Arne Noe-Nygaard, Danish geologist (1908‒1991) Noe-Nygaard was a Nestor in Danish and Scandinavian geology through a long and very productive academic career. He was a professor for 40 years, also widely involved in popularising geology and was intimately involved in…
Along the full lot-to-analysis pathway Analytical measurements comprise at least two error generating steps: delineating and extracting the primary sample, and analysis of the analytical aliquot. There may be several sub-sampling steps before having a sufficiently small aliquot (analytical sample)…
The purpose of sampling is to extract a representative amount of material from a ‘lot’ – the ‘sampling target’. It is clear that sampling must and can only be optimized before analysis. In a recent paper, we show how non-representative…
An aliquot is the ultimate sub-sample extracted in a 'Lot-to-Aliquot' pathway for analysis. By analogy, process analytical technology involves the extraction of virtual samples, which are defined volumes of matter interacting with a process analytical instrument.
Analysis is the systematic examination and evaluation of the ultimate sub-sample of chemical, biological, or physical substance (Aliquot) to determine its composition, structure, properties, or presence of specific components.
Analytical bias is a systematic deviation of measured values from true values. An analytical bias can arise from multiple sources, including instrument calibration errors, sample preparation techniques, operator method, or inherent methodological limitations. Unlike random errors, which fluctuate unpredictably, analytical bias consistently skews results in a particular direction. Identifying and correcting this bias is crucial to ensure the accuracy and reliability of analytical data (bias correction).
Analytical precision refers to the degree of agreement among repeated analyses of the same aliquot under identical conditions. It reflects the consistency and reproducibility of the results obtained by a given analytical method. High precision indicates minimal random analytical error and close clustering of analytical results around an average. Precision does not necessarily imply accuracy, as a method can be precise yet still yield systematically biased results.
Composite sampling extracts a number (Q) of Increments, established to capture the Lot Heterogeneity. Composite sampling is the only way to represent heterogeneous material. A composite sample is made by aggregating the Q increments subject to the Fundamental Sampling Principle (FSP). The required amount of increments for the requested Representativity Q can be carefully established to make sampling fit-for-purpose.
Compositional heterogeneity is the variation between individual fundamental units of a target material (particles, fragments, cells, ...). CH is an intrinsic characteristic of the target material to be sampled.
Data must be reported as the measurement results and the Measurement Uncertainties stemming from sampling and analysis. Note that MUAnalysis and MUSampling are expressed as variances.
Data = Measurement +/- (MUSampling ; MUAnalysis)
Example: 375 ppm +/- (85 ppm ; 18 ppm)
Note that the Uncertainties 85 ppm and 18 ppm are the square roots of MUSampling and MUAnalysis.
Distributional heterogeneity is the variation between groups of fundamental units of a target material. Groups of units manifest themselves as Increments used in sampling. DH is an expression of the spatial heterogeneity of a material to be sampled (Lot).
This standard is a matrix-independent standard for representative sampling, published by the Danish Standards Foundation. This standard sets out a minimum competence basis for reliable planning, performance and assessment of existing or new sampling procedures with respect to representativity. This standard invalidates grab sampling and other incorrect sampling operations, by requiring conformance with a universal set of six Governing Principles and five Sampling Unit Operations. This standard is based on the Theory of Sampling (TOS).
A dynamic lot is a moving material stream where sampling is carried out at a fixed location. For both Stationary Lots and Dynamic Lots, sampling procedures must be able to represent the entire lot volume guided by the Fundamental Sampling Principle.
FSE results from the impossibility to fully compensate for inherent Compositional Heterogeneity (CH) when sampling. FSE is always present in all sampling operations but can be reduced by adherence to TOS' principles. Even a fully representative, non-biased sampling process will be unable to materialise two samples with identical composition due to Lot Heterogeneity. FSE can only be reduced by Crushing (followed by Mixing / Blending) i.e. by transforming into a different material system with smaller particle sizes.
The Fundamental Sampling Principle (FSP) stipulates that all potential Lot Increments must have the same probability of being extracted to be aggregated as a Composite Sample. Sampling processes in which certain areas, volumes, parts of a Lot are not physically accessible cannot ensure Representativity.
The GEE is the total data estimation error, the sum of the Total Sampling Error (TSE) and the Total Analytical Error (TAE).
Six Governing Principles (GP) describe how to conduct representative sampling of heterogeneous materials:
1) Fundamental Sampling Principle (FSP)
2) Sampling Scale Invariance (SCI)
3) Principle of Sampling Correctness (PSC)
4) Principle of Sampling Simplicity (PSS)
5) Lot Dimensionality Transformation (LDT), and
6) Lot Heterogeneity Characterisation (LHC).
Process of extracting a singular portion of the Lot. Grab sampling cannot ensure Representativity for heterogeneous materials. Grab sampling results in a sample designated a Specimen.
The GSE originates from the inherent tendency of Lot particles, or fragments hereof, to segregate and/or to group together locally to varying degrees within the full lot volume. This spatial irregularity is called the Distributional Heterogeneity (DH). There will always be segregation and grouping of Lot particles at different scales. GSE plays a significant role in addition to the Fundamental Sampling Error FSE. Unlike FSE however, the effects from GSE can be reduced in a given system state by Composite Sampling and/or Mixing / Blending. GSE can in practice be reduced significantly but is seldomly fully eliminated.
Heterogeneity refers to the state of being varied in composition. It is often contrasted with homogeneity, which implies complete similarity among components, which is a rare case. For materials in science, technology and industry heterogeneity is the norm. Heterogeneity applies to various contexts, such as populations of non-identical units, bulk materials, powders, slurries, biological swhere multiple distinct components coexist.
Heterogeneity in context of the Theory of Sampling, is described using three distinct characteristics, Compositional Heterogeneity CH, Distributional Heterogeneity DH and Particle-Size Heterogeneity
Heterogeneity tests are used for optimizing sampling protocols for a variable of interest (analyte, feature) with regards to minimising the Fundamental Sampling Error (FSE).
Experimental approaches available are the 50-particle method, the heterogeneity test (HT), the sampling tree experiment (STE) or the duplicate series/sample analysis (DSA), and the segregation free analysis (SFA).
Recently, sensor-based heterogeneity tests have been introduced which bring the advantage of cost-effective analysis of large numbers of single particles.
An assemblage of material units with identical unit size, composition and characteristics. There are practically no homogenous materials in the realm of technology, industry and commerce (mineral resources, biology, pharmaceuticals, food, feed, environment, manufacturing and more) of interest for sampling. With respect to sampling, it is advantageous to consider that all materials are in practice heterogeneous.
The principle for extracting correct Increments from processes is to delineate a full planar-parallel slice across the full width and depth of a stream of matter (Dynamic Lot. IDE results from delineating any other volume shape. When a sampling system or procedure is not correct relative to the appropriate Increment delineation, a Sampling Bias will result. The resulting error is defined as the Increment Delimitation Error (IDE). Similar IDE definitions apply to delineation and extraction of increments from Stationary Lots.
Increments must not only be correctly delimitated but must also be extracted in full. The error incurred by not extracting all particles and fragments within the delimitated increment is the Increment Extraction Error (IEE). IDE and IEE are very often committed simultaneously because of inferior design, manufacturing, implementation or maintenance of sampling equipment and systems.
Adverse sampling bias effects may occur for example during sample transport and storage (e.g. mix-up, damage, spillage), preparation (contamination and/or losses), intentional (fraud, sabotage) or unintentional human error (careless actions; deliberate or ill-informed non-adherence to protocols). All such non-compliances with the criteria for representative sampling and good laboratory practices (GLP) are grouped under the umbrella term IPE. The IPE is part of the bias-generating errors ISE that must always be avoided.
There are four ISE, which result from an inferior sampling process. These ISE can and must be eliminated.
IWE reflects specific weighing errors associated with collecting Increments. For process sampling, IWE is incurred when extracted increments are not proportional to the contemporary flow rate (dynamic 1-dimensional lots), at the time or place of extraction. IWE is often a relatively easily dealt with appropriate engineering attention. Increments, and Samples, should preferentially represent a consistent mass (or volume).
Fundamental unit of sampling, defined by a specific mass or correctly delineated volume extracted by a specified sampling tool.
A legal person is any person or entity that can do the things a human person is usually able to do in law – such as entering into contracts or commit to specified obligations.
a) A Lot is made up of a specific target material to be subjected to a specified sampling procedure.
b) A Lot is the totality of the volume for which inferences are going to be made based on the final analytical results (for decision-making). Lot size can range from being extremely large (e.g. an ore body, a ship) to very small (e.g. a blood sample).
c) The term Lot refers both to the material as well as to lot size (volume/mass) and physical characteristics. Lots are distinguished as stationary or dynamic lots. A stationary lot is a non-moving volume of material, a dynamic lot is a material stream (Lot Dimensionality). For both stationary and dynamic lots, sampling procedures must address the entire lot volume guided by the Fundamental Sampling Principle (FSP).
Lot Definition describes the process of defining the target volume, which will be subjected to Sampling.
TOS distinguishes Lot volume according to the dimensions that must be covered by correct Increment extraction. This defines the concept of 'lot dimensionality', an attribute which is independent of the lot scale. Lot dimensionality is a characterisation to help understand and optimise sample extraction from any lot at any sampling stage. There are four main lot types: 0-, 1-, 2- and 3-dimensional lots (0-D, 1-D, 2-D and 3-D lots).
Lots are classified by subtracting the dimensions of the lot that are fully 'covered' be the salient sampling extraction tool in question. The higher the number of dimensions fully covered in the resulting sampling operation, the easier it is to reduce the Total Sampling Error TSE.
By the Governing Principle Lot Dimensionality Transformation LDT, stationary 0-D, 2-D and 3-D lots can in many cases advantageously be transformed into dynamic 1-D lots, enabling optimal sampling. However, the application of LDT has practical limits as some lots cannot be transformed (e.g. a body of soil, or a mine resource, biological cells). The optimal approach for such cases is penetrating one dimension with complete increment extraction (usually height) turning a 3-D lot into a 2-D lot.
The lot heterogeneity is the combination of Compositional Heterogeneity, Distributional Heterogeneity and Particle-Size-Heterogeneity.
CH + DH + PH
Mass-reduction is a physical process that divides a given quantity into manageable sub-samples. Mass-reduction must ensure that these sub-samples are representative of the original quantity (Representative Mass Reduction – Subsampling
The total process of producing numerical data about a Lot, including sampling and analysis is called Measurement. Simultaneously, sensor-based analytical technology combines virtual sampling and signal processing. For both types of measurements the principles and rules of the Theory of Sampling apply.
MU expresses the variability interval of values attributed to a quantity measured. MU is the effect of a particular error, e.g. a sampling error, or an analytical error or of combined effects (see MUTotal).
MUsampling reflects the variability stemming from sampling errors
MUanalysis reflects the variability stemming from analytical errors
MUtotal is the effective variability stemming from both sampling and analysis
MUtotal= MUsampling+ MUanalysis
Mixing and blending reduces Distributional Heterogeneity (DH) before sampling/sub-sampling. N.B. Forceful mixing is a much less effective process than commonly assumed.
PH is the compositional difference due to assemblages of units with different particle sizes (or particle-size classes).
The founder of the Theory of Sampling (TOS), Pierre Gy (1924--2015) single-handedly developed the TOS from 1950 to 1975 and spent the following 25 years applying it in key industrial sectors (mining, minerals, cement and metals processing). In the course of his career he wrote nine books and gave more than 250 international lectures on all subjects of sampling. In addition to developing TOS, he also carried out a significant amount of practical R&D. But he never worked at a university; he was an independent researcher and a consultant for nearly his entire career - a remarkable scientific life and achievement.
Precision is a measure of the variability of quantitative results. The larger the variability, the smaller the precision. In practice, precision is measured as the statistical variance s2 of the quantitative results (square of the standard deviation).
The initial mass extracted from the lot. The Primary Sample is the product of Composite Sampling and consists of Q Increments. Both the mass of the Primary Sample as well as the number of increments extracted influence the sampling variability. As the primary sampling stage often has by far the largest impact on MUTotal, optimisation always starts at this stage.
The Principle of Sampling Correctness (PSC) states that all TOS' Incorrect Sampling Errors (ISE) shall be eliminated, or a detrimental Sampling Bias will have been introduced.
PSS states that sampling along the Lot-to-Aliquot can be optimised separately for each (primary, secondary, tertiary ....) sampling stage. Since the Primary Sampling stage is often the dominant source of sampling error, optimization logically shall always begin at this stage.
PPE is incurred if short-, mid- or long-term periodic process behaviour is not corrected for, in which case it may contribute to a sampling bias.
A process sampling strategy must make use of a high enough sampling frequency to uncover such behaviours; the sampling frequency must as a minimum always be higher than twice the most frequent periodicity encountered.
PSE come into effect when Dynamic Lots are being sampled without compensating for process trends or periodicities (Process Trend Error and Process Periodicity Error).
PTE occurs if mid- to long-term process trends are not corrected for, in which case they may contribute to a Sampling Bias. PTE and Process Periodicity Error PPE may, or may not, occur simultaneously depending on the specific nature of the process to be sampled.
Number of Increments composited to a Sample.
R is the number of replications of a series of independent complete ‘Lot-to-Aliquot’ Measurements, made under identical conditions applied in a Replication Experiment.
Representative Mass Reduction (RMR) aka sub-sampling. TOS argues why Riffle-Splitting and Vezin-sampling are the only options leading to Representative Mass Reduction.
A sampling process is representative if it captures all intrinsic material features, e.g., composition, particle size distribution, physical properties (e.g. intrinsic moisture) of a Lot.Representativity is a characteristic of a sampling process in which the Total Sampling Error and Total Analytical Error have been reduced below a predefined threshold level, the acceptable Total Measurement Uncertainty.
Representativity is the prime objective of all sampling processes. The representativity status of an individual sample cannot be ascertained in isolation, if removed from the context of its full sampling-and-analysis pathway. The characteristic Representative can only be accorded a sampling process that complies with all demands specified by TOS (DS3077:2024).
Sampling is the process of collecting units from a Lot (sampling procedure; sampling process): Grab Sampling or Composite Sampling. There are only two principal types of sampling procedures: Grab Sampling or Composite Sampling.
Closeness of the analytical result of an Aliquot with regards to the true concentration of the Lot]/glossary]. NB. “sampling accuracy” = “sampling + analytical accuracy”
The Sampling Bias is the difference between the true Lot concentration and the average concentration from replicated sampling. Such a difference is a direct function of the Lot Heterogeneity and as such inconstant; it changes with each additional sampling and can therefore not be corrected for. This is the opposite to the Analytical Bias for which correction is often carried out.
SEM determines the priorities and tools for all sampling procedures in the following order:
The Sampling Manager is the Legal Person accountable for ensuring that all sampling activities are conducted in accordance with scientifically valid principles to achieve representative results. They are responsible for managing the design, implementation, and evaluation of sampling protocols while balancing constraints such as material variability, logistics, and resource limitations. This role requires expertise in the Theory of Sampling (TOS), leadership, project management and stakeholder communication skills.
The Sampling Precision is the variance of the series of analytical determinations, for example from a Replication Experiment (RE). Sampling precision always includes the Analytical Precision, since all analysis is always based on an analytical Aliquot, which is the result of a complete 'Lot-to-Aliquot' sampling pathway. Therefore sampling precision = sampling + analysis precision.
Document explaining the undertakings necessary for the sampling process. It contains the tools and procedures from Lot-to-Aliquot[/glossary].
The Principle of SSI states that all Sampling Unit Operations (SUO) can be applied identically to all sampling stages, only the scale of sampling tools differs.
Sampling Uncertainty is the difficulty of collecting a representative sample due to Lot Heterogeneity; the more heterogeneous the material, the higher the uncertainty associated with any sample attempting to represent the whole Lot.
A secondary sample is the product of Representative Mass Reduction - Subsampling from a Primary Sample. Identical nomenclature applies for further Representative Mass Reduction steps (Tertiary...).
A specimen is a portion of a larger mass/volume (Lot) extracted by a non-representative sampling process. Grab Sampling results in a specimen.
A Stakeholder is any entity interested in the result coming from sampling and analysis. Data representing stationary or flowing heterogeneous materials are requested by different parties with a multitude of differing objectives. Stakeholders can be internal, from commercial organisations, public authorities, research and academia or non-governmental organisations.
A Stationary Lot is a non-moving volume of material where sampling is carried at from multiple locations, each resulting in an Increment. For both Stationary Lots and Dynamic Lots, sampling procedures must address the entire Lot volume guided by the Fundamental Sampling Principle (FSP).
TOS Theory and Practice of Sampling: necessary-and-sufficient framework of Governing Principles (GP), Sampling Unit Operations (SUO), Sampling Error Management rules (SEM) together with normative practices and skills needed to ensure representative sampling procedures. TOS is codified in the universal standard DS3077:2024.
TAE is manifested as the Measurement Uncertainty resulting only from analysis (MUAnalysis). TAE includes all errors occurring during assaying and analysis (e.g. related to matrix effects, analytical instrument uncertainty, maintenance, calibration, other), as well as human error.
Whereas Measurement Uncertainty (MU) is traditionally only addressing analytical determination, e.g. concentration := 375 ppm +/- 18 ppm (MUanalysis), Theory of Sampling (TOS) stipulates reporting analytical results with uncertainty estimates from both sampling and analysis. This gives users of analytical data the possibility to evaluate the relative magnitudes of MUsampling vs. MUanalysis, enabling fully informed assessment of the true, effective data quality involved. A complete data uncertainty must have this format:
MUTotal = MUSampling + MUAnalysis
The attribute Total Measurement Uncertainty (MUTotal) is the most important factor determining the attribute data quality.
The Incorrect Sampling Errors (ISE) and Correct Sampling Errors (CSE) add up to the effective Total Sampling Error (TSE). TSE is causing the Total Uncertainty resulting from material extraction along the sampling pathway from-lot-to-aliquot (MUSampling).
The acceptable Total Measurement Uncertainty, which must include the Sampling Measurement Uncertainty (MUSampling) and Analytical Measurement Uncertainty (MUAnalysis).
Variography is a variability characterisation of a dynamic 1-dimensional dynamic lot. A variogram describes variability as a function of Increment pair spacing (in time). Variography is also applied in geostatisctics in describing the variability as a function of spacing/distance between analyses.
