We consider the problem of estimating the arithmetic average of a finite collection of real vectors stored in a distributed fashion across several compute nodes subject to a communication budget constraint. Our analysis does not rely on any statistical assumptions about the source of the vectors. This problem arises as a subproblem in many applications, including reduce-all operations within algorithms for distributed and federated optimization and learning. We propose a flexible family of randomized algorithms exploring the trade-off between expected communication cost and estimation error. Our family contains the full-communication and zero-error method on one extreme, and an ϵ-bit communication and (Formula presented.) error method on the opposite extreme. In the special case where we communicate, in expectation, a single bit per coordinate of each vector, we improve upon existing results by obtaining (Formula presented.) error, where r is the number of bits used to represent a floating point value.