Sure, let's break down the process step-by-step to find the median marks from the given data.
### Step 1: Understand the Data
We are given marks and the number of students who received those marks. The data is structured as:
| Marks | 24 | 36 | 50 | 65 | 78 |
|--------|-----|-----|-----|-----|-----|
| Students | 2 | 4 | 12 | 11 | 6 |
### Step 2: Expand the Data
First, we expand the marks based on the number of students who received those marks. This means we write each mark the number of times it was received.
- 24 appears 2 times.
- 36 appears 4 times.
- 50 appears 12 times.
- 65 appears 11 times.
- 78 appears 6 times.
When we write these out, we get:
[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: Sorting the Data
Even though the expanded list appears to be sorted, let's ensure it is, as sorting is crucial for finding the median:
[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 4: Finding the Median
The median is the middle value in the sorted list.
- First, we count the total number of observations. Here, the total number is 35.
Since the number of observations (35) is odd, the median is the value at the [tex]\( \left(\frac{35+1}{2}\right) \)[/tex]-th position. This simplifies to the [tex]\( 18 \)[/tex]-th position in the ordered list.
### Step 5: Locate the Median
Count to the 18th position in the sorted list:
[tex]\[ [24, 24, 36, 36, 36, 36, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, \textbf{50}, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 78, 78, 78, 78, 78, 78] \][/tex]
The 18th value is [tex]\( 50 \)[/tex].
### Conclusion
The median of the marks based on the given data is [tex]\( 50 \)[/tex].
