# What is the geometric mean?

Computes the geometric mean of a list of positive numbers, the nth root of their product, with the arithmetic and harmonic means for comparison.

- Page: https://www.acalculator.org/statistics/geometric-mean-calculator
- JSON spec: https://www.acalculator.org/statistics/geometric-mean-calculator.json
- Version: 6a960597fbab

## Default answer

Example with the default inputs (Your numbers [10, 51.2, 8]): The geometric mean of the 3 numbers is 16.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| data | Your numbers | Positive numbers (0 is allowed), separated by commas, spaces, semicolons, or new lines. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| gm | Geometric mean | The nth root of the product of the n numbers. |
| am | Arithmetic mean | The sum divided by the count. |
| hm | Harmonic mean | The count divided by the sum of the reciprocals. It is not shown when a number is 0. |
| n | Count (n) | How many numbers are in the list. |

## Method

GM = (x₁ × x₂ × … × xₙ)^(1/n) = exp((ln x₁ + … + ln xₙ) ÷ n); AM = Σx ÷ n; HM = n ÷ Σ(1/x).

## Assumptions

- Every number is 0 or more. With a 0 in the list, the geometric mean is 0 and the harmonic mean is not shown.
- For growth rates, enter growth factors: +5% is 1.05 and −3% is 0.97. The average rate is the geometric mean minus 1.

## Worked examples

1. data = 10 or 51.2 gives gm = 16, am = 23.066667, hm = 12.268371, n = 3. Source: OpenStax, Introductory Business Statistics, §2.5 Geometric Mean (x̃ = (x₁ × x₂ × … × xₙ)^(1/n); 10, 51.2 and 8 have product 4,096 and geometric mean 16), https://openstax.org/books/introductory-business-statistics/pages/2-5-geometric-mean (retrieved 2026-10-05).
2. data = 2 or 8 gives gm = 4, am = 5, hm = 3.2. Source: OpenStax, Introductory Business Statistics, §2.5 Geometric Mean (x̃ = (x₁ × x₂ × … × xₙ)^(1/n); 10, 51.2 and 8 have product 4,096 and geometric mean 16), https://openstax.org/books/introductory-business-statistics/pages/2-5-geometric-mean (retrieved 2026-10-05).
3. data = 1.1 or 0.9 gives gm = 1.059105, am = 1.066667. Source: OpenStax, Introductory Business Statistics, §2.5 Geometric Mean (x̃ = (x₁ × x₂ × … × xₙ)^(1/n); 10, 51.2 and 8 have product 4,096 and geometric mean 16), https://openstax.org/books/introductory-business-statistics/pages/2-5-geometric-mean (retrieved 2026-10-05).
4. data = 5 or 0 gives gm = 0, am = 2.666667. Source: OpenStax, Introductory Business Statistics, §2.5 Geometric Mean (x̃ = (x₁ × x₂ × … × xₙ)^(1/n); 10, 51.2 and 8 have product 4,096 and geometric mean 16), https://openstax.org/books/introductory-business-statistics/pages/2-5-geometric-mean (retrieved 2026-10-05).

## FAQ

### What is the geometric mean?

The geometric mean of n numbers is the nth root of their product. For 10, 51.2 and 8, the product is 4,096 and the cube root of 4,096 is 16. It is the number that, used n times in place of each value, gives the same product.

### When should I use the geometric mean instead of the average?

Use it for values that multiply together, such as growth factors, investment returns over several years, and ratios or indexes. The ordinary (arithmetic) average suits values that add together.

### How do I find the average growth rate with the geometric mean?

Turn each rate into a growth factor: +10% is 1.1, −10% is 0.9 and +20% is 1.2. The geometric mean of 1.1, 0.9 and 1.2 is 1.059105, so the average growth rate is about 5.91% a year. The arithmetic average of the rates, 6.67%, overstates it.

### Can the geometric mean have negative numbers or zero?

Not negative numbers: their product can be negative, and a negative number has no real even root. This page gives no answer for them. A zero is allowed, but it makes the product 0, so the geometric mean is 0.

### Why is the geometric mean smaller than the arithmetic mean?

For positive numbers, the geometric mean is never more than the arithmetic mean, and the harmonic mean is never more than the geometric mean. All three are equal only when every number is the same. The more spread out the numbers, the bigger the gap.

### How is the geometric mean calculated for a long list?

Multiplying many numbers can overflow, so the page adds their natural logarithms, divides by n and takes the exponential: GM = exp((ln x₁ + … + ln xₙ) ÷ n). This gives the same value as the nth root of the product.

## Sources

- OpenStax, Introductory Business Statistics, §2.5 Geometric Mean: x̃ = (x₁ × x₂ × … × xₙ)^(1/n); 10, 51.2 and 8 have product 4,096 and geometric mean 16. https://openstax.org/books/introductory-business-statistics/pages/2-5-geometric-mean (retrieved 2026-10-05)