  Stable distributions is an interesting and important class of probability
distributions. They were discovered explicitly by Paul L\'{e}vy in 1925
\cite{lk}. They possess many interesting properties, most importantly they are
by definiton invariant under addition, up to a scale. Noteworthly they have
power-law type of decay and therefore they are an excellent model for modelling
many natural phenomena, such as earthquakes, financial returns, and a multitude
of social phenomena such as size distributions of cities and firms
\cite{scaling}. The major problem concerning them is that they have an infinite
variance \cite{GK} and therefore their practical applicability is somewhat
limited. Also they generally do not possess a density expressible in an
analytic form. This study proposes a dispersion measure for them, drawing ideas
from Fisher information, differential geometry and most importantly, the
uncertainty principle for Fourier transform pairs \cite{Weyl}. The study begins
with a brief discussion on characteristic functions and their relation to
Fourier transforms and their properties, proceeds to a brief presentation of
stable distributions and accumulates in defining a concept of
\textit{characteristic curvature}, which is proposed as a suitable measure of
dispersion for class of stable distributions. Characteristic curvature
satisfies the familiar scaling property for stable processes, reduces to that
of standard deviation in the Gaussian case and is explicitly related to the
Euler Gamma function, the Hurst exponent H and the fractal dimension 2-H.
