# Geometric Mean Formula

The geometric mean is a type of mean that indicates the central tendency or typical value of a set of numbers by using the product of their values. It is defined as the $n^{th}$ root of the product of $n$ numbers. You cannot calculate geometric mean from the arithmetic mean. Geometric mean is well defined only for positive set of real numbers. Example calculating the central frequency $f_{0}$ of a bandwidth BW= $f_{2} - f_{1}$

Formula for evaluating geometric mean is as follows if we have n number of observations, then

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

### Solved Examples

Question: Find the geometric mean of 4 and 3 ?

Solution:

Geometric Mean = $\sqrt{4 \times 3}$ = $2\sqrt{3}$ = 3.46

Question: What is the geometric mean of 4, 8, 3, 9 and 17 ?
Solution:
Step 1: n = 5 is the total number of values. Find 1/n.
1/5 = 0.2
Step 2: Find geometric mean using the formula:
$\left [ 4*8*3*9*17 \right ]^{(0.2)}$
Geometric Mean = 6.81