Geometric Mean Calculator

The geometric mean is the nth root of the product of n positive numbers. It is the right average for growth rates, ratios and anything that multiplies. Enter values to see the product, the root and the equivalent log method.

4 values read.

Geometric mean
1.0535
Arithmetic mean
1.055
Count
4
Avg growth per period
5.3539%
0.9811.021.041.061.081.11.12
▲ mean 1.055 · ◆ median 1.065

Step-by-step working

  1. Multiply all the values1.05 × 1.12 × 0.97 × 1.08 = 1.2319776
  2. Take the 4th root of the productG = 1.2319776^(1/4) = 1.053539
  3. Same answer via logs (safer for long lists): average the natural logs, then exponentiatemean of ln(x) = 0.052155 → e^0.052155 = 1.053539

Formula G = (x₁ · x₂ · … · xₙ)^(1/n) = exp( Σ ln xᵢ ÷ n )

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An AI tutor reads your inputs and results above and explains them in plain English. The numbers come from the calculator; the explanation is AI-written, so check it against your notes.

Average growth rates

An investment grows 5%, 12%, −3% and 8% over four years. Write each year as a growth factor (1.05, 1.12, 0.97, 1.08). The geometric mean of the factors is about 1.0544, so the average annual growth is 5.44%. The arithmetic mean of the percentages (5.5%) overstates it, because gains and losses compound.

AM ≥ GM

For any positive numbers, the geometric mean is never larger than the arithmetic mean, and they are equal only when all values are the same. The bigger the spread, the bigger the gap. All values must be positive; a zero makes the product zero, and negatives make the root undefined.

Frequently asked questions

How do I calculate the geometric mean of two numbers?

Multiply them and take the square root. The geometric mean of 4 and 9 is √36 = 6.

Why use logs?

Multiplying many values can overflow or underflow a calculator. Averaging the natural logs and then exponentiating gives the same result safely.