Question 14

**Setting up:** Sorted values with frequencies: 4(5), 5(f₁), 6(1), 8(f₂), 9(2), 11(3), 12(1). Sum of frequencies: 12 + f₁ + f₂ = 19 ⇒ f₁ + f₂ = 7. Median = 6 with N = 19 means the 10th observation is 6. Cumulative frequencies: up to 4 → 5; up to 5 → 5 + f₁; up to 6 → 6 + f₁. Need 5 + f₁ < 10 ≤ 6 + f₁ ⇒ f₁ = 4, so f₂ = 3. **(P):** 7f₁ + 9f₂ = 28 + 27 = 55 → (5) **Mean:** Σfx = 20 + 20 + 6 + 24 + 18 + 33 + 12 = 133, so mean = 133/19 = 7. **(Q):** 19α = Σf|x − 7| = 5·3 + 4·2 + 1·1 + 3·1 + 2·2 + 3·4 + 1·5 = 48 → (3) **(R):** 19β = Σf|x − 6| = 5·2 + 4·1 + 0 + 3·2 + 2·3 + 3·5 + 1·6 = 47 → (2) **(S):** 19σ² = Σf(x − 7)² = 45 + 16 + 1 + 3 + 8 + 48 + 25 = 146 → (1) So (P)→(5), (Q)→(3), (R)→(2), (S)→(1), which is option C.

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