The device-independent approach to physics is one where conclusions are drawn directly and solely from the observed correlations between measurement outcomes. In quantum information, this approach allows one to make strong statements about the properties of the underlying devices via the observation of Bell-inequality-violating correlations. However, since one can only perform a finite number of experimental trials, statistical fluctuations necessarily accompany any estimation of these correlations. Consequently, an important gap remains between the many theoretical tools developed for the asymptotic scenario and the experimentally obtained raw data. In particular, a sensible way to estimate the underlying quantum distribution, and hence the corresponding quantum state, has so far remained elusive. Here, we provide a few algorithmic tools to bridge this gap and show that the resulting point estimates of the true distribution are physical, unique, computationally tractable and consistent. As an application, we demonstrate how such estimates of the underlying quantum distribution can be used to provide sensible estimates of the amount of entanglement present in the measured system. In stark contrast with existing approach to device-independent parameter estimations, our estimation does not require the prior knowledge of any Bell inequality tailored for the specific property and the specific correlation of interest.