Harmonic Mean
Harmonic mean is another measure of central tendency and is also based on mathematics like arithmetic mean and geometric mean. Like arithmetic mean and geometric mean, harmonic mean is also useful for quantitative data. Harmonic mean is defined as:
Harmonic mean is the quotient of the “number of the given values” and the“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 is grouped data and the variable involved is the ages of first year students, while the number of students represents 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

$$\sum f = 30$$

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

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