To find the median marks, we need to follow a step-by-step process.
### Step 1: Expand the Data
Firstly, we list all the marks received by the students based on the given frequencies.
- Marks of 24 are received by 2 students.
[tex]\[ 24, 24 \][/tex]
- Marks of 36 are received by 4 students.
[tex]\[ 36, 36, 36, 36 \][/tex]
- Marks of 50 are received by 12 students.
[tex]\[ 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50 \][/tex]
- Marks of 65 are received by 11 students.
[tex]\[ 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65 \][/tex]
- Marks of 78 are received by 6 students.
[tex]\[ 78, 78, 78, 78, 78, 78 \][/tex]
Combining all of these data points, we get the following list of marks:
[tex]\[ 24, 24, 36, 36, 36, 36, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 78, 78, 78, 78, 78, 78 \][/tex]
### Step 2: Sort the Data
Next, sort the marks from smallest to largest. In this case, the marks are already sorted in ascending order.
### Step 3: Find the Median
The median is the middle value in a list when the list is sorted in ascending order. If the number of data points is odd, the median is the middle number. If the number of data points is even, the median is the average of the two middle numbers.
Calculate the total number of students:
[tex]\[ 2 + 4 + 12 + 11 + 6 = 35 \][/tex]
Since we have 35 students (an odd number), the median is the [tex]\( \left(\frac{35 + 1}{2}\right)^{th} \)[/tex] value, which is the 18th value.
Looking at the expanded list of marks:
[tex]\[ 24, 24, 36, 36, 36, 36, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, \mathbf{50}, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 78, 78, 78, 78, 78, 78 \][/tex]
The 18th value in the list is 50.
### Step 4: Conclusion
Therefore, the median mark is:
[tex]\[
\boxed{50}
\][/tex]
