Statistics
Variance
Grade 11
Question:
<p>In an experiment with 15 observations on \(x\), the following results were available: \(\Sigma x^2 = 2830\), \(\Sigma x = 170\). One observation, 20, was found to be wrong and was replaced by the correct value 30. Then the corrected 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 Σx² and Σx, so when correcting a single observation, update both sums and recalculate using the formula var = Σx²/n - (Σx/n)². The key is properly adjusting both components of the variance formula.
<p><strong>Step 1:</strong> Identify the correction needed. Replace observation 20 with 30, so the correction is +10 to one value.</p><p><strong>Step 2:</strong> Update Σx: Originally Σx = 170. Remove 20 and add 30:</p><p>Corrected Σx = 170 - 20 + 30 = 180</p><p><strong>Step 3:</strong> Update Σx²: Originally Σx² = 2830. Remove 20² and add 30²:</p><p>Corrected Σx² = 2830 - 400 + 900 = 3330</p><p><strong>Step 4:</strong> Calculate corrected variance using var = (Σx²)/n - (Σx/n)²:</p><p>var = 3330/15 - (180/15)²</p><p>var = 222 - (12)²</p><p>var = 222 - 144</p><p>var = 78</p><p>∴ Answer: A</p>
Correct Answer: A