The prohibitive expense of automatic performance tuning at scale has largely limited the use of autotuning to libraries for shared-memory and GPU architectures. We introduce a framework for approximate autotuning that achieves a desired confidence in each algorithm configuration's performance by constructing confidence intervals to describe the performance of individual kernels (subroutines of benchmarked programs). Once a kernel's performance is deemed sufficiently predictable for a set of inputs, subsequent invocations are avoided and replaced with a predictive model of the execution time. We then leverage online execution path analysis to coordinate selective kernel execution and propagate each kernel's statistical profile. This strategy is effective in the presence of frequently-recurring computation and communication kernels, which is characteristic to algorithms in numerical linear algebra. We encapsulate this framework as part of a new profiling tool, Critter, that automates kernel execution decisions and propagates statistical profiles along critical paths of execution. We evaluate performance prediction accuracy obtained by our selective execution methods using state-of-the-art distributed-memory implementations of Cholesky and QR factorization on Stampede2, and demonstrate speed-ups of up to 7.1x with 98% prediction accuracy.