Geometric mean

Geometric mean is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. Unlike the arithmetic mean, which uses addition and division, the geometric mean is calculated by multiplying the numbers together and then taking the square root (for two numbers), cube root (for three numbers), etc.

Definition

The geometric mean of a data set {a1, a2, ..., an} is given by the nth root of the product of the numbers, i.e.,

(a1 * a2 * ... * an)^(1/n)

This can also be expressed in logarithmic form:

exp((1/n) * (ln(a1) + ln(a2) + ... + ln(an)))

Applications

The geometric mean has many applications in various fields, including mathematics, statistics, finance, biology, engineering, and computer science. It is particularly useful when dealing with data that varies exponentially or multiplicatively, such as growth rates, ratios, and geometric sequences.

In finance, the geometric mean is used to calculate the compound annual growth rate (CAGR), which represents the average annual growth rate over a specified period of time.

In biology, the geometric mean is often used in the analysis of proportional data and log-normal distributions.

Properties

The geometric mean has several important properties:

• It is always less than or equal to the arithmetic mean, with equality only when all the numbers are the same.
• It is invariant under change of scale, i.e., multiplying all the numbers by a constant factor does not change the geometric mean.
• It is the limit of the arithmetic-geometric mean as the two means converge.

See also

• Arithmetic mean
• Harmonic mean
• Arithmetic-geometric mean
• Log-normal distribution
• Compound annual growth rate

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  Geometric mean illustration
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  Derivation of 16:9 aspect ratio