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