Question 14

Step 1: Find $f_1$ and $f_2$. Total frequency $\sum f_i = 5 + f_1 + f_2 + 2 + 1 + 1 + 3 = 19 \Rightarrow f_1 + f_2 = 7$. Ordered data: Values $x_i$: 4, 5, 6, 8, 9, 11, 12 with corresponding frequencies $f_i$: 5, $f_1$, 1, $f_2$, 2, 3, 1. Median is the $10^{\text{th}}$ observation. Since Median = 6, the cumulative frequency before 6 must be less than 10, and including 6 must be $\ge 10$. $5 + f_1 < 10 \Rightarrow f_1 < 5$ and $5 + f_1 + 1 \ge 10 \Rightarrow f_1 \ge 4$. Thus $f_1 = 4$ and $f_2 = 3$.

Step 2: Match (P). $7f_1 + 9f_2 = 7(4) + 9(3) = 28 + 27 = 55$. Thus, P $\to$ 5.

Step 3: Calculate Mean ($\bar{x}$). $\bar{x} = \frac{4(5) + 5(4) + 6(1) + 8(3) + 9(2) + 11(3) + 12(1)}{19} = \frac{20+20+6+24+18+33+12}{19} = \frac{133}{19} = 7$.

Step 4: Match (Q). $19\alpha = \sum f_i |x_i - \bar{x}| = 5|4-7| + 4|5-7| + 1|6-7| + 3|8-7| + 2|9-7| + 3|11-7| + 1|12-7|$ $19\alpha = 15 + 8 + 1 + 3 + 4 + 12 + 5 = 48$. Thus, Q $\to$ 3.

Step 5: Match (R). $19\beta = \sum f_i |x_i - \text{Median}| = 5|4-6| + 4|5-6| + 1|6-6| + 3|8-6| + 2|9-6| + 3|11-6| + 1|12-6|$ $19\beta = 10 + 4 + 0 + 6 + 6 + 15 + 6 = 47$. Thus, R $\to$ 2.

Step 6: Match (S). $19\sigma^2 = \sum f_i (x_i - \bar{x})^2 = 5(3)^2 + 4(2)^2 + 1(1)^2 + 3(1)^2 + 2(2)^2 + 3(4)^2 + 1(5)^2$ $19\sigma^2 = 45 + 16 + 1 + 3 + 8 + 48 + 25 = 146$. Thus, S $\to$ 1.

Correct matching is P-5, Q-3, R-2, S-1. Correct Option: C