  The basic properties of the Fisher information allow to reveal the
statistical meaning of classical inequalities between mean functions. The
properties applied to scale mixtures of Gaussian distributions lead to a new
mean function of purely statistical origin, unrelated to the classical
arithmetic, geometric, and harmonic means. We call it the informational mean
and show that when the arguments of the mean functions are Hermitian positive
definite matrices, not necessarily commuting, the informational mean lies
between the arithmetic and harmonic means, playing, in a sense, the role of the
geometric mean that cannot be correctly defined in case of non-commuting
matrices.\\ Surprisingly the monotonicity and additivity properties of the
Fisher information lead to a new generalization of the classical inequality
between the arithmetic and harmonic means.
