Let S be a semigroup, and f a mapping of S into the class C of non-zero cardinal numbers. Then, the pair (S,f) is said to be representable by a groupoid if there exist a groupoid \(\zeta\) and a homomorphism g of \(\zeta\) onto S such that \(Ker(g)=r(\zeta)\), where Ker(g) is the kernel of g and r(\(\zeta)\) is the least semigroup congruence on \(\zeta\), and the cardinal number of \(g^{-1}(a)\) is equal to f(a) for every \(a\in S\). For a semigroup S, let \(L(S)=\{a\in S:\) \(a\in Sa\}\) and \(R(S)=\{a\in S:\) \(a\in aS\}\). In this paper, firstly the authors give some necessary conditions for a pair (S,f) to be representable by a groupoid. Secondly, they also give several sufficient conditions for a pair (S,f) to be representable by a groupoid. In particular, it is shown that if S is a semigroup such that \(S=L(S)\cup R(S)\), then the pair (S,f) is representable by a groupoid for any mapping f: \(S\to C\). The last part of this paper also contains a special treatment of a semigroup of order five.