Question

Find the mean of the following frequency distribution:

Class Interval	10-30	30-50	50-70	70-90	90-110	110-130
Frequency	5	8	12	20	3	2

Answer 1

Hint: First of all find the mid value of every class interval. Then make a separate column in the given table where you write the mid values corresponding to each class (and mark it as ${{x}_{i}}$). Then multiply every mid value with their corresponding frequency and add all of them. Now, to find the mean divide this multiplication result by the addition of all frequencies.

Complete step-by-step answer:
We have given a class interval corresponding with the frequencies which we are showing below:

Class Interval	10-30	30-50	50-70	70-90	90-110	110-130
Frequency	5	8	12	20	3	2

 Now, we are going to find the mid values of each class.
The mid value of class interval $\left( 10-30 \right)$ is calculated by adding the lower and upper values of the class interval followed by division by 2 in the following manner:
$\begin{align}
  & \dfrac{10+30}{2} \\
 & =\dfrac{40}{2} \\
 & =20 \\
\end{align}$
The mid value of class interval $\left( 30-50 \right)$ is:
$\begin{align}
  & \dfrac{30+50}{2} \\
 & =\dfrac{80}{2} \\
 & =40 \\
\end{align}$
Similarly, we can find the mid values of other class interval and tabulate in the table given above as:

Class interval	10-30	30-50	50-70	70-90	90-110	110-130
Frequency$\left( {{f}_{i}} \right)$	5 $\left( {{f}_{1}} \right)$	8 $\left( {{f}_{2}} \right)$	12 $\left( {{f}_{3}} \right)$	20 $\left( {{f}_{4}} \right)$	3 $\left( {{f}_{5}} \right)$	2 $\left( {{f}_{6}} \right)$
Mid value$\left( {{x}_{i}} \right)$	20$\left( {{x}_{1}} \right)$	40 $\left( {{x}_{2}} \right)$	60 $\left( {{x}_{3}} \right)$	80 $\left( {{x}_{4}} \right)$	100 $\left( {{x}_{5}} \right)$	120 $\left( {{x}_{6}} \right)$

We know that formula for mean in class distribution is:
$\dfrac{\sum\limits_{i=1}^{6}{{{f}_{i}}{{x}_{i}}}}{\sum\limits_{i=1}^{6}{{{f}_{i}}}}$ ……………….Eq. (1)
Now, we are going to add an extra row in the above table in which we are multiplying the second row with the third row.

Class interval	10-30	30-50	50-70	70-90	90-110	110-130
Frequency$\left( {{f}_{i}} \right)$	5	8	12	20	3	2
Mid value$\left( {{x}_{i}} \right)$	20	40	60	80	100	120
$\left( {{f}_{i}}{{x}_{i}} \right)$	100	320	720	1600	300	240

Summation of all the elements in the last row of the above table we get,
\[\begin{align}
  & \sum\limits_{i=1}^{6}{{{f}_{i}}{{x}_{i}}}={{f}_{1}}\left( {{x}_{1}} \right)+{{f}_{2}}\left( {{x}_{2}} \right)+{{f}_{3}}\left( {{x}_{3}} \right)+{{f}_{4}}\left( {{x}_{4}} \right)+{{f}_{5}}\left( {{x}_{5}} \right)+{{f}_{6}}\left( {{x}_{6}} \right) \\
 & \Rightarrow \sum\limits_{i=1}^{6}{{{f}_{i}}{{x}_{i}}}=100+320+720+1600+300+240 \\
 & \Rightarrow \sum\limits_{i=1}^{6}{{{f}_{i}}{{x}_{i}}}=3280 \\
\end{align}\]
Now, adding all the elements of second row in the above table we get,
$\begin{align}
  & \sum\limits_{i=1}^{6}{{{f}_{i}}}={{f}_{1}}+{{f}_{2}}+{{f}_{3}}+{{f}_{4}}+{{f}_{5}}+{{f}_{6}} \\
 & \Rightarrow \sum\limits_{i=1}^{6}{{{f}_{i}}}=5+8+12+20+3+2 \\
 & \Rightarrow \sum\limits_{i=1}^{6}{{{f}_{i}}}=50 \\
\end{align}$
Substituting the above summation values in eq. (1) we get,
$\begin{align}
  & \dfrac{\sum\limits_{i=1}^{6}{{{f}_{i}}{{x}_{i}}}}{\sum\limits_{i=1}^{6}{{{f}_{i}}}}=\dfrac{3280}{50} \\
 & \Rightarrow \dfrac{\sum\limits_{i=1}^{6}{{{f}_{i}}{{x}_{i}}}}{\sum\limits_{i=1}^{6}{{{f}_{i}}}}=65.6 \\
\end{align}$
Hence, the mean of the above class distribution is 65.6.

Note: Instead of solving the mid values, by symmetry and the observation you can find the mid values of all the class intervals. For instance, the mid value for class 10 – 30 is 20. If you carefully look the mid value (20) and the lower and upper limit of the class you will find that the first digit of the mid value (20) is the number 2 which lies between 1 and 3 with a gap of 1 (where 1 and 3 are the first digits of lower and upper limit) and the second digit of the mid value (20) is 0 which is same as the second digit of lower and upper limits of the class. If you can understand the analogy then the mid value for the next class interval i.e. 30 – 50 is 40. Similarly for the class interval with three digits like 110 – 130, the last digit of mid value is the same as the lower and upper limit (i.e. 0) and the first two digits are numbers lying between 11 and 13 i.e. 12 so the mid value is 120.