$$\begin{equation*} w(x,t)dx = \frac{1}{\sqrt{4\pi Dt}}\exp{(-\frac{x^2}{4Dt})}dx. \end{equation*}$$ At a time $t=2$s the new variance is $\sigma^2=4D$s, implying that the root mean square value is $\sqrt{\langle x^2\rangle-\langle x\rangle^2} = 2\sqrt{D}$. At a further time $t=8$ we have $\sqrt{\langle x^2\rangle-\langle x\rangle^2} = 4\sqrt{D}$. While time has elapsed by a factor of $4$, the root mean square has only changed by a factor of 2.