We give an intuitive though general explanation of the finitesize effect in scale-free networks in terms of the degree distribution of the starting network. This result clarifies the relevance of the starting network in the final degree distribution. We use two different approaches: the deterministic meanfield approximation used by Barabási and Albert (but taking into account the nodes of the starting network), and the probability distribution of the degree of each node, which considers the stochastic process. Numerical simulations show that the accuracy of the predictions of the mean-field approximation depend on the contribution of the dispersion in the final distribution. The results in terms of the probability distribution of the degree of each node are very accurate when compared to numerical simulations. The analysis of the standard deviation of the degree distribution allows us to assess the influence of the starting core when fitting the model to real data.