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Let's imagine that I sample 100 values from some probability distribution $Distribution$ over the real numbers. Out of these samples, I pick the maximum value $m$.

It seems intuitive (and apparently confirmed by some simulations) that $m$ will be "around" the 99th percentile of values of $Distribution$.

Is there some standard probability distribution that describes this? i.e. some well-characterized distribution for the percentile rank of the maximum value among $N$ samples from an arbitrary $Distribution$?

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The $k$-th largest value of a sample of size $n$ from a distribution is its $k$-th order statistic. This term should already be helpful in searching for results. We also have an tag. Specifically, the maximum is the $n$-th order statistic.

The distributions of the order statistics of certain sampled distributions are known, so you can compare the expected values of the order statistic distributions to the quantiles of the underlying distribution. As an example, the $k$-th order statistic from a $U[0,1]$ distribution is $\text{Beta}(k,n+1-k)$ distributed, so the $n$-th one has a $\text{Beta(n,1)}$ distribution. Its expectation is $\frac{n}{n+1}$, quite close to the $(n-1)$-th quantile at $\frac{n-1}{n}$.

The Wikipedia article on order statistics even has a section on using order statistics to estimate quantiles of the underlying distribution.

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