1. **Problem Statement:** Calculate the mean, median, mode, and standard deviation for the number of modules presented by 50 students based on the given frequency distribution. 2. **Given Data:** | Number of modules (x) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |-----------------------|---|---|---|---|---|---|---| | Number of students (f) | 5 | 3 | 7 | 8 | 9 | 10| 8 | 3. **Formulas:** - Mean: $$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$$ - Median: The middle value when data is ordered. - Mode: The value with the highest frequency. - Standard deviation: $$\sigma = \sqrt{\frac{\sum f_i (x_i - \bar{x})^2}{\sum f_i}}$$ 4. **Calculate Mean:** $$\sum f_i x_i = 5\times1 + 3\times2 + 7\times3 + 8\times4 + 9\times5 + 10\times6 + 8\times7 = 5 + 6 + 21 + 32 + 45 + 60 + 56 = 225$$ $$\sum f_i = 5 + 3 + 7 + 8 + 9 + 10 + 8 = 50$$ $$\bar{x} = \frac{225}{50} = 4.5$$ 5. **Calculate Median:** Total students = 50, median position = $$\frac{50+1}{2} = 25.5$$th student. Cumulative frequencies: - 1 module: 5 - 2 modules: 8 - 3 modules: 15 - 4 modules: 23 - 5 modules: 32 Median lies in 5 modules category (since 25.5 > 23 and \leq 32), so median = 5. 6. **Calculate Mode:** Highest frequency is 10 for 6 modules, so mode = 6. 7. **Calculate Standard Deviation:** Calculate $$f_i (x_i - \bar{x})^2$$: - For 1: $$5(1-4.5)^2 = 5 \times 12.25 = 61.25$$ - For 2: $$3(2-4.5)^2 = 3 \times 6.25 = 18.75$$ - For 3: $$7(3-4.5)^2 = 7 \times 2.25 = 15.75$$ - For 4: $$8(4-4.5)^2 = 8 \times 0.25 = 2$$ - For 5: $$9(5-4.5)^2 = 9 \times 0.25 = 2.25$$ - For 6: $$10(6-4.5)^2 = 10 \times 2.25 = 22.5$$ - For 7: $$8(7-4.5)^2 = 8 \times 6.25 = 50$$ Sum = 61.25 + 18.75 + 15.75 + 2 + 2.25 + 22.5 + 50 = 172.5 $$\sigma = \sqrt{\frac{172.5}{50}} = \sqrt{3.45} \approx 1.857$$ 8. **Interpretation:** - Mean (4.5) is the average number of modules presented. - Median (5) indicates half the students presented 5 or fewer modules. - Mode (6) shows the most common number of modules presented. - Standard deviation (~1.857) shows the spread of the data around the mean. Final answers: - Mean = 4.5 - Median = 5 - Mode = 6 - Standard deviation ≈ 1.857