Geometric mean

The geometric mean of a set of positive data is defined as the product of all the members of the set, raised to a power equal to the reciprocal of the number of members.

In a formula: the geometric mean of a1, a2, ..., an is [itex](a_1 \cdot a_2 \dotsb a_n)^{1/n}[itex], which is [itex]\sqrt[n]{a_1 \cdot a_2 \dotsb a_n}[itex].

The geometric mean is useful to determine "average factors". For example, if a stock rose 10% in the first year, 20% in the second year and fell 15% in the third year, then we compute the geometric mean of the factors 1.10, 1.20 and 0.85 as (1.10 × 1.20 × 0.85)1/3 = 1.0391... and we conclude that the stock rose 3.91 percent per year, on average.

The geometric mean of a data set is always smaller than or equal to the 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:

[itex]a_{n+1} = \frac{a_n + h_n}{2}, \quad a_1=\frac{x + y}{2}[itex]

and

[itex]h_{n+1} = \frac{2}{\frac{1}{a_n} + \frac{1}{h_n}}, \quad h_1=\frac{2}{\frac{1}{x} + \frac{1}{y}}[itex]

then an and hn will converge to the geometric mean of x and y.

Relationship with arithmetic mean of logarithms

The product form of the geometric mean computation is expressed as:

[itex]\left(\prod_{i=1}^nx_i\right)^{1/n}[itex]

By using logarithmic identities to transform the formula, we can express the multiplications as a sum and the power as a multiplication.

[itex]\exp\left[\frac1n\sum_{i=1}^n\ln x_i\right][itex].

This is simply computing the arithmetic mean of the logarithm transformed values of [itex]x_i[itex] (i.e. the arithmetic mean in log space) and then using the exponentiation to return the computation to real space. 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 mean of the log transformed values, e.g. emean(ln(X)).

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