What this calculator does
The geometric mean of a set of numbers is the nth root of their product: multiply every value together, then take the root that matches how many values there are. It answers a different question to the ordinary average. Where the arithmetic mean asks "what single number, added n times, gives the same total?", the geometric mean asks "what single number, multiplied by itself n times, gives the same product?".
That distinction matters most for figures that compound, such as investment returns, population growth rates or anything expressed as a ratio from one period to the next. Averaging a sequence of yearly growth factors with the arithmetic mean overstates the typical rate, because compounding is multiplicative, not additive; the geometric mean gives the rate that actually reproduces the same overall change.
The formula
Multiply all the numbers together to get their product, then take the nth root of that product, where n is how many numbers were entered. This calculator computes it as the exponential of the average of the natural logs of the numbers, which gives the identical result without the product overflowing for long lists.
| Term | Meaning |
|---|---|
| Geometric mean | The nth root of the product of n positive numbers: (x₁ × x₂ × … × xₙ)^(1/n). |
| Arithmetic mean | The ordinary average: the sum of the numbers divided by how many there are, shown for comparison. |
| n | The count of numbers entered, which determines which root is taken of the product. |
The inputs explained
| Field | What to enter |
|---|---|
| Numbers (comma or space separated) | A list of positive numbers, separated by commas or spaces, such as growth factors, ratios or measurements. |
When to use it
Averaging investment or growth returns
A sequence of yearly growth factors, such as 1.10 for a 10% gain, averages correctly only with the geometric mean, since returns compound onto each other rather than simply adding up.
Combining ratios or index numbers
Ratios and index values, such as price relatives or scaling factors, are naturally multiplicative, so the geometric mean gives a more representative typical value than the arithmetic mean.
Checking how far apart the two means are
The geometric mean is always less than or equal to the arithmetic mean for the same numbers, and the gap between them grows with how spread out the values are, which this calculator shows side by side.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does the geometric mean compare to the arithmetic mean across different data sets?
A few example data sets, from tightly clustered to widely spread, showing both measures together.
| Numbers entered | Geometric mean | Arithmetic mean (for comparison) |
|---|---|---|
| 5, 5, 5, 5 | 5.000 | 5.000 |
| 2, 8 | 4.000 | 5.000 |
| 1, 3, 9 | 3.000 | 4.333 |
| 4, 4, 4, 100 | 8.944 | 28.000 |
| 1, 100 | 10.000 | 50.500 |
What average growth rate does a sequence of yearly growth factors imply?
Growth factors above 1 mean growth, below 1 mean decline; the geometric mean gives the single steady rate that reproduces the same overall change.
| Yearly growth factors | Geometric mean | Arithmetic mean (for comparison) |
|---|---|---|
| 1.10, 1.20, 0.90 | 1.059 | 1.067 |
| 1.05, 1.05, 1.05 | 1.050 | 1.050 |
| 1.50, 0.50 | 0.8660 | 1.000 |
| 1.08, 1.08, 1.08, 1.08 | 1.080 | 1.080 |
Questions
How to calculate GM by hand?
Multiply all the numbers together to get one product, then take the nth root of that product, where n is the count of numbers. For example, the geometric mean of 2 and 8 is the square root of 16, which is 4.
Why is the geometric mean always smaller than the arithmetic mean?
This follows from a general mathematical inequality that holds for any set of positive numbers: the geometric mean can only equal the arithmetic mean when every number in the set is identical, and is strictly smaller whenever the numbers differ.
Can the geometric mean be used with negative numbers or zero?
Not in the ordinary real-number sense used here. A zero in the list makes the whole product zero, and negative numbers make the root undefined for even counts, so this calculator requires every entered number to be greater than zero.
When should I use the geometric mean instead of the average?
Use it whenever the numbers describe multiplicative change, such as growth rates, returns or ratios from one period to the next. For simple totals or typical sizes that add together naturally, the ordinary arithmetic mean remains the right tool.
For an ordinary sum-and-divide average instead, see the average calculator. For a sequence built from repeated multiplication by a fixed ratio, see the geometric sequence calculator.