Geometric Mean Calculator
By Santaji GadepatilinUpdated October 27, 2026
The geometric mean multiplies values together and takes the root. It is the right average for growth rates, returns and ratios.
How to use the geometric mean calculator
Enter positive numbers. For percentage changes, enter growth factors: +5% → 1.05, −5% → 0.95.
Geometric mean formula
GM = (x₁ × x₂ × … × xₙ)1/n
Worked example — four years of returns
Returns of +5%, +10%, −5%, +20% → factors 1.05, 1.10, 0.95, 1.20
- Geometric mean = 1.0712 → average growth 7.12% per year
- Arithmetic mean = 1.0750 → overstates growth at 7.50%
Arithmetic vs geometric mean for returns
| Year | Return | Growth factor |
|---|---|---|
| 1 | +50% | 1.50 |
| 2 | −50% | 0.50 |
- Arithmetic mean return: 0% — suggests you broke even.
- Geometric mean: √(1.5 × 0.5) − 1 = −13.4% a year.
- Actual result: 100 → 150 → 75, a 25% loss.
The geometric mean tells the truth about compound growth.
Using growth factors
For percentage changes, enter growth factors (1 + rate), find the geometric mean and subtract 1. For returns of 10%, −5% and 20%, enter 1.10, 0.95 and 1.20. The geometric mean is 1.0784, so the average yearly return is 7.84%.
The AM–GM inequality
For positive numbers, the geometric mean is always less than or equal to the arithmetic mean. They are equal only when all values are the same. The more volatile the data, the bigger the gap.
Harmonic mean
For rates such as speeds over equal distances, use the harmonic mean: n ÷ Σ(1/x). For the same positive data: harmonic ≤ geometric ≤ arithmetic.
Other uses
- Average growth of populations, sales or prices.
- Comparing items rated on different scales (indices).
- Aspect ratios and image scaling.
- Bacterial counts and other log-normally distributed data.
When to use it
- Investment returns and CAGR
- Population or sales growth rates
- Averaging ratios and indexes
Frequently asked questions
What is the geometric mean?
The nth root of the product of n numbers. It is the right average for quantities that multiply together, such as growth factors.
Why can't I use the normal average for returns?
Returns compound. +50% then −50% averages 0% arithmetically, but you actually lose 25%. The geometric mean shows the true rate.
Can I use zero or negative numbers?
No. Convert percentage returns to growth factors first (+5% → 1.05, −5% → 0.95).
Why can’t the geometric mean use zero or negative numbers?
It multiplies the values and takes a root. A zero makes the product zero, and negative values can make it undefined. Use growth factors (1 + rate) for returns.
Is CAGR a geometric mean?
Yes. CAGR is the geometric mean of the yearly growth factors, minus 1.