In date set of 5 points, the mean , median , and mode are each equal to 8 . Which of the following could be the date set ?

Options

A. `{ 5,7,8,8,12}`
B. `{7,7,8,8,12}`
C. `7,8,8,8,12}`
D. `{7,8,8,10,12}`

Answer

To solve the problem, we need to find a data set of 5 points where the mean, median, and mode are all equal to 8. Let's go through the steps to determine which of the provided options meets these criteria.

### Step-by-Step Solution:

1. **Understanding Mean, Median, and Mode**:
- **Mean**: The average of the numbers. It is calculated by summing all the numbers and dividing by the count of numbers.
- **Median**: The middle value when the numbers are arranged in ascending order. For a set of 5 numbers, it will be the 3rd number when sorted.
- **Mode**: The number that appears most frequently in the data set.

2. **Setting Up the Problem**:
- We need to find a data set of 5 numbers where:
- Mean = 8
- Median = 8
- Mode = 8

3. **Calculating Mean**:
- Since the mean is 8 and there are 5 numbers, the total sum of the numbers must be:
\[ \text{Total Sum} = \text{Mean} \times \text{Number of Points} = 8 \times 5 = 40 \]

4. **Finding the Median**:
- For the median to be 8, when the numbers are arranged in ascending order, the 3rd number must be 8.

5. **Finding the Mode**:
- For the mode to be 8, the number 8 must appear more frequently than any other number in the data set.

6. **Testing Options**:
- We will evaluate each option to see if they meet the criteria of mean, median, and mode being equal to 8.

**Option A**: 7, 8, 8, 8, 9
- Mean: \( \frac{7 + 8 + 8 + 8 + 9}{5} = \frac{40}{5} = 8 \)
- Median: 8 (3rd number in sorted order)
- Mode: 8 (appears 3 times)
- **Conclusion**: This option meets all criteria.

**Option B**: 7, 7, 8, 8, 12
- Mean: \( \frac{7 + 7 + 8 + 8 + 12}{5} = \frac{42}{5} = 8.4 \)
- **Conclusion**: Does not meet the criteria.

**Option C**: 7, 8, 8, 8, 12
- Mean: \( \frac{7 + 8 + 8 + 8 + 12}{5} = \frac{43}{5} = 8.6 \)
- **Conclusion**: Does not meet the criteria.

**Option D**: 7, 8, 8, 10, 12
- Mean: \( \frac{7 + 8 + 8 + 10 + 12}{5} = \frac{45}{5} = 9 \)
- **Conclusion**: Does not meet the criteria.

7. **Final Answer**:
- The only option that meets all criteria (mean, median, and mode all equal to 8) is **Option A: 7, 8, 8, 8, 9**.