To find the mean of the given data set, we will follow a systematic approach. The mean is calculated by summing all the values in the data set and then dividing that sum by the total number of values. ### Step-by-Step Calculation 1. **List the Data**: The data set provided is: \[ 7, -3, 4, -2, 5, -9, 4, 8, -6, 12 \] 2. **Sum the Values**: We will add all the numbers together. Let's break it down: - Start with the first number: \(7\) - Add the second number: \(7 + (-3) = 4\) - Add the third number: \(4 + 4 = 8\) - Add the fourth number: \(8 + (-2) = 6\) - Add the fifth number: \(6 + 5 = 11\) - Add the sixth number: \(11 + (-9) = 2\) - Add the seventh number: \(2 + 4 = 6\) - Add the eighth number: \(6 + 8 = 14\) - Add the ninth number: \(14 + (-6) = 8\) - Finally, add the tenth number: \(8 + 12 = 20\) So, the total sum of the data set is: \[ 20 \] 3. **Count the Number of Values**: There are 10 values in the data set. 4. **Calculate the Mean**: The mean is calculated using the formula: \[ \text{Mean} = \frac{\text{Sum of values}}{\text{Number of values}} = \frac{20}{10} = 2 \] ### Conclusion The mean of the data set is \(2\). ### Explanation of Options - **Option A: 3** - This is incorrect because the sum of the values does not yield a mean of 3 when divided by the number of values. - **Option B: 4** - This is also incorrect for the same reason; the calculated mean does not equal 4. - **Option C: 1** - This option is incorrect as well; the mean is not 1 based on our calculations. - **Option D: 2** - This is the correct answer, as we have shown through our calculations that the mean of the data set is indeed 2. ### Common Pitfalls - **Forgetting to include negative numbers**: When summing the values, it’s crucial to remember that negative numbers will decrease the total sum. - **Miscounting the number of values**: Ensure that you count all the numbers in the data set accurately. - **Incorrect division**: Double-check the division step to ensure that you are dividing the total sum by the correct number of values. ### Revision Summary - The mean is calculated by summing all values and dividing by the number of values. - The total sum of the data set is 20, and there are 10 values. - The mean is \( \frac{20}{10} = 2\). - The correct answer is option D: 2.