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# Practical applications of sampling theory

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s ink ty of T d form line 2 l, pro ith th pling liabili pling estimated, procedures are regularly audited, and the person- 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 Abstract 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. doi:10.1016/j.chemolab.2004.03.013 * Tel.: +358-5-621-2102 (office), +358-40-504-9413 (mobile); fax: +358-5-621-2199. E-mail address: Pentti.Minkkinen@lut.fi (P. Minkkinen). of sampling theory kinen* 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. www.elsevier.com/locate/chemolab 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 achieved. 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