In this thesis, we study the concepts of deficiency and ambiguity for a bijection on a finite Abelian group, in particular, for permutation polynomials over finite fields. Our focus is on all normalized permutation polynomials of degree ≤ 5. We obtain explicit numbers for deficiency and ambiguity of eleven classes of polynomials in Table A.0.1. We verified them by using SAGE. However, we can not do much for the remaining two cases, except one observation on a relation between ambiguity and deficiency in a special case for one of the remaining unsolved classes.