LIVINGSTON, Circuit Judge:

Plaintiffs-appellees, eleven Asian Americans currently or formerly employed as police officers by the Port Authority of New York and New Jersey ("Port Authority"), sued the Port Authority under Title VII of the Civil Rights Act of 1964, 42 U.S.C. § 2000e et seq., alleging that they were passed over for promotions because of their race. The plaintiffs asserted three theories of liability for discrimination: individual disparate treatment, pattern-or-practice disparate treatment, and disparate impact. After a nine-day trial, a unanimous jury found the Port Authority liable for discrimination against seven of the plaintiffs under all three theories and awarded back pay and compensatory damages to each of those seven plaintiffs. The district court (Miriam Goldman Cedarbaum, Judge) also granted equitable relief to certain of the prevailing plaintiffs in the form of retroactive promotions, seniority benefits, and salary and pension adjustments corresponding with the hypothetical promotion dates that the jury apparently selected as a basis for calculating these plaintiffs' back pay awards.

On appeal, the Port Authority argues: (1) that evidence predating the onset of the statute of limitations should not have been admitted; (2) that the evidence was insufficient to support the jury's verdict with respect to each of the plaintiffs' theories; and (3) that the damages and equitable relief were premised on time-barred claims and were otherwise excessive. With regard to the plaintiffs' individual disparate treatment allegations, we hold that the district court properly admitted background evidence predating the onset of the limitations period and that there was sufficient evidence for a reasonable juror to conclude that the Port Authority discriminated against the seven prevailing plaintiffs within the limitations period. The district court erred, however, in: (1) submitting the pattern-or-practice disparate treatment theory to the jury in this private, nonclass action; and (2) concluding that the "continuing violation" doctrine applied to the plaintiffs' disparate impact theory so that the jury could award back pay and compensatory damages for harms predating the onset of the statute of limitations. We therefore vacate the back pay for four of the plaintiffs, whose awards correspond with hypothetical promotion dates beyond the limitations period, as well as the injunctive relief for three of the same plaintiffs, and we also vacate the award of compensatory damages for all seven prevailing plaintiffs. We remand for a new trial on damages as to all seven prevailing plaintiffs and for reconsideration of equitable relief to the extent such relief was premised on failures to promote occurring outside the limitations period.

The four plaintiffs who did not prevail at trial cross-appeal, arguing that the district court erred by excluding expert testimony from an industrial psychologist. One of these plaintiffs, cross-appellant Howard Chin, further argues that the district court erred in denying the plaintiffs' motion for sanctions in the form of an adverse inference instruction due to the Port Authority's destruction of promotion records. Finding no abuse of discretion in the district court's determinations as to these matters, we affirm.

BACKGROUND

The Port Authority is a bi-state transportation agency whose facilities are policed by its Public Safety Department. The eleven plaintiffs-appellees in this case are Asian Americans who were employed by that department as police officers. Christian Eng was hired in 1977, David Lim in 1980, Richard Wong in 1983, Milton Fong in 1985, Howard Chin and Alan Lew in 1987, Stanley Chin in 1988, George Martinez and Nicholas Yum in 1993, and Michael Chung and Sanrit Booncome in 1999. All of the plaintiffs were members of the Port Authority Police Asian Jade Society of New York & New Jersey Inc. ("Asian Jade Society"), a nonprofit organization comprised of Port Authority police officers of Asian or Pacific Islander origin, whose stated goal is to "promot[e] understanding, friendship and cooperation among members of the Port Authority police department."

I. The Port Authority Police Department's Promotion Process

During the period relevant to this case, entry-level police officers in the Port Authority's police department could be promoted to the rank of Sergeant, the first level in a hierarchy of supervisory positions (followed by Lieutenant, Captain, Inspector, Chief, and finally Superintendent of Police). To become eligible for promotion to Sergeant, a police officer was required (among other requirements) to pass an examination, which would place him on an "eligibility list" for a period of time. When such a list expired, the officer would have to pass the examination again to be placed on the new list. Three lists were in effect during the period relevant to this case: the 1996 List, the 1999 List, and the 2002 List.1 These lists were "horizontal," which meant that the lists did not rank the officers, but merely established eligibility for promotion.

Each Port Authority facility's commanding officer (generally a Captain) was periodically asked to recommend eligible officers for promotion, at their discretion. The Port Authority did not dictate any criteria for recommendation. Moreover, commanding officers were free to make recommendation decisions themselves, solicit input from the police officers' direct supervisors (generally Sergeants and Lieutenants), or delegate the responsibility entirely to the direct supervisors. A promotion folder was prepared for each recommended officer, which included a performance evaluation by a supervisor, a photograph of the officer, and his record of absences, commendations, awards, and disciplinary history.

Officers recommended in this way were typically considered by the Chiefs' Board, in which the Chiefs would collectively decide which officers to recommend to the Superintendent. The Chiefs' Board did not operate under any written guidelines, and from 1996 through September 2001, took no minutes or notes. Each Chief would vote regarding each recommended officer, and any officer who received a majority of votes from the Chiefs' Board would then be recommended to the Superintendent. This step in the process was not always necessary to promotion, however; for example, Acting Superintendent Joseph Morris did not use the Chiefs' Board at all during his tenure from September 2001 through April 2002.

The ultimate decision to promote an officer to Sergeant belonged solely to the Superintendent. In fact, the Superintendent occasionally promoted officers whom the Chiefs' Board had declined to recommend ahead of those recommended by the Board.

As of January 31, 2001, no Asian American had ever been promoted to Sergeant.

II. Procedural History

On January 31, 2001, the Asian Jade Society filed a charge of discrimination with the Equal Employment Opportunity Commission ("EEOC") on behalf of its members, claiming that the Port Authority had denied Asian American police officers promotions because of their race. On August 29, 2003, the EEOC determined that there was reasonable cause to believe the Port Authority had violated Title VII, and on January 25, 2005, the Department of Justice issued a right-to-sue letter to the Asian Jade Society.2 The eleven plaintiffs in this case filed suit on April 15, 2005, alleging that the Port Authority had discriminated against Asian Americans in making promotions to Sergeant.

During discovery, the plaintiffs learned that the Port Authority had not implemented a document retention policy and that, as a result, at least thirty-two promotion folders used to make promotion decisions between August 1999 and August 2002 had been destroyed. The plaintiffs moved for sanctions, seeking an adverse inference against the Port Authority for spoliation. The district court denied the motion, reasoning that the plaintiffs had ample alternative evidence regarding the relative qualifications of the plaintiffs and that the Port Authority's destruction of the documents was "negligent, but not grossly so." Port Auth. Police Asian Jade Soc'y of N.Y. & N.J. Inc. v. Port Auth. of N.Y. & N.J. (Port Auth. I), 601 F.Supp.2d 566, 570 (S.D.N.Y.2009).

On the eve of trial, the district court granted the Port Authority's motion to exclude testimony from one of the plaintiffs' expert witnesses: Dr. Kathleen Lundquist, an industrial psychologist who specializes in analyzing the reliability and validity of employee-selection procedures. Dr. Lundquist had prepared a report opining on the effectiveness of the Port Authority's promotion process, on whether it included safeguards to prevent bias and discrimination, and on the comparative qualifications of the plaintiffs relative to the qualifications of the officers who had been promoted. Citing Rule 702 of the Federal Rules of Evidence,3 the district court concluded that Dr. Lundquist's testimony "would not assist the trier of fact" and was therefore excluded.

The nine-day trial began on March 11, 2009, and included testimony from twenty-two fact witnesses and four expert witnesses. All eleven of the plaintiffs testified regarding their personal backgrounds, education, experiences as police officers, attendance and disciplinary records, awards and commendations, and performance evaluations. Six Chiefs, one former Superintendent, the Superintendent at the time of trial, and three other Port Authority managers testified regarding the Port Authority's promotion procedure. Each side also presented a statistical expert and a damages expert.

Most relevant to this appeal, the plaintiffs' statistical expert, Dr. Christopher Cavanagh, presented two analyses that, in his view, demonstrated a high probability that Asian Americans had been discriminated against in the Port Authority's promotion process. In his first study, Cavanagh compared the percentage of white police officers who held a supervisory position (out of all white police officers) with the percentage of Asian Americans who held a supervisory position (out of all Asian American police officers) from 1996 through 2004. For each year, he used a Fisher Exact Test to calculate the likelihood that the difference between Asian American and white representation at the supervisory level (as compared to the representation of these groups at the nonsupervisory level) was due to chance.4 From 1996 through 2000, the likelihood that the disparities were due to chance was about two percent or less; from 2001 through 2004, the likelihoods that the disparities were due to chance were between about five and eleven percent.

Cavanagh's second analysis compared the promotion rate for whites who were on the eligible lists to the promotion rate for Asian Americans who were on the eligible lists over the period from August 1996 through January 31, 2001 (the date on which the EEOC charge was filed). Of the 259 white officers on the lists over this period, 36 were promoted; of the 12 Asian Americans on the lists, none were promoted. Using the Fisher Exact Test, Cavanagh calculated that the likelihood this disparity would occur due to chance was about thirteen percent.

The district court instructed the jury regarding three theories of discrimination: (1) disparate impact; (2) pattern-or-practice disparate treatment; and (3) individual disparate treatment. After two-and-a-half days of deliberation, the jury returned a unanimous verdict, finding that seven of the eleven plaintiffs — Christian Eng, Milton Fong, Alan Lew, Stanley Chin, Nicholas Yum, George Martinez, and David Lim — had proven all three of their theories of liability, and awarding more than $1.6 million in total to those seven plaintiffs. The back pay awards corresponded precisely to certain hypothetical promotion dates suggested by the plaintiffs' damages expert.5

On the plaintiffs' motion, the district court also granted the seven prevailing plaintiffs equitable relief, including salary adjustments for pension purposes for Milton Fong, Stanley Chin, Alan Lew, George Martinez, Nicholas Yum, and David Lim, and retroactive promotions for Alan Lew, George Martinez, and Nicholas Yum. The hypothetical dates the district court used were October 31, 1999, for Fong, Chin, and Lew, and September 30, 2002, for Martinez, Yum, and Lim — corresponding with the hypothetical dates the jury apparently used as a basis for computing back pay. The court also ordered the Port Authority to take certain specific actions to prevent future violations.

The Port Authority filed a motion pursuant to Rules 50 and 59 of the Federal Rules of Civil Procedure to set aside the jury's verdict or, alternatively, for a new trial and for remittitur. The Port Authority argued that: (1) the district court improperly admitted evidence pertaining to events prior to the onset of the statute of limitations period; (2) the jury was improperly instructed to consider events outside the limitations period for purposes of establishing liability; (3) there was insufficient evidence to find the Port Authority liable under any of the plaintiffs' three theories; (4) the jury instructions were erroneous and confusing with respect to the statute of limitations; and (5) the jury's damages included time-barred claims and were otherwise excessive.

The district court denied the Port Authority's motion in its entirety. See Port Auth. Police Asian Jade Soc'y of N.Y. & N.J. Inc. v. Port Auth. of N.Y. & N.J. (Port Auth. II), 681 F.Supp.2d 456 (S.D.N.Y.2010). As pertinent to this appeal, the district court first held that background evidence from beyond the statute of limitations is admissible in support of a timely claim. See id. at 462. Next, the court concluded that the plaintiffs' individual disparate treatment claims were premised on "discrete acts" and thus that the Port Authority could be liable only for those acts within the statute of limitations. See id at 463. The court concluded that the plaintiffs' disparate impact and pattern-or-practice disparate treatment theories of liability, however, were premised on the existence of an "ongoing discriminatory policy," and thus were subject to the "continuing violations" doctrine, so that the plaintiffs could recover for untimely discrete acts so long as they were the product of a discriminatory policy that continued into the statute-of-limitations period. See id. at 463-66. Third, the district court held that although Cavanagh's statistical evidence did not reach the conventional five-percent level of statistical significance, see Smith v. Xerox Corp., 196 F.3d 358, 366 (2d Cir.1999) (noting that statistical significance at the five-percent level is generally "sufficient to warrant an inference of discrimination"), the jury had before it other evidence of discrimination sufficient to find for the plaintiffs on each of the theories of liability. See Port Auth. II, 681 F.Supp.2d at 468-69. Finally, the district court declined to remit the jury's compensatory damages awards because other judges had upheld similar awards and because the awards did not "shock the judicial conscience." Id. at 470.

The Port Authority appeals, and argues before this Court that it is entitled to a new trial with respect to the seven prevailing plaintiffs because: (1) evidence predating the onset of the limitations period should not have been admitted; (2) the evidence at trial was insufficient to support the jury's verdict with respect to each of the plaintiffs' theories; and (3) the damages and equitable relief are premised on time-barred claims and are otherwise excessive.

The four plaintiffs who did not prevail at trial — Howard Chin, Richard Wong, Sanrit Booncome, and Michael Chung — cross-appeal, and argue here that they are entitled to a new trial because the district court erred by excluding Lundquist's testimony. Howard Chin further argues that he is entitled to a new trial because the district court improperly denied the plaintiffs an adverse inference instruction despite the Port Authority's destruction of promotion records.

DISCUSSION

The plaintiffs argue that they are entitled to damages for injuries that occurred before the onset of the statute of limitations period because the "continuing violations" doctrine applies to two of their three theories of liability — pattern-or-practice disparate treatment and disparate impact. We dispose of half of this argument at the outset of this opinion by holding that no such pattern-or-practice theory of liability is available to the private, non-class plaintiffs in this case. We next consider and affirm the district court's judgment with respect to the plaintiffs' two remaining theories of liability — individual disparate treatment and disparate impact — by holding that background evidence from outside the limitations period was admissible and that the evidence presented at trial was sufficient to sustain the jury's findings of liability on both theories. We then conclude, however, that the "continuing violations" doctrine does not apply to either theory in this case, and therefore vacate and remand for reconsideration of the damages and equitable relief granted by the district court to the prevailing plaintiffs whose awards correspond (or may correspond) to hypothetical promotion dates preceding the onset of the limitations period. Finally, we consider the plaintiffs' cross-appeal, and hold that the district court did not abuse its discretion by excluding Lundquist's testimony or by denying the plaintiffs an adverse inference instruction.

"A motion for a new trial should be granted when, in the opinion of the district court, `the jury has reached a seriously erroneous result or ... the verdict is a miscarriage of justice.'" Song v. Ives Labs., Inc., 957 F.2d 1041, 1047 (2d Cir. 1992) (quoting and altering Smith v. Lightning Bolt Prods., Inc., 861 F.2d 363, 370 (2d Cir.1988)). "The district court's denial of a Rule 59 motion for a new trial is reviewed for abuse of discretion." Manganiello v. City of New York, 612 F.3d 149, 165 (2d Cir.2010). "A district court has abused its discretion if it has (1) `based its ruling on an erroneous view of the law,' (2) made `a clearly erroneous assessment of the evidence,' or (3) `rendered a decision that cannot be located within the range of permissible decisions.'" Id. (quoting Sims v. Blot, 534 F.3d 117, 132 (2d Cir.2008)). We review the denial of a motion for judgment as a matter of law de novo. Lore v. City of Syracuse, 670 F.3d 127, 150 (2d Cir.2012). "[W]hether conducting review de novo or under a less sweeping standard, we must disregard all errors and defects.... if there is no likelihood that the error or defect affected the outcome of the case." Id. (internal quotation marks omitted).

As a prerequisite to filing suit under Title VII, a private plaintiff must first file a timely charge with the EEOC. See 42 U.S.C. § 2000e-5(e)(1), (f)(1). Both parties agree that in this case, the plaintiffs' charge was due "within one hundred and eighty days after the alleged unlawful employment practice occurred." 42 U.S.C. § 2000e-5(e)(1). Accordingly, because the EEOC charge in this case was filed on January 31, 2001, only an unlawful employment practice that "occurred" after August 2, 2000, may give rise to liability.6

I. The Pattern-or-Practice Method of Proof

As an initial matter, we address the question whether the method of proof described in International Brotherhood of Teamsters v. United States, 431 U.S. 324, 97 S.Ct. 1843, 52 L.Ed.2d 396 (1977), and known as the "Teamsters" or "pattern-or-practice" method, was available to the nonclass private plaintiffs in this case.7 We conclude that it was not and that the judgment as to pattern or practice must for this reason be reversed. We emphasize, however, that evidence that the Port Authority engaged in a pattern or practice of discrimination — in the ordinary sense of those words, rather than in the technical sense describing a theory of liability for discrimination — remains relevant in assessing whether the plaintiffs proved discrimination using the individual disparate treatment and disparate impact methods of proof.

The phrase "pattern or practice" appears only once in Title VII — in a section that authorizes the government to pursue injunctive relief against an employer "engaged in a pattern or practice of resistance to the full enjoyment of any of the rights secured by" the statute. 42 U.S.C. § 2000e-6. Notwithstanding the Supreme Court's recognition in Teamsters that this language "was not intended as a term of art, and the words reflect only their usual meaning," Teamsters, 431 U.S. at 336 n. 16, 97 S.Ct. 1843, the phrase is often used in a technical sense to refer either to this unique form of liability available in government actions under § 2000e-6, see, e.g., EEOC v. Shell Oil Co., 466 U.S. 54, 67-68 n. 19, 70, 80, 104 S.Ct. 1621, 80 L.Ed.2d 41 (1984), or to the burden-shifting framework set out in Teamsters and available both to the government in § 2000e-6 litigation and to class-action plaintiffs in private actions alleging discrimination, see, e.g., Wal-Mart Stores, Inc. v. Dukes, ___ U.S. ___, 131 S.Ct. 2541, 2552 n. 7, 180 L.Ed.2d 374 (2011).

We begin with § 2000e-6. The building blocks of liability pursuant to this provision — which provides for prospective injunctive relief where the government establishes that an employer is engaged in a "pattern or practice of resistance to the full enjoyment" of rights secured by Title VII — differ from those that provide the foundation for typical, private-party Title VII litigation. To establish an employer's liability for discrimination in violation of Title VII, a private plaintiff ordinarily must show that an employer took an adverse employment action against him or her because of his or her race, or on account of another protected ground. See McDonnell Douglas Corp. v. Green, 411 U.S. 792, 802, 93 S.Ct. 1817, 36 L.Ed.2d 668 (1973); Aulicino v. N.Y. City Dep't of Homeless Servs., 580 F.3d 73, 80 (2d Cir. 2009). In § 2000e-6 litigation, by contrast, the government need not demonstrate specific losses to specific individuals to establish that injunctive relief is appropriate. The government must "prove more than the mere occurrence of isolated or `accidental' or sporadic discriminatory acts": it must prove that unlawful discrimination "was the company's standard operating procedure." Teamsters, 431 U.S. at 336, 97 S.Ct. 1843. Once established, however, "a court's finding of a pattern or practice justifies an award of prospective relief" even absent proof of losses to specific individuals. Id. at 361, 97 S.Ct. 1843.

The parties here use the term "pattern or practice" to refer not to an element of a § 2000e-6 claim, but to the method of proof that the Supreme Court endorsed in Teamsters for the adjudication of such claims. This method of proof, however, originated in the class action context, in Franks v. Bowman Transportation Co., 424 U.S. 747, 96 S.Ct. 1251, 47 L.Ed.2d 444 (1976). The Supreme Court in Franks determined that once the private plaintiffs in the class action there "carried their burden of demonstrating the existence of a discriminatory hiring pattern and practice by the [employer] ..., the burden [was] upon [the employer] to prove that individuals who reappl[ied] were not in fact victims of previous hiring discrimination." Id. at 772, 96 S.Ct. 1251. The Court in Franks used the phrase "pattern and practice" to refer to the common question of fact (whether the employer had engaged in a practice of discriminatory hiring) to be litigated by class plaintiffs, and apparently viewed its holding as no more than an application of McDonnell Douglas' burden-shifting framework in the class-action context. See Franks, 424 U.S. at 773, 96 S.Ct. 1251 (citing McDonnell Douglas, 411 U.S. 792, 93 S.Ct. 1817).

The Teamsters Court thereafter determined that the Franks burden-shifting framework for certain class actions should also apply to government "pattern or practice" suits brought under § 2000e-6:

Teamsters, 431 U.S. at 360-62, 97 S.Ct. 1843 (internal citation and footnotes omitted). Since Teamsters, this burden-shifting framework has been known as the "Teamsters method of proof" or the "pattern-or-practice method." See, e.g., Celestine v. Petroleos de Venezuella SA, 266 F.3d 343, 355 (5th Cir.2001) ("A pattern or practice case is not a separate and free-standing cause of action ..., but is really merely another method by which disparate treatment can be shown." (internal quotation marks omitted)); Lowery v. Circuit City Stores, Inc., 158 F.3d 742, 760 (4th Cir.1998) ("The courts of appeals have ... permitted pattern or practice class action suits using the Teamsters method of proof."), vacated on other grounds, 527 U.S. 1031, 119 S.Ct. 2388, 144 L.Ed.2d 790 (1999).8 In sum, unlike in a typical individual disparate treatment suit, "a plaintiff's burden under the pattern-or-practice method requires the plaintiff to prove only the existence of a discriminatory policy rather than all elements of a prima facie case of discrimination" — but "under the pattern-or-practice method, only prospective relief [is] available, unless the plaintiffs offer[ ] additional proof." Semsroth v. City of Wichita, 304 Fed.Appx. 707, 716 (10th Cir.2008) (describing the reasoning in Lowery, 158 F.3d at 761).

Permitting private plaintiffs to use the pattern-or-practice method of proof outside the class action context would require us to extend this method beyond its current application. This we decline to do. Such an extension would allow nonclass private plaintiffs who have shown a pattern or practice of discrimination (but have not made out a disparate impact claim) to shift the burden to employers to prove that they did not discriminate against a particular individual. But this would conflict with the Supreme Court's oft-repeated holding in the context of disparate-treatment, private nonclass litigation that "[t]he ultimate burden of pursuading the trier of fact that the defendant intentionally discriminated against the plaintiff remains at all times with the plaintiff." Tex. Dep't of Cmty. Affairs v. Burdine, 450 U.S. 248, 253, 101 S.Ct. 1089, 67 L.Ed.2d 207 (1981). To be sure, proof that an employer engaged in a pattern or practice of discrimination may be of substantial help in demonstrating an employer's liability in the individual case. But such proof cannot relieve the plaintiff of the need to establish each element of his or her claim.

We note that the district court in this case did not instruct the jury that a finding of a pattern or practice of discrimination shifted the burden of persuasion. Rather, the verdict sheet instructed the jury that each individual plaintiff was required to prove by a preponderance of the evidence that he was discriminated against as part of the pattern or practice. This instruction only underscores, however, why there was no need for the jury to make a specific finding regarding a pattern or practice of discrimination in this private, nonclass suit, as opposed to determining directly whether each individual plaintiff had been intentionally discriminated against. Where, as here, there are only individual, nonclass disparate-treatment claims, a district court need not and should not instruct the jury that a common pattern of discrimination is an element of liability.

For these reasons, all of our sister circuits to consider the question have held that the pattern-or-practice method of proof is not available to private, nonclass plaintiffs. See Semsroth v. City of Wichita, 304 Fed.Appx. 707, 715 (10th Cir.2008); Davis v. Coca-Cola Bottling Co. Consol., 516 F.3d 955, 967-69 (11th Cir.2008); Bacon v. Honda of Am. Mfg., 370 F.3d 565, 575 (6th Cir.2004); Celestine v. Petroleos de Venezuella SA, 266 F.3d 343, 355-56 (5th Cir.2001); Gilty v. Vill. of Oak Park, 919 F.2d 1247, 1252 (7th Cir.1990); Lowery v. Circuit City Stores, Inc., 158 F.3d 742, 761 (4th Cir.1998), vacated on other grounds, 527 U.S. 1031, 119 S.Ct. 2388, 144 L.Ed.2d 790 (1999); see also Schuler v. PricewaterhouseCoopers, LLP, 739 F.Supp.2d 1, 6 n. 2 (D.D.C.2010) ("Courts in every other Circuit that has touched on this issue have indicated that an individual plaintiff cannot maintain a pattern and practice claim.") (collecting cases); 1 Lex Larson et al., Employment Discrimination § 8.01[3], at 8-13 (2d ed. 2011) ("[C]ourts have refused to permit individuals to use the pattern or practice proof structure for claims of individual discrimination...."). We have suggested as much, albeit in dicta. See Brown v. Coach Stores, Inc., 163 F.3d 706, 711 (2d Cir. 1998).

For the foregoing reasons, we now hold that the pattern-or-practice method of proof is not available to nonclass, private plaintiffs in cases such as the one before us. Evidence of an employer's general practice of discrimination may be highly relevant to an individual disparate treatment or to a disparate impact claim. Outside the class context, however, private plaintiffs may not invoke the Teamsters method of proof as an independent and distinct method of establishing liability. The district court erred in submitting this method of proof to the jury as a basis on which it could hold the Port Authority liable.

II. Admissibility and Sufficiency of Evidence

A. Admissibility of Evidence from Outside the Limitations Period

Turning to the plaintiffs' individual disparate treatment and disparate impact claims, the Port Authority argues that the district court improperly admitted evidence of events prior to August 2, 2000, for purposes of liability and damages. It is well established, however, that so long as at least "one alleged adverse employment action ... occurred within the applicable filing period[,] ... evidence of an earlier alleged retaliatory act may constitute relevant `background evidence in support of [that] timely claim.'" Jute v. Hamilton Sundstrand Corp., 420 F.3d 166, 176 (2d Cir.2005) (quoting and altering Nat'l R.R. Passenger Corp. v. Morgan, 536 U.S. 101, 113, 122 S.Ct. 2061, 153 L.Ed.2d 106 (2002)). Such background evidence "may be considered to assess liability on the timely alleged act." Id. at 177; see also Petrosino v. Bell Atl., 385 F.3d 210, 220 (2d Cir.2004) (applying this rule in the failure-to-promote context). In particular, we have noted that statistical studies may include data from outside the statute of limitations to prove timely discriminatory acts. See Rossini v. Ogilvy & Mather, Inc., 798 F.2d 590, 604 n. 5 (2d Cir.1986).9 Title VII's statute of limitations therefore did not prohibit admission of the plaintiffs' evidence of discrimination before August 2, 2000.

B. Sufficiency of the Evidence

The Port Authority next argues that the plaintiffs' evidence of individual disparate treatment and disparate impact was insufficient as a matter of law, and that the district court therefore abused its discretion in declining to set aside the verdict. "In reviewing the sufficiency of the evidence in support of a jury's verdict, we examine the evidence in the light most favorable to the party in whose favor the jury decided, drawing all reasonable inferences in the winning party's favor." Gronowski v. Spencer, 424 F.3d 285, 291 (2d Cir.2005). We will overturn the verdict here "only if there is `such a complete absence of evidence supporting the verdict that the jury's findings could only have been the result of sheer surmise and conjecture, or such an overwhelming amount of evidence in favor of the [Port Authority] that reasonable and fair minded men could not arrive at a verdict against [the Port Authority].'" Id. at 292 (quoting LeBlanc-Sternberg v. Fletcher, 67 F.3d 412, 429 (2d Cir.1995)) (some internal quotation marks omitted).

With respect to a Title VII individual disparate treatment claim, "[w]hether judgment as a matter of law is appropriate in any particular case will depend on a number of factors. Those include the strength of the plaintiff's prima facie case, the probative value of the proof that the employer's explanation is false, and any other evidence that supports the employer's case and that properly may be considered on a motion for judgment as a matter of law." Reeves v. Sanderson Plumbing Prods., Inc., 530 U.S. 133, 148-49, 120 S.Ct. 2097, 147 L.Ed.2d 105 (2000). A plaintiff establishes a prima facie case by showing "(1) that he belonged to a protected class; (2) that he was qualified for the position he held; (3) that he suffered an adverse employment action; and (4) that the adverse employment action occurred under circumstances giving rise to an inference of discriminatory intent." Feingold v. New York, 366 F.3d 138, 152 (2d Cir.2004). An employer may then rebut this prima facie case by offering a legitimate, nondiscriminatory business reason for its conduct. See id. A plaintiff ultimately prevails if he proves that the defendant's employment decision was based in whole or in part on intentional discrimination. See id.

To prevail under the disparate impact theory of liability, a plaintiff must show that the employer "uses a particular employment practice that causes a disparate impact on the basis of race, color, religion, sex, or national origin." 42 U.S.C. § 2000e-2(k)(1)(A)(i). This requires a plaintiff to (1) "identify a specific employment practice" or policy, Malave v. Potter, 320 F.3d 321, 326 (2d Cir.2003); "(2) demonstrate that a disparity exists; and (3) establish a causal relationship between the two." Robinson v. Metro-North Commuter R.R. Co., 267 F.3d 147, 160 (2d Cir.2001). "The statistics must reveal that the disparity is substantial or significant," and "must be of a kind and degree sufficient to reveal a causal relationship between the challenged practice and the disparity." Id. (internal quotation marks omitted). To rebut a plaintiff's statistics, a defendant may introduce evidence showing that "either no statistically significant disparity in fact exists or the challenged practice did not cause the disparity." Id. at 161.

If the trier of fact determines that the plaintiffs have established a disparate impact violation of Title VII, each person seeking individual relief such as back pay and compensatory damages "need only show that he or she suffered an adverse employment decision `and therefore was a potential victim of the proved discrimination.'" Id. at 159 (quoting Teamsters, 431 U.S. at 362, 97 S.Ct. 1843) (alteration omitted); see id. at 161-62. After such a showing, the employer bears the burden of persuading the trier of fact that its decision was made for lawful reasons; otherwise, the employee is entitled to individualized relief, which may include back pay, front pay, and compensatory damages for "emotional pain, suffering, inconvenience, mental anguish, loss of enjoyment of life, [or] other nonpecuniary losses." 42 U.S.C. § 1981a(b)(3); see Robinson, 267 F.3d at 159-60.

The Port Authority challenges three aspects of the plaintiffs' evidence. First, the Port Authority argues that the plaintiffs' statistical evidence was fatally flawed and that without it the plaintiffs lack sufficient evidence to prove a disparate impact. Second, the Port Authority contends that the plaintiffs did not show that the multiple-step promotion process was "not capable of separation for analysis," 42 U.S.C. § 2000e-2(k)(1)(B)(i), and therefore the plaintiffs were required but failed to identify the specific promotion practice that caused a disparate impact. Third, the Port Authority contends that the plaintiffs' anecdotal evidence of intentional discrimination consists of nothing more than personal affronts outside of the promotion context, and therefore that the plaintiffs' individual disparate treatment claims must fail for lack of evidence that any discrimination was intentional.

We disagree with all three of the Port Authority's arguments and hold that the plaintiffs introduced sufficient evidence to support the jury's verdict as to plaintiffs' disparate impact and individual disparate treatment claims.

1. Statistical Evidence

The Port Authority argues first that the statistical evidence presented by Dr. Cavanagh, the plaintiffs' expert witness, was insufficient to prove their disparate impact claim because Dr. Cavanagh's analyses impermissibly (1) relied on data predating the onset of the statute of limitations, (2) did not focus on the relevant pool of candidates eligible for promotion, and (3) failed to establish statistical significance. We address each of these contentions in turn.

First, the Port Authority is incorrect in asserting that Dr. Cavanagh's statistics were flawed because they relied in part on data predating the onset of the statute of limitations period. In Bazemore v. Friday, 478 U.S. 385, 106 S.Ct. 3000, 92 L.Ed.2d 315 (1986), the Supreme Court stated that evidence predating the 1972 enactment of Title VII was not only admissible but, "to the extent that proof is required to establish discrimination with respect to salary disparities created after 1972, evidence of pre-Act discrimination is quite probative." Id. at 402 n. 13, 106 S.Ct. 3000 (internal citation omitted). Moreover, we have made clear that a district court errs by "downgrading" statistical studies on the ground that they "relied in part on pre-statute of limitations data." Rossini, 798 F.2d at 604 n. 5.

The Port Authority next argues that Dr. Cavanagh's year-end demographic statistics were not sufficient to show a disparate impact because they simply compared the percentage of Asian Americans in supervisory positions with the percentage of Asian American officers, rather than looking to the relevant pool for promotion (i.e., the percentage of Asian Americans on the eligible lists). On this point, we agree.

As we have said, "plaintiffs must identify the correct population for analysis. In the typical disparate impact case the proper population for analysis is the applicant pool or the eligible labor pool." Smith v. Xerox Corp., 196 F.3d 358, 368 (2d Cir.1999) (emphasis added), overruled on other grounds by Meacham v. Knolls Atomic Power Lab., 461 F.3d 134, 140 (2d Cir.2006). "In the context of promotions, we have held that the appropriate comparison is customarily between the [racial] composition of candidates seeking to be promoted and the [racial] composition of those actually promoted," at least so long as the relevant data are available. Malave, 320 F.3d at 326 (emphasis added). The plaintiffs in this case did not allege that the eligibility test was discriminatory; rather, they alleged that discrimination entered the process at the discretionary stage after the eligible lists had already been drawn up. The relevant population for analysis, then, includes only those officers on the eligible lists. Dr. Cavanagh's year-end demographic analyses include all officers, and therefore do not meet the statistical standards prescribed by law.

Putting aside these demographic analyses, then, we are left with Dr. Cavanagh's statistical analysis comparing the percentage of Asian Americans on the eligibility lists with the percentage of Asian Americans promoted from 1996 to January 31, 2001 (the date that the EEOC complaint was filed). None of the 12 Asians on the eligible lists were promoted during this period, in contrast to 36 out of 259 whites; according to Dr. Cavanagh's calculations, this difference would occur due to chance "a bit under 13 percent" of the time. The Port Authority argues that a due-to-chance figure of 13 percent is not statistically significant because "it is generally accepted that statistical significance is at a 5% level or less."

It is true that "we have ... looked to whether the plaintiff can show a statistically significant disparity of two standard deviations," which (in a normal distribution) requires statistical significance at approximately the 5-percent level. Xerox, 196 F.3d at 365. However, we have also said that "[t]here is no minimum statistical threshold requiring a mandatory finding that a plaintiff has demonstrated a violation of Title VII. Courts should take a `case-by-case approach' in judging the significance or substantiality of disparities, one that considers not only statistics but also all the surrounding facts and circumstances." Waisome v. Port Auth. of N.Y. & N.J., 948 F.2d 1370, 1376 (2d Cir.1991) (quoting Ottaviani v. State Univ. of N.Y. at New Paltz, 875 F.2d 365, 372-73 (2d Cir.1991)); see also Watson v. Fort Worth Bank & Trust, 487 U.S. 977, 995 n. 3, 108 S.Ct. 2777, 101 L.Ed.2d 827 (1988) ("[W]e have not suggested that any particular number of `standard deviations' can determine whether a plaintiff has made out a prima facie case in the complex area of employment discrimination...."); accord Xerox, 196 F.3d at 366.

In many (perhaps most) cases, if there is a 13-percent likelihood that a disparity resulted from chance, it will not qualify as statistically significant. In this case, the plaintiffs offered other evidence that reasonable jurors could have relied upon to find that an 87-percent likelihood that the disparity was not due to chance qualified as significant. First, no Asian Americans were promoted during the relevant period; requiring a statistical showing of 95-percent confidence would make it mathematically impossible to rely upon statistics in a case like this one, in which the relevant population included so few Asian Americans. See Waisome, 948 F.2d at 1379 ("[T]he lack of statistical significance in the ultimate promotion reflects only the small sample size."). Second, as the Port Authority acknowledges, the plaintiffs presented a substantial amount of evidence that reasonable jurors could have relied on to conclude that the plaintiffs were more qualified than some of the white officers who were promoted, including comparing length of service, attendance records, and disciplinary histories. In the context of this case, it would not be unreasonable for a juror to find Dr. Cavanagh's statistics significant despite only being significant at the 13-percent level.

Finally, despite the Port Authority's argument to the contrary, Dr. Cavanagh's choice to limit his time frame to the period from 1996 through January 2001 (rather than, as defendant's expert did, extending the analysis into 2005) was not unreasonable. The plaintiffs' theory was that the Port Authority's failures to promote them caused a disparate impact through 2001, when the EEOC charge in this case was filed. Dr. Cavanagh's selected time frame was directly relevant to answering this question.

2. Specific Employment Practice

The Port Authority next argues that there was insufficient evidence to support the plaintiffs' disparate impact claim on the ground that plaintiffs either failed to identify a specific promotion practice resulting in a disparate impact on Asian Americans or failed to show that the Port Authority's promotion process could not be separated into component parts for analysis. According to the Port Authority, the promotion process involved three separate steps — recommendation by a commanding officer, approval by the Chiefs' Board, and selection by the Superintendent — and these steps were wholly capable of being separated from each other for the purpose of statistical analysis. For the following reasons, we disagree.

To make out a disparate impact claim (or, more generally, to rely on statistical evidence), a plaintiff must identify a specific discriminatory employment practice. See Wal-Mart Stores, Inc. v. Dukes, ___ U.S. ___, 131 S.Ct. 2541, 2555-56, 180 L.Ed.2d 374 (2011) ("[R]espondents have identified no `specific employment practice'.... Merely showing that Wal-Mart's policy of discretion has produced an overall sex-based disparity does not suffice."); Watson, 487 U.S. at 994, 108 S.Ct. 2777 ("Especially in cases where an employer combines subjective criteria with the use of more rigid standardized rules or tests, the plaintiff is in our view responsible for isolating and identifying the specific employment practices that are allegedly responsible for any observed statistical disparities."). Title VII, however, expressly provides that "if the complaining party can demonstrate to the court that the elements of a respondent's decisionmaking process are not capable of separation for analysis, the decisionmaking process may be analyzed as one employment practice." 42 U.S.C. § 2000e-2(k)(1)(B)(i). Whether a particular decisionmaking process is capable of separation for analysis largely turns on the details of the specific process and its implementation in a given case. See McClain v. Lufkin Indus., 519 F.3d 264, 278 (5th Cir.2008); cf. Meacham v. Knolls Atomic Power Lab., 381 F.3d 56, 74 (2d Cir.2004), vacated on other grounds, 544 U.S. 957, 125 S.Ct. 1731, 161 L.Ed.2d 596 (2005).

Here, the evidence amply demonstrated that recommendation by the Chief's Board could not be separated from the rest of the promotion process for the purpose of statistical analysis. Such recommendation was neither necessary nor sufficient for promotion, and the weight it carried in the process was both unclear and variable. For example, two candidates who were not recommended by the Chiefs' Board in January 2003 were nonetheless promoted by the Superintendent later that month, even as others who received unanimous recommendations from the Chiefs were not promoted for a year, or two years. Another Superintendent did not bother to use the Chiefs' Board at all. Recommendation by the Chiefs' Board was therefore not capable of separation from the rest of the promotion process.

The commanding officers' recommendations were similarly inseparable from the Superintendent's ultimate decisions regarding promotions because they played an indeterminate role in the integrated promotion process. For example, former Chief Thomas Farrell testified that he occasionally would ask for performance evaluations of everyone on the eligible list — not just those who were recommended by commanding officers — while other testimony indicated that commanding officers' recommendations were often important in the promotion process. We therefore agree with the district court that these "steps" in the promotion process were not capable of separation for analysis. See Port Auth. II, 681 F.Supp.2d at 464. Accordingly, the decisionmaking process involved in promotions to Sergeant was properly analyzed as one employment practice.

3. Proof of Intent

The Port Authority next argues that it was entitled to judgment as a matter of law on the plaintiffs' individual disparate treatment claims because many of the plaintiffs' anecdotes of intentional discrimination were merely "situations involving personal affront as opposed to examples of overt racism," and moreover, that "[n]one of the specific instances relied upon by plaintiffs took place in the context of promotion." Appellants' Reply Br. at 17. Even if we were to accept the Port Authority's characterization of these accounts of discrimination, however, the plaintiffs also provided evidence that they were better qualified for promotion than several white officers who were promoted instead. In conjunction with the plaintiffs' statistical evidence, we conclude that this anecdotal evidence of intent was sufficient for a reasonable jury to conclude that the Port Authority intentionally discriminated against the plaintiffs by failing to promote them.

III. Damages and the Statute of Limitations

The Port Authority argues, finally, that it was improperly assessed back pay and compensatory damages for harms that were suffered by the plaintiffs prior to August 2, 2000. The district court disagreed because it believed that the "continuing violation" doctrine applied in the context of plaintiffs' disparate impact allegations so that damages could properly be awarded for failures to promote that occurred outside the limitations period.10 We agree with the Port Authority and hold that the continuing violation doctrine does not apply to plaintiffs' disparate impact proof. As a result, we further conclude: (1) that the back pay awards to Eng, Lew, Stanley Chin, and Fong must be vacated, as well as the retroactive promotion of Lew and the salary and pension adjustments for Lew, Stanley Chin, and Fong; and (2) that the jury's compensatory damage awards with regard to all seven prevailing plaintiffs must also be vacated. We remand to the district court for a new trial on damages and for reconsideration of equitable relief to the extent such relief was premised on failures to promote occurring outside the limitations period.

A. The Continuing Violation Doctrine

It has been the law of this Circuit that "[u]nder the continuing violation exception to the Title VII limitations period, if a Title VII plaintiff files an EEOC charge that is timely as to any incident of discrimination in furtherance of an ongoing policy of discrimination, all claims of acts of discrimination under that policy will be timely even if they would be untimely standing alone." Lambert v. Genesee Hosp., 10 F.3d 46, 53 (2d Cir.1993), abrogated on other grounds by Kasten v. Saint-Gobain Performance Plastics Corp., ___ U.S. ___, 131 S.Ct. 1325, 1329-30, 179 L.Ed.2d 379 (2011); see also Patterson v. Cnty. of Oneida, 375 F.3d 206, 220 (2d Cir.2004); Fitzgerald v. Henderson, 251 F.3d 345, 359 (2d Cir.2001); Van Zant v. KLM Royal Dutch Airlines, 80 F.3d 708, 713 (2d Cir.1996); Cornwell v. Robinson, 23 F.3d 694, 703-04 (2d Cir.1994). Applying this principle, the district court in this case concluded that the Port Authority could be liable, and assessed damages, for discriminatory failures to promote outside the statute of limitations because, pursuant to the plaintiffs' disparate impact theory, those failures to promote were the product of an ongoing discriminatory policy that continued after August 2, 2000, thus triggering the continuing-violation doctrine. See Port Auth. II, 681 F.Supp.2d at 463.

* * *

The Port Authority argues that the continuing-violation doctrine does not apply in this case because (1) the plaintiffs did not identify a specific, ongoing discriminatory policy or custom; and (2) under the Supreme Court's decision in National Railroad Passenger Corp. v. Morgan, 536 U.S. 101, 122 S.Ct. 2061, 153 L.Ed.2d 106 (2002), failures to promote are "discrete acts" of discrimination and thus do not implicate the continuing-violation doctrine. Because we agree with the Port Authority's second argument, we do not address the first.

In Morgan, the Supreme Court unanimously rejected the Ninth Circuit's view that a series or pattern of "related discrete acts" could constitute one continuous "unlawful employment practice" for purposes of the statute of limitations. Id. at 111, 122 S.Ct. 2061. Rather, the Court held that "discrete discriminatory acts are not actionable if time barred, even when they are related to acts alleged in timely filed charges. Each discrete discriminatory act starts a new clock for filing charges alleging that act." Id. at 113, 122 S.Ct. 2061. By a divided vote, however, the Morgan Court distinguished such discrete acts from an allegedly hostile work environment, which it held could be a continuing violation because its "very nature involves repeated conduct." Id. at 115, 122 S.Ct. 2061. "Such claims are based on the cumulative effect of individual acts," the Court wrote, noting that "a single act of harassment may not be actionable on its own." Id.

The plaintiffs argue that Morgan's analysis of "discrete acts" cannot apply to disparate impact claims because such claims — like hostile work environment claims — are "necessarily based on the cumulative effect of a particular practice over time." Appellees' Br. at 28. It is true that Morgan involved only an individual disparate treatment claim premised on a series of related discrete acts, and therefore did not directly address whether the continuing-violation doctrine applies where an ongoing discriminatory policy results in discrete discriminatory acts both before and after the limitation date. See Morgan, 536 U.S. at 107, 122 S.Ct. 2061 (noting in passing that in the Ninth Circuit, pre-Morgan, another type of continuing violation could be established by showing "a systematic policy or practice of discrimination that operated, in part, within the limitations period," but neither endorsing nor repudiating that category of continuing violations); id. at 115 n. 9, 122 S.Ct. 2061 ("We have no occasion here to consider the timely filing question with respect to `pattern-or-practice' claims brought by private litigants as none are at issue here."). Morgan's reasoning, however, demonstrates that a plaintiff may recover for a failure to promote — regardless whether it was caused by an ongoing discriminatory policy — only if he files an EEOC charge within 180 or 300 days of that decision.11

Morgan established that an employer's failure to promote is by its very nature a discrete act. "Discrete acts such as termination, failure to promote, denial of transfer, or refusal to hire are easy to identify," the Court wrote. Id. at 114, 122 S.Ct. 2061 (emphasis added); see also Forsyth v. Fed'n Emp't & Guidance Serv., 409 F.3d 565, 572 (2d Cir.2005), abrogated on other grounds by Ledbetter v. Goodyear Tire & Rubber Co., 550 U.S. 618, 127 S.Ct. 2162, 167 L.Ed.2d 982 (2007). Moreover, each discrete act necessarily "constitutes a separate actionable `unlawful employment practice,'" Morgan, 536 U.S. at 114, 122 S.Ct. 2061 — unlike the incidents that comprise a hostile work environment claim, which may not be individually actionable, id. at 115, 122 S.Ct. 2061. Both the employer and the aggrieved party may therefore rely on the clear and predictable statute of limitations when contemplating prospective litigation regarding failures to promote or other discrete acts. As Justice Ginsburg has explained:

Ledbetter, 550 U.S. at 649, 127 S.Ct. 2162 (Ginsburg, J., dissenting). Accordingly, under Morgan, every failure to promote is a discrete act that potentially gives rise to a freestanding Title VII claim with its own filing deadline.

Discrete acts of this sort, which fall outside the limitations period, cannot be brought within it, even when undertaken pursuant to a general policy that results in other discrete acts occurring within the limitations period. This is the conclusion of every circuit to consider the question after Morgan. Each of our sister circuits has held that an allegation of an ongoing discriminatory policy does not extend the statute of limitations where the individual effects of the policy that give rise to the claim are merely discrete acts. See, e.g., Williams v. Giant Food Inc., 370 F.3d 423, 429 (4th Cir.2004) ("Nor does [the plaintiff's] allegation of a 20-year `pattern or practice' of discrimination extend the applicable limitations periods."); Davidson v. Am. Online, Inc., 337 F.3d 1179, 1185-86 (10th Cir.2003) (holding that claims cannot be premised on an untimely discrete act "even if the discrete act was part of a company-wide or systemic policy"); cf. Tademe v. St. Cloud State Univ., 328 F.3d 982, 988 (8th Cir.2003) ("Although [the plaintiff] argues that the district court failed to consider that he was asserting a pattern-or-practice of discrimination, Morgan makes clear that the failure to promote, refusal to hire, and termination are generally considered separate violations."); Lyons v. England, 307 F.3d 1092, 1107 & n. 8 (9th Cir.2002) (holding that an individual plaintiff's "assertion that this series of discrete acts flows from a company-wide, or systemic, discriminatory practice will not succeed in establishing the employer's liability for acts occurring outside the limitations period," but distinguishing and declining to address class-wide pattern-or-practice claims).

This conclusion is not altered by the fact that the plaintiffs employ the disparate impact method of proof. To prevail on a disparate impact claim, a plaintiff must "demonstrate[] that a respondent uses a particular employment practice that causes a disparate impact." 42 U.S.C. § 2000e-2(k)(1)(A)(i) (emphasis added). In Lewis v. City of Chicago, ___ U.S. ___, 130 S.Ct. 2191, 2197-99, 176 L.Ed.2d 967 (2010), the Supreme Court interpreted this language to mean that every "use" of an employment practice that causes a disparate impact is a separate actionable violation of Title VII with its own 180- or 300-day statute-of-limitations clock. See id. at 2197-99. Accordingly, under Lewis and Morgan, each time the Port Authority failed to promote one of the plaintiffs, that plaintiff had 180 days to challenge the decision.

In an attempt to distinguish Morgan, the plaintiffs argue that they "challenge the process by which the Port Authority made promotion decisions, rather than any specific promotion decision." Appellees' Br. at 29. But this argument hurts rather than helps them. In Lewis, the Supreme Court considered the case of an allegedly discriminatory examination used by the City of Chicago to make hiring decisions. The examination's scores and the City's plan to hire based on certain cutoff scores were announced outside the limitations period, but the actual hiring occurred within the limitations period. See Lewis, 130 S.Ct. at 2195-96. The Supreme Court explained that although "[i]t may be true that the City's ... decision to adopt the cutoff score (and to create a list of the applicants above it) gave rise to a freestanding disparate-impact claim[,] [i]f that is so, the City is correct that since no timely charge was filed attacking it, the City is now entitled to treat that past act as lawful." Id. at 2198-99 (citation and internal quotation marks omitted). If the process by which the Port Authority promoted police officers from its eligibility lists did not materially change within the limitations period, as the plaintiffs claim, then the Port Authority is entitled to treat the process as lawful. See id. at 2199. The process itself therefore cannot be challenged; rather, only specific failures to promote that occurred within the limitations period are actionable.

B. Damages & Equitable Relief

The district court properly instructed the jury regarding the statute of limitations for plaintiffs' individual disparate treatment claims, and the jury indicated on the verdict sheet its express findings that the Port Authority made discriminatory decisions not to promote Eng, Fong, Lew, Stanley Chin, Yum, Martinez, and Lim "after August 2, 2000." Pursuant to the district court's conclusion that the continuing violation doctrine was applicable to plaintiffs' disparate impact proof, however, the jury was permitted to assess damages for failures to promote occurring outside the limitations period.12 With this in mind, we turn to the Port Authority's claim that the damages and equitable awards here were premised on time-barred claims and were otherwise excessive.

1. Back Pay

The jury's back-pay awards correspond precisely to certain calculations of the plaintiffs' damages expert, such that both parties and the district court agreed below that the jury found that four of the prevailing plaintiffs (Eng, Lew, Stanley Chin, and Fong) would have been promoted on October 31, 1999, and the other three (Yum, Lim, and Martinez) would have been promoted on September 30, 2002.

The Port Authority argues first that the jury could not award back pay to multiple plaintiffs dating back to the same date when fewer than that number of plaintiffs were actually promoted on that date. It points out that there were only three promotions on October 31, 1999, but the jury awarded back pay to four plaintiffs corresponding to a failure to promote on that date. Likewise, there were only two promotions on September 30, 2002, but the jury awarded back pay to three plaintiffs extending back to that date. The Port Authority urges that the back pay awards for this reason "suffer from a fundamental error of law" and must be vacated. Appellants' Br. at 46.

We disagree. Although in many circumstances an employer may have only a fixed, limited number of possible promotion slots such that relief would be limited accordingly, see Dougherty v. Barry, 869 F.2d 605, 614-15 (D.C.Cir.1989) (R.B. Ginsburg, J.), that is not the case here. The plaintiffs presented evidence that the Port Authority could and did create new Sergeant-level vacancies. For example, during cross-examination, Chief Farrell conceded that the Superintendent occasionally would not specify the number of new Sergeants he was looking for, and that from time to time the Port Authority created new Sergeant-level vacancies based on staffing needs. A reasonable jury could therefore have concluded that the Port Authority could have promoted three officers rather than two on September 30, 2002, and four officers rather than three on October 31, 1999.

Nevertheless, the back pay awards to Christian Eng, Alan Lew, Stanley Chin, and Milton Fong were improper because they were premised on a hypothetical promotion date outside the statute of limitations. As explained earlier, see supra section III.A, the district court should have instructed the jury that the Port Authority could be liable only for discriminatory failures to promote after August 2, 2000, and that individual remedies were limited accordingly. We therefore vacate the back pay awards to these four plaintiffs and remand to the district court for determination of their proper back-pay awards.

2. Compensatory Damages

The Port Authority next argues that the jury's compensatory damages awards were based on discriminatory acts that predated the onset of the statute of limitations period. The plaintiffs do not contest this allegation, but rather embrace it, and defend the awards solely on the basis of the continuing violation theory. See Appellees' Br. at 48 ("The compensatory damages awards correlate to each Plaintiff's seniority on the job — and thus, the duration of each Plaintiff's distress — awarding $250,000 to the two Plaintiffs who each had more than twenty-nine years on the job, $100,000 and $75,000 to the three Plaintiffs who had between twenty and twenty-five years on the job, and $15,000 to the two Plaintiffs who had sixteen years on the job."). "When `[i]t is not possible to ascertain what portions of the compensatory and punitive damages awards were attributable' to claims that were time-barred, the damages awards must be vacated" and remanded for a new trial on damages. Annis v. Cnty. of Westchester, 136 F.3d 239, 248 (2d Cir.1998) (quoting Rush v. Scott Specialty Gases, Inc., 113 F.3d 476, 485 (3d Cir.1997)). Because the jury may have included time-barred claims with respect to each of the plaintiffs, we vacate all seven prevailing plaintiffs' compensatory damages awards and remand for a new trial on damages. On remand, the district court should instruct the jury to award damages only for injuries stemming from a discriminatory failure to promote after August 2, 2000.

3. Equitable Relief

The Port Authority next argues that the district court's equitable relief of retroactive promotions and salary and pension adjustments should have been granted only pro rata under the theory that only a limited number of promotions were available on each day. See Dougherty, 869 F.2d at 614-15. But this argument fails with respect to equitable relief for the same reason it fails regarding back pay, see supra section III.B.1; on the evidence presented, a reasonable jury could have concluded that the Port Authority could promote more officers on a given date than it chose to.

The equitable relief should not, however, have extended retroactive promotions or salary or pension adjustments beyond the limitations period. The district court's award of salary and pension adjustments for Milton Fong, Stanley Chin, and Alan Lew, as well as the retroactive promotion of Alan Lew, must be vacated and remanded for reconsideration because the award of such equitable relief was premised on a hypothetical promotion date of October 31, 1999. On remand, the district court should determine the date, after August 2, 2000, that each of these three plaintiffs would have been promoted absent discrimination and may grant appropriate equitable relief accordingly.

IV. Exclusion of Lundquist's Expert Testimony

We now turn to the cross-appeal. The four cross-appealing plaintiffs argue that the district court erred in excluding the expert testimony of Dr. Lundquist, who would have testified that the Port Authority's promotion procedure was so unstructured and subjective that it fell below professional standards, and who would have compared the qualifications of the plaintiffs with those of the officers who were actually promoted. Expert testimony is admissible if it "(a) will help the trier of fact to understand the evidence or to determine a fact in issue," so long as "(b) the testimony is based upon sufficient facts or data; (c) the testimony is the product of reliable principles and methods; and (d) the expert has reliably applied the principles and methods to the facts of the case." Fed.R.Evid. 702. A district court's exclusion of expert testimony is reviewed for abuse of discretion, and "[a] decision to admit or exclude expert scientific testimony is not an abuse of discretion unless it is `manifestly erroneous.'" Amorgianos v. Nat'l R.R. Passenger Corp., 303 F.3d 256, 265 (2d Cir.2002) (quoting McCullock v. H.B. Fuller Co., 61 F.3d 1038, 1042 (2d Cir.1995)). "Further, an erroneous evidentiary ruling warrants a new trial only when `a substantial right of a party is affected,' as when `a jury's judgment would be swayed in a material fashion by the error.'" Lore v. City of Syracuse, 670 F.3d 127, 155 (2d Cir.2012) (quoting Arlio v. Lively, 474 F.3d 46, 51 (2d Cir.2007)).

The district court did not abuse its discretion in concluding that it lacked evidence that Dr. Lundquist's testimony was based on established principles and methods and that, in any event, her testimony would not have provided assistance to the trier of fact beyond that afforded by the arguments of counsel, as required by Rule 702. On appeal, the plaintiffs argue that the district court failed to acknowledge the portion of Dr. Lundquist's testimony that compared the qualifications of the plaintiffs with those of the white officers who were promoted instead. But Dr. Lundquist's analysis as to the comparative qualifications of the plaintiffs was both brief and simple, relying mostly on various officers' years of experience, commendations, discipline, and absences. For each of the four plaintiffs who did not prevail, Dr. Lundquist merely summed up their qualifications in a few sentences and then compared each of them to two officers who were promoted instead but whose record suggested that they may have been less qualified. For example, she compared both Michael Chung and Sanrit Booncome to a promoted officer named Gary Griffith, whom she described only as having "sixty-seven absences in 2000 alone."

The district court did not abuse its discretion in concluding that expert analysis was not required to help the jury understand such evidence. Indeed, the plaintiffs' attorneys made the same points in argument that were made in Dr. Lundquist's report. Chung and Booncome's qualifications were established in detail while they were on the stand, and their attorney brought out Gary Griffith's relative lack of experience and his significant number of absences through questioning of a former Superintendent. The plaintiffs' attorneys, moreover, emphasized throughout the trial the relative qualifications of the plaintiffs when compared with officers who were promoted. At the trial's conclusion, the plaintiffs' summation detailed the qualifications of each of the plaintiffs in almost exactly the same way as Dr. Lundquist's testimony would have, including occasionally comparing a plaintiff with someone who had been promoted. The district court therefore did not abuse its discretion in determining that Dr. Lundquist's testimony was not relevant expert testimony that would help the jury understand the facts at issue.

V. Sanctions for Spoliation

Finally, cross-appealing plaintiff Howard Chin argues that the district court erred in denying the plaintiffs' motion requesting an adverse inference instruction due to the Port Authority's destruction of the promotion folders used to make promotions off of the 1999 eligible list.13 See Port Auth. I, 601 F.Supp.2d 566 (S.D.N.Y.2009). The Port Authority does not dispute that, upon receiving notice of the filing of plaintiffs' EEOC charge in February 2001, it had an obligation to preserve the promotion folders yet failed to do so. It argues, however, that the district court did not abuse its discretion in denying an adverse inference instruction. We agree.

"[A] party seeking an adverse inference instruction based on the destruction of evidence must establish (1) that the party having control over the evidence had an obligation to preserve it at the time it was destroyed; (2) that the records were destroyed with a culpable state of mind; and (3) that the destroyed evidence was relevant to the party's claim or defense such that a reasonable trier of fact could find that it would support that claim or defense." Residential Funding Corp. v. DeGeorge Fin. Corp., 306 F.3d 99, 107 (2d Cir.2002) (internal quotation marks omitted). If these elements are established, a district court may, at its discretion, grant an adverse inference jury instruction insofar as such a sanction would "serve[] [the] threefold purpose of (1) deterring parties from destroying evidence; (2) placing the risk of an erroneous evaluation of the content of the destroyed evidence on the party responsible for its destruction; and (3) restoring the party harmed by the loss of evidence helpful to its case to where the party would have been in the absence of spoliation." Byrnie v. Town of Cromwell, 243 F.3d 93, 107 (2d Cir.2001). Our review of a district court's decision on a motion for discovery sanctions is limited to abuse of discretion, which includes errors of law and clearly erroneous assessments of evidence. See Residential Funding Corp., 306 F.3d at 107. "[A]bsent a showing of prejudice, the jury's verdict should not be disturbed." Id. at 112.

Howard Chin argues that the Port Authority's failure even to issue a litigation hold regarding the promotion folders at any point between 2001 and 2007 amounted to gross, rather than simple, negligence. We reject the notion that a failure to institute a "litigation hold" constitutes gross negligence per se. Contra Pension Comm. of Univ. of Montreal Pension Plan v. Banc of Am. Secs., LLC, 685 F.Supp.2d 456, 464-65 (S.D.N.Y.2010). Rather, we agree that "the better approach is to consider [the failure to adopt good preservation practices] as one factor" in the determination of whether discovery sanctions should issue. Orbit Comm'ns, Inc. v. Numerex Corp., 271 F.R.D. 429, 441 (S.D.N.Y.2010). Moreover, as the district court recognized, see Port Auth. I, 601 F.Supp.2d at 570, a finding of gross negligence merely permits, rather than requires, a district court to give an adverse inference instruction. See Residential Funding Corp., 306 F.3d at 109; Byrnie, 243 F.3d at 108. Even if we assume arguendo both that the Port Authority was grossly negligent and that the documents here were "relevant," we have repeatedly held that a "case-by-case approach to the failure to produce relevant evidence," at the discretion of the district court, is appropriate. Residential Funding Corp., 306 F.3d at 108 (quoting Reilly v. Natwest Mkts. Grp., 181 F.3d 253, 267 (2d Cir. 1999)). In this case, the district court concluded that an adverse inference instruction was inappropriate in light of the limited role of the destroyed folders in the promotion process and the plaintiffs' ample evidence regarding their relative qualifications when compared with the officers who were actually promoted. See Port Auth. I, 601 F.Supp.2d at 570-71. At trial, Howard Chin was able to establish his service record and honors, and Chief Charles Torres testified that Howard Chin was very smart and a good employee. Under these circumstances, the district court did not abuse its discretion in concluding that an adverse inference instruction was inappropriate.

CONCLUSION

For the foregoing reasons, we affirm the district court's conclusion that the Port Authority is liable to Christian Eng, Nicholas Yum, Alan Lew, David Lim, George Martinez, Stanley Chin, and Milton Fong under both the individual disparate treatment and disparate impact theories. We also affirm the denial of individual relief to Howard Chin, Richard Wong, Sanrit Booncome, and Michael Chung. Because the district court erred in applying the continuing-violation exception to the plaintiffs' claims, however, we: (1) vacate the jury's back pay awards with respect to Christian Eng, Alan Lew, Stanley Chin, and Milton Fong; (2) vacate the jury's compensatory damage awards with respect to Christian Eng, Nicholas Yum, Alan Lew, David Lim, George Martinez, Stanley Chin, and Milton Fong; (3) vacate the retroactive promotion of Alan Lew; and (4) vacate the salary and pension adjustments for Alan Lew, Stanley Chin, and Milton Fong. We remand all of these remedies issues to the district court for a new trial solely on damages and for the reconsideration of equitable relief. On remand, individual relief should be awarded only insofar as it corresponds to discriminatory failures to promote committed after August 2, 2000.