$$ \begin{equation*} w(x,t)dx = \frac{1}{\sqrt{4\pi Dt}}\exp{(-\frac{x^2}{4Dt})}dx. \end{equation*} $$ 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.