It is far from obvious what would qualify as a “Supreme Court cases decided based on statistical evidence”, actually, it is far from obvious what constitutes “statistical evidence”. At the minimum, I expect that it is evidence which at least counts numbers of examples of something, and uses that number to reach a legal conclusion. For example “Of the two candidates, Smith received 10,000 votes and Jones received 10,001 votes. The wording of the statute declares to be the winner that candidate who receives ‘more than any other candidate’, therefore we hold that Jones won”. People often supplement raw counts with some measure of ‘significance’, which can be a significant (sorry about that) factor in discrimination cases, based on disparate impact theories; and also in evaluating forensic evidence (thus DNA evidence was poorly treated until the lawyers figured how to sell it).
Any appeal to statistical hypothesis testing is highly unlikely to ever arise at the level of SCOTUS. That is because (generally-speaking, perhaps there are obscure uses baked into law) the law and the legislature do not mandate particular alphas for acceptance of a statistical hypothesis. Instead, admissibility of testimony is (federally, and for half of the states) determined by the Daubert standard, encoded as FRE 702 which requires that “the testimony is based on sufficient facts or data; the testimony is the product of reliable principles and methods; and the expert has reliably applied the principles and methods to the facts of the case”, implicitly subsuming some domain-specific standards of hypothesis testing.
To the extent that DNA evidence is admissible in court and the expert can make a compelling, unrebutted argument for identity (“1 chance in 70 billion” as opposed to “49 chances in 100), statistical evidence is routinely accepted (or excluded as inconclusive). SCOTUS does not commonly have original jurisdiction so it is very unlikely that they would have ever been presented with trial-court level disputes involving words like “Bonferroni” or “Pearson's chi-squared test”.
This article looks at the role of statistics in disparate impact cases in the US. The ruling in Griggs v. Duke Power
does not directly appeal to any statistical reasoning, it indicates simply that an pair of intelligence tests were used as a substitute for a requirement of a high school education, but the tests were not intended “to measure the ability to learn to perform a particular job or category of jobs”. A claim rejected by the lower court was that “these two requirements operated to render ineligible a markedly disproportionate number of Negroes, they were unlawful under Title VII unless shown to be job-related”.
The dicta of Griggs provides numerous numeric statements such as
“while 34% of white males had completed high school, only 12% of Negro males had done so”, “use of a battery of tests… resulted in 58% of whites passing the tests, as compared with only 6% of the blacks”, “the percentage of white employees who were promoted but who were not high school graduates was nearly identical to the percentage of nongraduates in the entire white workforce”.
A striking mention of statistical concepts cited in Griggs is found in CFR § 1607, 35 Fed.Reg. 12333 (Aug. 1, 1970) which is a requirement that employers using tests have available “data demonstrating that the test is predictive of or significantly correlated with important elements of work behavior which comprise or are relevant to the job or jobs for which candidates are being evaluated”. SCOTUS is absolutely silent on the question of actual numbers in this case, that is taken to be a matter of fact determined by the lower court, and SCOTUS’s only job is to decide if employers can be required to show that a particular test is job-related. There is no discussion of what level of asymmetry in numbers is proof of discrimination.
Another relevant SCOTUS case is Wards Cove Packing v. Antonio
which might be seen as a case of SCOTUS being “unfriendly” to statistics, but the ruling simply holds that proving statistical significant between variables is insufficient. In fact in the holding SCOTUS identifies the Court of Appeals error as being
a comparison of the percentage of cannery workers who are nonwhite and
the percentage of noncannery workers who are nonwhite makes out a
prima facie disparate-impact case. Rather, the proper comparison is
generally between the racial composition of the at-issue jobs and the
racial composition of the qualified population in the relevant labor
In the above statistical article, other cases are cited where the courts engaged in magnitude inquiries of practical significance. 29 CFR 1607.4 states a “rule” known as the ‘four-fifths” rule that
A selection rate for any race, sex, or ethnic group which is less than
four-fifths (4/5) (or eighty percent) of the rate for the group with
the highest rate will generally be regarded by the Federal enforcement
agencies as evidence of adverse impact, while a greater than
four-fifths rate will generally not be regarded by Federal enforcement
agencies as evidence of adverse impact
however the Supreme Court in Watson v. Fort Worth Bank & Tr., 487 U.S. 977 finds that this is no more that “a rule of thumb”. Ricci v. Destefano finds that “[A] prima facie case of disparate-impact liability [is] essentially, a threshold showing of a significant statistical disparity . . . and nothing more . . . .”, and a prima facie demonstration does not require a disparity of any particular magnitude.
In short, SCOTUS cases frequently have a “statistical” underpinning, any time a scientific question arises. By the nature of expert testimony and the Daubert standard, the court does not rule on what constitutes a technically-correct statistical test, that is a matter for the trier of fact to decide. Disparate impact cases provide the best opportunity for higher courts to be confronted with statistical arguments. If some law were passed that specifically requires a particular p-value on some body of numbers and demographics, the court would have to discern the legislative intent of such a law, and not the majority opinion of some body of scientists as to what is “statistically correct”.