Geometric Mean and Harmonic Mean

Lesson 09 of STA301 completes the study of central tendency by examining specialized mathematical averages: the Geometric Mean (GM) and the Harmonic Mean (HM). The Geometric Mean of a set of n positive values is defined as the nth root of their product: GM = (X1 * X2 * ... * Xn)^(1/n). Because direct multiplication of numerous observations leads to unwieldy magnitudes, the computation is executed using logarithms: log(GM) = sum(log X) / n for ungrouped data, leading to GM = antilog[sum(log X) / n]. For grouped frequency data, the weighted logarithmic formula GM = antilog[sum(f * log X) / sum(f)] is applied. The geometric mean is uniquely suited for averaging ratios, percentages, index numbers, and compounding growth rates across chronological periods. However, it cannot be computed if any observation is zero or negative. The Harmonic Mean is defined as the reciprocal of the arithmetic mean of the reciprocals of the values. For ungrouped observations, HM = n / sum(1 / X), and for grouped frequency distributions, HM = sum(f) / sum(f / X). The harmonic mean is the mathematically rigorous average to deploy when dealing with rates, speeds, and prices where values are expressed per unit of time, distance, or currency. For example, calculating average vehicular speed over equal distances requires the harmonic mean, as using the arithmetic mean would produce an upwardly biased overestimate. The lecture analyzes the Fundamental Inequality connecting the three mathematical averages: for any dataset of positive numbers, Arithmetic Mean >= Geometric Mean >= Harmonic Mean (AM >= GM >= HM). Equality holds strictly when all observations are identical; otherwise, the arithmetic mean yields the highest magnitude and the harmonic mean the lowest. For two positive numbers, the relationship GM^2 = AM * HM also holds. In Virtual University midterm exams, students are regularly tested on the conditions under which GM or HM is appropriate, solving logarithmic GM problems, and verifying the AM >= GM >= HM inequality. Consulting sta301 short notes and solved past papers guarantees success.