Geometric mean

The geometric mean is the nth root of the product of n values: GM = (x₁ × x₂ × ... × xₙ)^(1/n). Unlike the arithmetic mean, the geometric mean is the right tool for averaging ratios, percentages, and compounding growth rates.

Why it matters: imagine an investment that returns +50% in year 1 and −50% in year 2. The arithmetic mean of [1.5, 0.5] is 1.0 — suggesting the investment broke even. But $100 invested becomes $150 after year 1 and $75 after year 2 — a 25% loss. The geometric mean of [1.5, 0.5] is √(1.5 × 0.5) = √0.75 ≈ 0.866, correctly indicating a 13.4% annualised loss.

Standard use cases:

• Compound annual growth rate (CAGR). The geometric mean of the year-over-year ratios.
• Index numbers. Stock indices weighted by ratios use geometric means to avoid arithmetic-mean distortion.
• Ratio-of-ratios comparisons. When the inputs are multiplicative rather than additive (gains compounding, dosing).

The geometric mean is always ≤ the arithmetic mean. Equality holds only when all values are identical. For widely varying values, the gap can be large — which is precisely why using the right one matters.