In this paper we quantify the rate of convergence in Selberg's central limit theorem for $\log|ζ(1/2+it)|$ based on the method of proof given by Radziwill and Soundararajan. We achieve the same rate of convergence of $(\log\log\log T)^2/\sqrt{\log\log T}$ as Selberg in the Kolmogorov distance by using the Dudley distance instead. We also prove the theorem for the multivariate case given by Bourgade with the same rate of convergence as in the single variable case.