How to use the geometric mean calculator
- Choose Plain values to average positive numbers, or Percent growth to average a list of percent changes such as yearly returns.
- Enter the numbers separated by commas, spaces or new lines. In plain mode every number must be above 0. In growth mode every rate must be above -100.
- Read the geometric mean. In plain mode the arithmetic and harmonic means are shown beside it for comparison.
- In growth mode the result is the steady growth per period that would give the same total growth.
Formula
Geometric mean = (x1 x x2 x ... x xn)^(1/n)
Equivalent: exp( (ln x1 + ln x2 + ... + ln xn) / n )
Average growth per period = (product of (1 + r/100))^(1/n) - 1
The tool uses the logarithm form, which gives the same answer without overflowing on long lists or large numbers.
The geometric mean of 2 and 8
The product of 2 and 8 is 16, and the square root of 16 is 4, so the geometric mean is 4. The arithmetic mean is 5 and the harmonic mean is 3.2. As always for positive values that are not all equal, the order is harmonic below geometric below arithmetic.
For yearly returns of +10%, -5% and +20% the growth factors are 1.10, 0.95 and 1.20, which multiply to 1.254. The cube root is 1.0784, so the average growth is 7.84% a year, compared with a simple average of 8.33%. A steady 7.84% for three years gives the same ending amount as the three real returns.
When the geometric mean is the right average
Use it when values combine by multiplication: growth rates, ratios, index numbers and compounding returns. The ordinary average of growth rates overstates the true result whenever the rates vary, because a 20 percent gain after a 20 percent loss does not get you back to even.
For a single start and end value, the same idea gives the compound growth rate, which the CAGR calculator computes. The percent change calculator handles one step at a time.
Why zero and negative values are rejected
A zero makes the whole product zero, and a negative number makes the root undefined or complex for an even count. Rather than return a misleading answer, the tool asks for values above zero. For growth rates the equivalent limit is -100 percent, which would wipe out the starting amount.
To compare against other kinds of average, see the harmonic mean calculator and the average calculator.
Assumptions and limits
- All values are above zero in plain mode, and all growth rates are above -100 percent in growth mode.
- Growth periods are of equal length and the rates are applied one after another to the same balance.
- Results are shown to 8 significant digits.
- No contributions, withdrawals, fees or taxes are included in growth mode.
Frequently asked questions
What is the geometric mean?
It is the nth root of the product of n positive numbers. It is the average that fits quantities that multiply, such as growth factors.
How do I find an average growth rate?
Turn each percent change into a factor by adding it to 1, multiply the factors, take the root that matches the number of periods, then subtract 1. The growth option does this for you.
Why is the geometric mean lower than the arithmetic mean?
For positive numbers that are not all equal, the arithmetic mean is always the largest of the three common averages. The geometric mean is pulled down because it responds to proportions rather than to raw differences.
Can the geometric mean include zero?
Not in this tool. With a zero in the list the product is zero, which says nothing about the other values, so the tool asks for numbers above zero.