To find the variance of the given set of numbers (2, 6, 8, 6, 2, and 6), we will follow a systematic approach. Let's break it down step-by-step. ### Step 1: Calculate the Mean The first step in finding the variance is to calculate the mean (average) of the data set. **Formula for Mean:** \[ \text{Mean} (\mu) = \frac{\text{Sum of all data points}}{\text{Number of data points}} \] **Calculating the Mean:** - Sum of the data points: \(2 + 6 + 8 + 6 + 2 + 6 = 30\) - Number of data points: \(6\) Now, we can calculate the mean: \[ \mu = \frac{30}{6} = 5 \] ### Step 2: Calculate the Variance Variance measures how far each number in the set is from the mean and thus from every other number in the set. **Formula for Variance:** \[ \text{Variance} (\sigma^2) = \frac{\sum (x_i - \mu)^2}{N} \] where \(x_i\) represents each data point, \(\mu\) is the mean, and \(N\) is the number of data points. **Calculating Each Squared Deviation:** Now, we will calculate the squared deviation of each data point from the mean: 1. For \(x_1 = 2\): \[ (2 - 5)^2 = (-3)^2 = 9 \] 2. For \(x_2 = 6\): \[ (6 - 5)^2 = (1)^2 = 1 \] 3. For \(x_3 = 8\): \[ (8 - 5)^2 = (3)^2 = 9 \] 4. For \(x_4 = 6\): \[ (6 - 5)^2 = (1)^2 = 1 \] 5. For \(x_5 = 2\): \[ (2 - 5)^2 = (-3)^2 = 9 \] 6. For \(x_6 = 6\): \[ (6 - 5)^2 = (1)^2 = 1 \] ### Step 3: Sum of Squared Deviations Now, we sum all the squared deviations: \[ 9 + 1 + 9 + 1 + 9 + 1 = 30 \] ### Step 4: Calculate the Variance Now we can calculate the variance using the sum of squared deviations: \[ \sigma^2 = \frac{30}{6} = 5 \] ### Conclusion The variance of the data set (2, 6, 8, 6, 2, 6) is **5**. ### Explanation of Options - **Option A (6)**: This is incorrect because it does not match our calculated variance of 5. - **Option B (5)**: This is the correct answer as we calculated the variance to be 5. - **Option C (√6)**: This is incorrect because it represents the square root of a number, not the variance itself. - **Option D (√5)**: This is also incorrect as it represents the square root of the variance, not the variance itself. ### Revision Summary - The variance is calculated by finding the mean, then determining the squared deviations from the mean, summing those, and dividing by the number of data points. - The mean of the data set (2, 6, 8, 6, 2, 6) is 5. - The variance of the data set is 5. - Always ensure to differentiate between variance and its square root (standard deviation).