To find the median, we need to arrange the data in ascending order, considering both the marks and the number of students in each group.

Step 1: Organize the data.

You have the following marks and corresponding number of students:

• Marks: 20, 35, 28, 29, 33, 38, 42, 45
• Number of students: 6, 20, 24, 28, 15, 4, 2, 1

Let’s express the dataset in a way that shows how many students got each mark:

• 6 students got 20 marks.
• 20 students got 35 marks.
• 24 students got 28 marks.
• 28 students got 29 marks.
• 15 students got 33 marks.
• 4 students got 38 marks.
• 2 students got 42 marks.
• 1 student got 45 marks.

Step 2: Arrange the data in increasing order of marks (already ordered here).

Step 3: Find the total number of students.

The total number of students is: $6 + 20 + 24 + 28 + 15 + 4 + 2 + 1 = 100$

Step 4: Locate the median position.

Since the total number of students is 100, the median is the average of the 50th and 51st values in the ordered dataset.

Step 5: Find the 50th and 51st values.

To locate the 50th and 51st values:

• The first 6 students have 20 marks.
• The next 20 students (7th to 26th) have 35 marks.
• The next 24 students (27th to 50th) have 28 marks. So, the 50th student has 28 marks.
• The next 28 students (51st to 78th) have 29 marks. So, the 51st student has 29 marks.

Step 6: Calculate the median.

The median is the average of the 50th and 51st values, which are 28 and 29, respectively. So: $\text{Median} = \frac{28 + 29}{2} = \frac{57}{2} = 28.5$

Thus, the median mark is 28.5.

Would you like more details or have any questions?

Here are five related questions you might find useful:

1. How is the median affected by outliers in a data set?
2. What is the difference between mean, median, and mode?
3. How can you find the quartiles of this data set?
4. What is the significance of the median in statistical analysis?
5. How would the median change if you added more data points?

Tip: The median is less sensitive to extreme values (outliers) than the mean, making it a better measure of central tendency for skewed data.