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:
Show Hint
- 48
- 44
- 40
- 52
The Correct Option is B
Approach Solution - 1
To solve this problem, we need to use the concept of the median in a grouped frequency distribution. Given that:
- Median = 14
- Median Class Interval = 12-18
- Median Class Frequency = 12
- Number of students with marks less than 12 = 18
The formula for finding the median in a grouped data set is:
\(Median = L + \left( \frac{N/2 - F}{f} \right) \times h\)
- \(L\) is the lower limit of the median class (12 in this case).
- \(N\) is the total number of observations.
- \(F\) is the cumulative frequency of the class preceding the median class (students getting marks less than 12, which is 18).
- \(f\) is the frequency of the median class (12).
- \(h\) is the class width (18 - 12 = 6).
Substitute the known values into the formula:
\(14 = 12 + \left( \frac{N/2 - 18}{12} \right) \times 6\)
Solving for \(N\):
\(14 - 12 = \left( \frac{N/2 - 18}{12} \right) \times 6\)
\(2 = \left( \frac{N/2 - 18}{12} \right) \times 6\)
\(2/6 = \frac{N/2 - 18}{12}\)
\(1/3 = \frac{N/2 - 18}{12}\)
Convert this to fraction and solve:
\(12 \times 1/3 = N/2 - 18\)
\(4 = N/2 - 18\)
\(N/2 = 4 + 18\)
\(N/2 = 22\)
\(N = 44\)
Therefore, the total number of students is 44. This matches with the given correct answer option.
Approach Solution -2
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: \[ 14 - 12 = \left( \frac{\frac{N}{2} - 18}{12} \right) \times 6 \] \[ 2 = \left( \frac{\frac{N}{2} - 18}{12} \right) \times 6 \] \[ 2 = \frac{\frac{N}{2} - 18}{2} \] \[ 4 = \frac{N}{2} - 18 \] \[ \frac{N}{2} = 22 \quad \Rightarrow \quad N = 44 \]
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