Scalability is a major challenge for existing behavioral log analysis algorithms, which extract finite-state automaton models or temporal properties from logs generated by run-ning systems. In this paper we present statistical log anal-ysis, which addresses scalability using statistical tools. The key to our approach is to consider behavioral log analysis as a statistical experiment. Rather than analyzing the entire log, we suggest to analyze only a sample of traces from the log and, most importantly, provide means to compute sta-tistical guarantees for the correctness of the analysis result. We present the theoretical foundations of our approach and describe two example applications, to the classic k-Tails algorithm and to the recently presented BEAR algorithm. Finally, based on experiments with logs generated from real-world models and with real-world logs provided to us by our industrial partners, we present extensive evidence for the need for scalable log analysis and for the effectiveness of statistical log analysis.