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| '''Gy's sampling theory''' is a [[scientific theory|theory]] about the sampling of materials, developed by [[Pierre Gy]] from the 1950s to beginning 2000s<ref name="Gy2004"> Gy, P (2004), Chemometrics and Intelligent Laboratory Systems, 74, 61-70.</ref> in articles and books including:
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| *(1960) Sampling nomogram
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| *(1979) Sampling of particulate materials; theory and practice
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| *(1982) Sampling of particulate materials; theory and practice; 2nd edition
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| *(1992) Sampling of [[Heterogeneous]] and Dynamic Material Systems: Theories of Heterogeneity, Sampling and Homogenizing
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| *(1998) Sampling for Analytical Purposes
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| The abbreviation "TOS" is also used to denote Gy's sampling theory.<ref name="Esbensen">K.H. Esbensen. 50 years of Pierre Gy's “Theory of Sampling”—WCSB1: a tribute. Chemometrics and Intelligent Laboratory Systems. Volume 74, Issue 1, 28 November 2004, pages 3–6.</ref>
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| Gy's sampling theory uses a [[scientific model|model]] in which the sample taking is represented by [[statistical independence|independent]] [[Bernoulli trials]] for every particle in the parent population from which the sample is drawn. The two possible outcomes of each Bernoulli trial are: (1) the particle is selected and (2) the particle is not selected. The probability of selecting a particle may be different during each Bernoulli trial. The model used by Gy is mathematically equivalent to [[Poisson sampling]].<ref name="Geelhoed">{{cite journal |first=B. |last=Geelhoed |first2=H. J. |last2=Glass |title=Comparison of theories for the variance caused by the sampling of random mixtures of non-identical particles |journal=Geostandards and Geoanalytical Research |volume=28 |issue=2 |year=2004 |pages=263–276 |doi=10.1111/j.1751-908X.2004.tb00742.x }}</ref> Using this model, the following equation for the [[variance]] of the [[sampling error]] in the mass concentration in a sample was derived by Gy:
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| :<math>V = \frac{1}{(\sum_{i=1}^N q_i m_i)^2} \sum_{i=1}^N q_i(1-q_i) m_{i}^{2} \left(a_i - \frac{\sum_{j=1}^N q_j a_j m_j}{\sum_{j=1}^N q_j m_j}\right)^2 .</math> | |
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| in which ''V'' is the variance of the sampling error, ''N'' is the number of particles in the population (before the sample was taken), ''q''<sub> ''i''</sub> is the probability of including the ''i''th particle of the population in the sample (i.e. the [[first-order inclusion probability]] of the ''i''th particle), ''m''<sub> ''i''</sub> is the mass of the ''i''th particle of the population and ''a''<sub> ''i''</sub> is the mass concentration of the property of interest in the ''i''th particle of the population.
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| It is noted that the above equation for the variance of the sampling error is an approximation based on a [[linearization]] of the mass concentration in a sample.
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| In the theory of Gy, [[correct sampling]] is defined as a sampling scenario in which all particles have the same probability of being included in the sample. This implies that ''q''<sub> ''i''</sub> no longer depends on ''i'', and can therefore be replaced by the symbol ''q''. Gy's equation for the variance of the sampling error becomes:
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| :<math>V = \frac{1-q}{q M_\text{batch}^2} \sum_{i=1}^N m_{i}^{2} \left(a_i - a_\text{batch} \right)^2 .</math>
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| where ''a''<sub>batch</sub> is the concentration of the property of interest in the population from which the sample is to be drawn and ''M''<sub>batch</sub> is the mass of the population from which the sample is to be drawn. It has been noted that a similar equation had already been derived in 1935 by Kassel and Guy.<ref name="Kassel">{{cite journal |last=Kassel |first=L. S. |last2=Guy |first2=T. W. |title=Determining the correct weight of sample in coal sampling |journal=Industrial and Engineering Chemistry Analytical Edition |year=1935 |volume=7 |issue=2 |pages=112–115 |doi= }}</ref><ref name="Cheng">{{cite journal |last=Cheng |first=H. |last2=Geelhoed |first2=B. |last3=Bode |first3=P. |year=2011 |title=A Markov Chain Monte Carlo comparison of variance estimators for the sampling of particulate mixtures |journal=Applied Stochastic Models in Business and Industry |volume= |issue= |pages= |doi=10.1002/asmb.878 }}</ref>
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| ==Criticism on Gy's sampling theory==
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| Recent experimental results and novel insights show that parts of Gy's theory need to be updated or revised.<ref name="Critique">http://www.vixra.org/abs/1203.0081</ref> The main points of criticism are that Gy's theory fails to show that certain of its parts are in fact compatible with each other, that the theory relies sometimes on the use of [[Wiktionary:fudge factor|fudge factors]], and that the theory is at various points in its presentation unnecessarily complex.<ref name="Critique"/>
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| == See also ==
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| *[[Statistical sampling]]
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| == References ==
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| {{Reflist}}
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| {{DEFAULTSORT:Gy's Sampling Theory}}
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| [[Category:Sampling (statistics)]]
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