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How well does sample range estimate range?

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How well does sample range estimate range?

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I’ve been doing some work with Focused Objective lately, and today the following question came up in our discussion. If you’re sampling from a uniform distribution, how many samples do you need before your sample range has an even chance of covering 90% of the population range?

This is a variation on a problem I’ve blogged about before. As I pointed out there, we can assume without loss of generality that the samples come from the unit interval. Then the sample range has a beta(n – 1, 2) distribution. So the probability that the sample range is greater than a value c is

\int_c^1 n(n-1) x^{n-2} (1-x) \,dx = 1 - c^{n-1} (n - c(n-1))

Setting c = 0.9, here’s a plot of the probability that the sample range contains at least 90% of the population range, as a function of sample size.

The answer to the question at the top of the post is 16 or 17. These two values of n yield probabilities 0.485 and 0.518 respectively. This means that a fairly small sample is likely to give you a fairly good estimate of the range.

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