With this geometric mean calculator, all calculations will be a pleasure! Just type the numbers of which you want to calculate the geometric mean, and the result will appear in no time. Remember that you may enter up to 30 numbers — extra boxes will appear as you go. If you're wondering what a geometric mean is and you're looking for a definition and formula of the geometric mean, then keep reading.
What is a geometric mean? Geometric mean definition and formula
A geometric mean — along with the arithmetic, weighted, and harmonic means — is one of the most useful and common types of averages. The geometric mean may be defined as:
the n-th root of the product of n values.
The n-th root is at the same time the power of one divided by n. Thus, we may state the geometric mean definition as the product of the values to the power of 1/n. The general geometric mean formula looks then as follows:
G = ⁿ√(x₁ × x₂ × … × xₙ) = (x₁ × x₂ × … × xₙ)^(1/n)
Simply speaking, if you are wondering how to find the geometric mean, just multiply your values and take a square root (for two numbers), a cube root (for three numbers), a fourth root (for four numbers), and so on. Generally, the geometric mean applies only to positive numbers.
💡 You might also want to check out our harmonic mean and arithmetic mean tools — all three are available in one place in our mean calculator.
Geometric mean vs. arithmetic mean
In many cases, the arithmetic mean is the average we're looking for. However, there are some cases when the geometric mean is the preferable mean:
- Different numeric ranges — e.g., we want to take into account two values assuming they are equally important: one from the 0–5 range and another from the 900–1000 range. A change of the smaller number, even a significant one (e.g., from 1 to 5), will result in a tiny difference in the arithmetic mean. The arithmetic mean neglects the smaller number, but that's the perfect case for the geometric mean.
- Skewed data with large positive outliers — the geometric mean can take care of that, in contrast to the arithmetic average.
Have a look at the table for a direct comparison of the geometric mean vs. the arithmetic mean:
| Geometric mean | Arithmetic mean | |
|---|---|---|
| Definition | The n-th root of the product of n values. | The sum of n values divided by n. |
| Formula | (x₁ × … × xₙ)^(1/n) | (x₁ + … + xₙ) / n |
| Works with | Positive numbers only | Any real numbers |
| Best for | Growth rates, ratios, values across very different ranges, skewed data | Quantities that add up, symmetric data |
For any set of positive numbers, the geometric mean is always less than or equal to the arithmetic mean — this is the famous arithmetic–geometric means (AM–GM) inequality:
geometric mean ≤ arithmetic mean
The only way to make the two means equal is to have a list of identical numbers.
Geometric mean triangles and other applications in geometry
The geometric mean has an elegant geometric interpretation. In a right triangle, the altitude drawn from the right angle to the hypotenuse is the geometric mean of the two segments it creates on the hypotenuse (the geometric mean theorem, also called the altitude-on-hypotenuse theorem):
h = √(p × q)
Here p and q are the two segments of the hypotenuse, and h is the altitude. Similarly, the geometric mean of two numbers a and b equals the side of a square whose area is the same as an a × b rectangle — which is exactly why the geometric mean is sometimes called the "mean proportional." Beyond triangles, the geometric mean appears wherever a quantity scales multiplicatively: average investment returns, population growth rates, aspect ratios, and index numbers.
How to find a geometric mean? An example on how to use the geometric mean calculator
Finding the geometric mean by hand is straightforward:
- Multiply the values and denote their product by p. For 2, 4, and 8: p = 2 × 4 × 8 = 64.
- Take the n-th root of the product. Here there are three numbers, so we take the cube root: G = ∛64 = 4.
Using the geometric mean calculator is even easier:
- Select the value type — a plain number, currency (USD, RUB, EUR, GBP and 20+ more), US imperial, or metric units.
- Enter each number into a separate field. New rows appear automatically as you type, up to 30 numbers. All values must be positive.
- Watch the live preview update the geometric mean (and the arithmetic mean for comparison) as you go.
- Press Calculate to see the full step-by-step solution, the product, the dataset table, and the AM–GM comparison.
FAQs
What is the geometric mean of 2 and 8?
Multiply the values (2 × 8 = 16) and take the square root because there are two numbers: √16 = 4. So the geometric mean of 2 and 8 is 4.
Why does the geometric mean only work with positive numbers?
The geometric mean is the n-th root of a product. If any value is zero, the product — and therefore the mean — becomes zero, and if any value is negative, the root may not be a real number. For these reasons the geometric mean is defined only for strictly positive values.
When should I use the geometric mean instead of the arithmetic mean?
Use the geometric mean for quantities that multiply rather than add — growth rates, investment returns, ratios, and index numbers — or when your data spans very different orders of magnitude or is skewed by large outliers.
Can I compute the geometric mean of currency or measurement values?
Yes. Choose the Currency option for 25 world currencies including USD, RUB, EUR, GBP, JPY, INR, BRL, and CAD, or pick US imperial (in, ft, lb, gal…) or metric (mm, cm, kg, L…) units. The formula works the same regardless of the unit.
How many numbers can I enter?
You can enter up to 30 numbers. New input fields appear automatically as you fill out the form, and a live preview shows the geometric and arithmetic means as you type.