Geometric Mean Calculator
Find the geometric mean of positive values or percentage growth rates, and compare it with the arithmetic and harmonic means.
Geometric mean formula
G = (x₁ · x₂ · … · xₙ)^(1/n) = exp( mean of ln xᵢ )
Why the arithmetic mean misleads for growth
An investment that gains 50% one year and loses 50% the next has an arithmetic mean return of 0%, yet you end with 1.5 × 0.5 = 0.75 of your money. The geometric mean of 1.5 and 0.5 is √0.75 ≈ 0.866, an average loss of 13.4% a year, which matches reality.
The three Pythagorean means always satisfy harmonic ≤ geometric ≤ arithmetic, with equality only when every value is the same. The wider the spread, the bigger the gap.
Frequently asked questions
When should I use the geometric mean?
Use it for quantities that multiply together over time: investment returns, population growth, bacterial doubling, or ratios and index numbers. The geometric mean growth factor, applied every period, reproduces the same final result as the actual sequence.
How do I find the geometric mean of percentages?
Convert each percentage change to a growth factor (+10% → 1.10, −5% → 0.95), take the geometric mean of the factors, then subtract 1. Switch the calculator to “percent changes” and it does this for you.
Why can’t the geometric mean include zero or negative numbers?
It is the nth root of a product. A zero makes the product zero, and negative values can make it negative, so the root is undefined or meaningless. Convert percentage losses to growth factors first.
What is the harmonic mean used for?
Averaging rates where the numerator is fixed, such as average speed over equal distances, or price-to-earnings ratios. For 60 km/h out and 40 km/h back, the average speed is the harmonic mean, 48 km/h, not 50.