We derive some new results on the k-th barycentric Olson constants of abelian groups (mainly cyclic). This quantity, for a finite abelian (additive) group (G,+), is defined as the smallest integer l such that each subset A of G with at least l elements contains a subset with k elements {g_1, ... , g_k} satisfying g_1 + ... + g_k = k g_j for some 1 <= j <= k.