Many operational weather services use ensembles of forecasts to generate probabilistic predictions. Computational costs generally limit the size of the ensemble to fewer than 100 members, although the large number of degrees of freedom in the forecast model would suggest that a vastly larger ensemble would be required to represent the forecast probability distribution accurately. In this study, we use a computationally efficient idealised model that replicates key properties of the dynamics and statistics of cumulus convection to identify how the sampling uncertainty of statistical quantities converges with ensemble size. Convergence is quantified by computing the width of the 95% confidence interval of the sampling distribution of random variables, using bootstrapping on the ensemble distributions at individual time and grid points. Using ensemble sizes of up to 100,000 members, it was found that for all computed distribution properties, including mean, variance, skew, kurtosis, and several quantiles, the sampling uncertainty scaled as