Geometric mean

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Geometric construction to find arithmetic media (A), quadratic (Q), geometric (G) and harmonic (H) of two numbers a and b.

In mathematics and statistics, the geometric mean of an arbitrary number of numbers (say n numbers) is the nth root of the product of all numbers; It is recommended for geometric progression data, for averaging ratios, compound interest, and index numbers.

${displaystyle {bar {x}}={sqrt[{n}]{prod _{i=1}^{n}{x_{i}}}}={sqrt[{n}]{x_{1}cdot x_{2}cdots x_{n}}}}$

For example, the geometric mean of 2 and 18 is the square root of the product of both ${displaystyle {sqrt[{2}]{2cdot 18}}={sqrt[{2}]{36}}=6}$ . Another example, the geometric mean of 1, 3 and 9 would be the cubic root of the product of the three numbers ${displaystyle {sqrt[{3}]{1cdot 3cdot 9}}={sqrt[{3}]{27}}=3}$ .

A geometric mean is often used when comparing different aspects, whose performances have units of measure in different numerical ranges. For example, the geometric mean can give a serious value to compare two companies that have a score between 0 to 5 for their environmental sustainability, and a score between 0 to 100 for their financial viability. If the arithmetic mean were used instead of the geometric mean, financial viability would carry more weight because its numerical range is greater. That is, a small percentage change in financial rating (for example, going from 80 to 90) would make a much bigger difference to the arithmetic mean than a large percentage change in environmental sustainability (for example, going from 2 to 5). Using the geometric mean normalizes values of different range, which means that a given percentage change in any of the properties has the same effect on the geometric mean. So a 20% change in environmental sustainability from 4 to 4.8 has the same effect on the geometric mean as a 20% change in financial viability from 60 to 72.

This average can be understood in geometric terms. The average of two numbers, $a$ and $b$ , is the length of the side of a square whose area is equal to the area of a rectangle with sides of lengths $a$ and $b$ . Similarly, the average of three numbers, $a$ , $b$ and $c$ , is the length of the edge of a cube whose volume is the same as that of an orthoedron whose sides are equal to the three given numbers.

The geometric mean is also one of the three Pythagorean means, along with the arithmetic mean, mentioned above, and the harmonic mean. For all positive data sets that contain at least one pair of unequal values, the harmonic mean is always the smallest of the three means, while the arithmetic mean is always the largest of the three, and the geometric mean is always in the middle. (See Inequality of Arithmetic and Geometric Means.)

Properties

The logarithm of the geometric mean is equal to the arithmetic mean of the logarithms of the variable values.
The geometrical mean of a set of positive numbers is always less or equal to the arithmetic mean:

${displaystyle (x_{1}x_{2}dots x_{n})^{frac {1}{n}}leq {frac {x_{1}+x_{2}+dots +x_{n}}{n}}}$

Equality is only achieved if ${displaystyle x_{1}=x_{2}=dots =x_{n}}$ .

Advantages

Consider all distribution values
It is less sensitive than arithmetic mean to extreme values.

Disadvantages

It is statistically less intuitive than arithmetic mean.
Your calculation is more difficult.
If a value ${displaystyle x_{i}=0,}$ then the geometric mean is overrided or not determined.

The geometric mean is only relevant if all numbers are positive. As we have seen, if one of them is 0, then the result is 0. If there were a negative number (or an odd number of them) then the geometric mean would be either negative or non-existent in the real numbers.

In many occasions its transformation is used in the statistical management of variables with non-normal distribution.

The geometric mean is relevant when multiple quantities are multiplied to produce a total.

Weighted geometric mean

Just as in an arithmetic mean weights can be entered as multiplicative values for each of the values in order to weight or weigh certain values more in the final result, in the geometric mean weights can be entered as exponents:

${displaystyle {bar {x}}=left({prod _{i=1}^{n}{x_{i}}^{alpha _{i}}}right)^{frac {1}{sum _{i}{alpha _{i}}}}=left({x_{1}}^{alpha _{1}}{x_{2}}^{alpha _{2}}dots {x_{n}}^{alpha _{n}}right)^{frac {1}{alpha _{1}+dots +alpha _{n}}}}$

Where the ${displaystyle alpha _{i},}$ They are the “weights”.

Illustrative case

A chain of gasoline vending machines last year increased its income compared to the previous year by 21%; and they have projected that this year they will reach an increase of 28% compared to last year. What is the average annual percentage increase?

Definitely not (21% + 28%):2 = 24.5%.

The amount of production, at the end of two years, is 100(1.21)(1.28)= 154.88. If in each year we have an annual rate of increase of i%, it results

100 → 100(1+i) → 100(1 +i)².

100(1 +i)² = 154.88

(1 +i)² = 1.5488

1 + i =

{displaystyle {sqrt {1,5488}}}

=1,244507 (this is the value of

{displaystyle {bar {x}}={sqrt {(1,21cdot 1,28)}},}

)

i = 0,244507 = 24.451%

Where it appears

Geometry

in all triangle rectangle:
- the height meets ${displaystyle h={sqrt {mn}}}$ , being m and n the projections of the caetos on the hypotenuse.
- a nut. b birthday ${displaystyle b={sqrt {ma}}}$ , being m its projection and a the hypotenuse.
the tangent t a circumference ${displaystyle t={sqrt {sk}}}$ s It's dry and k the internal part.
the side of a square equivalent to a rectangle is the geometrical mean of the sides of this; the radius of a circle equivalent to an ellipse is the geometric mean of the semiejes of this. The same is the case of the sphere with the ellipsoid.
the side (arist) d of a cube equivalent to a side orthoedron a, b, c is ${displaystyle d={sqrt[{3}]{abc}}}$

Weights

The weight w of a substance that has weights found by two scales or v results ${displaystyle w={sqrt {uv}}}$

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