How to use this geometric mean calculator
Enter positive numbers above (growth factors, ratios or rates in decimal form) and press Calculate. Alongside the geometric mean you will see the arithmetic mean, so the difference is visible rather than theoretical.
The right average for growth
The arithmetic mean assumes things add. Growth multiplies: money compounds, populations multiply, areas scale. Averaging multipliers with an adding tool gives an answer that is always too high: a mathematical guarantee (the AM–GM inequality), not a rounding issue. The geometric mean fixes it: multiply all the values, then take the n-th root.
Three years of an investment: +10%, +20%, then −5%. As factors: 1.10, 1.20, 0.95. Their product is 1.254, and the cube root is ≈ 1.0784, an average growth of 7.84% per year. Check it: Rs 100 growing at exactly 7.84% for three years becomes ≈ Rs 125.4, matching the real path (100 becomes 110, then 132, then 125.4). The arithmetic mean would have promised 8.33%, and a final value that never happened.
Rules of the geometric mean
- All values must be positive: a single zero collapses the product to zero, and negatives break the root. Convert percentage changes to factors (add 1) before averaging.
- It is always ≤ the arithmetic mean for the same data, with equality only when all values are identical.
- It answers "what constant rate would have produced the same overall change?", the right question for returns, growth and ratios.
Frequently asked questions
When should I use the geometric mean instead of the arithmetic mean?
When the values multiply rather than add: investment returns, growth rates, ratios and index changes. For ordinary additive quantities, the arithmetic mean is correct.
Why can't the geometric mean handle zero or negative values?
Because it multiplies the values and takes a root: a zero makes the whole product zero, and an even root of a negative product is not a real number. Convert percentage changes into growth factors (1 + rate) first.
What is the geometric mean of 2 and 8?
√(2 × 8) = √16 = 4, noticeably less than the arithmetic mean of 5.