1. **Problem 1: Statistical Measures for Dataset 1** Given dataset: 45, 62, 75, 85, 90, 93, 95, 95, 100 2. **Range** is the difference between the maximum and minimum values: $$\text{Range} = 100 - 45 = 55$$ 3. **Quartiles and Percentiles:** - Sort the data (already sorted). - The 1st quartile (Q1) is the 25th percentile, the 3rd quartile (Q3) is the 75th percentile. - To find Q1 (25th percentile), locate the position: $$P = \frac{25}{100} \times (n+1) = 0.25 \times 10 = 2.5$$ - Q1 is the average of the 2nd and 3rd values: $$\frac{62 + 75}{2} = 68.5$$ - To find Q3 (75th percentile), position: $$0.75 \times 10 = 7.5$$ - Q3 is average of 7th and 8th values: $$\frac{95 + 95}{2} = 95$$ - The 50th percentile is the median, position: $$0.5 \times 10 = 5$$ - Median is the 5th value: 90 4. **Mean** (average): $$\text{Mean} = \frac{45 + 62 + 75 + 85 + 90 + 93 + 95 + 95 + 100}{9} = \frac{740}{9} \approx 82.22$$ 5. **Median** is the middle value in sorted data: 90 6. **Mode** is the most frequent value: 95 (appears twice) 7. **Variance** measures spread: - Calculate each deviation squared: $$\sum (x_i - \bar{x})^2 = (45-82.22)^2 + (62-82.22)^2 + ... + (100-82.22)^2$$ - Calculated sum of squares: approximately 3346.22 - Variance: $$\frac{3346.22}{9} \approx 371.80$$ 8. **Standard Deviation** is the square root of variance: $$\sqrt{371.80} \approx 19.29$$ 9. **Comments on equality:** - The 25th percentile equals the 1st quartile by definition. - The 50th percentile equals the median, which is expected. - The 3rd quartile (Q3) is the 75th percentile, but here we only calculated Q3; the 75th percentile would be the same as Q3. - The equality between 1st quartile and 25th percentile, and median and 50th percentile, is due to their definitions. 10. **Problem 2: Data Visualization for Dataset 2** Given dataset: 72, 88, 94, 119, 85, 91, 77, 84, 75, 79, 83, 80, 87, 70, 76, 82, 93, 78, 89, 95, 86, 90, 73, 92, 81 11. **Stem-and-Leaf Plot:** - Stems represent tens, leaves represent units. - Example: 70 | 0 3 5 6 7 8 9 80 | 0 1 2 3 4 5 6 7 8 9 90 | 1 2 3 4 5 100 | 110 | 9 12. **Histogram:** - Group data into bins (e.g., 70-79, 80-89, 90-99, 100-109, 110-119). - Count frequencies: 70-79: 7 80-89: 10 90-99: 6 100-109: 0 110-119: 1 13. **Box Plot:** - Calculate quartiles: Sorted data: 70, 72, 73, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 119 - Median (Q2): 85 - Q1: median of lower half (70 to 84): 77 - Q3: median of upper half (86 to 119): 91 - Identify outliers using 1.5*IQR: IQR = Q3 - Q1 = 91 - 77 = 14 Lower bound = 77 - 1.5*14 = 56 Upper bound = 91 + 1.5*14 = 112 - Outlier: 119 (above upper bound) 14. **Labels and Titles:** - Stem-and-leaf: "Stem-and-Leaf Plot of Test Scores" - Histogram: "Histogram of Test Scores with Frequency" - Box plot: "Box Plot of Test Scores with Outlier" 15. **Summary:** - The stem-and-leaf plot shows distribution and individual values. - The histogram visualizes frequency distribution. - The box plot summarizes spread, center, and outliers. - 119 is a potential outlier in the dataset.