Geometric Mean Calculator
Calculate the geometric mean of a list of positive numbers.
How to use this tool
- Enter numbers (comma-separated) in the fields above.
- Results update instantly as you type — or click Calculate.
- Read your geometric mean and the full breakdown beneath it.
Geometric mean = (x₁ × x₂ × … × xₙ)^(1/n). Computed as exp(mean of log values) to avoid overflow.
Formula
GM = exp( (1/n) × Σ ln(xᵢ) ) = (x₁ × x₂ × … × xₙ)1/n
The calculator sums the natural logarithms of all positive values, divides by n, and exponentiates the result. Non-positive values are excluded.
How it works
This calculator computes the geometric mean — the n-th root of the product of n positive numbers — using logarithms for numerical stability with large datasets.
The geometric mean is appropriate for data that are multiplicative or exponential in nature, such as growth rates, ratios, and financial returns. It is always less than or equal to the arithmetic mean and is undefined for datasets containing zero or negative values.
Worked example
Worked example
- Input: 1, 2, 4, 8 (n = 4).
- Natural logarithms: ln(1)=0, ln(2)≈0.6931, ln(4)≈1.3863, ln(8)≈2.0794.
- Sum of logs: 0 + 0.6931 + 1.3863 + 2.0794 = 4.1589.
- Mean of logs: 4.1589 / 4 = 1.0397.
- Exponentiate: e^1.0397 ≈ 2.8284.
Geometric mean = 2.8284
Key terms
- Geometric mean (GM)
- The n-th root of the product of n positive numbers; represents the central tendency for multiplicative data and is less sensitive to large outliers than the arithmetic mean.
- Multiplicative data
- Data where values relate to each other through multiplication rather than addition, such as compound growth rates, ratios, or index numbers.
- Log-normal distribution
- A distribution where the logarithm of the variable is normally distributed; the geometric mean is the natural measure of centre for log-normal data.
- AM–GM inequality
- For any set of positive numbers, the arithmetic mean is always greater than or equal to the geometric mean, with equality only when all values are identical.
- Compound annual growth rate (CAGR)
- The geometric mean of annual growth factors over multiple periods; shows the equivalent constant annual rate that produces the same total growth.
Frequently asked questions
- When should I use the geometric mean?
- The geometric mean is appropriate for averaging rates of change, ratios, or growth factors. For example, if an investment grows by 10%, 50%, and −20% in three years, the geometric mean gives the equivalent constant annual rate.