It is well known that the Gauss dynamical system of regular continued fractions satisfies many strong mixing properties, in fact, it is Bernoulli. In addition to the standard mixing properties in dynamical system, the Gauss dynamical system also satisfies the $\psi$-mixing property from the viewpoint of stochastic processes. This $\psi$-mixing property, introduced by J.R. Blum, D.L. Hanson and L.H. Koopmans (1963), implies strong mixing, weak mixing and ergodicity in the sense of dynamical system. More importantly, the $\psi$-mixing property gives an upper bound for the $L^2$-convergence of the ergodic average. We use this upper bound to develop a quantitative $L^2$-ergodic theorem and apply the theorem to refine classical results of metric theory of continued fractions. For example, we show that the geometric mean of the partial quotients of the continued fraction expansion converges to the Khinchin's constant with an error term of order $o(n^{-1/2}(\log n)^{3/2}(\log(\log n))^{1/2 +\epsilon})$. It is worth noting that our method gives an improvement on the error term derived from the classical method of I.S. Gal and J.F. Koksma (1950). This is a joint work with J. Hancl, A. Haddley and R. Nair.