Geometric Mean Calculator

Find the geometric mean of positive numbers, or the true average growth rate from a series of percentage changes.

Your data

Positive numbers, separated by commas, spaces, or new lines. 4 values read.

Geometric mean

5.65685

GM = (2 × 8 × 4 × 16)^1/4 = 1024^1/4 = 5.65685

Arithmetic mean = 7.5; harmonic mean = 4.26667

For positive values that are not all equal, harmonic mean < geometric mean < arithmetic mean.

How to use this tool

Choose Numbers to find the geometric mean of a list of positive values, or Growth rates to average percentage changes over several periods.
Enter your values, separated by commas, spaces, or new lines.
Read the result and the working, including the comparison with the arithmetic mean.
For growth rates, note how the simple average overstates growth.

Formula Reference

Geometric mean

GM = (x₁ × x₂ × … × xₙ)^(1/n)

Via logarithms

GM = exp[ (Σ ln xᵢ) ÷ n ]

Average growth rate

(Π (1 + rᵢ))^(1/n) − 1

Harmonic mean

n ÷ Σ(1/xᵢ)

ArticleWeighted and Geometric Mean ArticleMean, Median and Mode ArticleThe Log-Normal Distribution CalcWeighted Mean Calculator CalcDescriptive Statistics Calculator GlossaryGlossary: Geometric Mean

Frequently Asked Questions

When should I use the geometric mean?+

Use it for quantities that combine by multiplication, such as growth rates, ratios, and index numbers. The arithmetic mean of growth rates overstates the true average growth, while the geometric mean gives the constant rate that would produce the same total.

Why must every value be positive?+

The geometric mean multiplies the values together and takes a root, so a zero makes the product zero and a negative value can make the root undefined. For growth rates the calculator works with growth factors (1 + rate), which must be positive.

Is the geometric mean always smaller than the arithmetic mean?+

It is never larger. The two are equal only when all values are identical. The more spread out the values are, the bigger the gap.

What is the geometric mean of 2, 8, 4 and 16?+

Multiply them to get 1,024, then take the fourth root: about 5.657. The arithmetic mean of the same numbers is 7.5.

How to use this tool

Choose Numbers to find the geometric mean of a list of positive values, or Growth rates to average percentage changes over several periods.

Enter your values, separated by commas, spaces, or new lines.

Read the result and the working, including the comparison with the arithmetic mean.

For growth rates, note how the simple average overstates growth.