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Observing and counting sunspots constitutes one of the longest-running scientific experiment, with first observations dating back to Galileo and the invention of the telescope around 1610. Today the sunspot number (SN) time series acts as a benchmark of solar activity in a large range of physical models. An appropriate statistical modelling, adapted to the time series' complex nature, is however still lacking. In this work, we provide the first comprehensive uncertainty quantification analysis of sunspot counts. Our interest lies in the following three components: the number of spots ($N_s$), the number of sunspot groups ($N_g$), and the composite $N_c$, defined as $N_c:=N_s+10N_g$. Those are reported by a network of observatories around the world, and are corrupted by errors of various types. We use a multiplicative framework to provide, for each of the three components, an estimation of their error distribution in various regimes (short-term, long-term, minima of solar activity). We also propose a robust estimator for the underlying solar signal and fit a density distribution that takes into account intrinsic characteristics such as over-dispersion, excess of zeros, and multiple modes. The estimation of the solar signal underlying the composite $N_c$ may be seen as a robust version of the International Sunspot Number (ISN), a quantity widely used as a proxy of solar activity. Therefore our results on $N_c$ may serve to characterize the uncertainty on ISN as well. Our results paves the way for a future monitoring of the observatories in quasi-real time, with the aim to alert the observers when they start deviating from the network and prevent large drifts from occurring in the network.