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# Introduction to the theory of sampling

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Such low concentrations cannot therefore be ignored in a theoretical study such as this. A well-known pro- fessor publicly claimed, a few years ago, to be able to detect and quantify traces in the ppq range, i.e. lo-l5 g/g. This practically amounts to detecting and quantifying some three molecules in one gram of water. Let us assume that the analytical result is trends in analytical chemistry, vol. 14, no. 2, 1995 three. This would mean nothing because, owing to the very small number of atoms involved, the sta- tistical distribution is so skewed that the actual unknown number of atoms in one ml of water may be anywhere between zero and a very large number. Several papers have recently been published on single-molecule detection [ 6,7]. We wonder whether and how the sampling problem has been taken into account. This illustrates the limitations of trace anaivsis. Analysts who are interested in concentrations expressed as percentages probably believe they have nothing to do with such subtleties, but the researcher who is building up a theory is compelled to be very strict: there simply cannot be one defi- nition of heterogeneity in the percent range and another in the trace range. The theory of hetero- geneity has to be valid, irrespective of the concen- tration in the element we are looking for. Hence, for statistical reasons, the distribution of ions and molecules throughout a glass of water or beaker of solution, or that of any constitutive ele- ment throughout any batch of matter, must always be regarded as heterogeneous. To summarize the conclusions of the last two sections, no material may ever be regarded as con- stitution- or distribution-homogeneous. All mate- rials are more or less constitution- and dis- tribution-heterogeneous and must be dealt with as such. Homogeneity can and will be defined mathe- matically but these definitions require hypotheses that are unrealistic in the physical world. Homo- geneity can never be observed because we are deal- ing with matter which is made of discrete constitutive elements, irrespective of the observa- tion scale - molecules, ions, atoms, solid frag- ments, etc. or groups of such elements. To define and quantify constitution- and distri- bution-heterogeneity mathematically we must introduce the important distinction between two cases according to whether the batch of constitutive units makes up a population or a time-series. 0 The batch of constitutive units must be regarded as a \u2018statistical population\u2019 when the unit order is irrelevant or disregarded. This is what we call a \u2018zero-dimensional set\u2019. This case only will be dealt with in the current article. 0 The batch of constitutive units must be regarded as a \u2018series\u2019 when the unit order is relevant. This is what we call a \u2018one-dimen- sional set\u2019. The dimension involved is usually 69 the time and the series is a \u2018time-series\u2019. This model applies to every kind of flowing stream, a practically important case. 5. Heterogeneity carried by a constitutive unit of a set 5.1. fntroduction and nomenclature Our purpose is to quantify the heterogeneity of a batch (or lot) of matter in regard to all problems related to its heterogeneity and more specifically its sampling. We shall use the following notations. L Batch (lot) of matter composed of a set of discrete units which can make up either a population or a time-series. N, Number of these units in L, U,,, Unspecified unit of L with m= 1, 2 ,... N,. In this article L is regarded as a population. The subscript m is assumed to be attributed at random. The case of a time series, where m characterizes the chronological order, will be dealt with in another article. The unit U,, can be: 0 a well-defined element such as a particle (mineral fragment, vegetable grain, ion or molecule of a liquid, etc.) ; 0 a group of adjoining elements; 0 a transportation or handling unit (wagon, truck, drum, bag; hand or mechanical shov- elful, scoopful, etc.). For a given critical component A, the properties of unit U,,Z are completely defined by three para- meters which will be called its \u2018describers\u2019. Mass of unit U,,,, Mass of critical component A in unit Cr,,,, Critical content of unit U,,, (proportion of component A), a,,, =A,,IM,, or A,,, = a,, M,,, (1) The identities ( 1) are structural so that two describers only (usually a,,, and M,,,) can define U,, completely with respect to A. We shall denote by Ml<, AL and aL the describers of batch L. with m = 1,2 ,..., Ncr. 70 6. Heterogeneity of a population of unspecified units U,,, We assume that each unit of L is an indivisible, unalterable whole whose internal heterogeneity, at this point at least, is regarded as irrelevant. To use statistical nomenclature, we are interested here in the heterogeneity between-units, not in the heter- ogeneity within-units. 6.7. Parameter characterizing the heterogeneity carried by U,,, Our purpose is to devise a single describer which will completely define the role of unit U,, (content urn and mass M,,) . Homogeneity of lot L relative to component A A batch L would be said to be homogeneous if (with m\u2019 different from m) : urn = arnl = uL, irrespective of m, and m\u2019 a with m, m\u2019 = 1,2,...N, (3) Heterogeneity carried by unit U,,, We define \u2018the heterogeneity carried by U,,\u2019 as factor proportional to the deviation (am - aL) from the state of homogeneity and to the mass M,,, and therefore to the product (am - a,)M,,. On the other hand, experience shows it is always easier to deal with relative, dimensionless quantities. For this reason, we shall divide the product (a,-a,)M,, by the mass of critical component contained in the average unit U,: with: U,:: average unit of L, defined by the following parameters. 0 its mass M,: which is equal to the average unit mass : M; =MLINU (4) 0 its critical content a,: which is equal to that of L aLEaL (5) 0 the mass Ai of critical component in U,: is therefore We are now in a position to define: h,, Heterogeneity carried by U,, (within the lot L) trends in analytical chemistry, vol. 14, no. 2, 1995 (7) Remarks Our attention had been turned to the quantity h,,, as early as 1951 because the sampling variance (today the \u2018fundamental variance\u2019, see Chapter 19 of [ 1 ] ), was proportional to the variance of the population of h,, in L. Note also that, thanks to the relative definition of h,,, masses and contents can be expressed in any units, provided the same units are used everywhere. 6.2. Expected value m(h,,,) of the population of h, in batch L It can easily be shown that the expected value m( h,,) is zero. m(h,J = h,=O (8) 6.3. Variance s\u2019(h,,,) of the population of h, in batch L (9) 6.4. Heterogeneity HL of the batch L We shall now define HL, the heterogeneity of the batch or lot L of NU units as the variance of the distribution of h,,. The quantity HL is an intrinsic property of the matter of which L is made. HL=S2(h,,) 20 ( 10) 6.5. Heterogeneity invariant HI, of batch L From a practical standpoint the factor HL suffers from serious shortcomings. Although always per- fectly defined, it cannot be estimated as soon as the number of units becomes too large. This is the case with particulate solid or liquid batches, the most important in all practical problems. As soon as we try to implement the theoretical results, we need a factor, derived from HL, that remains computable, even when NU is no longer enumerable. We define: trends in analytical chemistry vol. 14, no. 2, 1995 71 HI,, the heterogeneity invariant of the matter making up batch L, where (11) HI, has the dimension of a mass while HL was dimensionless. We have shown, however, that HI, is characteristic of the matter making up the batch L, irrespective of the bulk of L. An example of the computation of HI, is given in the Appendix (see also Ref. [ 11).