Aims‚ Estimates of geographic range size derived from natural history museum specimens are probably biased for many species. We aim to determine how bias in these estimates relates to range size.\r\n\r\nLocation‚ We conducted computer simulations based on herbarium specimen records from localities ranging from the southern United States to northern Argentina.\r\n\r\nMethods‚ We used theory on the sampling distribution of the mean and variance to develop working hypotheses about how range size defined as area of occupancy (AOO) was related to the inter-specific distribution of: (1) mean collection effort per area across the range of a species (MC); (2) variance in collection effort per area across the range of a species (VC); and (3) proportional bias in AOO estimates (PBias: the difference between the expected value of the estimate of AOO and true AOO divided by true AOO). We tested predictions from these hypotheses using computer simulations based on a dataset of more than 29000 herbarium specimen records documenting occurrences of 377 plant species in the tribe Bignonieae (Bignoniaceae).\r\n\r\nResults‚ The working hypotheses predicted that the mean of the inter-specific distribution of MC VC and PBias were independent of AOO but that the respective variance and skewness decreased with increasing AOO. Computer simulations supported all but one prediction: the variance of the inter-specific distribution of VC did not decrease with increasing AOO.\r\n\r\nMain conclusions‚ Our results suggest that despite an invariant mean the dispersion and symmetry of the inter-specific distribution of PBias decreases as AOO increases. As AOO increased range size was less severely underestimated for a large proportion of simulated species. However as AOO increased range size estimates having extremely low bias were less common.