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

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

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