About the Geometric Mean Calculator
This geometric mean calculator finds the “multiplicative average” of a list. In numbers mode, enter positive values and it returns the geometric mean — the n-th root of their product — alongside the arithmetic and harmonic means for comparison. In growth-rate mode, enter percentage changes such as yearly investment returns (+10, −5, 20 …) and it converts each to a growth factor, takes the geometric mean and gives the average compound growth rate and the total growth over the whole period.
Use it whenever values multiply rather than add: investment and sales growth, population growth, inflation over several years, bacterial growth, ratios and index numbers, or averaging quantities that span several orders of magnitude. Investors use it because the arithmetic mean of returns overstates what a portfolio actually earned, while the geometric mean is the constant rate that produces the same final value.
The geometric mean is only defined for positive numbers, so in numbers mode zeros and negatives are ignored with a warning. In growth-rate mode each rate must be greater than −100% (a −100% return wipes out the investment).
With the default inputs, the geometric mean (or average growth rate, %) is 7.79612. Change any value above to recalculate instantly.
How to use the geometric mean calculator
- 1Choose “numbers” for plain values or ratios, or “growth rates” for percentage changes.
- 2Paste your values, separated by commas, spaces or new lines.
- 3Read the geometric mean (or average compound growth rate).
- 4Compare it with the arithmetic mean to see how much compounding matters.
Formula and method
The geometric mean is the n-th root of the product of n positive numbers. To avoid overflow with long lists, the calculator averages the natural logarithms and exponentiates the result, which gives exactly the same answer. It is always less than or equal to the arithmetic mean, with equality only when every value is the same.
For growth rates, each percentage change r is converted to a growth factor 1 + r/100. The geometric mean of the factors, minus 1, is the constant rate per period that would turn the starting value into the same ending value — the compound annual growth rate (CAGR) when periods are years. The arithmetic mean of the rates ignores compounding and overstates growth whenever returns vary.
- xᵢ
- Positive values to average
- rᵢ
- Growth rate or return in period i (as a decimal)
- n
- Number of values or periods
- Π
- Product of all terms
Worked examples
Geometric mean of 3, 5, 8, 12, 20
The product is 3 × 5 × 8 × 12 × 20 = 28,800 and its fifth root is about 7.796. That is well below the arithmetic mean of 9.6 because the larger values pull the ordinary average up.
Four years of returns: +10%, −5%, +20%, +3%
The growth factors 1.10 × 0.95 × 1.20 × 1.03 multiply to 1.29162, a total gain of 29.16%. The fourth root is 1.066065, so the average compound return is about 6.61% per year — less than the 7% arithmetic average.
Volatile returns: +25%, −20%, +15%, −10%, +30%
The arithmetic mean says 8% a year, but the factors multiply to 1.3455, so the investment grew only 34.55% in total — equivalent to 6.11% compounded annually. The bigger the swings, the bigger the gap.
Frequently asked questions
How do you calculate the geometric mean?+
Multiply all n values together and take the n-th root. For 4 and 9 the product is 36 and the square root is 6, so the geometric mean is 6 (the arithmetic mean would be 6.5).
When should I use the geometric mean instead of the arithmetic mean?+
Use it when values combine by multiplication: growth rates, investment returns, inflation, ratios and index numbers, or data spanning several orders of magnitude. The arithmetic mean suits quantities that add up, such as heights or prices.
How do I find the geometric mean of percentage returns?+
Convert each return to a growth factor (1 + r), take the geometric mean of the factors and subtract 1. Returns of +10% and −10% give √(1.1 × 0.9) − 1 ≈ −0.50%, not 0%.
Can the geometric mean be calculated with zero or negative numbers?+
Not in the usual sense. A zero makes the product zero, and negative numbers can make the root undefined. For returns, work with growth factors (1 + r), which stay positive as long as no loss reaches 100%.
Why is the geometric mean always less than the arithmetic mean?+
This is the AM–GM inequality: for positive numbers the arithmetic mean is always at least the geometric mean, with equality only when all values are identical. The more spread out the data, the larger the gap.