Introduction to the theory of sampling
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Introduction to the theory of sampling

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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 
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 
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 
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. 
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 
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 
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 
aLEaL (5) 
0 the mass Ai of critical component in U,: is 
We are now in a position to define: 
h,, Heterogeneity carried by U,, (within the lot 
trends in analytical chemistry, vol. 14, no. 2, 1995 
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 
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 
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).