Geometric & Harmonic Mean
Geometric, harmonic, arithmetic and quadratic mean of a list — the right average for growth rates, for speeds over equal distances, and for quantities that add.
All four Pythagorean means with the AM ≥ GM ≥ HM ordering and the gap between them, the average growth rate when the values are factors, and a note on when each mean is the correct one.
Example: Growth factors 1.10, 1.05, 0.95, 1.20 average 1.0712 geometrically — 7.12 % a period, not the 7.5 % the arithmetic mean suggests; 60 and 40 km/h over equal distances average 48 km/h harmonically.
Three averages,
three different jobs.
How each mean is computed, which one belongs to which kind of data, and when they are undefined.
The three means
The arithmetic mean is Σx ÷ n. The geometric mean is the nth root of the product, computed here as exp(mean of ln x) so that a hundred values do not overflow into infinity before the root is taken. The harmonic mean is n ÷ Σ(1/x) — the reciprocal of the mean of the reciprocals. The quadratic mean (RMS) √(mean of x²) is shown too, since it is the one that treats a −3 like a +3.
Which one to use
Use the arithmetic mean for quantities that add: lengths, masses, marks. Use the geometric mean for quantities that multiply: growth factors, index numbers, ratios, rates of return — it is the constant factor that would turn the first value into the last over the same number of steps, which the arithmetic mean never is. Use the harmonic mean for rates measured over a fixed quantity: two speeds over equal distances, P/E ratios in an index, resistors in parallel. Averaging 60 and 40 km/h arithmetically gives 50; the trip actually averages 48, and only the harmonic mean says so.
Order and edge cases
For positive numbers AM ≥ GM ≥ HM always, with equality only when every value is identical — the gap between them grows with the spread, so AM − GM is itself a crude measure of dispersion. A zero makes the harmonic mean undefined (division by zero) and the geometric mean zero-or-undefined; a negative value makes the geometric mean undefined, because the root of a negative product is not real. Convert percentage rates to factors (5 % → 1.05) before averaging them geometrically. Nothing leaves the browser; the same four anonymous usage counts as the rest of the site apply.
SOURCES
- GM = exp(mean of ln x) (stable for long lists), HM = n ÷ Σ(1/x), QM = √(mean of x²); AM ≥ GM ≥ HM for positive values, with equality only when all are equal
Last reviewed 22 September 2026. How results are checked: How we verify.