Significant Numbers: A Survey of the “Statistical Significance” in Securities Fraud Cases

Introduction to Statistical Significance

In securities fraud litigation, fierce fights, at the most critical stages of an action, are sometimes fought over a relatively esoteric (and often misunderstood) concept: statistical significance. At its most basic level, plaintiffs in securities fraud cases often use statistical significance to show that it is highly unlikely that a stock price movement happened by chance. If a stock price movement is unlikely to simply be a random change, then the stock price must have moved for a reason. This is another way of saying it must have been caused by something. Parties in civil securities fraud litigation care about randomness because claims may rise and fall on the question of causation, specifically whether an alleged misrepresentation caused economic losses in the form of stock price changes.

Of course, scholars and investors have directed a keen eye to the causes of stock price movements for centuries, but it was not until the middle of the twentieth century that researchers really began to understand that some of the movements in stock prices are, in fact, random—that, all things being equal, we should expect stock prices to move by chance even without a specific causal event. This reality—the “random walk” of stock prices—presents a critical question for securities fraud litigators, who often seek to prove or challenge allegations that the company’s stock prices moved in response to information or misinformation.

Applying the Statistical Significance Threshold in Securities Fraud Cases

Statisticians, economists, litigators, and courts thus need a way to estimate the likelihood that a specific stock price movement was caused by some external event as opposed to the expected random movements in prices. For decades, therefore, securities fraud litigators have proposed that courts apply a threshold for statistical significance: If a stock price change is statistically significant at the 95 percent confidence level (suggesting a 5 percent likelihood that the movement was random), then forensic economists, litigators, and the courts may take comfort that the stock price movement at issue was almost certainly caused by some event, usually information about the company. Given the clarity of this bright line, and the sense that it is a generally applied scientific measure, parties and courts alike often cite this 95 percent threshold as dispositive of whether stock prices moved in response to information about the company at issue in the case.

Whether a court accepts a strict 95 percent cutoff may have important implications in securities fraud cases at two critical stages. The first is at class certification, when plaintiffs seek to invoke (and defendants to rebut) the fraud-on-the-market presumption to establish that reliance can be decided on a class-wide basis. Plaintiffs often seek to show that investors relied on the stock price, which in turn incorporated the allegedly misrepresented information. Defendants may seek to rebut that showing by proving that the stock price did not tend to move based on company-specific information. In making these arguments, plaintiffs may try to show that the stock price moved in a statistically significant manner when company-specific news was released, and defendants may try to show that any price change after the news was not statistically significant and thus it was unlikely that the price moved because of the news. Second, securities fraud plaintiffs often attempt to prove—and a defendant to oppose—that the specific stock price changes at issue in the case were caused by an alleged misrepresentation or company-specific news that corrected or revealed that misrepresentation. To prove this, a plaintiff might introduce evidence that the stock price change at issue was statistically significant, and the defendant might try to introduce evidence that it was not.

During the same decades over which securities fraud parties have put statistical significance at the center of these debates in litigation, statisticians, scientists, and researchers have increasingly questioned the application of the strict 95 percent cutoff for statistical significance in scientific research. In 2016, the American Statistical Association published an official statement, citing other sources, rejecting the rigid cutoff. See Ronald L. Wasserstein & Nicole A. Lazar, “The ASA Statement on p-Values: Context, Process, and Purpose,” 70Am. Statistician, no. 2, 129–33 (2016).

Recently, some parties have started to ask courts to consider questions similar to those that statisticians and researchers have been grappling with: whether to apply the strict 95 percent threshold for statistical significance, and how to assess evidence that a stock price change is not statistically significant at that or lower thresholds. To offer a general understanding of the latest discussion on this issue in securities fraud litigation, this article surveys recent, relevant decisions in civil securities fraud litigation.

A Survey of the Cases

The survey below employs a few set parameters. First, the survey includes decisions in cases brought under section 10(b) of the Exchange Act and Securities and Exchange Commission (SEC) Rule 10b-5, in which the court addressed statistical significance in assessing either the fraud-on-the-market presumption or loss causation. Second, the survey includes decisions published on WestLaw from January 1, 2023, through February 7, 2025. The results are intended to be comprehensive, but it is possible some decisions were missed.

In total, the survey identified 46 decisions. Of those, 34 decisions referenced the parties’ arguments on whether a stock price movement was statistically significant, but the court did not address the appropriate threshold for significance or whether statistical significance is dispositive of an issue in the case. In each of these decisions, the parties and court expressly or impliedly applied the 95 percent threshold for statistical significance, without party dispute or argument.

Of the remaining 12 decisions, one stated in dicta that a lack of statistical significance necessarily means that price movements were not caused by an alleged misrepresentation or corrective disclosure. See Ark. Teacher Ret. Sys. v. Goldman Sachs Grp., Inc., 77 F. 4th 74, 86 n.5 (2d Cir. 2023) (“If the stock price movement is indistinguishable from random price fluctuations, it cannot be attributed to company-specific information announced on the event date.”). One decision used the term “statistical significance” to describe the absolute size of a stock price change (rather than using the term to correctly refer to a measure using regression), mistakenly concluding that if a stock price drops more than 10 percent in total, the drop is necessarily “statistically significant” and lower absolute drops are not. See Ramos v. Comerica Inc., 2024 WL 2104398, at *4 (C.D. Cal. Apr. 12, 2024). The analysis in the Ramos case appears to be an outlier among recent cases. One decision noted the presence of a disagreement among courts about the import of the 95 percent threshold but did not reach a holding on the issue. See Ind. Pub. Ret. Sys. v. AAC Holdings, Inc., 2023 WL 2592134, at *14 n.18 (M.D. Tenn. Feb. 24, 2023) (“[T]he Court notes that there is disagreement among the courts as to whether non-statistically significant stock price movement can evidence lack of price impact.”)

Finally, nine decisions explicitly find that statistical significance at the 95 percent threshold is not a required showing to demonstrate that a stock price movement was caused by company- or case-relevant information. One decision held that statistical significance may be established at a lower threshold, suggesting the court still considered some showing of statistical significance relevant to the court’s analysis. See Sjunde AP-Fonden v. Goldman Sachs Grp., Inc., 2024 WL 1497110, at *19 (S.D.N.Y. Apr. 5, 2024) (“There is no bright line legal or statistical rule that a result below the 95% confidence level disproves price impact—particularly at the class certification stage.”).

Notably, all nine of the decisions held that a lack of statistical significance is not alone dispositive, reasoning, for example, that “the absence of a statistically significant price adjustment does not show that the stock price was unaffected by the misrepresentation.” Del. Cnty. Emps. Ret. Sys. v. Cabot Oil & Gas Corp., 2023 WL 6300569, at *8 (S.D. Tex. Sept. 27, 2023); see also In re Cassava Scis., Inc. Sec. Litig., 2024 WL 4824243, at *14 (W.D. Tex. Nov. 15, 2024) (same); Gelt Trading Ltd. v. Co-Diagnostics, Inc., 2023 WL 5334623, at *5 (D. Utah Aug. 18, 2023) (finding “compelling” an expert report that stated “[s]tatistical significance generally proves price impact, but lack of statistical significance does not prove there was no price impact”); Spence v. Am. Airlines, Inc., 2025 WL 225127 (N.D. Tex. Jan. 10, 2025) (denying motion to exclude expert testimony that used an event study that did not apply the strict 95 percent / 5 percent statistical significance cutoff); Boston Ret. Sys. v. Alexion Pharm., Inc., 2023 WL 2932485, at *11 (D. Conn. Apr. 13, 2023) (“Although defendants may attempt to show lack of price impact by a statistically-based event study, courts have cautioned that ‘the failure of an event study to disprove the null hypothesis with respect to an event does not prove that the event had no impact on the stock price.’”); Hall v. Johnson & Johnson, 2023 WL 9017023, at *13 (D.N.J. Dec. 29, 2023) (“Here, even though the price decline was not at the traditionally ‘statistically significant’ standard, when considering the price decline in conjunction with the information published in the Release and that Defendants have not identified another explanation for the price decline, the Court concludes that Defendants have not rebutted the [fraud-on-the-market] presumption.”); Crews v. Rivian Auto., Inc., 2024 WL 3447988, at *15 (C.D. Cal. July 17, 2024) (“[A]s other courts have noted, while a statistically significant price drop after a corrective disclosure is evidence of price impact, the converse is not necessarily true.”); Sjunde, 2024 WL 1497110, at *19 (“[T]he absence of a statistically significant price drop does not disprove price impact.”); In re Apache Corp. Sec. Litig., 2024 WL 532315, at *11 (S.D. Tex. Feb. 9, 2024) (finding lack of statistical significance “a relevant fact to consider alongside Defendants’ other evidence concerning price impact” but “not dispositive of price impact”).

Conclusion

As courts continue to consider evidence on statistical significance in the context of the fraud-on-the-market presumption and loss causation in securities fraud cases, litigants and courts can expect to see parties continuing to raise disputes over the necessary threshold and impact of the statistical significance evidence.