To find the median marks from the given data, we will follow these steps:
### Step 1: Organize the Data
First, let's organize the data:
- Marks: 24, 36, 50, 65, 78
- Number of students: 2, 4, 12, 11, 6
### Step 2: Expand the Data
We need to represent each mark with its corresponding number of students:
- Two students have marks 24: [tex]\( 24, 24 \)[/tex]
- Four students have marks 36: [tex]\( 36, 36, 36, 36 \)[/tex]
- Twelve students have marks 50: [tex]\( 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50 \)[/tex]
- Eleven students have marks 65: [tex]\( 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65 \)[/tex]
- Six students have marks 78: [tex]\( 78, 78, 78, 78, 78, 78 \)[/tex]
When we put all these numbers together, we get the following list of marks in sorted order:
[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 3: Calculate the Total Number of Students
Let's count the total number of students:
[tex]\[ 2 + 4 + 12 + 11 + 6 = 35 \][/tex]
### Step 4: Determine the Median Position
Since the number of students (35) is odd, the median will be the middle value. The position of the median is given by:
[tex]\[ \text{Median position} = \frac{n + 1}{2} \][/tex]
where [tex]\( n \)[/tex] is the total number of students.
Substitute [tex]\( n = 35 \)[/tex]:
[tex]\[ \text{Median position} = \frac{35 + 1}{2} = \frac{36}{2} = 18 \][/tex]
So, the median is the 18th value in the ordered list of marks.
### Step 5: Identify the Median Value
Looking at our expanded 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]
The 18th value is 50.
### Conclusion
Therefore, the median marks are [tex]\( \boxed{50} \)[/tex].
