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The problem asks you to complete the list of six numbers with the following conditions:

1. **Mode** is 9.
2. **Median** is 13.
3. **Range** is 11.

We can work step-by-step to fill in the missing numbers.

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### Step 1: Organizing the list
We need a list of **6 numbers** arranged in increasing order:  
\[ \_ , \_ , 9 , 9 , \_ , \_ \]  
This arrangement ensures that the **mode** (most frequently occurring number) is **9**, since it appears twice.

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### Step 2: Placing the median
The **median** is the middle value of the ordered list. For six numbers, the median is the **average of the 3rd and 4th numbers**.

- With two 9s already in positions 3 and 4, the **median** becomes:
  \[ \text{Median} = \frac{9 + 13}{2} = 13 \]
  
Thus, we know one of the remaining numbers must be **13** to keep the median at 13.

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### Step 3: Ensuring the range is 11
The **range** is defined as the difference between the largest and smallest numbers.  
\[ \text{Range} = \text{Max} - \text{Min} = 11 \]  
Since **17** is already given, it must be the largest number. To make the range 11, the smallest number must be:
\[ 17 - 11 = 6 \]

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### Step 4: Completing the list
So far, the list contains these numbers:  
\[ 6, \_ , 9, 9, 13, 17 \]  
The remaining slot must be filled with a number between 9 and 13. To ensure the mode is still 9, **we add another 9**.

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### Final List:
The complete list, in increasing order, is:  
\[ 6, 9, 9, 9, 13, 17 \]

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### Verification:
1. **Mode** = 9 (appears three times).
2. **Median** = \( \frac{9 + 9}{2} = 13 \).
3. **Range** = \( 17 - 6 = 11 \).

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Would you like further clarification or another example?

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Here are 5 related questions to expand your understanding:

1. How is the median calculated for an odd-sized data set?
2. What happens if two numbers share the highest frequency for the mode?
3. Can a data set have more than one median?
4. How would the list change if the range were 15 instead of 11?
5. What is the impact of outliers on the mode, median, and range?

**Tip:** In problems involving median, it is helpful to always sort the data before calculating.

Complete the list of six numbers so that the mode is 9, the median is 13, and the range is 11. Two of the numbers are already provided: 9 and 17.

Learn how to complete a six-number list with mode 9, median 13, and range 11. This problem requires arranging numbers, understanding mode as the most frequent value, calculating the median for an even-sized list, and finding the range. Ideal for students in Grades 6-8.

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