Geometric Mean Calculator

The geometric mean of n positive numbers is the nth root of their product: for 4 and 9, \(\sqrt{4 \times 9} = 6\). It is the right average for growth rates: returns of +10%, −20% and +30% average 4.59% a year, not 6.67%.

How to use the geometric mean calculator

1. Enter the values separated by commas or spaces.
2. Choose Numbers for ordinary values, or Percent returns to find an average growth rate.
3. Press Calculate to see the geometric mean, the steps and the other two means.

Formula

\(G = \sqrt[n]{x_{1} x_{2} \cdots x_{n}} = \exp\left(\frac{1}{n}\sum_{i=1}^{n} \ln x_{i}\right)\)

For rates, convert each percent r to a growth factor 1 + r/100, take the geometric mean of the factors, and subtract 1.

Worked example: average return

Returns of +10%, −20%, +30% give factors 1.1, 0.8, 1.3. Product = 1.144. \(\sqrt[3]{1.144} \approx 1.04586\), so the average return is about 4.59% a year. Check: \(1.04586^{3} \approx 1.144\), the same total as the actual returns. The arithmetic average, 6.67%, would overstate the result.

Arithmetic, geometric and harmonic means

For positive numbers, harmonic ≤ geometric ≤ arithmetic, with equality only when all values are equal. Use the geometric mean for ratios, growth rates and values that multiply; the arithmetic mean for amounts that add; the harmonic mean for rates like speeds over equal distances.

Geometric mean practice quiz: 10 questions

Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.

Frequently asked questions

How do you calculate the geometric mean?

Multiply the n values and take the nth root. For many or large values, average their natural logarithms and take e to that power.

When should I use the geometric mean?

For quantities that multiply together, such as investment returns, growth rates, ratios and index numbers.

Can the geometric mean be negative?

No. It is defined for positive values. For returns, use growth factors (1 + r), which must be positive.

Why is the geometric mean smaller than the average?

By the AM–GM inequality, the geometric mean never exceeds the arithmetic mean, and is strictly smaller unless all values are equal.

What is the CAGR?

The compound annual growth rate is the geometric mean of the yearly growth factors minus 1, the same as the Percent returns mode here.

Sources

• OpenStax Introductory Statistics 2e
• NIST/SEMATECH e-Handbook of Statistical Methods

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