Species accumulation curves are found throughout the paleoecological literature. However, the curves are usually not analyzed beyond basic description. This may in part be due to the jagged nature of many hand-constructed curves, which obscures the underlying relationship between species richness and cumulative number of specimens or samples. A technique for smoothing species accumulation curves proposed here offers a more lucid insight into functions underlying empirical curves and allows for more accurate extrapolations and interpolations based on those curves.

Four Cenozoic data sets were analyzed including two molluscan data sets from the Pliocene Yorktown Formation, one molluscan data set from the Eocene Gosport Sand, and one foraminiferan data set from the Miocene Calvert Cliffs Formation. All studies used bulk collection techniques and sampled constrained stratigraphic intervals.

Species accumulation curves were constructed using a randomization protocol. The samples were selected in a random order for each of 1,000 runs, yielding 1,000 independently derived curves. For each step (sample added), a mean species richness and mean abundance were determined, and a mean species accumulation curve constructed, and 99% ranges were determined using the distribution of 990 data points around the species richness and abundance means.

For all curves, species richness was linearly related to the logarithm of the accumulated specimens. The inverse of the first derivative of each curve was estimated by using the simple run/rise equation (change in x/change in y). This is a measure of how many specimens need to be counted before a new species is found, and thus an estimate of sampling efficiency. In each case, the inverse of the first derivative was linearly related to the abundance of specimens, as expected for a logarithmic relationship. So, the efficiency with which new species are accumulated decreases linearly with the number of specimens. The slope of the line is an indicator of the underlying distribution of species cumulative abundance and thus a type of diversity index. The species efficiency can also be used to standardize sample size when comparing two data sets by setting the same sampling efficiency, and determining the species richness for those efficiency levels.