# Statistics - Harmonic Mean

## 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.

## 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}$
410.2500
720.2857
1210.0833
1910.0526
2510.0400
$\sum W$$\sum \frac{W}{X}$= 0.7116

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

$H.M. = \frac{W}{\sum (\frac{W}{X})} \\[7pt] \, = \frac{6}{0.7116} \\[7pt] \, = 8.4317$

∴ Weighted H.M = 8.4317

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

## Individual Data Series

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 Frequency 5 10 20 30 40 50 60 70 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 Frequency 0-5 5-10 10-20 20-30 30-40 2 5 1 3 12