Geometric mean
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The geometric mean of a collection of positive data is defined as the nth root of the product of all the members of the data set, where n is the number of members.
Calculation
The geometric mean of a data set [a1, a2, ..., an] is given by
![\bigg(\prod_{i=1}^n a_i \bigg)^{1/n} = \sqrt[n]{a_1 \cdot a_2 \cdot \dots \cdot a_n}](/images/math/c/b/3/cb3f235f9d699ee54c24b04d09a63e95.png)
.
The geometric mean of a data set is smaller than or equal to the data set's arithmetic mean (the two means are equal if and only if all members of the data set are equal). This allows the definition of the arithmetic-geometric mean, a mixture of the two which always lies in between.
The geometric mean is also the arithmetic-harmonic mean in the sense that if two sequences (an) and (hn) are defined:
and
then an and hn will converge to the geometric mean of x and y.
Relationship with arithmetic mean of logarithms
By using logarithmic identities to transform the formula, we can express the multiplications as a sum and the power as a multiplication.
![\bigg(\prod_{i=1}^nx_i \bigg)^{1/n} = \exp\left[\frac1n\sum_{i=1}^n\ln x_i\right]](/images/math/e/3/4/e34ce55c414723aa2e9a7400893b1433.png)
This is simply computing the arithmetic mean of the logarithm transformed values of xi (i.e. the arithmetic mean on the log scale) and then using the exponentiation to return the computation to the original scale. I.e., it is the generalised f-mean with f(x) = ln x.
Therefore the geometric mean is related to the log-normal distribution. The log-normal distribution is a distribution which is normal for the logarithm transformed values. We see that the geometric mean is the exponentiated value of the arithmetic mean of the log transformed values, i.e. emean(ln(X)).
See also
- Arithmetic mean
- Arithmetic-geometric mean
- Average
- Generalized mean
- Geometric standard deviation
- Harmonic mean
- Heronian mean
- Hyperbolic coordinates
- Inequality of arithmetic and geometric means
- Log-normal distribution
- Muirhead's inequality
- Product
- Rate of return
- Weighted geometric mean
External links
- Geometric mean calculator
- Calculation of the geometric mean of two numbers in comparison to the arithmetic solution
- Arithmetic and geometric means at cut-the-knot
- When to use the geometric mean
- Practical solutions for calculating geometric mean with different kinds of data
- Geometric Mean on MathWorld
- Geometric Meaning of the Geometric Mean
- Geometric Mean Calculator for larger data sets
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