Statistics
Variance and Standard Deviation
Grade 11
Question:
<p>In an experiment with 15 observations on \(x\), then following results were available: \(\sum x^2 = 2830\), \(\sum x = 170\). One observation that was 20 was found to be wrong and was replaced by the correct value 30. Then the correct variance is</p>
<p>78.00</p>
<p>188.66</p>
<p>177.33</p>
<p>8.33</p>
Step-by-Step Solution
Key Concept: Variance depends on both Σx and Σx². When an observation is corrected, recalculate both sums, then apply the variance formula: σ² = (Σx²/n) - (Σx/n)². The key is updating both components systematically.
<p><strong>Step 1:</strong> Identify the correction needed. The incorrect observation was 20, replaced by correct value 30. The change is: 30 - 20 = 10.</p><p><strong>Step 2:</strong> Update Σx (sum of observations):<br/>Correct Σx = 170 - 20 + 30 = 180</p><p><strong>Step 3:</strong> Update Σx² (sum of squares):<br/>Σx² = 2830 - (20)² + (30)² = 2830 - 400 + 900 = 3330</p><p><strong>Step 4:</strong> Apply variance formula σ² = (Σx²/n) - (Σx/n)²:<br/>σ² = (3330/15) - (180/15)²<br/>σ² = 222 - (12)²<br/>σ² = 222 - 144<br/>σ² = 78</p><p>∴ Answer: A (The correct variance is 78)</p>
Correct Answer: A