Question

Marks obtained by all the students of class 12 are presented in a frequency distribution with classes of equal width. Let the median of this grouped data be 14 with median class interval 12-18 and median class frequency 12. If the number of students whose marks are less than 12 is 18, then the total number of students is:

• A. 48
• B. 44
• C. 40
• D. 52

Hint:

When solving for the total number of students using the median formula, always ensure the proper use of the class width and cumulative frequency before the median class.

Answer 1

The Correct Option is B

Step 1: The median of a grouped data is given by the formula: \[ \text{Median} = \ell + \left( \frac{\frac{N}{2} - F}{f} \right) \times h \] where: - \( \ell \) is the lower boundary of the median class, - \( N \) is the total number of observations, - \( F \) is the cumulative frequency before the median class, - \( f \) is the frequency of the median class, - \( h \) is the class width. From the problem, we are given: - Median class interval: 12-18, - Median class frequency \( f = 12 \), - \( \ell = 12 \), - Median = 14, - Number of students with marks less than 12 is 18. Step 2: Using the formula: \[ 14 = 12 + \left( \frac{\frac{N}{2} - 18}{12} \right) \times 6 \] Simplifying the equation: \[ 14 - 12 = \left( \frac{\frac{N}{2} - 18}{12} \right) \times 6 \] \[ 2 = \left( \frac{\frac{N}{2} - 18}{12} \right) \times 6 \] \[ 2 = \frac{\frac{N}{2} - 18}{2} \] \[ 4 = \frac{N}{2} - 18 \] \[ \frac{N}{2} = 22 \quad \Rightarrow \quad N = 44 \]