Geometric Mean Calculator
The average of a set of growth rates or ratios — the mean that compounds correctly, where the arithmetic mean does not.
The geometric mean is the nth root of the product: multiply the values and take the nth root.
How the geometric mean calculator works
The geometric mean is the nth root of the product: multiply the values and take the nth root. Equivalently, average their logarithms and exponentiate, which is how it is computed here to avoid overflow.
It is the right average whenever the quantities multiply rather than add. Growing 100% one year and −50% the next leaves you exactly where you started; the arithmetic mean says +25%, the geometric mean says 0%.
Formula: GM = (x₁ x₂ … xₙ)^(1/n) = exp(mean of ln x)
Worked examples
| Inputs | Geometric mean | Note |
|---|---|---|
| 2, 8, 4, 16 | 5.656854 | GM = 5.657 |
| Growth of 1.10 and 0.90 | 0.994987 | GM = 0.9950 — a net loss |
| All the same | 7 | GM = 7 |
FAQFrequently asked questions
When should I use the geometric mean?
Whenever the numbers multiply: growth rates, investment returns, ratios, index numbers, rates of change.
Why not the arithmetic mean for returns?
Because it overstates them. +100% then −50% leaves you level, but the arithmetic mean says +25%.
Can it handle zero or negative values?
No — the product would be zero or the root undefined. For returns, use growth factors (1.10, 0.90) rather than percentages.
How does it compare to the other means?
Always harmonic ≤ geometric ≤ arithmetic, with equality only when every value is identical.
What is it used for outside finance?
Aspect ratios, image scaling, the Human Development Index, and averaging anything measured on different scales.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- National curriculum in England — Mathematics — the terms and methods taught in UK schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.