We prove that if G=(VG,EG) is a finite, simple, and undirected graph with κ components and independence number α(G), then there exist a positive integer k∈N and a function f:VG→N0 with non-negative integer values such that f(u)≤dG(u) for u∈VG, α(G)≥k≥∑u∈VG1dG(u)+1−f(u), and ∑u∈VGf(u)≥2(k−κ). This result is a best possible improvement of a result due to Harant and Schiermeyer [J. Harant, I. Schiermeyer, On the independence number of a graph in terms of order and size, Discrete Math. 232 (2001) 131–138] and implies that α(G)n(G)≥2(d(G)+1+2n(G))+(d(G)+1+2n(G))2−8 for connected graphs G of order n(G), average degree d(G), and independence number α(G).