1. **State the problem:** We are given three sets of data and need to calculate the mean ($\bar{x}$), the sum of squared differences from the mean ($\sum (x_i - \bar{x})^2$), and the standard deviation ($\sigma$) for each set. 2. **Formula for mean:** $$\bar{x} = \frac{\sum x_i}{n}$$ where $n$ is the number of data points. 3. **Formula for variance and standard deviation:** Variance $s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}$ Standard deviation $\sigma = \sqrt{s^2} = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$ 4. **Step-by-step for the first data set:** - Data points: 95, 92, 98, 95, 90, 90 - Calculate mean: $$\bar{x} = \frac{95 + 92 + 98 + 95 + 90 + 90}{6} = \frac{560}{6} = 90$$ - Calculate each difference from mean ($x_i - \bar{x}$) and square it: - 95 - 90 = 5, $5^2 = 25$ - 92 - 90 = 2, $2^2 = 4$ - 98 - 90 = 8, $8^2 = 64$ - 95 - 90 = 5, $5^2 = 25$ - 90 - 90 = 0, $0^2 = 0$ - 90 - 90 = 0, $0^2 = 0$ - Sum of squared differences: $$\sum (x_i - \bar{x})^2 = 25 + 4 + 64 + 25 + 0 + 0 = 118$$ - Calculate variance: $$s^2 = \frac{118}{6-1} = \frac{118}{5} = 23.6$$ - Calculate standard deviation: $$\sigma = \sqrt{23.6} \approx 4.86$$ 5. **Step-by-step for the second data set:** - Data points: 70, 95, 80, 100, 75, 90 - Calculate mean: $$\bar{x} = \frac{70 + 95 + 80 + 100 + 75 + 90}{6} = \frac{510}{6} = 85$$ - Differences and squares: - 70 - 85 = -15, $(-15)^2 = 225$ - 95 - 85 = 10, $10^2 = 100$ - 80 - 85 = -5, $(-5)^2 = 25$ - 100 - 85 = 15, $15^2 = 225$ - 75 - 85 = -10, $(-10)^2 = 100$ - 90 - 85 = 5, $5^2 = 25$ - Sum of squared differences: $$225 + 100 + 25 + 225 + 100 + 25 = 700$$ - Variance: $$s^2 = \frac{700}{5} = 140$$ - Standard deviation: $$\sigma = \sqrt{140} \approx 11.83$$ 6. **Step-by-step for the third data set:** - Data points: 89, 90, 88, 91, 87, 89 - Calculate mean: $$\bar{x} = \frac{89 + 90 + 88 + 91 + 87 + 89}{6} = \frac{534}{6} = 89$$ - Differences and squares: - 89 - 89 = 0, $0^2 = 0$ - 90 - 89 = 1, $1^2 = 1$ - 88 - 89 = -1, $(-1)^2 = 1$ - 91 - 89 = 2, $2^2 = 4$ - 87 - 89 = -2, $(-2)^2 = 4$ - 89 - 89 = 0, $0^2 = 0$ - Sum of squared differences: $$0 + 1 + 1 + 4 + 4 + 0 = 10$$ - Variance: $$s^2 = \frac{10}{5} = 2$$ - Standard deviation: $$\sigma = \sqrt{2} \approx 1.41$$ **Summary:** - First set: $\bar{x} = 90$, $\sigma \approx 4.86$ - Second set: $\bar{x} = 85$, $\sigma \approx 11.83$ - Third set: $\bar{x} = 89$, $\sigma \approx 1.41$ This method helps you understand how spread out the data is around the mean by calculating the standard deviation.