# Statistics - Geometric Mean

Geometric mean of n numbers is defined as the nth root of the product of n numbers.

## Formula

${GM = \sqrt[n]{x_1 \times x_2 \times x_3 ... x_n}}$

Where −

• ${n}$ = Total numbers.

• ${x_i}$ = numbers.

### Example

Problem Statement:

Determine the geometric mean of following set of numbers.

 1 3 9 27 81

Solution:

Step 1: Here n = 5

${GM = \sqrt[n]{x_1 \times x_2 \times x_3 ... x_n} \\[7pt] \, = \sqrt[5]{1 \times 3 \times 9 \times 27 \times 81} \\[7pt] \, = \sqrt[5]{3^3 \times 3^3 \times 3^4} \\[7pt] \, = \sqrt[5]{3^{10}} \\[7pt] \, = \sqrt[5]{{3^2}^5} \\[7pt] \, = \sqrt[5]{9^5} \\[7pt] \, = 9 }$

Thus geometric mean of given numbers is $9$.