  We show that a sequence has effective Hausdorff dimension 1 if and only if it
is coarsely similar to a Martin-L\"{o}f random sequence. More generally, a
sequence has effective dimension $s$ if and only if it is coarsely similar to a
weakly $s$-random sequence. Further, for any $s<t$, every sequence of effective
dimension $s$ can be changed on density at most $H^{-1}(t)-H^{-1}(s)$ of its
bits to produce a sequence of effective dimension $t$, and this bound is
optimal.
