## What is Harmonic Mean?

Harmonic Mean is also a mathematical average but is limited in its application. It is generally used to find average of variables that are expressed as a ratio of two different measuring units e. g. speed is measured in km/hr or miles/sec etc.

## Weighted Harmonic Mean

## Formula

$H.M. = \frac{W}{\sum (\frac{W}{X})}$

Where −

- ${H.M.}$ = Harmonic Mean
- ${W}$ = Weight
- ${X}$ = Variable value

### Example

**Problem Statement:**

Find the weighted H.M. of the items 4, 7,12,19,25 with weights 1, 2,1,1,1 respectively.

**Solution:**

${X}$ | ${W}$ | $\frac{W}{X}$ |
---|---|---|

4 | 1 | 0.2500 |

7 | 2 | 0.2857 |

12 | 1 | 0.0833 |

19 | 1 | 0.0526 |

25 | 1 | 0.0400 |

$\sum W$ | $\sum \frac{W}{X}$= 0.7116 |

Based on the above mentioned formula, Harmonic Mean $G.M.$ will be:

\, = \frac{6}{0.7116} \\[7pt]

\, = 8.4317 $

∴ Weighted H.M = 8.4317

We’re going to discuss methods to compute the **Harmonic Mean** for three types of series:

- Individual Data Series
- Discrete Data Series
- Continuous Data Series

## Individual Data Series

When data is given on individual basis. Following is an example of individual series:

Items | 5 | 10 | 20 | 30 | 40 | 50 | 60 | 70 |
---|

## Discrete Data Series

When data is given alongwith their frequencies. Following is an example of discrete series:

Items | 5 | 10 | 20 | 30 | 40 | 50 | 60 | 70 |
---|---|---|---|---|---|---|---|---|

Frequency | 2 | 5 | 1 | 3 | 12 | 0 | 5 | 7 |

## Continuous Data Series

When data is given based on ranges alongwith their frequencies. Following is an example of continous series:

Items | 0-5 | 5-10 | 10-20 | 20-30 | 30-40 |
---|---|---|---|---|---|

Frequency | 2 | 5 | 1 | 3 | 12 |

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