Arithmetic, geometric and harmonic mean

"Average" isn't a single thing. The three most common — arithmetic, geometric and harmonic — fit different situations, and using the wrong one leads to bad conclusions.

Arithmetic mean

The sum divided by the count. Good for values that add up (grades, heights):

(a + b + … ) / n

Example: grades 6, 8, 10 → (6+8+10)/3 = 8.

Geometric mean

The nth root of the product. Good for values that multiply (growth rates, returns over several years):

(a · b · … )1/n

Example: +50% then −50% doesn't break even. The geometric mean of 1.5 and 0.5 is √0.75 ≈ 0.866 → ~13% loss per year.

Harmonic mean

The reciprocal of the mean of reciprocals. Good for rates over the same distance (average round-trip speed):

n / (1/a + 1/b + …)

Example: 60 km/h out, 40 km/h back → harmonic mean = 48 km/h (not 50).

The relationship

For positive numbers, always: harmonic ≤ geometric ≤ arithmetic. Equal only when all values are equal.

FAQ

Why use the geometric mean for returns? Because returns multiply year over year. The arithmetic mean overstates the real gain.