Statistics
Variance
Grade 11
Question:
<p>Out of 100 observations, \(\displaystyle\sum_{i=1}^{100} x_i = 400\) and \(\displaystyle\sum_{i=1}^{100} x_i^2 = 2425\). If two observations 8 and 12 are removed (so \(\sum x_i\) becomes 388 and \(N = 97\) ... wait, \(N = 98\)), find the variance of the remaining observations using \(\sigma^2 = \dfrac{\sum x_i^2}{N} - \left(\dfrac{\sum x_i}{N}\right)^2 = \dfrac{2425}{97} - \left(\dfrac{388}{97}\right)^2\).</p>
<p>\(\dfrac{2425}{97} - \left(\dfrac{388}{97}\right)^2\)</p>
<p>\(\dfrac{2425}{98} - \left(\dfrac{388}{98}\right)^2\)</p>
<p>\(\dfrac{2301}{97} - \left(\dfrac{376}{97}\right)^2\)</p>
<p>\(\dfrac{2301}{98} - \left(\dfrac{376}{98}\right)^2\)</p>
Step-by-Step Solution
Key Concept: Variance formula requires recalculating both the sum of values and sum of squares after removing observations, then applying σ² = (Σx²ᵢ/N) - (Σxᵢ/N)² with the updated count.
<p><strong>Step 1:</strong> After removing observations 8 and 12:</p><p>New sum: Σxᵢ = 400 - 8 - 12 = 380</p><p>New sum of squares: Σxᵢ² = 2425 - 8² - 12² = 2425 - 64 - 144 = 2217</p><p>New count: N = 100 - 2 = 98</p><p><strong>Step 2:</strong> Calculate variance using the formula σ² = (Σxᵢ²/N) - (Σxᵢ/N)²</p><p>σ² = (2217/98) - (380/98)²</p><p>σ² = 22.621 - (3.878)²</p><p>σ² = 22.621 - 15.04 = 7.581 ≈ 7.5 or exact: (2217×98 - 380²)/(98²) = (217266 - 144400)/9604 = 72866/9604 ≈ 7.58</p><p>∴ Answer: A</p>
Correct Answer: A