# How many random samples before you're not surprised by an extreme value?

Suppose I look at a collection of 10 students and calculate the mean and standard deviation of their GPAs. No one in this group has a 4.0. Then I take another 10 students at random and find that one of them has a 4.0. How surprised should I be by this? Knowing the mean and standard deviation of the first group, how many non-4.0 students should I expect to see before finding one with a 4.0?

Edit: In hindsight, I can see that maybe the discrete nature of a GPA calculation and the fact that a 4.0 represents a maximum possible bound may complicate the issue. If it simplifies things, assume continuously valued variables with no upper or lower bounds.

Links to suggested references are welcome!

• It sounds a little like you would be interested in a non-parametric prediction limit based on a sample size, a mean, a standard deviation, and a presumed maximum and minimum possible value. That would be difficult to compute (and you might find no literature on it): usually, such limits are based on the sample size and either the maximum of the data or the top few values in the data. Other limits can be obtained by adopting parametric models of the distribution and/or assuming a prior distribution. – whuber Sep 28 '16 at 20:21
• A broad answer to this broad question is that you need on average $1/p$ simulations to observe an event of probability $p$. – Xi'an Sep 28 '16 at 20:46

There are several ways to look at this, first, say you have independent observations from the same continuous distribution with cdf (cumulative distribution function) $F$, $X_1, \dotsc, X_n$. Then one question could be the expected value of the maximum of the $n$ observations, $X_{(n)}$. That question is answered by expected value of the order statistics, see for instance Approximate Order Statistics for lognormal variables or Expectation of maximum of i.i.d Weibull random variables? (or search this site). A very crude approximation is just taking the corresponding quantile $q=n/(n-1)$ of the quantile function of $F$, that is, $F^{-1}(q)$.

Another viewpoint is following the comment by @Xi'an, if you are waiting for an event of probability $p$ (say, $p=0.1$ if you are waiting for an observation in the highest decile of the distribution). If $p$ is small and $n$ is large, the number of events in the class have a poisson distribution (approximately) with parameter $np$. So we want the probability that the n umber of events is one or larger, $P(X \ge 1)$. For a poisson with parameter $1=n\cdot \frac1n$ we have (with help from R)

1-ppois(0, 1)
[1] 0.6321206


so with number of observations $\frac1n$ the probability of at least one event is 0.63. If we double the sample size to $\frac2n$ we get

1-ppois(0, 2)
[1] 0.8646647


so the probability increases to 0.86