Geometric Mean Calculator

The geometric mean is the n-th root of the product of n numbers. You should use it when working with values that are multiplied together rather than added — such as growth rates, investment returns, ratios, or percentages. It is the most appropriate average when data spans several orders of magnitude or represents proportional change over time.

Multiply all the values in your dataset together, then take the n-th root of that product, where n is the count of values. For example, the geometric mean of 4, 16, and 64 is ∛(4 × 16 × 64) = ∛4096 = 16. Alternatively, you can compute the average of the natural logarithms and then exponentiate the result.

The arithmetic mean adds all values and divides by the count, making it suitable for additive data. The geometric mean multiplies values and takes the root, making it better for multiplicative or exponential data like growth rates. The geometric mean is always less than or equal to the arithmetic mean (they are equal only when all values are identical).

If any value in the dataset is zero, the product of all values becomes zero, making the geometric mean zero as well. In practice, zeros are often replaced with a small positive constant or excluded from the dataset before calculating the geometric mean, depending on the context of the analysis.

Strictly speaking, the geometric mean is defined for positive numbers. However, it can be extended to datasets of all-negative numbers by computing the geometric mean of their absolute values and then applying the original sign. Mixed positive and negative datasets cannot be handled with the standard formula, as the product may be negative and the root undefined in real numbers.

If an investment grows by 10% in year one, shrinks by 5% in year two, and grows by 20% in year three, you express each as a growth factor (1.10, 0.95, 1.20) and compute their geometric mean: ∛(1.10 × 0.95 × 1.20) ≈ 1.081. This means the investment grew at an average rate of about 8.1% per year — a more accurate figure than the arithmetic mean of the percentage returns.

The harmonic mean is the reciprocal of the arithmetic mean of reciprocals, best used for rates and speeds (e.g. average speed over equal distances). The geometric mean is the n-th root of the product, best used for growth rates and ratios. For any dataset, harmonic mean ≤ geometric mean ≤ arithmetic mean.

This calculator uses the log-sum method internally (averaging logarithms and exponentiating) to handle very large products without numeric overflow. This makes it accurate for large datasets and large numbers. Results are displayed to six decimal places by default.