What Benford's Law says
Benford's Law (also called the first-digit law or the Newcomb–Benford Law) describes how often each digit 1 through 9 appears as the leading digit in many real-world collections of numbers. Counterintuitively, the leading digit is not uniform: instead of each digit appearing about 11% of the time, the number 1 leads roughly 30% of the time, and each successive digit is progressively less likely. The exact probability that the leading digit equals d is:
P(d) = log₁₀(1 + 1/d), for d = 1, 2, …, 9
The expected first-digit distribution
Plugging every digit into the formula gives the benchmark this calculator compares your data against:
| Leading digit d | P(d) = log₁₀(1 + 1/d) |
|---|---|
| 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% |
These nine probabilities sum to 1, because log₁₀(2/1) + log₁₀(3/2) + … + log₁₀(10/9) telescopes to log₁₀(10) = 1.
Why does data follow this pattern?
Benford's Law tends to hold when data spans several orders of magnitude and is not artificially bounded — think populations of towns, river lengths, stock prices, street addresses, or accounting entries. The intuition is scale invariance: if a quantity is just as likely to sit anywhere on a logarithmic scale, then the span from 1 to 2 (a doubling) covers far more of that scale than the span from 8 to 9 (a ~12% increase), so small leading digits are simply more common. Any distribution that is genuinely scale-invariant must follow Benford's Law exactly.
How this calculator tests your data
The tool reads every number you paste, extracts each value's first non-zero digit (ignoring signs, currency symbols, decimal points, and thousands separators), and tallies how often digits 1–9 appear. It then computes the expected count for each digit as N × log₁₀(1 + 1/d) and runs a chi-square goodness-of-fit test:
χ² = Σ (Observedₖ − Expectedₖ)² / Expectedₖ
With nine categories the test has 8 degrees of freedom. If χ² is at or below the critical value of 15.51 (α = 0.05), the data is statistically consistent with Benford's Law; a larger χ² signals a significant departure worth investigating.
Where it is used
Benford's Law is a standard first-pass screen in forensic accounting and auditing (fabricated invoices and expense reports often over-represent certain leading digits), tax-fraud detection, election-forensics research, and scientific data-integrity checks. It flags anomalies rather than proving fraud — a failing test is a prompt to look closer, not a verdict.
Frequently Asked Questions
What is the exact formula for Benford's Law?
Does a failed test prove fraud or manipulation?
What kind of data should follow Benford's Law?
How many data points do I need?
Common Tip Percentages
| Service | Typical Tip Range | Notes |
|---|---|---|
| Restaurant (sit-down) | 15–20% | 20%+ for exceptional service |
| Food delivery | 10–20% | Consider extra for long distances |
| Taxi / rideshare | 10–20% | More for assistance with luggage |
| Hotel housekeeping | $2–5/night | Leave daily, not just at checkout |
| Barber / hair salon | 15–20% | On pre-discount total |
| Pizza delivery | 10–15% or $3–5 minimum | Higher in bad weather |
General Household Budget Ratios (50/30/20 Rule)
| Category | % of Net Income | Examples |
|---|---|---|
| Needs | 50% | Rent, groceries, utilities, insurance |
| Wants | 30% | Dining out, entertainment, hobbies |
| Savings / Debt | 20% | Emergency fund, retirement, extra payments |
Common Measurement Conversions for Everyday Use
| Quantity | US / Imperial | Metric Equivalent |
|---|---|---|
| 1 cup | 8 fl oz | 237 mL |
| 1 pint | 16 fl oz | 473 mL |
| 1 quart | 32 fl oz | 946 mL |
| 1 gallon | 128 fl oz | 3.785 L |
| 1 lb | 16 oz | 453.6 g |
| 1 ft | 12 in | 30.48 cm |