The formula: nth root of the product
For the numbers 4 and 9: multiply them together (4 × 9 = 36), then take the square root (since there are 2 numbers): √36 = 6. The geometric mean of 4 and 9 is exactly 6.
A four-number example
For 1, 3, 9, and 27: multiply all four together (1 × 3 × 9 × 27 = 729), then take the 4th root: 729^(1/4) ≈ 5.196152.
Why it's the right average for compounding quantities
When values represent successive multiplicative changes — like a series of annual growth rates — the geometric mean answers "what single constant rate, applied repeatedly, would produce the same overall result?" This is exactly what CAGR (Compound Annual Growth Rate) computes for the specific case of an initial and final investment value.
Why all numbers must be positive
The calculation involves multiplying every number together and then taking a root — if any value were zero, the whole product would be zero; if a value were negative, the root of the product may not be a real number, depending on how many negatives are in the list.