Statistics
Standard Deviation
Grade 11
Question:
<p>In a series of \(2n\) observations, half of them equal \(a\) and remaining half equal \(-a\). If the standard deviation of the observations is 2, then \(|a|\) equals</p>
<p>\(\dfrac{1}{n}\)</p>
<p>\(\sqrt{2}\)</p>
<p>2</p>
<p>\(\dfrac{\sqrt{2}}{n}\)</p>
Step-by-Step Solution
Key Concept: Standard deviation measures spread around the mean. With n observations of value 'a' and n observations of '-a', the mean is zero, so SD² = (sum of squared deviations)/2n directly gives us the variance formula.
<p><strong>Step 1:</strong> Identify the dataset: n observations equal to <em>a</em>, and n observations equal to <em>-a</em>. Total observations = 2n.</p><p><strong>Step 2:</strong> Calculate the mean: μ = (n·a + n·(-a))/2n = 0</p><p><strong>Step 3:</strong> Apply standard deviation formula: SD = √[(Σ(x - μ)²)/2n]</p><p><strong>Step 4:</strong> Calculate sum of squared deviations: Σ(x - μ)² = n(a - 0)² + n(-a - 0)² = na² + na² = 2na²</p><p><strong>Step 5:</strong> Substitute into SD formula: SD = √(2na²/2n) = √(a²) = |a|</p><p><strong>Step 6:</strong> Given SD = 2, we have |a| = 2</p><p>∴ Answer: C</p>
Correct Answer: C