# Geometric mean

The **geometric** **mean** of a collection of positive data is defined as the *n*th 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 [*a _{1}*,

*a*, ...,

_{2}*a*] is given by

_{n}- .

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 (*a*_{n}) and (*h*_{n}) are defined:

and

then *a*_{n} and *h*_{n} 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.

This is simply computing the arithmetic mean of the logarithm transformed values of (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. e^{mean(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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