Geometric Mean Calculator

About the Geometric Mean Calculator

The Geometric Mean Calculator computes the geometric mean of positive numbers. The geometric mean is calculated by multiplying all n values together and taking the nth root. It is the correct average for rates of change, growth rates, and ratios.

While the arithmetic mean adds values, the geometric mean multiplies them. This makes it ideal for compound interest calculations, average investment returns, and any situation where percentages are compounded over time.

If your investment grows 50% one year and loses 33% the next, the arithmetic mean suggests 8.5% average growth, but the geometric mean correctly shows 0% (you ended where you started).

When This Page Helps

The arithmetic mean overestimates compound growth. The geometric mean provides the true average rate of change for multiplicative processes like investment returns.

How to Use the Inputs

1. Enter positive numbers separated by commas.
2. View the geometric mean in the result panel.
3. Compare it with the arithmetic mean.
4. Use percent changes by entering growth factors (e.g. 1.10 for +10%).
5. The result represents the equivalent constant growth rate.

Example Calculation

Tips & Best Practices

• The geometric mean is always ≤ the arithmetic mean (AM-GM inequality).
• For investment returns, convert percentages to growth factors (1 + r/100).
• The geometric mean is undefined for negative or zero values.
• CAGR (compound annual growth rate) is a geometric mean.
• Use the log method for very large products to avoid overflow.
• The geometric mean of 1.10 and 0.90 is about 0.995, showing a slight net loss.

Geometric Mean in Finance

The geometric mean return is the only correct average for compounded returns. If a fund gains 100% then loses 50%, the arithmetic mean return is +25% but the geometric mean is 0%—you ended where you started.

Geometric Mean in Science

Biologists use the geometric mean for population growth rates. Chemists use it for reaction rate averages. The geometric mean is natural whenever data spans multiple orders of magnitude.

Computing Large Products

To avoid numerical overflow, compute the geometric mean in log space: exp(average of ln(xᵢ)). This is numerically stable even for very large or very small numbers.

Frequently Asked Questions

• What is the geometric mean?

  The geometric mean is the nth root of the product of n values. It represents the central tendency of multiplicative data like growth rates and ratios.

• When should I use geometric instead of arithmetic mean?

  Use the geometric mean for compound growth rates, investment returns, inflation rates, population growth, and any data that is multiplied together over time. Documenting the assumptions behind your calculation makes it easier to update the analysis when input conditions change in the future.

• Why can't geometric mean include zero or negatives?

  A zero in the product makes the result zero regardless of other values. Negative values make even roots of the product undefined in real numbers.

• What is the AM-GM inequality?

  The Arithmetic Mean is always ≥ the Geometric Mean for positive numbers. They are equal only when all values are identical. This is a fundamental inequality in mathematics.

• How is CAGR related to geometric mean?

  CAGR is the geometric mean of annual growth factors minus 1. If an investment goes from $100 to $200 in 5 years, CAGR = (200/100)^(1/5) − 1 ≈ 14.87%.

• How do I handle percentages?

  Convert percentage changes to multipliers first. A 10% increase is 1.10, a 5% decrease is 0.95. Take the geometric mean of these multipliers, then subtract 1 for the average rate.