Estimation of the mean and reproducibility of aggregate compositions based on constituent mass data

Given a set of representative samples of aggregate source material (e.g. from a stockpile or production line), it is shown that if only percentaged data is reported, the mean composition of the percentaged data is best calculated using a robust estimate of the logratio mean. However, if raw mass data is also available (as will generally be the case), then it is best to first calculate the average of the constituent masses, and then to convert these into a percentage. The difference between confidence, prediction and tolerance intervals is reviewed and it is shown that the repeatability of aggregate samples and an effective long-term ‘detection limit’ for constituents can be reliably estimated using duplicates of run-of-the-mill samples; an upper boundary can be placed on the detection limit by application of, say, a two-sided {95%, 99%} tolerance interval. Numerical approximations are given to enable a variety of normal two-sided tolerance factors to be computed.

[1]  J. Aitchison.  A new approach to null correlations of proportions , 1981 .

[2]  Luis A. Escobar,et al.  Statistical Intervals: A Guide for Practitioners , 1991 .

[3]  F. Chayes.  Ratio Correlation: A Manual for Students of Petrology and Geochemistry , 1971 .

[4]  P. Rousseeuw.  Robust estimation and identifying outliers , 1990 .

[5]  Richard J. Howarth,et al.  Duplicate analysis in geochemical practice. Part II. Examination of proposed method and examples of its use , 1976 .

[6]  John Aitchison,et al.  The statistical analysis of geochemical compositions , 1984 .

[7]  Richard J. Howarth,et al.  The rapid estimation and control of precision by duplicate determinations , 1973 .

[8]  David Hanratty,et al.  Aggregates: sand, gravel and crushed rock aggregates for construction purposes (3rd ed.) 1 , 2002 .

[9]  R. Howarth.  Improved estimators of uncertainty in proportions, point-counting, and pass-fail test results , 1998 .

[10]  Michael J. Thompson,et al.  Duplicate analysis in geochemical practice. Part I. Theoretical approach and estimation of analytical reproducibility , 1976 .

[11]  John Aitchison,et al.  Measures of location of compositional data sets , 1989 .

[12]  J. Aitchison.  Logratios and Natural Laws in Compositional Data Analysis , 1999 .

[13]  J. Kruskal.  Nonmetric multidimensional scaling: A numerical method , 1964 .

[14]  S. Shen,et al.  Measurement error in compositional data , 1984 .

[15]  L Collis,et al.  Aggregates: Sand, gravel and crushed rock aggregates for construction purposes , 1985 .

[16]  John Aitchison,et al.  The Statistical Analysis of Compositional Data , 1986 .

[17]  D. F. Andrews,et al.  Robust Estimates of Location , 1972 .

[18]  Nicholas M.S. Rock,et al.  ROBUST: An interactive FORTRAN-77 package for exploratory data analysis using parametric, ROBUST and nonparametric location and scale estimates, data transformation , 1987 .

[19]  K. Pearson.  Mathematical contributions to the theory of evolution.—On a form of spurious correlation which may arise when indices are used in the measurement of organs , 1897, Proceedings of the Royal Society of London.