Based on Gumbel's lectures the following function will reproduce the values in the table found here:
f[n_] := {n, Mean[Table[-Log[-Log[m/(n + 1)]], {m, 1, n}]] // N,
StandardDeviation[Table[-Log[-Log[m/(n + 1)]], {m, 1, n}]] Sqrt[(n - 1)/n] // N}
For example:
f[10]
(* {10, 0.495207, 0.949625} *)
