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 from the mean. With n observations of value 'a' and n observations of '-a', the mean is 0, so SD² = average of squared deviations = (na² + na²)/(2n) = a². Therefore a² = 4, giving |a| = 2.
<p><strong>Step 1:</strong> Identify the dataset structure: n observations equal to a, and n observations equal to -a (total 2n observations).</p><p><strong>Step 2:</strong> Calculate the mean: Mean = (na + n(-a))/(2n) = 0</p><p><strong>Step 3:</strong> Calculate variance using the formula Var = Σ(xᵢ - mean)²/(2n)</p><p>Var = [n(a - 0)² + n(-a - 0)²]/(2n) = [na² + na²]/(2n) = 2na²/(2n) = a²</p><p><strong>Step 4:</strong> Use the given standard deviation: SD = √(Var) = 2</p><p>Therefore: √(a²) = 2 ⟹ |a| = 2</p><p>∴ Answer: C</p>
Correct Answer: C