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.

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:

$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	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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