What is the geometric mean?

The geometric mean of n positive numbers is the nth root of their product. For example, the geometric mean of 2, 8, and 32 is the cube root of (2 × 8 × 32) = the cube root of 512 = 8.

When should I use the geometric mean instead of the arithmetic mean?

Use the geometric mean when your values represent multiplicative factors, ratios, or percentage growth rates. If you averaged 10% and 50% annual returns arithmetically you would get 30%, but geometrically the multipliers 1.10 and 1.50 give a true CAGR of about 28.5%. The geometric mean is more accurate for compounding scenarios.

Why must all numbers be positive?

The geometric mean requires taking the nth root of a product. If any value is zero the result is always zero; if values are negative the product of an even count of negatives is positive but the interpretation is meaningless for most real-world use cases. This calculator requires all values to be strictly greater than zero.

How is the geometric mean calculated?

Multiply all n values together to get a product P, then compute P^(1/n). Equivalently, take the arithmetic mean of the natural logarithms of the values, then exponentiate: exp( (ln x₁ + ln x₂ + … + ln xₙ) / n ). The log-sum approach is used here to avoid overflow with large products.

What is the geometric mean used for in finance?

In finance, the geometric mean gives the compound annual growth rate (CAGR). If an investment grew by 10%, shrank by 5%, and grew by 20% over three years, the CAGR is the geometric mean of 1.10, 0.95, and 1.20 minus 1. This gives a single representative annual return that, if applied each year, would yield the same final value.

Can I enter percentages?

Enter growth multipliers, not percentages. A 10% gain is 1.10, a 5% loss is 0.95, and flat (0%) is 1.00. The calculator works with the raw multiplier values. If you enter percentages like 10 and 5, the result is the geometric mean of those numbers — which may not equal the CAGR you intended.

Geometric Mean Calculator

Enter a list of positive numbers and instantly get their geometric mean — the multiplicative average used for growth rates and ratios.

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Summary

The geometric mean is the nth root of the product of n numbers. Unlike the arithmetic mean, it gives the "central tendency" of multiplicative data — making it the correct average for growth rates, percentage changes, and financial returns. For example, the geometric mean of annual returns gives the compound annual growth rate (CAGR). This calculator accepts any set of positive numbers (comma or newline separated) and computes the result instantly in your browser.

How it works

Enter your numbers in the input box, separated by commas or new lines.
The tool parses and validates each value — all numbers must be positive.
It computes the product of all values, then takes the nth root (where n is the count).
The geometric mean, count, product, and step-by-step formula are displayed immediately.
Click "Copy Result" to copy the geometric mean to your clipboard.

Use cases

Calculate compound annual growth rate (CAGR) from a series of yearly returns.
Average ratios or index values without distortion from large outliers.
Find the geometric mean of population growth multipliers over several periods.
Compare investment portfolio performance across multiple assets.
Average biological or scientific measurements that span multiple orders of magnitude.
Normalize price relatives and index numbers in economics.
Compute the geometric mean in statistics homework or exams.
Benchmark normalized test scores across categories.

Frequently Asked Questions

Last updated: 2026-06-04 · Reviewed by Nham Vu