Statistics & Probability

Instant, private, and free

Benford's Law Calculator.

Find the expected probability for a leading digit.

On-device calculationNo signup
01

Set your values

Results update as you type.

probabilityPercent: 30.103

probabilityPercent

30.103000

Results update automatically as you type.

Use Cases

Data Auditing and Fraud Detection

Compare the distribution of leading digits in financial or accounting data to Benford's Law to identify anomalies that may indicate fraud or errors.

Example: An auditor checks if the first digits of invoice amounts follow the expected probabilities.

Educational Demonstrations

Use the calculator to illustrate Benford's Law in statistics classes, showing how real-world data often deviates from uniform distribution.

Example: A teacher shows that population numbers of countries follow Benford's Law.

Frequently Asked Questions

What is Benford's Law?
Benford's Law, also known as the first-digit law, states that in many naturally occurring sets of numerical data, the leading digit is more likely to be small. For example, the number 1 appears as the first digit about 30% of the time, while 9 appears less than 5% of the time.
How do I use this calculator?
Simply select a first digit from 1 to 9 using the dropdown or input field. The calculator will instantly display the expected probability of that digit appearing as the leading digit in a dataset that follows Benford's Law.
What is the formula used?
The probability is calculated using the formula P(d) = log10(1 + 1/d), where d is the digit from 1 to 9. This formula gives the expected proportion of numbers with that leading digit.

Tips & Common Mistakes

Tips

  • Benford's Law applies best to datasets that span several orders of magnitude, such as populations, stock prices, or river lengths.
  • For a quick check, compare the observed frequency of each digit in your data to the expected probabilities from this calculator.
  • Remember that Benford's Law is a statistical tendency, not a strict rule; small datasets may not conform.
  • Use this calculator to get the expected probability for any digit, then compare it to your data's actual frequency.

Common Mistakes to Avoid

  • Assuming Benford's Law applies to all datasets; it does not apply to data with a limited range or artificially constrained numbers.
  • Using the calculator for digits outside 1-9; the calculator only supports first digits from 1 to 9.
  • Misinterpreting the probability as a percentage of total numbers, when it is specifically the proportion of numbers with that leading digit.

Last updated: August 13, 2026