Continuing to develop the theory of weighted n-dimensional means, this article, presents two theorems that follow Theorem 1 ( Glaser, 2002 ) and, hence, are labeled Theorems 2 and 3. Theorem 2 gives a bound on the distance between two one dimensional weighted means ( Cashwell-Everett, 1969 ). Theorem 3 generalizes this bound for the distance between two n -dimensional means.