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Thompson–Howarth error analysis: unbiased alternatives to the large-sample method for assessing non-normally distributed measurement error in geochemical samples

¹ Department of Earth and Environmental Science, Acadia University, Wolfville, Nova Scotia, B4P 2R6, Canada(cliff.stanley{at}acadiau.ca)
² ioGlobal, Level 3, IBM Building, 1060 Hay Street, West Perth, WA 6005, Australia(dave.lawie{at}ioglobal.net)

The Thompson–Howarth error analysis procedures have become common in geochemical applications for assessing the magnitude of measurement error at any stage of determination (initial sampling, sample preparation, geochemical analysis). However, the large-sample method, as defined by Thompson and Howarth, which relies on an assumption that the measurement errors are normally distributed, produces significantly biased results when the errors are not normally distributed. Four examples of quality control data-sets from a variety of mineral deposit types illustrate that normally distributed errors are probably the exception rather than the rule in ore deposits, and non-normally distributed geochemical data-sets may exist in other geological materials. As a result, using Thompson and Howarth's large-sample error analysis approach, geoscientists may obtain a significantly inaccurate estimate of the quality of their geochemical concentration data.

Two new methods, which are modifications to the Thompson–Howarth large-sample technique, eliminate this bias because they ensure that the results are independent of a normally distributed error assumption. Regression of group root mean square standard deviations produces accurate error estimates, at least provided that the concentrations are distributed relatively evenly across their range. Similarly, regression of duplicate variances against duplicate means using a quadratic model, and then taking the square root of this model, also results in an unbiased estimate of measurement error. Furthermore, if this quadratic model is a perfect square, then the square root of the quadratic model will be linear on a mean versus standard deviation scatterplot. This approach is further independent of the distribution of the error in the data. Using these modified approaches, geoscientists can now obtain unbiased Thompson–Howarth estimates of measurement error that is not normally distributed in quality control/quality assessment programmes.

KEYWORDS: measurement error, precision, relative error, Thompson–Howarth, duplicate, replicate, bias, standard deviation, variance