Market Microstructure is the investigation of the process and protocols that govern the exchange of assets with the objective of reducing frictions that can impede the transfer. In financial markets, where there is an abundance of recorded information, this translates to the study of the dynamic relationships between observed variables, such as price, volume and spread, and hidden constituents, such as transaction costs and volatility, that hold sway over the efficient functioning of the system. We consider a measure of similarity, the Bhattacharyya distance, across distributions of these variables. We illustrate a novel methodology based on the marriage between the Bhattacharyya distance and the Johnson Lindenstrauss Lemma, a technique for dimension reduction, providing us with a simple yet powerful tool that allows comparisons between data-sets representing any two distributions. We demonstrate a relationship between covariance and distance measures based on a generic extension of Stein's Lemma. The degree to which different markets or sub groups of securities have different measures of their corresponding distributions tells us the extent to which they are different. This can aid investors looking for diversification or looking for more of the same thing. We briefly discuss how this methodology lends itself to numerous Marketstructure studies and even applications outside the realm of finance / social sciences.