We present experimental and computational analyses of three-dimensional chaotic laminar mixing in one of the most widely utilized mixing devices, the stirred tank. We find that 98% of the power expended in mechanical stirring produces no detectable mixing at all, and in fact convective mixing is generated solely by tiny perturbations to a dominant non-chaotic flow. By analyzing the role of these perturbations, dramatic improvements in performance are easily achieved: for example, minor judicious, changes in agitator design produce order of magnitude improvements in mixing effectiveness.