<p>The standard deviation of the following frequency distribution is</p><table border='1'><tr><th>X</th><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td></tr><tr><th>f</th><td>4</td><td>9</td><td>16</td><td>14</td><td>11</td><td>6</td></tr></table><p>Find the standard deviation.</p>
Step-by-Step Solution
Key Concept: Standard deviation requires calculating the mean first, then finding the sum of squared deviations weighted by frequencies. The formula is σ = √[Σf(x-x̄)²/Σf], where x̄ is the weighted mean.
<p><strong>Step 1:</strong> Organize the data: X = {2,3,4,5,6,7}, f = {4,9,16,14,11,6}</p><p><strong>Step 2:</strong> Calculate total frequency: Σf = 4+9+16+14+11+6 = 60</p><p><strong>Step 3:</strong> Calculate weighted mean: x̄ = Σfx/Σf = (2·4+3·9+4·16+5·14+6·11+7·6)/60 = (8+27+64+70+66+42)/60 = 277/60 ≈ 4.617</p><p><strong>Step 4:</strong> Calculate Σf(x-x̄)²:</p><p>• (2-4.617)²·4 = 27.28</p><p>• (3-4.617)²·9 = 23.52</p><p>• (4-4.617)²·16 = 6.11</p><p>• (5-4.617)²·14 = 3.26</p><p>• (6-4.617)²·11 = 22.27</p><p>• (7-4.617)²·6 = 43.06</p><p>Σf(x-x̄)² ≈ 125.5</p><p><strong>Step 5:</strong> Standard deviation σ = √(125.5/60) = √2.092 ≈ 1.45</p><p>∴ Answer: A</p>
Correct Answer: A