The randomness of disc packings, generated by random sequential adsorption (RSA), random packing under gravity (RPG) and Mason packing (MP) which gives a packing density close to that of the RSA packing, has been analysed, based on the Delaunay tessellation, and is evaluated at two levels, i.e. the randomness at individual subunit level which relates to the construction of a triangle from a given edge length distribution and the randomness at network level which relates to the connection between triangles from a given triangle frequency distribution. The Delaunay tessellation itself is also analysed and its almost perfect randomness at the two levels is demonstrated, which verifies the proposed approach and provides a random reference system for the present analysis. It is found that (i) the construction of a triangle subunit is not random for the RSA, MP and RPG packings, with the degree of randomness decreasing from the RSA to MP and then to RPG packing; (ii) the connection of triangular subunits in the network is almost perfectly random for the RSA packing, acceptable for the MP packing and not good for the RPG packing. Packing method is an important factor governing the randomness of disc packings.