Harmonic Mean

Harmonic mean is another measure of central tendency and also based on mathematic footing like arithmetic mean and geometric mean. Like arithmetic mean and geometric mean, harmonic mean is also useful for quantitative data. Harmonic mean is defined in following terms:

Harmonic mean is quotient of “number of the given values” and “sum of the reciprocals of the given values”.

Harmonic mean in mathematical terms is defined as follows:

For Ungrouped Data

For Grouped Data

H.M{\text{ of }}X = \overline X = \frac{n}{{\sum \left( {\frac{1}{x}} \right)}}

H.M{\text{ of }}X = \overline X = \frac{{\sum f}}{{\sum \left( {\frac{f}{x}} \right)}}

Example:

Calculate the harmonic mean of the numbers: 13.5, 14.5, 14.8, 15.2 and 16.1

Solution:

The harmonic mean is calculated as below:

x

\frac{1}{x}

13.2

0.0758

14.2

0.0704

14.8

0.0676

15.2

0.0658

16.1

0.0621

Total

\sum \left(    {\frac{1}{x}} \right) = 0.3417

H.M{\text{ of }}X = \overline X = \frac{n}{{\sum \left( {\frac{1}{x}}  \right)}}

H.M{\text{ of }}X = \overline X = \frac{5}{{0.3417}} = 14.63





Example:

Given the following frequency distribution of first year students of a particular college. Calculate the Harmonic Mean.

Age (Years)

13

14

15

16

17

Number of Students

2

5

13

7

3

Solution:

The given distribution belongs to a grouped data and the variable involved is ages of first year students. While the number of students Represent frequencies.

Ages (Years)
x

Number of Students
f

\frac{1}{x}

13

2

0.1538

14

5

0.3571

15

13

0.8667

16

7

0.4375

17

3

0.1765

Total

\sum f = 30

\sum \left( {\frac{1}{x}} \right) = 1.9916

Now we will find the Harmonic Mean as
            \overline X = \frac{{\sum  f}}{{\sum \left( {\frac{f}{x}} \right)}} = \frac{{30}}{{1.9916}} = 15.0631  \approx 15 years.

Example:

Calculate the harmonic mean for the given below:    

Marks

30 - 39

40 - 49

50 - 59

60 - 69

70 - 79

80 - 89

90 - 99

f

2

3

11

20

32

25

7

Solution:

The necessary calculations are given below:

Marks

x

f

\frac{f}{x}

30 - 39

34.5

2

0.0580

40 - 49

44.5

3

0.0674

50 - 59

54.5

11

0.2018

60 - 69

64.5

20

0.3101

70 - 79

74.5

32

0.4295

80 - 89

84.5

25

0.2959

90 - 99

94.5

7

0.0741

Total

 

\sum f = 100

\sum \left( {\frac{f}{x}} \right) = 1.4368

Now we will find the Harmonic Mean as

\overline X = \frac{{\sum  f}}{{\sum \left( {\frac{f}{x}} \right)}} = \frac{{100}}{{1.4368}} = 69.60

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