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