Math
Geometric Mean Calculator
Compute the geometric mean of positive numbers and compare it to the arithmetic and harmonic mean. Built for growth rates, ratios, and returns.
Enter positive numbers
Only positive numbers. Use growth factors (1.10 for +10%, 0.95 for -5%) for compounded returns.
Compare mode also shows the arithmetic and harmonic mean so you can verify AM >= GM >= HM.
Common use cases
- Investment returns: Compound growth factors over multiple periods. A portfolio that gained 10%, 5%, then lost 5% has GM = 1.0314, i.e. ~3.14% per period.
- Growth rates: Average year-over-year revenue or population growth. The geometric mean is the constant rate that produces the same end result.
- Aspect ratios and dimensions: Average ratios like 16:9 and 4:3 by multiplying and taking the n-th root, rather than averaging the parts separately.
Geometric mean
1.031419
From 3 values.
Count
3
Sum of ln(x)
0.092807
Product
1.09725
AM, GM, HM (must satisfy AM >= GM >= HM)
Arithmetic
1.033333
sum / n
Geometric
1.031419
(prod)^(1/n)
Harmonic
1.029476
n / sum(1/x)
Notes
- Values look like growth factors. The geometric mean gives the equivalent constant period-over-period rate.
- Useful for averaging rates, ratios, and growth factors that compound multiplicatively.
Frequently Asked Questions about the Geometric Mean Calculator
What is the geometric mean?
The geometric mean is the nth root of the product of n positive numbers. For 2, 8: it is sqrt(2 x 8) = 4. It is the right average for growth rates, ratios, and anything that compounds: think investment returns over multiple years.
When should I use geometric mean instead of arithmetic mean?
Use it when values compound multiplicatively. A portfolio that gains 100% then loses 50% has an arithmetic mean return of +25% (which is misleading, you broke even) but a geometric mean return of 0%, which is the real story.
Can I include negative numbers or zero?
No. The geometric mean is only defined for positive numbers because it involves an nth root of a product. A zero makes the product zero, and negative values make even roots undefined. The calculator rejects non-positive inputs.
How does the calculator avoid overflow with many values?
Instead of multiplying everything (which overflows quickly), it computes exp(mean(ln(values))). The math is identical but works for huge lists. You can paste 1000 values without precision problems.
What is the relationship to arithmetic and harmonic mean?
For any list of positive numbers, harmonic mean <= geometric mean <= arithmetic mean. They are equal only when every value is the same. The calculator shows all three so you can see the gap, which itself is a useful signal of how spread out the values are.