Which of the following is more surprising?
- In a group of 100 people, the tallest person is one inch taller than the second tallest person.
- 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.