Metadata about: Here we present an improved mathematical analysis of the time evolution of the Covid-19 pandemic in Italy and a statistical error analyses of its evolution, including Monte Carlo simulations with a very large number of runs to evaluate the uncertainties in its evolution. A previous analysis was based on the assumption that the number of nasopharyngeal swabs would be constant, however the number of daily swabs has been increasing with an average factor of about five with respect to our previous analysis. Therefore, here we consider the time evolution of the ratio of the diagnosed positive cases to number of swabs, which is more representative of the evolution of the pandemic when the number of swabs is increasing or changing in time. We consider a number of possible distributions representing the evolution of the pandemic in Italy and we test their prediction capability over a period up to four weeks. The results show that a distribution of the type of Planck's black body radiation law provides very good forecasting. The use of different distributions provides an independent estimate of the uncertainty. We then consider five possible cases for the number of daily swabs and we then estimate the potential dates of a substantial reduction in the number of diagnosed positive cases. We then perform Monte Carlo simulations with 25000 runs to evaluate the uncertainty in the prediction of the date of a substantial reduction in the number of diagnosed daily cases. Finally, we present an alternative method to evaluate the uncertainty in our mathematical predictions based on the study of each region of Italy and we present an application of the Central Limit Theorem with 100000 runs to display the uncertainty in our mathematical predictions based on the analysis of each region.

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