Some statistical properties of power averages for lognormal samples

Abstract

Power averages are often used to estimate the effective permeability of a region based on sample measurements. Since the average is computed from a limited number of samples, it may suffer bias and sampling variation. Expressions for bias and sampling variation are derived here for averages having any exponent value between +1 (arithmetic average) and -1 (harmonic average). The derivations assume independent, lognormally distributed samples. The expressions agree with known results for the arithmetic, geometric, and harmonic averages. Estimator bias is generally smaller than 15% for common levels of permeability variability and sample numbers. Sampling variability, however, may be a considerable fraction of the true mean. If sampling schemes aim to collect sufficient data to keep the harmonic average within acceptable tolerances, other averages will meet or better that tolerance.