Consider this set of raw data: $$ \{9 ; 8 ; 7 ; 9 ; 7 ; 11 ; 10 ; 8 ; 7 ; 11\} $$ 1. What is the mode of the data set? 2. What is the mean value of the data set? 3. What is the median value of the data set? List the mode(s), mean, and median. INSTRUCTIONS: - Round your answer to two decimal places where necessary. - If there are two or more modes, give all of the mode values separated example, if 3 and 4 are both modes, submit your answer as $$ 3 ; 4 $$

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Let's analyze the given data set and compute the mode, mean, and median as requested.

**Data Set:**
$$ \{9 ; 8 ; 7 ; 9 ; 7 ; 11 ; 10 ; 8 ; 7 ; 11\} $$

1. **Mode:**
   - **Definition:** The mode is the number that appears most frequently in the data set.
   - **Calculation:**
     - Count of each number:
       - 7 appears **3** times
       - 8 appears **2** times
       - 9 appears **2** times
       - 10 appears **1** time
       - 11 appears **2** times
     - The number **7** has the highest frequency.
   - **Mode:** $$ 7 $$

2. **Mean:**
   - **Definition:** The mean is the average of all numbers in the data set.
   - **Calculation:**
     $$
     \text{Mean} = \frac{9 + 8 + 7 + 9 + 7 + 11 + 10 + 8 + 7 + 11}{10} = \frac{87}{10} = 8.70
     $$
   - **Mean:** $$ 8.70 $$

3. **Median:**
   - **Definition:** The median is the middle value when the data set is ordered in ascending order. If there's an even number of observations, it's the average of the two middle numbers.
   - **Ordered Data Set:**
     $$
     \{7, 7, 7, 8, 8, 9, 9, 10, 11, 11\}
     $$
   - **Calculation:**
     - Number of data points: 10 (even)
     - Middle positions: 5th and 6th values (both indices 5 and 6)
     $$
     \text{Median} = \frac{8 + 9}{2} = 8.50
     $$
   - **Median:** $$ 8.50 $$

**Summary:**
- **Mode:** $$ 7 $$
- **Mean:** $$ 8.70 $$
- **Median:** $$ 8.50 $$
- **Mode:** $$ 7 $$ - **Mean:** $$ 8.70 $$ - **Median:** $$ 8.50 $$

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