# Geometric Mean Formula

Geometric mean formula, as the name suggests, is used to calculate the geometric mean of a set of numbers. To recall, the geometric mean (or GM) is a type of mean that indicates the central tendency of a set of numbers by using the product of their values. It is defined as the nth root of the product of n numbers. It should be noted that you cannot calculate geometric mean from the arithmetic mean. The geometric mean is well defined only for a positive set of real numbers. Example of using the formula for the geometric mean is to calculate the central frequency f0 of a bandwidth BW= f2–f1.

Formula to Calculate Geometric Mean

## Formula for Geometric Mean

For GM formula, multiply all the “n” numbers together and take the “nth root of them. The formula for evaluating geometric mean is as follows if we have “n” number of observations.

 $\bar{x}_{geom}=\sqrt[n]{\prod_{i=1}^{n}x_{i}}=\sqrt[n]{x_{1}\cdot x_{2}\cdot\cdot\cdot x_{n}}$

### Notation in the GM Formula

• $\bar{x}_{geom}$ is the geometric mean
• “n” is the total number of observations
• $\sqrt[n]{\prod_{i=1}^{n}x_{i}}$ is the nth square of the product of the given numbers.

### Example Question Using Geometric Mean Formula

Question 1: Find the geometric mean of 4 and 3?

Solution: Using the formula for G.M., the geometric mean of 4 and 3 will be:

Geometric Mean = $\sqrt{4\times3}=2\sqrt{3}$

So, GM = 3.46

Question 2: What is the geometric mean of 4, 8, 3, 9 and 17?

Solution:

Step 1: n = 5 is the total number of values. Now, find 1/n.

⅕ = 0.2.

Step 2: Find geometric mean using the formula:

$(4\times8\times3\times9\times17)^{0.2}$

So, geometric mean = 6.81

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