Benford’s Law Applied to Hydrology Data—Results and Relevance to Other Geophysical Data

Abstract

Benford’s Law gives the expected frequencies of the digits in tabulated data and asserts that the lower digits (1, 2, and 3) are expected to occur more frequently than the higher digits. This study tested whether the law applied to two large earth science data sets. The first test analyzed streamflow statistics and the finding was a close conformity to Benford’s Law. The second test analyzed the sizes of lakes and wetlands, and the finding was that the data did not conform to Benford’s Law. Further analysis showed that the lake and wetland data followed a power law. The expected digit frequencies for data following a power law were derived, and the lake data had a close fit to these expected digit frequencies.

The use of Benford’s Law could serve as a quality check for streamflow data subsets, perhaps related to time or geographical area. Also, with the importance of lakes as essential components of the water cycle, either Benford’s Law or the expected digit frequencies of data following a power law could be used as an authenticity and validity check on future databases dealing with water bodies. We give several applications and avenues for future research, including an assessment of whether the digit frequencies of data could be used to derive the power law exponent, and whether the digit frequencies could be used to verify the range over which a power law applies. Our results indicate that data related to water bodies should conform to Benford’s Law and that nonconformity could be indicators of (a) an incomplete data set, (b) the sample not being representative of the population, (c) excessive rounding of the data, (d) data errors, inconsistencies, or anomalies, and/or (e) conformity to a power law with a large exponent.