HARMONIC MEAN FORMULA

Harmonic mean is used to calculate the average of a set of numbers. The number of elements will be averaged and divided by the sum of the reciprocals of the elements.

It is calculated by dividing the number of observations by the sum of reciprocal of the observation.

The formula to find the harmonic mean is given by:

For Ungrouped Data:

H.M of X = $\bar{X}$ = $\frac{n}{\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+...............+\frac{1}{x_{n}}}=\frac{n}{\sum (\frac{1}{x})}$

Where,

n = Total number of numbers or terms.

x1, x2, x3, …. xn = Individual terms or individual values.

 Lets Work Out- Example: Find the harmonic mean of the following data {8, 9, 6, 11, 10, 5} ? Solution: Given data: {8, 9, 6, 11, 10, 5} So Harmonic mean = $\frac{6}{\frac{1}{8}+\frac{1}{9}+\frac{1}{6}+\frac{1}{11}+\frac{1}{10}+\frac{1}{5}}$ H =$\frac{6}{0.7936 }$= 7.560 Harmonic mean(H) = 7.560

For Grouped Data:

H.M of X = $\bar{X}$ = $\bar{X}=\frac{f_{1}+f_{2}+f_{3}+................+f_{n}}{\frac{f_{1}}{x_{1}}+\frac{f_{2}}{x_{2}}+\frac{f_{3}}{x_{3}}+................+\frac{f_{n}}{x_{n}}}=\frac{\sum f}{\sum (\frac{f}{x})}$

Where,

f = Individual entries.

x1, x2, x3, …. xn = Individual terms or individual values.

Lets Work Out-

Example:The table given below represent the frequency-distribution of ages for Standard 1st students.

 Ages 4 5 6 7 Number of Students 6 4 10 8

Find the Harmonic Mean of the given class.

Solution:

Here the data given are distributed data. So the ages are the variables and the number of student is considered as the frequency.

 Ages (x) Number of Students (f) $\frac{f}{x}$ 4 10 2.5 5 6 1.2 6 8 1.33 7 4 0.57 Total $\sum f$=28 $\sum(\frac{f}{x})$= 5.6

So Harmonic mean = $\frac{\sum f}{\sum(\frac{f}{x})}=\frac{28}{5.6}$ = 5 years

Therefore, Harmonic mean(H) = 5 years