Geometric Mean
The nth root of the product of n values.
Formula
Variables
Example
The geometric mean of 2 and 8 is √(2·8) = 4.
How it works
The geometric mean combines several positive values by multiplying them and then taking the n-th root, which gives a single “typical” factor rather than a typical sum. It is especially useful when the values represent ratios, growth rates, or anything that compounds multiplicatively, because it treats each value as a multiplier with equal weight. This works because the product of n equal numbers is reversed by the n-th root, so the result is the common value whose repeated multiplication would match the overall product.
When to use it
Use the geometric mean when averaging percentages, growth factors, rates of return, or measurements that multiply over time. It is also common in science and engineering for comparing quantities that span several orders of magnitude, such as intensities or concentrations.
Worked example, step by step
Find the geometric mean of 3, 12, and 27. GM = ∛(3 × 12 × 27) GM = ∛972 GM = 9.9 (approximately) So, the geometric mean is about 9.9.
Common mistakes to avoid
- Using the geometric mean for values that should be added, such as test scores or heights
- Including zero or negative values without checking whether the context allows them
- Forgetting to take the n-th root after multiplying all the values
Did You Know?
The geometric mean is the right average for growth rates and ratios — use it for compounding returns.
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