## Harmonic mean for grouped data

Let $(x_i,f_i), i=1,2, \cdots , n$ be given frequency distribution. Then the harmonic mean of $X$ is denoted by $HM$ and is given by

```
$$
\begin{aligned}
HM &= \dfrac{N}{\sum_{i=1}^{n} \dfrac{f_i}{x_i}}
\end{aligned}
$$
```

where

- $N=\sum f$ total no. of observations

## Example 1

Compute harmonic mean for the following frequency distribution.

x | 10 | 15 | 20 | 25 | 30 |
---|---|---|---|---|---|

f | 3 | 12 | 25 | 10 | 5 |

### Solution

$x$ | Freq ($f$) | $f/x$ | |
---|---|---|---|

10 | 3 | 0.3 | |

15 | 12 | 0.8 | |

20 | 25 | 1.25 | |

25 | 10 | 0.4 | |

30 | 5 | 0.1667 | |

Total | 55 | 2.9167 |

The harmonic mean of $X$ is

```
$$
\begin{aligned}
HM &= \dfrac{N}{\sum_{i=1}^{n} \dfrac{f_i}{x_i}}\\
&= \frac{55}{\frac{3}{10}+\frac{12}{15}+\frac{25}{20}+\frac{10}{25}+\frac{5}{30}}\\
&= \frac{55}{2.9167}\\
&= 18.8571
\end{aligned}
$$
```

## Example 2

Compute harmonic mean for the following frequency distribution.

x | 10-15 | 15-20 | 20-25 | 25-30 | 30-35 |
---|---|---|---|---|---|

f | 2 | 13 | 21 | 14 | 5 |

### Solution

Class | mid-value ($x$) | Freq ($f$) | $f/x$ | |
---|---|---|---|---|

10-15 | 12.5 | 2 | 0.16 | |

15-20 | 17.5 | 13 | 0.7429 | |

20-25 | 22.5 | 21 | 0.9333 | |

25-30 | 27.5 | 14 | 0.5091 | |

30-35 | 32.5 | 5 | 0.1538 | |

Total | 55 | 2.4991 |

The harmonic mean of $X$ is

```
$$
\begin{aligned}
HM &= \dfrac{N}{\sum_{i=1}^{n} \dfrac{f_i}{x_i}}\\
&= \frac{55}{\frac{2}{12.5}+\frac{13}{17.5}+\frac{21}{22.5}+\frac{14}{27.5}+\frac{5}{32.5}}\\
&= \frac{55}{2.4991}\\
&= 22.0077
\end{aligned}
$$
```