Both state propagation and sensor measurements are often corrupted by unmodeled non-Gaussian or heavy-tailed noise. Without dealing with such outliers, the accuracy of a estimator significantly degrades, and control systems that rely on high-quality estimation lose stability. To estimate the states of dynamic systems in which both types of outliers occur, we propose a novel approach that combines a real-time outlier detection technique with an extended version of an outlier robust Kalman filter (ORKF). Unlike the ORKF for only measurement outliers, the technique, the extended ORKF (EORKF), also handles situations in which propagation outliers arise; that is, to approximately compute the optimal precision matrices of process outliers, we derive equations and algorithms using the variational inference method. Hence, the EORKF does not restrict noise at either a constant or Gaussian level. Furthermore, for lower computational effort and memory uses, our approach employs the typicality and eccentricity data analysis (TEDA), which provides information about the time when outliers occur and runs the EORKF whenever the TEDA detects outliers. The results of Monte Carlo simulations show that our approach leads to greater improvement in robustness and lower computational complexity than existing methods.