Geometric Mean Calculator
Paste or type any list of positive numbers. The calculator returns the geometric mean — the right average for growth rates, compounding returns, and ratios — alongside the arithmetic and harmonic means for comparison.
Geometric mean
4
- Arithmetic mean
- 5
- Harmonic mean
- 3.2
- Product
- 16
- Minimum
- 2
- Maximum
- 8
- Count
- 2
Calculated from 2 values. The geometric mean is the nth root of the product — the right average for growth rates, compounding returns, and ratios.
How to use this calculator
Type or paste positive numbers into the input. Separators can be commas, spaces, tabs, semicolons or new lines — mix them however you like. The headline result is the geometric mean; the breakdown shows the arithmetic mean, harmonic mean, product, minimum, maximum and count so you can see how the three classical means relate. The geometric mean is only defined for strictly positive values, so any zero or negative number will halt the calculation with a clear error. To average annual growth rates or investment returns, convert each rate r to a growth factor (1 + r), take the geometric mean of the factors, then subtract one — the result is the compound annual growth rate.
How the calculation works
The geometric mean of n positive numbers is defined as the nth root of their product: GM = (x₁ · x₂ · … · xₙ)^(1/n). Equivalently, it is the exponential of the average of the natural logarithms: GM = exp((ln x₁ + ln x₂ + … + ln xₙ) / n). The calculator uses the log-sum form internally to avoid overflow on long lists or large values. For any positive dataset the AM-GM-HM inequality holds: HM ≤ GM ≤ AM, with equality only when every value is identical. The geometric mean is the correct average whenever the quantity is multiplicative — successive growth factors, ratios such as price-to-earnings, index numbers, or anything that compounds — because it preserves the property that applying the mean n times reproduces the original product.
Worked example
For the values 1, 2, 4: the product is 1 × 2 × 4 = 8 and there are three numbers, so the geometric mean is ∛8 = 2 exactly. The arithmetic mean is (1 + 2 + 4) ÷ 3 ≈ 2.333 and the harmonic mean is 3 ÷ (1 + 0.5 + 0.25) = 12/7 ≈ 1.7143 — illustrating HM ≤ GM ≤ AM. As a practical application: a portfolio returns +10%, −5% and +20% over three years. The growth factors are 1.10, 0.95 and 1.20, with product 1.254. The geometric mean factor is ∛1.254 ≈ 1.0784, so the compound annual growth rate is about 7.84% — meaningfully different from the arithmetic mean of the rates, which is 8.33% and overstates the true annualised return.
Frequently asked questions
When should I use the geometric mean instead of the arithmetic mean?
Use the geometric mean whenever the values are multiplicative — growth rates, investment returns, index numbers, dilution factors, price ratios, or anything where applying the mean repeatedly should reproduce the cumulative effect. The arithmetic mean of percentage returns will systematically overstate the true compound annual return; the geometric mean of the growth factors (1 + r) gives the CAGR exactly. For additive quantities such as test scores or temperatures, the arithmetic mean remains the correct choice.
Why does the calculator reject zero and negative numbers?
The geometric mean is defined only for strictly positive real numbers, because it is the nth root of a product. A single zero collapses the product to zero and the geometric mean to zero, which obscures the rest of the data. Negative values can make the root undefined (the cube root of a negative is real, but the square root is not) and turn the answer meaningless for the growth-rate and ratio applications geometric means are used for. By convention the calculator halts and asks you to clean the input rather than silently produce a misleading result.
How does the geometric mean relate to the arithmetic and harmonic means?
For any set of positive numbers, HM ≤ GM ≤ AM — the harmonic mean is the smallest, the arithmetic mean is the largest, and the geometric mean sits between them. The three are equal only when every value in the dataset is the same. The further apart the values, the larger the gap between the three means. The calculator shows all three so you can compare them and pick the one that fits the question you are actually asking.
How do I compute the compound annual growth rate (CAGR)?
Convert each periodic return r to a growth factor (1 + r). For example, +10% becomes 1.10, −5% becomes 0.95. Paste those factors into the calculator. The geometric mean of the factors is the average growth factor per period; subtract one to express it as a rate. For returns of +10%, −5% and +20%, the factors 1.10, 0.95, 1.20 have a geometric mean of about 1.0784, so the CAGR is about 7.84%. Equivalently, CAGR = (end value ÷ start value)^(1/years) − 1.
Why use the log-sum formulation internally?
Mathematically, (x₁·x₂·…·xₙ)^(1/n) and exp(Σ ln xᵢ / n) are identical. Numerically, the product of many values can overflow or underflow long before the final root brings it back into range — even a list of a hundred 10s would push past the limits of a normal floating-point number. Summing logarithms keeps every intermediate value bounded, so the calculator stays accurate for long lists or extreme values without losing precision on small examples.
Is the geometric mean used in finance and economics?
Yes. CAGR, time-weighted return, and most index-construction methods rely on the geometric mean. The Financial Times All-Share index uses arithmetic averaging for historical reasons, but most modern equal-weighted return indices and inflation-adjusted return calculations use geometric averaging because it correctly reflects compounding. The geometric mean is also the right way to combine ratios such as price-to-earnings across periods.
What separators can I use to paste a list of numbers?
Commas, spaces, tabs, semicolons or new lines all work, and you can mix them. A column copied straight from a spreadsheet, a comma-separated row, or just numbers separated by spaces all parse correctly. Tokens that are not valid numbers are reported under the result so you can see what was ignored.