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Let r denote the maximum pairwise distance among the observations, i.e. r = max |x_i - x_j|, and let the sample variance be defined by S² = (1)/(n-1)∑_(i=1)ⁿ(x_i-x̄)². Using the fact that each deviation satisfies (x_i-x̄)² < r², which of the following bounds for S² is correct?

1. (a) S² ≤ (nr²)/(n-1)
2. (b) S² ≥ (nr²)/(n-1)
3. (c) S² < (r²)/(n-1)
4. (d) S² ≤ r²

Correct answer: (a) S² ≤ (nr²)/(n-1)

Solution

The correct option is based on the fact that each squared deviation from the mean is less than the maximum pairwise distance squared, which leads to the conclusion that the average of these deviations, when scaled by the sample size, cannot exceed the upper bound derived from the maximum distance.

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