<p>Consider the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10. If 1 is added to each number, the variance of the numbers so obtained is</p>
Step-by-Step Solution
Key Concept: Variance is invariant under translation (adding a constant to all data points). When a constant is added to each observation, the variance remains unchanged because variance measures dispersion about the mean, and both the data and mean shift equally.
<p><strong>Step 1:</strong> Recall the property of variance: V(X + c) = V(X), where c is any constant.</p><p><strong>Step 2:</strong> Original numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10</p><p>After adding 1 to each: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11</p><p><strong>Step 3:</strong> Calculate variance of original set:</p><p>Mean = (1+2+3+...+10)/10 = 55/10 = 5.5</p><p>Variance = Σ(xᵢ - 5.5)²/10 = [(-4.5)² + (-3.5)² + (-2.5)² + (-1.5)² + (-0.5)² + (0.5)² + (1.5)² + (2.5)² + (3.5)² + (4.5)²]/10</p><p>= [20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25 + 20.25]/10 = 82.5/10 = <strong>8.25</strong></p><p><strong>Step 4:</strong> Since adding a constant doesn't change variance, the variance of the new set (2, 3, 4, ..., 11) is also <strong>8.25</strong></p><p>∴ Answer: D</p>
Correct Answer: D