1. **Stating the problem:** We want to calculate the variance of a data set, which measures how spread out the numbers are around the mean. 2. **Formula for variance:** The variance $s^2$ is given by $$s^2 = \frac{\sum (x_i - \bar{x})^2}{n}$$ where $x_i$ are the data points, $\bar{x}$ is the mean (average) of the data, and $n$ is the number of data points. 3. **Explanation:** - First, calculate the mean $\bar{x} = \frac{\sum x_i}{n}$. - Then subtract the mean from each data point to find the deviation: $x_i - \bar{x}$. - Square each deviation to make them positive: $(x_i - \bar{x})^2$. - Sum all squared deviations: $\sum (x_i - \bar{x})^2$. - Finally, divide by the number of data points $n$ to get the variance. 4. **Important notes:** - Variance is always non-negative. - It gives a measure of how data points vary from the mean. - The square root of variance is the standard deviation, which is in the same units as the data. 5. **Example:** Suppose data points are 2, 4, 6. - Mean: $\bar{x} = \frac{2+4+6}{3} = 4$ - Deviations: $2-4 = -2$, $4-4=0$, $6-4=2$ - Squared deviations: $4, 0, 4$ - Sum: $4+0+4=8$ - Variance: $s^2 = \frac{8}{3} \approx 2.67$ This completes the calculation of variance.