A fullerene graph is a cubic 3-connected plane graph with exact 12 pentagons and other hexagons.In the present work, by Rayleighs short-cut method and Fosters formulas, we show that in a fullerene graph resistance distances between pairs of adjacent vertices lie within a rather narrow interval ranging from 48/79 to 5/7 with there value being 2(n-1/)/3n;and resistance distances between pairs of vertices at distance two lie within the small interval [66/79,8/7] with their average value being n-2/n.