1. **Stating the problem:** We need to calculate the range and standard deviation for rainfall data of City A and City B, and determine which city has more variability in rainfall. We will also discuss how these measures help us understand rainfall patterns. 2. **Given data:** - City A rainfall (in inches): $\{3.2, 4.1, 3.8, 4.0, 3.5, 3.9\}$ - City B rainfall (in inches): $\{2.5, 5.0, 3.0, 4.5, 2.8, 4.2\}$ 3. **Calculate the Range:** - Range is the difference between the maximum and minimum values. - For City A: $\max = 4.1$, $\min = 3.2$, so Range = $4.1 - 3.2 = 0.9$ - For City B: $\max = 5.0$, $\min = 2.5$, so Range = $5.0 - 2.5 = 2.5$ 4. **Calculate the Standard Deviation:** - Standard deviation measures how spread out the data is from the mean. **City A:** - Mean $\mu_A = \frac{3.2 + 4.1 + 3.8 + 4.0 + 3.5 + 3.9}{6} = \frac{22.5}{6} = 3.75$ - Calculate squared deviations: $$ (3.2 - 3.75)^2 = 0.3025 $$ $$ (4.1 - 3.75)^2 = 0.1225 $$ $$ (3.8 - 3.75)^2 = 0.0025 $$ $$ (4.0 - 3.75)^2 = 0.0625 $$ $$ (3.5 - 3.75)^2 = 0.0625 $$ $$ (3.9 - 3.75)^2 = 0.0225 $$ - Sum of squares = $0.3025 + 0.1225 + 0.0025 + 0.0625 + 0.0625 + 0.0225 = 0.575$ - Variance $\sigma_A^2 = \frac{0.575}{6} \approx 0.0958$ - Standard deviation $\sigma_A = \sqrt{0.0958} \approx 0.31$ **City B:** - Mean $\mu_B = \frac{2.5 + 5.0 + 3.0 + 4.5 + 2.8 + 4.2}{6} = \frac{22.0}{6} \approx 3.67$ - Calculate squared deviations: $$ (2.5 - 3.67)^2 = 1.3689 $$ $$ (5.0 - 3.67)^2 = 1.7689 $$ $$ (3.0 - 3.67)^2 = 0.4489 $$ $$ (4.5 - 3.67)^2 = 0.6889 $$ $$ (2.8 - 3.67)^2 = 0.7569 $$ $$ (4.2 - 3.67)^2 = 0.2809 $$ - Sum of squares = $1.3689 + 1.7689 + 0.4489 + 0.6889 + 0.7569 + 0.2809 = 5.3124$ - Variance $\sigma_B^2 = \frac{5.3124}{6} \approx 0.8854$ - Standard deviation $\sigma_B = \sqrt{0.8854} \approx 0.94$ 5. **Comparison and discussion:** - City A range = 0.9, standard deviation = 0.31 - City B range = 2.5, standard deviation = 0.94 Since City B has both higher range and standard deviation, it experiences more variability in rainfall. These measures help us understand how consistent or variable the rainfall is month-to-month: - Range gives the spread from lowest to highest rainfall. - Standard deviation gives information on how much rainfall values deviate around the mean, reflecting typical fluctuations. Understanding variability helps in planning for agriculture, water resources, and flood control.