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If the sum of (x_i - 5) for i=1 to 10 equals 5, and the sum of (x_i - 5)² for i=1 to 10 equals 125, then the variance of x₁, x₂,..., x₁₀ is:

1. 9/4
2. 25/4
3. 49/4
4. 81/4

Correct answer: 49/4

Solution

Setting y_i = x_i - 5, mean(y) = 5/10 = 0.5. Variance = mean(y²) - [mean(y)]² = 125/10 - (1/2)² = 25/2 - 1/4 = 50/4 - 1/4 = 49/4. Since variance is translation-invariant, Var(x) = 49/4.

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