Question

Following tale gives a frequency distribution of an amount of bonus paid to the workers in a certain factor.

Bonus paid (in Rs.)	Below $500$	Below $600$	Below $700$	Below $800$	Below $900$	Below $1000$	Below $1100$
Number of workers	$4$	$12$	$24$	$41$	$51$	$58$	$60$

Find the median amount of bonus paid to the workers.

Answer 1

Hint: In these types of questions first we will find a median class or by using the median formula which we use for finding mean value in ungrouped data then we will calculate the mean value for the particular class.

Complete step by step Solution:
We know the formula for finding the value of the Median for Class data which is
$Median = l + \dfrac{{(\dfrac{N}{2} - cf)}}{{{f_m}}} \times c$
Where $l = $ Lower boundary of the median class, $N = \sum\limits_{}^{} {{f_i}} = $ sum of all frequency which is given as number of workers, $cf = $ cumulative frequency of previous class of median class, $c = $ class interval
${f_m} = $ frequency of median class.
Now we will draw a table in which we will draw three columns one is for class and second for the frequency and the third one for the cumulative frequency
For these types of questions in which the class interval is repeated, Cumulative frequency is nothing; it is only the subtraction of previous class frequency from current class frequency then we will get the current class frequency and then add it to previous cumulative frequency.
We will understand it by the table
So now we will draw table -

Class Interval$ = ci$	Number of workers-frequency$ = {f_i}$	Cumulative frequency$ = cf$
$below - 500$	$4$	$04$
$500 - 600$	$12$	$12$
$600 - 700$	$24$	$24$
$700 - 800$	$41$	$41$
$800 - 900$	$51$	$51$
$900 - 1000$	$58$	$58$
$1000 - 1100$	$60$$\sum\limits_{}^{} {{f_i} = 60} $	$60$

Now first we will calculate that which class interval is the median class interval for our given data-
We know that for finding the median class interval we have to calculate the value of the $\dfrac{{\sum\limits_{}^{} {{f_i}} }}{2} = \dfrac{N}{2}$
So the value of $\dfrac{{\sum\limits_{}^{} {{f_i}} }}{2} = \dfrac{N}{2} = \dfrac{{60}}{2} = 30$
So the our class interval is the $ci = 700 - 800$ now we will calculate the median
Now the values for our formula is
$l = 700 = $ Lower boundary of the median class,
$N = \sum\limits_{}^{} {{f_i}} = 60$ sum of all frequency which is given as number of workers,
And the value of $\dfrac{N}{2} = \dfrac{{\sum\limits_{}^{} {{f_i}} }}{2} = \dfrac{{60}}{2} = 30$
$cf = 24 = $ cumulative frequency of previous class of median class,
$c = 100 = $ class interval,
${f_m} = 17 = $ frequency of median class
Now we will put all value in the formula of median so we will calculate the median value of bonus which paid to the workers
$Median = l + \dfrac{{(\dfrac{N}{2} - cf)}}{{{f_m}}} \times c$
Now after putting all value in the formula we will get
$ \Rightarrow Median = 700 + \dfrac{{(30 - 24)}}{{17}} \times 100$
After calculating values we will get
$ \Rightarrow Median = 700 + \dfrac{{(6)}}{{17}} \times 100$
After completely solving above equation we will get
$ \Rightarrow Median = 700 + 35.29 = 735.29$

Therefore the median amount of bonus paid to the workers is $735.29$ which is our required value for our question.

Note:
For finding the value first order in the form of increasing value or in the form of decreasing value according to given frequency then we will find the cumulative frequency then we can directly see the value of the mean class then calculate it further only this way we can solve these types of questions.