g
0.00017
0.00039
0.0021
0.022
0.049
0.26
0.17
0.39
2.1
22
49
260
170
390
2,100
M. = 50g
0.000043
0.00097
0.00053
0.0054
0.012
0.066
0.04
0.10
0.53
5.4
12
66
43
97
530
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Section 4
Subsampling Techniques
There are many ways one can generate subsamples. Many of them are familiar and widely used but a
few of them are less common. Not all of these methods are recommended, including some commonly
used methods that are subject to large errors. Each subsampling method is described and comments
summarizing their advantages and disadvantages are provided. The nature of the uncertainty for each
method is related to both the method and the sample characteristics. For example, one method may
perform as good as another method if the sample is a fine powder, but wholly different results may occur
for a dried stream sediment with large density differences between the analyte and the inert particles.
Sampling that provides unbiased, low variable, quantitative estimates becomes more difficult as the
heterogeneity of a sample increases. Suppose the range of particle sizes is increased. The effect of
including or excluding a larger particle will increase the range of possible results. If the density of one
particle type is much greater or much less than the others, then that particle type will tend to self-select or
automatically exclude itself from the final subsample (due to gravity). The higher the density difference,
the larger the expected segregation.
Table 8 (at the end of this section) lists the relative rankings and performance of the various
subsampling methods described below.
4.1 Subsampling Methods
4.1.1 Sectorial Splitter (Pitard, 1993; p. 268)
A sectorial splitter consists of a rotating metal cone with ridges and valleys. The sectors should be
radially symmetric and of equal size. The sample is placed in a hopper with adjustable vibration levels
set so the sample particles slowly emerge and fall onto the side of the rotating cone (see Figure 14). As
the particles fall from the hopper, they are sometimes channeled through a funnel before dropping onto
the side of the rotating cone. The hopper should be just above the funnel or cone to minimize loss from
bouncing off the apparatus. Particles fall into containers placed under each valley. The receiving vessels
depend on the size and design of the splitter. Small splitters may use a test tube while large splitters may
require beakers or jars. The best results are obtained when operating the sectorial splitter with a constant
rotational velocity and feeding it at a constant rate. Slow feed rates increase the number of increments
and help to minimize the grouping and segregation error.
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4.1.1.1 Advantages
Sectoral splitters have several positive attributes. There is little extraneous between-particle
correlation allowed to propagate from the sample to the subsample. The increment size is small
irrespective of any segregation due to processes at work in the hopper and container, and only a small
amount of sample is presented to the splitter at any time. Thus, because of those many increments, the
grouping and segregation errors are small. Since the entire sample is processed, the delimitation and
extraction errors are negligible as well. The only significant error is the fundamental error (assuming no
gross operational errors). Related to the lack of correlation, one finds that different particle sizes tend to
emerge independently of each other. Sectorial splitters remove virtually all of the uncertainty associated
with operator bias and require very little time from the analyst.
4.1.1.2 Disadvantages
The user needs to pay careful attention to the fate of very fine powders when using mid-to-large sized
sectorial splitters. In our limited experience, it appears that manufacturers have a tendency toward
allowing surface roughness in the sectorial splitter to be proportional to size. If the analyte is associated
with very fine particle levels, a significant fraction may become lodged on the surface of the rotating
splitter head.
4.1.2 Paper Cone Sectorial Splitting (Geriach et a/., 2002)
This method uses a sheet of paper folded to resemble the ridges and valleys of a sectorial splitter (see
Figure 15). Each valley is positioned above a container and the sample is poured so the particles drop just
off-center. The source stream is rotated
around the vertical axis of the cone
during the splitting process so that each
container receives approximately the
same amount of sample. This
procedure results in a large number of
increments. If the sample is slowly
poured to maximize the increment
number and there is no transfer loss,
then this method mimics the processes
involved with sectorial splitters. Paper
cone sectorial splitting subsampling has
been shown to rival the performance of
standard riffle splitters and perform
better than alternate shoveling, coning
and quartering, and grab sampling
(Geriach, etal., 2002).
Figure 15. A paper cone sectorial splitter with eight sectors.
Paper cone sectorial splitting is inexpensive and the materials needed are commonly available. On
the other hand, it is more prone to operator error than mechanical sectorial splitters. Paper cones take a
few minutes to make, but have the advantage of being disposable.
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4.13 Incremental Sampling (Pitard, 1993; pp. 128, 207 ff.)
An increment is a group of particles or material physically extracted from the lot (or sample) with a
single operation of the sampling device. The sample (or subsample) is made from the reunion of many
increments (N > 30 increments is recommended) taken at random locations across the lot (or sample) to
be represented. Incremental sampling increases the probability of sampling each location of the lot.
When used with a correct sampling device, it can provide correct subsamples. However, the
materialization error can inflate the uncertainty if the increments are biased due to poorly designed
sampling tools. Incremental sampling relies somewhat on the skill and experience of the sampler to avoid
bias and acquire a representative sample. The sampling device must allow for correct sampling. For
example (see Figure 8), a scoop with a rounded bottom is not properly delimited to obtain a correct
sample because it prevents one from sampling the particles at the bottom of a sample with the same
probability as the particles at the top. For a correct sample, the scoop needs to have a flat bottom and
parallel sides (e.g., square or rectangular).
To sample a one-dimensional lot (Pitard, 1993; p. 239), form the material into an even long pile made
from many layers (the more, the better) and take at least 30 increments, each taken entirely across the pile
at several equally spaced points until the sample mass, M^ is acquired. Thus, if 30 increments are being
taken to make the subsample, then the mass of each increment should be about M/30. Taking each
increment requires a correct sampling device to remove all of the material from top to bottom in the pile.
The one-dimensional pile is recommended for laboratory incremental subsampling. If the mass required
for the subsample is so small that taking 30 increments is physically constraining, then fewer increments
may be taken, although the error may increase with fewer increments. The relative variance due to the
grouping and segregation error, s^2, can be made relatively small compared to the relative variance due
to the fundamental error, s^2, by increasing the number of random increments, N. That is, for N
increments, taken with correct sampling practices (Pitard, 1993, p. 388)
and sTE2 > sCE12 ~ SpE2 if N is made large enough. At least N = 30 increments are recommended as a rule of
thumb to reduce s^2 compared to s^2 (Pitard, 1993; p. 187). Grinding the sample or using a sectorial
splitter may be a better option in that case.
To sample a two-dimensional lot (Pitard, 1993; p. 230), form the material into a flat pancake and take
at least 30 increments at random locations using a cylindrical sampler with a constant cross section, with
the sampler's cutter perpendicular to the pile and take the increments all of the way through the pile.
Two-dimensional incremental subsampling is usually not as reliable as one-dimensional incremental
subsampling for the laboratory, since the material from the increment can easily fall back onto the sample
surface, especially if the material is very dry.
Incremental sampling with at least 30 increments reduces the relative variance of the grouping and
segregation error (s^2). However, there are a few cases where this is ineffective, such as when the
number of particles with high levels of analyte is small. In this case, incremental subsampling is about as
effective as taking the entire subsample at one spot. Each increment is still subject to the materialization
error (ME). For incremental sampling, one must use a correct sampling device that minimizes that error
(for example, see Figure 8). Under most circumstances, incremental sampling should allow the analyst to
meet the study requirements (e.g., the DQOs).
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4.1.4 Riffle Splitting (Pitard, 1993; p. 266)
A riffle splitter (sometimes known as a Jones splitter) is a mechanical device
with a series of alternating chutes that deposit one-half of the sample
into one discharge bin and the other half into a second bin (see
Figure 16). (One should avoid any riffle splitter with an
odd number of chutes.) The method is limited to free-
flowing samples. Riffle splitters utilize multiple
fractions (chutes), increasing the number of
increments in each round, more so than methods
such as coning and quartering. However, riffle
splitters have much larger increments than sectorial
splitters. Several varieties of riffle splitters have
been developed. They are available in many
sizes and some provide splitting of a sample into
more than halves, such as fourths or eighths, in
one operation.
Riffle splitters can perform well, but the
results rely on the skill and training of the operator.
The sample needs to be presented to the riffle
splitter such that each chute gets a similar
amount, and there should be no bias in
presenting the sample to the chutes (Pitard,
1993). Schumacher et al. (1990) demonstrated
that up to six passes were needed to minimize the uncertainty added with this procedure. When properly
run, riffle splitters are excellent mass reduction tools (Allen and Kahn, 1970; Mullins and Hutchison,
1982; and Gerlach, et al., 2002).
While riffle splitters may have only 10 to 30 chutes, depending on the size and model, repeated passes
result in each chute receiving particles from additional locations in the original sample. This causes the
number of increments to be much larger than the number of chutes for the portion of the sample being
processed. However, since one discards one-half of the sample in the first stage of a multiple-step mass
reduction procedure, the grouping and segregation error associated with the first pass may dominate the
uncertainty associated from the entire splitting process. This means that there may be very little
improvement in reducing the grouping and segregation error after the first pass as the rest of the sample
splitting procedure takes place.
Figure 16. A riffle splitter with 20 chutes and two
collection pans.
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4.1.5 Alternate Shoveling (Pitard, 1993; p. 271)
Alternate shoveling involves taking a series of
scoops (increments) selected randomly from the entire
sample and depositing the alternate scoops in two piles
containing an equal number of scoops (Figure 17).
The increments should be the same size and the
minimum number of increments should be around nine
for each pile. However, increasing the number of
increments should minimize the effect of the grouping
and segregation error. Each scoop tends to select
particles adjacent to each other, maintaining much of
the naturally occurring grouping and segregation error.
The analyst must balance the extra time required for
small scoop sizes to achieve a lower grouping and
segregation error with being able to accomplish all the
shoveling steps in the time available for sample
processing. Another drawback with this method is that
the procedure may need to be repeated until the sample
is reduced to the mass required for chemical or
physical analysis.
Figure 17. The alternate shoveling procedure.
4.1.6 Coning and Quartering (Pitard, 1993; p. 270)
Coning and quartering involves mixing and then pouring the sample into the shape of a cone
(Figure 18). The cone is flattened, divided into four sections with a cross cutter having 90° angles, or by
first cutting it in half with a stiff piece of material (e.g., a sheet of plastic or paper), and then dividing each
half to get quarters. Alternate quarters (splits) are combined to make a subsample, and one subsample is
Coning & quartering
Discard
Figure 18. The coning and quartering procedure.
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chosen at random for any additional mass reduction until the desired subsample mass (M,.) is reached. As
with alternate shoveling, this method can take considerable time to obtain a subsample.
Coning and quartering is a process that preserves particle correlations within each quadrant, and this
effect is worsened by combining quarters. Essentially, one has obtained a two-increment sample
containing one-half of the original mass. Thus, there is some tendency to maintain the variability due to
grouping and segregation problems. In coning and quartering, there is also the difficulty of creating the
initial pile so that all of the particles are randomly distributed across each quadrant. Coning and
quartering is not recommended since it is a lot of effort just to reduce the grouping and segregation error
by two (the number of increments, N = 2) each time that the method is performed.
4.1.7 Rolling and Quartering (Benedetti-Pichler, 1956; p. 215)
Rolling and quartering is variation on coning and quartering. The sample is placed in a conical pile
on a large sheet of material with a smooth surface, such as: glazed paper, a thin plastic sheet, or
rubberized cloth. The cone is then flattened and the material is mixed by rolling it back and forth. The
material is returned to the center by lifting all four corners of the sheet, flattened out, and then split in half
using the quartering procedure. One-half of the material is selected at random for any further mass
reduction or for analysis.
The only difference between this method and coning and quartering is that some mixing takes place
when the sample is rolled. This activity will cause any loosely bound clumps of particles to break apart.
It may also serve to reduce some bias in the sample pile associated with pouring out the sample. Overall,
there is little difference between rolling and quartering, and coning and quartering. Therefore, this
method cannot be recommended, either.
4.1.8 Fractional Shoveling (Pitard, 1993; p. 272)
Fractional shoveling involves processing the sample into several subsamples. One increment (shovel-
full or scoop-full) of material at a time is removed from the lot or sample (as a pile) and added in turn to
form each of the subsample piles (see Figure 19). This method has characteristics very similar to
alternate shoveling and the minimum number of increments should be 10 per pile. In this case, the
sample is divided into several piles. The number of increments will be inversely related to the number of
subsample piles and to the particle correlation level. Once the original lot or sample pile is exhausted,
one of the new subsample piles is selected at random. If the subsample pile is still too large (greater than
MJ, then it can be reduced with another round of fractional shoveling, and the process can be repeated
until the desired subsample mass (Mj) is obtained.
Fractional shoveling can be time-consuming for large samples. In addition, segregation effects
related to density may result in trends in composition as the pile is reduced. For small samples, this may
result in an increased bias between piles, as the last increment may be enriched (or depleted) compared to
previous increments. If the resulting piles do not appear to be visually similar, then the sample
characteristics may not be amenable to splitting by fractional shoveling.
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Fractional shoveling into 5 fractions
F3 = 3 + 8 + 13 + . ..
F4 = 4 + 9 + 14 •*• . . .
Figure 19. The fractional shoveling procedure. In this case, there are five
fractions, fractions (samples) F1 through F5, each made by
sequentially combining every fifth increment.
4.1.9 Degenerate Fractional Shoveling (Pitard, 1993; p. 272)
Degenerate fractional shoveling is similar to fractional shoveling. The only
modification is that one out of every so many (e.g.,
every fifth scoop) increments is placed in the
subsample pile while the other scoops are place in a
discard pile (Figure 20).
Degenerate fractional shoveling is
expected to perform slightly worse that
fractional shoveling, as the operator has only
one pile designated for a subsample. This
situation lends itself to deliberate or
inadvertent bias when choosing part of the
sample with the scoop. This makes degenerate
fractional shoveling a biased method and it should be avoided.
For fractional shoveling methods, the variability from the
grouping and segregation error will increase with larger, but fewer,
increments (scoop sizes), and the variability
will increase with a larger number of
subsamples (containing fewer increments)
considered at a time. For example, if there
are just two subsamples, then only 50% of
the sample can be selected each time, and
subsequent subsampling is expected to
Figure 20. The degenerate fractional shoveling procedure.
Every fifth increment taken becomes part of the
fraction (sample) while the other increments
become part of the rejects.
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select portions from any of those subsamples for final analysis. However, for 10 subsaraples, or 1/10
degenerate fractions, the operator discards 90% of the sample. If a second phase of subsampling is
carried out, then only 10% of the original material is available at the beginning of the second phase; thus,
much of the grouping and segregation error from the first phase will remain and could influence the
overall error from the remaining sample reduction steps. That is, whatever bias was introduced at the first
stage will most likely influence all of the subsequent steps of the sample mass reduction process. This
type of problem is similar to the one discussed for riffle splitters.
4.1.10 Table Sampler (Allen, 1997, p. 20)
Sample
A table sampler consists of an inclined surface
with various triangular prisms placed to divide the
sample as the surface is vibrated and the particles
move from the top to the bottom (see Figure 21).
Like the riffle splitter, this device splits the sample
in one pass. However, the number of increments
associated with this technique is small with a
tendency to allow large grouping and segregation
errors to remain. The device is also bulky and not readily
available. This method has little to offer for recommendation.
The method should suffer from operator bias in pouring the sample
into the apparatus in the same manner that degrades the performance for
coning and quartering, and for riffle splitting. The initial location on the table
with respect to the other parts of the sample will bias certain particles toward a
particular subsample. In effect, some of the initial correlation derived
from the placement of the sample is maintained, and that means that some
of the grouping error is maintained.
4.1.11 V-Blender (Pitard, 1993; p. 190)
V-blenders are devices for
homogenizing samples (see
Figure 22). However, material in
a V-blender tends to segregate in
the time it takes to discharge the
sample, and the material will most
likely segregate again as it is
discharged from the bottom of the
blender. Obtaining unbiased
subsamples using this method
would require "correct" sampling
practices prior to discharging the
material from the blender (this
may be impossible).
Figure 21. A table sampler.
Sample
Discharge
Figure 22. A V-blender.
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4.1.12 Vibratory Spatula (Pitard, 1993; pp. 197, 239)
This device is sometimes used to feed material when dividing samples. Its use should be avoided.
The result is more likely to enhance segregation than to reduce it. It is well known that vibrating a
quantity of particles tends to enhance segregation based on size, shape, and weight.
4.1.13 Grab Sampling (Pitard, 1993; pp. 80, 205)
Grab sampling almost always involves taking the subsample off the top of the sample. One or more
scoops are typically placed on a balance. The scoop, most likely an incorrectly designed sampling device,
is often used to return some of the material to the sample container if the mass is larger than needed for
the analytical procedure. Grab sampling does not meet the criteria of a correct sampling procedure for
heterogeneous particles because it does not give each particle the same probability of being sampled.
Grab sampling does absolutely nothing to reduce the sampling errors (i.e., GE, DE, and EE) that we can
minimize through correct sampling techniques. If the analyte is associated with particles that have
physically different characteristics, such as size, shape, or density, then a biased result is expected. If
there is a tendency for the particles to segregate, then the bias and uncertainty may be very large. Grab
sampling is particularly prone to generating biased results due to gravity effects and to sampler bias.
Grab sampling is judgmental and nonprobabilistic, and should be avoided since it is not designed to take a
representative subsample. As a sampling method, the grab sampling procedure appears to be designed to
provide biased results. Grab sampling should only be considered if one has previously shown that the
matrix and particle size have no significant effect upon sampling error. Even then, the analyst is at risk of
reporting unreliable results if any of the samples fail to meet those assumptions. Avoid grab sampling!
4.2 Minimizing Particle and Mass Fragment Correlation
Minimizing the correlation among all of the mass fragments is similar to the criteria of sampling each
mass fragment with equal probability. Several examples will show how this concept can guide the
selection of sampling methods. If the selection probability is defined in units of mass fractions rather than
individual particles, then the effect of large particles in any sampling scheme is now obvious. A large
particle can be thought of as consisting of numerous smaller mass fraction units that behave as a single
larger unit. Selecting one mass fraction that is part of the large particle causes the other smaller mass
fractions to get selected too. This 100% correlation between mass fragments, whether selected into a
subsample or remaining in the original matrix, results in an increased variability for any measurement.
Now suppose one is preparing a subsample by fractional shoveling. The size of the scoop, or increment,
is related to the level of correlation between particles that is common between the sample and the
subsample. The smaller the increment, the lower the level of correlation between the particles in both the
sample and the subsample. Also, the smaller the increment mass, the more random increments that can be
taken and combined to make up the subsample mass (Ms), thus, decreasing sGE2. This explains why
sectorial sampling is invariably better than fractional shoveling.
However, there is a practical limit on how small the increment can be for a correctly designed
sampling device to remain correct, and it has to do with the extraction error (EE) and the size of the space
of the inner walls of the sampling device. Obviously, this space must be large enough to accommodate
the diameter of the largest fragments (d) to be sampled. Not immediately obvious, however, is that the
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action of the cutter moving through the sample could cause some of the fragments that should be part of
the increment (that is, those fragments that have their center of gravity within the increment extended
boundary) to move from the leading cutting edge making contact with those fragments to past the
opposite trailing cutting edge if that space is too small. Therefore, those fragments do not become part of
the sample and the quest for a fragment to have an equal chance to be part of the sample or to not be part
of the sample is lost, and a sampling bias is introduced.
Pitard (1993, pp. 292 ff) gives several rules for cutter speed and inner wall sampler width. Those
rules are generally summarized giving: a maximum cutter speed of 0.6 m/s; an inner wall sampler width
of 3d for coarse materials (d ^3 mm) and 3d + 10 mm for very fine materials, thus giving a minimum of
10 mm; and a sampler depth of at least 3d. The cutting angle should be either zero (the cutting edge is
perpendicular to the extended increment) or greater than or equal to 45 degrees. Those rules may be quite
constraining for the very small increments that may be needed for laboratory subsampling. If a width of
less than 10 mm is used, we can only suggest (without our own empirical evidence) that the cutter speed
should be much slower.
For riffle splitters, Pitard (1993, p. 302) recommends a correct riffle chute width of 2d + 5 mm. No
minimum width is recommended for sectorial splitters, although an opening somewhat larger than d is
obvious; however, Pitard (1993, p. 303) does recommend that the sector slope should be at least 45° for
dry materials and 60° for slightly moist materials. Since true splitting methods select the splitting
increments at random, the extraction bias (EE) should cancel out for those methods.
One can roughly rank the expected performance of a sample mass reduction scheme with the
probability that one particle will get selected given that an adjacent particle is selected. The lower that
probability (that is, the lower the correlation between the fragments), the better the expected performance.
Thus, by careful review of the available options for subsampling, one should be able to rank their
performance. Gerlach et al. (2002) demonstrated this concept with several common subsampling
methods. Knowing the performance with one sampling method will allow the investigator to narrow or
broaden the list of subsampling methods providing acceptable performance. If adequate performance
levels are still not met, then reducing the fragment size may be needed.
4.3 Ranking Subsampiing Methods
To rank any two methods, one must have the explicit details for the process (Mullins and Hutchinson,
1982; Gerlach et al., 2002). To compare coning and quartering to riffle splitting, for instance, one needs a
few operational details. For coning and quartering, the process is briefly described as:
1) pour the sample into a single pile
2) flatten the pile
3) divide the pile into fourths
4) retain two opposing quarter fractions (i.e., only 2 increments)
5) discard the other two opposing quarter fractions (see Figure 18)
For example, to obtain a 6.25 g subsample (MJ from a 100 g (ML) sample, the process is carried out 4
times. Each application of coning and quartering divides the sample in half by combining 2 splits (i.e., N
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= 2 increments to make the subsample) of the 4 splits. This hardly seems worth the effort for potentially
reducing s^2 by 2 for each time that the process is repeated!
Now consider dividing the sample with a 12-chute riffle splitter (an example of a riffle splitter is
shown in Figure 16). Each pass through the riffle splitter results in a division into 12 splits, 6 of which
are recombined in each receiving pan. Thus, N = 6 increments make up each subsample per pass,
potentially reducing sGE2 by 6 for each time that this process is repeated. Again, for the 100 g sample, the
process is repeated 4 times to give a final 6.25 g subsample.
The coning and quartering method and riffle splitter method each produced a 6.25 g subsample; but
which sampling protocol is most likely to have the lowest variability? The answer is the 12 chute riffle
splitter, which had more increments or splits, each with a smaller mass, per step than the coning and
quartering procedure.
The reason that sectorial splitters outperform riffle splitters is similar to the reason that riffle splitters
normally outperform coning and quartering methods. The explanation lies in the details of the method.
For the riffle splitter, the operator pours the sample across a pan that gets dumped into the riffle splitter.
The sectorial splitter is spinning and only a small amount of the sample exits the feed hopper during the
time needed to turn the splitter from one sector to the next. The result is that very small, but very many,
sample increments are utilized to get the equivalent final subsample mass, thus, greatly reducing the
grouping and segregation error for the sectorial splitter. This dramatically decreases the odds of particle
correlation (that one particle will be sampled if an adjacent particle is sampled).
Table 8 lists the relative rankings and performance of the various subsampling methods that were
described in this section.
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Table 8. Authors' relative rankings (from best to worst) for subsampling methods. N is the
number of increments, GE is the grouping and segregation error, N.R. means not
recommended, and N.A. means not applicable.
Method
Sectorial
Splitter
Paper Cone
Sectorial Splitter
Incremental
Sampling
Riffle Splitting
Alternate
Shoveling
Fractional
Shoveling
Table Sampler
Degenerate
Fractional
Shoveling
Rolling and
Quartering
Coning and
Quartering
V-Blender
Vibratory
Spatula
Grab Sampler
Typical
Increment
Size
Very Small
Small
Small to
Medium
Small to
Medium
Medium
Medium to
Large
Medium
Medium to
Large
Large
Large
N.A.
Small
Variable
Sensitivity to
Grouping &
Segregation
Low
Low
Moderate; Low
with Many (N
>30) Correct
Increments
Low to
Moderate
Low to
Moderate
Moderate
Moderate to
High
Moderate to
High
High
High
High
Very High
Very High
Moisture
Content
Dry
Dry
Dry to
Moist
Dry
Dry to
Moist
Dry to
Moist
Dry
Dry to
Moist
Dry
Dry
Dry
Dry
Dry to
Moist
Correct
Sampling
Possible
Yes
Yes
Yes,
with1-d
Pite
Yes,
Takes Skill
Yes,
if Careful
Yes,
if Careful
Very
Difficult;
Depends
on Design
Yes,
if Careful
Yes,
if Careful
Yes,
if Careful
NA
No
No
Agreement with
Calculated sfe2
Very Close
Close
Close
Good to Fair
Good to Fair
Fair
Not Close
Unlikely
Usually Not
Close
Usually Not
Close
Very Unlikely
Not Close
Not Close
Comments
Method of
Choice
Performs Well
Bias Still
Possible,
Decreases
with N and
Correct
Sampling
Possible Loss
of Fines
Takes Time to
Prepare from
a Large Lot
Performance
Tied to Lot
Mass
High
variability;
N.R.
Performance
Tied to Lot
Mass; Subject
to Bias; N.R.
Highly
Variable; NR
Usually
Biased; NR
Problems with
GE; N.R.
Problems with
GE;NR
Biased and
Variable; N.R.
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Section 5
Comminution (Particle Size Reduction) Methods
There are a large number of products available to reduce a sample by comminution (grinding or
crushing). One must be careful to match the equipment to the sample, and to the study goals. Sample
mass reduction equipment should be constructed of materials that are not of interest to the study. One
should not use stainless steel grinding equipment if the analysis includes chromium. Also, some samples
should not be processed with specific equipment. Pitard (1993) notes that samples with pure gold
inclusions should not be reduced in a mill that pounds the sample. Since gold is very malleable, the result
would be equivalent to preparing and applying gold leaf to the equipment (fashionable, but not
practical!). Similarly, in one of our early studies, an attempt was made to crush a sugar sample laced with
salt crystals. The result was a very fine powdered sugar, but there was no reduction in variability
associated with the salt particles. The reason was that the sugar was preferentially and more easily
crushed, and the much larger mass of the sugar also protected the salt particles from being broken up. If
there is any doubt as to the effectiveness of a sample comminution step, several test samples should be
run to verify that a lower uncertainty level was reached. Several general types of sample particle size
reduction devices are listed below.
Ball Mill/Rod Mill: Ball mills come in various sizes, and can be used to convert small rock fragments to
smaller mesh sizes.
Jaw Crusher: Jaw crushers can be used to reduce small rock fragments to the size of very coarse sand (3
cm to 2 mm).
Disk Mill: Disk mills can reduce particle size diameters from several mm to less than 0.5 mm.
Ring and Puck Mill: A ring and puck mill (also known as a shatterbox or a rotary mill) uses an orbital
motion to force a circular center piece of steel (reminiscent of a hockey puck) against an outer steel
ring. Particles are crushed between the edges of the ring and puck, and between the bottom of the pan
and the puck. The end result is a fine powder (< 0.05 mm).
Vibratory Ball Mill, Planetary Ball Mill, and Ring Mill: These are all examples of completely
enclosed techniques appropriate for small sample sizes. One must be careful to select (ball or ring)
materials which are not made up of, or will not interfere with, the analyte of interest.
Mixer Mill: A mixer mill can be used to reduce small (< 20 g) samples into a fine powder. The sample
is placed in a capsule with a ball or rod. The capsule and ball or rod are all made of high strength
materials. The filled capsule is shaken at a high speed to convert the sample into a powder. Mixer
mills are used instead of ring and puck mills to limit the effect of contamination when the sample
mass is small.
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Section 6
Sample Characterization and Assessment
Every environmental site is unique and each matrix will have different characteristics. The attributes
of the matrix must be known before the sampling requirements can be identified. A site sampling design
that results in an accurate summary of environmental conditions relies on knowing the sample
characteristics across a site. Likewise, someone taking a subsample from a heterogenous particulate
sample sent to a laboratory needs to evaluate the characteristics of the entire sample, not just the top layer.
Frequently, the analyte of interest is highly concentrated in the fine particulate fraction of the sample.
This can result in fractionation and a bias if correct sampling techniques are not used since the small
particles can drop to the bottom of a sample no matter how much the sample is shaken.
Information about the particles containing the analyte of interest is particularly useful in assessing
whether a proposed sampling practice is acceptable. Is the analyte concentrated in several particle types
or in just one type of particle? Are there particles highly concentrated in the analyte resulting in a density
effect? Do the analyte, or analyte-rich, particles have different densities, shapes, texture, or other features
that distinguish them from the other particles in the sample? How does the size distribution of analyte
particles compare to the size distribution for the entire sample?
6.1 Identifying Important Sample Characteristics
Almost every characteristic of a particle is directly or indirectly related to some factor affecting
sampling uncertainty. The directly related factors are considered in Gy sampling theory. Indirect, or
proxy, factors are features that are correlated to the actual factors. For instance, the color or hue of a
particle may be related to the amount of contaminant, and the texture might be related to density. While
Gy sampling theory was developed with consideration of the contaminant levels, there has been no
mention of color or texture. However, when sample assessment is the goal, any direct or indirect
indicator of the nature and complexity of the sample should be used to formulate decisions about how to
process the sample. The following list includes examples of both direct and proxy factors related to
sampling uncertainty.
Color (hue and intensity) Hardness Mottling Structure
Composition Mass (weight) Opacity Texture
Concentration Mineral type Porosity Variegation pattern (stripes,
Density Moisture (and other liquid) Shape streaks, or speckles)
Friability content Size
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The analyst should consider all of those factors in assessing the sample. The above descriptors refer
to the properties of individual particles. A description of the sample also includes information related to
groups of particles with common characteristics; for instance, the size distribution of the analyte-
containing particles. A complete description with respect to the possible factors could result in a large
multivariate data set.
The ways to identify the sample characteristics include visual inspection, and chemical and physical
analysis. These are not mutually exclusive, though many characteristics that are not subject to visual
assessment are only known through measurement and analysis.
6.2 Visual Characteristics
Sometimes a reasonable sampling design can be proposed once the visual characteristics are known.
Many particle characteristics can be determined through careful inspection. A magnifying glass or a
microscope may be useful in identifying whether or not multiple particle types are present. Visual
assessments are quick, inexpensive, and provide sufficient information for many sampling decisions. A
visual assessment may reveal if the sample is composed of different types of particles by looking at their
color, shape, size, texture, and other distinguishing features (be sure to look at different parts of the
sample, or at randomly selected increments, because of grouping and segregation effects). The features
that one is most interested in are the features important to the Gy sampling theory.
6.2.1 Analyte Particles: Color, Texture, Shape, and Number
Color, texture, and shape are distinctive visual clues related to important characteristics in Gy
sampling theory. While these factors are not useful in generating quantified uncertainty estimates, they
could be excellent proxies for many sample properties mat are directly related to uncertainty. Different
source materials usually have distinctive features. They also tend to have individual size distributions.
The most important visual feature is one that distinguishes analyte containing particles from non-
analyte particles. Identifying these features might be difficult because the sample may contain many
particle types and several analyses may be required to determine if they are associated with the analyte of
interest (such as a hazardous material at a site). However, sometimes prior information about the lot (e.g.,
a site) provides the desired information. If so, then one should consider whether or not the expected
number of analyte particles is large enough to approximate their distribution with normal distribution
statistics.
A rough rule of thumb is that an average of 6 or more particles per sample is required before normal
distribution statistics can make a reasonable approximation. If one is splitting a sample, then the
minimum number of each type of particle in the parent sample should be more than 6 times the number of
possible subsamples because the sample splitting process may not uniformly distribute the particles of
interest among the subsamples. Some subsamples may get more than 6 particles and some will get less
than 6. We suggest a factor of 10 times the number of subsamples so that most of the subsamples will
have at least 6 analyte particles. For example, for a single pass of a parent sample through a riffle splitter,
the number of possible subsamples is 2 and, therefore, the number of contaminant particles in the parent
sample should be at least 20 (to approximate a normal-shaped distribution so that the expected
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proportionate number of contaminant particles in the subsample is the same as the proportion of
contaminant particles in the parent sample).
6.2.2 Unique or Special Features
Some samples will contain particles with rare or unusual characteristics. These particles may be
related directly or indirectly to the amount of contamination in the sample. The analyst should note any
distinguishing characteristics for the different component particles in the sample as this may prove to be
valuable information at a later date. If there are any visually distinctive features, then they should be
present in all of the sample splits with the same proportion as the original sample. If one can determine a
difference visually, then it is likely that there will be a significant measured difference between the
subsample and original sample.
A sample may also contain unexpected material (e.g., gum wrapper, bottle top, shotgun shell casing,
etc.). The sample processing plan should address how the sample should be treated when unexpected
components are present. For instance, if soils are being sampled, there should be a standard protocol with
respect to dealing with organic objects such as large bark chips or root fragments. If the study design
allows objects larger than the particle size associated with acceptable uncertainty, then the sample must be
modified prior to subsampling.
6.2.3 Density
Density is the principal cause of segregation of different particle types and is often associated with
visual cues, such as color, shape, and texture. If a particle type has a significant density difference and
another distinctive visual characteristic, then the analyst will be able to make a cursory check that
fractionation is affecting the subsampling process. Density differences are usually inferred from visual
observation, with occasional measurement confirmation. If one observes obvious density differences,
then subsampling methods should be chosen that are least affected by it.
6.3 Moisture Content and Thermally Sensitive Materials
Wet samples are often air- or oven-dried to reduce the moisture content prior to any sample
modification. Under such conditions, volatile, and perhaps semi-volatile, components cannot be
determined (those components may also escape) when particle size reduction is required. If the sample is
dry and flows freely, then the analyst should continue the assessment process by evaluating the particle
size.
Thermally sensitive contaminants require a sampling design compatible with the compounds of
interest. If wet samples are anticipated, then the design should identify whether the samples are to be
processed wet or to be dried first. This guidance adequately addresses only dry samples.
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If the sample is wet or does not flow freely due to the presence of moisture, then one must decide to
either:
*• Dry the sample, restore the particle nature of the sample (gently roll or physically break up the
sample without changing the particle distribution), and continue
or
*• Take a subsample for analysis, then dry it
The first decision is preferred if one is trying to characterize the analyte proportion (concentration) in the
original sample by analyzing a subsample. The second decision is preferred if one is splitting a sample to
get representative subsamples, including a representative moisture content, since moisture content can
also vary over the lot or sample to be represented. In any case, the analytical subsample must mimic the
particle size distribution of the sample. In addition to one-dimensional incremental sampling, two-
dimensional incremental sampling may also be appropriate for moist samples (see subsection "4.1.3
Incremental Sampling," page 63).
It is assumed that there is sufficient water in the sample to cause the particles to adhere to each other
but not to flow. Moisture could affect a number of mechanisms involved with particulate heterogeneity,
such as the distribution heterogeneity (grouping and segregation effects). Also, note that wet samples can
cause problems with comminution devices; hence, it may be difficult to reduce %/ by comminution. The
moisture content may also interfere with the analysis.
Drying is an alteration of the chemical and physical composition of the sample; therefore, drying is
considered a materialization error; specifically a preparation error. Obviously, drying will affect the
estimation of the proportion of the analyte in the analytical subsample as compared to the sample, and this
can greatly affect decisions. Therefore, it is imperative that the moisture content is documented. The
moisture content must be accounted for in drying operations, which should be well documented (as
should be the calculations of the results). The analytical results must be descriptive about the moisture
content, such as "80 ppm Pb, dry weight sample" or "7 ppm Cd, 20% water by weight in sample." It is
important that the moisture content does not change between the sampling step and the weighing step and
the analysis step (protect samples with a desiccator). Determining the moisture content can be tricky with
respect to sampling and the reader is directed to read the sections given by Pitard (1993), "the
simultaneous drying method" (pp. 322-323) and "the method of the single sample" (pp. 323-326).
6.4 Particle Size, Classification, and Screening Decisions
Sieving a sample provides information about particle size distributions, and the analysis of various
fractions can aid in determining whether or not analyte levels are dependent on particle size. If a sample
matrix does have unique particle types, then the general mix of particles in the subsample should be
similar to the mixture seen in the original sample. However, this is only a check for gross differences,
and analytical measurements are required to determine whether the subsampling procedure is performing
adequately.
Particle size is a key decision characteristic for the application of Gy sampling theory. The size of the
largest particles is a critical factor in determining the expected uncertainty. The long axis particle length
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that Gy sampling theory associates with a sample is the screen mesh size that retains 5% of the particles.
Table 9 provides a description of soil particle classes by particle size. It is adapted from the Soil Survey
Manual, Handbook Number 18, USDA, Washington, DC, 1993. An extensive discussion of various soils
by percent composition of the various soil classes is also provided in the Soil Survey Manual. The
classification listings that include mixtures of the soil classes are known as soil types. A sample with
particles about 2 mm in diameter has maximum particle sizes near very coarse sand. Particles somewhat
larger than 2 mm are considered fine pebbles and even larger particles can be classified as rock fragments.
If particles of this size contain the analyte of interest, then one can expect significant uncertainty when
selecting subsamples on the order of 1 g.
Table 9. Soil particle classes by particle size, and classification names for soil fragments from
clay to boulders.
International
Class
clay
silt
fine sand
coarse sand
gravel
stones
Size (mm)
< 0.002
0.002-0.02
0.02 - 0.25
0.25-2.0
2-20
>20
USDA
Major Category
fine earth
rock fragments
Size (mm)
<2
>2
Class
clay
silt
very find sand
fine sand
medium sand
coarse sand
very coarse sand
fine pebbles
medium pebbles
coarse pebbles
cobbles
stones
boulders
Size (mm)
< 0.002
0.002-0.05
0.05-0.10
0.10-0.25
0.25-0.5
0.5-1.0
1.0-2.0
2-5
5-20
20-75
75-250
250 - 600
>600
If the analyte is known to only be present as fine grains and the sample consists of a small particle
component and an analyte-free, large particle component, then the sample can be screened to remove the
large particles prior to subsampling. The fine particle component can then be subsampled with,
presumably, much higher accuracy. For example, consider a 500 g sample where 362 g consists of small
pebbles that were screened out and the other 138 g consists of fine particles and includes all (42 g) of the
contaminant. The chemical analysis of the fine-particle component reveals 40 g of contaminant out of the
138 g (29% by weight) with a standard deviation of 8.3 g (6% by weight). One can conclude that the
analyte level in the sample is 40 g out of 500 g (8% by weight) with a standard deviation of 8.3 g out of
500 g, or 1.7% (by weight), provided that the uncertainty in the gravimetric analysis is appropriately
relatively low. Note that this example does not address the uncertainty associated with selecting the
original sample. The variability from sample to sample may be very high depending on the variability of
the ratio of the large particles to the small particles. Only the analyses of truly replicate samples can
address that issue.
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6.5 Concentration Distribution
A variety of uncertainty components are associated with the distribution of the analyte among the
particles. Identifying the analyte proportion (or concentration levels) versus the particle size distribution
would be very valuable; however, this is difficult to determine visually or experimentally. Occasionally,
one can identify the presence of a few large or highly concentrated particles, which suggests that a nugget
effect may be expected (a nugget effect includes the random variance in estimating the parameters: the
mean proportion of the contaminant within a sampling unit, the mean mass of a sampling unit, and the
small scale heterogeneity within a sampling unit (Pitard, 1993, pp. 109 ff.). If analyte particles are
identifiable and rare (such as an occasional lead shot in a soil matrix), then the analysis results may be
distributed as a Poisson distribution or some similar, highly skewed probability distribution, and the level
of uncertainty reported for any analysis should include this possibility.
If large inert particles (e.g., pebbles and rocks) containing little or no analyte, then the variability of
any subsample analysis will increase. Even if all of the analyte is present as a fine powder, the sampling
variability may increase dramatically due only to the uncertainty associated with the larger inert
components.
6.6 Comminution (Grinding and Crushing)
If the sample consists of small enough particles to achieve an acceptable error, then no particle size
changes are needed. Otherwise, the size of the particles needs to be reduced, increasing the number of
particles and reducing the relative variance of the fundamental error. If grinding is not an acceptable
option, then the mass of the subsample, M^ should be increased.
Quick and rough estimates of the sample mass, M3, required for the corresponding minimum
participate sizes to obtain a CV (%RSD) of 15% (i.e., a relative variance of 0.0225) for Spg are
summarized in Table 10 (from ASTM D 605 1 ; see also Ramsey et a/., 1989; and Pitard, 1993). Note that
about a factor of 10 in mass is associated with about a factor of 2 in particle size. The dominant cause of
this relationship is that the required mass is proportional to the cube of the particle diameter. For most
situations, this means that particle size is the key consideration. Screening the sample (or a representative
subsample) to identify the particle size distribution is a quick way to estimate whether particle size
reduction is needed. For more accurate estimates of the required minimum sample mass, the equation
used for estimating the relative variance for the fundamental error should be used; a sampling strategy
based on the sampling nomograph should also prove useful.
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Table 10. The minimum sample mass, Ms,
and the maximum particle size, d,
for SFE £ 15% (density = 2.5,
analyte weight proportion = 0.05).
Minimum Ms [g]
5
50
100
500
1000
5000
Maximum d [cm]
0.170
0.37
0.46
0.79
1.0
1.7
6.6.7 Caveats
Size reduction strategies, such as grinding, are inappropriate when the analyte is soft and not
amenable to comminution, or if the analysis is intended to determine only the exposed or easily
recoverable component. For example, only the easily extractable metals might be of interest since
components imbedded within the sample matrix would not be considered hazardous.
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Section 7
Proposed Strategies
A number of strategies and lists of sampling criteria related to the subsampling of particulate
materials have been published (Pitard, 1993; Kern et al, 1997; and Mishalanie and Ramsey, 1999).
Please refer to those references and the text presented in this guidance for details. The proposed strategies
presented here build on previous suggestions, but are more comprehensive. The extra features are the
result of realizing that sampling particulate material is a multifaceted problem that cannot be easily
summarized. The reader is reminded that the following is only guidance and the best strategy will likely
be some variation on what is presented here. The best strategies will only be developed when the analyst
has an understanding of the fundamental principles of the sampling theory for particulate materials and an
actual sample to observe and test. Although common strategies can be developed and used as a general
guidance, sampling plans are not generic and a sampling plan should be developed for each unique case.
The goal of our sampling strategy is to reduce data uncertainty by selecting a representative analytical
subsample (representative of the laboratory sample). The analytical subsample is that mass which
undergoes analysis. A representative sample can only be attained through correct sampling practices.
Recall (refer to the text or the glossary) that the total sampling error is
TE = FE + GE + CE2 + CE3 + DE + EE + PE
If correct sampling practices are used, then the terms, GE, DE, EE, andPE are minimized; that is, GE +
DE +EE +PE * 0. Assuming that CE2 + CE3 - 0 for laboratory subsampling, and if correct sampling
practices are used, then the total sampling error becomes
a< - a1
TE = FE = —
aL
This is the minimum sampling error due simply to the nature of the material being sampled (the
constitution heterogeneity) and represents a goal ofGy's correct sampling practices.
The mean of the total sampling error then becomes the mean of the fundamental error under the
conditions of correct sampling practices and negligible effects from CE2 and CE3, and is expected to be
negligible; that is,
m(af) - a,
m(FE) = -—^- £ - 0
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The relative variance of the total sampling error is
SJE = SpE + SGE + SCEJ + SQEJ + SDE + SEE + SpE .
If correct sampling practices are used, then the terms s^2, s^2, sEE2, and sPE2 are minimized; that is, s^2 +
SoE2 + SEE2 + SpE2 ~ 0- Assuming that s^2 + s^2 = 0 for laboratory subsampling, and if correct sampling
practices are used, then the relative variance of the total sampling error becomes
STE ~ %E
This is the minimum sampling relative variance due simply to the nature of the material being sampled
(the constitution heterogeneity) and is the basis for developing a representative sampling strategy using
Gy 's correct sampling practices.
The degree of representativeness, rTE2, is given by
FTE ~ mTE + STE
where r^2 is the mean square of the total sampling error, m^2 is the square of the mean of the total
sampling error, and s^2 is the relative variance of the total sampling error. A sample is representative
when
r 2 2
rTE s roTE
where roTE2 is a specified and quantitative measure of a representative sample (the smaller this number, the
more representative is the sample); that is, it is a level of representativeness regarded as acceptable as
defined by the study objectives (e.g., data quality objectives, DQOs) and the sampling plan.
We can estimate how representative our analytical subsample is of the laboratory sample by
developing a strategy such that the bias is negligible (m^2 = 0) and the "controllable" relative variances
are minimized when the sampling practices are perfectly correct (that is, when GE = DE = EE = PE = 0).
It is desirable to keep s^2 <. soTE2, where soTE2 is a level of the relative variance of the total sampling error
within user specifications. If sampling practices are perfectly correct (that is, s^2 = s^,2 = sEE2 = SpE2 = 0),
then Sjj,2 = Spjj2 (the relative variance of the fundamental error). Thus, if correct sampling practices are
applied and all of those "controllable" errors are minimized, then a representative sample could be
characterized by keeping the relative variance of the fundamental error below a specified level, s^2 <.
soFE2. Under such conditions,
r 2 ~ t_2 f c 2 ~ r 2
rTE ~ **FE s NoFE ~ roTE
JTz/s equation will serve as the basis of our sampling strategy to obtain a representative subsample for
analysis. This is fortuitous for our sample planning purposes, and for formulating our study objectives
(e.g., DQOs), since Spg2 is the only error that can be calculated, based on the physical and chemical
properties oftheparticulate material, a priori', that is, before sampling even takes place! Thus, the user
now has a logical approach to specifying the tolerable amounts of bias and variability as set by the study
objective (e.g., DQOs), and a basis for a sampling strategy.
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The above discussion on the theory behind developing a strategy to get a representative analytical
subsample may seem like a lot to take in, and the ensuing discussion on developing a practical sampling
strategy may seem like too much effort just to analyze a small amount of material. When one is in a hurry
and has a large case load, it may seem downright threatening and overwhelming. But, remember that the
seemingly simple task of taking a small amount of material out of a laboratory sample bottle could
possibly be the largest source of error in the whole measurement process. Not taking a representative
subsample could produce meaningless results, which is at the very least a waste of resources, and at the
very most, could lead to disastrous decisions and consequences.
Sampling is one of those endeavors that you "get what you pay for," at least in terms of effort. But
the effort is not necessarily that much. It pays to have a basic understanding of the theory. Become
familiar with what causes the different sampling errors and how to minimize them through correct
sampling practices. Be able to specify what constitutes a representative subsample. Know what your
sampling tools are capable of doing and if they can correctly select an increment. Do a sample
characterization (at least a visual inspection) first. At a minimum, always have study objectives and a
sampling plan for each particular case. One should determine the maximum sra2 and s^2 that will be
tolerated, and the final MS for analysis. If possible, a team approach should be taken for developing the
study objectives and the sampling plan. Historical data or previous studies should help here. Be sure to
record the entire process.
The generic strategy proposed below is meant to be fairly comprehensive for developing plans for
most situations (see the text for some exceptions; also, see Pitard, 1993). If the generic strategy seems a
bit too overwhelming for the intended purposes, then at least try to use correct sampling practices and
correct sampling devices the best that you can under your circumstances. Try to take as many random
increments as you can (a sectorial splitter is a good tool for this). If you can only take a few, say five,
increments rather than the recommended 30, then you are still better off than taking a grab sample "off
the top" from the sample bottle. And now, you are at least aware of the consequences of only reducing
SQE2 by N = 5 increments relative to s^2; recall that
SGE2 = SFE2/N
and
STE - SCEI ^ SFE ~*~ [SFE /N].
Thus, Syg2 > sCE12 ~ SpE2 if N is made large enough. (At least N = 30 increments are recommended as a
rule of thumb to reduce sGE2 compared to s^2 (Pitard, 1993; p. 187).) Grinding the sample to a fine
powder followed by taking many increments may be another "quick" alternative. Also, the sampling
nomograph provides an effective visual aide for quickly developing a subsampling plan.
7.1 The Importance of Historical and Preliminary Information
If practical, preliminary trial samples, analyzed in advance of the study, may prove useful so that the
study designers can identify an appropriate subsampling method that meets the study DQOs.
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To determine which subsampling methods are important, an assessment of the possible sampling
errors should be made and compared to the study DQOs. Based on the sample matrix, the analytical
procedure, and the study DQOs, one can assess the relative importance of the sampling methods
following the guidance in this document. The test samples may be analyzed using any sampling
technique shown to meet the study criteria in terms of relative error. Several representative subsamples
should be analyzed to determine if variability is an issue. With data from six or more representative
subsamples, one can compare the variability and the recovery between sampling techniques. Variability
is much more important at this stage.
If the variability is acceptable, then no particle size reduction may be necessary. If not, then a
different spitting method is needed or the sample characteristics must be altered before subsampling.
Unfortunately, some samples may have characteristics that are difficult to detect, such as rare, but high
level, contamination patterns. The analysis of test sample results is merely an alternate way of identifying
conditions in the particulate matrix. If all of the particles are fairly homogeneous in the analyte levels,
then the variance due to the particle size distribution should be small, and low uncertainties should be
expected. The same particle size distribution may give large uncertainties if the analyte is confined to one
of many particle types in the sample. This demonstrates the need for prior information concerning the
sampling site. Information about the site provides crucial information needed to efficiently evaluate
possible sampling strategies.
If several representative laboratory samples are present, then one can identify the minimum errors
expected from simple preparation and subsampling efforts. These can be compared to the study
requirements to determine if more intensive processing is desirable. Alternatively, one can evaluate
individual characteristics of the sample. For instance, one might determine if the analyte is confined to
the fine particulate fraction. Perhaps it is possible to reduce the mass by merely screening the sample as
part of an initial processing step.
7.2 A Generic Strategy to Formulate an Analytical Subsampling Plan
The initial assumption is that the analyst must obtain a representative laboratory analytical subsample
from a laboratory sample composed of a heterogeneous collection of particles. The laboratory analytical
subsample is the subsample that undergoes laboratory analysis to estimate the analyte concentration, a,,.
The laboratory sample is the sample (and, in this case, the lot) received by the laboratory containing an
analyte concentration, aL (which is estimated by a^.
At this point, we may not know how the sample received by the laboratory was collected. For
instance, the laboratory sample may not have been selected using correct sampling methods and may not
be representative of the original lot (e.g., a hazardous waste site). Therefore, our attempt in the laboratory
is to select the analytical subsample so that it is representative of the laboratory sample that was received
and we should not be tempted to extrapolate claims beyond the laboratory sample (e.g., about the site) for
which we may have no knowledge or control.
The particles are of various sizes, shapes, and composition, and the analyst may not know which
particle(s) is important in terms of the analyte of interest. The lack of initial information can be a limiting
factor in being able to identify the best strategy to use when producing a laboratory analytical
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subsampling plan. The following steps allow one to gather information and develop strategies for
preparing representative laboratory analytical subsamples that meet a study's uncertainty requirements.
1. Examine the sample as outlined in the section on "Sample Characterization and Assessment."
2. This step provides a bound on the variability contributed by sampling and other factors in the
measurement process. The following should be identified up front in the planning stage when
determining the study requirements (e.g., DQOs):
>• Determine the acceptable level of error for the laboratory analytical subsampling step; that is,
determine the acceptable level of the bias for the total sampling error, 1%^, and the relative
variance for the total sampling error, sOTE2. Determine the degree of acceptable
representativeness, rOTE2, given by rOTE2 = m^2 + sOTE2.
»• Ascertain the error limits for the study; that is, determine the acceptable level of the bias for
the overall error, m^, and the relative variance of the overall error, which is a linear
combination of the relative variance of the total sampling error and the relative variance of
the analytical error (which should already be known): sOE2 = s^2 + sAE2.
>• Try to identify any other error contributions from other steps in the measurement process.
Determine if the sample has any unique features that require a special treatment prior to
subsampling. Those features and treatments, and how to correct for them, should be identified
up front in the planning stage when determining the study requirements (e.g., DQOs). For
example:
•• If the sample has a high moisture content, then it may need to be dried.
*• Unexpected items (e.g., twigs) may need to be removed.
»• If large inert (no contaminant) particles are present, then they may need to be removed or
reduced in size. In that case, the new diameter of the largest remaining particles becomes the
diameter, d, used in the equation to estimate sFE2
2
4. Estimate the laboratory sample analyte concentration level, aL, and (if possible, but not
necessary) try to determine how the analyte is distributed in the sample.
*• Knowing the sample analyte concentration is of course necessary for the laboratory analysis
(e.g., choosing the correct analytical method and the range of calibration standards);
however, it can also help in refining the estimates of the constant terms used in the relative
variance for the fundamental error equation. Knowing if the analyte (contaminant) is
expected throughout the sample or is present as isolated particles provides information
relevant to subsampling methods and decisions about reducing the particle size. Try to
determine if the analyte is distributed across most of the particles or if the analyte is
distributed in a highly heterogeneous manner, occurring at high levels in a few particles and
at low levels in the other particles.
* If the analyte of interest is confined to a limited range of particle sizes or types, then (if
consistent with the study requirements identified in step 3) remove and analyze only the
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portion of the sample containing the analyte of interest. Be sure to record all of the actions
and the weights of the fractions.
»• Disadvantages of removing and analyzing only the active particle size fractions: Requires
screening each sample to identify the size fraction containing the analyte of interest.
Requires sample separation by size fraction. Hence the sample must be fairly dry prior to
separation, possible affecting volatile and liquid compounds.
»• Advantages of removing and analyzing only the active particle size fractions: The analysis is
targeted only at the contaminated fractions, limiting the sample amount required for
processing.
*• When it is known that the target compound is not present in the large particles, screen out all
of the material (weigh this material) larger than the analyte particles size level. The
remaining sample is processed with the standard protocol and the reported concentration is
corrected for the amount of matrix that was removed.
> Disadvantages of removing the larger inert material: Volatile compounds may be lost or
diminished in concentration during processing. A sample large enough to meet the
uncertainty requirements for both the screened material and the screened out material must be
available.
> Advantages of removing the larger inert material: The number of required sample analyses is
minimized.
5. Estimate the size of the largest particles (d).
»• Using the diameter of the largest particles in the relative variance for the fundamental error
equation gives the most conservative estimate of s^2 based on a given diameter. The
equation for the relative variance for the fundamental error, %/, is more dependent on
particle size (because it is cubed) than any other factor. A very rough rule of thumb is that if
the diameter of the largest particles is less than around 2 mm, then the sample variability is
likely to be small. Samples with larger particles may require a size reduction step.
6. Estimate the density of the analyte, A,M [gcm~ 3], and the density of the gangue (matrix),
7. Estimate the constant terms (c, /, f, and g) in the equation for the relative variance for the
fundamental error.
*• The constant terms, / (the liberation factor), f (the shape factor), and g (the granulometric
factor), can be estimated by measurements (see this text and Pitard, 1993) or by observations
and using Tables 4, 5, and 6.
>• The mineralogical factor, c, is given by
C— A
fv ft f
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where aL is the decimal proportion [unitless] of the analyte in the sample, AM is the density of
particles containing the analyte [g/cm3], and A.g is the density of the gangue [in g/cm3]
8. Estimate the relative variance of the fundamental error (SpE2) based on the required subsample
mass (M^ for the laboratory analysis.
*• If one knows the particle size, d, the required laboratory analytical subsample mass, M,,, and
the constant terms in the equation for the relative variance for the fundamental error, then an
estimate of the relative variance of the fundamental error, s^2, can be made:
3 3
where Ms is the sample weight [g], ML is the mass of the lot [g], c is the mineralogical (or
composition) factor [g cm"3], / is the dimensionless liberation factor, f is the dimensionless
particle shape factor, g is the dimensionless particle size range (or granulometric) factor, d is
the nominal size of the particles [cm], C = cflg is the sampling constant [g cm"3], and IHL is
the constant factor of constitution heterogeneity (also called the invariant heterogeneity).
• If the laboratory sample (the lot) is large compared to the analytical subsample, that is, ML »
Ms, then the term, 1/ML, is negligible and the relative variance of the fundamental error is:
M
s
• Those constant terms can be estimated by measurement or by observation, and by using
tables (see Tables 4, 5, and 6). A quick and very rough estimate of the relative variance of
the fundamental error can be made using some "typical" values for the constant terms:
SFE ~
Cd3 18
-------
»• Reducing the variance of the fundamental error (s^2) can be achieved by increasing the
laboratory subsample mass (M^; however, this increased subsample mass must be compatible
with the analytical method.
* Disadvantages of increasing the laboratory subsample mass (Mc): It is expensive and
cumbersome to process large samples. The extraction facilities may not be large enough.
*• Advantages increasing the laboratory subsample mass (M,.): Besides reducing the
fundamental error, this action may be effective if the analytical process can be easily adapted
to large sample masses. For instance, if the analyte can be extracted prior to analysis, the
issue of subsampling variability disappears at the laboratory stage.
10. Consider particle size reduction (comminution - see text) if the required laboratory subsample
mass (MJ is too large.
>• Use the sampling nomograph (see text) to develop a subsampling strategy with the fewest
comminution (grinding or crushing methods) steps to obtain the laboratory subsample mass
(M^ that meets the desired sra2. Be sure to allow for the propagation of error if there is more
than one mass reduction (subsampling) step (see the "Hypothetical Example" and Figure 13
in the text under the "Sampling Nomograph" section). The size reduction (comminution)
steps are designed to reduce uncertainty due to particle size and variable composition effects.
•• Disadvantages of comminution: Comminution is not appropriate if volatile compounds are of
interest, as a significant bias would be expected. Comminution is not appropriate for wet or
oily samples, or when the analyte is a soft metal (which is not amenable to grinding). For
large samples, the effort may be correspondingly large.
» Advantages of comminution: Besides reducing the fundamental error, grinding can also help
to reduce the grouping and segregation error. For small samples, the cost-to-benefit ratio
may be relatively low.
11. Collect (select) the laboratory analytical subsample using correct sampling equipment (see text)
and correct sampling practices (see text).
* Correct sampling means to minimize the effects of all of the sampling errors that we have
control over through our sampling techniques. This includes all of the sampling errors except
for the relative variance of the fundamental error (spj,2), which can only be reduced by
increasing the sample mass, Ms, or by reducing the particle size, d, though comminution
(crushing or grinding). Consider either incremental subsampling or splitting subsampling
methods (see text).
»• For incremental or splitting subsampling, the relative variance of the grouping and
segregation error (SoE2) can be reduced by taking at least 30 random increments to make up
the subsample. To correctly select an increment with respect to the increment delimitation
error (DE), the sides of the sampling device should be parallel with a flat bottom (e.g., a
scoop) for a one-dimensional (that is, one long dimension) pile or the sampling device should
have a constant diameter (e.g., a cylinder) for a two-dimensional (that is, two long
dimensions) surface or cake, and the sampling device should go completely through the pile
or surface and at a slow, even rate. A rule of thumb for correctly collecting an increment
Page 90 of 134
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with respect to the increment extraction error (EE) is that the inside diameter of the sampling
device should be at least 3 tunes the diameter of the largest particle. Splitting methods are
not generally affected by DE or EE if the splitters have a good design and good technique is
used. Some splitting subsampling methods are recommended somewhat over incremental
methods, with the sectorial splitter being the method of choice. Last, but not least, the
preparation error (PE) is reduced through common sense, honesty, and awareness.
7.3 An Example Quick Estimate Protocol
Any large rocks in a sample should be broken up into fragments no larger than about 2 to 3 cm in
diameter. For most samples, this is easily accomplished with a hammer or small sledge hammer. Then,
the samples with fragments up to about 3 cm in diameter should be crushed to the level of very coarse
sand, with grain sizes about 2 mm in diameter. This step requires a mechanical crusher. The particle size
can be reduced again in one or two stages:
Two stage process
The entire sample can be reduced in size from very coarse sand to a medium or fine sand with a disk
mill. A subsample of about 100 g (20 to 200 g range limit) is then taken with a riffle or sectorial splitter.
The subsample is then ground in a rotary ring and puck mill to a fine powder (< 0.05 mm diameter). The
two-stage process has the advantage of processing the entire sample until subsampling just before the
final particle size reduction stage.
One stage process
A subsample of about 100 g is taken with a riffle or sectorial splitter without having gone through the
disk mill. The fundamental error is larger for this method because it affects the variability of the
subsampling step.
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Section 8
Case Studies
8.1 Case Study: Increment Subsampling and Sectorial Splitting
Subsampling
A simple example illustrating the difficulty associated with particle sampling is presented by Gerlach
et al, (2002). Eight, 5 g subsamples from a 40 g mixture consisting of 0.2 g NaCl and 39.8 g of screened
(0.600 mm to 0.850 mm) sand were prepared by incremental sampling as follows. The mixture was first
poured into a flat Pyrex pan. To make each subsample, 10 increments were taken in a random systematic
pattern across the pan and then the increments were combined to give a total weight of 5 g. A total of 8
subsamples (each weighing 5 g and each containing 10 random increments) were thus obtained. Eight, 5
g subsamples from a 40 g mixture consisting of 0.2 g NaCl and 39.8 g of screened (0.600 mm to 0.850
mm) sand were also prepared by a sectorial splitter by evenly pouring the mixture into the center of a
rotating sectorial splitter containing 8 receiving sectors (see Figure 14).
Figure 23a and 23b show the individual estimate bias from each single subsample as a function of the
sample acquisition index for the incremental sampling and for the sectorial splitter sampling experiments,
respectively. Figure 23a and b also show the cumulative bias, comparing the true value to the running
mean calculated from all of the subsamples up to, and including, that run.
For the incremental sampling experiment, the individual values are biased low for the first 6 of 8
samples (see Figure 23a). A Wilk-Shapiro normality test showed less than a 5% chance that the
distribution of the results was from a normal distribution. Dixon's outlier test, based on the range,
resulted in the high value being declared an outlier (PO.01) (Dixon and Massey, 1969). However, the
most interesting feature of the incrementally sampled data set is that the cumulative estimate of the bias
remains over 16% low until the very last sample when the bias from the exhaustive analysis is reduced to
1.9%. For a discussion on outliers, please refer to Barnett and Lewis (1995) or to Singh and Nocerino
(1995). Software has been developed for the methods discussed in the latter reference by Singh and
Nocerino; the software is called Scout (Scout, 1999).
The incremental sampling results can be compared to the results from splitting a similar sample using
a sectorial splitter. The sectorial splitter also produced eight, 5-g samples. Five subsamples were biased
high and 3 were biased low, with the cumulative bias estimate always less than ± 5%, except for the
second run, which was biased low at only - 8.76% (see Figure 23b). A final cumulative bias of 3.2% was
found after exhaustive analysis.
Page 93 of 134
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inn
IDU
125-
100 -
U Kf\ -
C 50
OL O£
25
0..
-25 -
_j a ]
Analyte Estimate by Run Sequence
- - • - - Sample Estimate
— A — Cumulative Bias
/
'
t
*
J
ft
A- A .•* --^^^
tr* -»' -^
1
234
Subsample
5678
Run Number
Analyte Estimate by Run Sequence
* Kf\ nn
lOU.UU
125.00 -
1 00 00 -
yc nn -
§cn nn -
OU.UU
25.00
o.oo-
•>K nn
-iO.UU
— [JbJ - • - - Sample Estimate
— ^s — Cumulative Bias
J§TJII"'A_
"*'^"J
**"* • A
-— ---r-— IT^TTTI ii , j-.. I1M * ' t\ ~(^
v /
w
\JU.vU I | I I
1234
Subsample
I t E 1
5678
Run Number
Figure 23. (a) Sample estimate bias and cumulative bias versus run number for the
Incremental subsampling runs, (b) Sample estimate bias and cumulative
bias versus run number for the sectorial sampling runs.
The particle sizes in this example were similar for both NaCl and sand. The sampler was experienced
at taking incremental samples. The sample did not include any larger particles (greater than 0.850 mm).
Despite all of these favorable features, there was a systematic bias with the incremental sampling. The
"correct" answer was not approached until an exhaustive analysis was completed. The distribution and
variability of the results were similar between both the incremental sampling and the sectorial splitting
methods, except for one apparent outlier in the incremental method data set. But this statistical "outlier"
was a real part of the data set and was useful for understanding the data. While both methods resulted in
Page 94 of 134
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similar results after exhaustive analysis, the sectorial splitter would clearly be the method of choice in this
study.
The errors associated with the incremental sampling study could probably have been minimized with
"correct" sampling techniques. The error should have been reduced if more increments were taken for
each subsample (see the discussion on "The Importance of Correctly Selected Increments"). At least N =
30 increments are recommended as a rule of thumb to reduce sGE2 compared to Sj^2 (Pitard, 1993; p. 187).
Some materialization error (ME; note that ME = DE + EE) may also be confounded with (thus
inflating) the incremental sampling study error due to using a less than "correct" sampling device and
performing a less than perfect delimitation as the sampler moved across the pan to retrieve an increment.
In practice, with incremental sampling, the selection of a random subset may not be a pragmatic option.
A sectorial splitter is sometimes an easier and faster option over taking many increments correctly.
However, a sectorial splitter must also be used correctly (see Pitard, 1993). Also, note that if three, 5-g
subsamples were required by the incremental sampling plan for this case study mixture, only the first 3
subsamples would have been taken, and the average result would have been a consistently larger negative
bias compared to the results from the 3 randomly chosen samples from the sectorial splitter.
Important considerations and conclusions from this study include:
*• even a simple matrix can be subject to a significant subsampling error
»• statistical outliers are not necessarily poor data (in this case, the statistical outlier was critical
in determining the true average)
*• the splitting subsampling method producing fractions independent of order is preferred to the
incremental subsampling method where bias or variability changes as the subsamples are
prepared from the ordered increments
»• "correct" sampling techniques should always be considered to reduce subsampling errors -
namely, use at least 30 increments to compose each subsample to reduce the GE and use a
"correct" sampling device "correctly" (proper technique) to minimize the ME.
8.2 Case Study: The Effect of a Few Large Particles on the Uncertainty
from Sampling
To illustrate the effect of a few large particles on the uncertainty from sampling, a sample was
prepared with 1 gNaCl (the analyte), 23 g sand (inert matrix), and 72 large particles of sandstone (inert
matrix) collectively weighing 12 g. The 36 g sample was split 6-fold with a sectorial splitter, one split
was selected at random and split 6-fold again. The resulting subsamples were expected to have, on
average, 2 large particles (72 particles / 6 subsamples =12 particles / subsample; 12 particles per
subsample / 6 subsamples = 2 particles / subsample). The frequency of the large particles in a subsample
was found to be in agreement with a Poisson distribution. The particles appeared to be subsampled at
random irrespective of their particle size.
The nominal 1 g samples were weighed with and without the large particles and analyzed for NaCl.
The resulting CV was 17% for salt in sand. Theoretically, if the large particles were ground to sand, then
the crushed sample would have had about 1.5 times ([12 g + 23 g] / 23 g = 1.52) the mass of sand. Since
Page 95 of 134
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the salt variability should be the same, the theoretical error for a sample where all the large particles are
crushed to sand is predicted to be 17% / 1.5 = 11%. However, the measured CV for the samples with
large particles was 35%. Failure to crush the large particles resulted in a greater than 3-fold increase in
the expected CV due primarily to the uncertainty associated with sampling a small number of large
particles.
This example showed that having a few large inert particles in the sample may inflate the variance.
The variability in mass was sufficient enough to affect the outcome in a significant way. These results
also demonstrated that the particulate materials were distributed by the sectorial splitter independently of
the mass. The independent selection of particles via the sectorial splitter demonstrates the principal
strength of this splitting subsampling method.
8.3 Case Study: Sampling Uncertainty Due to Contaminated Particles with
Different Size Fractions
This case study illustrates the contributions to sampling uncertainty due to contaminated particles
with different size fractions. Several 25-to-75 mm diameter quartz rocks were processed with a rock
crusher until the particles would pass through a 4 mm sieve. The aggregate was then separated to obtain
large, medium, and small fractions using 2 mm, 0.710 mm, and 0.180 mm mesh sieves. A 0.34 M NaCl
solution was poured over each fraction, decanted after 10 minutes, and the samples were then air dried for
24 hours. For each size fraction, 12 g were subsampled with a 6-fold sectorial splitter. The results (see
Table 11) show that, despite having the least amount of NaCl, the large particles fraction had a higher
relative uncertainty compared to the smaller particles fraction. The large fragment fraction has 5 times as
much uncertainty (as standard deviation) as the small fragment fraction, despite that it contains only one
third as much NaCl. This example demonstrates the effect of particle size on sampling uncertainty.
Table 11. The influence of particle size on uncertainty.
Fraction
Large
Medium
Small
Fragment Size (mm)
2 to 4
0.71 to 2
0.18 to 0.71
Mean NaCl (g)
0.00107
0.00247
0.00324
%RSD
24
6.5
14
SD(g)
0.00026
0.00016
0.00005
8.4 Case Study: The Relative Variance of the Fundamental Error and Two
Components
The requirements for estimating the relative variance of the fundamental error depend on the matrix.
Suppose that there are just two components, A and B, in the laboratory sample. In addition, the
hazardous component is assumed to be present in component A as 25% by weight with no hazardous
contribution to component B. To estimate the relative variance of the fundamental error for the analytical
subsample, one must obtain or estimate values for:
»• d, the nominal particle size
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*• Ms, the mass of the analytical subsample and ML, the mass of the laboratory sample (the lot)
>• f, the shape factor
>• /, the liberation factor
•• g, the granulometric factor.
»• AM»tne density of the hazardous component particles, and Xp the density of the nonhazardous
components.
> a,^, the decimal proportion of the analyte in component A.
Already one notices a difficulty. The last bullet item requires the concentration of the analyte in a
fraction of the material. But, at this point, one does not know which component of the sample might
contain the hazardous material. Even if this piece of information was available, one does not know the
concentration.
The flaw in the above discussion is that one cannot model the sample, predict the uncertainty, or
identify an appropriate sampling strategy without a complete understanding of the sample. This level of
understanding is, to say the least, rare. If part of the required information is unknown, then one must
either go about obtaining it or utilize a less accurate method.
8.5 Case Study: IHL Example
Suppose that one has a soil sample screened to 2 mm that is contaminated with an organic pesticide
and information about the sample mass and the variability is desired. For the organic component, an
assumption is made that X = 1 [g cm"3]. Using
c - KM - ltecm~3l
C- ———— —
aL
and the IHL parameters listed in Table 12 gives
IHL = clfgd3 = | 1[gC/"'3]
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If the organic component is 1% (a^ = 0.01), then IHL = 0.16 g. Recall that the value of IHL is the mass
associated with a relative variance of the fundamental error of Spg = 1.0. That is, SFE = JHLI Ms or
IHL = Mg when the relative variance of the fundamental error is 1 .0, of which the square root, or the
relative standard deviation of the fundamental error is 1.0. One can use this relationship to identify the
sample mass needed for a particular variance or standard deviation. For a relative variance of 0 . 1 ,
Ms =
= 0.16g/0.1 =
Table 12. Case study: IHL example parameters.
Parameter
c = A/aL
I
f
9
d
Description
Mineralogy Factor
Liberation Factor
Shape Factor
Size Range Factor
Fragment Dimension
Value
1/aL [gem'3]
1
0.5
0.4
0.2cm
Comment
Relative concentration
100% available
Typical particle shapes
Typical range of sizes
Large fragment dimension
Suppose in the above example that the concentration of the pesticide was 1 ppm, a proportion (or
decimal fraction) of 1Q-*. Then fflL - 0.0016 / aL = 0.0016 / 10"6 = 1600 g, corresponding to a sample
mass requirement of 1.6 kg to get a relative fundamental error of 1.0. This demonstrates the effect of the
sample concentration on variability. The relative variance of the fundamental error is proportional to d3
and inversely proportional to M,,. For example, if one wants either a smaller variance with the same
sample mass or a smaller sample mass with the same variance, the particle size must be reduced.
8.6 Case Study: Selecting a Fraction Between Two Screens
Suppose that one is selecting a fraction between two screens (a particle size class, Lc) of 100 g of
classified soil with an average particle size 0.5 cm from a 2 kg lot and the contamination proportion in
this particle size class is estimated to be at the low level of a^ = 0.03. The particle size of 0.5 cm is
between fine and medium sized pebbles. The equation for the relative variance of the fundamental error
equation can be used to estimate the expected relative variance. Parameters for this example are shown in
Table 13. Recall that for low analyte levels with Ms < MJ10
SFE ~
a
Lc
Then,
**
loo
- 2)(0.53) = 0.049
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This corresponds to a relative standard deviation of 0.22, or a CV of 22%. If this value is higher than the
DQO value, then one can use the same formula to estimate the required mass. However, one should first
estimate the mass required for a given particle size. One of Pitard's rule-of-thumb formulas (Pitard, 1993;
pp. 337 and 389) states that, even when the critical component has not been identified or aL has not been
estimated, the sample must be representative of all of the particle size fractions. Since the largest particle
size, d, will be the most representative particle size by giving the most conservative (largest) estimate of
the relative variance of the fundamental error, then d can be substituted for d^ (Pitard, 1993; p. 160). For
f = 0.5, A = 2.5, aL = 0.03, and a study goal of SFE = 0.05 (sFE2 = 0.0025), the mass is estimated to be:
***
Lc
0.0025 0.03
- 2)(0.5cm)3 = l,958g
Table 13. Case study: Parameters for selecting a fraction between two screens.
Parameter
f
A
Ms
aLC
dFLC
Description
Shape factor
Density of material
Mass of the sample
Proportion in the critical size fraction, LFc
Average fragment diameter of the
particle size class of interest
Value
0.5
2.5 g/cm3
100 g
0.03
0.50 cm
Comment
Standard sample
Typical soil density
Total analysis mass
Pre-analysis estimate
Conservative estimation of
average for Lc
This suggests that the initial particle size is too coarse to achieve the study goals. If a fundamental error
with 15% relative standard deviation is acceptable (that is, SpE = 0.15 or s^2 = 0.0225), then the sample
mass required is:
"FE
a
'Lc
= ((0.5)(2.5g/cm3)) J_ _ 2)(05cm)3 = 21g
0.0225 0.03
This is higher than the 100 g that the analytical method allows. However, by rearranging the above
formula one can estimate the maximum particle size needed to obtain a relative standard deviation of 15%
for a 100 g sample:
4FLc
(ax— - 2)
a
Lc
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4.
- 2)
This corresponds to a diameter d^ - (0.057 cm3)1/3 = 0.386 COT. This value is just a little lower
than the originally assumed conservative maximum diameter of 0.5 cm. The conclusion would be to
process the samples as they arrived rather than go through an extra step for particle size reduction.
8.7 Case Study: Subsampling Designs
(This case study is adapted from: ASTM D 5956, Standard Guide for Sampling Strategies for
Heterogeneous Wastes, Second Edition, 1997.) A laboratory received a 1 kg sample of potentially
hazardous waste in which one-gram nuggets of cadmium are randomly distributed. The average level of
cadmium is to be determined based on the analysis of 10 subsamples (the required analytical mass
depends on the design given below). The gangue (matrix) is free of cadmium and has a much smaller
particle size than the cadmium. The sample is 33% cadmium by weight and the gangue is similar to fine
sand (< 0.25 mm particle diameter) with a density about a third of that for Cd. Cadmium has a density of
8.65 g/cm3, and one can estimate the volume (V^), radius (r), and diameter (d) of a sphere of 1 g of Cd
as:
= i = . v , o.ll6c/»
rr 3 Cd i 3 cd
V cm \cm
cd
r = (IT)3 = 0.302cm ; d = 2r = 0.604cm
4n
Since the Cd weight fraction of the sample is 33%:
0.33 g Cd + 0.67 g gaungue = I g sample
Since the density of Cd is about 3 times that of the gangue, then the volume of one gram of sample is
0-33 g Ce^ + ( 0.67 g gangi^ = 1 g sample =
8.65 gem'3 8.65 gem'3 \ g(cm)"3
and the density of the sample is:
/ \-3 1 g sample ., -A -3
g(c/«) = — - - ~: = 3.70 gem
0.2705 c«3
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The volume fraction of Cd is, therefore:
,0.33 g Cdl 8.65
0.2705 cm3
8.7.1 Subsampling Design A
The subsample mass required for analysis: 0.1 g.
Subsampling method: Select the subsample using a small spatula.
Discussion: None of the Cd nuggets are selected since their mass (one-gram nuggets) is much larger than
required by the analytical method, and the Cd nuggets are too big for the size of the spatula.
They literally roll off before the spatula leaves the container. Sample analysis results are all
less than 1% w/w Cd. The results are not incompatible with assuming a normal distribution,
and one falsely concludes the population is uniformly low in Cd.
Conclusion: No significant Cd present; all of the subsamples < 1% w/w Cd with CV < 1%.
Truth: Cd is 33% w/w.
8.7.2 Subsampling Design B
The subsample mass required for analysis: l.Og.
Subsampling method: Select the subsample with a much larger spatula with at least an inner diameter of
3 times the size of the Cd nuggets (3d = 1.812 cm); thus, a spatula (parallel side
and a square bottom) with an inner diameter of 2 cm should suffice.
Discussion: Some of the subsamples will consist primarily of Cd nuggets, but most will have cadmium
levels like those found in Design A. The probability of randomly selecting a Cd nugget (if
there is no GE; that is, the sample is perfectly mixed, which is highly unlikely) will be
proportional to the volume fraction that is Cd. The probability (p = 0.141) is that 14.1% v/v
of the subsamples would be expected to be a Cd nugget. Based on a binomial
approximation for the probabilities associated with selecting Cd as the volume fraction, the
expected number of the 10 one-gram analytical subsamples being a one-gram Cd nugget is
X = np = (10)(0.141) = 1.41, with a standard deviation of s = (npqf = [(10)(0.141)(0.859)]'/2
= 1.10, where q= 1 - p. This gives a CV of (100%)(1.10/1.41) = 78%. That is, on average,
1.41 of the 10 analytical subsamples would contain a one-gram Cd nugget. In practice, one
cannot get a fractional nugget; therefore, the most likely result will be that for the 10 one-
gram subsamples taken for analysis, 1 or 2 samples will be a one-gram nugget of Cd (10%
v/v or 20% v/v). This discussion will assume that only 1 of the 10 analytical subsamples
will be a Cd nugget, resulting in an average Cd level of 0.1 v/v (10% v/v).
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Given that most of the subsamples show no Cd (9 of the 10 subsamples taken), the one high Cd value
may be mistakenly declared to be an outlier. Suppose that 4 additional analytical subsamples were run to
verify the low Cd levels. The probability that none of the 4 subsamples contain a Cd nugget is
(100%)(0.859)4, or 54%. Hence, there is a better than even chance that none of the additional 4
subsamples will show high levels of Cd, helping to confirm the outlier designation. For a discussion on
outliers, please refer to Barnett and Lewis (1995) or to Singh and Nocerino (1995). Software has been
developed for the methods discussed in the latter reference by Singh and Nocerino; the software is called
Scout (Scout, 1999).
Conclusion: The average Cd level is 10% v/v (23% w/w) with a CV = 78% with no outliers claimed; or,
the average Cd level is 0% v/v with a CV = 78%, but an occasional hot spot may exist.
Truth: Cd is 14% v/v (33% w/w).
8.7.3 Subsampling Design C
The sample mass required for analysis: 31.25g.
Subsampling method: Use a sectorial splitter in a two-step process.
Step 1: Divide the 1 kg sample into 8 fractions of 125 g.
Step 2: Divide each 125 g fraction into 4 fractions of 31.25 g.
Discussion: A 1 kg sample with 33% w/w Cd in the form of 1 g nuggets has 333 nuggets. If the sample
is equitably partitioned into 32 subsample splits, then the number of nuggets per subsample
split should average approximately 10. If an average of 10 nuggets per subsample is
expected, then the results should follow a normal probability distribution. The average
result from the analysis of 10 subsamples is found to be 32% w/w Cd with a CV of 3%.
Conclusion: The average Cd level is 32% w/w Cd with a CV of 3%.
Truth: Cd is 33% w/w.
While the correct answer is obtained with a large sample mass, one may not be able to obtain it if the
analysis is constrained to use only small mass sizes. To get the same or better results when using a small
sample mass, the particle distribution of the analyte needs to be altered so that Subsampling behaves like
Design C instead of Design A. That means that one needs to have a much larger number of small analyte
particles. The goal is that each subsample will, on average, have enough analyte particles (greater than 5)
to be modeled as a normal distribution.
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8.7.4 Subsampling Design D
The sample mass required for analysis: 0.1 g
Subsampling method: Subsampling requires a two-step procedure.
Step 1: The sample is ground to pass through a 0.25 mm mesh.
Step 2: Small spatula (>3d = 0.75 mm is easily achieved) is used, taking many random increments,
to obtain each of the 10 subsamples for analysis.
Discussion: For this design the sample must be ground to reduce the maximum particle size from
3.0 mm down to 0.25 mm. For the Cd particles, this is a size reduction of about a factor of
10 in the linear dimension. The mass reduction for Cd particles follows the change in
volume and decreases by approximately a factor of 1000 (down to 0.001 g per particle), and,
therefore, the number of analyte particles will increase by about 1000-fold. The number of
Cd particles per 0.1 g sample can be estimated by noting that 33% of 0.1 g = 0.033 g. If this
is the average amount of Cd in the sample, then 33 particles per subsample would be
expected and the uncertainty should be well approximated by a normal distribution function,
with a CV less than 1%.
Conclusion: Based on a 0.1 g sample, the average level of Cd is 33% w/w with CV < 1%.
Truth: Cd is 33% w/w.
8.8 Case Study: Subsampling Designs Summary
This case study is somewhat simplified in that all of the analyte is in the form of pure particles and no
analyte exists in the remaining gangue (matrix). However, it demonstrates that a variety of results may be
reported for the average analyte level (see Table 14), and each conclusion appears justified by the
supporting data. The difference between correct and incorrect conclusions is due to the analyst's
assumptions about how the analyte is distributed throughout the sample. As noted by Pitard (1993),
obtaining the correct answer requires the analyst to either increase the sample mass (Design C) or to
decrease the particle size (Design D). Given the added constraint of a small subsample for the analysis,
the only design that will give the correct answer is Design D.
All of the procedures would have given the correct answer if Cd was fairly homogeneously
distributed across all of the particle types. Overcoming the variability from various heterogeneous factors
requires more knowledge about the analyte distribution and the sampling process than initially exists.
Determining what information is required is the key to the development of a correct sample mass
reduction procedure.
Page 103 of 134
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Table 14. A summary of the results from the case study designs.
Case Study Design
A. Small sample mass
B1. No outlier
B2. Outlier claimed
C. Larger sample mass
D. Smaller particle size
Analytical
Sample
Mass, M, (g)
0.1
1.0
1.0
31.25
0.1
Cd
(w/w %)
<1
23
<1
32
33
CdCV
(%RSD)
<1
78
<1
3
< 1
Accuracy
Low
Moderate
Low
High
High
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Section 9
Reporting Results
9.1 Introduction
Summarizing the results from a study usually requires more than listing simple concentration
estimates for individual samples or producing a summary statistic, like the average analyte concentration.
Additional information is necessary at several levels if the reported values are to be correctly interpreted.
Scientifically sound decisions require the critical assessment of a number of factors describing the study.
As a general guideline for writing the report, give the information that would be produced by
developing and following the sampling plan (see the guidance given in the section on "Proposed
Strategies"). Give an historical account leading to the study. Include historical and preliminary study
information and results. If known, discuss how the laboratory sample was obtained (including the
transportation and the chain-of-custody of the sample). Document the sample characterization and
assessment observations. List the study objectives (e.g., DQOs) and the rational for their development.
Present the sampling plan (include the sampling nomograph if it was used). Describe the analytical
method and discuss why it was appropriate for this study. Give the required analytical mass, M^
Describe any pre-sampling treatments (drying, removing inert debris) and discuss the actions to take
(including mass calculation corrections). Report the steps taken to ensure correct sampling practices;
give: (1) a description of the subsampling method, (2) a description of the equipment used for correct
materialization, (3) the number of increments selected, (4) a description of how those increments were
selected, (5) a description of any comminution steps, and (6) a discussion on minimizing the preparation
error. List the IHL factors used to calculate %E2. Document how the data will be treated (e.g., the
propagation of errors), including calculations and graphs. Give estimates of the analyte proportion (a,,)
and all of the sampling and analytical errors.
The information needed to evaluate the study is usually presented so that readers with different levels
of expertise are not required to read the entire report to find the information that they are interested in.
This section describes the type of information, in addition to the analytical and statistical results, that
would be of interest to the four previously identified individual types targeted as the intended audience
for this guidance document: the analyst, the scientist or statistician, the manager, and the decision maker.
Those four individual types may each participate in writing the report or they may just be part of the
intended audience. In either case, each has the responsibility to make sure that certain information is
presented in the report. Much of that information may be of interest to more than one type of reader;
although, most readers will only be interested in a small part of the study. Certainly, most readers will be
interested in additional details if unexpected results or procedures were followed.
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The report must be written to address the interests and input from each person involved in the
measurement process - from designing the sampling plan to taking the samples to making any decisions
from the results. Thus, the report must discuss and document all aspects of the study, including: the
planning, the execution (sample selection and analysis), the interpretation, and the decision making.
9.2 For the Analyst
The analyst is concerned with accurately documenting the technical execution of the study
(subsampling and analysis). Deviation from the sampling plan or the analytical procedure must be noted.
The analyst must demonstrate that correct sampling practices were followed and that a suitable analytical
method was followed. The analyst should document the data manipulation steps used to process and
generate the final analytical results of the study. The analyst should also report the following.
I. State the objectives to be achieved. For example, "analyze for the average concentration of
arsenic." If uncertainty limits were identified for the study, then those should be provided, too.
II. Describe the sample, including: size; mass; physical characteristics, such as particle size range;
color; contaminants, such as bottle caps or glass shards, etc.
HI. Reference or describe the subsampling process. This includes the protocol for selecting the
analytical subsample from the laboratory sample and any sample processing activity.
IV. Describe the subsample, including: The mass of the subsample used in the physical analysis; the
maximum particle size in the subsample; any unusual physical characteristics; any differences from
the description of the sample; if replicate analyses were used, then report the results (from replicate
extractions, replicates of a single extraction, or other type of replicates).
V. Summarize the analytical method. The analytical method should be described, either by reference
or, for custom methods or study-dependent sample treatments, with a specific description of the
process. The range of the analytical method, historical analytical performance data, known
analytical interferences, corrections for analytical bias, and any deviations from the sampling plan or
the analytical method must be reported.
Performance criteria for the method should be listed or referenced. However, the study report should
contain at least summary results related to performance. If interlaboratory performance data are available,
men those figures-of-merit are of interest. However, individual laboratories have unique responses;
therefore, the results from the laboratory's internal calibration, QA, and QC work are of primary interest
in demonstrating the level of uncertainty associated with the physical analysis of a subsample.
Those analytical results should provide estimates of the detection limit, quantification limit, relative
variability (s^2) as a function of concentration, and bias (m(AE)). The number and type of each
performance statistic depend on the experimental design. The closer the results are to the decision
criteria, the more important the performance characterization results will become.
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Report the analytical range for the standard curve. This may be different from the range of sample
concentrations. An analytical method with a lower concentration range could be used to determine
samples with a higher analyte concentration through sample dilution.
VI. Document the procedures affecting the data base. Report the estimated analytical uncertainty and
a description of how it was determined.
Give the lower concentration limit for reporting a quantified value. If minimum quantified values are
estimated on a sample-by-sample basis, then the report should indicate that.
Describe the laboratory reporting procedure for results below the limit-of-quantification. Often
results below the quantification level are reported as sample-specific values. Each sample with low
analyte levels may have a different value associated with it in the data base, giving the superficial
appearance that a quantified result is present. If an estimated value is provided, it should be clearly
marked as such. The algorithm for estimating any non-quantified value should be provided or referenced.
If a value of zero is reported, then the report should indicate if this result is due to the round off of
significant figures or if zero was substituted for the measured response. Some samples, such as blanks,
may occasionally result in negative concentration estimates. While these are perfectly valid as the
measured response, they are often rounded up to zero because it is impossible to have a negative
concentration. While this is acceptable when interpreting a data set, it is not acceptable when creating
one.
Hypothetical Example: AACME Laboratory confidently converts all of their negative responses to zero
prior to entering the analytical results into a data base. No negative values are allowed because no
sample has a negative concentration. An analysis for the average response from a set of blanks is
later calculated, and the result is a value of 5 ppm Cr; and, based on the number of results, one can
claim with great confidence that this value is greater than zero. Unfortunately, this result is a biased
estimate. For a true (unbiased, zero) blank, half of the responses should be above zero and half of the
responses should be below zero. The average should be indistinguishable from zero. Converting all
of the negative numbers to zero artificially biases the data.
If all of the original values from the standard curve were reported, then one could compare the level
of Cr from the field samples to those of the blank samples. When negative values are converted to zero,
then the comparison is only acceptable if the estimated bias in the standards is small compared to the
difference between the samples and the blank. Unfortunately, this is just the situation when accurate
statistical models are not needed. When the differences are large, there is little reason to check for
statistical relevance. The random low variation near zero is statistically just as important as any random
high estimate in the samples with concentrations above the detection level.
VII. Summarize the QA and QC results related to the sample integrity to verify performance. Report
the results with regard to any loss or gain from contamination or sample carryover during analysis.
Previous performance for the analysis at another laboratory or summarized from an interlaboratory
study only suggests that the method will perform as designed. A review of the method performance
as provided by the QA and QC results will provide the only appropriate accuracy and bias estimates
for the data set.
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9.3 For the Scientists and Statisticians
The scientists and statisticians are concerned with interpreting, evaluating, validating, and
summarizing the data generated and reported by the analyst. A list (if practical) and a summary of the
data should be presented. How the data were processed should be documented. The report should
include an assessment of the effect that sampling and analysis had on the study. The importance of
sampling should be assessed in the context of all the other factors that might affect the conclusions as part
of a standard sensitivity analysis. Estimates of m^2, s^2, r^2, m^2, s^2, niog2, and sOT2 should be given.
The propagation of errors (e.g., for each subsampling step) must be addressed.
I. Give a summary of all of the sample and data manipulation actions that occurred when the
samples were taken, and during the sample processing and analysis stages. For instance, particle
size information for particles containing the analyte of interest and for non-analyte containing
particles should be reported.
II. Describe the sample history and origin, if known, including any division of the original lot into
multiple lots, strata, or batches, such as considering a warehouse or packaging building as one lot,
an exterior storage area as a second lot, and an adjacent chemical manufacturing facility as a third
lot.
HI. Report the results from any preliminary characterization measurements that were used to justify a
particular analysis protocol. Give descriptions of the samples, subsamples, analytical method,
subsampling method, and the limit-of-quantitation.
IV. Give the performance characteristics of the method. Report the limit-of-quantification (or list
representative examples if the limit-of-quantification is determined on a sample-by-sample fashion).
V. Report the percentage of non-quantitative results. For example, report the fraction of results that
were below the contract or the method quantitative analysis limits. If more than one analyte is
measured, then that limit should be reported on a per analyte basis.
VL Describe all of the protocols followed in the preparation of the data base. This would include an
explanation describing the values inserted into a data base when non-quantitative values are present.
A similar explanation is needed if alternate values are substituted for those data base entries when
calculating summary statistics. For example, the reader should be informed if the data base contains
one-half of the estimated detection limit whenever the concentration was found to be below the
quantitation limit. Data bases often contain sample-specific estimates of the detection or
quantitation limits that, for all practical appearances, look as if they are quantitative estimates. In
reality those non-quantitative artifacts of the data base recording protocol are often an arbitrary
guess of the analyte level and may be difficult to recognize.
If the results from below the limit-of-quantitation are included in the summary statistics, then a
description of what values were used for those results should be included. For example, report if the
concentrations were set to zero, to the limit-of-quantitation, to one-half of the limit-of-quantitation, to a
random number between zero and the quantitative limit, or to some other value.
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Include equations and example calculations. Explain why, and describe (may be referenced), any
uncommon statistical methods that are used (e.g., robust outlier detection methods).
9.4 For the Managers
The manager is concerned with documenting if the study was performed within the design and
planning specifications, and if the study objectives were met. Anomalies and deviations must be
documented, discussed, and explained. The results provided by the analyst, and the scientist or
statistician, should be summarized and verified. The manager must be sure that the mass data have been
corrected for any pre-sampling actions, such as drying or removing inert objects, as identified in the
sampling plan. The manager should be able to validate if correct sampling practices were followed (e.g.,
does Sj/ = sFE2?) and that a representative sample was obtained within the limits set by study objectives
(rj/ < rOTE2). The manager must verify that the study interpretations are defensible. Besides reporting if
the results meet the technical requirements, the manager may also discuss cost and benefit issues. It is the
manager's responsibility to make sure that the information that the decision maker sees is informative,
accurate, and reliable. Managers should see or report the same information as the decision makers, but
with the added level of detail summarized below.
I. Report whether or not the sample description is compatible with the sample treatment. If the
samples have particle sizes that suggest that particle size reduction is needed but the sample
treatment did not include that step, then the sampling protocol may be incorrect.
II. Describe and discuss the statistical results. If the distribution of the data is highly skewed and the
fraction of samples with below detection values is large, then there should be a discussion of the
sample mass and subsampling practices. This becomes more important as the skewness increases or
as the fraction of the samples with below the quantitative levels of the analyte increases. In either
case, the estimate of the mean analyte concentrations) becomes less certain and more dependent on
the sample characteristics.
HI. The concentration (or concentration ranges) dividing the non-quantitative from the quantitative
results should be identified. Flags should identify any substituted values. There should also be an
assessment if the assumptions made to fill in values for an analysis are influential with respect to the
decisions being made. If the results are influenced by the fill-in procedure, then they are more than
likely to be unstable and probably will not justify a decision.
IV. Review the fraction of the samples with low, non-quantitative results. If that fraction is large, then
the treatment of below quantitative data may play an important role in determining the true analyte
level.
V. Discuss any outliers identified in the data base. If any outliers were not included in any of the
calculations or in the data base, then give a justification for their removal. For a discussion on
outliers, please refer to Barnett and Lewis (1995) or to Singh and Nocerino (1995). Software has
been developed for the methods discussed in the latter reference by Singh and Nocerino; the
software is called Scout (Scout, 1999).
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VI. Give any estimates of measurement reproducibitity and sources of variation. If possible, those
estimates should be partitioned into sources of variation.
9.5 For the Decision Makers
The decision maker is concerned with the implications of the study for what to do next. The decision
maker needs to be sure that the information reported or verified by the manager is informative, accurate,
and reliable so that correct (informative, accurate, and reliable) decisions can be made. Supporting
evidence should be summarized showing that correct and representative sampling took place, and that it is
reasonable to believe that the average proportion of the analyte in the analytical subsample is the same,
within the user-specified study objectives, as the average proportion of the analyte in the laboratory
sample; that is, a^a aL. At a minimum, the decision maker should note whether or not sampling concerns
are addressed, but should extend to evaluating whether or not the study activities met the cost and benefit
goals, resulted in acceptable risks, and met legal and policy requirements.
I. Give a clear statement of the issues and the factors that were considered in setting the study
objectives. Determine if the study results are accompanied by some measure of sensitivity that
identifies if there is a clearly supported action suggested by the data or whether the data did not
meet the pre-study requirements. Give any evidence to suggest that the response is associated with
enough variability to justify delaying action.
II. Document the number and location of the samples. If the samples were taken during several
sampling events, then an indication of the time sequence should be present. If different laboratories
or analysis times were used to generate data, then the procedure and results for ensuring the
comparability of different data sets should be provided as well as key summary statistics.
III. Include the key features of the data, including the minimum concentration for quantitation, the
fraction of samples (by analyte) with results below this level, and the concentration of concern. A
description of how a typical sample was taken, the sample processing step(s), and the analytical
method used should be given.
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Section 10
Summary and Conclusions
The characteristics of samples collected from a lot are used to make estimates of the characteristics of
that lot. Thus, samples are used to infer properties about the lot in order to make correct decisions
concerning that lot. Therefore, for sampling to be meaningful, it is imperative that a sample is as
representative as possible of the lot, and more generally, each subsample must be as representative as
possible of the parent sample from which it is derived. Subsampling errors propagate down the chain
from the largest primary sample to the smallest laboratory analytical subsample. If a collection of
samples does not represent the population from which they are drawn, then the statistical analyses of the
generated data may lead to misinformed conclusions and perhaps costly decisions.
Sampling can be the major source of error in the measurement process and it can be an especially
overwhelming source of error for heterogenous particulate materials, such as soils. Since classical
statistical sampling theory is not adequate for such samples, the Pierre Gy statistical sampling theory was
introduced in this document as a viable alternative that takes into account the nature of particulate
materials. Although this theory, developed in the mid-1950s, has proven itself in practice in the mining
industry, very little has been published with substantiating experimental evidence for this theory. An
ongoing research program has been established to experimentally verify the application of the Gy theory
to environmental samples, which served as a supporting basis for the material presented in this guidance.
Research results from studies by the U.S. EPA have confirmed that the application of the Gy sampling
theory to environmental heterogeneous particulate materials is the appropriate state-of-the-science
approach for obtaining representative laboratory subsamples.
This document provided general guidelines for obtaining representative subsamples for the laboratory
analysis of particulate materials using "correct" sampling practices and "correct" sampling devices.
Besides providing background and theory, this document gave guidance on: sampling and comminution
tools, sample characterization and assessment, developing a sampling plan using a general sampling
strategy, and reporting recommendations. Considerations were given to: the constitution and the degree
of heterogeneity of the material being sampled, the methods used for sample collection (including what
proper tools to use), what it is that the sample is supposed to represent, the mass (sample support) of the
sample needed to be representative, and the bounds of what "representative" actually mean. A glossary
and a comprehensive bibliography have been provided and should be consulted for more details.
The basic strategic theme developed in this document concluded that if "correct" sampling practices
are followed and "correct" sampling devices are used, then all of the sampling errors should become
negligible, except for the minimum sampling error that is fundamental to the physical and chemical
composition of the material being sampled. That is, the mean of the total sampling error becomes
negligible (no sampling bias) and the relative variance of the total sampling error is reduced to the
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relative variance of the fundamental error (srE2 = Spg2). Since this minimum fundamental sampling error
can be estimated before any sampling takes place, s^2 can be used to develop a sampling plan.
It was shown that correct sampling (or selection) can be associated with three practices: 1) correctly
taking many increments (greater than about 30 are recommended) to make up the subsample to minimize
the grouping and segregation error (GE), 2) using correctly designed sampling tools to minimize the
materialization error (ME), and 3) using common sense and vigilance to minimize the preparation error
(PE).
The bounds of what are acceptable for a sample to be "representative" of the lot are set by the study
objectives (e.g., the data quality objectives or DQOs). The degree of representativeness, rre2, has been
defined in this document by r^2 = m^2 + s^2, where TE is the total sampling error, r^2 is the mean
square of the total sampling error, m^2 is the square of the mean of the total sampling error, and s^2 is the
relative variance of the total sampling error. A sample is representative when rTC2 < roTE2, where roTE2 is a
specified and quantitative measure of a representative sample (the smaller this number, the more
representative is the sample); that is, it is a level of representativeness regarded as acceptable (usually set
by the study objectives). If correct sampling practices are applied and all of the "controllable" errors are
minimized, then a representative sample could be characterized by keeping the relative variance of the
fundamental error below a specified level, s,^2 z s^2. Under such conditions, rTE2 » SpE2 < soFE2 a roTE2.
This equation served as the basis of our strategy to obtain a representative analytical subsample. This is
fortuitous for our planning purposes, and for formulating our study objectives (e.g., DQOs), since Sj^2 is
the only error that can be calculated, based on the physical and chemical properties of the paniculate
material, a priori: that is, before sampling even takes place!
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Recommendations
For those familiar with most of the information given in this guidance, it was suggested that such a
user, searching for the guidance to develop a sampling plan to obtain a representative analytical
subsample, go to the section on "Proposed Strategies," which gives a general and somewhat extensive
sampling strategy guide.
The guidance presented is general and is not limited to environmental samples, and is applicable to
samples analyzed in the field. The information given in this guidance should also prove to be useful in
making reference standards as well as taking samples from reference standards. Similarly, the presented
guidance should be of value in: Monitoring laboratory performance, creating performance evaluation
materials (and how to sample them), certifying laboratories, running collaborative trials, and performing
method validations.
Remember that sampling is one of those endeavors that you "get what you pay for," at least in terms
of effort. But, with the right knowledge and a good sampling plan, the effort is not necessarily that much.
It pays to have a basic understanding of the theory. Become familiar with what causes the different
sampling errors and how to minimize them through correct sampling practices. Be able to specify what
constitutes a representative subsample. Know what your sampling tools are capable of doing and if they
can correctly select an increment. Always do a sample characterization (at least a visual inspection) first.
At a minimum, always have study objectives and a sampling plan for each particular case. If possible,
take a team approach when developing the study objectives and the sampling plan. Historical data or
previous studies should be reviewed. And be sure to record the entire process!
This guidance focused on the issues and actions related to samples composed of particulate materials.
It is not intended for samples selected for analysis of volatile or reactive constituents, and it does not
extend to sampling biological materials, aqueous samples, or viscous materials, such as grease or oil
trapped in a particulate matrix. Such cases warrant more research and guidance. Correct sampling
practices to obtain representative field samples should also be the subject of ongoing research and a future
guidance document.
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References
Allen,!. 1997. Particle Size Measurement, Vol. 1, Fifth Edition. Chapman & Hall, London.
Allen, T., Khan, A. A. 1970. "Critical Evaluation of Powder Sampling Procedures," Chemical Engineer
(London), Vol. 238, pp. CE108-CE112.
ASTM. 1997. ASTM Standards on Environmental Sampling. Second Edition. ASTM,
West Conshohocken, PA.
Barnett, V., Lewis, T. 1994. Outliers in Statistical Data. Wiley, NY.
Barth, D.S., Mason, B.J., Starks, T.H., Brown, K.W. 1989. Soil Sampling Quality Assurance User's
Guide, Second Edition, EPA/600/8-89/046, U.S. Environmental Protection Agency, Las Vegas, NV.
Dixon, W.J., Massey, F.J., Jr. 1969. Introduction to Statistical Analysis, Third Edition, McGraw-Hill,
New York.
Francois-Bongarcon, D., Gy, P. 1999. The most common error in applying "Gy's Formula" in the theory
of mineral sampling. Australian Journal of Mining.
Gerlach, R.W., Dobb, D.E., Raab, G.A., Nocerino, J.N. 2002. Gy Sampling Theory in Environmental
Studies I: Assessing Soil Splitting Protocols. Journal ofChemometrics, 16(7), 871-878.
Gy, P.M. 1982. Sampling of Particulates - Theory and Practice, Second Edition, Elsevier, Amsterdam.
Gy, P.M. 1998. Sampling for Analytical Purposes. Wiley, Chichester, UK.
Ingamells, C.O. 1976. Derivation of the sampling constant equation. Talanta, 23, 263-264.
Ingamells, C.O., Pitard, F.F. 1986. Applied Geochemical Analysis. Wiley, NY.
Ingamells, C.O., Switzer, P. 1973. A proposed sampling constant for use in geochemical analysis.
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Kern, A.M., Evans, J.D., Hayes, S.L., Partymiller, K.G., Swanson, G.R. 1997. Representative sampling
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Mason, B.J. 1992. Preparation of Soil Sampling Protocols: Sampling Techniques and Strategies.
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Mishalanie, E., Ramsey, C. 1999. Obtaining Trustworthy Environmental Data: Sampling and Analysis
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Smith, P.L. 2001. A Primer for Sampling Solids, Liquids, and Gases Based on the Seven Sampling
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20. Gilbert, R.O. 1987. Statistical Methods for Environmental Pollution Monitoring, VanNostrand
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21. Gy, P.M. 1982. Sampling ofParticulates- Theory and Practice, Second Edition. Elsevier,
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22. Gy, P.M. 1992. Sampling of Heterogeneous and Dynamic Material Systems. In: Theories of
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23. Gy, P.M. 1998. Sampling for Analytical Purposes. Wiley, Chichester, UK.
24. Hatton, T.A. 1977. Representative sampling of particles with a spinning riffler: A stochastic
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25. Horwitz, W. 1988. Sampling and preparation of sample for chemical examination. Journal of the
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26. Ingamells, C.O. 1974. Control of geochemical error through sampling and subsampling designs.
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27. Ingamells, C.O. 1976. Derivation of the sampling constant equation. Talanta, 23,263-264.
28. Ingamells, C.O. 1978. A further note on the sampling constant equation. Talanta, 25, 731-732.
29. Ingamells, C.O. 1981. Evaluation of skewed exploration data-The nugget effect. Geochimica et
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30. Ingamells, C.O., Pitard, F.F. 1986. Applied Geochemicat Analysis. Wiley, New York.
31. Ingamells, C.O., Switzer, P. 1973. A proposed sampling constant for use in geochemical analysis,
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32. Jenkins, J.F., Grant, C.L., Brar, G.S., Thome, P.O., Schumacher, P.W., Rannaey, T.A. 1997.
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33. Journel, A.G., Huijbregts, CJ. 1978. Mining Geostatistics. Academic Press, London.
34. Keith, L.H. 1988. Principles of Environmental Sampling. American Chemical Society,
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sampling and analysis of heterogeneous soils. Environmental Testing & Analysis, pp. 20-22, 27.
36. Lame, F.P.L., Deflze, P.R. 1993. Sampling of contaminated soil: Sampling error in relation to
sample size and segregation. Environmental Science & Technology, 27(10), 2035-2044.
37. Lopez-Avila, V. (Ed.) 1998. Current Protocols in Field Analytical Chemistry. Wiley, New York.
38. Mason, BJ. 1992. Preparation of Soil Sampling Pro tocols: Sampling Techniques and Strategies.
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Agency, Las Vegas, NV.
39. Mishalanie, E., Ramsey, C. 1999. Obtaining Trustworthy Environmental Data: Sampling and
Analysis Issues. Natural Resources & Environment, Vol. 13(4), 522-527.
40. Mullins, C.E., Hutchinson, BJ. 1982. The variability introduced by various subsampling
techniques. Journal of Soil Science, 33, 547-561.
41. Myers, J.C. 1996. Geostatistical Error Management: Quantifying Uncertainty for Environmental
Sampling and Mapping. Van Nostrand Reinhold, New York.
42. NAVFAC andU.S.EPA. 1998. Field Sampling and Analysis Technologies Matrix and Reference
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43. Pitard, F.F. 1989. Pierre Gy's Sampling Theory and Sampling Practice, Vol. 1, Heterogeneity and
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44. Pitard, F.F. 1989. Pierre Gy's Sampling Theory and Sampling Practice, Vol. 2, Sampling
Correctness and Sampling Practice. CRC Press, Boca Raton, FL.
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46. Ramsey, CA., Ketterer, M.E., Lowery, J.H. 1989. Application of Gy's Sampling Theory to the
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Errors of Pierre Gy. Society for Industrial and Applied Mathematics, Philadelphia, PA.
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51. Thompson, S.K. 1992. Sampling. Wiley, New York.
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EPA/540/R-93/071. U.S. Environmental Protection Agency, Office of Emergency and Remedial
Response. Washington, DC.
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96/056, U.S. Environmental Protection Agency, Washington, DC.
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(EPA QA/G-9). EPA/600/R-96/084, U.S. Environmental Protection Agency, Office of Research
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EPA/600/R-96/085, U.S. Environmental Protection Agency, Office of Research and Development,
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98/018), U.S. Environmental Protection Agency, Washington, DC.
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96/055, U.S. Environmental Protection Agency, Office of Environmental Information, Washington,
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64. U.S. EPA. 2000b. Guidance for Choosing a Sampling Design for Environmental Data Collection,
EPA QA/G-5S, Peer Review Draft, Office of Environmental Information, U.S. Environmental
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Environmental Protection Agency, Washington, DC.
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QAJG-4HW). EPA/600/R-00/007, U.S. Environmental Protection Agency, Office of Environmental
Information, Washington, DC.
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of soils. EPA/600/4-90/013, U.S. Environmental Protection Agency, Environmental Monitoring
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68. Visman, J. 1969. A general theory of sampling. Materials Research and Standards, 9(11), 8-13.
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Mathematical Geology, 22(1), 107-121.
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Glossary of Terms
Words highlighted in bold letters within the glossary definitions can be found elsewhere in the glossary
with their own definitions.
accuracy — Unfortunately, this term suffers from an extraordinary number of definitions, which can lead
to confusion and outright disdain for the word. However, it is still a viable word with correct usage.
Accuracy is a degree, or measurement, of closeness to a target value. Accuracy should not be
confused with precision. A sample is accurate if the absolute value of the bias of the total sampling
error is within a specified acceptable level of accuracy:
\mTE\ - mOTE
analytical error (AE) - This error component arises from imperfections in the analysis (chemical or
physical) operation. It includes errors associated with such activities as chemically extracting the
analyte from the sample matrix, instrumentation error, operator errors, moisture analysis, gravimetric
errors, and other measurement errors.
bias - The systematic or persistent distortion of a measurement process that causes errors in one direction
along a metric away the true value; that is, bias is a function of systematic error (e.g., the average
measured mass differs from the true mass by +0.034 g). In the context of sampling, bias (B^) is the
mean of the total sampling error, m^, or
Ba. =
where m^ is the mean estimate of the proportion of the analyte in the sample and aL is the true
proportion of the analyte in the lot.
coefficient of cubicity (f) - See shape factor.
comminution - Comminution is a crushing or grinding process used to decrease the particle size of a lot
or a sample.
composite sample - A sample created by combining several distinct subsamples.
composition factor (c) — See mineralogical factor.
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continuous selection error (CE) - This error is generated by the immaterial selection process and is the
sum of three errors, the short-range fluctuation error (CEt), the long-range heterogeneity
fluctuation error (CE2), and the periodic heterogeneity fluctuation error (CE3):
CE = CE, + CE2 + CE3.
correct sampling practice - Correct sampling gives each item (particle, fragment) an equal and constant
probability of being selected from the lot to be part of the sample. Likewise, any item that is not
considered to be part of the lot (that is, should not be represented by the sample) should have a zero
probability of being selected. Note that the bias of the total sampling error should be zero (m^ -
0) when the sampling practices are perfectly correct (that is, when GE = DE = EE = PE = 0). If
sampling practices are perfectly correct (that is, s^2 = s^,2 = s^2 = sPE2 = 0), then s-jE2 = %E2 (the
relative variance of the fundamental error). Thus, correct sampling practices minimize those
"controllable" errors through correctly designed sampling devices (to minimize DE, s^2, EE, and
SE^), common sense (to minimize PE and SpE2), and by correctly taking many random increments (N
2:30) combined (to minimize GE and s^2) to make up the sample. Correct sampling practices allow a
representative sample to be taken.
data quality objectives (DQQs) - Qualitative and quantitative statements derived from the DQO process
that clarify study objectives, define the appropriate types of data, and specify tolerable levels of
potential decision errors that will be used as the basis for establishing the quality and quantity of the
data needed to support decisions.
data quality objective (POO) process - A systematic planning tool to facilitate the planning of
environmental data collection activities (see: U.S. EPA, 1994,1996a, 1996c, 1997, 2000a, and
2000d). The DQO process allows planners to focus their planning efforts by specifying the intended
use of the data, the decision criteria, and the decision maker's tolerable decision error rates. Data
quality objects are the qualitative and quantitative outputs from the DQO process.
defensible - The ability to withstand a reasonable challenge related to the veracity, integrity, or quality of
the logical, technical, or scientific approach taken in a decision making process.
dimension (of a lot) - If the components of a lot are related by location in space or time, then they are
associated with a dimension. The dimension of a lot is the number of major axes having an ordered
metric. Dimensions in the characterization of a lot become reduced when one or more dimensions
become negligible when compared to the other dimensions). Sampling units, such as bags of
charcoal or bottles of beer from a production line, represent one dimensional lots, with the time of
production giving order to the individual items. A zero-dimensional lot has no order to the sampling
units. An elongated pile is a one-dimensional lot since it has one major ordered direction. Surface
contamination at an electrical transformer storage site represents a two-dimensional lot. Higher (two-,
and especially three-) dimensional lots tend to be much more difficult to sample in producing
laboratory subsamples.
estimate - A measured value that approximates the true value. In this text, the proportion of the analyte
in the sample, a^ is an estimate of the true proportion of the analyte in the lot, a^ Measurements are
always subject to errors and one can never claim that the measured value is exactly correct. This is
also true of any statistic based on measured data. Thus, statistics and the data used to calculate them
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can both be classified as estimates. The statistic (e.g., the sample mean) from a sample (or samples)
of a population is an estimate of the true value of the parameter (the true mean) of the population.
That is, for an estimate to have merit, the sample must mimic (be representative of) the population in
every way, including the distribution of the individual items or members (particles, analytes, and
other fragments or materials) of that population.
fundamental error (FE) - This error is fundamental to the composition of the particles (or other items or
fractions) of the lot being chemically or physically different; that is, it is a result of the constitution
heterogeneity (CH) of the lot. Thus, this is the only sampling error that can never cancel out. To get
an accurate representation of this constitutional heterogeneity, one must be sure that the samples are
always representative of all particle size fractions that are part of the lot. This relative variance of
the fundamental error can be estimated before sample selection and may be reduced by decreasing the
diameter of the largest particles to be represented or by increasing the mass of the sample. The
relative variance of the fundamental error is estimated by:
sra2 = [(l/Ms)-(l/ML)]clfgd3
grab sample - A nonprobabilistic selection of a sample (really, a specimen), usually chosen on the basis
of being the most accessible or by some judgement of the operator. A grab sample is taken with no
consideration for obtaining a representative sample. Grab sampling has been shown to be
associated with very high uncertainty and bias. It should only be used on sample matrices that have
been extensively studied and shown to provide adequate data quality. Even then, great caution should
be exercised as grab sampling may not provide an indication of matrix changes resulting in non-
representative sampling.
granulometric factor (g) - Sometimes called the particle size distribution factor, the granulometric
factor is a particle size factor based on the particle size distribution. The size of each particle is not a
constant. This factor accounts for the varying particle sizes when estimating the fundamental error.
grouping and segregation error (GE) - This error is due to the distributional heterogeneity (DH) of the
particles (or other items) of the lot. The relative variance of GE, sGE2, is due to the constitution
heterogeneity, as well as to grouping and segregation (usually because of gravity) - i.e., incremental
samples are different. (Note that the short-range heterogeneity fluctuation error, CEl = FE + GE.)
The relative variance of the short-range fluctuation error, sCE12, can be minimized to be about the
magnitude of the fundamental error if many increments are taken and combined to make the
sample. Since
2 2 , „ 2
SCE1 SFE :*GE •
and for most cases, one can assume (Pitard, 1993; p. 189) that
s 2 < s 2
then
2 2 , „ 2 . 9 „ 2
JFF ' ^nF — ** OFF •
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(Note that this is not always true; for example, a sample made of only one increment containing a
highly segregated fine material may have a very small s^2 but a much larger sGE2). Then, for N
increments, taken with correct sampling practices, we can write (Pitard, 1993, p. 388)
2 _
~
Under such conditions, the relative variance of the total sampling error becomes
Thus, ST/ £ sCE12 « SpE2 if N is made large enough. At least N = 30 increments are lecommended as a
rule of thumb to reduce SGE2 compared to s^2 (Pitard, 1993; p. 187).
hazardous waste - Any waste material that satisfies the definition of hazardous waste given in 40 CFR
261, "Identification and Listing of Hazardous Waste."
heterogeneity - The condition of a population (or a lot) when all of the individual items are not identical
with respect to the characteristic of interest. For this guidance, the focus is on the differences in the
chemical and physical properties (which are responsible for the constitution heterogeneity, CH) of the
paniculate material and the distribution of the particles (which leads to the distribution heterogeneity,
DH).
homogeneity - The condition of a population (or a lot) when all of the individual items are identical
with respect to the characteristic of interest. It is the lower bound of heterogeneity as the difference
between the individual items of a population approach zero (which cannot be practically achieved).
increment - A segment, section, or small volume of material removed in a single operation of the
sampling device from the lot or from a subset of the lot (that is, the material to be represented); many
increments (N ^30 increments are recommended) taken randomly are combined to form the sample
(or subsample).
increment delimitation error (DE) - This increment materialization error (ME) involves the physical
aspects of selecting the increment using a correctly designed sampling device. The volume
boundaries of a correct sampling device must give all fractions collected an equal and constant chance
of being part of the sample. For example, a "one-dimensional" pile should be completely transected
perpendicularly by a scoop with parallel sides. The increment delimitation error occurs when an
incorrectly designed sampling device delimits (boundary limits of the extended increment) the
volume of the increment giving a nonuniform probability for each item (fraction or particle) to be
collected within the boundaries of the sampling device.
increment extraction error (EE) - This increment materialization error (ME) also involves the
physical aspects of selecting the increment using a correctly designed sampling device. For a
correctly designed sampling device, there must be an equal chance for all of the parts of the extended
increment to be part of the sample or part of the rejects. The shape of the sampling device's edges is
important to the center of gravity of each the particles and for each particle's chance to be part of the
sample or part of the rejects.
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increment materialization error (ME) - This error involves the physical process of selecting and
combining the increments in preparing the sample (or subsample), and is technically the sum of
three errors: the increment delimitation error (DE), the increment extraction error (EE), and the
preparation error (PE); that is: ME = DE + EE + PE. However, since DE and EE are the result of
the increment selection process and PE is a result of the sample preparation process, those errors are
treated separately and we will use ME = DE + EE.
liberation factor (/) - An estimate of the fraction of analyte that is separated (liberated) as a pure
constituent from the gangue (matrix). The liberation factor takes on values 0 < / <> 1; when / = 1, the
analyte is completely liberated. The liberation factor is estimated as:
"max - °L
where aL is the true critical content of the constituent of interest (analyte or contaminant) in the lot,
and amaii is the maximum critical content of the constituent of interest in the coarsest fragments of the
lot. If one has a mineral-like analyte, then the liberation factor can also be estimated by:
"4
where dj is the diameter needed to liberate the contaminant and d is the diameter of the largest
contaminated particle.
limit of quantification (Ln) — The analyte concentration below which there is an unacceptable error in
determining a quantitative value. It is commonly set at
where yg is the measurement of the average value of the blank (no analyte) and sg is the standard
deviation of the blank measurements.
long-range heterogeneity fluctuation error (CE7) — The essentially nonrandom error associated with
long-range local trends (e.g., local concentration trends) across the lot. The relative variance of this
error is identified by variographic experiments and may be better characterized by reducing the size
of the strata or taking many increments to form the sample. (Note that the continuous selection
error, CE = CEl + CE2 + CE3.)
lot - All of the material (the population) being characterized or studied. Anything from a single sample
to a truck load of material to an entire Superfund site could be a single lot. A lot is the material that is
to be represented by the sample.
mean- See sample mean.
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method - A body of procedures and techniques for performing an activity.
mineralogical factor (c) - Also known as the composition factor. This factor represents the maximum
degree of heterogeneity that the analyte can produce and is attained when the analyte is completely
liberated. The mineralogical factor can be estimated as:
c =
where aL is the decimal proportion [unitless] of the analyte in the sample, AM is the density of
particles containing the analyte [g/cm3], and Ag is the density of the gangue [in g/cm3].
outlier - A statistical outlier is an observation that can be shown to belong to a population distribution
other than the population distribution in question (usually the underlying dominant population) or
shown not to belong to the population distribution in question. An outlier sample is a sample that is
not representative of the distribution of the results from a particular population of samples.
particle size distribution factor - See granulometric factor.
percentile - The specific value of a distribution that divides the distribution such that p percent of the
distribution is equal to or below that value. For p = 95, "The 95th percentile is X" means that 95% of
the values of the population (or sample) are less or equal to X.
periodic heterogeneity fluctuation error CCE?) - The error due to large-scale periodic or cyclical, but
essentially nonrandom, fluctuations across the lot (e.g., periodic analyte concentrations due to a
process). The relative variance of this error is identified by variographic experiments and may be
"smoothed out" by reducing the size of the strata or taking many increments to form the sample.
(Note that the continuous selection error, CE = CE, + CE2 + CE3.)
population - The total collection of objects (the lot) to be studied. A population comprises those objects
(material) that are to be represented by the sample. The sample statistics (e.g., the sample mean, x,
and the sample variance, s2) are estimates of the population parameters (e.g., the population mean,
|i, and the population variance, a2).
population mean - The true mean, u-, for all N items, \iv in the population.
N
precision - Sometimes referred to as reproducibility, precision is a measure of the mutual agreement
among individual measurements of the same property, usually under prescribed similar conditions
expressed generally in terms of the standard deviation or variance, which is a function of random
error.
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preparation error (PE) - This error involves gross errors such as losses, contamination, and alteration
(e.g., sample degradation).
quality assurance (QA) - An integrated system of activities involving planning, quality control, quality
assessment, reporting, and quality improvement. It is the activity of providing, to all concerned, the
evidence needed to establish confidence that the quality function is being performed adequately to
provide fitness for use.
quality control (QC) - The overall system of activities that measure the attributes (quality
characteristics) and performance of a process. For example, a measurement of the collection of data
(or data analysis) is compared with standards and an action may be taken depending on the magnitude
of the difference.
representative sample (or subsample) - To be truly representative, the sample must mimic (be
representative of) the population in every way, including the distribution of the individual items or
members (particles, analytes, and other fragments or materials) of that population. However,
depending on predefined specifications, the sample may only have to be representative of only one
(or more) characteristics of the population, and estimated within acceptable bounds. The
characteristic most often sought after is an estimate of the population mean analyte concentration
(aL) and an estimate of the overall error (overall relative variance) for the lot. A representative
sample is defined as a sample that is both accurate (within a specified level of bias) and precise
(within a specified level of relative variance) at the same time (Pitard, 1993; p. 415). The degree of
representativeness, r^2, is given by
rTE = mTE + STE
where TE is the total sampling error, rTE2 is the mean square of the total sampling error, m^2 is the
square of the mean of the total sampling error relative, and s^2 is the relative variance of the total
sampling error. A subsample should be representative of the sample it was taken from and be an
estimation of the original lot, as defined by study objectives (e.g., data quality objectives, DQOs)
and the sampling plan. In the case of laboratory subsampling, the analytical subsample (e.g., for
chemical analysis) must be representative of the entire contents of the laboratory sample bottle. A
sample is representative when
r 2 < r 2
rTE s roTE
where roTE2 is a specified and quantitative measure of a representative sample (the smaller this
number, the more representative is the sample); that is, it is a level of representativeness regarded as
acceptable. Note that mTE2 = 0 when the sampling practices are perfectly correct (that is, when GE =
DE = EE = PE = 0). It is desirable to keep sTE2 < soTE2, where soTE2 is a level of the relative variance of
the total sampling error within user specifications. If sampling practices are perfectly correct (that is,
sGE2 = sDE2 = sEE2 = SpE2 = 0), then s^2 = s^2 (the relative variance of the fundamental error). Thus, if
correct sampling practices are applied and all of those "controllable" errors are minimized, then a
representative sample could be characterized by keeping the relative variance of the fundamental error
below a specified level, SpE2 < soFE2. Under such conditions,
T 2 ~ <^_Z < H 2 ~ T 2
1TE ~ ^fE, ~ \¥E ~ 1oTE
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The above equation is the basis of our strategy to obtain a representative subsample for chemical
analysis.
sample - Since it is often too difficult to analyze an entire population (or lot), a sample (a portion) is
taken from the population in order to make estimations about the characteristics of that population;
e.g., sample statistics, such as the sample mean and sample variance, are used to estimate
population parameters, such as the population mean and population variance. Because sampling is
never perfect and because there is always some degree of heterogeneity in the population, there is
always a sampling error. To get accurate estimates of the population by the sample(s) and to
minimize the total sampling error, a representative sample is sought using correct sampling
practices. Thus, a sample is made from the combination of many correctly selected increments. To
be truly representative, the sample must mimic (be representative of) the population in every way,
including the distribution of the individual items or members (particles, analytes, and other fragments
or materials) of that population. However, depending on predefined specifications, the sample may
only have to be representative of only one (or more) characteristics of the population, and estimated
within acceptable bounds.
sample mean - The statistic, x , used to estimate the population mean from the n sample observations,
*/•
n
£ xi
~ = '=1
n
The mean of the selection error, m(SE) (see bias (Ba,.)) is
m(a ) - a
= B(a,)
where m(as) is the mean estimate of the proportion of the analyte in the sample and aL is the true
proportion of the analyte in the lot.
sampling or selection error (SE) — The sampling or selection error refers to the relative difference
between the expected (true or assumed) value of the proportion of the analyte in the lot, aL, and the
estimated value of the proportion of the analyte in the sample, a,, (when there is no preparation
error, PE):
a, - a,
5 i
Note that SE = CE + ME = FE + GE + CE2 + CE, + DE + EE.
sampling unit- A volume, mass, or item of material being sampled, in part or in total. If one is
characterizing a pile of 55-gal drums, then the sampling unit might be an individual drum. Sampling
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units are not necessarily identical with the samples themselves, but an (often naturally) delineated
fraction of the lot.
short-range fluctuation error (CEt) - This short scale error is the sum of the fundamental error (FE)
and the grouping and segregation error (GE):
CEj = FE + GE.
shape factor ffl - Also known as the coefficient of cubicity, the shape factor summarizes the average
shape of the particles with respect to a cube (which has a shape factor of 1.0).
specimen - A specimen is a portion of the lot taken without regard to correct sampling practices and
therefore should never be used as a representative sample of the lot. It is a nonprobabilistic sample;
that is, each item (particle, fragment) does not have an equal and constant probability of being
selected from the lot to be part of the sample. Likewise, any item that is not considered to be part of
the lot (that is, should not be represented by the sample) does not have a zero probability of being
selected. A specimen is sometimes called apurposive or judgement sample. An example of a
specimen is a grab sample.
standard deviation - A measure of the dispersion or imprecision of a sample or population distribution
expressed as the positive square root of the variance and has the same unit of measurement as the
mean.
statistic - A summary value calculated from a sample, usually as an estimator (e.g., the sample mean or
the sample variance) of a population parameter (e.g., the population mean or the population
variance).
subsample - A subsample is simply a sample of a sample. This term is used when one wants to
distinguish the parent sample (from which the subsample is taken) from the primary lot. An example
would be that the site is the primary lot, the bottle coming to the laboratory is the (parent) sample, and
the portion taken for analysis is the subsample. In correct sampling practices, the parent sample is
considered the (new) lot from which a respesentative sample is to be taken. Less confusing may be
to use terms for successive sampling stages: lot, primary sample, secondary sample, etc.
support - The support is the size (mass or volume), shape, and orientation of the sampling unit or that
portion of the lot that the sample is selected from. If a soil sample is taken as section of a 2.5 cm
diameter core sample from 10 to 15 cm depth, then it will have a slightly different support compared
to first digging a trench and then acquiring 10 increments of soil along a 2 m traverse at 10 to 15 cm
depth. In one case the support is a compact geometric shape while in the other case the support
represents an average behavior over a short distance. The support affects the estimation of the
population (or lot) parameters.
total sampling error fTE) - This sampling error refers to the relative difference between the expected
(true or assumed) value of the proportion of the analyte in the lot, aL, and the estimated value of the
proportion of the analyte in the sample, as, when there is preparation error, PE:
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TE-
Note that TE = SE + PE = CE + ME = FE + GE
traceabilitv - This is the ability to trace the history, application, or location of an entity to its origin, such
as a field sample or a calibration sample.
uncertainty - This is a term with multiple meanings. As such, it should always be defined when used.
For this document, uncertainty refers to variation from the correct or expected value due to all factors
affecting the measurement process. It is often characterized as the combination of effects that cause
bias or variance components.
variability - Observed differences attributable to the heterogeneity or diversity in a population (or lot),
the influence of outliers, or in the measurements made to estimate population (or lot) parameters.
Sources of variability are the results of random or systematic processes.
variance - A measure of the dispersion of a set of n values, x;; i = 1, 2, ..., n. The population variance is
indicated by o2 and the sample variance is indicated by s2.
E (Mi -
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U.S. Environmental Protection Agency
Region 5, Library (Pt-12J)
77 West Jackson Boulevard, 12tfc floor
Chicago. IL 60604-3590 -
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vvEPA
United States
Environmental Protection
Agency
Office of Research and Development
National Exposure Research Laboratory
Environmental Sciences Division
P.O. Box 93478
Las Vegas, Nevada 89193-3478
Official Business
Penalty for Private Use
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EPA/600/R-03/027
November 2003
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