Geometric Mean Calculator
Find the geometric mean of a set of positive numbers by entering them as a comma-separated list. The geometric mean is the n-th root of the product of all values — ideal for averaging growth rates, ratios and percentages.
(x₁ × x₂ × … × xₙ)^(1/n) — the n-th root of the product of all values
- 1
Count n
3 - 2
Sum of natural logs
ln(x₁) + … + ln(xₙ) = 6.238325Log-sum avoids overflow when multiplying many large numbers. - 3
Average log
6.238325 ÷ 3 = 2.079442 - 4
Geometric mean
e^2.079442 = 8
How does this calculator work?
The geometric mean of n positive numbers is their n-th root: GM = (x₁ × x₂ × ⋯ × xₙ)^(1/n). For {2, 8, 32} the GM = (2 × 8 × 32)^(1/3) = 512^(1/3) = 8. It is the right average for growth rates and ratios — always ≤ the arithmetic mean.
Formula
How this is calculated
The geometric mean of n positive numbers is the n-th root of their product: GM = (x₁ × x₂ × ⋯ × xₙ)^(1/n). It can also be computed as the exponential of the arithmetic mean of the logarithms — exp(Σ ln xᵢ / n) — which is numerically stable for large datasets and avoids the overflow you would get from multiplying many large numbers directly. This calculator uses the log-sum method for accuracy regardless of input magnitude.
The geometric mean is the correct average for quantities that multiply together or that are expressed as ratios or percentages. For example, if an investment grows by 10%, 50% and −20% in three consecutive years, the true compound annual growth rate (CAGR) is not (10+50−20)/3 = 13.3% but the geometric mean of (1.10 × 1.50 × 0.80)^(1/3) − 1 ≈ 9.66%. Similarly, the geometric mean of index values, pixel intensities, or aspect ratios is typically more meaningful than the arithmetic mean.
All input values must be strictly positive because the geometric mean of non-positive numbers is undefined (the logarithm of zero or a negative number is not a real number). If your dataset contains zeros or negatives you may need to shift the values or use a different average. The arithmetic mean is also shown for comparison — by the AM-GM inequality, the arithmetic mean is always ≥ the geometric mean, with equality only when all values are identical.
Frequently asked questions
Use the geometric mean when averaging rates of change, ratios, percentages, or any data that compounds multiplicatively — such as annual returns, inflation rates, population growth rates, or signal amplitudes on a log scale. Use the arithmetic mean for additive quantities like temperatures or distances.
No. The geometric mean always lies between the minimum and maximum of the dataset (inclusive). It is always ≤ the arithmetic mean. If all values are equal, GM = AM.
The geometric mean is defined as the n-th root of a product. A product of an even number of negatives is positive, but the geometric mean would then not represent the data meaningfully. More fundamentally, the log-based computation requires ln(x), which is undefined for x ≤ 0.
Also known as
TG we-Calculate Editorial Team. (2026). Geometric Mean Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/geometric-mean-calculator
TG we-Calculate Editorial Team. "Geometric Mean Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/geometric-mean-calculator.
TG we-Calculate Editorial Team, "Geometric Mean Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/geometric-mean-calculator
@misc{wecalculate_geometric_mean_calculator, title = {Geometric Mean Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/geometric-mean-calculator}}, year = {2026}, note = {TG we-Calculate} }
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