Question

Consider the data :

Class	$65-85$	$85-105$	$105-125$	$125-145$	$145-165$	$165-185$	$185-205$
Frequency	$4$	$5$	$13$	$20$	$14$	$7$	$4$

The difference of the upper limit of the median class and the lower limit of the modal class is

• A. $0$
• B. $19$
• C. $20$
• D. $38$

Answer 1

The correct option is C

$20$

Step 1: Create cumulative frequency table

To find the median and mode of the given grouped data, we need the cumulative frequency.

The cumulative frequency table is:

Class	Frequency $\left(f\right)$	Cumulative Frequency $\left(N\right)$
$65-85$	$4$	$4$
$85-105$	$5$	$9$
$105-125$	$13$	$22$
$125-145$	$20$	$42$
$145-165$	$14$	$56$
$165-185$	$7$	$63$
$185-205$	$4$	$67$

Step 2: Finding the median class

Median class: The class whose cumulative frequency is greater than and near to $\left(\frac{N}{2}\right)th$ term is called the median class of grouped data.

From the above data

$\left(\frac{N}{2}\right)th\mathrm{term}=\left(\frac{67}{2}\right)th\mathrm{term}=\left(33.5\right)th\mathrm{term}$ that lies in the interval $125-145$.

Hence, upper limit of median class is $145$.

Step 3: Finding the modal class

Modal Class: Modal class refers to the class having the highest frequency.

Here, we observe that the highest frequency is $20$ which lies in $125-145$.

Hence, the lower limit of modal class is $125$.

Step 4: Find the Required differences

$\therefore$ $\mathrm{Required}\mathrm{difference}=\mathrm{Upper}\mathrm{limit}\mathrm{of}\mathrm{median}\mathrm{class}-\mathrm{Lower}\mathrm{limit}\mathrm{of}\mathrm{modal}\mathrm{class}$

$$\begin{gathered} =145-125 \\ =20 \end{gathered}$$

Thus, the difference of the upper limit of the median class and the lower limit of the modal class is $20$.

Hence, the correct option is C.