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Geometric Mean Calculator — Step-by-Step Solutions

The Geometric Mean Calculator computes the central tendency of a set of positive numbers using multiplication rather than addition. It is ideal for averaging rates of return, growth rates, and ratios.

Enter a dataset of positive values for the geometric mean, input percentage growth rates to find the average growth rate, or compare the geometric mean against the arithmetic and harmonic means.

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How to Use the Geometric Mean

  1. Geometric Mean: enter positive numbers separated by commas, such as 2, 8, 4, 16. The result is the geometric mean along with the count and the product.
  2. Growth Rates: enter percentage growth rates for consecutive periods, for example 10, -20, 30. A negative rate is a decline. The calculator gives the average growth rate per period, the overall growth factor and the geometric mean of the growth factors.
  3. GM vs AM: enter positive numbers to compare the arithmetic, geometric and harmonic means, and see how far the arithmetic mean is above the geometric mean.
  4. All values must be greater than zero in the Geometric Mean and GM vs AM modes. In Growth Rates mode each rate must be above -100%.

Formula and Theory

The geometric mean of n positive numbers is the n-th root of their product. It is the right average for quantities that combine by multiplication, such as growth rates, ratios, and index numbers. Its meaning is the single constant factor that would give the same overall result as the actual sequence of factors.

For growth rates, convert each rate r% to the growth factor 1 + r/100, take the geometric mean of the factors, and subtract 1. This gives the compound average growth per period. The arithmetic mean of the rates overstates it whenever the rates vary.

In practice the calculator works with logarithms, GM = exp(mean of ln x), which avoids overflow with long lists. For any positive numbers the three Pythagorean means satisfy HM <= GM <= AM, with equality only when all values are equal.

The geometric mean is undefined if any value is zero or negative, because the product can then be zero or negative and the root does not describe a growth factor.

Worked Examples

Example 1: Geometric mean of a list

Problem: Find the geometric mean of 2, 8, 4, 16.

Solution: Product = 2 x 8 x 4 x 16 = 1,024. There are 4 numbers, so take the fourth root: 1,024^(1/4). Since 1,024 = 2^10, this is 2^(10/4) = 2^2.5 = 5.6569.

Answer: 5.6569

Example 2: Average growth over three years

Problem: An investment changes by +10%, -20% and +30% in three consecutive years. What is the average annual growth rate?

Solution: Growth factors: 1.10, 0.80, 1.30. Product = 1.10 x 0.80 x 1.30 = 1.144, so the total growth is 14.4%. Geometric mean = 1.144^(1/3) = 1.04586. Average growth rate = 4.586% per year. The arithmetic mean of the rates is (10 - 20 + 30) / 3 = 6.67%, which would imply 1.0667^3 = 21.4% total, far above the actual 14.4%.

Answer: About 4.59% per year (overall factor 1.144)

Example 3: Comparing the three means

Problem: Compare the arithmetic, geometric and harmonic means of 2 and 8.

Solution: AM = (2 + 8) / 2 = 5. GM = sqrt(2 x 8) = sqrt(16) = 4. HM = 2 / (1/2 + 1/8) = 2 / 0.625 = 3.2. So HM <= GM <= AM: 3.2 <= 4 <= 5. Also GM^2 = AM x HM for two numbers: 16 = 5 x 3.2.

Answer: AM 5, GM 4, HM 3.2

Frequently Asked Questions

When should I use the geometric mean instead of the average?
Use it for numbers that multiply: growth rates, investment returns, ratios and index values. The arithmetic mean overstates the typical result for those because it ignores compounding.
How do I calculate the geometric mean by hand?
Multiply the numbers together and take the n-th root, where n is how many numbers you have. For long lists, average the natural logs and exponentiate.
Why can it not include zero or negative numbers?
A zero makes the whole product zero, and negative numbers can make the product negative so that no real n-th root exists. For growth rates, use factors (1 + rate), which stay positive unless the loss is 100% or more.
Is the geometric mean always smaller than the arithmetic mean?
It is never larger. The geometric mean equals the arithmetic mean only when all the numbers are equal, and is smaller otherwise.
How do I find the average annual growth rate (CAGR)?
It is the geometric mean of the yearly growth factors minus 1. The Growth Rates mode does exactly this from a list of yearly percentage changes.
What is the overall growth factor?
It is the product of all period growth factors. A value of 1.144 means the quantity ended 14.4% above where it started.

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