Statistics
Variance
Grade 11
Question:
<p>If mean and standard deviation of 5 observations \(x_1, x_2, x_3, x_4, x_5\) are 10 and 3, respectively, then the variance of 6 observations \(x_1, x_2, \ldots, x_5\) and \(-50\) is equal to ______.</p>
Step-by-Step Solution
Key Concept: When adding a new observation to a dataset, recalculate the combined mean and sum of squares to find the new variance. The variance formula requires Σ(xᵢ - new_mean)² divided by the new count.
<p><strong>Step 1:</strong> Find sum and sum of squares from original 5 observations.</p><p>Given: n = 5, mean = 10, SD = 3</p><p>∴ Variance = 3² = 9</p><p>Sum: Σx₁ = 5 × 10 = 50</p><p>Sum of squares: Σx₁² = n(variance) + (mean)² × n = 5(9) + (10)² × 5 = 45 + 500 = 545</p><p><strong>Step 2:</strong> Add the sixth observation (-50) and recalculate.</p><p>New sum: 50 + (-50) = 0</p><p>New sum of squares: 545 + (-50)² = 545 + 2500 = 3045</p><p>New mean: 0/6 = 0</p><p><strong>Step 3:</strong> Calculate new variance.</p><p>Variance = (Σx₁² - n × mean²)/n = (3045 - 6 × 0²)/6 = 3045/6 = 507.5</p><p>∴ Answer: <strong>507.5</strong></p>
Correct Answer: 507.5