We determine p53 protein abundances and cell to cell variation in two human cancer cell lines with single cell resolution, and show that the fractional width of the distributions is the same in both cases despite a large difference in average protein copy number. We developed a computational framework to identify dominant mechanisms controlling the variation of protein abundance in a simple model of gene expression from the summary statistics of single cell steady state protein expression distributions. Our results, based on single cell data analysed in a Bayesian framework, lends strong support to a model in which variation in the basal p53 protein abundance may be best explained by variations in the rate of p53 protein degradation. This is supported by measurements of the relative average levels of mRNA which are very similar despite large variation in the level of protein.