An involution is a permutation that is its own inverse. Given a permutation $σ$ of $[n],$ let $\mathbf{N}_{n}(σ)$ denote the number of ways to write $σ$ as a product of two involutions of $[n].$ If we endow the symmetric groups $S_{n}$ with uniform probability measures, then the random variables ${\mathbf N}_{n}$ are asymptotically lognormal. The proof is based upon the observation that, for most permutations $σ$, $\mathbf{N}_{n}(σ)$ can be well approximated by $\mathbf{B}_{n}(σ),$ the product of the cycle lengths of $σ$. Asymptotic lognormality of $\mathbf{N}_{n}$ can therefore be deduced from Erdős and Turán's theorem that $\mathbf{B}_{n}$ is itself asymptotically lognormal.