## Geometric mean for ungrouped data

Let $x_i, i=1,2, \cdots , n$ be $n$ positive observations then the geometric mean of $X$ is denoted by $GM$.

## Formula

The geometric mean is given by

`$GM =\big(\prod_{i=1}^n x_i\big)^{1/n}=\sqrt[n]{x_1\times x_2\times \cdots \times x_n}$`

## Example

The marks obtained by a sample of 5 students in a class test are as follows:

20,30,21,10 and 15.

Find the geometric mean of the marks.

### Solution

The geometric mean of $X$ is

`$$ \begin{aligned} GM &=\big(\prod_{i=1}^n x_i\big)^{1/n}\\ &= \big(20*30*21*10*15\big)^{1/5}\\ &= \big(1890000\big)^{1/5}\\ &= 18.0008 \end{aligned} $$`