Why check a lottery against Benford's Law?

The check allows you to assess the naturalness of the result distribution. If draw sums conform to Benford's Law, this indirectly confirms the randomness and fairness of the lottery number generator.

What does the chi-squared test mean in the Benford context?

The χ² test compares the observed distribution of first digits with the expected distribution according to Benford's Law. If χ² is less than the critical value of 15.507 (at α=0.05), we cannot reject the hypothesis of conformity to the law.

Can Benford's Law predict lottery numbers?

No, Benford's Law is an analysis tool, not a prediction tool. It describes the distribution of first digits in large data sets but cannot predict specific numbers in future draws.

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Benford's Law — First Digit Analysis

Q: What is Benford's Law?

Benford's Law is a mathematical law stating that first digits in natural data sets are distributed unevenly: digit 1 appears in approximately 30% of cases, while 9 appears in only 4.6%. Formula: P(d) = log₁₀(1 + 1/d).

Check whether lottery results conform to the first digit distribution law

Benford's Law states that first digits in natural data sets are distributed unevenly: digit 1 appears ~30% of the time, while 9 appears only ~4.6%. We check how the first digits of draw sums for "Keno" lottery relate to this fundamental mathematical law.

Analysis based on 20 draws from 02/07/2026, 06:00 PM to 02/26/2026, 07:00 PM

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How to Use Benford's Law Analysis

Step-by-step guide to checking lottery results against Benford's Law

Choose the data source

Switch between "Ball sums" and "Draw numbers" modes. Ball sums are better suited for Benford analysis as they cover a wider range of values.

Evaluate the chi-squared test result

If the χ² value is less than the critical value (15.507), the data conforms to Benford's Law. This indicates a natural distribution of first digits.

Study the histogram

Compare the expected and actual value bars. Significant discrepancies may indicate anomalies in the data.

Analyze the deviations

The deviation chart shows which digits appear more or less frequently than expected. Positive deviations mean the digit appears more often, negative — less often.

What is Benford's Law?

History and mathematical foundations

Benford's Law (or the first digit law) is an observation from probability theory about the distribution of leading significant digits in numerical data sets. It was discovered by astronomer Simon Newcomb in 1881 and independently rediscovered by physicist Frank Benford in 1938.

Benford's Formula

P(d) = log₁₀(1 + 1/d)

where d is the first digit (from 1 to 9). This gives: P(1) ≈ 30.1%, P(2) ≈ 17.6%, ..., P(9) ≈ 4.6%.

Application to lotteries

Although individual lottery numbers are uniformly distributed, ball sums follow a normal distribution. Analyzing first digits of these sums can reveal hidden patterns and confirm the fairness of the random number generator.

Chi-squared test

To check conformity with Benford's Law, the χ² criterion with 8 degrees of freedom is used. At a significance level of 0.05, the critical value is 15.507. If the calculated χ² is less than the critical value, the data conforms to the law.

Analysis Tips

Practical recommendations for correct interpretation of results

Benford's Law works best for data spanning several orders of magnitude. Ball sums are a more suitable source than draw numbers.

For reliable analysis, use at least 100 draws. The more data, the more accurate the result.

Non-conformity with Benford's Law does not mean the lottery is unfair — small number ranges may naturally deviate from the law.

Use Benford analysis in combination with other statistical methods for a more complete picture.

Frequently Asked Questions

Answers to common questions about Benford's Law in lotteries