Geometric Mean Calculator
Free geometric mean calculator: paste a list of positive values to get the n-th root of their product, with steps and the arithmetic mean for comparison.
About the Geometric Mean Calculator
The geometric mean of n positive values is the n-th root of their product: GM = (x₁ × x₂ × … × xₙ)^(1/n). It is the correct average for quantities that multiply — growth rates, investment returns, ratios and index numbers — where the ordinary arithmetic mean overstates the result. This free geometric mean calculator accepts values separated by commas, spaces or new lines and shows the product steps plus the arithmetic mean for comparison.
Internally it uses logarithms (GM = exp(mean of ln x)), so long lists and large numbers never overflow. Students meet the geometric mean in statistics and sequences; investors use it every time they annualise multi-year returns — the CAGR is exactly a geometric mean. All values must be positive; the tool explains why if any are not.
How to Use the Geometric Mean Calculator
- 1Paste or type your positive values, separated by commas, spaces or new lines.
- 2Read the geometric mean, the count and the product formula with your numbers.
- 3Compare against the arithmetic mean shown alongside.
Frequently Asked Questions
How is the geometric mean calculated with an example?
Multiply the values and take the n-th root. For 4 and 9: √(4 × 9) = √36 = 6. For 2, 4 and 8: ∛(2 × 4 × 8) = ∛64 = 4. Compare the arithmetic means (6.5 and 4.67) — the geometric mean is always less or equal.
When should I use the geometric mean instead of the arithmetic mean?
Whenever values compound or multiply. An investment that grows 50% then falls 20% has factors 1.5 and 0.8; the geometric mean √(1.5 × 0.8) = √1.2 ≈ 1.0954 means 9.54% average annual growth. The arithmetic mean of +50% and −20% (+15%) badly overstates it.
Why must all values be positive?
A zero makes the whole product zero regardless of the other values, and a negative value can make the n-th root undefined for even n (√ of a negative). For percentage changes, convert to growth factors first: +20% → 1.20, −10% → 0.90, then take the geometric mean.
How is the geometric mean related to CAGR?
CAGR is the geometric mean of annual growth factors, minus 1. If a fund returns +10%, +30% and −5% over three years, CAGR = (1.10 × 1.30 × 0.95)^(1/3) − 1 = (1.3585)^(1/3) − 1 ≈ 10.75% per year.