Statistics
Standard Deviation
Grade None
Question:
<p>If \(\displaystyle\sum_{i=1}^{9}(x_i - 5) = 9\) and \(\displaystyle\sum_{i=1}^{9}(x_i - 5)^2 = 45\), then the standard deviation of the 9 items \(x_1, x_2, \ldots, x_9\) is</p>
<p>3</p>
<p>9</p>
<p>4</p>
<p>2</p>
Step-by-Step Solution
Key Concept: Standard deviation depends only on deviations from the mean, not the mean itself. Use the given conditions to find the mean first, then apply the variance formula σ² = Σ(xᵢ - mean)²/n.
<p><strong>Step 1:</strong> Find the mean of the data.</p><p>From Σ(xᵢ - 5) = 9, we have: Σxᵢ - 9(5) = 9</p><p>Therefore: Σxᵢ = 45 + 9 = 54</p><p>Mean x̄ = 54/9 = 6</p><p><strong>Step 2:</strong> Use the variance formula relating deviations from different points.</p><p>We know: Σ(xᵢ - 5)² = 45</p><p>Expanding: Σ[(xᵢ - x̄) + (x̄ - 5)]² = 45</p><p>Σ[(xᵢ - x̄)² + 2(xᵢ - x̄)(x̄ - 5) + (x̄ - 5)²] = 45</p><p>Σ(xᵢ - x̄)² + 2(x̄ - 5)Σ(xᵢ - x̄) + 9(x̄ - 5)² = 45</p><p><strong>Step 3:</strong> Simplify using properties.</p><p>Since x̄ = 6: Σ(xᵢ - x̄)² + 2(1)(0) + 9(1)² = 45</p><p>Σ(xᵢ - x̄)² + 9 = 45</p><p>Σ(xᵢ - x̄)² = 36</p><p><strong>Step 4:</strong> Calculate standard deviation.</p><p>Variance σ² = 36/9 = 4</p><p>Standard deviation σ = √4 = 2</p><p>∴ Answer: D</p>
Correct Answer: D