Geometric Mean Calculator

For n positive values \(x_1, x_2, \ldots, x_n\), the geometric mean is:

$$G = \left(\prod_{i=1}^{n} x_i\right)^{1/n} = \sqrt[n]{x_1 \cdot x_2 \cdots x_n}$$

To avoid numerical overflow with large products, the calculation uses logarithms:

$$G = \exp\left(\frac{1}{n}\sum_{i=1}^{n} \ln(x_i)\right)$$

This log-transform approach is computationally stable and works for any positive values. The geometric mean requires all values to be strictly positive — if any value is zero, the product is zero and the geometric mean is zero; negative values would produce complex logarithms.

The geometric mean has a key property: it is the value that, if substituted for every item, would produce the same product as the original values. This makes it the natural average for multiplicative processes.

A geometric mean of 8.0 for the values {2, 4, 8, 16, 32} means that multiplying five 8's together gives the same product as 2 * 4 * 8 * 16 * 32. The arithmetic mean of the same data is 12.4, which is higher — this gap demonstrates the AM-GM inequality and reflects the right-skewed nature of this dataset.

In finance, if your portfolio returns 10%, -5%, 20%, and 15% over four years, the geometric mean return correctly captures the compounding effect, while the arithmetic mean would overstate performance.