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The given data presents the marks of 12 students, arranged in ascending order as:

\[ 2, 3, 3, 3.4, x, x+2, 8, p, q, 8, 8, 9 \]

We are provided with the following conditions:
- Median = 6
- Mode = 8

### Step 1: Understanding the Median
Since there are 12 data points, the median is the average of the 6th and 7th values. 

Thus, \[
\text{Median} = \frac{(x + (x+2))}{2} = 6
\]
This simplifies to:
\[
\frac{2x + 2}{2} = 6
\]
\[
x + 1 = 6 \quad \Rightarrow \quad x = 5
\]

### Step 2: Understanding the Mode
The mode is the value that appears most frequently. Since the mode is 8, it indicates that 8 must appear at least twice. From the data, we already see two occurrences of 8. Therefore, \( p = 8 \) to satisfy the condition that the mode is 8.

### Step 3: Determining the Value of \( q \)
We have the updated data as:
\[ 2, 3, 3, 3.4, 5, 7, 8, 8, q, 8, 8, 9 \]

Since the list is in ascending order, and \( q \) is placed after two 8's and before 9, we conclude:
\[
q = 8
\]

### Final Answer:
The values of \( x \), \( p \), and \( q \) are:
- \( x = 5 \)
- \( p = 8 \)
- \( q = 8 \)

Would you like more details or have any questions? Here are some follow-up questions:

1. How do we calculate the median if the number of data points is odd?
2. What is the importance of knowing the mode in a data set?
3. How would the solution change if the mode was a different number, like 7?
4. What if \( x \) was smaller, how would it affect the median?
5. Can a data set have more than one mode?

**Tip:** The median divides the data set into two equal halves, while the mode reflects the most frequent value. Understanding both helps in interpreting the central tendency of data.

The data given shows the marks of 12 students in a test, arranged in ascending order: 2, 3, 3, 3.4, x, x+2, 8, p, q, 8, 8, 9. If the median is 6 and the mode is 8, find the values of x, p, and q.

Solve for x, p, and q in a set of 12 student marks where the median is 6 and the mode is 8. Explore how to use the median and mode formulas to find unknown values in a data set. Suitable for Grades 8-10.

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