Comparison of arithmetic and geometric means as measures of a central tendency in cattle nematode populations

Abstract

Efficacy calculations in anthelmintic studies require estimates of the central tendency for the nematode populations. Confusion exists among practitioners regarding which measures of central tendency are most appropriate; although the arithmetic mean is frequently used, there are theoretical reasons for preferring the geometric mean. To investigate this controversy, arithmetic and geometric means were compared for their suitability for use in measuring efficacy. Arithmetic and geometric means were compared as measures of central tendency for skewed distributions. The following criteria were developed to facilitate the comparison: (1) probability around the parameter, (2) influence of extreme values, and (3) proximity to the median. Under log-normality, theoretical results demonstrated the superiority of the geometric mean. Modified-bootstrap simulations using empirical data from cattle were used to confirm theoretical expectations. Simulations on log-normal data supported the geometric mean as the better indicator of the central tendency. Additionally, for data not confirmed as log-normal, the superiority of geometric means was demonstrated. In a comparison of precision, it was shown that mean squared error was always smaller for sample geometric means than for arithmetic means when n ≥2. Simulation results added support to that conclusion.