This paper introduces the concept of negatively associated (NA) random variables and explores their properties and applications. NA random variables are defined as those where for any disjoint subsets of indices, the covariance of increasing functions of the variables in these subsets is non-positive. Key properties of NA include that increasing functions of disjoint subsets of NA variables are also NA, and that NA is closed under the formation of increasing functions of disjoint sets. The paper also shows that NA is not simply the dual of positive association but differs in important respects. Several well-known multivariate distributions are shown to possess the NA property, including the multinomial, convolution of unlike multinomials, multivariate hypergeometric, Dirichlet, and Dirichlet compound multinomial. The paper also demonstrates that negatively correlated normal random variables are NA. Applications of NA include sampling without replacement, multiple ranking and selection procedures, and categorical data analysis. In particular, it is shown that NA and positive association can coexist in the same model, as illustrated by contingency tables. The paper also discusses other types of negative dependence, such as negative upper and lower orthant dependence, reverse regular of order two (RR₂), conditionally decreasing in sequence (CDS), and negatively dependent in sequence (NDS). It is shown that only NA is closed under the formation of increasing functions of disjoint sets. The paper provides theoretical results, including a theorem that shows that the conditional distribution of NA variables given their sum is NA. It also discusses permutation distributions and their properties, showing that permutation distributions are NA. The paper concludes with applications of NA in various statistical models and procedures.This paper introduces the concept of negatively associated (NA) random variables and explores their properties and applications. NA random variables are defined as those where for any disjoint subsets of indices, the covariance of increasing functions of the variables in these subsets is non-positive. Key properties of NA include that increasing functions of disjoint subsets of NA variables are also NA, and that NA is closed under the formation of increasing functions of disjoint sets. The paper also shows that NA is not simply the dual of positive association but differs in important respects. Several well-known multivariate distributions are shown to possess the NA property, including the multinomial, convolution of unlike multinomials, multivariate hypergeometric, Dirichlet, and Dirichlet compound multinomial. The paper also demonstrates that negatively correlated normal random variables are NA. Applications of NA include sampling without replacement, multiple ranking and selection procedures, and categorical data analysis. In particular, it is shown that NA and positive association can coexist in the same model, as illustrated by contingency tables. The paper also discusses other types of negative dependence, such as negative upper and lower orthant dependence, reverse regular of order two (RR₂), conditionally decreasing in sequence (CDS), and negatively dependent in sequence (NDS). It is shown that only NA is closed under the formation of increasing functions of disjoint sets. The paper provides theoretical results, including a theorem that shows that the conditional distribution of NA variables given their sum is NA. It also discusses permutation distributions and their properties, showing that permutation distributions are NA. The paper concludes with applications of NA in various statistical models and procedures.