In this note, we prove a characterization formula for biharmonic maps in the Euclidean spheres of radius R, whose image lies in a small hypersphere. This formula represents a generalization of a result in E. Loubeau, C. Oniciuc, On the biharmonic and harmonic indices of the Hopf map, Trans. Amer. Math. Soc. 359 (2007). Then we apply it for quad-ratic maps between spheres.