Fossil samples almost always accumulate on timescales much longer than the life spans of individual organisms. This time-averaging has the potential to inflate the variability of fossil samples because phenotypic changes may occur during the interval of sample accumulation. Although many have realized that this effect may increase the variance of fossil samples, only qualitative predictions have been possible thus far. In this paper, I assume a simple but general Markovian model of evolution to derive expressions that predict the effects of time-averaging on trait variance and covariance. For lineages evolving as an unbiased random walk, phenotypic variance in samples increases linearly with the duration of time-averaging, at a slope that is proportional to the evolutionary rate. Although based on a very simple model of specimen input into time-averaged samples, the expressions relating variance, time-averaging, and evolutionary rate prove to be robust or adaptable to more realistic assumptions.
The theoretical findings are applied to analyze variation in a set of samples of the deep-sea ostracode Poseidonamicus miocenicus that vary greatly in temporal acuity. The relationship between duration of time-averaging and morphological variance is used to estimate evolutionary rates of two morphological traits in these ostracodes. These rate estimates are similar to those calculated independently from differences between presumed ancestor-descendant pairs of populations. Consistent with other studies of variability and time-averaging, these data suggest that phenotypic variance tends to increase rather slowly with the duration of time-averaging, indicating that greatly inflated variance is expected only in samples that have accumulated over many tens of thousands of years.