Synthesis tools for approximate sequential circuits require the ability to quickly, efficiently, and automatically characterize and bound the errors produced by the circuits. Previous approaches to characterize errors in approximate sequential circuits have been based on simulations spanning all cycles of a sequential computation. These approaches, however, are not scalable and only accommodate small circuit modules and short computation times. In this paper, we observe that the statistical properties of errors in many approximate sequential circuits follow patterns that can be easily learned. Therefore, statistical error characteristics of these approximate sequential circuits can be predicted with high accuracy using only a few cycles of characterization data. Based on this novel observation, we propose a methodology for predicting error statistics in approximate sequential circuits that is accurate, fully automated, and has significantly lower overhead than prior approaches. Our methodology is robust to changes in predicted error metrics, circuit input distributions, and types of approximate hardware modules used in approximate circuits. We demonstrate the accuracy and scalability of our approach over a range of sequential circuits. On average, prediction inaccuracy is less than 2% and error characterization time is reduced by 99% compared to a simulation-based approach.