Defendants-appellants Local 28 of the Sheet Metal Workers’ International Association (the “Union” or “Local 28”), the Sheet Metal and Air Conditioning Contractors’ Association of New York City, Inc., and the Sheet Metal and Air Conditioning Contractors’ National Association of Long Island, Inc. (together the “Contractors”) appeal from a June 24, 1998 order of the United States District Court for the Southern District of New York. Judge Robert L. Carter held the Union in contempt for violating previous orders of the district court and imposed various remedies on the Union.
On appeal, Local 28 challenges the district court’s finding of contempt as well as its award of back pay and appointment of a statistical expert at Union expense. The Contractors join in contesting the appointment of a statistical expert. They also contend that the district court’s order that a special master investigate the hiring of certain categories of employees extends beyond the “minor and ancillary” relief that a district court is empowered to impose on parties, such as the Contractors, who have not been held liable for any violation of law.
BACKGROUND
These parties appear before us for the second time in connection with these contempt proceedings, and for the fifth time over the life of this litigation. The facts and procedural history of this case have been set out at length in our prior opinions, see, e.g., EEOC v. Local 638, 81 F.3d 1162, 1168-71 (2d Cir.1996), and we recite here only its most important features and recent developments.
I. Early History
On July 18, 1975, after a bench trial, the district court found that Local 28 had discriminated against nonwhites in violation of Title VII of the Civil Rights Act of 1964, 42 U.S.C. §§ 2000e et seq. See EEOC v. Local 638, 401 F.Supp. 467, 487 (S.D.N.Y.1975) (“EEOC I”). Among other remedies, the district court ordered the Union to meet a remedial goal of 29 percent nonwhite membership by July 1, 1981. See id. at 489.
The district court entered an Order and Judgment (“O & J”) on August 28, 1975. The O & J permanently enjoined Local 28 from discriminating in recruitment or admission to the Union, and it required the parties to work with a court-appointed Administrator to establish an affirmative action program. The Administrator then submitted an Affirmative Action Program and Order (“AAPO”) that was adopted by the district court. See EEOC v. Local 638, 421 F.Supp. 603, 617-20 (S.D.N.Y.1975) (“EEOC II”).
On appeal, we affirmed the district court’s determination of liability for discrimination. See EEOC v. Local 638, 532 F.2d 821, 825-27 (2d Cir.1976) (“EEOC III”). We also affirmed the appointment of the Administrator, the permanent injunction against the Union, the 29 percent nonwhite membership goal, and, with some modifications, the AAPO. See id. at 829-33. On remand, the district court adopted these modifications in a Revised Affirmative Action Plan and Order (“RAAPO”), and we affirmed the RAAPO. See EEOC v. Local 638, 565 F.2d 31, 36 (2d Cir.1977) (“EEOC IV”).
The July 1,1981 target date for 29 percent nonwhite membership came and went. In *2821982 and again in 1983, the district court held the defendants in civil contempt for violating its prior orders. The district court imposed various remedies and adopted an Amended Affirmative Action Plan and Order (“AAA-PO”) proposed by the Administrator. On appeal, we affirmed the findings of contempt as to the Union, but reversed the contempt finding against the Contractors. See EEOC v. Local 638, 753 F.2d 1172, 1181-82 (2d Cir.1985) (“EEOC V”). We affirmed the AAAPO, with certain modifications, and affirmed its new membership target of 29.23 percent on the ground that it was a permissible goal, not a permanent quota. See id. at 1185-89. The Supreme Court affirmed in turn. See Local 28 v. EEOC, 478 U.S. 421, 106 S.Ct. 3019, 92 L.Ed.2d 344 (1986).
II. Current Contempt Proceedings
In July of 1993, the City of New York launched a second round of contempt proceedings by moving that Local 28 once again be held in contempt of the district court’s orders. After settlement attempts broke down, Local 28 filed opposition papers in March of 1994. In 1995, the district court held the Union in contempt for 1) violating recordkeeping requirements; 2) failing to meet its membership goal; 3) failing to provide equal work opportunities to nonwhite journeypersons; and 4) maintaining a discriminatory reinstatement policy. See EEOC v. Local 638, 889 F.Supp. 642, 652-68 (S.D.N.Y.1995) (“EEOC VI”). The district court ordered, inter alia, back pay, the creation of a hiring hall and job rotation system, the elimination of certain reinstatement requirements, and the appointment of a field monitor to ensure compliance. See id. at 668-87.
On defendants’ subsequent appeal, we affirmed the contempt findings based on the Union’s recordkeeping violations and failure to provide equal employment opportunities to nonwhites. We also affirmed the contempt finding based on the Union’s two-year limitation on reinstatement, but we vacated the contempt holding based on the Union’s other reinstatement conditions on the ground of lack of notice to defendants. See EEOC v. Local 638, 81 F.3d 1162, 1171-76 (2d Cir. 1996) (“EEOC VII ”). As for the remedies, we 1) vacated the back pay award and remanded to the district court to refashion a remedy that would not allow back pay damages for journeypersons against whom the Union had not discriminated; 2) affirmed the abolition of the two-year rule for reinstatement, but vacated the order based on other conditions for reinstatement; 3) reversed the imposition of a hiring hall and job rotation system; and 4) affirmed the appointment of a field monitor, except for the clause in the appointment order empowering him to investigate and conciliate job-related complaints. See id. at 1176-81.
On remand, the district court made a new finding of contempt based on the Union’s imposition of fees and back dues upon reinstatement. Among other remedies, the district court 1) enjoined the Union from charging fees and back dues upon the reinstatement of certain members; 2) fashioned a new back pay procedure that allowed the Union to produce evidence of non-discrimination; 3) established a referral hall; and 4) appointed a referral hall monitor and statistical expert to oversee and analyze the hiring process. See EEOC v. Local 638, 13 F.Supp.2d 453, 459, 462-69 (S.D.N.Y.1998) (“EEOC VIII”). Predictably, this appeal followed.
DISCUSSION
The Union and the Contractors raise numerous challenges to the district court’s decision. Many of these challenges are based on an inaccurate reading of the court’s opinion. Assessing what the opinion actually holds, we affirm most of the rulings, but reverse the finding of contempt based on the Union’s reinstatement policy; reverse, subject to a qualification, the remedy of prohibiting the collection of back dues; and modify slightly one aspect of the remedy concerning the Special Master.
I. Liability for Contempt
A party may be held in contempt only if it is proven by “clear and convincing” evidence that the party violated a “clear and unambiguous” order of the court. See EEOC VII, 81 F.3d at 1171 (quoting United States *283v. Local 1804-1, 44 F.3d 1091, 1096 (2d Cir. 1995)). The violation need not be willful, but it must be demonstrated that “the contemnor was not reasonably diligent in attempting to comply.” Id. (quoting Local 1804-1, 44 F.3d at 1096). We review a finding of contempt for abuse of discretion, but because of the formidable and potentially harmful nature of the contempt power, this review is more rigorous than in other contexts. See Local 1804-1, 44 F.3d at 1095-96.
The district court held Local 28 in contempt based on its reinstatement policy. The Union had required a member to pay a $25 reinstatement fee as well as all unpaid dues in order to be reinstated after termination for failure to pay dues. The district court found this policy to be contumacious “to the extent that nonwhite journeymen who were underemployed between 1984 to 1991 because of the union’s discrimination were 1) suspended or terminated from the union, and 2) subsequently required ... to pay fees and back dues for reinitiation of their membership.” EEOC VIII, 13 F.Supp.2d at 459.
Local 28 argues that this contempt finding was an abuse of discretion. We agree, and reverse the district court’s finding of contempt.
The district court based its finding of contempt on the O & J’s prohibition from “engaging in any act or practice which has the purpose or the effect of discriminating in ... admission to membership in Local 28 ... on the basis of race, color, or national origin,” and the AAAPO’s mandate to make “regular and substantial progress” toward the 29.23 percent nonwhite membership goal. Id. at 460. While these orders are unquestionably clear and unambiguous, see EEOC VII, 81 F.3d at 1171, there is no clear and convincing evidence that the reinstatement policy violated them. Although, as we explain below, the reinstatement fee policy, as applied to jour-neypersons entitled to back pay, was properly prohibited as a remedy for the previously adjudicated contempt, this policy was not itself shown to be a new contempt.
In finding a new contempt, the district court properly noted that nonwhites were overrepresented among journeypersons terminated between 1984 and 1991 for failure to pay dues. However, this wrong has already been the subject of contempt proceedings: the Union was held in contempt for its discrimination that was (at least in part) responsible for these terminations, see id. at 1172— 74, and it was also held in contempt for its refusal to reinstate members who had been suspended for more than two years for nonpayment of dues, see id. at 1175-76. These actions by Local 28 had the proven effect of discriminating in Union admission and impeding the membership goal.
In contrast, there is no clear and convincing evidence in the record that the imposition of back dues and fees on candidates for reinstatement had a disparate impact on nonwhites, such as would support an independent finding of contempt. It was never demonstrated, for example, that nonwhites whose nonpayment of dues resulted from discriminatory Union practices were impeded from reinstatement due to a $25 fee and the collection of dues. There is no evidence in the record to connect the discriminatory termination of nonwhite journeypersons to the conclusion that a requirement of fees and dues upon reinstatement had a discriminatory impact on Local 28 membership. The new contempt finding was based on conduct that had already been the basis for contempt. Accordingly, the contempt finding based on these parts of the reinstatement policy must be reversed.1
II. Remedies for Previously Affirmed Contempt
We now turn to the remedies the district court imposed pursuant to its previously affirmed contempt findings. See EEOC VII, 81 F.3d at 1171-76. These remedies include the award of back pay, the appointment of a statistical expert, and an order that the Special Master investigate whether fan mainte*284nance and target agreement workers should be subject to a mandatory referral hall.
A. Back Pay
As a compensatory remedy, a back pay award “should reimburse the injured party for its actual damages.” New York State Nat’l Org. for Women v. Terry, 886 F.2d 1339, 1353 (2d Cir.1989) (citing United States v. United Mine Workers, 330 U.S. 258, 304, 67 S.Ct. 677, 91 L.Ed. 884 (1947)). In EEOC VII, we vacated the district court’s award of back pay because it would have allowed journeypersons to receive damages “even if there was a legitimate non-discriminatory reason that would explain why the journeyperson was not hired.” 81 F.3d at 1177. On remand, the district court modified the procedure for awarding back pay as follows: first, a nonwhite journeyperson may make a prima facie case of entitlement to back pay by proving that his hours worked were two or more standard deviations below the mean for the period of 1984 to 1991; following this showing, the burden shifts to Local 28 to produce evidence that the disparity in the particular journeyperson’s hours was not due to Union discrimination; the applicant must then present evidence of actual discrimination against him within 30 days; and finally, the Union has 30 days in which to rebut the proffered evidence of discrimination. See EEOC VIII, 13 F.Supp.2d at 465-66.
Local 28 contends that this procedure is penal, not compensatory, because it forces the Union to compensate nonwhite journey-persons who were not victims of Union discrimination. The Union also argues that the procedure improperly shifts the burden of proof of non-discrimination to Local 28. We reject both contentions.
The district court’s order shifts only the burden of production to the Union after the presentation of appropriate statistical evidence by a journeyperson. At no time, however, is the ultimate burden of persuasion transferred to the Union. The procedure ordered by the district court entails nothing more than the familiar burden-shifting that has long been a feature of Title VII litigation. See McDonnell Douglas Corp. v. Green, 411 U.S. 792, 800-05, 93 S.Ct. 1817, 36 L.Ed.2d 668 (1973).
As we explained in EEOC VII, there are many evidentiary avenues that Local 28 can pursue to avoid liability in an individual case:
For instance, a journeyperson could have a poor work history of which employers became aware after checking his references, or the journeyperson might only have sought work from contractors who were looking for someone more skilled. Indeed, it may even be the case that the Union can show that a particular journeyperson was unemployed, not because of its own discrimination, but rather, because the contractors were illegally discriminating against the journeyperson. In the foregoing situations, a back pay award would be inappropriate.
81 F.3d at 1177. These and other showings could satisfy the Union’s burden of production, at which point the burden shifts back to the applicant seeking back pay.
The Union’s insistence that this procedure will force it to compensate journeypersons who did not suffer Union discrimination is without merit. Its argument that there is no basis for holding it responsible for the hours disparity of many nonwhites flies in the face of repeated judicial findings of intentional, systematic racial discrimination. See, e.g., EEOC I, 401 F.Supp. at 487 (“Local 28 has maintained clearly discernable discriminatory practices in recruitment, selection, training and admission to membership of non-white workers.”); EEOC III, 532 F.2d at 825 (Local 28 has “consistently and egregiously violated Title VII”); EEOC IV, 565 F.2d at 36 n. 8 (noting “overwhelming evidence of purposeful racial discrimination”); EEOC VII, 81 F.3d at 1173-74 (affirming district court’s finding of discriminatory practices by Union business agents). As the Supreme Court has stated, “[t]he employer cannot, therefore, claim that there is no reason to believe that its individual employment decisions were dis-criminatorily based; it has already been shown to have maintained a policy of discriminatory decisionmaking.” International Bhd. of Teamsters v. United States, 431 U.S. 324, 362, 97 S.Ct. 1843, 52 L.Ed.2d 396 (1977). The evidence of Union discrimination has *285supported a verdict of liability under Title VII and several subsequent contempt findings, and the statistical evidence of an extreme disparity in hours worked is sufficient to shift the burden to the Union to produce a non-discriminatory explanation. The back pay remedy as structured by the district court is affirmed.
While the reinstatement policy of requiring back pay and fees from journey-persons who had faced discrimination cannot sustain a finding of contempt, we must consider the waiver of fees and back dues separately as part of a back pay remedy for the previously adjudicated contempt. We easily affirm the district court’s order that the reinstatement fee be waived for those journey-persons who are entitled to back pay. Had there been no discrimination, and had these members never been terminated for inability to pay dues due to lack of work, they would never have been forced to pay the reinstatement fee. However, returning the members to their ex ante position does not require a waiver of all back dues. If the nonwhite journeypersons had been working steadily from 1984 to 1991, they would have paid dues in the normal course. As a compensatory contempt remedy, the back pay award should not place the journeypersons in a better financial position than they would have been in if the Union had not discriminated against them. Accordingly, we reverse the order of the district court waiving the payment of back dues from back pay awards, with one caveat. Due to the Union’s financial condition, journeypersons discriminated against by the Union are likely to receive only a portion of their lost back pay. See EEOC VIII, 13 F.Supp.2d at 471. Back dues should therefore not be deducted from a back pay award for a given year except to the extent that the actual award for that year exceeds an amount equal to the full back pay for that year minus back dues for the year.
B. The Statistical Expert
The district court appointed a statistical expert, at Union expense, to “periodically analyze the work records of journeypersons to determine whether or not a white/nonwhite hours disparity persists in violation of the 0 & J and AAAPO.” Id. at 469. The district court also empowered the expert to analyze “the role [of] the Contractors in contributing to the white/nonwhite hours disparity.” Id. Local 28 seeks to avoid financial responsibility for this latter task of the statistical expert, arguing that this forces it to pay for the investigation of a party that has never been found liable. The Contractors object to being the subject of any expert analysis.
The statistical expert’s role appears to be limited to running regression analyses of the referral, hiring and work hours of journeypersons. To the extent that the data gleaned from this project may reveal certain practices (discriminatory or otherwise) of the Contractors, the Union can make no objection. However, to the extent the district court’s order requires the expert to perform any additional analysis that goes beyond what is necessary to monitor the Union’s activity, that aspect of the order shall have no legal effect.
The Contractors, having never been found liable for any violation of Title VII, may only be subjected to “minor and ancillary” remedial relief. General Bldg. Contractors Ass’n v. Pennsylvania, 458 U.S. 375, 399, 102 S.Ct. 3141, 73 L.Ed.2d 835 (1982); see also EEOC VII, 81 F.3d at 1179-81. Although the Contractors protest that the statistical expert’s analysis is neither minor nor ancillary, at oral argument they conceded that the district court’s order imposed no new discovery requirement on them, and could point to no other tangible effect. Given that the district court’s order imposed no concrete burden on the Contractors, we affirm the appointment of a statistical expert. If in the future the Contractors are met with court-ordered obligations that are more than minor or ancillary, the appropriate avenue of redress is in the first instance with the district court.
C. Fan Maintenance and Target Agreement Workers
The district court, in considering the categories of workers for which the court-ordered referral hall should be mandatory rather than voluntary, received objections *286from the Union and the Contractors regarding fan maintenance and target agreement employees, for whom the defendants argue the referral hall should not be mandatory. The district court found that it had “insufficient evidence to render a decision about the propriety of the referral hall’s use with respect to these two job categories.” EEOC VIII, 13 F.Supp.2d at 467. In particular, the district court was concerned about a new amendment to the target agreement. See id. As a result, the district court ordered the Special Master to “investigate the circumstances surrounding the change in policy with respect to target agreement jobs, including its possible impact on the outstanding orders in this case.” Id. at 468.
The Contractors argue that this investigation is improper and exceeds the judicial authority of the district court under Article III of the United States Constitution. Without addressing the merits of this constitutional claim, it is evident that any challenge to this aspect of the district court’s order is premature. The word “investigate” in the district court’s order may have caused the Contractors unfortunate discomfort, but it is clear that the district court is doing no more here than gathering evidence from the parties so that it can make an informed decision with regard to these categories of employees and their potential relationship to the hiring hall. If, after the district court reaches a decision, the Contractors take issue with its ensuing order, they may seek appropriate relief in the district court.
CONCLUSION
For the foregoing reasons, 1) the finding of contempt based on the Union’s reinstatement policy is reversed; 2) the award of back pay and the burden-shifting procedure established by the district court are affirmed; 3) the remedy of waiving back dues is reversed, except insofar as the diminution of back pay awards due to the Union’s financial condition would result in an award of less than the back pay that would have been awarded, absent the Union’s adverse financial condition, reduced by the back dues for a given year; 4) the district court’s remedial waiver of reinstatement fees for those journeyper-sons entitled to back pay is affirmed; 5) the district court’s appointment of a statistical expert, as far as the expert’s analysis is related in some direct way to monitoring Union activity, is affirmed; and 6) the district court’s order that the Special Master gather evidence as to the fan maintenance and target agreement employees is affirmed. We take no position as to their relationship to the referral hall.
The case is hereby remanded for further proceedings in accordance with this opinion. Each party shall bear its own costs.
. Because we reverse the contempt finding on other grounds, we need not consider the Union’s argument, citing Adarand Constructors, Inc. v. Pena, 515 U.S. 200, 115 S.Ct. 2097, 132 L.Ed.2d 158 (1995), that the contempt finding requires a construction of the district court’s order that imposes an unconstitutional race-conscious remedy.