📐 Geometric & Harmonic Mean Calculator
Paste your numbers to get the geometric mean, the harmonic mean and the arithmetic mean side by side.
Geometric mean
4
Harmonic mean
3.428571
Arithmetic mean
4.666667
For positive data, arithmetic ≥ geometric ≥ harmonic mean.
The geometric mean is the nth root of the product — the right average for growth rates, ratios and things that multiply (like investment returns). The harmonic mean is n divided by the sum of reciprocals — the right average for rates over a fixed distance (like average speed) and for ratios such as P/E. The arithmetic mean is shown for comparison. 🔒 In your browser.
How the geometric & harmonic mean calculator works
Different situations need different averages. The geometric mean is the nth root of the product of the values (equivalently, exp of the mean of their logarithms) — it's the correct average for things that multiply, like year-on-year growth rates or investment returns. The harmonic mean is the number of values divided by the sum of their reciprocals — the correct average for rates measured over a fixed quantity, like average speed over equal distances. The tool computes both, plus the familiar arithmetic mean, so you can compare them.
Both the geometric and harmonic means require every value to be positive (they use logarithms and reciprocals). For any set of positive numbers, the arithmetic mean is always at least the geometric mean, which is at least the harmonic mean — they're equal only when all the values are identical.
Frequently asked questions
What is the geometric mean?
The nth root of the product of n numbers: for 2, 4, 8 it's (2·4·8)^(1/3) = 4. It's the right average for growth rates and ratios — averaging annual returns of +50% and −50% gives a geometric mean below zero, correctly reflecting the loss.
What is the harmonic mean?
The number of values divided by the sum of their reciprocals: for 1, 2, 4 it's 3 ÷ (1 + ½ + ¼) ≈ 1.71. It's the correct average for rates over a fixed distance, such as average speed, and for ratios like the P/E of a portfolio.
When should I use the geometric mean instead of the arithmetic mean?
Use the geometric mean for quantities that compound or multiply — investment returns, growth rates, index numbers, ratios. The arithmetic mean overstates the average return of a volatile series; the geometric mean gives the true compound rate.
When is the harmonic mean the right average?
When averaging rates over a constant numerator — average speed over equal distances, average price-to-earnings, or rates like miles per gallon across equal-distance trips. Using the arithmetic mean there gives too high a figure.
Why do the means need positive numbers?
The geometric mean takes logarithms (undefined for zero or negatives) and the harmonic mean takes reciprocals (undefined for zero). Both are defined only for strictly positive data; the tool flags it if any value isn't.