<p>The mean of 5 observations is 5 and their variance is 124. If three of the observations are 1, 2 and 6; then the mean deviation from the mean of the data is</p>
Step-by-Step Solution
Key Concept: Use the mean and variance formulas to find the two unknown observations, then calculate mean deviation as the average of absolute deviations from the mean.
<p><strong>Step 1: Find the two unknown observations</strong></p><p>Given: Mean = 5, so Σxᵢ = 5 × 5 = 25</p><p>Three observations are 1, 2, 6. Let the unknown observations be a and b.</p><p>Therefore: 1 + 2 + 6 + a + b = 25</p><p>⟹ a + b = 16 ... (i)</p><p><strong>Step 2: Use variance formula</strong></p><p>Variance = Σ(xᵢ - x̄)²/n = 124</p><p>So: Σ(xᵢ - 5)² = 124 × 5 = 620</p><p>(1-5)² + (2-5)² + (6-5)² + (a-5)² + (b-5)² = 620</p><p>16 + 9 + 1 + (a-5)² + (b-5)² = 620</p><p>(a-5)² + (b-5)² = 594</p><p><strong>Step 3: Solve for a and b</strong></p><p>Expanding: a² - 10a + 25 + b² - 10b + 25 = 594</p><p>a² + b² - 10(a+b) + 50 = 594</p><p>a² + b² - 10(16) + 50 = 594</p><p>a² + b² = 704</p><p>From (a+b)² = a² + b² + 2ab: 256 = 704 + 2ab ⟹ ab = -224</p><p>So a and b are roots of: t² - 16t - 224 = 0</p><p>t = (16 ± √(256 + 896))/2 = (16 ± √1152)/2 = (16 ± 24√2)/2 = 8 ± 12√2</p><p><strong>Step 4: Calculate Mean Deviation</strong></p><p>MD = [|1-5| + |2-5| + |6-5| + |a-5| + |b-5|]/5</p><p>= [4 + 3 + 1 + |3-12√2| + |3+12√2|]/5</p><p>= [8 + (12√2-3) + (12√2+3)]/5</p><p>= [8 + 24√2]/5 = <strong>8/5 + 24√2/5</strong> or approximately <strong>4.88</strong></p><p>∴ Answer: D (Check your options for the form: likely (8 + 24√2)/5 or equivalent)</p>
Correct Answer: D