Geometric mean for grouped data

Let $(x_i,f_i), i=1,2, \cdots , n$ be the given frequency distribution then the geometric mean of $X$ is denoted by $GM$.

Formula

The geometric mean of grouped data is given by

$GM=\bigg(\prod_{i=1}^n x_i^{f_i}\bigg)^{1/N}$

OR

`$\log (GM) =\frac{1}{N}\sum_

where,

  • $N=\sum_i f_i$ total number of observations

Example

Compute geometric mean for the following frequency distribution.

x 5-8 9-12 13-16 17-20 21-24
f 2 13 21 14 5

Solution

Class Interval Class Boundries mid-value ($x_i$) Freq ($f_i$) $log(x_i)$ $f_i*log(x_i)$
5-8 4.5-8.5 6.5 2 1.8718 3.7436
9-12 8.5-12.5 10.5 13 2.3514 30.5682
13-16 12.5-16.5 14.5 21 2.6741 56.1561
17-20 16.5-20.5 18.5 14 2.9178 40.8492
21-24 20.5-24.5 22.5 5 3.1135 15.5675
Total 55 146.8846

The log of geometric mean is

$$ \begin{aligned} \log (GM) &=\frac{1}{N}\sum_{i=1}^n f_i\log(x_i)\\ &=\frac{146.8846}{55}\\ &=2.6706 \end{aligned} $$ The geometric mean is $$ \begin{aligned} GM & =\exp(\log (GM))\\ &=\exp(2.6706)\\ &=14.4486 \end{aligned} $$

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