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