1. **Problem Statement:** We have the masses of 22 students: 46, 49, 50, 50, 52, 53, 55, 55, 56, 58, 59, 60, 61, 61, 63, 64, 66, 67, 68, 70, 72, 75. We need to find: - a) Mean and mode - b) Variance and standard deviation - c) Box-and-whisker plot 2. **Formulas and Rules:** - Mean: $\bar{x} = \frac{\sum x_i}{n}$ where $n$ is the number of data points. - Mode: The value(s) that appear most frequently. - Variance: $s^2 = \frac{1}{n-1} \sum (x_i - \bar{x})^2$ - Standard deviation: $s = \sqrt{s^2}$ - For box plot: find minimum, Q1 (first quartile), median (Q2), Q3 (third quartile), and maximum. 3. **Calculations:** **a) Mean and Mode:** - Sum of data: $46 + 49 + 50 + 50 + 52 + 53 + 55 + 55 + 56 + 58 + 59 + 60 + 61 + 61 + 63 + 64 + 66 + 67 + 68 + 70 + 72 + 75 = 1270$ - Number of data points: $n=22$ - Mean: $\bar{x} = \frac{1270}{22} = 57.727$ (rounded to 3 decimals) - Mode: Values appearing most frequently are 50, 55, and 61 (each appears twice). So modes are 50, 55, and 61. **b) Variance and Standard Deviation:** - Calculate each squared deviation $(x_i - \bar{x})^2$ and sum: $$\sum (x_i - 57.727)^2 = (46-57.727)^2 + (49-57.727)^2 + ... + (75-57.727)^2$$ - Computing each term: $(46-57.727)^2 = 137.98$ $(49-57.727)^2 = 75.17$ $(50-57.727)^2 = 59.69$ (twice for two 50s) $(52-57.727)^2 = 32.78$ $(53-57.727)^2 = 22.36$ $(55-57.727)^2 = 7.44$ (twice for two 55s) $(56-57.727)^2 = 2.98$ $(58-57.727)^2 = 0.07$ $(59-57.727)^2 = 1.62$ $(60-57.727)^2 = 5.17$ $(61-57.727)^2 = 10.67$ (twice for two 61s) $(63-57.727)^2 = 27.76$ $(64-57.727)^2 = 39.34$ $(66-57.727)^2 = 68.56$ $(67-57.727)^2 = 85.98$ $(68-57.727)^2 = 105.40$ $(70-57.727)^2 = 150.68$ $(72-57.727)^2 = 204.68$ $(75-57.727)^2 = 299.44$ - Sum all these: $137.98 + 75.17 + 59.69*2 + 32.78 + 22.36 + 7.44*2 + 2.98 + 0.07 + 1.62 + 5.17 + 10.67*2 + 27.76 + 39.34 + 68.56 + 85.98 + 105.40 + 150.68 + 204.68 + 299.44 = 1393.68$ - Variance: $s^2 = \frac{1393.68}{22-1} = \frac{1393.68}{21} = 66.37$ - Standard deviation: $s = \sqrt{66.37} = 8.15$ **c) Box-and-Whisker Plot:** - Sorted data: already sorted. - Minimum: 46 - Maximum: 75 - Median (Q2): middle value between 11th and 12th data points: - 11th: 59 - 12th: 60 - Median = $\frac{59 + 60}{2} = 59.5$ - First quartile (Q1): median of first 11 data points: - 6th data point: 53 - Third quartile (Q3): median of last 11 data points: - 17th data point: 66 4. **Final answers:** - Mean: $57.727$ - Mode: 50, 55, 61 - Variance: $66.37$ - Standard deviation: $8.15$ - Box plot values: Min = 46, Q1 = 53, Median = 59.5, Q3 = 66, Max = 75 This completes the solution.