Theory of Sampling

The quality of an analytical result is not only about laboratory analysis – it is just as much about what comes before: The Theory and Practice of Sampling of heterogeneous materials and processes (TOS) to the rescue.

The common characteristic of all naturally occurring as well as all technological and industrial resource materials (rocks, alloys, biomass and environmental samples, aggregates, mineralisations, ores, concentrates) is heterogeneity, which has a far more complex spatial distribution than can be captured by classical statistics. While lots and materials seem to differ without restriction, it turns out that exactly the same representative sampling principles can be applied to lots of all sizes, shapes, compositions and at all scales, in fact only based on TOS’ proven and effective means of counteracting the effects of heterogeneity in the sampling process. TOS offers a superior introduction to the detrimental aspects of heterogeneity – and what can be done about it. You can find out more in the article “Materials Properties: Heterogeneity and Appropriate Sampling Modes” (PDF).

1. TOS – A complete foundation

The Theory of Sampling (TOS) provides a complete description of heterogeneity and all error types occur in sampling of materials and processes as well as the necessary tools for their evaluation, elimination and/or minimization.

The foundation of the Theory of Sampling (TOS) provides the necessary-and-sufficient principles to understand all critical sampling error components from lot-to-analysis. These principles provide a complete basis to identifying faulty, inefficient, or suboptimal sampling procedures – and what opportunities exist for improvement.

Theory of Sampling (TOS) Systems approach. TOS is comprised by six Governing Principles (GP) (top grey panel), four Sampling Unit Operations, SUO (bottom yellow panel) and a rule set for sampling error management (blue/maroon). Various sub-sampling procedures that usually take place in the analytical laboratory are enclosed in a rectangle. SIX-S’ system’s approach allows these to be understood as but scaled-down SUO in the context of the full lot-to-aliquot pathway.

Correct theoretical competence (TOS) and appropriate sampling equipment are imperative for guaranteed representative sampling at all stages from lot to aliquot, whether by extracting physical samples or by PAT sensor signal acquisition – both cases are ruled and governed by the Theory of Sampling (TOS).

Correct delineation and extraction of sampling increments is the primary success factor for all process sampling, shown here, together with illustrations of many incorrect increment definitions which give rise to biased sampling.

2. The Sampling Process

A set of six Governing Principles (GP) and four Sampling Unit Operations (SUO) cover all necessary practical aspects of representative sampling and provides a comprehensive framework/toolbox for frontline plant and field staff, process engineers, laboratory personnel, quality units as well as supervisors and management who need to make critical decisions based on valid analytical results.

The key technical issue is that a sampling bias is of a fundamentally different nature to the well-known analytical bias, which unfortunately defeats any attempts to ‘bias-correction’ in sampling. Instead, TOS offers a number of practical ways to achieve “sampling correctness” (unbiasedness) through sound understanding, design and implementation of the correct types of equipment in the sampling procedures. TOS provides a comprehensive insight into how to ensure that all primary sampling and all subsequent sub-sampling (splitting) and sample preparation prior to analysis, can be documented as representative (procedures, equipment, maintenance).

After the critical primary sampling step, correct (unbiased) mass reduction/sub-sampling in the laboratory is equally important in order to ensure valid analysis analytical results. It is often unknown and therefore often neglected, that the Total Sampling Error (TSE) is by far the largest contributor to the total Measurement Uncertainty (MU) and is often 10-50 times larger than the Total Analytical Error (TAE). TOS deals with both steady-state lots as well as dynamic process lots and places particular emphasis on conducting experiments to estimate and characterise heterogeneity (replication experiments and variographic experiments). The most recent professional introduction to TOS is a well-received textbook “Introduction to the Theory and Practice of Sampling”.

3. Replication experiment

See Article “Sampling quality assessment: the replication experiment” (PDF).

4. DS 3077

See Article “There are standards—and there is the standard” (PDF).