What is the geometric mean?
Enter a list of positive numbers to get their geometric mean, the nth root of their product. The arithmetic and harmonic means show beside it for comparison.
- Geometric mean
- 16
The geometric mean of the 3 numbers is 16.
- Arithmetic mean
- 23.066667
- Harmonic mean
- 12.268371
- Count (n)
- 3
Geometric mean: 16. The geometric mean of the 3 numbers is 16.
Where do your numbers fall?
How to calculate
Computes the geometric mean of a list of positive numbers, the nth root of their product, with the arithmetic and harmonic means for comparison.
Example with the default inputs (Your numbers [10, 51.2, 8]): The geometric mean of the 3 numbers is 16.
Method: GM = (x₁ × x₂ × … × xₙ)^(1/n) = exp((ln x₁ + … + ln xₙ) ÷ n); AM = Σx ÷ n; HM = n ÷ Σ(1/x).
- 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
Each example is checked against the calculator on every build.
- Your numbers 10, 51.2, 8 gives Geometric mean 16, Arithmetic mean 23.066667, Harmonic mean 12.268371, Count (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)
- Your numbers 2, 8 gives Geometric mean 4, Arithmetic mean 5, Harmonic mean 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)
- Your numbers 1.1, 0.9, 1.2 gives Geometric mean 1.059105, Arithmetic mean 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)
- Your numbers 5, 0, 3 gives Geometric mean 0, Arithmetic mean 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)
How it works
For the n numbers x₁, x₂, …, xₙ, each 0 or more:
- Geometric mean GM = (x₁ × x₂ × … × xₙ)^(1/n), worked out as exp((ln x₁ + ln x₂ + … + ln xₙ) ÷ n), so a long product never overflows. For up to 50 numbers, when a decimal of at most 12 significant digits c has cⁿ exactly equal to the product, GM is c (so 10, 51.2 and 8 give exactly 16).
- Arithmetic mean AM = (x₁ + … + xₙ) ÷ n, exact on the decimals you type.
- Harmonic mean HM = n ÷ (1/x₁ + … + 1/xₙ), exact on the decimals you type.
Rules:
- The list has 1 to 10,000 numbers. A negative number gives no answer, with a message.
- A 0 in the list makes GM = 0; HM is then not shown, because 1/0 is not defined.
- One number is its own geometric mean.
Assumptions
- For growth rates, enter growth factors (1 + rate): +5% is 1.05, −3% is 0.97. The average rate per period is GM − 1.
Worked examples by hand
10, 51.2, 8 (the default). The product is 10 × 51.2 × 8 = 4,096 = 16³, so GM = 16. AM = 69.2 ÷ 3 = 23.066667. HM = 3 ÷ (0.1 + 0.01953125 + 0.125) = 3 ÷ 0.24453125 = 12.268371.
2, 8. GM = √16 = 4, AM = 5, HM = 2 ÷ (0.5 + 0.125) = 3.2.
1.1, 0.9, 1.2 (growth of +10%, −10%, +20%). The product is 1.188, and its cube root is 1.059105, an average growth of about 5.91% per period. AM = 1.066667.
5, 0, 3. The product is 0, so GM = 0. AM = 8 ÷ 3 = 2.666667.
Other questions people ask
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.