Benford’s Law states that in many naturally occurring datasets, the leading digit d (1–9) follows: \[ P(d) = \log_{10}(1 + \frac{1}{d}) \]
This pattern helps detect anomalies or fraud in data such as accounting records, tax data, or election results.
Mathematically, the probability that a number has the first digit \(d\) is:
\[ P(d) = \log_{10}\left(1 + \frac{1}{d}\right) \]
for \(d = 1, 2, 3, \ldots, 9\).
Generates and displays the theoretical probability for each first digit (1–9) according to Benford’s Law. This “benford” object will be used throughout the presentation to generate graphs.
library(ggplot2) library(plotly) library(dplyr) benford <- data.frame( digit = 1:9, probability = log10(1 + 1 / (1:9)) ) benford
## digit probability ## 1 1 0.30103000 ## 2 2 0.17609126 ## 3 3 0.12493874 ## 4 4 0.09691001 ## 5 5 0.07918125 ## 6 6 0.06694679 ## 7 7 0.05799195 ## 8 8 0.05115252 ## 9 9 0.04575749
Displays the simulated counts and proportions of digits (1–9) from 1,000 random draws using Benford probabilities.
## digit count prop ## 1 1 312 0.312 ## 2 2 183 0.183 ## 3 3 130 0.130 ## 4 4 82 0.082 ## 5 5 77 0.077 ## 6 6 64 0.064 ## 7 7 46 0.046 ## 8 8 67 0.067 ## 9 9 39 0.039
An interactive 3D plot comparing digits, their theoretical probabilities, and simulated random variation. This helps visualize how real-world data might slightly deviate from the perfect Benford curve.
A 2D visualization comparing observed frequencies (from simulated data) with expected probabilities (from Benford’s Law).
Shows cumulative probabilities for both observed and expected distributions. This highlights whether the simulated data follows the same cumulative pattern as Benford’s prediction — useful for detecting subtle deviations or anomalies.
