Sparse random linear network coding (SRLNC) is a promising solution for reducing the complexity of random linear network coding (RLNC). RLNC can be modeled as a linear operator channel (LOC). It is well known that the normalized channel capacity of LOC is characterized by the rank distribution of the transfer matrix. In this paper, we study the rank distribution of SRLNC. By exploiting the definition of linear dependence of the vectors, we first derive a novel approximation to the probability of a sparse random matrix being non-full rank. By using the Gauss coefficient, we then provide a closed approximation to the rank distribution of a sparse random matrix over a finite field. The simulation and numerical results show that our proposed approximation to the rank distribution of sparse matrices is very tight and outperforms the state-of-the-art results, except for the finite field size and the number of input packets are small, and the sparsity of the matrices is large.