Statistics
Variance
Grade 11
Question:
<p>The sum of 100 observations and the sum of their squares are 400 and 2474, respectively. Later on, three observations, 3, 4 and 5, were found to be incorrect. If the incorrect observations are omitted, then the variance of the remaining observations is</p>
<p>8.00</p>
<p>8.25</p>
<p>9.00</p>
<p>8.50</p>
Step-by-Step Solution
Key Concept: Variance is calculated as E(X²) - [E(X)]², where E(X²) = (Σx²)/n and E(X) = (Σx)/n. We must remove the three incorrect observations from both the sum and sum of squares before computing variance.
<p><strong>Step 1:</strong> Original data: n = 100, Σx = 400, Σx² = 2474</p><p><strong>Step 2:</strong> Remove incorrect observations (3, 4, 5):<br/>New sum: Σx' = 400 - (3 + 4 + 5) = 400 - 12 = 388<br/>New sum of squares: Σx'² = 2474 - (3² + 4² + 5²) = 2474 - (9 + 16 + 25) = 2474 - 50 = 2424<br/>New count: n' = 100 - 3 = 97</p><p><strong>Step 3:</strong> Calculate mean of remaining observations:<br/>Mean = Σx'/n' = 388/97 = 4</p><p><strong>Step 4:</strong> Calculate variance using formula Variance = (Σx'²)/n' - (Mean)²:<br/>Variance = 2424/97 - (4)² = 24.99... - 16 ≈ 25 - 16 = 9<br/>Or more precisely: 2424/97 - 16 = (2424 - 1552)/97 = 872/97 ≈ 9</p><p>∴ Answer: A (Variance = 9)</p>
Correct Answer: A