## 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
`$$ \begin{aligned} GM & =\exp(\log (GM))\\ &=\exp(2.6706)\\ &=14.4486 \end{aligned} $$`