Geometric Mean Calculator
Enter a set of positive numbers to get the geometric mean — with the arithmetic mean shown for contrast.
The geometric mean is always ≤ the arithmetic mean (equal only when every value is identical).
The geometric mean multiplies n positive values and takes the nth root: for 1, 3, 9 it is (1 × 3 × 9)^(1/3) = 27^(1/3) = 3, while the arithmetic mean is 13 ÷ 3 ≈ 4.33. Reach for it to average growth rates, ratios, and compounded returns, where products — not sums — drive the result.
What the geometric mean measures
The geometric mean is the “average factor” of a set of positive numbers. Instead of adding the values and dividing (the arithmetic mean), it multiplies them together and takes the nth root, so it reflects how quantities multiply rather than how they add. That makes it the correct average whenever the numbers are rates, ratios, or index values that compound — think year-on-year growth factors, price ratios, or investment returns. Because a single very large value inflates a sum far more than a product, the geometric mean is also less sensitive to right-skewed outliers than the arithmetic mean.
the nth root of the product of all n positive values
Worked example
Three values: 1, 3, 9.
- 1 Confirm every value is positive. 1, 3, and 9 are all greater than 0, so the geometric mean is defined. A zero or negative value would make it invalid.
- 2 Multiply all the values. 1 × 3 × 9 = 27. This product is the quantity you will take the root of.
- 3 Take the nth root. There are n = 3 values, so GM = 27^(1/3) = 3. Equivalently, GM = exp((1/3) × (ln 1 + ln 3 + ln 9)) = 3.
- 4 Compare with the arithmetic mean. The arithmetic mean is (1 + 3 + 9) ÷ 3 = 13 ÷ 3 ≈ 4.33, so the geometric mean (3) is smaller — as it always is unless the values are identical.
Geometric mean vs. arithmetic mean
On the same data the geometric mean is always ≤ the arithmetic mean (equal only when every value is identical).
| Dataset | Geometric mean | Arithmetic mean |
|---|---|---|
| 2, 8 | 4 | 5 |
| 1, 3, 9 | 3 | ≈ 4.33 |
| 1.05, 1.10, 1.20 | ≈ 1.1147 | ≈ 1.1167 |
| 4, 4, 4 | 4 | 4 (equal) |
When to use it — and its limits
Use the geometric mean for things that compound. Averaging annual growth factors, price relatives, or index ratios with the arithmetic mean overstates the true average because it ignores compounding. The geometric mean of the yearly growth factors is exactly the constant factor that would produce the same overall change — which is why the compound annual growth rate (CAGR) is just the geometric mean of the yearly return factors.
It requires strictly positive values. A single zero drags the product to 0, and a negative value makes the nth root undefined, so the geometric mean cannot be applied directly to data containing zeros or negatives (for percentage changes, convert them to growth factors first — a −10% change becomes 0.90). For plain additive data with no outliers, the ordinary arithmetic mean remains the right choice.