Benford’s Law
According to Benford’s law, a variety of different data sets include numbers with leading (first) digits that follow the distribution shown in the table below. In Exercises 21–24, test for goodness-of-fit with the distribution described by Benford’s law.





Detecting Fraud When working for the Brooklyn district attorney, investigator Robert Burton analyzed the leading digits of the amounts from 784 checks issued by seven suspect companies. The frequencies were found to be 0, 15, 0, 76, 479, 183, 8, 23, and 0, and those digits correspond to the leading digits of 1, 2, 3, 4, 5, 6, 7, 8, and 9, respectively. If the observed frequencies are substantially different from the frequencies expected with Benford’s law, the check amounts appear to result from fraud. Use a 0.01 significance level to test for goodness-of-fit with Benford’s law. Does it appear that the checks are the result of fraud?

Verified step by step guidance

1. Step 1: State the null and alternative hypotheses. The null hypothesis (H0) is that the observed frequencies of the leading digits follow Benford's Law distribution. The alternative hypothesis (H1) is that the observed frequencies do not follow Benford's Law distribution.
2. Step 2: Calculate the expected frequencies for each leading digit based on Benford's Law. Multiply the total number of checks (784) by the percentages given in the table for each leading digit. For example, the expected frequency for the leading digit '1' is 784 × 0.301.
3. Step 3: Use the chi-square goodness-of-fit test formula: χ² = Σ((Oᵢ - Eᵢ)² / Eᵢ), where Oᵢ represents the observed frequency and Eᵢ represents the expected frequency for each leading digit. Compute this value for all leading digits (1 through 9).
4. Step 4: Determine the degrees of freedom (df) for the chi-square test. The degrees of freedom are calculated as (number of categories - 1). In this case, there are 9 leading digits, so df = 9 - 1 = 8.
5. Step 5: Compare the calculated chi-square statistic to the critical value from the chi-square distribution table at a 0.01 significance level and 8 degrees of freedom. If the calculated value exceeds the critical value, reject the null hypothesis and conclude that the checks are likely the result of fraud. Otherwise, fail to reject the null hypothesis.

Key Concepts

Benford's Law

Benford's Law states that in many naturally occurring datasets, the leading digits are not uniformly distributed. Instead, smaller digits appear more frequently as the leading digit. For example, the digit '1' appears as the leading digit about 30.1% of the time, while larger digits like '9' appear only about 4.6% of the time. This phenomenon is used in various fields, including fraud detection, to identify anomalies in data.

Goodness-of-Fit Test

A goodness-of-fit test is a statistical hypothesis test used to determine how well a set of observed data matches a specific distribution. In this context, it assesses whether the observed frequencies of leading digits from the checks align with the expected frequencies according to Benford's Law. The Chi-square test is commonly used for this purpose, comparing the observed counts to the expected counts to evaluate the fit.

Significance Level

The significance level, often denoted as alpha (α), is the threshold used to determine whether to reject the null hypothesis in hypothesis testing. In this case, a significance level of 0.01 indicates a 1% risk of concluding that a difference exists when there is none. If the p-value from the goodness-of-fit test is less than 0.01, it suggests that the observed data significantly deviates from what Benford's Law would predict, potentially indicating fraudulent activity.