What is the geometric mean?
The geometric mean of n numbers is the nth root of their product. Unlike the arithmetic mean, which adds values together and divides by the count, the geometric mean multiplies the values and takes a root. This makes it useful for quantities that change multiplicatively, such as growth rates, ratios, and investment returns.
Geometric Mean = (x₁ × x₂ × ... × xₙ)1/n Why use it instead of the arithmetic mean?
The arithmetic mean works well when values are naturally additive, but it can give a misleading result for compounded changes. For example, if an investment grows 10% one year and shrinks 10% the next, the arithmetic average of the two percentage changes is 0%. But the investment does not break even: a 10% gain followed by a 10% loss leaves you down 1% overall.
The geometric mean accounts for this compounding because it averages the growth factors rather than the percentage changes themselves. This is why geometric averaging is useful for compounded rates, average returns, and other multiplicative series.
Standard mode: geometric mean of non-negative numbers
Enter a list of non-negative numbers to calculate their geometric mean directly. For example, the geometric mean of 4 and 9 is exactly 6, because 6 × 6 = 4 × 9 = 36.
If any value is 0, the geometric mean is 0 because the product of the dataset is 0. Negative raw values aren't supported in Standard mode; if you're working with negative percentage changes, such as −10%, use the Growth Rates tab instead.
Growth Rates mode: average compounded growth rate
Growth Rates mode is designed for percentage changes, which can legitimately be negative. A return of −10% corresponds to a positive growth factor of 0.90, so it can be included in a geometric average even though a raw value of −10 isn't accepted in Standard mode.
Each percentage change is converted to a growth factor by adding 1. For example, 10% becomes 1.10 and −5% becomes 0.95. The calculator finds the geometric mean of those factors and converts the result back to a percentage.
For example, with growth rates of 10%, −5%, and 20%:
- Convert to factors: 1.10, 0.95, 1.20
- Geometric mean of factors: (1.10 × 0.95 × 1.20)1/3 ≈ 1.078
- Convert back to a percentage: (1.078 − 1) × 100 ≈ 7.84%
So the average compounded growth rate across the three periods is about 7.84%. A simple arithmetic average of 10%, −5%, and 20% would instead give about 8.33%, which does not account for compounding.
Frequently asked questions
Why can Growth Rates mode accept negative numbers but Standard mode can't?
A negative percentage change, such as −10%, is converted to a positive growth factor of 0.90 before the geometric mean is calculated. Standard mode works directly with the numbers you enter and this calculator does not support negative raw values.
What happens if I enter a percentage change below −100%?
It is rejected. A change of exactly −100% produces a growth factor of 0 and is allowed. A change below −100% would produce a negative growth factor, which is not supported by the Growth Rates calculation.
Is the geometric mean the same as the geometric average?
Yes. "Geometric average" is another name for the geometric mean. Both terms refer to an average found by multiplying values together and taking the appropriate root.
Is this the same as CAGR (compound annual growth rate)?
They are closely related but not identical. CAGR is typically calculated from a starting value and an ending value over a specified period. This calculator's Growth Rates mode averages a series of individual period-by-period percentage changes. Both use the same underlying idea that compounded changes should be combined multiplicatively.