Geometric mean

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The geometric mean of a set of positive data is defined as the nth root of the product of all the members of the set, where n is the number of members.

1 Calculation
2 Relationship with arithmetic mean of logarithms
3 When to use the Geometric Mean
4 See also
5 External links

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Calculation

The geometric mean of a set {a₁, a₂, ..., a_n} is:

<math>\bigg(\prod_{i=1}^nx_i \bigg)^{1/n} = (a_1 \cdot a_2 \dotsb a_n)^{1/n} = \sqrt[n]{a_1 \cdot a_2 \dotsb a_n}</math>.

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 (a_n) and (h_n) are defined:

and

then a_n and h_n will converge to the geometric mean of x and y.

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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.

<math>\bigg(\prod_{i=1}^nx_i \bigg)^{1/n} = \exp\left[\frac1n\sum_{i=1}^n\ln x_i\right]</math>.

This is simply computing the arithmetic mean of the logarithm transformed values of <math>x_i</math> (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 mean of the log transformed values, i.e. e^mean(ln(X)).

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When to use the Geometric Mean

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.

Put another way...

The arithmetic mean is relevant any time several quantities add together to produce a total. The arithmetic mean answers the question, "if all the quantities had the same value, what would that value have to be in order to achieve the same total?"

In the same way, the geometric mean is relevant any time several quantities multiply together to produce a product. The geometric mean answers the question, "if all the quantities had the same value, what would that value have to be in order to achieve the same product?"

For example, suppose you have an investment which earns 10% the first year, 50% the second year, and 30% the third year. What is its average rate of return? It is not the arithmetic mean, because what these numbers mean is that on the first year your investment was multiplied (not added to) by 1.10, on the second year it was multiplied by 1.50, and the third year it was multiplied by 1.30. The relevant quantity is the geometric mean of these three numbers. Source.

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