Geometric Mean Calculator
Enter positive values for a plain geometric mean, or percentage returns to find the compound average growth rate.
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Formula
- Values:
geometric mean = (x1 × x2 × ... × xn)^(1/n), computed asexp(average of ln x) - Returns: growth factor
g = 1 + r ÷ 100;average return = (g1 × g2 × ... × gn)^(1/n) - 1;total growth = g1 × g2 × ... × gn - 1
Worked example
Values 2 and 8: product 16, square root 4 (arithmetic mean 5).
Returns of +10%, -20% and +30%:
- Growth factors 1.10, 0.80 and 1.30, product 1.144
- Total growth = 1.144 - 1 = 14.4%
- Average return = 1.144^(1/3) - 1 = 4.586% per period
- The arithmetic average, 6.667%, overstates what compounding delivers.
Frequently asked questions
How do you calculate the geometric mean?
Multiply the n numbers together and take the nth root. For 2 and 8 the product is 16 and the square root is 4. The arithmetic mean of the same numbers is 5.
How do I find the average return over several years?
Use the returns mode. Turn each return into a growth factor (10% becomes 1.10), take the geometric mean of the factors and subtract 1. Returns of +10%, -20% and +30% give 1.144^(1/3) - 1 = 4.59% a period, with a total gain of 14.4%.
Why is the geometric mean not the same as the average return?
The arithmetic mean of +10%, -20% and +30% is 6.67%, but compounding that every period would overstate your result. The geometric mean of 4.59% reproduces the actual 14.4% total.
Can the geometric mean handle zero or negative numbers?
Not in values mode: every number must be above zero, because a zero product makes the mean zero and a negative one has no real root in general. In returns mode each return must be above -100%.
Is the geometric mean always smaller than the arithmetic mean?
It is never larger. They are equal only when all numbers are identical. For 4, 9, 16 and 1.5 the geometric mean is 5.42 and the arithmetic mean is 7.63.