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# Sampling of discrete materials a new introduction to the theory of sampling

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flowing streams. It is the only one that is probabilistic and easily rendered correct (Fig. 7). Figs. 8 illustrates the taking of a fraction of the stream for the whole of the time. It is obviously not probabilistic. m of particulate matter. Preview of Part III, here only used to facilitate a ch is shown here in three different alternatives: stopped-belt, uni-directional proportion is always unknown. The grade aS is an estimator of aL. Fig. 7. The principle of correct cross-stream sampling: taking all of the flow for a fraction of the time; shown here are two correct increments. P. Gy / Chemometrics and Intelligent Laboratory Systems 74 (2004) 7\u201324 13 5 Analytical result aR: this is an estimate of aS and hence, in turn, also an estimator for aL. The analytical result is used as the final estimate of aL. proportion of component A in sample S. This Figs. 9 illustrates the taking of a fraction of the stream for a fraction of the time. It also is obviously neither probabilistic nor correct. 8. Sampling errors\u2014the Global Estimation Error (GEE) 5 Component A: Let A be a certain component of interest. This must be a physical entity that can be isolated or observed without any alteration of the material, e.g. a mineral\u2014as opposed to the entity a bchemical componentQ, e.g. such as a metal in the same mineral, the proportion of which is estimated by analysis. In the size analysis of an aggregate for example, component A can be a certain size fraction of interest, e.g. the material which remains between, say, the 10- and 5-mm screens. In a moisture analysis, component A is the free, adsorbed water as opposed to the constitutional water that belongs to the lattice structure of the minerals present. 5 Grade a: mass proportion, defined as follows (for solids): a ¼ Mass of component A in a given object Total mass of dry solids in the same object (when a is the moisture content of the object, this proportion is usually expressed with the mass of wet solids in the denominator instead of that of dry solids). o Grade aL of lot L: true, well-definable proportion of component A in lot L. This proportion is always unknown. It is also sometimes called the bcritical content of LQ. The purpose of the entire sampling evaluation process is to estimate aL. o Grade aS of sample S: true, well-definable Fig. 8. Example of non-probabilistic, hence incorrect sampling of a one-dimen section) of the flow, even though extracted all of the time. This has a critic Technologies), which often suffer from exactly this incorrectness. Relative errors are often easier to use and to compare than absolute errors, and this fact will be used extensively in the exposition of the theory of sampling in all of what follows below. Several essential defi- nitions now need to be introduced (below a star introduces the name and notation of a new sampling error): * Global Estimation Error (GEE): this is the relative difference between the analytical result aR and the actual, well-defined but unknown, value of aL GEE ¼ aR \ufffd aL aL ð1Þ 5 Properties of GEE: Above, it was mentioned that quality estimation is a sequence of two error-generating groups of operations: sampling and analysis, which implies: Global Estimator Error GEE ¼ Total Sampling Error TSE þ Total Analytical Error TAE ð2Þ * Total Sampling Error (TSE), defined as follows: TSE ¼ aS \ufffd aL aL ð3Þ * Total Analytical Error (TAE) defined as follows: TAE ¼ aR \ufffd aS aL ð4Þ 5 Properties of TSE: For all purposes, sampling oper- ations can in general be broken up into two main stages, primary and secondary, which generate two main groups of errors: the primary error\u2014extraction of a laboratory sample S1 from the lot L; and the secondary error\u2014extraction of an assay portion from sample S1 (potential subdivision of the sampling process in more stages than two, if needed in practice, sional body or stream. It is incorrect only to take a fraction (of the cross- al bearing on many currently popular PAT schemes (Process Analytical a one- m a f nt La is trivial and not needed for the present introductory theoretical understanding): Total Sampling Error TSE ¼ Primary Sampling Error PSE þ Secondary Sampling Error SSE ð5Þ * Primary Sampling Error (PSE), a random error defined as follows: PSE ¼ aS1 \ufffd aL aL ð6Þ * Secondary Sampling Error (SSE), a random error defined as follows: SSE ¼ aS2 \ufffd aS1 aL ð7Þ 5 Additivity of sampling and analytical errors. From the above definitions, we can easily deduce the following relationships, well-known from statistics: Global Estimation Error GEE ¼ PSEþ SSEþ TAE Additivity of Errors ð8Þ The three random error components of GEE are independent of each other with the following consequences (the mean, or expected value of a random error is the bias): m GEEð Þ ¼ m PSEð Þ þ m SSEð Þ þ m TAEð Þ Additivity of Biases ð9Þ r2 GEEð Þ ¼ r2 PSEð Þ þ r2 SSEð Þ þ r2 TAEð Þ Additivity of Variances Fig. 9. Another example of non-probabilistic, hence incorrect sampling of Fig. 8, in which sampling is only taking place for a fraction of the time fro P. Gy / Chemometrics and Intellige14 ð10Þ 5 Remark: Experience shows that sampling biases and variances, especially those resulting from the prac- tical implementation of the theoretical model, can be much, much larger than analytical biases and variances. In our activities as a consultant and troubleshooter we have met primary sampling biases as large as 1000% (relative) and secondary sam- pling biases as large as 50%, whereas analytical biases do not usually exceed 0.1\u20131%. Exceptions is both a professional as well as a scientific insult to the integrity of the overall quality estimation process! n Proper sampling of discrete materials and more specifically particulate solids is, however, the object of a complete theory that has never been are especially in the domain of trace and ultra-trace concentrations, etc. 5 Sampling vs. analysis-first conclusions: In view of the above definitions, we hope that the reader will accept the following first conclusions: n Sampling requires at least as much care (and investment) as analysis, which is unfortunately all too often overlooked. Why? Analytical chemistry is taught extensively at practically all universities worldwide\u2014but the theory of sampling is not. (There exist only a few notable exceptions that the author is aware of, such as at Lappeenranta Technological University, Finland and 2lborg University Esbjerg, Denmark, as well as at the Powder Science and Technology group [POSTEC], Porsgrunn in Norway). Sampling would seem to fall in a no-man\u2019s land between several university departments and thus remains ignored by practically all. When confronted, all bresponsibleQ authorities give the same, universal answer\u2014that sampling is somebody else\u2019s prob- lem. Whose problem? Analysts (chemical, other) are all brought up academically to be totally convinced that their task first begins when the proverbial blaboratory sampleQ is received and then processed as best as they can, which usually is very well. When specifically asked however, they usually don\u2019t know how it has been obtained and, which is worse, they usually don\u2019t care. This dimensional body or stream. This is but a variant of the situation depicted in raction of the stream. boratory Systems 74 (2004) 7\u201324 seriously contested in 50 years! More than 120 articles in various European languages and 9 books in French and English have been published on the theory of sampling by this author (clandestine translations into Russian and Japanese also exist). Various other authors have also written books and numerous articles about this theory. Is the theory ignored because it is new? Hardly\u2014the first paper was written in 1950! A chronology of sampling theory development (for the first time ever) is presented in part IV of this series. course. n The same person should be in charge of ent La sampling both in bthe fieldQ, e.g. in