Harmonic mean, Formula, definition, Uses, Merits and demerits| Statistics

Harmonic mean

The harmonic mean is a one of the type of numerical average. It is calculated by dividing the number of Values (observations) by the reciprocal of each number in the series. Thus, we can say that harmonic mean is the reciprocal(n/value) of the arithmetic mean of the reciprocals(1/value).

Harmonic mean definition

The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals

Yes, that is a lot of reciprocals!

("Reciprocal" simply means 1/value)

Harmonic mean formula

Harmonic mean Formula 

Where a,b,c,... are the values, and n is how many values.

Steps to calculate Harmonic mean

Step 1: Calculate the reciprocal (1/value) for every value.

Step 2: Find the average of those reciprocals (just add them and divide by no. of values

Step 3: Then do the reciprocal of that average (=1/average)

Example: What is the harmonic mean of 4, 5 and 10?

The reciprocals of 4, 5 and 10 are:

1/4 = 0.25,   1/5 = 0.20,   1/10 = 0.10

Now add them up:

 0.25 + 0.20 + 0.10 = 0.55

Divide by how many:

Average = 0.55/3

The reciprocal of that average is our answer:

Harmonic Mean = 3/0.55 = 5.454 (to 3 places)

Application of Harmonic Mean/ Uses of Harmonic Mean

Harmonic Mean is useful in computation of average prices average speed etc under certain conditions.

Merits of Harmonic mean

1. It is rigidly defined

2. It is based on all observation

3. It is least affected by fluctuation in sampling

4. It gives to greater importance to small items

5. It is capable of further algebraic treatment.

Demerits of Harmonic mean

1. It is difficult to calculate

2. It does not give equal weight to every item

3. It may not be represented in the actual data

4. It is not defined for negative value.

In this video I have explained how to solve Harmonic mean problem with example in Hindi.


Harmonic mean
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Harmonic mean definition