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Let x1, x2,..., x100 be 100 observations with sum(xi) = 0, sum over all i < j of |xi * xj| = 80000, and mean deviation from the mean equal to 5. Find the standard deviation.

1. 10
2. 30
3. 40
4. 50

Correct answer: 10

Solution

Since sum(xi) = 0, mean = 0. Mean deviation = (1/100)*sum|xi| = 5 => sum|xi| = 500. Note (sum|xi|)² = sum xi² + 2*sum_(i<j)|xi||xj| = sum xi² + 2*80000 = sum xi² + 160000. So sum xi² = 500² - 160000 = 250000 - 160000 = 90000. Wait — but the given is sum|xi*xj| not sum|xi||xj|. Since |xi*xj| = |xi|*|xj|, we get sum xi² = 250000 - 160000 = 90000. Variance = sum xi² / 100 = 900. SD = 30.

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