  Most factorization invariants in the literature extract extremal
factorization behavior, such as the maximum and minimum factorization lengths.
Invariants of intermediate size, such as the mean, median, and mode
factorization lengths are more subtle. We use techniques from analysis and
probability to describe the asymptotic behavior of these invariants.
Surprisingly, the asymptotic median factorization length is described by a
number that is usually irrational.
