1. **Problem Statement:** Given a frequency distribution with 120 observations and median 30, find: a) Frequencies X and Y b) The mode c) The value below which 50% of observations lie 2. **Given Data:** Class Limits: 0-9, 10-19, 20-29, 30-39, 40-49, 50-59 Frequencies: 10, X, 25, 30, Y, 10 Total observations: 120 Median = 30 3. **Step a) Find X and Y:** - Sum of frequencies: $10 + X + 25 + 30 + Y + 10 = 120$ - Simplify: $75 + X + Y = 120 \Rightarrow X + Y = 45$ 4. **Median class:** Median is 30, so median class is 30-39 with frequency 30. 5. **Cumulative frequencies before median class:** - $CF = 10 + X + 25 = 35 + X$ 6. **Median formula:** $$\text{Median} = L + \left(\frac{\frac{N}{2} - F}{f_m}\right) \times h$$ where $L = 30$ (lower boundary of median class), $N = 120$ (total frequency), $F = CF$ before median class, $f_m = 30$ (frequency of median class), $h = 10$ (class width) 7. Substitute values: $$30 = 30 + \left(\frac{60 - (35 + X)}{30}\right) \times 10$$ 8. Simplify: $$0 = \left(\frac{60 - 35 - X}{30}\right) \times 10$$ $$0 = \frac{25 - X}{3}$$ 9. Solve for $X$: $$25 - X = 0 \Rightarrow X = 25$$ 10. Recall $X + Y = 45$, so: $$25 + Y = 45 \Rightarrow Y = 20$$ 11. **Step b) Find the mode:** Mode class is the class with highest frequency. Frequencies: 10, 25, 25, 30, 20, 10 Highest frequency is 30 in class 30-39. 12. **Mode formula:** $$\text{Mode} = L + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h$$ where $L = 30$ (lower boundary of modal class), $f_1 = 30$ (frequency of modal class), $f_0 = 25$ (frequency before modal class), $f_2 = 20$ (frequency after modal class), $h = 10$ 13. Substitute values: $$\text{Mode} = 30 + \left(\frac{30 - 25}{2 \times 30 - 25 - 20}\right) \times 10 = 30 + \left(\frac{5}{60 - 45}\right) \times 10 = 30 + \left(\frac{5}{15}\right) \times 10$$ 14. Simplify: $$\text{Mode} = 30 + \frac{1}{3} \times 10 = 30 + 3.33 = 33.33$$ 15. **Step c) Find the value below which 50% observations lie (Median):** Given median is 30, so the value below which 50% observations lie is 30. **Final answers:** - $X = 25$ - $Y = 20$ - Mode $= 33.33$ - Median (50% observations below) $= 30$