Statistics
Standard Deviation
Grade 11
Question:
<p>If the standard deviation of the numbers \(-1, 0, 1, k\) is \(\sqrt{5}\) where \(k > 0\), then \(k\) is equal to:</p>
<p>\(2\sqrt{6}\)</p>
<p>\(2\sqrt{\dfrac{10}{3}}\)</p>
<p>\(4\sqrt{\dfrac{5}{3}}\)</p>
<p>\(\sqrt{6}\)</p>
Step-by-Step Solution
Key Concept: Standard deviation formula requires finding the mean first, then using σ² = Σ(xᵢ - mean)²/n. Set up the equation σ² = 5 and solve the resulting quadratic for k > 0.
<p><strong>Step 1:</strong> Find the mean of the numbers {-1, 0, 1, k}</p><p>Mean = (-1 + 0 + 1 + k)/4 = k/4</p><p><strong>Step 2:</strong> Apply the standard deviation formula: σ² = Σ(xᵢ - mean)²/n</p><p>σ² = [(-1 - k/4)² + (0 - k/4)² + (1 - k/4)² + (k - k/4)²]/4</p><p><strong>Step 3:</strong> Simplify each term:</p><p>= [(-4 - k)²/16 + k²/16 + (4 - k)²/16 + (3k)²/16]/4</p><p>= [(16 + 8k + k²) + k² + (16 - 8k + k²) + 9k²]/(16 × 4)</p><p>= [32 + 12k²]/64 = [8 + 3k²]/16</p><p><strong>Step 4:</strong> Set σ² = 5 (since σ = √5)</p><p>[8 + 3k²]/16 = 5</p><p>8 + 3k² = 80</p><p>3k² = 72</p><p>k² = 24</p><p>k = 2√6 (taking k > 0)</p><p>∴ Answer: A (k = 2√6)</p>
Correct Answer: A