To determine the values of "a," "b," and "c," we need to understand the given information about the data set.
Given data set: 5 3 a 12 2 b 9 3 c 11
1. **The range is 12:**
The range is the difference between the largest and smallest values in the data set. In this case, the largest value is 12, and the smallest value is not given explicitly. However, since the mode is 3, which is the smallest value that appears more frequently, we can assume that the smallest value is 3.
Range = Largest Value - Smallest Value
12 = 12 - 3
Smallest Value = 3
2. **The mode is 3:**
The mode is the value that appears most frequently in the data set, and in this case, it is 3.
3. **The median is 8:**
The median is the middle value when the data set is arranged in ascending order. The data set is: 2, 3, 3, 5, 9, 11, 12
Since the number of data points is odd (7 data points), the median is the fourth value, which is 5.
Now, let's solve for the values of "a," "b," and "c":
1. **a:**
Since the median is 5, and "a" is between 3 and 5, we can conclude that "a" must be 4. (It is the value that comes right after the median.)
2. **b:**
The data set does not provide any direct information about the value of "b." We cannot determine its exact value with the given information.
3. **c:**
Since the largest value in the data set is 12, and "c" comes right after that, we can conclude that "c" must be 13.