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Let r be the range and S be the standard deviation of a set of n observations x1, x2,..., xn. Which option correctly states the relationship between S, r, and n, and also gives the bound on S when r = 2 and n = 9?

1. S <= r * sqrt(n/(n-1))
2. S <= 3/sqrt(2)
3. S >= sqrt(n/(n-1)) and S >= 2/sqrt(3)
4. S >= sqrt(n/(n+1)) and S >= 3/sqrt(2)

Correct answer: S <= r * sqrt(n/(n-1))

Solution

The standard result is S <= r*sqrt(n/(n-1)) (or equivalently for the sample SD). With r=2 and n=9: S <= 2*sqrt(9/8) = 3/sqrt(2), confirming option A and B are consistent (B gives the numerical value).

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