Wiki Page, Math 13 - Perez Sandbox : Module 11 - Probability Distributions for a Quantitative Variable (1 of 3)

Wiki Page
2026-01-17T21:15:10-08:00

Previously, we examined probability distributions for variables that are categorical (e.g. coin toss or blood type) and quantitative (e.g. eggs in a Boreal owl nest).

If the variable is quantitative, then we can describe the probability distribution using shape, center and spread. In the next example we will see how a comparison of probability distributions can help detect tax fraud.

EXAMPLE Using probability distributions to detect tax fraud

In this example we will compare probability distributions for legitimate tax records and for fraudulent records using shape, center (mean) and spread (standard deviation). We will also use empirical probability distributions (from data) and a theoretical probability distribution (from a mathematical formula.)

The variable is the first digit in the reported income on tax return. The variable values are the digits 1 to 9. For example, suppose the income reported on a tax return is $20,712. The first digit is 2. Tax returns with reported income of $4,320 or $43,000 or $492,565 all have a first digit of 4.

When we examine a very large number of tax returns, we can calculate the relative frequency of the first digits to determine an empirical probability distribution. A Professor of Accounting, Mark Nigrini used data from 169,662 IRS files to calculate relative frequencies (aka empirical probabilities) in the table below.

Distribution of first digits from 169,662 federal tax returns

First digit from legitimate tax return | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
Relative frequency of 1st digit (legit tax return) | 0.305 | 0.178 | 0.126 | 0.096 | 0.078 | 0.066 | 0.056 | 0.050 | 0.045

Let’s check our understanding of the probability distribution. For example, if we randomly select a tax return, there is a 30.5% chance that the reported income has a first digit of 1. A reported income beginning with the digit 9 is much less likely with a probability of 4.5%.

Surprisingly, the empirical probability distribution from Nigrini’s data is almost identical to a theoretical probability distribution generated by a mathematical formula called Benford’s law. We can see this in the table below.

Distribution of first digits from 169,662 federal tax returns

First digit from legitimate tax return | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
Relative frequency of 1st digit (legit tax return) | 0.305 | 0.178 | 0.126 | 0.096 | 0.078 | 0.066 | 0.056 | 0.050 | 0.045
Probability predicted by Benford's law (theoretical) | 0.301 | 0.176 | 0.125 | 0.097 | 0.079 | 0.067 | 0.058 | 0.051 | 0.046

(Here is the formula for Benford’s Law if you are interested P(d) = log(1+1/d). Don’t worry if you are not interested.)

Below is a histogram of the theoretical probability distribution for first digits based on Benford’s law. We can see that the probabilities are right skewed. We used technology to calculate a mean of the first digits and a standard deviation. The mean is approximately 3.4 (marked with the blue line), with a standard deviation of about 2.5. The lines on the left and right indicate one standard deviation below and one standard deviation above the mean.

OK, now let’s look at fraudulent tax returns. Will fraudulent tax returns display a similar probability distribution?

By comparison, below is the probability distribution for first digits in fraudulent tax records from a study of fraudulent cash disbursement and payroll expenditures conducted in 1995 by the district attorney’s office in Kings County, New York. For fraudulent data, the probability distribution is more symmetric. The mean is approximately 5.2, with a standard deviation of about 0.9.

Obviously, the relative frequencies of first digits from the fraudulent data do not follow Benford’s law. This probability distribution has a very different shape, center (mean), and spread (standard deviation.) Compared to legitimate tax records, in fraudulent data, we are much more likely to see numbers with a first digit of 5 and much less likely to see numbers with a first digit of 1, 2, or 3.

Mark this page as done, then click the "Next" or > button to continue.

Content by the Open Learning Initiative and licensed under CC BY.