Geometric Mean
The nth root of the product of n numbers.
Formula
Variables
Example
GM of 4 and 9 = 6
How it works
The geometric mean is the single "balanced" value that represents a set of positive numbers when they multiply together rather than add together. It is found by multiplying all the values and then taking the nth root so the result has the same number of factors as the original list. This makes it especially useful when the numbers grow by factors or percentages, because it reflects the typical multiplicative scale rather than the simple average.
When to use it
Use the geometric mean when combining rates, growth factors, ratios, or percentages over several stages, such as investment returns or population growth. It is also helpful when comparing quantities that vary across orders of magnitude and you want a central value that respects multiplication.
Worked example, step by step
Find the geometric mean of 2, 8, and 16. GM = \sqrt[3]{2 \cdot 8 \cdot 16} GM = \sqrt[3]{256} GM = 6.35 (approximately) So, the geometric mean is about 6.35.
Common mistakes to avoid
- Using the arithmetic mean instead of multiplying the values and taking the root
- Including zero or negative numbers without checking whether the geometric mean is defined in the given context
- Forgetting that the root degree must match the number of values being combined
Did You Know?
The geometric mean is the right average for growth rates and is never larger than the ordinary arithmetic mean.
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