Conformance of Public Water Use Data to Benford's Law

Abstract

Water use data are essential to managing public water systems, but because most of such data in the United States are self-reported, it is difficult to assess their accuracy. Benford's law, which gives the expected frequency of leading digits in numerical data, could serve as one validation tool. This analysis tests whether Benford's law applies to observations of potable water use by US public water systems. Data at system, county, and state levels were examined using the Kolmogorov-Smirnov test and visual inspection. Almost all were found to conform to Benford's law, although statistical power is generally low, and the tests may favor conclusions of conformance. This finding could serve as a quality check for historical water use data as well as projections. In one example application, two sets of data were compared, and the one known to be more accurate more closely matched Benford's law.

Leading-Digit Patterns from Smart Water Meters

“…The findings for this particular dataset suggest that readings from smart water meters-and, by extension, water use by individual customers-could be expected to follow a particular nonuniform pattern of leading digits, whether Benford's law, a steep power law, or simply the pattern where 1 is the most frequent leading digit (regardless of the frequencies of subsequent digits). The results accord with those of Sowby (2018) for coarser water use data and prompt further exploration of other datasets and how this pattern might be exploited.…”

“…Sambridge et al (2010) applied leading-digit analysis to validate observations of exoplanet masses, seismic wave speeds, and global temperature anomalies. Sowby (2018) found a leading-digit pattern in annual potable water use data aggregated by U.S. public water suppliers, counties, and states and suggested that the pattern could help validate water use observations. The study recommended examination of patterns in higher-resolution data and water end-uses.…”

“…Conformance to Benford's law was evaluated with the Kolmogorov-Smirnov (KS) test (Massey 1951). The test is a common choice for Benford analysis, including water-related applications by Sowby (2018) and Nigrini and Miller (2007). The null hypothesis is that the sample conforms to Benford's law, i.e., that the cumulative distribution of leading digits in the empirical dataset is the same as that in Benford's law.…”

“…To complement existing validation methods, water use data could be screened by a leading-digit analysis before being finalized. As in accounting (Durtschi et al 2004;Nigrini and Mittermaier 1997), water use data samples that deviate from the expected pattern could be flagged for further attention since they may be incomplete, be rounded too much, contain errors, or indicate abnormal water use (Sowby 2018;Nigrini and Miller 2007). Such functions could be programmed as quality control measures into smart meter software backends and water utility databases, or stand alone for ad-hoc processing.…”

“…Leading-digit patterns may also help one compare the accuracy of two or more datasets on the same water uses and inform a choice about which dataset to use. Sowby (2018) illustrated such a case where, given two datasets on public water suppliers' use, the dataset known to be more accurate also more closely followed Benford's law. The same may be true of smart meters and individual water users.…”