The Implications of Discretizing Continuous Random Variables: An Example Using the U.S. Geological Survey Reporting Standards for Streamflow Data

One of the most fundamental assumptions in statistical analysis is whether the random variable of interest is continuous or discrete. Natural processes such as streamflow are considered to be continuous random variables, able to take on an unlimited number of possible values. However, to reflect data accuracy, reported streamflow values are generally rounded. For example, the U.S. Geological Survey (USGS) reports mean daily streamflow values less than 1 cubic foot per second (ft³/s) to the nearest hundredth place (0.01), resulting in only 99 possible values for streamflows between 0 and 1 ft³/s available for statistical analyses. The limits of the data subsequently result in the discretization of a continuous variable; the effects of discretization on the statistical properties of the data could affect applications such as moment estimation, regional frequency analysis, and distribution fitting, particularly for low flows where rounding could substantially reduce the number of possible values. In an effort to determine the effects of discretization, large samples of independent data were generated from a generalized probability distribution to the sixteenth decimal place. The synthesized data was then rounded according to USGS streamflow-reporting standards. Initial comparisons of the original synthesized data and the rounded data show that estimates of the sample mean and standard deviation only differed slightly. Additional experiments will explore the effects of rounded data in other statistical analyses as well as for autocorrelated data.