A positive Lyapunov exponent is the most convincing signature of chaos. However, existing methods for estimating the Lyapunov exponent from a time series often give unreliable estimates because they trace the time evolution of the distance between a pair of initially neighboring trajectories in phase space. Here, we propose a mathematical method for estimating the degree of dynamical instability, as a surrogate for the Lyapunov exponent, without tracing initially neighboring trajectories on the basis of the information entropy from a symbolic time series. We apply the proposed method to numerical time series generated by well-known chaotic systems and experimental time series and verify its validity.