Geometric Mean Calculator

Convert percentage changes to growth factors before averaging: +10% becomes 1.10, −5% becomes 0.95, +20% becomes 1.20.

All inputs must be strictly positive — a zero or negative value has no logarithm, and a single zero would force any product to zero. The calculator flags such values instead of silently dropping them.

The textbook definition multiplies all n values together and takes the n-th root. This calculator uses the mathematically identical logarithmic form:

The two are equal because a root of a product is the exponential of an average of logarithms. Computationally, though, they behave very differently: multiplying hundreds of values can overflow to infinity or underflow to zero in floating-point arithmetic long before the final root is taken. Summing logarithms never leaves a safe numeric range, which is why serious statistical software — and this page — computes the geometric mean as exp(mean(ln x)).

Investment returns compound: each year's result multiplies what the previous years left behind. Averaging the percentages arithmetically ignores that chaining, and the error always lands on the flattering side. The culprit is volatility — a 50% loss needs a 100% gain to break even, so up-and-down swings destroy more value than a simple average of the percentages suggests. Finance calls the gap between the two averages volatility drag.

The geometric mean of the growth factors is, by construction, the single constant rate that reproduces the actual ending value. That is exactly what a compound annual growth rate (CAGR) is. The more the returns bounce around, the further the arithmetic average drifts above this true compounded rate — for steady returns the two agree almost exactly.

For any set of positive numbers, the arithmetic mean is greater than or equal to the geometric mean, with equality only when every value is identical:

This is one of the oldest and most useful inequalities in mathematics, and the calculator lets you watch it in action: the reported gap is always zero or positive, and it grows with the spread of your data. Practically, the inequality is a guarantee that quoting an arithmetic average of growth rates never understates performance — which is precisely why fund marketing prefers it and financial regulators pay attention to which average is quoted. The mean, median, mode calculator covers the arithmetic side of the story, and the harmonic mean calculator completes the classical chain: AM ≥ GM ≥ HM.

A portfolio returns +10%, −5%, and +20% in three consecutive years. What was the average annual growth?

1. Convert to growth factors: 1.10, 0.95, 1.20.
2. Multiply: 1.10 × 0.95 × 1.20 = 1.2540 — the portfolio grew 25.4% in total.
3. Geometric mean: 1.2540^(1/3) ≈ 1.0784, an average growth of 7.84% per year.
4. Check: 1.0784³ ≈ 1.254, exactly the observed total growth.
5. Arithmetic mean for comparison: (10 − 5 + 20) ÷ 3 = 8.33% per year.

The arithmetic average of 8.33% overstates reality: compounding at 8.33% for three years would produce 1.0833³ ≈ 1.2714, a 27.1% total gain the portfolio never achieved. The geometric mean of 7.84% is the only constant rate consistent with the money actually in the account — the down year costs more than the simple average admits.