Benford's Law Calculator - Test First-Digit Distribution
Analyze first-digit frequency distribution and test if your dataset follows Benford's Law for fraud detection and data validation
Enter Your Data
Analysis Results
Frequency Distribution
| Digit | Observed | Expected (Benford) | Difference |
|---|
What is a Benford's Law Calculator?
A Benford's Law Calculator is a statistical tool that analyzes the frequency distribution of leading digits in a dataset and compares it to the expected distribution predicted by Benford's Law.
This calculator is used for:
- Fraud Detection - Identifying manipulated financial data
- Data Validation - Verifying authenticity of numerical datasets
- Accounting Audits - Detecting anomalies in financial records
- Research Analysis - Testing natural occurrence patterns
To perform the statistical test used in Benford's Law analysis, check out our Chi-Square Calculator to understand how chi-square tests measure the difference between observed and expected frequencies.
To analyze how data is distributed across different ranges, explore our Frequency Distribution Calculator to organize and visualize your dataset patterns.
To visualize the distribution of first digits graphically, use our Histogram Calculator to create bar charts comparing observed versus expected frequencies.
To understand probability distributions in discrete data, visit our Binomial Distribution Calculator to calculate probabilities for binary outcomes in statistical analysis.
How Benford's Law Works
Benford's Law states that the probability of digit d being the first digit is:
Expected frequencies:
- 1 = 30.1%
- 2 = 17.6%
- 3 = 12.5%
- 4 = 9.7%
- 5 = 7.9%
- 6 = 6.7%
- 7 = 5.8%
- 8 = 5.1%
- 9 = 4.6%
Key Concepts Explained
First-Digit Law
The phenomenon where leading digits are not uniformly distributed, with smaller digits appearing more frequently.
Chi-Square Test
Statistical test measuring the difference between observed and expected frequencies. Lower values indicate better conformity.
Degrees of Freedom
For Benford's Law analysis, df = 8 (9 digits - 1). This determines the critical value for significance testing.
How to Use This Calculator
Enter Your Data
Input numbers separated by commas or spaces (minimum 20 numbers recommended)
Click Analyze
The calculator will extract first digits and compare to Benford distribution
Review Results
Check chi-square statistic and distribution table for conformity assessment
Benefits of This Calculator
- Fraud Detection - Quickly identify suspicious data patterns
- Instant Analysis - Get immediate chi-square test results
- Visual Comparison - Compare observed vs. expected frequencies
- Statistical Rigor - Uses standard chi-square goodness-of-fit test
- Easy Interpretation - Clear compliance indicators
- Professional Tool - Suitable for auditing and research
Factors Affecting Results
- Sample Size - Larger datasets provide more reliable results
- Data Range - Should span multiple orders of magnitude
- Data Type - Natural data follows Benford better than assigned numbers
- Constraints - Maximum/minimum limits can distort distribution
- Rounding - Excessive rounding may affect leading digits
- Data Source - Fabricated data typically deviates from Benford's Law
Frequently Asked Questions
What is Benford's Law?
Benford's Law, also called the First-Digit Law, states that in many naturally occurring datasets, the leading digit is more likely to be small. The digit 1 appears about 30.1% of the time, while 9 appears only about 4.6% of the time.
How is Benford's Law used in fraud detection?
Benford's Law is used to detect fraud in financial data, tax returns, and accounting records. When data significantly deviates from the expected Benford distribution, it may indicate manipulation or fabrication.
What is the formula for Benford's Law?
The probability that the first digit is d is given by: P(d) = log₁₀(1 + 1/d), where d is a digit from 1 to 9.
What types of data follow Benford's Law?
Data that follows Benford's Law includes population numbers, financial data, physical constants, transaction amounts, stock prices, and many naturally occurring datasets that span multiple orders of magnitude.
How do you test if data follows Benford's Law?
To test if data follows Benford's Law, calculate the frequency of each leading digit in your dataset and compare it to the expected Benford frequencies using chi-square test or visual comparison.
What is a good chi-square value for Benford's Law?
For Benford's Law with 8 degrees of freedom at 95% confidence, a chi-square value less than 15.507 suggests the data follows Benford's Law. Higher values indicate deviation from the expected distribution.