It is well known, by the Birkhoff's pointwise ergodic theorem, that the time average (ergodic average) converges to the space average (expectation) for almost all points in an ergodic dynamical system. What can be said about the rate of the convergence? In this talk, we shall derive an error term of the difference between the ergodic average and the expectation for some ergodic dynamical systems under certain condition. The required condition involves an upper bound for the $L^2$-convergence of the ergodic average. In particular, this requirement can be attained by the strong mixing condition ($\alpha$-mixing property) from the viewpoint of stochastic processes. For example, it is known that the regular continued fractions satisfy the $\psi$-mixing property, which implies the strong mixing condition, so we can apply our quantitative ergodic theorem to refine classical results of the metric theory of continued fractions. That is, we shall see 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).