The first step in topological analysis of data, where the input represents a set of points in euclidean space, is the construction of a simplicial complex on these points. In this thesis we will focus on simplicial complexes constructed on a random sample of points in the plane or 3D space. As the basic model for reconstruction we have chosen the Čech complex and its aproximation, the Alpha complex, which is more appropriate for experiments on random samples. The shape of a simplicial complex is reflected by its topological invariants like the number of connected components and the Euler characteristic. An analysis of the average number of connected components and the average Euler characteristic of a larger number of random samples of points in the plane and in 3D space depends on the number of points in the sample and the parameter of the complex which determines the resolution of the reconstruction. A comparison between results and the theoretical expectation is given.