I will discuss an ensemble sampling scheme based on a decomposition of the target average of interest into subproblems that are each individually easier to solve and can be solved in parallel. Versions of this basic philosophy go by the name stratification in the statistics literature and have been an important part of experimental design for many decades. A version of the idea, there referred to as umbrella sampling, was developed independently starting in the 70a**s in the chemical physics community to approximate (by computer simulation) free energy differences and has been an important tool in understanding the stability of molecular structures and reaction pathways ever since. I will present a simple derivation of the scheme in the context of computing averages with respect to a given probability density. We have developed a careful understanding of the accuracy of the scheme that is sufficiently detailed to explain the success of the approach in practice and to suggest improvements including adaptivity. Our error bounds reveal that the existing understanding of the method is incomplete and leads to a number of erroneous conclusions. Our bounds are motivated by new perturbation bounds for Markov Chains that we recently established and that are substantially more detailed than existing perturbation bounds for Markov chains. They demonstrate, for example, that stratification is robust in the sense that in limits in which the straightforward approach to sampling from a density becomes exponentially expensive, the cost to achieve a fixed accuracy with umbrella sampling can increase only polynomially. If time permits I will also share a formulation of this stratification approach for computing fairly general dynamic quantities with respect to a given Markov process.