Let G = (V, E) be a simple connected graph of order n with m edges. Also let eG(vi) be the eccentricity of a vertex vi in G. We can assume that eG(v1) ≥ eG(v2) ≥ · · · ≥ eG(vn−1) ≥ eG(vn). The average eccentricity of a graph G is the mean value of eccentricities of vertices of G, avec(G) = 1 n n∑ i=1 eG(vi) . Let γ = γG be the largest positive integer such that eG(vγG ) ≥ avec(G). In this paper, we study the value of γG of a graph G. For any tree T of order n, we prove that 2 ≤ γT ≤ n− 1 and we characterize the extremal graphs. Moreover, we prove that for any graph G of order n, 2 ≤ γG ≤ n and we characterize the extremal graphs. Finally some Nordhaus-Gaddum type results are obtained on γG of general graphs G.