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There are two questions in analytic number theory which have attracted muchattention over the years. The first one is about the asymptotic formula for thevariance associated with the distribution of a real sequence in arithmeticprogressions, which origins in the work of Barban. The second one is about thefunction $r_3(n)$, the number of ordered representation of $n$ in the sum ofthree positive cubes, and several results are recently proved by Robert C.Vaughan. The paper is concerned with the conjunction of these questions, withthe application of the Hardy-Littlewood Method and the Farey sequence.

On a variance associated with the distribution of $r_3(n)$ in arithmetic progressions