Sampling for Analytical Purposes

Transcription

1 Sampling for Analytical Purposes Pierre Gy The Paris School ofphysics and Chemistry Translated by A. G. Royle JOHN WILEY & SONS Chichester New York Weinheim Brisbane Singapore Toronto

2 Preface to French Edition Preface to English Edition Acknowledgements General Plan ofthis Book xi xiii xv xvii 1 Introduction The paradox of the diamond and the sample The role of sampling in valuation The role of sampling in research Sampling: an unavoidable precursor to analysis The parties involved in sampling Sampling errors and the risks incurred Examples ofcostly sampling errors The complementary nature of sampling and analysis The relative importance given to sampling and to analysis A caricaturist 's lookat the bibliography of sampling The future of analysis Beginnings of a general sampling theory Conclusions 14 PART I THE QUALITATIVE APPROACH: ACHIEVING ACCURACY IN SAMPLING 2 How Can the Mass of a Lot L be Reduced? Incremental sampling Sampling by Splitting 18 3 Heterogeneity and Sampling Constituent elements and units Concepts of homogeneity and heterogeneity ofa set 24 4 Sampling Defmitions 27

3 Vlll 4.1 Introduction The role of the theoretician Definitions of selection; of sampling; and of a sample as functions of the conditions under which selection took place How to make a correct selection Definitions of selection; of sampling: and of a sample as functions of the sampling error Relationships between sampling conditions and the Statistical properties of the sampling error Biases introduced by incorrect sampling Accuracy and reproducibility; the difference Non-probabilistic sampling; 'picking' The dangerous assumption of homogeneous distribution: heterogeneity from fraud or by gravitational segregation 35 5 Objects in Three, Two, One or Zero Dimension(s) Objects in three dimensions Objects in two dimensions Objects in one dimension Objects of zero dimension Meeting point of the zero-and one-dimensional modeis 40 6 The Practical Sampling ofmoving Streams of Material Three ways of reducing the mass of a moving stream Sampling all the stream fora fraction of the time Taking a fraction of the stream for the whole of the time Taking a fraction of the stream for a fraction of the time The only correct method for sampling fiowing streams 55 7 Conclusions on the Qualitative Approach Sampling is not a simple technique Howcan the userjudge a sampling device? How should a manufacturer design a sampler? How to use a correct sampler correctly Education, standardisation and equipment manufacture Practical conclusions 59 PART II THE QUANTITATIVE APPROACH: TAKING REPRODUCIBLE SAMPLES FROM A POPULATION OF NON-ORDERED OBJECTS Preface to Parts II and III 61 8 The Heterogeneity of a Population A reminder on heterogeneity and sampling Quantifying population heterogeneity The characteristic parameters of a substance 66

4 9 Sampling: The Zero-dimensional Model: The Fundamental 69 Error FE 9.1 Definition of the fundamental error FE The distribution lawof FE The mean m(fe) of the fundamental error FE The variance s 2 (FE) of the fundamental error FE Conditions for nullifying the variance s 2 (FE) Conditions for minimising the variance s 2 (FE) Solution of problems involving the variance 5 2 (FE) Calculating the confidence limits ±2s(a E ) Resume of the properties of the fundamental model Sampling: The Zero-dimensional Model: The Segregation and Grouping Error SGE Definition of the minimum practical error MPE The variance ^2(MPE) of the minimum practical error MPE Definition of the segregation and grouping error SGE The mean of the segregation and grouping error SGE The variance of the segregation and grouping error SGE Nullifying the segregation and grouping error SGE Minimising the variance of the segregation and grouping error SGE Homogenisation and bed-blending An instructive example drawn from card games Hyperhomogeneity Strategy regarding the segregation and grouping error SGE 81 ix PART III THE QUANTITATIVE APPROACH: ENSURING THE REPRODUCIBILITYOF SAMPLES TAKEN FROM A SERIES OF CHRONOLOGICALLYORDERED OBJECTS 11 Heterogeneity: The One-dimensional Model Review of the conditions for the model tobe valid Linking the zero- and one-dimensional modeis Characterising the heterogeneity of lot L Nature of the function x(t) [either a(t) or h(t)] Introduction to the characterisation of the function x(t) Characterising the function x(t): the variogram Examples of variograms The variogram as a tool for the analysis of heterogeneity Designing a variographic experiment Sampling: The One-dimensional Model Introduction 101

5 X 12.2 The variogram's auxiliary functions The meanvalue w(j) ofthe single integral ofthe variogram The mean w'(f) ofthe double integral ofthe variogram The usual intervals at which increments are taken The error-generating functions W(j) The variance s 2 (IE) ofthe integration error IE Practical calculation ofthe auxiliary functions Estimation ofthe ordinate v(0) ofthe variogram (or intercept) Point-by-point calculation of an integral Point-by-point calculation ofthe auxiliary functions and error-generating functions (egf) The variogram as a detector of autocorrelation Consequences ofthe error made by treating an ordered series as a non-ordered population Decomposition of a stochastic series into its two components Variography and the control of treatment plants 114 PART IV THE QUANTITATIVE APPROACH: OTHER APPLICATIONS OF SAMPLING THEORY 13 Measurement of Mass by Proportional Sampling Justification for weighing by proportional sampling Principle and accuracy of proportional sampling The theoretical reproducibility of proportional sampling Estimation ofthe time sampling rate r'\ continuous sampling Systems Industrial-scale and pilot-plant regimes Estimation ofthe time sampling rate r': discontinuous sampling devices An experimental study of proportional sampling Practical reproducibility of proportional sampling Hypothetical application to a lot of kg of water In conclusion 131 Appendix 1: Brief Resume of Classical Statistics (Non-weighted) 133 Appendix 2: Three Branches of Statistics 141 Appendix 3: Integral Calculus Nomenclature 145 References 147 Index 149