  We extend the concept of Wigner-Yanase-Dyson skew information to something we
call ``metric adjusted skew information'' (of a state with respect to a
conserved observable). This ``skew information'' is intended to be a
non-negative quantity bounded by the variance (of an observable in a state)
that vanishes for observables commuting with the state. We show that the skew
information is a convex function on the manifold of states. It also satisfies
other requirements, proposed by Wigner and Yanase, for an effective
measure-of-information content of a state relative to a conserved observable.
We establish a connection between the geometrical formulation of quantum
statistics as proposed by Chentsov and Morozova and measures of quantum
information as introduced by Wigner and Yanase and extended in this article. We
show that the set of normalized Morozova-Chentsov functions describing the
possible quantum statistics is a Bauer simplex and determine its extreme
points. We determine a particularly simple skew information, the ``lambda-skew
information,'' parametrized by a lambda in (0,1], and show that the convex cone
this family generates coincides with the set of all metric adjusted skew
informations.
  Key words: Skew information, convexity, monotone metric, Morozova-Chentsov
function, lambda-skew information.
