Formula
G = \left(\prod_{i=1}^{n} x_i\right)^{\frac{1}{n}}Used to calculate: The average of a set of positive values based on their product rather than their sum.
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What Is The Geometric Mean Formula?
The geometric mean formula calculates a type of average by multiplying all values in a data set and taking the nth root of the product, where n is the number of values. It is especially useful for quantities that change multiplicatively, such as growth factors and proportional changes.
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Formula and Variables
G = \left(\prod_{i=1}^{n} x_i\right)^{\frac{1}{n}}| Symbol | Meaning | Unit |
| Geometric mean | Same unit as the values when they share a common unit | |
| Individual value in the data set | Depends on the data | |
| Number of values in the data set | None | |
| Product of the specified values | Depends on the values |
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How to Use the Formula
Multiply all of the values in the data set together. Determine the number of values, then take the nth root of the resulting product. The result is the geometric mean.
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When Is This Formula Used?
The geometric mean is used when values combine multiplicatively or when proportional changes are more meaningful than additive differences.
Common uses:
– Calculating average growth factors or rates
– Comparing proportional changes
– Analyzing financial, scientific, and statistical data involving multiplicative relationships
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Important Notes
– For the standard real-valued geometric mean used in statistics, the values are typically required to be positive.
– The number of values n must be greater than zero.
– When all values have the same physical unit, the geometric mean has that same unit.
– A common point of confusion is using the arithmetic mean for multiplicative growth. The geometric mean is generally more appropriate when averaging growth factors or proportional changes.
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Alternate Forms
Alternate form:
G = \sqrt[n]{x_1x_2\cdots x_n}Alternate form:
\ln G = \frac{1}{n}\sum_{i=1}^{n}\ln x_i—
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