This paper proposes a means to characterize the variability in multivariate data. To this end, we propose the Sliced-Normal (SN) class of distributions. This class is called SN because their joint densities result from evaluating a multivariate Gaussian on a polynomial manifold in a high-dimensional feature space. The versatility of SNs enables characterizing complex parameter interdependencies with minimal modeling effort. Optimization-based strategies for the estimation of SNs from data in both physical and feature space are proposed. The formulations in physical space yield non-convex optimization programs whose solutions often outperform the solutions in feature space. However, the formulations in feature space yield convex optimization programs thereby facilitating their application to problems in high dimensions. The superlevel sets of a SN density, which take on a closed semi-algebraic form, are then used to prescribe regions where future data are likely to fall. The probability of this outcome is bounded within a distribution-free framework using scenario theory.