Let G=(V, E) be a simple connected graph of order [mid ]V[mid ]=n[ges ]2 and minimum degree δ, and let 2[les ]s[les ]n. We define two parameters, the s-average distance μs(G) and the s-average nearest neighbour distance Λs(G), with respect to each of which V contains an extremal subset X of order s with vertices ‘as spread out as possible’ in G. We compute the exact values of both parameters when G is the cycle Cn, and show how to obtain the corresponding optimal arrangements of X. Sharp upper and lower bounds are then established for Λs(G), as functions of s, n and δ, and the extremal graphs described.