About the Geometric Mean Calculator
The ordinary average adds numbers up and divides. That works for many things, but not for growth rates, ratios or anything that multiplies. If an investment gains 10 percent one year and loses 10 percent the next, the simple average says it broke even, yet you ended up with 1 percent less. The geometric mean fixes this by multiplying the values and taking the nth root. This calculator gives the geometric mean of positive numbers alongside the arithmetic and harmonic means, and it has a growth mode: enter yearly percentage changes and get the true average yearly growth rate.
Updated October 2026
How to use the Geometric Mean Calculator
- 1Choose ordinary positive numbers or yearly growth rates.
- 2Enter your numbers, separated by commas, spaces or new lines.
- 3For growth rates, enter losses as negative numbers, such as −10.
- 4Read the geometric mean or the average growth rate.
- 5Compare with the arithmetic and harmonic means in the table.
Example
Suppose a fund returns +20 percent, −10 percent and +15 percent over three years. The simple average is 8.33 percent, but the geometric mean is the cube root of 1.2 × 0.9 × 1.15, minus 1, which is about 7.5 percent a year. Over three years, 7.5 percent a year turns 1,000 into about 1,242, which is exactly what the three returns give. The simple average would predict 1,271, which is too high.
Three means compared
- Arithmetic mean: add and divide. Best for amounts that add.
- Geometric mean: multiply and take the root. Best for growth and ratios.
- Harmonic mean: reciprocal of the average reciprocal. Best for rates and speeds.
- For positive numbers that differ, harmonic ≤ geometric ≤ arithmetic.
- All three agree when every number is equal.
Practical tips
Use growth mode when your numbers are percent changes. Use ordinary mode for index values or ratios like price relatives. If you only know the start and end values, the CAGR Calculator is quicker. Beware of the sign: a drop of 10 percent is entered as −10, not 10.
Related tools
The Average Calculator gives the arithmetic mean, median and mode, the CAGR Calculator finds growth between two values, the Standard Deviation Calculator measures spread and the Compound Interest Calculator projects growth into the future.
Features
- Geometric, arithmetic and harmonic mean
- Average growth rate from percentage changes
- Total growth over all periods
- Product of all numbers shown
- Rejects zero and negative values for the means
- Copy the result
Frequently asked questions
What is the geometric mean?
The nth root of the product of n numbers. For 2, 8, 4 and 16, the product is 1,024 and the fourth root is about 5.657. It is always less than or equal to the arithmetic mean.
When should I use the geometric mean?
For quantities that multiply or compound: growth rates, investment returns, ratios, index numbers and sizes of populations. The arithmetic mean overstates growth when values vary.
How do I find average annual growth?
Convert each yearly change to a factor (10 percent is 1.10, −10 percent is 0.90), multiply them, take the nth root and subtract 1. Growth mode does it for you: +10 and −10 give −0.5 percent a year.
What is the harmonic mean?
The reciprocal of the average of reciprocals. It is the right average for rates over the same distance, like speeds. Travelling a distance at 40 and returning at 60 gives an average speed of 48, the harmonic mean.
Why can't I include zero or negative numbers?
The geometric mean uses logarithms or roots of the product, which are undefined or unreliable for zero and negative values. For growth rates, the calculation works with the factor 1 plus the rate, so only losses of 100 percent or more are a problem.
How is this different from CAGR?
CAGR (compound annual growth rate) uses the starting and ending values over a number of years. The geometric mean of the yearly changes gives the same number, so both routes lead to the same answer.