A quantitative notion of statistical distinguishability led R.A. Fisher to his idea of statistical distance which has since been developped into Riemannian geometries on the space of statistical ensembles. Parallel to though independently of this progress, Riemannian geometries were being proposed on spaces of quantum states and also of thermodynamic states. Riemannian geometries in various fields have found various applications as different as in population dynamics and fractional distillation, just to mention the first and the most recent ones. For decades, however, little attention was being paid to the common theoretical basis of these geometric methods. This Chapter intends to fill the gap. We present an elementary introduction to the concept and mathematics of statistical distance in order to help understand the emergence of Riemannian-geometrical structures. While we put more emphasis on the thermodynamic aspects, the main goal is still the interpretation of different applications on an equal footing and using a unified framework.
        1. Introduction
The Riemannian metric structure of thermodynamic theory, initiated by Weinhold (1975) and Ruppeiner (1979), contains important and hitherto barely tapped information concerning a physical system. The structure runs deep; its presence can be felt at all levels of physical description. The Riemannian metric of thermodynamics is, as shown first by Diósi et al. (1984), in fact a realization of R. A. Fisher's concept of statistical distinguishability (1922). He had applied it in 1922 to measure genetic drift and later it became the basis for the mathematical theory of information geometry. The corresponding notion of statistical distance has since been introduced for various statistical systems. At the quantum level the distance measures the reliability of experiments designed to optimally distinguish between the two states along a one-parameter family of density operators (Braunstein and Caves 1994). At the statistical mechanical level, distance is the number of statistically distinguishable intermediate states as we transform one state into another (Wootters 1981). This leads to a natural Riemannian metric on the space of distributions in the thermodynamic limit of Gibbs statistical ensembles. Numerous authors have speculated about the meaning of the curvature defined by this geometry as a measure of stability or interaction strength (c.f. Ruppeiner's recent review 1995). The requirement of covariance with respect to this geometry can be used to give an important correction to thermodynamic fluctuation theory. Finally, at the macroscopic level, the square of this same distance between two equilibrium states of a thermodynamic system equals the minimum entropy produced in a process that transforms one state into the other, multiplied by the number of relaxations during the transformation (Salamon and Berry 1983, Nulton et al. 1985). This result has become known as the horse-carrot theorem. In this Chapter, we recapitulate basic ideas and results concerning the Riemannian metric structure of thermodynamics while we attempt to shed light on the underlying concept of statistical distance used in a much broader context.
        2. Empirical Statistical Distance
The class of continuous variables spans from typical continuous quantities of physics to approximate continuous quantities e.g. in population statistics. Consider a continuous variable corresponding to a measurement of a real number x to a certain precision Dx. The true value of x lies in the confidence interval (x-Dx, x+Dx) with a probability which amounts to 68% when Dx is, as we generally assume, the standard deviation of x. The values Dx provide a measure of distinguishability between different values of x. Two values, say x and x', are statistically indistinguishable if |x-x'| < Dx + Dx'. In the opposite case they are well-distinguishable. By convention, we shall say that x and x' are statistically  distinguishable if the equality holds: