Divergence among estimators of the population mean under right-skewed distributions and outliers – a case study of metal accumulation in Cu-Ag deposits of the LGCD

Abstract

The accumulation of metals in the sediment‑hosted Cu-Ag deposits in the Lubin–Głogów Copper District (LGCD, SW Poland) are typically characterized by strongly right‑skewed empirical distributions containing numerous outliers. This study quantifies the divergence among estimates of the population mean of metal accumulation obtained using five commonly used measures of central tendency, treated here as alternative estimators for the target mean relevant to additive resource estimation, under conditions of strong skewness and the presence of outliers.

Large empirical datasets of Cu, Co, Ni, and Pb accumulation values, obtained from ore-deposit sampling in the Rudna mine workings, were treated as reference populations, with their arithmetic means adopted as reference approximations of the population mean relevant to additive resource estimation. Monte Carlo simulations were used to generate 1,000 independent random samples of sizes from empirical distributions: n = 10, 20, 50, 100 and 200.

Two evaluation criteria were applied:

the frequency with which a given estimator produced values closest to the reference population mean,

the mean relative deviation of the estimator from the reference population mean.

The results show that the arithmetic mean yields estimates with negligible bias (from −0.1% to +0.4%), whereas average relative underestimation of the reference population mean by the geometric mean reaches −16.6% for Cu, −29.4% for Co, −18.3% for Ni and −66.7% for Pb.

Within the adopted inferential framework, in which the target parameter is the population mean relevant to additive resource estimation and is approximated by the arithmetic mean of the reference population, the arithmetic mean most frequently yielded estimates closest to this reference value. Its advantage increased with sample size. The Winsorized and trimmed means showed intermediate performance, whereas the median and geometric mean tended to underestimate the reference population mean, particularly under strong skewness and in the presence of numerous outliers.

This conclusion applies specifically within the framework of additive resource estimation and does not necessarily extend to other descriptive uses of central tendency measures. Alternative measures such as the median or geometric mean may occasionally be useful for very small data sets; however, in the present study, they did not outperform the arithmetic mean. The study also highlights the limited usefulness of individual theoretical distributions in the description of empirical metal accumulation data in the LGCD, mainly due to the large number of outliers.