The Mean: Axiomatics, Generalizations,Applications

We present an axiomatic approach to the mean and discuss generalizations of the mean, including one due to Kolmogorov based on the Weak Law of Large Numbers. We offer examples and counterexamples, describe conventional and unconventional uses of the mean in statistical mechanics, and resolve an anomaly in quantum theory concerning apparent simultaneous coexistence of means and variances of observables. These issues all arise from the familiar definition of the mean.