1. The problem is to find an alternative formula for the sample variance. 2. The sample variance $s^2$ measures the spread of data points in a sample and is usually calculated by the formula: $$s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2$$ where $n$ is the sample size, $x_i$ are the data points, and $\bar{x}$ is the sample mean. 3. An alternative formula for sample variance, often called the computational formula, is: $$s^2 = \frac{1}{n-1} \left( \sum_{i=1}^n x_i^2 - \frac{(\sum_{i=1}^n x_i)^2}{n} \right)$$ 4. This formula is useful because it avoids calculating the mean first and then subtracting it from each data point, which can be computationally simpler. 5. To use this formula: - Calculate the sum of the squares of each data point: $\sum x_i^2$ - Calculate the square of the sum of the data points: $(\sum x_i)^2$ - Divide the square of the sum by $n$ - Subtract this from the sum of squares - Finally, divide by $n-1$ to get the sample variance 6. This formula gives the same result as the original definition but can be more efficient for large datasets or when data is streamed. Final answer: $$s^2 = \frac{1}{n-1} \left( \sum_{i=1}^n x_i^2 - \frac{(\sum_{i=1}^n x_i)^2}{n} \right)$$