Benford’s Law

Power Laws

Power-law distributions describe patterns where small occurrences are extremely common, while large instances are rare but significantly impactful. These distributions follow the general form:

\[ P(x) \propto x^{-\alpha} \]

where \(\alpha\) is a constant typically greater than 1.

Power laws appear across a wide range of both manmade and natural phenomena, such as: - City populations (e.g. Tokyo vs. smaller towns) - Income and wealth (Pareto Principle or 80/20 Rule) - Word frequencies in human language - Internet traffic and website popularity - Earthquake magnitudes - Network connectivity (social media, power grids, etc.)

Notable Power-Law-Related Laws:

• Zipf’s Law: In a given body of text, the frequency of any word is inversely proportional to its rank.
• Benford’s Law: Predicts the frequency of digits in naturally occurring numerical datasets.
• Bradford’s Law: Describes how articles on a given topic are distributed across journals.

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Benford’s Law

Benford’s Law, also known as the First-Digit Law, reveals that in many real-world datasets, lower digits occur as the leading digit more frequently than higher ones.

Leading Digit Probabilities:

• Digit 1: ~30.1%
• Digit 2: ~17.6%
• …
• Digit 9: ~4.6%

This distribution contrasts sharply with the intuition that all digits 1–9 should appear equally often (~11.1% each).

Why It Works:

Benford’s Law holds especially well for datasets that: - Span several orders of magnitude - Are not bounded artificially - Emerge from multiplicative processes, such as growth rates or natural measurements

It applies across fields—physics, economics, social sciences, and even biology—whenever these conditions are met.

Historical Background:

• First noted by Simon Newcomb in 1881 when he observed that logarithmic tables’ earlier pages were more worn.
• Later formalized and empirically tested by Frank Benford in 1938 using over 20 datasets.

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Applications of Benford’s Law

Benford’s Law is a powerful diagnostic tool for detecting unnatural patterns and suspicious data. It’s especially valuable in: - Forensic accounting and financial audits - Tax fraud detection - Scientific data integrity checks - Election result validation - Ecological and environmental reporting

When fabricated numbers or manipulated datasets diverge from the expected Benford distribution, red flags are raised for further investigation.

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Benford Distribution in R

Here’s how to compute and visualize the Benford distribution in R using the VGAM package:

# Load the required library
library(VGAM)

# Compute probabilities for digits 1 through 9
benford_probs <- dbenf(1:9)

# Visualize the distribution
barplot(benford_probs,
        names.arg = 1:9,
        col = "steelblue",
        main = "Benford's First Digit Distribution",
        xlab = "Leading Digit",
        ylab = "Probability")