PART ONE INTRODUCTION AND A MANAGEMENT SRATEGY Introduction Historical Summary Subdivisions of the text Definition of basic terms and symbols Basic terms List of notations and symbols Latin Letters Greek Letters The word error versus the word uncertainty controversy Introduction Going back to Matheron and Gy''s fundamentals Jumping from uncertainty to error Sampling correctness the mandatory path to predictable uncertainty A management strategy Structural Property Circumstantial Property Concepts of Primary and Secondary Properties The primary structural property of sampling correctness Relationship between correctness and accuracy Practical advantages of controlling sampling correctness Disadvantages and risks of controlling accuracy Conclusions PART TWO FUNDAMENTAL STATISTICAL CONCEPTS USED IN THE THEORY OF SAMPLING Fundamental statistical concepts Notion of probability Probability law of a random variables Notion of Random Variable From a random Variable to its probability law Definition of a probability law Graphic representation of a probability law (Discrete variable) Dependence between random variables Position parameters and characterization of a probability distribution The arithmetic average of a discrete set The quadratic average of a discrete set The geometric average of a discrete set The harmonic average of a discrete set The weighted average of a discrete set The median The mode Dispersion parameters The Pearson variation factor the relative standard deviation Permutations and combinations The Gaussian model the normal probability distribution The Binomial model The Poisson model Limitation of normal and lognormal statistical models Poisson processes Preventive recommendations relative to sampling The grade of a sample obeys a normal distribution The grade of a sample obeys a Poisson distribution Capital notion of probabilistic selection process Random and systematic errors Notion of precision Notion of accuracy Notion of representativeness Graphic representation of the notion of accuracy and precision Summation of random and systematic errors A logical introduction to the components of the Overall Estimation Error Constitution heterogeneity Distribution heterogeneity Classification of lots Number of dimensions characterizing a lot Continuity or discontinuity of the selected model representing the lot Order or disorder of the constituents of the lot Heterogeneity of a zero-dimensional lot Heterogeneity of a one-dimensional lot Heterogeneity of two-dimensional lots Heterogeneity of three-dimensional lots Qualitative and quantitative components of the Heterogeneity Fluctuation Error Materialization of the sampling operation the Increment Materialization Error The Sampling Selection Error The Total Sampling Error The Analytical Error The Overall Estimation Error PART THREE HETEROGENEITY AND HOMOGENEITY A logical introduction to the notion of heterogeneity Qualitative analysis of the duality homogeneity versus heterogeneity Constitution Heterogeneity Classification of the lots submitted to quality or quantity control Number of dimensions of the selected model intended to represent a lot Continuity or discontinuity of the selected model Order or disorder of the units making up a lot Functional, random, and stochastic relations Functional relation Random relation Stochastic relation Caption of the various cases Zero-dimensional and discontinuous lots One-dimensional and continuous lots Two- and three-dimensional lots Heterogeneity of a zero-dimensional lot Introduction Definitions and basic relationships Population of specified units population of fragments Heterogeneity carried by a fragment within the lot Average of the heterogeneities carried by the fragments of a lot Variance of the heterogeneities carried by the fragments of a lot Definition of the Constitution Heterogeneity of a lot Constitution Heterogeneity of a composited lot The Intrinsic Heterogeneity of the fragments making up the lot The Intrinsic Heterogeneity of a composited lot Respective properties of CHL and IHL Constitution homogeneity of a lot Population of specified units population of groups of fragments Introduction to the notion of Distribution Heterogeneity Heterogeneity carried by a fragment within a group of fragments Heterogeneity carried by a group of fragments within the lot Definition of the Distribution Heterogeneity of a lot Relationship between Constitution and Distribution Heterogeneities Definition of the Constitution Homogeneity Definition of the Distribution Homogeneity Natural Distribution Homogeneity within a lot Different kinds of Distribution Homogeneity Experimental verification of the homogeneity Maximum Distribution Heterogeneity within a lot Definition and properties of the Grouping Factor Definition and properties of the Segregation Factor Effect of the observation scale on the value of the Distribution Heterogeneity Effect of the size distribution of the fragments on the value of the Constitution Heterogeneity General expression of the Distribution Heterogeneity Illustration of the definition of heterogeneity Transformation of a set of units with two descriptors into an equivalent set of units with one descriptor Practical use of the proposed definitions of the heterogeneity Practical example of the calculation of heterogeneity characteristics in a counted population Heterogeneity of a one-dimensional lot notion of variography Introduction Total heterogeneity supported by a one-dimensional lot Definition of the heterogeneity carried by the unit Um Characterization of a chronological series Overall characterization of the heterogeneity of a one-dimensional lot Sequential characterization of the heterogeneity of a one-dimensional lot Order and correlation Basic definition of the semi-variogram Actual presentation of the variogram Problem associated with the central values of the chronological series Problem associated with the precision of the variographic variance Problem associated with the main characteristics of the processing stream Comparison between the variogram of hm with the variograms of the two descriptors am and Mm Definition of the relative variogram Relevance of the chronological order of the units Modeling of the experimental variogram Description of the heterogeneity of a one-dimensional lot in terms of a variogram The short-range heterogeneity fluctuation The long-range heterogeneity fluctuation The periodic heterogeneity fluctuation Properties of the residual component Properties of the variogram before and beyond the range Area of influence of one increment Stationarity of the information provided by a variogram Auxiliary functions of the variogram The first order average integral of the variogram The second order average integral of the variogram The moving average of the variogram From heterogeneity to the continuous Heterogeneity Fluctuation Error Definition of error generators Point-by-point interpretation of the variogram Graphical integration of a variogram Significance of the term V () Practical estimation of the term V () using a separate experiment Point-by-point calculation of the first order average integral W ( j ) Point-by-point calculation of the second order average integral W ''( j ) Calculation of the Heterogeneity Fluctuation Error Step-by-step interpretation of a variogram Investigation of the random term V () Investigation of the continuous term V ( j ) Practical interest of the variographic experiment in Quality Control Stability of a variogram Average estimates versus instantaneous estimates Practical applications of the auxiliary functions of the variogram PART FOUR SAMPLING ERRORS INTRODUCED BY VARIOUS FORMS OF HETEROGENEITY Sampling of one-dimensional lots the continuous model Theoretical use of the one-dimensional model The Analytical Error The one-dimensional model Definitions and basic notations Definition of the real lot L Definition of an imaginary lot L '' Characterization of the heterogeneity of a one-dimensional lot Characterization of the increment sampling process Characterization of the sample The continuous Heterogeneity Fluctuation Error HFE Variance of the continuous Heterogeneity Fluctuation Error HFE Variance of HFE for random systematic sampling Variance of HFE for stratified random sampling Variance of HFE for strict random sampling Components of the con.