Introduction to the Theory and Practice of Sampling

Score: 5.00 (votes: 4)
Reviews: 4

This book presents the Theory and Practice of Sampling (TOS) starting from level zero in a novel didactic framework without excessive mathematics and statistics. It represents 20 years of teaching experience which has developed into a unique conceptual framework with which the TOS’ six principles and four unit operations can be understood in a unifying manner, enabling the reader to start sampling in a correct fashion right away. The book covers sampling from stationary lots, from moving, dynamic lots (process sampling) and has a vital focus on sampling in the analytical laboratory. It contains a wealth of complementing cases, examples and references (most of which are accessible on-line) meant to inspire and motivate the reader to individual skills-building and further self-study.

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“Sampling is not gambling”. Analytical results forming the basis for decision making in science, technology, industry and society must be relevant, valid and reliable. However, analytical results cannot be detached from the specific conditions under which they originated. Sampling comes to the fore as a critical success factor before analysis, which should only be made on documented representative samples. There is a very long and complex pathway from heterogeneous materials in “lots” such as satchels, bags, drums, vessels, truck loads, railroad cars, shiploads, stockpiles (in the kg–ton range) to the miniscule laboratory aliquot (in the g–µg range), which is what is actually analysed. Exactly how to acquire a documented, representative analytical result across mass-reduction of up to six orders of magnitudeof heterogeneous materials is far from a simple materials handling issue. There are specific principles and rules behind representativity. The TOS to the fore!

This book presents the Theory and Practice of Sampling (TOS) starting from level zero in a novel didactic framework without excessive mathematics and statistics. It represents 20 years of teaching experience which has developed into a unique conceptual framework with which the TOS’ six principles and four unit operations can be understood in a unifying manner, enabling the reader to start sampling in a correct fashion right away. The book covers sampling from stationary lots, from moving, dynamic lots (process sampling) and has a vital focus on sampling in the analytical laboratory. It contains a wealth of complementing cases, examples and references (most of which are accessible on-line) meant to inspire and motivate the reader to individual skills-­building and further self-study.

The book will teach you:

  • WHY sampling is much more than materials handling
  • WHY we need the Theory of Sampling (TOS)
  • HETEROGENEITY—the root of all evil
  • The Fundamental Sampling Principle (FSP)—which must never be broken
  • WHAT—and HOW TO conduct representative sampling
  • The TOS in the laboratory—sample splitting without errors 
  • The sampling bias—a fatal enemy that can be avoided
  • The TOS for sampling of stationary lots and materials
  • The TOS for sampling of moving lots and materials—process sampling
  • The TOS and business ethics, buyer–seller relationships, OEM obligations, societal needs
  • The TOS—Pro’s and Con’s
  • “The TOS will save you a lot of money”

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Customer reviews
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5.00 (Votes: 4)
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  • Peter Appel
    31-01-2020, 11:23
    Review by Peter Appel
    AppelGlobal (

    It is interesting to observe how the science of analysis is continually developing more and more precise methods that can analyse smaller and smaller amounts of a target compound. Historically determinations obtained from analytical laboratories were first in percent, later in ppm, followed by ppb, and most recently in ppt. However, these scientific and technical advancements were not followed up by thoughts and demonstrations on how we can obtain reliable results of concentrations in the large amounts of original material, the so-called lots.

    Kim Esbensen’s book is presenting a new, valuable introduction in the form of a comprehensive overview of the principles and unit operations behind documentable, representative sampling from heterogeneous lots typically 1:103–1:106 larger than the primary samples, and far larger still of the analytical aliquots. This book introduces the Theory of Sampling (TOS) in a novel didactic fashion, which will be of wide interest.

    My first encounter with Kim Esbensen was several years ago when he invited himself to give a presentation of Theory and Practice of Sampling (TOS) in the department Economic Geology of the Geological Survey of Denmark and Greenland where I worked then. Together with my colleagues I listened politely to his presentation, but found that this whole Theory of Sampling (TOS) approach to sampling was way too time consuming and thus not really appropriate for our work as practical field geologists (but yes, perhaps in the sample-processing stages in the laboratory, which was, luckily, “somebody else’s problem”).

    However, some years after this presentation, I embarked on a project where I realised that TOS was in fact the only way to obtain scientifically reliable results. I participated in a global project finding ways to clean millions of tons of tailings from small-scale gold mining. These tailings contain large amounts of mercury and gold, but contents are in the very low ppm range, e.g. 3–150 ppm, while tailings tonnages are enormous, 3–5 tonnes. Mercury released from these tailings account for 37 percent of the global mercury pollution! In order to evaluate the efficiency of a newly developed mobile processing plant, I had to know the exact ultra-trace concentrations in the tailings. This was crucial for evaluating the efficiency of success extraction these two metals from the overwhelmingly large tailing tonnages. This was when I suddenly found that the usual geological sampling approach was totally inadequate for this purpose, but where only the TOS could outline a valid procedure. The results of this project, and others, have been published,1 where the details of this “barefoot sampling” approach can be found.

    This new introductory book has many excellent chapters explaining why and how to obtain proper representative results of the content of a particular element or group of elements in significantly heterogeneous materials, rocks, ores and tailings (typical lots in my line of work), but one of the book’s strongpoints is that is shows the common nature of proper sampling. For example, TOS is equally important in investigating the pollution impacts from mining operations or from dumping of industrial tailings of various sorts. In fact, at first it was shocking to read that “from the point of view of sampling, all materials are identical, it is only a matter of whether their heterogeneity is small, medium or large”, after which TOS’ claimed universal principles made much more sense. I am now slightly embarrassed over my first dismissal of TOS, which sadly, is rather typical within geology. Take notice!

    Thus my subsequent work using TOS was really an eye-opener. Upon reflection, and the above critical use of TOS in practice, this book should be compulsory in educating not only geology students at universities, but also scientists and technicians in a number of other disciplines, not least analytical chemistry (there is a whole world outside the four walls of the laboratory). All will benefit fundamentally from addressing this book.

  • Leo Simon Morf
    31-01-2020, 11:22
    Review by Dr Leo Simon Morf
    Deputy Head of Waste Management Department, Canton Zurich, Switzerland
    Head of Technical Advisory Committee, Swiss foundation development center for sustainable management of recyclable waste and resources (ZAR)
    Lecturer for waste management techniques at ETH Zurich, Switzerland

    Investment decisions in the mining industry, or raw material delivery prices, depend critically on reliable knowledge of content of the substances that determine the value of the products and goods. In this industry sector, it has since long been known that the precision of an analytical result, whether in the exploitation of a deposit or the characterisation of a ship’s cargo, is less dependent on the analytical modality (e.g. high-precision gravimetric methods or high-resolution, accurate analytical instruments) than on the question of whether it is based on representative samples. For this reason, representative (accurate and precise) sampling and sample processing techniques have been developed by the sampling community, specifically by the Founder of the Theory of Sampling (TOS) Pierre Gy. This development started no longer ago than in 1950.

    In the second half of the previous century, a large proportion of the valuable substances in geogenic deposits has been extracted and are now to be found in the material inventory of industrialised countries, whether it be in infrastructure, consumer and investment goods (sometimes characterised as “Urban Mines”). On the other hand, our modern, highly-developed society demands ever more complex products with more and more components; these include not only valuable, recyclable materials such as precious metals, rare earths, but also increasing amounts of new, potential pollutants, often also toxic, e.g., flame retardants, nanomaterials. It is therefore not surprising that waste flows out of these mines contain both potentially hazardous substances and potentially recyclable materials that warrant development of new technologies for their utilisation. In this contemporary waste management context, in essence quite similar to geogenic raw materials, these resources should be explored and characterised using suitable analytical procedures, and the finished preparation products derived therefrom (such as metal fractions from municipal solid waste incineration plant MWIP bottom ashes) should absolutely be considered for emerging commercialisation, while hazardous substances should get destroyed or immobilised. In this challenging field of interrelated urban mining, sorting, refinement, recycling and safe waste deposition representative sampling play a critical key role also in the emerging paradigm of circular economy.

    Surprisingly, in the global waste management sector, the Theory of Sampling has yet barely been considered or applied consistently. The Danish Standard DS 3077 for representative sampling, the facto international standard, is not known or applied in practice in this sector so far. Combined with potential errors due to incorrect sampling techniques and inadequate sample preparation (as may arise from “simplifications” in both steps), this will unavoidably lead to unnecessary errors in the determination of the quantities of pollutants, toxic components or recyclable materials, and will thus hamper optimal estimation and assessment of both ecological risks and economic opportunities. Applying TOS approaches it has recently been demonstrated that highly significant, but quite unnecessary, uncertainties accumulate along the “lot-to-aliquot” pathway, which is extraordinarily complex within waste management and circulatory economy.

    Prof. Esbensen’s new comprehensive book Introduction to Theory and Practice of Sampling closes an important general gap, also for users in the waste management sector. This book is critically necessary for application of professional sampling procedures concepts and techniques in modern waste management systems—and beyond. This book shows in an unusually refreshing, simple and clear way all the relevant concepts and connections between the General Principles of sampling and the powerful concept of Sampling Unit Operations, and it demonstrates the often large quantitative influences on the final uncertainty of the tiny analytical samples that ultimately are delivered to the laboratory. A very useful principle is that all mass-reduction, sample splitting, sub-sampling and sample preparation operations can be treated identically as primary sampling, only at smaller and smaller scales. This understanding helps building a necessary overview of TOS.

    One of the book’s major advantages is the lavish use of carefully designed didactic diagrams which help the reader to form his/her own understanding of what turns out not to be so “complex” subject-matter as many claim. The many practical examples are vivid and valuable with which also better to understand TOS’ systematic interconnections.

    I am convinced this book will find great interest and application among individual actors, companies, organisations and other stakeholders in many application sectors in science, technology and society.
  • Allan Trebbien
    31-01-2020, 11:22
    Review by Allan Trebbien
    B.Tech Management & Marine Engineering

    For a reader like me, a newcomer to the field, this new book gives the reader a well-crafted overview of how to get your sampling right from the start—and what it will cost you if you don’t. Combined with a first foray of practical highlights from the theory behind it all, the Theory of Sampling (TOS), it also supplies a lot of interesting background.

    From the start, the author delivers a book written in a somewhat unorthodox fashion that makes it very easy to read, complete with many detailed examples and case histories. This book may well end up being the standard introduction sourcebook for representative sampling.

    Having read the entire book, the reader is well equipped to start performing sampling in an effective and professional way. The reader will be fully able to appreciate the Danish standard DS3077 “Representative sampling – Horizontal standard (2013)”, the only international standard on this complex subject and a very large trove of literature (and internet) sources as well. This book delivers the necessary understanding for how to perform representative sampling across a great many types of materials, under nearly all conditions of interest in technology and industry.

    It comes highly recommended.
  • Gary Ritchie
    31-01-2020, 11:20
    Review by Gary Ritchie, MS
    Principal Consultant, GER COMPLIANCE
    Council for Near Infrared Spectroscopy News Editor
    QC/QA GMP Remediation, Training, Audits, Data Analytics, Chemometric/Spectroscopy Remote Analysis
    Former Director of Operations and Regulatory Affairs Analyst for Dynalabs, LLC, St Louis, MO, USA
    Former Director of Scientific Affairs for Infratrac, Silver Spring, MD, USA
    Former Scientific Fellow for Process Analytical Technology (PAT) with the United States Pharmacopeia (USP)
    International expert in multivariate spectroscopic analysis, pharmaceutical analysis, Good Manufacturing Practices compliance, laboratory management, design, quality, and process analytical technology using spectroscopic methods and multivariate analysis

    How to read this book?
    The reader is encouraged to read the first two chapters, followed by Chapters 22–24, and only then continuing from Chapter 3. In this fashion the reader is promised guaranteed optimal conditions with which to achieve all the promised learning goals. Having read the book in the manner recommended above, I have the following comments.

    Does this book work as intended?
    I thoroughly enjoyed the conversational style in which the book is written. I imagine that I could very well be listening to Pierre Gy himself as he presents his Theory of Sampling for the very first time. If the author intended for the reader to experience an introduction to the TOS as if he/she was the first to ever hear it, the book has achieved its objective well.

    What is good?
    Far from being a conventional cut and dry textbook, the informal and didactic style adopted allows the student-reader to learn at an easy and suitable pace. I followed the recommendation on how to read the book; I read the first two chapters, followed by Chapters 23–25, and only then continuing from Chapter 3; again. This works well because once read in this manner, a self-paced cadence for further instruction and insight into the references provided was easily achieved.

    There is a vast amount of additional text offered in the book (directly clickable sidebars),which can at first make it somewhat difficult to determine what is essential versus what is more for extra information. However as one moves along, these sidebars prove to be important locations for necessary reinforcement and references for further reading. The main text provides much scientific insight and color and additionalities as well, but it is precisely this mix that gives the reader-student the sense of discovering the TOS for the very first time.

    General assessment
    The take away lesson from this book is clear, and very unsettling.

    “...hidden economic losses due to faulty decisions based on faulty analytical results due to non-representative sampling.”

    These words should give every analytical chemist pause for concern. After all, we have been taught and we believe that our analytical results are based on a sample that must be representative, for why else are we testing it? And then, when/if the result fails the acceptance criteria, we are told, after a proper investigation, to test it again. It has become the focus of discussions: just what is a sample really and what is it representative of? This delves right into the realm of the Theory of Sampling (TOS). While analytical testing is in and of itself a science rooted in the natural sciences and mathematics, here now comes the TOS that teaches us that homogeneity of the aliquot isn’t the goal, but representativity of the primary sample (and all sub-samples hereof) is, and that sample size matters and sampling error goes beyond conventional statistics. It is all about the sampling process, and here the book’s strongpoint is that all major points made can easily be understood within the framework of TOS’ six fundamental sampling principles and four sampling unit operations.

    In Introduction to the Theory and Practice of Sampling, by Professor Esbensen, with contributions from several well-known and respected experts on the subject, the TOS is presented in an easy, comprehensible style that is assessible to everyone at all academic levels that allows readers, students and practitioners to learn very quickly a fundamental fact of nature that can no longer afford to be unknown or ignored: homogeneous materials do not exist in the real world; every material in technology and industry is heterogeneous. Hence, the flaw in our current thinking and practice of treating “analytical grab samples” as if they always are homogeneous and representative, turns out to be the single biggest source of economic loss to most of the modern world’s industrial endeavors.

    I recommend this book to all newcomers to TOS, but especially also to those who want to go beyond the TOS basics and further explore its numerous literature sidebars and background references. For in depth coverage of the quantitative nature of sampling theory and practice, this is the place to start. I feel the opportunity to review this book is an extension, and culmination, of my role when as a scientist at the United States Pharmacopeia about ten years ago, as the perfect time and place to get the TOS in front of the key committee members who would go onto incorporate the Theory of Sampling into the United States Drug Compendium.

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1 Theory of Sampling (TOS)—the missing link before analysis
1.1 A framework for representative sampling
1.2 What comes before analysis—the TOS!
1.3 What this book promises…
1.4 References
2 Theory of Sampling (TOS)—fundamental definitions and concepts
2.1 Lot dimensionality
2.2 Sampling terminology—the tower of Babel
2.3 References
3 Heterogeneity—the root of all evil (part 1)
3.1 Introduction to the concept of sampling errors (excerpt from DS3077)
3.2 Heterogeneity—the basics
3.3 Materials, sampling targets and lots
3.4 Homogeneity–heterogeneity
3.5 Scale
3.6 Heterogeneity vs sampling
3.7 References
4 Heterogeneity—the root of all evil (part 2)
4.1 Introduction
4.2 Constitutional Heterogeneity (CH)
4.3 Distributional Heterogeneity (DH)
4.4 Heterogeneity vs practical sampling
4.5 “Structured heterogeneity”
4.6 The fundamental insight on how to counteract heterogeneity
4.7 References
5 “Sampling—is not gambling”
5.1 Introduction
5.2 Enough analogy
6 Pierre Gy’s key concept of sampling errors
6.1 Rational understanding of heterogeneity and appropriate sampling
6.2 Although complex, there is hope
6.3 How to sample representatively: the TOS
6.4 References
7 Composite sampling I: the Fundamental Sampling Principle
7.1 WHAT TO DO with all this heterogeneity?
8 Composite sampling II: lot dimensionality transformation
8.1 1-D lots: conveniently elongated lots
8.2 Process sampling
8.3 Process sampling generalised
8.4 Q
8.5 Lot dimensionality transformation (LDT)
8.6 References
9 Sampling quality assessment: the replication experiment
9.1 Background
9.2 Clarification
9.3 Quantifying total empirical variability—the replication experiment
9.4 Relative sampling variability
9.5 Notes and references
10 Sampling quality criteria
10.1 Sampling quality criteria
10.2 First SQC component—definition of analyte(s)
10.3 Second SQC component—delineating the decision unit (DU)
10.4 Third SQC component—inference and confidence
10.5 Perspectives
10.6 Summary
10.7 References
11 There are standards—and there is the standard
11.1 First light
11.2 Analysis of sampling standards for solid biofuels
11.3 Analysis of grain sampling guide
11.4 Sampling for GMO risk assessment
11.5 Examples of too glib recommendations
11.6 TOS competence is crucial
11.7 Que faire?
11.8 DS 3077 Horizontal—a new standard for representative sampling. Design, history and acknowledgements
11.9 Chapter references
12 Spear sampling: a bane at all scales
12.1 Introduction
12.2 Spear sampling—at all scales
12.3 Not always bad—there is hope
12.4 Conclusions
12.5 References
13 Into the laboratory… the TOS still reigns supreme
13.1 Representative sampling—a scale invariant endeavour
13.2 Size does not matter—only heterogeneity, and how to counteract it
13.3 And there is more to be done in the lab ...
13.4 Further reading
14 Representative mass reduction in the laboratory: riffle splitting galore
14.1 Introduction
14.2 Riffle splitting
14.3 Automation—enter the rotary divider
14.4 Benchmark study
14.5 The ultimate method/equipment ranking for the laboratory
14.6 Conclusions
14.7 References
15 Introduction to process sampling
15.1 Lot dimensionality: ease of practical sampling
15.2 Lot dimensionality transformation
15.3 Process sampling
15.4 1-D lot heterogeneity
15.5 Variographic analysis: a first brief
15.6 Interpretation of variograms
15.7 References
16 Process sampling: the importance of correct increment extraction
16.1 Moving, or static, 1-D lots: increment cutting must be TOS-correct
16.2 “Sooner or later”…
16.3 References
17 The variographic experiment
17.1 The variogram
17.2 References
18 Experimental validation of a primary sampling system for iron ore pellets
18.1 Introduction: status of current ISO standards
18.2 Fundamental Sampling Principle and basic requirements for iron ore sampling systems
18.3 Principles and general requirements for checking sampling bias
18.4 Validation experiment
18.5 Experimental results
18.6 Discussion
18.7 References
19 Industrial variographic analysis for continuous sampling system validation
19.1 Variographic analysis
19.2 Continuous control of sampling systems
19.3 9–12.5 mm size fraction of iron ore pellets
19.4 Specific surface area of magnetite slurry
19.5 Iron grade in magnetite slurry
19.6 Conclusions
19.7 Acknowledgements
19.8 References
20 Theory of Sampling (TOS): pro et contra
20.1 A powerful case for the TOS in trade and commerce
20.2 Cases against the TOS (science, technology, commerce, trade)
20.3 Important reading with which to catch the attention of newcomers to the TOS
21 Following the TOS will save you a lot of money (pun intended)
21.1 Case 1: Always mind analysis
21.2 Case 2: Saving a client from a wrong, expensive investment
21.3 Case 3: The hidden costs—profit gained by using the TOS
21.4 Case 4: The cost of assuming standard normality for serial data
21.5 Lessons learned
21.6 References
22 A tale of two laboratories I: the challenge
22.1 Introduction (scientific, technological)
22.2 There is analysis… and there is analysis+
22.3 The core issue
22.4 The crux of the matter
22.5 The complete argument
22.6 The meaning of it all
22.7 Inside and outside the complacent four walls of the analytical laboratory
22.8 “One fine day”…
22.9 The really important aspect: costs or gains
22.10 What in the world?
22.11 References
23 A tale of two laboratories II: resolution
23.1 Epiphany interpretation I: knowingly closing one’s eyes or not?
23.2 Epiphany interpretation II: the economic dilemma
23.3 Epiphany interpretation III: the moral resolution
23.4 Laboratory B’s new vision and mission
23.5 Can this really lead to increased commercial success?
23.6 Acknowledgements
23.7 References
24 Sampling commitment—and what it takes…
24.1 Historical context
24.2 Awareness
24.3 Minimum competence level
24.4 Vade mecum
24.5 Trouble with some standards
24.6 In practice…
24.7 What could be argument(s) against …
24.8 Practice, practice, practice…
24.9 The last word
24.10 References
24.11 Further reading (a first selection)
25 Representative sampling and society
25.1 Sampling: from the point of view of buyers, consumers, citizens
25.2 The way forward: some proposals
25.3 Beyond traditional application fields
25.4 Conclusions
25.5 References
26 Epilogue: what’s next?

About the author

Kim H. Esbensen, PhD, Dr (hon), research professor in Geoscience Data Analysis and Sampling at GEUS, the National Geological Surveys of Denmark and Greenland (2010–2015), chemometrics & sampling professor at Aalborg University, Denmark (2001–2015), professor (Process Analytical Technologies) at Telemark Institute of Technology, Norway (1990–2000 and 2010–2015) and professeur associé, Université du Québec à Chicoutimi (2013–2016). From 2015 he phased out a more than 35-year academic career for a quest as an independent consultant: However, as he could not terminate his love for teaching, he is still on a roll as an international visiting-, guest- and affiliate professor around the world.

A geologist/geochemist/data analyst by training, he has been working for three decades at the forefront of chemometrics, but since 2000 he has devoted most of his scientific R&D to the theme of representative sampling of heterogeneous materials, processes and systems (Theory of Sampling, TOS), and PAT (Process Analytical Technology). He is a member of several scientific societies, has published over 250 peer-reviewed papers and is the author of a widely used textbook, Multivariate Data Analysis (35,000 copies), published in its 6th edition in 2018. He was the originator and chairman of the taskforce behind the world’s first horizontal (matrix-independent) sampling standard DS 3077 (2013). He is editor of the science magazine TOS forum ( and for the Sampling Column in Spectroscopy Europe/Asia (

Esbensen is fond of the right kind of friends and dogs, swinging jazz, fine cuisine, good wine, contemporary art and classical music. His has been collecting science fiction novels for more decades than he is comfortable contemplating, still, as ever, it’s all in the future…

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