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EspañolGEOMETRIC 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.
The geometric mean of a data set [''a1'', ''a2'', ..., ''an''] is given by
:.
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''.
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. emean(ln(X)).
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 signify 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, which is about 1.28966 or about 29% annual interest.
In the scientific community, when reporting experimental results, it is also important to know whether arithmetic mean or geometric mean should be used. If, for example, you are averaging ratios (i.e. ratio = new method/old method) over many experiments, geometric mean should be used. This becomes evident when considering the two extremes. If one experiment yields a ratio of 10,000 and the next yields a ratio of 0.0001, an arithmetic mean would misleadingly report that the average ratio was near 5000. Taking a geometric mean will more honestly represent the fact that the average ratio was 1.
★ 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
★ 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
| Contents |
| Calculation |
| Relationship with arithmetic mean of logarithms |
| When to use the geometric mean |
| See also |
| External links |
Calculation
The geometric mean of a data set [''a1'', ''a2'', ..., ''an''] is given by
:.
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. emean(ln(X)).
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 signify 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, which is about 1.28966 or about 29% annual interest.
In the scientific community, when reporting experimental results, it is also important to know whether arithmetic mean or geometric mean should be used. If, for example, you are averaging ratios (i.e. ratio = new method/old method) over many experiments, geometric mean should be used. This becomes evident when considering the two extremes. If one experiment yields a ratio of 10,000 and the next yields a ratio of 0.0001, an arithmetic mean would misleadingly report that the average ratio was near 5000. Taking a geometric mean will more honestly represent the fact that the average ratio was 1.
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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