Question

Find the mean of the following frequency distribution:

Class Interval	Frequency (${f_i}$)
$0 - 8$	$6$
$8 - 16$	$7$
$16 - 24$	$10$
$24 - 32$	$8$
$32 - 40$	$9$

Answer 1

According to given in the question we have to determine the mean for the given frequency distribution. So, first of all to find the mean we have to understand mean as explained below:
Mean: The mean is basically a way to find the average of the given data but before that we have to arrange the all data or given number into ascending order means we have to arrange all the given data from the smaller number to the largest number and after that we have to add up all the given data or number and same as we have to determine the total number of given data or terms.

Formula used:
$\overline x = \dfrac{{\sum {{f_i}{x_i}} }}{{\sum {{f_i}} }}........................(a)$
Obtained by But first of all we have to determine the class which can be obtained by dividing the sum of upper limit and lower limit by 2.
Now, we have to calculate ${f_i}{x_i}$ with the help of the table obtained, and after applying the formula (a) as given above we can obtain the required mean.

Complete step-by-step solution:
Step 1: First of all we have to determine the value of class $({x_i})$ which can be obtained by dividing the sum of lower limit and upper limit for the given class interval by 2 as mentioned in the solution step.

Class interval	Frequency (${f_i}$)	$({x_i})$
$0 - 8$	$6$	$\dfrac{{0 + 8}}{2} = 4$
$8 - 16$	$7$	$\dfrac{{8 + 16}}{2} = 12$
$16 - 24$	$10$	$\dfrac{{16 + 24}}{2} = 20$
$24 - 32$	$8$	$\dfrac{{24 + 32}}{2} = 28$
$32 - 40$	$9$	$\dfrac{{32 + 40}}{2} = 36$

Step 2: Now, we have to find the value of ${f_i}{x_i}$ which can be obtained by the multiplication of ${f_i}$ and $({x_i})$ as mentioned in the solution hint. Hence,

Class interval	Frequency (${f_i}$)	$({x_i})$	${f_i}{x_i}$
$0 - 8$	$6$	$\dfrac{{0 + 8}}{2} = 4$	$6 \times 4 = 24$
$8 - 16$	$7$	$\dfrac{{8 + 16}}{2} = 12$	$7 \times 12 = 84$
$16 - 24$	$10$	$\dfrac{{16 + 24}}{2} = 20$	$10 \times 20 = 200$
$24 - 32$	$8$	$\dfrac{{24 + 32}}{2} = 28$	$8 \times 28 = 224$
$32 - 40$	$9$	$\dfrac{{32 + 40}}{2} = 36$	$9 \times 36 = 324$

Step 3: Now, to find the value of mean we have to use the formula (a) as mentioned in the solution hint. Hence, on substituting all the values in the formula (a),
$\Rightarrow (\overline x ) = \dfrac{{24 + 84 + 200 + 224 + 324}}{{6 + 7 + 10 + 8 + 9}} \\\ \Rightarrow (\overline x ) = \dfrac{{856}}{{40}} \\\ \Rightarrow (\overline x ) = 21.4$

Hence, with the help of the formula (a) we have obtained the value of mean for the given data which is $(\overline x ) = 21.4$

Note: Mean can be determined by arranging the given data in ascending order and then we have to divide the sum of all the given data with the total number of given data.
The mean of the absolute values of the numerical differences between the numbers of a set such as, a static data and their mean and median.