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

To address this problem, the median concept for a grouped frequency distribution is applied. The provided data are:

• Median value: 14
• Median class interval: 12-18
• Frequency of the median class: 12
• Number of observations below the median class (marks<12): 18

The standard formula for calculating the median in grouped data is:

\(Median = L + \left( \frac{N/2 - F}{f} \right) \times h\)

• \(L\) represents the lower boundary of the median class, which is 12.
• \(N\) denotes the total count of observations.
• \(F\) is the cumulative frequency of the class immediately preceding the median class, given as 18.
• \(f\) is the frequency of the median class, which is 12.
• \(h\) signifies the width of the class interval, calculated as 18 - 12 = 6.

Substituting the known values into the formula yields:

\(14 = 12 + \left( \frac{N/2 - 18}{12} \right) \times 6\)

The objective is to solve for \(N\):

\(14 - 12 = \left( \frac{N/2 - 18}{12} \right) \times 6\)

\(2 = \left( \frac{N/2 - 18}{12} \right) \times 6\)

\(2/6 = \frac{N/2 - 18}{12}\)

\(1/3 = \frac{N/2 - 18}{12}\)

Proceeding with fractional calculations:

\(12 \times 1/3 = N/2 - 18\)

\(4 = N/2 - 18\)

\(N/2 = 4 + 18\)

\(N/2 = 22\)

\(N = 44\)

Consequently, the total number of students is 44. This result aligns with the provided correct option.