Geometric Mean Calculator
Calculate the geometric mean (nth root of the product) of any set of positive numbers. Ideal for growth rates, ratios, and financial returns.
Results
Geometric Mean
The geometric mean (GM) of n positive numbers is the nth root of their product: GM = (x₁ × x₂ × … × xₙ)^(1/n). An equivalent formula using logarithms is GM = exp(mean of ln(xᵢ)), which is numerically more stable for large datasets.
Growth Rates
If an investment returned +50%, −33%, +20% in three years, the growth factors are 1.50, 0.67, 1.20. The geometric mean = (1.50 × 0.67 × 1.20)^(1/3) ≈ 1.0, meaning roughly 0% average annual growth. The arithmetic mean of the percentages (+12.3%) would be misleadingly positive.
AM-GM Inequality
For any set of positive numbers, the arithmetic mean is always ≥ the geometric mean (equality only when all values are identical). This fundamental inequality appears in optimization, economics, and inequalities proofs throughout mathematics.
Worked Example: Investment Returns
A portfolio had these annual returns over 4 years: +20%, −10%, +30%, +5%. Convert to growth factors: 1.20, 0.90, 1.30, 1.05. Geometric mean = (1.20 × 0.90 × 1.30 × 1.05)^(1/4) = (1.4742)^(0.25) ≈ 1.1020.
This means the average annual return was about 10.2% — the single rate that, compounded four times, gives the same total growth. The arithmetic mean of the percentages (+11.25%) is slightly higher but misleading because it ignores compounding effects.
Comparing the Three Means
| Mean Type | Formula | Best For |
|---|---|---|
| Arithmetic | (x₁+x₂+...+xₙ)/n | Additive data: scores, lengths |
| Geometric | (x₁×x₂×...×xₙ)^(1/n) | Multiplicative data: growth rates, ratios |
| Harmonic | n / (1/x₁+1/x₂+...+1/xₙ) | Rates and speeds averaged over distance |
For positive numbers, Harmonic Mean ≤ Geometric Mean ≤ Arithmetic Mean. The three are equal only when all values are identical.
Frequently Asked Questions
GM = (x₁ × x₂ × … × xₙ)^(1/n). Equivalently, GM = exp((ln x₁ + ln x₂ + … + ln xₙ)/n). Both give the same result; the log form is preferred computationally for large n to avoid overflow.
Use geometric mean for multiplicative or ratio quantities: investment returns, population growth rates, price indices. Use arithmetic mean for additive quantities: test scores, temperatures, distances.
The nth root of a negative product may be complex (imaginary). And zero makes the product zero, giving GM=0 regardless of other values. All inputs must be strictly positive (> 0).
For two numbers a and b, the geometric mean √(ab) is the side of a square with the same area as a rectangle of sides a and b. For three numbers, it's the side of a cube with the same volume as a box with those dimensions.
Financial analysts use the geometric mean to calculate compound annual growth rate (CAGR). If an investment grows from $1,000 to $1,500 over 4 years, CAGR = (1500/1000)^(1/4) − 1 = 1.5^0.25 − 1 ≈ 10.67%. This geometric mean of the growth factor accurately represents the constant annual rate that would produce the same total growth.
Yes, as long as all values are strictly positive. Numbers between 0 and 1 are valid inputs (e.g., proportions, ratios less than one). The geometric mean of 0.5, 0.8, and 0.4 = (0.5 × 0.8 × 0.4)^(1/3) = (0.16)^(1/3) ≈ 0.543. The result will also be between 0 and 1.