  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.
