Statistics
Variance and Standard Deviation
Grade 11
Question:
<p>The outcome of each of 30 items was observed; 10 items gave an outcome \(1/2 - d\) each, 10 items gave outcome \(1/2\) each and the remaining 10 items gave outcome \(1/2 + d\) each. If the variance of this outcome data is \(4/3\) then \(|d|\) equals:</p>
<p>\(\dfrac{\sqrt{5}}{2}\)</p>
<p>\(2\)</p>
<p>\(\sqrt{2}\)</p>
<p>\(\dfrac{2}{3}\)</p>
Step-by-Step Solution
Key Concept: Variance formula with symmetric distribution: For data with values equally distributed around the mean, variance = (sum of squared deviations from mean) / total frequency. Here the mean is 1/2, so calculate variance using the three distinct values and their frequencies.
<p><strong>Step 1:</strong> Find the mean of the data.</p><p>Mean = (1/30)[10(1/2 - d) + 10(1/2) + 10(1/2 + d)] = (1/30)[5 - 10d + 5 + 5 + 10d] = 15/30 = 1/2</p><p><strong>Step 2:</strong> Calculate variance using the formula: Variance = Σf(x - mean)²/N</p><p>Variance = (1/30)[10(1/2 - d - 1/2)² + 10(1/2 - 1/2)² + 10(1/2 + d - 1/2)²]</p><p>= (1/30)[10(-d)² + 10(0)² + 10(d)²]</p><p>= (1/30)[10d² + 0 + 10d²]</p><p>= (1/30)[20d²]</p><p>= (2d²)/3</p><p><strong>Step 3:</strong> Set variance equal to 4/3 and solve for |d|.</p><p>(2d²)/3 = 4/3</p><p>2d² = 4</p><p>d² = 2</p><p>|d| = √2</p><p>∴ Answer: C</p>
Correct Answer: C