Table 1.
First significant digit relative frequencies for the Benford distribution.
Fig 1.
The Benford probability distribution of first significant digits derived from a set of naturally occurring empirical data.
Probability (d) = log10 [1 + (1/d)], where (d ∈ {1...9}).
Table 2.
Observed first digit counts of mean annual biomass for fifteen functional groups vs Benford’s law expected counts. Data from combined SCH and HUL sample locations. Raw data from Neutel et al., [42].
Fig 2.
Comparison between observed first digit counts and Benford’s law expected counts for stable below ground food-webs during (A) early and (B) late developmental stages of succession.
Graphs of data from Table 2.
Table 3.
Observed first digit diatom community cell counts vs Benford’s law expected counts for colonization experiments. Data from MacArthur and Wilson [43], their Table 6.
Fig 3.
Comparison between observed first digit counts and Benford’s law expected counts for diatom species colonizing glass slide islands at (A) small and (B) large spatial scales.
Graphs of data from Table 3.
Table 4.
Observed first digit counts of Plethodon salamander encounters vs Benford’s law expected counts for persistent (pre-1990) and declining (post-1990) populations from eastern United States forests. (Highton [44], his Table 8–1).
Fig 4.
Comparison between observed first digit counts and Benford’s law expected counts for encounters of Plethodon salamanders from forested habitats with (A) pre-1990 persistent populations and (B) post-1990 declining populations.
Graphs of data from Table 4.
Table 5.
Relative weight (kg/km) of 141 fish species encounters from five streams in Ohio designated Exceptional Warmwater Habitat (EWH) for protection of aquatic life. Individual species measurements sorted by first digit categories. Taylor’s power law relating first digit mean and variance: Y = 3.36 X 2.07, r = 0.97, n = 9, p = 0.001 (www.statology.org/power-regression-calculator/).
Table 6.
Observed first digit biomass counts vs Benford’ law expected counts for 141 fish species encounters from five streams in Ohio designated Exceptional Warmwater Habitat (EWH). Summary of Table 5 data.
Fig 5.
Comparison between observed first digit biomass counts and Benford’s law expected counts for species of fish collected from five geographically isolated streams with persistent and exceptional biotic integrity.
Graph of data from Table 6.
Table 7.
Summary of analytical test results for case reports.
Fig 6.
Canonical representation of the Benford probability distribution for naturally occurring multi-scale ecological systems with minimal anthropogenic disturbance in states of balanced dynamic equilibrium (case comparisons left to right: 1A, 1B, 3A, 4; Table 7).
Table 8.
Observed first digit counts vs Benford’s law expected counts for multi-scale ecological systems in reported steady state dynamic equilibrium. Data compiled from case comparisons (1A, 1B, 3A, 4; Table 7).