Question

**Step 1: Find the remaining two observations $a$ and $b$.** Sum of the 8 given observations: $$2+3+5+10+11+13+15+21 = 80$$ Using the mean condition: $$\frac{80 + a + b}{10} = 9 \implies a + b = 10 \quad \cdots (1)$$ **Step 2: Apply the variance condition.** Sum of squares of the 8 given observations: $$4+9+25+100+121+169+225+441 = 1094$$ $$\frac{\sum x_i^2}{10} - \bar{x}^2 = 34.2 \implies \frac{1094 + a^2 + b^2}{10} - 81 = 34.2$$ $$\implies a^2 + b^2 = 1152 - 1094 = 58 \quad \cdots (2)$$ **Step 3: Solve for $a$ and $b$.** From (1) and (2): $$(a+b)^2 - 2ab = 58 \implies 100 - 2ab = 58 \implies ab = 21$$ So $a$ and $b$ are roots of: $$t^2 - 10t + 21 = 0 \implies (t-3)(t-7) = 0$$ Thus $\{a, b\} = \{3, 7\}$. **Step 4: Arrange all 10 observations in ascending order.** $$2,\ 3,\ 3,\ 5,\ 7,\ 10,\ 11,\ 13,\ 15,\ 21$$ **Step 5: Find the median.** For 10 observations, median $= \dfrac{5^{\text{th}} + 6^{\text{th}}}{2} = \dfrac{7 + 10}{2} = 8.5$ **Step 6: Compute the mean deviation about the median.** $$\text{MD} = \frac{1}{10}\sum_{i=1}^{10}|x_i - 8.5|$$ $$= \frac{1}{10}\big(6.5 + 5.5 + 5.5 + 3.5 + 1.5 + 1.5 + 2.5 + 4.5 + 6.5 + 12.5\big)$$ $$= \frac{50}{10} = 5$$ **Answer: (A) $5$**