Geometric Mean Calculator - nth Root of a Product
Calculate the geometric mean of a list of positive numbers, with a comparison to the arithmetic mean.
Numbers (comma, space, or line separated, all positive)
Geometric Mean
5.6569
Arithmetic mean for comparison: 7.5
4 values
How to Calculate the Geometric Mean
The geometric mean of n numbers is the nth root of their product: multiply all the values together, then take the nth root of the result. This calculator computes it as exp(average of the natural logs of each number), which is mathematically equivalent but avoids numerical overflow when multiplying many large values directly.
Example
For 4, 8, 16, and 2, the product is 4 × 8 × 16 × 2 = 1,024, and the geometric mean is the 4th root of 1,024, which is 5.657. Compare this to the arithmetic mean of the same numbers, (4 + 8 + 16 + 2) ÷ 4 = 7.5 — the geometric mean is always less than or equal to the arithmetic mean for positive numbers that aren't all identical.
Common Use Cases
- Averaging growth rates or investment returns over multiple periods.
- Averaging ratios or index values, where the arithmetic mean would be misleading.
- Computing average factors of change, like population growth rate or compound interest rate.
FAQs
When should I use geometric mean instead of arithmetic mean?
Geometric mean is the right choice when averaging values that are multiplied together over time or combined multiplicatively, such as annual growth rates, investment returns, or ratios. Arithmetic mean works best for values that are simply added together, like test scores or measurements from repeated trials.
Why does the calculator require all positive numbers?
The geometric mean involves taking a root of the product of the values. If any value is zero, the entire product becomes zero. If any value is negative, the result can become undefined or complex for certain combinations, so this calculator requires strictly positive inputs.
Is geometric mean always smaller than arithmetic mean?
Yes, for any set of positive numbers that aren't all exactly equal, the geometric mean is always less than or equal to the arithmetic mean — this is a well-known mathematical inequality. They are equal only when every number in the set is identical.