# Examples of when statistical distributions like Binomial or Normal distribution was critical in a law/policy decision, in a court case or otherwise

This was closed as off-topic on math.se, and it was suggested I post this here, so here goes.

Firstly, I am aware that this thread exists, and I'll definitely be ordering a copy of the book, "Math on Trial...", in the top answer. My question, while related, is more specific.

I imagine there are many, many examples of the normal distribution being useful in decision-making of processes, such as whether or not to modify a bolt-making factory machine depending on whether it passes quality control standards, and this is found out by taking a sample of bolts and approximating the mean and variance of the length of the bolt, and seeing if the mean is off or if the variance is too large.

Another example could be in medicine: if the gaps between heart beats follows some sort of modified normal distribution; that could tell doctors if someone's heart-rate variability is high enough so that medical intervention is required.

But my question here today is about how statistical distributions - Binomial, Normal, Poisson or otherwise was critical in the decision of the ruling of a law, be it in a court case or otherwise.

Something I have in mind is if an individual, or a gambling site/entity, was accused of cheating and a hypothesis test with the normal distribution was a key factor in a trial. If it was the individual accused of cheating, then maybe the evidence was that it was that their winnings were statistically too unlikely to be coincidental. On the other hand, if it was the gambling company accused of being rigged, then maybe statistical evidence was found that that the probabilities in whatever game was played, was found to not be equal to the probabilities if the game were truly fair.

But also, it doesn't have to be a court case: it could be how calculations from statistical distributions affected government policy (law), e.g. decision making when it came to the placement of speed cameras on a road or the height of a building or whatever decision that policy-makers have to make.

Although the applications are not universally accepted, statistical distributions have found use in employment discrimination cases. The gist of the usage is that in some cases, US courts have found that statistical evidence can count as prima facie evidence of discrimination, i.e. the statistical evidence is obvious or stark enough that it warrants further examination.

Quoting from this paper:

Ikuta, Ben. "Why Binomial Distributions Do Not Work as Proof of Employment Discrimination." Hastings LJ 59 (2007): 1235.

A. INTERNATIONAL BROTHERHOOD OF TEAMSTERS V. UNITED STATES

In Teamsters, the central claim was that the employer had engaged in a pattern or practice of discriminating against minorities in "line driving" positions. The company had three main employee positions: "line drivers," "servicemen," and "city operations." Although none of these positions required a formal education, "line drivers" were paid a higher salary than the other positions. In analyzing the statistical data, the Court observed that of 571 minority employees, less than 3% held "line driving" positions and only 39% of the non minority employees held the two lower paying positions. Rejecting the defendant's argument that "statistics can never in and of themselves prove the existence of a pattern or practice of discrimination, or even establish a prima facie case," the Court held that discrimination should be inferred "where it reached proportions comparable to those in this case."

However, the most widely followed section is the analysis in footnote seventeen, which compares the ratio of minorities to non minorities in the "line driver" positions to the ratio of minorities to non minorities in the general population of the cities where each of the terminals were located. The Court reasoned that if 17.88% of Los Angeles is African-American, then the percentage of African-Americans in a low-education job like "line driving" should be comparable to 17.88%. Although these comparisons were extremely significant in illustrating what the appropriate statistical comparisons should be, the Court did not address the degree of statistical discrepancy needed to find discrimination because the statistical evidence was so one-sided.

B. HAZELWOOD SCHOOL DISTRICT V. UNITED STATES

The problem left open in Teamsters concerning the amount of statistical discrepancy needed to raise an inference of discrimination was"solved" in Hazelwood. Hazelwood involved the hiring of minorities for teaching positions in a public school. Similarly to Teamsters, the Court in Hazelwood used the percentage of minorities in teaching positions in the surrounding areas as a basis of comparison. The Court recognized that the percentage of minority teachers in the surrounding areas could either be 5.7% or 15.4%, depending on whether the court thought that it was appropriate to include one surrounding school district that had chosen to pursue a goal of hiring 50% minority teachers. Observing that the school at issue in the case had only a 3.7% hiring rate for minority teachers, the Court relied on Castaneda when it held that using "statistical methodology.., involving the calculation of the standard deviation as a measure of predicted fluctuations [shows] the difference between using 15.4% and 5.7% as the area-wide figure would be significant. However, Hazelwood did not fully describe the method or the pitfalls of using standard deviation based on binomial distributions.

More broadly there is a long history of using statistics in court cases.There is a long history of using statistics in court cases (not necessarily normal or binomial distributions). Shelby County v. Holder is a more recent example.

Two other articles that might be of interest:

• Rubinfeld, Daniel L. "Econometrics in the Courtroom." Columbia Law Review 85, no. 5 (1985): 1048-1097.

• Meier, Paul, Jerome Sacks, and Sandy L. Zabell. "What happened in Hazelwood: Statistics, employment discrimination, and the 80% rule." Law & Social Inquiry 9, no. 1 (1984): 139-186.

I'm sure more recent sources are available too.