Calculating the Median of Student Marks

The problem asks us to find the median mark of all 15 students in a class. We are given the marks of 10 students who passed and told that 5 students failed. To find the median, we need the marks of all 15 students and arrange them in ascending order.

Understanding the Data

• Total number of students: 15
• Number of students who failed: 5
• Number of students who passed: 10
• Marks of the 10 students who passed: 9, 6, 8, 7, 8, 9, 5, 6, 7, 4

Scores of Failed Students

The marks of the 5 students who failed are not explicitly given. However, since they failed and the others passed, it is reasonable to assume their marks are lower than the lowest passing mark. The lowest mark among the 10 passing students is 4. Therefore, the 5 failed students must have scored less than 4. For the purpose of finding the median, the exact value of their low marks doesn't matter, only that they are the lowest 5 scores in the entire set of 15.

Arranging the Marks in Ascending Order

First, let's sort the marks of the 10 students who passed in ascending order:

• 4, 5, 6, 6, 7, 7, 8, 8, 9, 9

Now, we include the 5 failed students. Their scores will be the lowest 5 scores. Let's represent them as unknown values lower than 4. When all 15 scores are sorted, the list will begin with the 5 failed scores, followed by the 10 sorted passing scores.

The combined sorted list of 15 marks looks like this:

[Failed Score 1, Failed Score 2, Failed Score 3, Failed Score 4, Failed Score 5, 4, 5, 6, 6, 7, 7, 8, 8, 9, 9]

Remember that the 5 failed scores are all less than 4 and are also sorted amongst themselves, though their specific values don't impact the position of the median here.

Finding the Median Position

For a dataset with an odd number of observations (N), the median is the value at the position $\frac{N+1}{2}$ in the sorted list.

In this case, N = 15. The median position is:

\(\text{Median Position} = \frac{15 + 1}{2} = \frac{16}{2} = 8\)

The median is the 8th value in the sorted list of all 15 students' marks.

Identifying the Median Value

Let's look at the sorted list and find the 8th value:

[Failed Score 1, Failed Score 2, Failed Score 3, Failed Score 4, Failed Score 5, 4, 5, 6, 6, 7, 7, 8, 8, 9, 9]

Counting the positions:

• 1st: Failed Score 1
• 2nd: Failed Score 2
• 3rd: Failed Score 3
• 4th: Failed Score 4
• 5th: Failed Score 5
• 6th: 4
• 7th: 5
• 8th: 6

The 8th value in the sorted list is 6.

Therefore, the median mark of all 15 students is 6.

Summary of Steps to Find Median

1. Gather all data points (marks of 15 students).
2. Include placeholder low scores for the failed students.
3. Sort all 15 scores in ascending order.
4. Calculate the median position using the formula \(\frac{N+1}{2}\) for an odd number of data points.
5. Identify the value at the calculated median position in the sorted list.

The sorted list of all 15 scores is conceptually: (5 scores < 4), 4, 5, 6, 6, 7, 7, 8, 8, 9, 9.

The 8th score in this sorted list is 6.

Revision Table: Key Concepts for Median

Additional Information on Measures of Central Tendency

The median is one of the measures of central tendency, which describe the center point of a dataset. Other common measures include the mean and the mode.

• Mean: The average of all data points. Calculated by summing all values and dividing by the number of values. It is affected by extreme values (outliers).
• Mode: The value that appears most frequently in the dataset. A dataset can have one mode (unimodal), multiple modes (multimodal), or no mode.
• Median: The middle value. It is less affected by extreme values compared to the mean, making it a good measure of central tendency for skewed distributions.

In this problem, knowing the marks of the failed students weren't specified but were implied to be lower than the passing marks allowed us to determine their relative position in the sorted list, which was crucial for finding the median position correctly.