The geometric mean?

Abstract

The sample geometric mean (SGM) introduced by Cauchy in 1821, is a measure of central tendency with many applications in the natural and social sciences including environmental monitoring, scientometrics, nuclear medicine, infometrics, economics, finance, ecology, surface and groundwater hydrology, geoscience, geomechanics, machine learning, chemical engineering, poverty and human development, to name a few. Remarkably, it was not until 2013 that a theoretical definition of the population geometric mean (GM) was introduced. Analytic expressions for the GM are derived for many common probability distributions, including: lognormal, Gamma, exponential, uniform, Chi-square, F, Beta, Weibull, Power law, Pareto, generalized Pareto and Rayleigh. Many previous applications of SGM assumed lognormal data, though investigators were unaware that for that case, the GM is the median and SGM is a maximum likelihood estimator of the median. Unlike other measures of central tendency such as the mean, median, and mode, the GM lacks a clear physical interpretation and its estimator SGM exhibits considerable bias and mean square error, which depends significantly on sample size, pd, and skewness. A review of the literature reveals that there is little justification for use of the GM in many applications. Recommendations for future research and application of the GM are provided.

KEYWORDS:

• Central tendency
• arithmetic mean
• median
• log transformation
• lognormal
• multiplicative aggregation
• effective

Acknowledgements

I wish to thank Attilio Castellarin, Dennis Lettenmaier, David Parkhurst, Jonathan Lamontagne, Nick Matalas and Robert Hirsch for their comments on an early version of this manuscript. I am also indebted to the two anonymous reviewers whose review comments led to considerable improvements. I am also indebted to J.R.M. Hosking for his assistance in deriving the geometric mean of a generalized Pareto variable.