Geometric Mean and Harmonic Mean

Not every situation calls for the arithmetic mean, even when a single "average" figure is genuinely needed — the geometric mean and harmonic mean exist precisely because certain kinds of business data are distorted, sometimes badly, by a simple arithmetic average.

Geometric Mean (GM). For $n$ positive observations $x_1, x_2, \ldots, x_n$:

For more than two or three values this is computed via logarithms, since taking an $n$-th root directly by hand is impractical:

The geometric mean is the correct average whenever values combine multiplicatively rather than additively — most importantly, rates of growth (population growth, compound interest, index numbers, and year-on-year business growth rates), where a simple arithmetic mean of successive growth rates systematically overstates the true average rate.

Harmonic Mean (HM). For $n$ positive observations:

For just two values $x$ and $y$, this simplifies to:

$HM = \dfrac{2xy}{x+y}$ …