Statistics
Standard Deviation
Grade 11
Question:
<p>If the standard deviation of 0, 1, 2, 3, ..., 9 is \(K\), then the standard deviation of 10, 11, 12, 13, ..., 19 is</p>
<p>\(K\)</p>
<p>\(K + 10\)</p>
<p>\(K + \sqrt{10}\)</p>
<p>\(10K\)</p>
Step-by-Step Solution
Key Concept: Standard deviation is invariant under translation (adding a constant to all data points). When each data point is shifted by the same amount, the spread around the mean remains unchanged, so SD stays the same.
<p><strong>Step 1:</strong> Find mean and SD of first dataset (0,1,2,...,9)</p><p>Mean₁ = (0+1+2+...+9)/10 = 45/10 = 4.5</p><p>Variance₁ = E[X²] - (E[X])² = (0²+1²+...+9²)/10 - (4.5)²</p><p>= 285/10 - 20.25 = 28.5 - 20.25 = 8.25</p><p>SD = √8.25 = K</p><p><strong>Step 2:</strong> Analyze second dataset (10,11,12,...,19)</p><p>Each value = previous value + 10</p><p>Mean₂ = 4.5 + 10 = 14.5</p><p><strong>Step 3:</strong> Apply translation property</p><p>Variance₂ = E[(Y)²] - (E[Y])² where Y = X + 10</p><p>= E[(X+10)²] - (E[X]+10)²</p><p>= E[X²+20X+100] - (E[X]² + 20·E[X] + 100)</p><p>= E[X²] + 20·E[X] + 100 - E[X]² - 20·E[X] - 100</p><p>= E[X²] - (E[X])² = Variance₁ = 8.25</p><p>∴ SD of second dataset = √8.25 = <strong>K</strong></p><p><strong>Answer: A</strong></p>
Correct Answer: A