Practical applications of sampling theory
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Practical applications of sampling theory

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Chemometrics and Intelligent Laboratory S
analysis, that takes into account both the technical and
statistical aspects of sampling, has been developed by Pierre
Gy. Gy\u2019s theory is presented in his books [1\u20133] and the
latest developments in papers of this issue. Pitard [4] has
also written a book about Gy\u2019s sampling theory. A useful
account covering the theory of stratified sampling and
equipment and procedures with correct ones.
Correct sampling largely eliminates the materializa-
tion and preparation errors. Weighting error is made
if the lot consists of sublots of different sizes or if
the flow rate varies during the sampling periods in
process streams, and simple average is calculated to
The most complete theory on sampling for chemical
nel have been adequately trained for their jobs. Step 1 Check that all sampling equipment and procedures
obey the rules of correct sampling. Replace incorrect
is correct, the uncertainties of the methods have been
is good. It is still largely unknown that there exists a useful
sampling theory developed for chemical analysis. The
situation is, hopefully, slowly changing. Laboratories and
consultants who are carrying out sampling as part of their
business have started to accredit their sampling procedures,
at least in Finland, probably elsewhere too. Basic require-
ments for the accreditation are that the sampling equipment
2. Design and audit of sampling procedures
The classification of errors of sampling forms a logical
framework for designing and auditing sampling procedures.
The classification is shown in Fig. 1 (see Gy\u2019s papers in this
issue for explanations of the different boxes of the figure).
Auditing and designing sampling procedures normally in-
volve the following steps:
1. Introduction
In many textbooks of analytical chemistry, it is stated that
the result is not better than the sample on which it is based.
Very little is however said on how to assure that the sample
sampling theory can help to develop cost-optimal proce-
dures. The optimization procedures described in this paper
are based on Sommer\u2019s work.
Practical application
Pentti M
Department of Chemical Technology, Lappeenranta Universi
Received 1 August 2003; received in revise
Available on
A large number of analyses is carried out, e.g., for process contro
purposes. The sampling theory developed by Pierre Gy, together w
analytical measurement protocols. A careful optimization of the sam
result in considerable savings in costs or in improvement of the re
D 2004 Elsevier B.V. All rights reserved.
Keywords: Gy\u2019s sampling theory; Stratified sampling; Optimization of sam
optimization of sampling procedures has been written by
Sommer [5]. The purpose of this paper is to elucidate how
0169-7439/$ - see front matter D 2004 Elsevier B.V. All rights reserved.
* Tel.: +358-5-621-2102 (office), +358-40-504-9413 (mobile); fax:
E-mail address: (P. Minkkinen).
of sampling theory
echnology, P.O. Box 20, FIN-53851 Lappeenranta, Finland
1 January 2004; accepted 12 March 2004
8 July 2004
duct quality control for consumer safety, and environmental control
e theory of stratified sampling, can be used to audit and optimize
and measurement steps of the complete analytical procedure may
ty of results.
ystems 74 (2004) 85\u201394
estimate the lot mean. This error is eliminated if
proportional cross-stream sampling can be carried
out, and the average is calculated as the weighted
mean weighted by the sample sizes.
Step 2 Estimate the remaining errors (fundamental samp-
ling error, grouping and segregation error, and point
selection error) and what is their dependence on
P. Minkkinen / Chemometrics and Intelligent Laboratory Systems 74 (2004) 85\u20139486
Fig. 1. Gy\u2019s classification of sampling errors according to the origin of
errors. Errors can be classified into two main groups, to those that originate
from incorrect design or operation of the sampling equipment (material-
increment size and sampling frequency. If the
necessary data are not available, design pilot studies
to obtain the data.
Step 3 Define the acceptable overall uncertainty level or
cost of the investigation and optimize the method,
i.e., the increment sizes, selection strategy (system-
atic or stratified), and the sampling frequency so
that the required uncertainty or cost level is
Step 1 is crucial. Normally it is difficult and expensive to
estimate the uncertainties of incorrect sampling. It is also
futile, because sampling biases are never constant due to the
fact that stream segregation is a transient phenomenon
which changes all the time. Therefore, sampling correctness
must be preventively implemented.
3. Applications of fundamental sampling error model
Fundamental sampling error is the minimum error of an
ideal sampling procedure. Ultimately, it depends on the
number of critical particles in the samples. For homoge-
neous gases and liquids, it is very small, but for solids,
powders, and particulate materials, especially at low con-
centrations of critical particles, the fundamental error can
be very large. If the lot to be sampled can be treated as
ization errors) and to statistical errors.
one-dimensional object, fundamental sampling error mod-
els can be used to estimate the uncertainty of the sampling.
If the lot cannot be treated as one-dimensional object, at
least the point selection error has to be taken into account
when the variance of primary samples is estimated. If the
sample preparation and size reduction by splitting are
carried out correctly, fundamental sampling error models
can be used for estimating the variance components
generated by these steps. If the expectance value for the
number of critical particles, in the sample can be estimated
easily as function of sample size, Poisson distribution or
binomial distribution can be used as sampling models to
estimate the uncertainty of the sample. In most cases, the
fundamental sampling error model developed by Gy has to
be used.
3.1. Estimation of fundamental sampling error by using
Poisson distribution
Poisson distribution describes the random distribution of
rare events in a given time or space interval. If the average
number of the critical particles expected in the sample can
be estimated, the standard deviation of the sample can be
estimated. Poisson distribution, as the model for sampling
error, has been treated, e.g., by Ingamells and Pitard [6]. The
important property of the Poisson distribution is that the
variance and the mean of the occurrences or the events in
the interval inspected are identical (ln, average number of
critical particles in the sample in our case). Standard
deviation expressed as the number of particles is
rn ¼ \ufb03\ufb03\ufb03\ufb03\ufb03lnp ð1Þ
The relative standard deviation is just as easy to estimate
rr ¼ 1\ufb03\ufb03\ufb03\ufb03\ufb03lnp ð2Þ
If ln is large (say, larger than 25), the confidence interval
can be estimated by replacing Poisson distribution by
normal distribution with the same standard deviation and
mean. If ln is small, the confidence intervals have to be
estimated from Poisson distribution. Example 1 describes a
typical situation where Poisson distribution can be used as
the model for sampling error estimation.
Example 1.
Plant Manager: I am producing fine-ground limestone that
is used in paper mills for coating printing paper. According
to their specification, my product must not contain more
than 5 particles/tonne particles larger than 5 Am. How
should I sample my product?
Sampling Expert: That is a bit too general a question. Let\u2019s
first define our goal. Would 20% relative standard deviation
for the coarse particles be sufficient?
Plant Manager: Yes.
available for the user, the usefulness