Geometric Mean Formula

Before beginning with the geometric mean formula, let us recall what is the geometric mean. The geometric mean is the central tendency of a set of numbers calculated using the product of their values. It can be defined as the n^th root of the product of n numbers. The geometric mean is also known as geometric average. Let us learn the geometric mean formula with a few solved examples.

What Is Geometric Mean Formula?

The geometric mean formula can be used to find the geometric mean or geometric average of the given data. It is one of the important measures of the central tendency of a given set of observations.

Geometric Mean Formula

Suppose we have n observation values: \(x_1, x_2, ..., x_n\). Geometric mean formula can be calculated as,

Geometric Mean = \(\sqrt[n]{x_1 x_2 ... x_n}\)
or
GM = \(\sqrt[n]{\prod}_{i = 1}^n x_i\)

Another way to represent the formula to calculate the geometric mean of a data is given as, 
Log GM = 1/n (x\(_1\), x\(_2\), x\(_3\), . . ., x\(_n\))
Log GM = 1/n(log x\(_1\) + log x\(_2\) + log\(_3\) + . . + log x\(_n\) )
Log GM = Σlog x\(_i\)/n
⇒ GM = antilog Σlog x\(_i\)/n
where, n = f\(_1\) + f\(_2\) + f\(_3\) + . . . + f\(_n\)

Geometric Mean Formula for Grouped Data

For a given set of data having n values \((x_1, x_2, ..., x_n)\), such that the frequency of these values are \((f_1, f_2, ..., f_n)\), then the GM can be given as,
GM = \((x_1^{f_1} • x_2^{f_2} • x_3^{f_3} • . . . • x_1^{f_n})^{1/n}\) = antilog Σlog x\(_i\)/n
where, n = f\(_1\) + f\(_2\) + f\(_3\) + . . . + f\(_n\)

Geometric Mean Formula for Ungrouped Data

If we have n observation values: \(x_1, x_2, ..., x_n\). Geometric mean formula for this case is given as,
Geometric Mean = \(\sqrt[n]{x_1 x_2 ... x_n}\)
or
GM = \(\sqrt[n]{\prod}_{i = 1}^n x_i\)

Applications of Geometric Mean Formula

The geometric mean formula finds application in the calculation of geometric mean, which is an important measure used in many fields. Its applications are:

• The geometric mean is used for describing proportional growth, in business the geometric mean of growth rates is known as the compound annual growth rate (CAGR). 
• It is used in the calculation of financial indices.
• In film and video, it is used to choose a compromise aspect ratio.
• It is used in signal processing to find the spectral flatness of a power spectrum.

Examples Using Geometric Mean Formula

Example 1: Find the geometric average of 2, 4, and 8?

Solution:

The number of observations, n = 3

Using the geometric mean formula,

Geometric Mean = \(\sqrt[3]{2 \times 4 \times 8}\\=\sqrt[3]{64}\\=4\)

Answer: Geometric mean of given observations = 4.

Example 2: Find the sum of geometric mean and the arithmetic mean of 9 and 4.

Solution:

Using the geometric mean formula,

Geometric Mean = √(9 × 4) = √36 = 6

Using the arithmetic mean formula,

Arithmetic Mean =(9 + 4)/2 = 13/2 = 6.5

Geometric Mean + Arithmetic Mean = 6 + 6.5 = 12.5

Answer: Sum of geometric mean and the arithmetic mean of 9 and 4 = 12.5.

Example 3: Find the geometric mean of the first 3 even natural numbers.

Solution:

First 3 even natural numbers = 2, 4, 6

Using the geometric mean formula,

Geometric Mean = \(\sqrt[3]{2 \times 4 \times 6}\\=3.63\)

Answer: Geometric mean of first 3 even natural numbers = 3.63.