Like the question title reads, in the Collatz conjecture, why are $\max(\textrm{collatz}(n))$ and $\textrm{var}(\textrm{collatz}(n))$ so closely related? See the Figure below for a log-log plot.
I refer with $\textrm{collatz}(n)$ to a sequence starting from $n$ (so $\max(\textrm{collatz}(n))$ is the maximum value of the sequence etc). EDIT: oops, I just noticed thet x- and y-labels are in wrong order. Maximum value is on the x-axis. The plot shows the data for sequences starting with $n = 1, ..., 100000$.
It's natural that the variance of a chaotic function of this nature will positively correlate with its maximum, particularly a function in which every known sample, shares the same minimum (1). The maximum therefore measures the spread of the sequence, which is exactly what the variance does.