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Which of the following is more surprising?

  1. In a group of 100 people, the tallest person is one inch taller than the second tallest person.
  2. In a group of one billion people, the tallest person is one inch taller than the second tallest person.

Put more precisely, suppose we have a normal distribution with given mean $\mu$ and standard deviation $\sigma$. If we sample from this distribution $N$ times, what is the expected difference between the largest and second largest values in our sample? In particular, does this expected difference go to zero as $N$ grows?

In another question, it is explained how to compute the distribution $MAX_N$ of the maximum, but I don't see how to extract an estimate for the expected value of the maximum from that answer. Though $E(MAX_N)-E(MAX_{N-1})$ isn't the number I'm looking for, it might be a good enough estimate to determine if the value goes to zero as $N$ gets large.

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    For continuous distributions, 2 is *much* more surprising. For discrete distributions with spacing over one inch, less so, particularly if there is only one item at each height.2011-03-03
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    @Ross: why is continuity important? The answer I most believe right now (Michael's) suggests that 2 is indeed much more surprising, but that this is special to the normal distribution. That is, if you sample from a continuous distribution with a longer tail, 1 is much more surprising.2011-03-03
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    I was thinking continuity is only important because samples can be close. Think of taking one billion samples from a distribution of the even naturals from 0 to 10^10^10. You wouldn't be surprised that the largest was more than 1 because you would have to get two from the same bin. But over that range, even with a continuous distribution you would expect the gap from the top to the second to be >1.2011-03-03

6 Answers 6