Geometric Mean Calculator
Calculate the geometric mean of a set of numbers.
Used in finance for average investment returns, population growth rates, and other multiplicative processes.
What Is the Geometric Mean?
The geometric mean is a type of average that is calculated by multiplying all values together and then taking the nth root, where n is the number of values. It is especially useful when values are multiplicative — such as growth rates or ratios.
The Formula
For n values:
Geometric Mean = (x₁ × x₂ × x₃ × … × xₙ)^(1/n)
Or equivalently using logarithms (better for large datasets):
Geometric Mean = exp[(ln x₁ + ln x₂ + … + ln xₙ) ÷ n]
When to Use the Geometric Mean vs Arithmetic Mean
| Situation | Use |
|---|---|
| Investment returns over multiple years | Geometric mean |
| Population growth rates | Geometric mean |
| Simple averages (test scores, heights) | Arithmetic mean |
| Ratios and indices | Geometric mean |
| Lengths, weights, ages | Arithmetic mean |
Practical Example — Investment Returns
An investment returns: +50% in year 1, −20% in year 2, +30% in year 3.
As multipliers: 1.50, 0.80, 1.30
Arithmetic average = (1.50 + 0.80 + 1.30) ÷ 3 = 1.20 (suggests 20% average return — misleading)
Geometric mean = (1.50 × 0.80 × 1.30)^(1/3) = (1.56)^(1/3) = 1.1597 (actual 15.97% average — correct)
The arithmetic mean overstates the actual compound growth rate. The geometric mean gives the true compound annual growth rate (CAGR).
Another Example — Aspect Ratios
What is the geometric mean screen size between a 9-inch and 16-inch dimension?
Geometric mean = √(9 × 16) = √144 = 12 inches
This is why 12-inch screens fit naturally between standard 9:16 content.
Limitations
- All values must be positive (cannot take the log of zero or a negative number)
- For investment returns entered as percentages (e.g. +50%, −20%), convert to multipliers first: add 1 to each percentage value