Question

There are 15 students in a class. Out of these 6 students failed. The marks of the passed students are 51, 67, 48, 71, 46, 45, 81, 90 and 50. The median of all the observations is:

• A. 46
• B. 50
• C. 51
• D. 67

Hint:

Exam Tip:
For finding the median:

• Arrange all observations in ascending order.
• If \(n\) is odd, median = value at position \((n+1)/2\).
• If \(n\) is even, median = average of values at positions \(n/2\) and \(n/2 + 1\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We have 15 students in total. 6 failed, so their marks are not given. We need to arrange all 15 observations in ascending order to find the median.

Step 2: Key Formula or Approach:
Since the number of observations is odd (15), the median is the middle value, i.e., the 8th observation when arranged in ascending order.
We know the marks of the 9 passed students. The failed students must have marks less than the passing marks.

Step 3: Detailed Explanation:
The passed students' marks are: 51, 67, 48, 71, 46, 45, 81, 90, 50.
Arranging these in ascending order: 45, 46, 48, 50, 51, 67, 71, 81, 90.
The 6 failed students must have marks less than the lowest passing mark (45).
So, all the failed students' marks are less than 45.
Now, combine the failed students' marks (6 values, all less than 45) with the passed students' marks.
The combined sorted list will have 6 values less than 45, followed by the 9 passed marks in ascending order.
So, the sorted list is: \(f_1, f_2, f_3, f_4, f_5, f_6, 45, 46, 48, 50, 51, 67, 71, 81, 90\).
The median is the 8th value.
The 8th value in this list is 50?
Let's count:
1st: \(f_1\)
2nd: \(f_2\)
3rd: \(f_3\)
4th: \(f_4\)
5th: \(f_5\)
6th: \(f_6\)
7th: 45
8th: 46
9th: 48
10th: 50
11th: 51
12th: 67
So, the 8th value is 46.
Wait, the 8th value is 46.
But the options include 46.
Let's re-count:
The passed marks sorted: 45, 46, 48, 50, 51, 67, 71, 81, 90.
The failed marks (6 values) are less than 45.
So, the combined list: 6 values (all less than 45), then 45, 46, 48, 50, 51, 67, 71, 81, 90.
The 15 values:
1: failed1
2: failed2
3: failed3
4: failed4
5: failed5
6: failed6
7: 45
8: 46
9: 48
10: 50
11: 51
12: 67
13: 71
14: 81
15: 90
The median is the 8th value, which is 46.
So, the median is 46.
Option (A) is 46.

Step 4: Final Answer:
Therefore, option (A) is correct.