Geometric Mean Calculator
Calculate the geometric mean of a set of numbers.
Used in finance for average investment returns, population growth rates, and other multiplicative processes.
What Is the Geometric Mean?
The geometric mean is a type of average that is calculated by multiplying all values together and then taking the nth root, where n is the number of values. It is especially useful when values are multiplicative, such as growth rates or ratios.
The Formula
For n values:
Geometric Mean = (x₁ × x₂ × x₃ × … × xₙ)^(1/n)
Or equivalently using logarithms (better for large datasets):
Geometric Mean = exp[(ln x₁ + ln x₂ + … + ln xₙ) ÷ n]
When to Use the Geometric Mean vs Arithmetic Mean
| Situation | Use |
|---|---|
| Investment returns over multiple years | Geometric mean |
| Population growth rates | Geometric mean |
| Simple averages (test scores, heights) | Arithmetic mean |
| Ratios and indices | Geometric mean |
| Lengths, weights, ages | Arithmetic mean |
Practical Example: Investment Returns
An investment returns: +50% in year 1, −20% in year 2, +30% in year 3.
As multipliers: 1.50, 0.80, 1.30
Arithmetic average = (1.50 + 0.80 + 1.30) ÷ 3 = 1.20, which suggests a 20% average return. It is wrong.
Geometric mean = (1.50 × 0.80 × 1.30)^(1/3) = (1.56)^(1/3) = 1.1598, an actual 15.98% per year.
Check it against the money. Put in $1,000: year 1 ends at $1,500, year 2 at $1,200, year 3 at $1,560. Now grow $1,000 at 15.98% three times over and you land on $1,560 again. Grow it at the arithmetic 20% and you get $1,728, which is $168 that never existed. That gap is the whole reason fund factsheets quote the compound annual growth rate (CAGR) rather than an average of yearly returns.
Another Example: Rectangles and Squares
The geometric mean of two lengths is the side of the square with the same area. A 9 by 16 rectangle covers 144 square units, so:
Geometric mean = √(9 × 16) = √144 = 12
A 12 by 12 square covers exactly the same area. That is the honest geometric reading, and it is also why the geometric mean is the right average for anything measured as a ratio rather than as a quantity: photographic f-stops, screen contrast ratios, and index numbers all behave this way.
Limitations
- All values must be positive (cannot take the log of zero or a negative number)
- For investment returns entered as percentages (e.g. +50%, −20%), convert to multipliers first: add 1 to each percentage value
- A single bad year drags the result down harder than it drags the arithmetic mean. That is not a flaw, it is the point: a −100% year takes the multiplier to zero, and no amount of good years afterwards recovers anything
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
SuperGlobalCalculator is independently built and maintained. See how we build and verify our calculators.