1. **Stating the problem:** We want to understand why the sample variance formula $$S_x^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \bar{x})^2 = \frac{1}{n} \sum_{i=1}^n x_i^2 - \bar{x}^2$$ can be rewritten and simplified step-by-step. 2. **Start with the definition:** $$S_x^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \bar{x})^2$$ This means the variance is the average of the squared differences between each data point $x_i$ and the mean $\bar{x}$. 3. **Expand the square inside the sum:** Using the algebraic identity $(a-b)^2 = a^2 - 2ab + b^2$, we get $$\frac{1}{n} \sum_{i=1}^n (x_i^2 - 2 x_i \bar{x} + \bar{x}^2)$$ 4. **Split the sum into three separate sums:** $$= \frac{1}{n} \sum_{i=1}^n x_i^2 - \frac{1}{n} \sum_{i=1}^n 2 x_i \bar{x} + \frac{1}{n} \sum_{i=1}^n \bar{x}^2$$ 5. **Factor out constants from sums:** Since $2$ and $\bar{x}$ are constants with respect to $i$, and $\bar{x}^2$ is also constant, $$= \frac{1}{n} \sum_{i=1}^n x_i^2 - \frac{2 \bar{x}}{n} \sum_{i=1}^n x_i + \frac{1}{n} \sum_{i=1}^n \bar{x}^2$$ 6. **Simplify the sums of constants:** The sum $\sum_{i=1}^n \bar{x}^2 = n \bar{x}^2$ because $\bar{x}^2$ is constant and added $n$ times. So, $$= \frac{1}{n} \sum_{i=1}^n x_i^2 - \frac{2 \bar{x}}{n} \sum_{i=1}^n x_i + \frac{1}{n} n \bar{x}^2$$ 7. **Recognize the mean in the sum:** By definition, $\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i$, so $$\sum_{i=1}^n x_i = n \bar{x}$$ Substitute this: $$= \frac{1}{n} \sum_{i=1}^n x_i^2 - \frac{2 \bar{x}}{n} (n \bar{x}) + \bar{x}^2$$ 8. **Cancel terms:** $$= \frac{1}{n} \sum_{i=1}^n x_i^2 - 2 \bar{x}^2 + \bar{x}^2$$ 9. **Combine like terms:** $$= \frac{1}{n} \sum_{i=1}^n x_i^2 - \bar{x}^2$$ **Summary:** The variance formula simplifies to the mean of the squares minus the square of the mean. This step-by-step expansion and simplification shows how the original formula is equivalent to the simpler expression.