Geometric Mean Calculator
Find the geometric mean — the nth root of the product of n values. The geometric mean is the correct average for rates of growth, ratios, and any data that compounds multiplicatively.
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Formula
GM = (x₁ × x₂ × x₃ × x₄ × x₅)^(1/5)
Multiply all five values together and take the fifth root of the product. Equivalently, exponentiate the arithmetic mean of the logarithms of the values. All values must be positive — zero or negative numbers make the geometric mean undefined or complex. For the defaults {2, 8, 4, 16, 1}, the product is 1,024 and the fifth root is 4.0.
How to use the Geometric Mean Calculator
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Enter your value 1
- 2
Enter your value 2
- 3
Enter your value 3
- 4
Enter your value 4
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Enter your value 5
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Read your results instantly
Results update in real time as you type.
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When geometric mean beats arithmetic mean
The geometric mean is the appropriate average whenever values multiply rather than add. Annual investment returns are a perfect example: a portfolio that gains 50% one year and loses 33% the next has an arithmetic mean return of 8.5%, but the actual compound return is 0% (1.5 × 0.667 = 1.0). The geometric mean captures this correctly.
Any time you have growth rates, ratios, index numbers, or multiplicative processes, geometric mean gives the correct central tendency. Arithmetic mean will overstate performance in volatile series.
The AM-GM inequality
The arithmetic mean is always greater than or equal to the geometric mean for positive numbers (they are equal only when all values are identical). This is the AM-GM inequality, one of the most important results in mathematics.
For the default values {2, 8, 4, 16, 1}, the arithmetic mean is 6.2 while the geometric mean is exactly 4.0. The gap between them reflects the variability of the data. As values spread further apart, the gap between AM and GM widens. This gap can be used as a measure of volatility in financial time series.
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Geometric mean in finance and science
In finance, compound annual growth rate (CAGR) is the geometric mean of annual return multipliers (1 + return). If a stock returns +20%, −10%, +30%, the CAGR uses the geometric mean of {1.20, 0.90, 1.30}.
In science, geometric means are used for pH (which is logarithmic), sound intensity (decibels), earthquake magnitude (Richter scale), and biological growth rates. Any measurement on a log scale should use geometric mean for averaging rather than arithmetic mean.
Tips & Insights
All inputs must be positive
The geometric mean is undefined for zero or negative values. If your data includes zeros, consider adding a small constant before calculating, and note this adjustment in your analysis.
Geometric mean for investment returns
For multi-year investment returns, always use geometric mean. Add 1 to each return percentage (so +10% becomes 1.10), compute the geometric mean, then subtract 1 for the CAGR.
Log transform trick
The geometric mean equals exp(arithmetic mean of the natural logs). For large datasets, computing log means is numerically more stable than multiplying many numbers together.
Worked Examples
Annual investment returns
Geometric mean: ≈ 1.116. Subtract 1 for CAGR of ≈ 11.6% per year. The arithmetic mean (1.12) would have slightly overstated performance.
Biological growth factors
Geometric mean: ≈ 2.44. The population multiplied by an average factor of 2.44× per generation over five generations.
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Frequently Asked Questions
What is the geometric mean?
The geometric mean is the nth root of the product of n positive values. It is the appropriate average for multiplicative processes like investment returns, growth rates, and ratios.
Why is geometric mean always less than or equal to arithmetic mean?
The AM-GM inequality guarantees this for all positive values. They are equal only when all values are identical. Greater spread between values creates a larger gap between the two means.
Can I use geometric mean with negative numbers?
No. The geometric mean requires all values to be positive. If you include a negative number, the product may be negative, and taking an even root of a negative number is not a real number.
When should I use geometric mean instead of arithmetic mean?
Use geometric mean for rates of change, ratios, index numbers, and any data where compounding occurs. Use arithmetic mean for additive quantities like heights, temperatures, or raw counts.
What is CAGR and how does geometric mean relate to it?
CAGR (compound annual growth rate) is the geometric mean of annual return multipliers minus 1. It answers: what constant annual rate would produce the same final value as the actual variable returns?
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