
    170 F.3d 1111
    Jessie BERGER, et al., Appellees/CrossAppellants, v. IRON WORKERS REINFORCED RODMEN, LOCAL 201, et al., Appellees/Cross-Appellants. International Association of Bridge, Structural and Ornamental Iron Workers, Appellant/Cross-Appellee
    Nos. 97-7019, 97-7020, 97-7021, 97-7027, 97-7029, 97-7031 and 97-7124.
    United States Court of Appeals, District of Columbia Circuit.
    Argued Nov. 30, 1998.
    Decided March 30, 1999.
    
      Laurence E. Gold and Victor Van Bourg argued the cause for appellants/cross-appellees. With them on the briefs were Sally M. Tedrow and Ellen O. Boardman, Washington, DC.
    St. John Barrett, Washington DC, argued the cause and filed the briefs for appellants Albert Berger, Alfonzia Berger, and Wordia W. Parks.
    John F. Dienelt and John L. Oberdorfer, Washington, DC, argued the cause for appellees/cross-appellants. With them on the brief was Christopher L. Killion. Melvin White, Joseph M. Sellers, James A. Treanor, III, Douglas M. Mangel, Michael R. Good-stein, Jonathan M. Malis, David J. Farber, David G. Leitch, Bonnie H. Rothell, Robert B. Duncan, William C. Edgar, Samuel G. Rubenstein, and Mark E. Martin entered appearances.
    Before: SILBERMAN, SENTELLE, and GARLAND, Circuit Judges.
   Opinion for the Court filed PER CURIAM.

Concurring opinion filed by Circuit Judge SILBERMAN.

Opinion concurring and dissenting in part filed by Circuit Judge SENTELLE.

Opinion concurring and dissenting in part filed by Circuit Judge GARLAND.

PER CURIAM:

This ease presents what we hope to be the penultimate chapter in a 23-year-old litigation involving racial discrimination by iron workers’ unions against a class of African-American construction workers. We upheld the unions’ liability a decade ago, and all of the remaining issues in the ease concern the remedy due, if any, to those claimants who have thus far not settled with the unions. Although we are reluctant to prolong this unduly protracted litigation any longer, the district court’s failure adequately to resolve the questions presented on appeal compels us to remand many of these challenges to the district court for further factual findings and supporting explanation. In those instances in which the district court’s findings and explanations make it possible for us to resolve an issue definitively, we affirm or reverse the district court’s award.

I. Background

The background of this case is set out in full in our prior opinion, see Berger v. Iron Workers Reinforced Rodmen Local 201, 843 F.2d 1395, 1405-07 (D.C.Cir.1988) (Berger I), and we see little need to repeat the details here. Suffice it to say that in 1975 a class of African-American rodmen — construction workers who handle and position steel rods for reinforcing concrete and other building materials — sued Local 201 of the Iron Workers Reinforced Rodmen and the International Association of Bridge, Structural and Ornamental Iron Workers for discriminatorily denying them union membership in violation of Title VII, 42 U.S.C. § 2000e et seq., and 42 U.S.C. § 1981. Rodmen obtained work for construction employers in the Washington, D.C. area through referrals distributed at Local 201’s hiring hall, and although referrals were available to non-union “permit men,” priority, along with the other benefits of union membership, went to the union members. See Berger I, 843 F.2d at 1405. The class pursued and succeeded on several theories of liability at trial, but we essentially upheld the district court’s liability determination on one theory alone. We held that the unions were liable for imposing training and apprenticeship prerequisites to taking the journeyman’s examination — the entrance examination for union membership. The class demonstrated with statistical evidence, to which the unions offered no rebuttal, that the educational prerequisites to taking the entrance examination worked to discriminate against “experienced” African-American rod-men (those rodmen with at least two-years’ experience which, according to the class’ expert, approximated 2,150 rodmen hours). See Berger I, 843 F.2d at 1414-15. We reversed the district court’s finding that the unions’ various entrance prerequisites from 1967 to the filing of the suit in 1975 constituted a single, continuing pattern of intentional discrimination. Central to this holding was our conclusion that the so-called “Open Period” from February to June 1971, during which all experienced rodmen were permitted to take the union entrance examination (though a more difficult one), marked a sharp break in the unions’ admissions practices. See id. at 1422-23. We thus limited the liability period to the time between June 1971, the close of the “Open Period” and the beginning of the Training and Apprenticeship prerequisites, and the filing of the suit on October 21,1975. See id. at 1422.

Since the trial bifurcated liability and damages, the district court on February 15, 1989, referred the case to a Special Master, Magistrate Patrick J. Attridge, and directed him in an “Order of Reference” to conduct proceedings to calculate the amount of back pay to be awarded to class members and to determine whether class members were entitled to compensatory and punitive damages and any other relief that might be appropriate. The Order stated that the class consisted of the eight named plaintiffs, and any other claimant who could make a prima facie case that he was a member of the class — subject to the unions’ rebuttal by clear and convincing evidence. It specified the applicable back pay period as follows:

Each individual class member may present a claim for back pay for the period commencing on the date when he first attempted to become, or was deterred or discouraged from becoming, a member of Local 201 and/or the International, and concluding on the date when he first was allowed to take the journeyman examination, passage of which is required for mem- ' bership in Local 201 and the International, or was given a bona, fide opportunity to take the examination. However, in no event shall the back pay period of any class member commence earlier than October 21, 1972, which is three years prior to the filing of the complaint in this case.

The Order also set forth procedures governing the burdens of proof for establishing the amount of back pay and other relief (pnma facie case by claimants subject to the unions’ rebuttal by clear and convincing evidence), notice to the class, a schedule for submitting claims, the formula for determining back pay awards, creation of a Relief Account in which the unions would deposit awards for each successful claimant, adjustment of pension records, legal representation of claimants at individual hearings, and status reports to be filed by the Special Master every six months.

The parties conducted discovery in 1989, and in 1990 the Special Master held individual trials for the 64 remaining claimants and heard the parties’ respective expert witnesses. By March of 1991, 47 claimants remained, and the parties submitted proposed findings to the Special Master. Nearly two years later, the class filed a request for a ruling from the district court. The district court did not respond to this request, nor to a renewed request by the class filed in April 1993. In July 1993, the class sought a writ of mandamus from this court compelling the Special Master to rule, which we denied, expressing confidence (unfortunately unjustified) that the district court would promptly issue a final order resolving all matters covered by the Order of Reference. When the Special Master still had not filed his report by March of 1994 (and thus the district court obviously had not issued a final order either), the class filed a second petition for a writ of mandamus with this court. Finally, on April 14, 1994, three years after the parties submitted proposed findings, the Special Master issued his report resolving the claims of the 35 remaining claimants.

In making the class membership determinations, the Special Master defined “experience” on a case-by-case basis, rejecting the unions’ contention that 2,150 hours of Local . 201 rodmen experience was a prerequisite to class membership, as well as their position that a claimant’s failure to pass the journeyman’s examination is a per se bar to class membership. The Special Master concluded that 11 claimants failed to prove membership in the class, and he awarded the remaining 24 claimants back pay, based on a formula multiplying the hourly rodmen wage rate for each year times the difference between the number of hours the claimant actually worked each year as a rodman and the number of hours he would have worked as a union member (less any non-rodwork interim earnings pursuant to 42 U.S.C. § 2000e-5(g)). The Special Master derived the number representing the average hours a claimant “would have worked” — the so-called “benchmark proxy” — from the pension records of Local 201, and rejected alternative benchmark figures proposed by experts for both the class and the unions. He also awarded prejudgment interest at a rate of 6% compounded annually, awarded 22 claimants compensatory damages for mental or emotional distress, and denied all of the requests for punitive damages. The parties filed objections to the report, and before the district court could rule, five of the successful claimants and two of the dismissed claimants settled.

We denied the class’ second petition for mandamus after the Special Master issued his report in April 1994, and assumed in that order that the district court would act promptly on the report “in light of the long delays in this case.” The district court issued its opinion and order on January 26, 1995. The court adopted the Special Master’s report with respect to the class-wide issues and the awards to the 19 remaining successful claimants, making small corrections in the amount of the award where appropriate, but specifically noting that its order would not constitute a final, appealable order until the court’s subsequent order addressing the excluded claimants’ claims issued. That March 16, 1995 order upheld the Special Master’s exclusion of the nine remaining claimants from the class, and amended the January 26 order by finding clear error, in light of the parties’ 1990 stipulation to the contrary, in the Special Master’s failure to include overtime in the back pay calculations of the successful claimants.

The next two years of this litigation involved premature appeals by the unions and six of the excluded claimants; although the district court, indicated in its January 1995 order that the order would become final upon issuance of the March 1995 order, the court ordered the plaintiffs to submit proposed judgment orders and did not certify any of the claims for appeal pursuant to Federal Rule of Civil Procedure 54(b). Accordingly, this court dismissed the unions’ appeal pursuant to the class’ motion, and dismissed the remainder of the excluded claimants’ appeals on its own motion. The district court entered an order of judgment on January 3, 1997, from which the unions, the class on behalf of the 19 successful claimants, and three excluded claimants, appealed to this court.

The unions challenge a number of legal conclusions and factual findings, both class-wide and with respect to individual claimants, in the district court’s opinion (and the Special Master’s report that the opinion adopted), including: the method for calculating the “benchmark proxy” from which individual awards were derived; the standard of review used to determine class membership; the conclusion that several individuals were properly included in the class; the calculation of several class members’ awards; the conclusion that some class members did not fail to mitigate their damages; and the award of compensatory damages and prejudgment interest. The class members obviously defend all of those decisions. In addition, the class argues that the district court erroneously calculated several back pay awards in the unions’ favor; incorrectly concluded that four claimants abandoned their efforts to join the union and forfeited their right to back pay subsequent to their abandonment; erred by failing to award punitive damages; and should have awarded even more prejudgment interest. Three individual class members also appeal the district court’s decision to exclude them from the class.

II. The Benchmark Determination

The unions advance numerous arguments against the benchmarks chosen by the Special Master, and the consequent awards of back pay to the members of the plaintiff class. We are convinced that the Special Master, and thus the district court, did commit clear error in two respects. The Special Master failed to include “zero-hour” workers (workers who for a number of years worked zero hours as union, rodmen) in the determination of the average number of hours worked by a union rodman in the relevant time period, and he failed entirely to address the “fixed-pie” issue raised by the unions’ expert, Dr. Farrell Bloch. First, the zero-hour workers reflect the inherent risk in the work, and failure to adequately account for their absence is clear error. If the risk of injury is calculated back into the equation when individual back pay awards are determined, it needs to be done explicitly, and the specific experiences of individual rodmen can be measured against the baseline risk of injury to see if they surpass it. Individual claimants whose injury-time exceeds the statistical average should then be adjusted downward to reflect the difference between their actual experience and the average. Second, as we explain below, the fixed-pie analysis permits the .court to determine what would have happened in the absence of the discrimination, International Bhd. of Teamsters v. United States, 431 U.S. 324, 372, 97 S.Ct. 1843, 52 L.Ed.2d 396 (1977), and the burden for showing what those conditions would have been falls on the plaintiff, who is responsible for proving damages.

A. The Special Master’s Methodology

We review the findings of fact by the district court, including the findings of the Special Master to the extent that they were adopted by the district court, under a clearly erroneous standard. See Cuddy v. Carmen, 762 F.2d 119, 123-24 (D.C.Cir.), cert. denied, 474 U.S. 1034, 106 S.Ct. 597, 88 L.Ed.2d 576 (1985); 28 U.S.C. § 636(b)(2); Fed. R. Civ. P. 53(e)(2). “The findings of a master, to the extent that the court adopts them, shall be considered as the findings of the court.” Fed. R. Civ. P. 52(a). ’ The basic standard for devising back pay awards in a Title VII case is undisputed by the parties. A court must, “as nearly as possible, recreate the conditions and relationships that would have been, had there been no unlawful discrimination,” International Bhd. of Teamsters v. United States, 431 U.S. at 372, 97 S.Ct. 1843 (quoting Franks v. Bowman Transp. Co., 424 U.S. 747, 769, 96 S.Ct. 1251, 47 L.Ed.2d 444 (1976) (internal quotations omitted)). The Order of Reference directed the Special Master to determine how many additional hours class members would have worked in the absence of discrimination, and award back pay by multiplying the expected hours worked by the average wage rate in effect during the year that the class member would have worked. The back pay period began for each individual class member when he was denied access to the examination or deterred from applying for the examination for union membership by the educational requirement, and ended when he either took or had a bona fide opportunity to take the entrance examination. No back pay award could be granted for periods before October 21, 1972, three years before the suit was filed, or after April 10, 1986, when the district court issued its remedial order granting comprehensive injunctive relief. The amount of back pay awarded to any given member of the plaintiff class is the product of the average wage rate in effect during the time during which the union discriminated against him multiplied by the number of hours worked by the average rodman during that time period.

Each side put on expert testimony and presented documentary evidence sponsoring a method for computing the number of hours worked by the average rodman for purposes of determining the proper amount of a back pay award. The Special Master noted that one of the primary difficulties with the calculation was the fact that the union did not employ people, but referred them to employers, who individually determined the terms and length of employment. The Special Master rejected the methods proposed by the expert witnesses in this case, Marc Ben-dick, Jr. for the claimants and Daniel Quinn Mills and Dr. Bloch for the unions, and created a method of his own based on the documentary evidence in the record. The Special Master determined that the best proxy for hours worked in the absence of discrimination was the hours actually worked by union members during the years in question. He therefore examined the pension records of Local 201 and calculated how many hours fully employed rodmen worked on average. The Special Master noted that the “pension records automatically take into account the unemployment levels of District of Columbia union rodmen and the. average number of days lost due to injury, sickness and attrition.” Report of Special Master, J.A. 341. He explained his choice as follows:

From the Local 201 pension records, a representative group of workers is readily identifiable. The representative group are those workers who received steady referrals during the relevant time period, as evidenced by a consistent number of hours worked per year. Workers with consistent referrals worked remarkably similar total numbers of hours for any given year. Excluded from this group are non-representative workers, i.e., those who for several years during the relevant time period worked no hours at all.
By taking an average of the number of hours worked by those engaged in full time employment and checking that figure for ball-park accuracy against certain indicators of local iron worker productivity during the relevant time periods, the undersigned arrived at the representative or “proxy” number of hours per year that an iron worker could be expected to work. Based upon these indicia, and taking into account the testimony regarding the relatively recent concept of “double breasting” [in which contractors worked both union and non-union crews], and having also considered all the testimony and exhibits received in evidence, the Special Master finds that the annual hours reasonably expected to be worked by a member of Local 201 is as follows:
Year Hours Worked
1972 1711
1973 1557
1974 1627
1975 1447
1976 1419
1977 1253
1978 1179
1979 1230
1980 1210
1981 1263
1982 1168
1983 1126
1984 953
1985 1397
1986 1549

See Report of the Special Master at 50-51, J.A. 342-43. Based on his calculations, he arrived at an average of 1,339 hours per year-

B. Alleged Errors in the Master’s Method

The unions argue that the Special Master committed clear error in the method he adopted for devising a benchmark for purposes of awarding back pay. Specifically, they argue that he did not sufficiently explain his choice, or show why he failed to adopt the methodology of their expert, Dr. Bloch, which has been endorsed for Title VII remedies in the past. Like the Special Master, Dr. Bloch used a cohort analysis based on pension records, but he used a smaller category of workers, those who were admitted to the union during the period when the union discriminated against the plaintiff class. Dr. Bloch’s calculations also differed from the Special Master’s in that they included the zero-hour workers, on the reasoning that a longitudinal analysis of the cohort should incorporate the risk of disabling injury. Finally, Dr. Bloch limited the number of hours that' could be awarded with a “fixed-pie” analysis. He adjusted the benchmark by adding the total number of hours worked by all rodmen, including union members, traveling members from another local, and permit men (non-union workers who were referred to jobs from the union hall). Members of the plaintiff class, he assumed, would have become union members if the discrimination had not occurred, and would have replaced non-union members on jobs that were referred by the union. In some years, however, there were limited referrals to nonunion members, which Dr. Bloch assumed was due to limited employment in the industry generally. In such years, since only a limited number of hours were available to non-union workers, class members could not possibly have replaced only nonunion workers, but would also have displaced other union members. Reflecting this, the total hours available for such years was placed in a “fixed-pie,” and the hours were divided by the total number of union workers plus the number of proven members of the class. The assumption is that the hours would have been divided equally between all members. See Declaration of Farrell Bloch Ph.D., at lili, J.A. 1383-85.

The unions argue that the reason the Special Master gave for rejecting Dr. Bloch’s analysis — that the cohort group for 1973, consisting of 38 individuals, was a statistically insignificant sample — was wrong as a matter of fact and as a matter of law. The unions point out that Dr. Bloch’s proxy group included the aggregate of all 135 journeymen admitted between 1971 and 1975, and that the size of the group used by Dr.. Bloch exceeds the sample size of union and nonunion workers used by the class to establish liability in the first instance. See Berger, 843 F.2d at 1415. The unions also argue that the Special Master misunderstood Segar v. Smith, 738 F.2d 1249 (D.C.Cir.1984), which he cited for the proposition that the sample was too small. They distinguish Segar by noting that it applied to a liability decision, not a damages decision. Further, the Segar panel’s analysis took issue with breaking cohorts into smaller and smaller subgroups, until they did become insignificant. The unions claim that Dr. Bloch’s much larger cohort does not resemble those found objectionable in Segar.

The unions cite a line of cases beginning with Pettway v. American Cast Iron Pipe Co., 494 F.2d 211, 262 (5th Cir.1974), for the proposition that matching the plaintiff class with a comparable group is a favored method of determining what would have happened absent discrimination. See also Green v. United States Steel Corp., 640 F.Supp. 1521, 1526-29 (E.D.Pa.1986) (calculating damages based on comparable class, and adjusting cohort class for attrition rate), modified on other grounds, Green v. USX Corp., 843 F.2d 1511 (3d Cir.1988), vacated USX Corp. v. Green, 490 U.S. 1103, 109 S.Ct. 3151, 104 L.Ed.2d 1015 (1989); Stewart v. General Motors Corp., 542 F.2d 445, 453-54 (7th Cir. 1976), cert. denied, 433 U.S. 919, 97 S.Ct. 2995, 53 L.Ed.2d 1105 (1977) (same). They also cite a series of cases that have adopted the comparison group methodology in determining Title VII remedies in the construction industry. See Hameed v. International Ass’n of Bridge, Structural and Ornamental Iron Workers, Local Union No. 396, 637 F.2d 506, 520-21 (8th Cir.1980); Rios v. Enterprise Ass’n Steamfitters Local 638 of U.A., 651 F.Supp. 109, 111-12 (S.D.N.Y.1986), modified and remanded, 860 F.2d 1168, 1177 (2d Cir.1988).

The class counters that we should adopt the Special Master’s analysis, and argues that if he erred, he did so on the low side, because the total he reached, 1,339 hours, was well below the 1,400-hour estimate published by the Institute of Ironworking Industry. The Special Master as trier of fact was free to accept or reject expert testimony, and was free to draw his own conclusion. See United States v. Jackson, 425 F.2d 574, 577 (D.C.Cir.1970); Powers v. Bayliner Marine Corp., 83 F.3d 789, 797-98 (6th Cir.1996); Michel v. Total Transp., Inc., 957 F.2d 186, 192 (5th Cir.1992); United States v. 0.161 Acres of Land, 837 F.2d 1036, 1040-41 (11th Cir.1988). The Special Master rejected Bloch’s figures because his cohort group was too small, and because the Special Master did not credit the fixed-pie theory. The Special Master found that the union could have found sufficient work for its members, so the pie was not fixed. The class suggests that this court can recreate the Special Master’s results from the record, and should affirm because he cannot be found to have committed clear error.

While we agree that the use of a cohort methodology substantially similar to that used by Dr. Bloch has met with approval in the cases cited by the unions, those cases do not require us to hold that the methodology adopted by the Special Master for defining the cohort to include all union members was clear error. The cases relied upon by the unions establish neither a specific minimum size of the cohort nor a particular degree of similitude needed to meet the mandates of Title VII. We therefore hold that rejecting the specific cohort methodology urged by Dr. Bloch was not clear error.

C. Zero-Hour Workers

Nonetheless, the Special Master is required to establish, as nearly as possible, what would have occurred in the absence of discrimination. International Bhd. of Teamsters, 431 U.S. at 372, 97 S.Ct. 1843. After reading the full report submitted by Dr. Bloch, it becomes, clear why he included the zero-hour rodmen in his calculation. “The. mean based on all journeymen including those working zero hours in a given year is appropriate to use in the back pay calculations because it incorporates individual attrition from Local 201 resulting from such factors as temporary illness or injury, aversion to rodmen [sic] work, a geographic move away from Local 201’s jurisdiction, or the decision to become a contractor.” Report of Dr. Farrell Bloch at 6-7. Dr. Bloch did not include zero-hour rodmen who had died, retired, were incarcerated or permanently disabled, all conditions which would have limited the unions’ liability. Id. at 7. The more cursory explanation by the Special Master does not make it clear why he removed the zero-hour workers from the calculation, other than his feeling that they are not representative. See Report of Special Master at 50. By removing them from the calculus, the Special Master removes from the equation the risk of disabling injury, or of finding another more desirable job, or whatever other reason a person might not work full time. It is a false assumption that all of the members of the plaintiff class would have remained full time in the industry, given the dangers and disincentives inherent in the work. The very real risk that they would have been unable or unwilling to continue working has been improperly removed. The amount an individual would work at full employment should be multiplied by the likelihood that they would remain fully employed. By leaving the zero-hour workers in, Dr. Bloch removed the need for coming up with a figure to approximate that risk. The historic value of that risk is represented by the zero-hour rodmen. Discounting of this type is a common practice when attempting to fix speculative damages. The failure to account for the risks inherent in the industry does amount to clear error. The Special Master did limit awards where he found that claimants had been disabled for lengthy periods. However, as we explain in Part IV, because there is no quantification of the amount of lost time built into the hours calculation, it is impossible to determine when a claimant is injured for a period that exceeds the anticipated lost time, and the Special Master’s inconsistent treatment of claimants when determining individual back pay awards shows the need for further analysis on this factual issue. A specific number can be calculated that quantifies the risk of injury, and then individuals’ actual experience should be compared against that figure.

We are not deciding today that each claimant’s award must be reduced for excessive injury-time. It is only because the district court appeared to operate on that assumption, yet applied that assumption without sufficient facts and in an inconsistent manner, that we are compelled to remand these questions. In this regard, we think it is necessary to note the interrelationship between the inclusion or exclusion of the “zero-hour” workers in the benchmark proxy and the injury-time calculation for each claimant. Both of course are means of discounting back pay awards to reflect unavailability for work during the liability period. The district court might conclude, if he includes the “zero-hour” workers in the benchmark proxy to reflect the overall risk of injury, that individual re-adjustments for individual claimants with “excessive” injury-time would not be necessary. In other words, treating all claimants like hypothetically average claimants might obviate the need to analyze unavailability for work due to injury for individual claimants. We do not in any way endorse that outcome, and indeed think the analytical differences between the overall discounting of the proxy caused by inclusion of the “zero-hour” workers and the specific discounting of an individual’s award caused by excessive injury-time reductions easily could support doing both. We wish only to emphasize the interrelationship between these two inquiries, and to direct the district court to consider that relationship when it resolves these issues on remand.

D. “Fixed-Pie” Analysis

We now turn to the Special Master’s ■ fixed-pie analysis, and the flawed assumption that underlies it. The Special Master assumed that the union could have found additional work for its members if it wanted to, had the numbers been increased by admitting the plaintiff class. See Report of the Special Master at 47-48. There is no evidence in the record to suggest that there was additional work in the D.C. area for Local 201 rodmen. In fact, the evidence on the issue of double-breasting suggests that the portion of rodwork available to unions generally was declining, as evidenced by the limited work available for permit men. There is no evidentiary basis for the assumption made by the Special Master. Common sense and experience suggest that a union will attempt to bring as many projects as possible under union control, and that it will not slow its attempts when the additional work will be allocated to non-members, i.e., permit men, much less when the number of hours available to union members is declining. Nonetheless, the Special Master cited such a determination on the part of Dr. Bloch as speculative. The Special Master impermissibly switched the burden of proof on this issue. If the class wished the court to award damages on the basis of hours not referred through the hall, it follows that they would have the burden of at least making a prima facie showing that additional hours were available to Local 201. Local 201 cannot be faulted for not allocating work that was never brought under its control. The Special Master placed the burden on the unions to prove that those hours were not available: “There was no evidence presented that the union would not have solicited other employment opportunities for its members had it been faced with an influx of members, or taken other actions to expand its piece of the rod work ‘pie.’ ” Report of the Special Master at 47. The unions made a prima facie showing that the hours were not available based on the declining hours referred out of the hall and the evidence they introduced of the declining market share available to the union due to double-breasting. As the unions noted in their brief, the record showed that “[t]he percentage of union jobs in the metropolitan Washington D.C. area was 77% in calendar years 1973 and 1974; peaked at 89% in 1975; and then dropped precipitously to 66% in 1976 and 47% in 1977, and then gradually declined to 22% in 1984 before rising to 33% in 1986.” Def. Br. at 16. We remand to the district court to consider the effect of the fixed-pie on the number of hours available to be allocated to the plaintiff class, and to apply the correct burden of proof. The district court should bear in mind the goal of recreating as nearly as possible the situation that would have occurred in the absence of discrimination. International Bhd. of Teamsters, 431 U.S. at 372, 97 S.Ct. 1843. That means that it should make findings as to the number of members of the original plaintiff class who actually would have been awarded union membership, and for any given year calculate how many had actually gained membership, and how many remained to be added into the fixed-pie calculation. It may well be that the difference by the end of the period is insignificant. Nonetheless, the failure to consider the issue, based as it was on the flawed and factually unsupported assumption that additional work was necessarily available to the union, leaves us with no option but to remand.

Judge Garland’s dissent takes issue with our holding that the Special Master’s failure to consider the “fixed-pie” when calculating damages, and the district court’s subsequent adoption of his report, amounts to clear error. First, he argues that the unions did not argue that the district court erred in failing to consider this issue. The unions argued that the district court erred in failing to adopt Dr. Bloch’s methodology. Among the points they enumerate in favor of Dr. Bloch’s analysis is the following: “Once the actual number of claimants in each year was known, Dr. Bloch would adjust each annual average so that the recalculated claimant hours did not produce a total hours figure for all workers that exceeded the actual total hours worked through Local 201, as derived from the Local 201 pension records.” Def. Br at 28. While it does not use the specific term “fixed pie,” as Judge Garland notes, the argument raised by the unions’ brief is the argument described by that term. The fact that the brief argued that all of the methodology used by their expert be adopted does not mean that we may examine either all or none of the points raised. As we note in our opinion, it is the ’ Special Master’s outright rejection of the issue, based on the unsupported assumption that the union could have found more jobs if it had so chosen, that we find to be clearly erroneous.

Judge Garland also misconstrues our reasoning -with regard to the assumption that the union could bring more rod work under its control if it so chose. The fact that some 1,649 job requests went unfilled over a 10-year period does not mean that the union had enough work for the 173 putative class members to be fully employed over that same 10-year period. Fluctuations on a given day that would result in a specific job referral being listed as unfilled are not the same as a finding that hundreds of thousands of hours were available. As we note, the evidence on double-breasting and the steadily declining share of work available to permit men, annual drop-offs of tens of thousands of hours, shows just the opposite. Moreover, the work available, according even to the Special Master’s calculations, showed significant disparities over the period. For instance, in 1972, the average rodman would have worked 1,711 hours, in 1973, 1557 hours, and in 1974, 1627 • hours. By 1977, that number had dropped to 1,253 hours, and stayed below that level until 1985, bottoming out at 953 hours in 1984. It makes no sense to conclude a fortiori that a union could readily have found full-time work for 173 additional union members when its existing members were working some 400 fewer hoúrs per year than during the full employment period. We agree with Judge Garland that it is the gross hours available for referral, not trends, that are relevant to the validity of the fixed-pie theory. While a more detailed examination of the record, considering such factors as an increasing percentage of class members gaining union membership in the years in question, may once again yield the same conclusion reached below, we cannot affirm, on the record before us, on the basis of the court’s stated reasoning, which we find to be clearly erroneous.

III. Class Membership

A. Burden of Proof

We now turn to the issue of the burden of proof at the remedial phase of a Title VII class action suit. Class action lawsuits brought under Title VII are typically bifurcated into two phases, a liability phase and a damages phase, as was done in this case. The first phase establishes whether the employer is liable to the class because of a pattern or practice of discrimination. See International Bhd. of Teamsters, 431 U.S. at 359, 97 S.Ct. 1843. The second phase addresses questions of class membership and the degree of damage suffered by individual class members. The district court, in its February 1989 Order of Reference, required each claimant to make a prima facie showing of class membership, which could in turn be rebutted by the defendants by “clear and convincing evidence.” This instruction was in keeping with D.C. Circuit precedent, requiring the defendant to disprove class membership by clear and convincing evidence at the second phase of a Title VII class-action suit. See Trout v. Lehman, 702 F.2d 1094, 1107 (D.C.Cir.1983), rev’d on other grounds, Lehman v. Trout, 465 U.S. 1056, 104 S.Ct. 1404, 79 L.Ed.2d 732 (1984); McKenzie v. Sawyer, 684 F.2d 62, 75-78 (D.C.Cir.1982). The unions argue that a supervening Supreme Court case, Price Waterhouse v. Hopkins, 490 U.S. 228, 109 S.Ct. 1775, 104 L.Ed.2d 268 (1989), decided weeks after the Order of Reference, established that .the proper standard of proof in Title VII cases is a preponderance, for both plaintiffs and defendants. After careful consideration, we agree. We review this question of law de novo. See United States of America v. Perkins, 161. F.3d 66, 69 (D.C.Cir.1998).

The unions acknowledge that Hopkins involved a mixed-motives case finding discrimination against an individual, not a disparate-impact class action. However, they note that “in both situations it remains for a particular individual to prove the defendant’s liability to him____ Each claimant is relieved of the burden of proving that defendants discriminated against the class, not that he is part of the class.” Def. Br. at 33. This is because in the remedial stage of a class action, “as to the individual members of the class, the liability phase of the litigation is not complete.” Hopkins, 490 U.S. at 266, 109 S.Ct. 1775 (O’Connor, J., concurring).

The class counters that the district court applied the correct burden of proof for three reasons. First, in the damages phase of a class action suit, the defendant is already a proven discriminator, therefore, they argue, increasing the defendant’s burden of proof as to class membership is appropriate. Second, the class argues that the Hopkins case is distinguishable because it was the liability phase of a mixed-motives case, not the damages phase of a class action suit. Finally, they argue that the unions made no attempt to show a business justification for their testing requirement, and the court is not weighing liability. Therefore they contend, the court should retain the clear and convincing standard established in McKenzie.

In McKenzie, we read the Supreme Court’s precedent as requiring that once the employer was a proven discriminator, “‘all doubts are to be resolved against the proven discriminator rather than the innocent employee.’ ” McKenzie, 684 F.2d at 77 (quoting International Bhd. of Teamsters, 431 U.S. at 372, 97 S.Ct. 1843). We went on to hold that the finding of liability in the first phase of the trial established the prima facie ease against the employer, and that the employer “should be required to rebut the plaintiffs’ individual showings by clear and convincing evidence.” Id. at 77-78.

Since our holding, the Supreme Court has revisited the issue of the burden of proof in Title VII lawsuits. In Hopkins, the Court overturned this circuit’s holding that an employer in a Title VII sex discrimination case who had allowed a discriminatory motive to play a motivating part in an employment decision was required to show by clear and convincing evidence that it would have reached the same decision in the absence of the discriminatory'motivation. Hopkins, 490 U.S. at 238 n. 2, 109 S.Ct. 1775, reversing Hopkins v. Price Waterhouse, 825 F.2d 458, 470-71 (D.C.Cir.1987). The proposition specifically applicable to this case states that no heightened burden is required in Title VII cases, even where a burden shift has occurred.

Conventional rules of civil litigation generally apply in Title VII cases, and one of these rules is that parties to civil litigation need only prove their case by a preponderance of the evidence. Exceptions to this standard are uncommon, and in fact are ordinarily recognized only when the government seeks to take an unusual coercive action — action more dramatic than entering an award of money damages or other conventional relief — against an individual. Only rarely have we required clear and convincing proof where the action defended against seeks only conventional relief, and we find it significant that in such cases it was the defendant rather than the plaintiff who sought the elevated standard of proof — suggesting that this standard ordinarily serves as a shield rather than, as Hopkins seeks to use it, as a sword.

Hopkins, 490 U.S. at 253, 109 S.Ct. 1775 (plurality opinion) (internal citations omitted). While Justice Brennan was writing for himself and Justices Marshall, Blackmun, and Stevens, Justice White concurred in the judgment “[bjecause the Court of Appeals required Price Waterhouse to prove by clear and convincing evidence that it would have reached the same employment decision in the absence of improper motive, rather than merely requiring proof by a preponderance of the evidence.” Id. at 260, 109 S.Ct. 1775.

We think Hopkins mandates that in this case the clear and convincing standard is inappropriate, and the ordinary preponderance of the evidence standard must apply. Hopkins makes it clear that the heightened burden should not apply in Title VII cases where the heightened burden would be used as a sword not a shield. While this ease may not be on all fours with Hopkins, as the distinctions noted by the class demonstrate, the basic principle stated by the Court applies. Raising the burden of proof to clear and convincing evidence is not justified in Title VII cases; instead a preponderance applies as to all factual issues, regardless of which party bears the burden, as in other civil actions. The Court relied on the basic principle it articulated — in Title VII cases, the standard burden of proof in civil cases will apply — to decide Hopkins. The distinctions to which the class points therefore make no difference to our determination that the preponderance of the evidence standard, and not a clear and convincing standard, should apply in this case. The burden shift itself is sufficient to meet the Court’s admonition that doubts be resolved in favor of the employee, because the party that bears the burden also bears the risk that he will be unable to carry that burden due to doubts on the part of the factfinder. With these principles firmly in mind, we now go on to consider how that decision affects the facts in this case.

B. Disputed Findings

The district court’s adoption of an incorrect standard for the unions’ rebuttal burden requires remand of the Special Master’s findings of class membership with respect to two claimants, because we are unable to determine whether the Master would have made the same findings if he had applied the correct burden of proof. In the remaining disputed cases, the validity of the Master’s findings turns not on the quantum of the parties’ evidence, but on issues that can be resolved without reference to the burden of proof. Our disposition of'the findings disputed by the parties is as follows:

1. O.C. Brown. The plaintiff class includes those experienced rodmen who attempted to become, or were deterred or discouraged from becoming, members of Local 201 during the liability period-ie., between June 1971 and October 21, 1975. See Order of Reference, J.A. 216; see also Berger I, 843. F.2d at 1411. Although there was uncontested evidence to support the Special Master’s finding that O.C. Brown was “discouraged and deterred from admission” to Local 201, J.A. 86, the conclusion that this occurred during the liability period depends upon disputed inferences from circumstantial evidence. Because we are unable to determine whether the Master would have reached the same conclusion had he applied the correct burden of proof, we remand Brown’s case for redetermination.

2. Silburn Francis. There was conflicting testimony and other evidence as to whether Silburn Francis sought union membership during, rather than after, the liability period. See J.A. 400-04. Because the Special Master weighed this evidence according to an incorrect standard, we remand for re-determination under the correct burden of proof.

3. John Offer. The unions do not dispute that John Offer sought union membership in June 1972. They contend, how¡ever, that this “predat[es] the critical period,” which they define as October 21, 1972 to October 21, 1975. The unions are mistaken as to the start of the liability period. October 21, 1972 starts the period for which the remedy of back pay is available under 42 U.S.C. § 1981. See Order of Reference, J.A. 216. However, as Berger I held, membership in the class is established by having sought (or having been deterred from seeking) entry into the union “from the end of the Open Period [June, 1971] until the filing of suit on October 21, 1975”—notwithstanding that back pay may not be awarded for the early part of that period. See 843 F.2d at 1422. See generally Thomas v. Denny’s, Inc., 111 F.3d 1506, 1513-14 (10th Cir.1997) (discussing distinction between a liability limitation period, which may effectively be extended by a continuing violation, and “the period within which damages can be recovered,” which is fixed by statute).

The unions also contend that Offer was ineligible for the Apprenticeship Program for the “lawful reasons” that he could not meet that Program’s educational (high school diploma) and age requirements. However, because Berger I established that the Apprenticeship Program itself was an unlawful prerequisite to union membership for experienced rodmen, see 843 F.2d at 1414, 1421, that Program’s own prerequisites are irrelevant. Offer’s membership in the class is affirmed.

4. Wordia Parks. Wordia Parks appeals from the Special Master’s finding that he abandoned efforts to join Local 201 prior to the eligibility period, and that he therefore neither sought nor was discouraged from seeking membership during that period. See J.A. 461. The Special Master also found that Parks’ evidence was “inconsistent and contradictory,” and that he had “repeatedly impeached his own responses to interrogatory questions.” J.A. 462. The Master’s findings regarding Parks are not clearly erroneous and therefore are affirmed.

5. Charles Dean and Eldridge Harmon. To be a member of the class, a claimant must have been an “experienced” rodman. The unions challenge the Special Master’s finding that Charles Dean and Eldridge Harmon were sufficiently experienced, on the ground that the Master counted nonunion rodwork toward the number of hours required to be regarded as “experienced.” This, the unions contend, is contrary to the “law of the case,” because Berger I assertedly established that only union-referred rodwork could be counted as experience. We reject this contention because nothing in Berger I limited the definition of experience to union-referred rod-work. See 843 F.2d at 1414-15, 1421-22.

Counting non-union-referred experience, the Special Master accepted the deposition testimony that Charles Dean had the 2,150 hours.of experience that all agree is sufficient to establish the necessary experience. See J.A. 990. The Master further noted in his report that “by 1974, Dean had performed rodwork for seven years,” J.A. 391, which is far in excess of the two-year figure from which the 2,150-hours number was extrapolated. See Berger I, 843 F.2d at 1414. Because the unions offer no evidence to rebut this prima facie case—no evidence at all that Dean’s total hours were less than 2,150—Dean’s membership in the class is affirmed.

With regard to Eldridge Harmon, the Special Master expressly credited Harmon’s testimony that he had worked over 2,150 hours by December 1972. J.A. 407. Although the unions complain that the Master should have required Harmon to provide documentation to substantiate his testimony, they offer no evidence to rebut Harmon’s prima facie case. Accordingly, Harmon’s membership is affirmed as well.

6. Alfonzia Berger. Claimant Alfonzia Berger appeals the Special Master’s decision to deny him class membership on the ground that he had no rodwork experience prior to 1974. Although Berger now argues that he was discouraged from seeking the very experience that would have made him a class member, the holding in Berger I was limited to discrimination against experienced rod-men. 843 F.2d at 1419. Alfonzia Berger is not a member of the class of experienced rodmen, and the Special Master’s determination is therefore affirmed.

7. Paul Brown, James Hicks, and James Brown. The unions contend that the Special Master should have excluded from the class two claimants who failed the Open Period exam (Paul Brown and James Hicks), and one claimant who assertedly failed to take that exam despite being given an opportunity to do so (James Brown), because those failures allegedly demonstrate that these claimants were not qualified to be union journeymen. Berger I, however, made clear that the relevant question is not whether a claimant was qualified during the Open Period, but whether he was qualified during the liability period—which did not begin until the Open Period ended. The union is liable, we said, “to those class members who were experienced workers, but were delayed entry to union ranks by the particular educational prerequisites affecting them from the end of the Open Period until the filing of suit on October 21, 1975.” 843 F.2d at 1422. That is the issue upon which the Special Master properly focused. See, e.g., J.A. 383, 412.

Although a failure on the Open Period exam may have rendered a claimant unqualified to enter the union during that period, contrary to Judge Sentelle’s dissent it did not by itself render him unqualified to. do so during the liability period. The Local did not have a rule that an applicant who failed the Open Period exam (or any other preliability period exam) was ineligible to gain entry by subsequently taking and passing the exam during the liability period. Indeed, claimant Hicks was permitted to do just that, and passed the exam in 1974. J.A. 412. Failing to pass the exam during the Open Period is no different than having had less than 2,150 hours of rodwork experience during that period. It may mean a claimant was unqualified to be a journeyman at that time; it does not mean he could not become qualified by the time of the liability period.

Nor was a failure on the Open Period exam conclusive evidence that a claimant would have failed had he been permitted to take the exam during the liability period. As we noted in Berger I, the Open Period exam was different from and notably more difficult than the exam offered during the liability period. During the Open Period, only 70.6% of white examinees and 35.3% of black examinees passed the exam. By contrast, 100% of white rodmen and 9.7.6% of black rodmen who took the exam given during the liability period passed. 843 F.2d at 1405-06 n. 2.

The Special Master’s determination of class membership for these claimants is affirmed.

8. Albert Berger. Finally, claimant Albert Berger appeals the Special Master’s decision to deny him class membership on the ground that he failed the exam and then failed to avail himself of an opportunity to retak'e the exam during the Open Period. This denial is inconsistent with the Master’s correct decision not to exclude the preceding claimants for the same reason. Berger’s failure prior to the liability period neither rendered him unqualified to retake the exam during the liability period, nor indicated he would fail again if permitted to do so. Indeed, like Hicks, Berger ultimately did retake and pass the exam in 1974. J.A. 361. He was not permitted to do so, however, until he completed the Training Program — a requirement we held unlawfully discriminatory in Berger I, 843 F.2d at 1414, 1421. Accordingly, Albert Berger’s exclusion from the class is reversed.

IV. Back Pay

The unions challenge as clearly erroneous the back pay awards to several class members, contending that the district court failed to deduct from these claimants’ awards for certain periods of time during which, for one reason or another, they were not entitled to recover back pay. The class disagrees, but also challenges as clearly erroneous the district court’s decision to reduce several other awards for a period of time during which, the class argues, the claimant was entitled to recover. We consider each contested claim below.

1. James Brown. As we noted above, the district court’s Order of Reference entitles a claimant to back pay “for the period commencing on the date when [the claimant] first attempted to become, or was deterred or discouraged from becoming, a member of Local 201 and/or the International,” such period not to begin prior to October 21, 1972. The unions contest the $242 back pay award to James Brown in 1973 because, according to Brown’s own testimony, J.A. 797, Brown first attempted to join the union in 1974. The class counters that, although Brown’s attempt to join the union in 1971 predates the applicable liability period, the 1971 attempt gave the unions knowledge, or should have given them knowledge, that Brown wanted to join the union. According to the unions, Brown presented no evidence below that he was discouraged from joining the union in 1972 or 19'73, and the class does not point to any such evidence in their brief. The Special Master awarded Brown 1973 back pay without addressing this question. J.A. 375. In its discussion of class-wide issues, the district court approved of the class’ knowledge theory, J.A. 528 n.10, and appeared to agree that it was unnecessary for the Special Master specifically to identify the date on which a claimant first attempted to become a union member, or was discouraged from doing so, since the Special Master implicitly did so when he determined the years of applicable back pay for each claimant. In its discussion of Brown’s award, the district court adopted the Special Master’s findings without further comment. J.A. 547.

We think the 1973 back pay award to James Brown is clearly erroneous. Despite the district court’s apparent acceptance of the class’ “knowledge” theory, the district court’s Order of Reference authorizes back pay only for those claimants who attempted to join the union, or were discouraged from doing so, within the relevant period — not, as the class would have it, those who did nothing during that period, but whom the union knew or should have known wanted to join because of prior attempts. Although evidence of discouragement in joining would be sufficient, the class points to no such evidence as tó Brown. And we think the district court’s suggestion that the Special Master implicitly found discouragement from whatever date that he began the back pay award is too much of a stretch, even for deferential review. We note that the district court’s resolution of this award is not affected by our instruction to the court on remand to apply the correct burden of proof as to class membership, since the complete absence of evidence supporting Brown’s position entitled the unions to prevail even under the more stringent standard.

2. Sherman Johnson. The unions challenge the 1972 award to Sherman Johnson for substantially the same reason that they challenge James Brown’s award. In Johnson’s case, however, we affirm the back pay award because the Special Master specifically found that Johnson sought to join the union in 1972. J.A. 434. It is true that Johnson testified that he only sought entrance' to the union in 1970, 1971, and 1973, but the Special Master acknowledged that testimonial omission and pointed instead to Johnson’s certification form, which stated that Johnson sought to join in 1972. J.A. 431 n.100. The unions do not challenge the Special Master’s findings on the certification form and thus have waived any objection to it. We therefore affirm the 1972 award to Johnson, subject óf course to any necessary re-calculation if the district court alters the benchmark figures on remand.

3. Robert Posley. The unions challenge the back pay award to Robert Posley for the portion of 1974 (29%) in which he did not have 2,150 hours of Local 201 union experience. The class’ only response is that the Special Master did not err by measuring experience in union and non-union hours. As we held above, although we agree with the class that experience can be measured in union and non-union hours, a showing of 2,150 hours of experience is a prerequisite to class membership. Because of the Special Master’s contrary position on this latter point, we cannot be confident at this juncture that his conclusion that.by October 1972 Posley “had been doing iron work for both union and non-union contractors for over four years” is consistent with the 2,150 hour prerequisite. Indeed, the Master said nothing about the number of hours Posley worked at all. This uncertainty's complicated by the stringent burden of proof that the Special Master erroneously imposed upon the unions to rebut Posle/s testimony. We leave it to the district court on remand to decide whether Posley’s award for 1974 is consistent with the principles we have outlined in this opinion.

4. Randolph Jackson and Ernest Bellamy. The district court’s Order of Reference directed that a class member’s entitlement to back pay ends on “the date when he first was allowed to take the journeyman examination ... or was given a bona fide opportunity to take the examination.” J.A. 216-17. The unions challenge the 1975 back pay award to Randolph Jackson to the extent the award postdates Jackson’s failure of the exam in March 1975. The class challenges the back pay award to Ernest Bellamy for the opposite reason; they claim that the Special Master erroneously denied Bellamy back pay for the period after September 30, 1974, the date on which Bellamy failed the journeyman’s examination.

We think it obvious that the Special Master’s findings, which the district court adopted without comment, are inconsistent. In discussing Jackson’s award, the Special Master awarded Jackson back pay for all of 1975, even though Jackson failed the exam in March of that year. But in discussing Ernest Bellamy’s award, the Special Master cut off back pay after the date on which Bellamy failed the exam, noting the Order of Reference and the fact that there were no challenges to the validity of the examination itself at the merits stage. J.A. 365 & n.61. We reconcile the inconsistency by reversing the 1975 award to Jackson and affirming the truncated award to Bellamy.

The class’ sole argument in support of the contrary result is that the Order of Reference should be read to cut off back pay when a claimant is given a bona fide opportunity to take the exam. The class reads “bona fide opportunity” to mean “bona fide exam,” and argues that neither Jackson’s nor Bellamy’s exam was a bona fide one. The class further argues that the Special Master implicitly credited Jackson’s contention that Ronnie Vermillion, the union business manager, was lying when he claimed that Jackson failed the exam in March 1975 because Jackson, an experienced rodman, did not know the steel tubing sizes. J.A. 424-25. In addition, the class contends that the Special Master erroneously excluded evidence that Bellamy intended to use to demonstrate that his exam too was not a bona fide one.

We think the unions are quite correct in contending that these arguments are really challenges to the administration of the journeyman’s exam — challenges which were not made at the merits stage and which we cannot, and will not, entertain at this late stage. The Special Master recognized this point in his discussion of Bellamy’s award, and in his discussion of another claimant not part of this appeal, J.A. 398-99, but awarded Jackson a full award because he did not believe that Jackson’s exam was bona fide. Moreover, as the unions point out, this court has repeatedly noted that the Title VII liability of the unions in this ease is not related to the journeyman’s exam itself or to its administration, but solely to the educational prerequisites to taking the examination. Berger I, 843 F.2d at 1440; Berger II, 852 F.2d at 621. And although we do not east doubt on the Special Master’s factual finding that Vermillion lied to Jackson about his failure of the exam, the Order of Reference speaks only to the fact of taking the exam, not to the validity of the union’s determination that a claimant passed or failed the exam. And no one contests that Jackson or Bellamy actually took the exam. Finally, the term “bona fide” in the order of reference clearly modifies the term “opportunity,” both of which are set off by a disjunctive from the phrase “allowed to take the ... exam.”' The class is thus wrong when it argues that the unions’ interpretation would render the term “bona fide opportunity” meaningless. Rather, to accept the class’ reading would be to excise “allowed to take the ... exam” from the Order of Reference, converting the remedial inquiry into the altogether distinct liability question of the bona fides of the exam. For back pay purposes, the inquiry into bona fides in this case is limited to examining, where appropriate, whether a claimant passed up a legitimate opportunity to take the exam.

We therefore affirm this aspect of Bellamy’s award and reverse the district court’s decision to grant Jackson an award for the period following his 1975 failure. We instruct the district court to reduce Jackson’s 1975 award by the appropriate amount, considering of course any alterations to the benchmarks that the court might make on remand. We note finally that although the district court’s error was harmless for 1976, since Jackson was not entitled to any recovery for 1976 under the current benchmark proxy, any alteration of the benchmarks must not result in a 1976 award to Jackson given our holding that his entitlement to back pay ended when he took the exam.

5. Eldridge Harmon. The unions challenge the back pay award to Eldridge Harmon for the years 1985 and 1986 because Harmon forewent the opportunity to join Iron Workers Local 84 in Houston, where Harmon resided from 1976 to 1985. Local 84 is, like Local 201, an affiliate of the International, and the International’s governing constitution provides that a two-year member of any Iron Workers local may obtain a “clearance card” from his local union to apply for a transfer of' membership to any other local. Harmon completed Local 84’s two-year training program, but did not obtain membership in that union because he failed to pay the initiation fee. The unions assert that Harmon’s back pay awards for 1985 and 1986 are clearly erroneous because Harmon failed to avail himself of the opportunity to become a member of Local 84, which would have enabled him to join Local 201 without having to complete Local 201’s discriminatory prerequisites. The unions further argue that the Special Master’s refusal to accept this argument is inconsistent with his treatment of Edgar James, another claimant who was denied class membership in part because of his failure to avail himself of membership in Local 201 through a “clearance card” procedure. J.A. 427-30. The class counters that the unions’ argument is highly speculative since the International’s governing constitution gives Local 201 the discretion to reject a clearance card from another local union.

We affirm this aspect of Harmon’s award. The Order of Reference only requires back pay termination when a claimant takes the journeyman’s exam, or has a bona fide opportunity to do so; it has no provision for terminating back pay in light of a failure to avail oneself of an alternative mechanism for becoming a Local 201 member. And although it is true that the Special Master discussed Edgar James’ failure to take advantage of a “clearance card” procedure to gain entrance to Local 201, that discussion focused on James’ inability to prove membership in the class. James had never attempted to join Local 201, and the Special Master found that James could not have been discouraged from doing so (the alternative means of proving class membership) since James could have joined Local 201 through the “clearance card” procedure. J.A. 430. Harmon, on the other hand, is clearly a member of the class because he actually applied for membership in Local 201 and was denied (rendering the discouragement issue irrelevant). There is thus no inconsistency between the Special Master’s treatment of Harmon and James, and no basis under the Order of Reference to reverse the award to Harmon as clearly erroneous. This is so even though the unions were subjected to the incorrect clear and convincing evidence standard since, under the more lenient preponderance of the evidence standard, Harmon still would be entitled to his award.

6. Jessie Berger, Silburn Francis, Eldridge Harmon, Thomas Kirkland, and Sherman Johnson. The unions assert that these five claimants were erroneously given back pay awards for periods during which they had injuries and were unavailable to work. The class counters that the benchmark proxy figure already takes into account time off due to minor injuries and bad weather, making it unnecessary to reduce an individual claimant’s award for those reasons. For the same reason, the class challenges the Special Master’s reduction of Sherman Johnson’s award for the one month in 1975 during which Johnson had asthma and could not work. The district court upheld all of these awards without comment.

The Special Master recognized its obligation in adopting a back pay formula to “as nearly as possible, recreate the conditions and relationships that would have been had there been no unlawful discrimination.” International Bhd. of Teamsters, 431 U.S. at 372, 97 S.Ct. 1843. The Special Master’s proxy does, as the class contends, factor in the “average number of days lost - due to injury, sickness, and attrition.” J.A. 341. But the Teamsters obligation arguably requires the district court to modify the benchmark proxy for any claimant whose absenteeism is so extreme as to be beyond the proxy’s statistical average. The Special Master recognized as much in the case of Van Edward Lewis, whose three-year shoulder injury represented an “extensive period of unemployment due to injury [that] falls outside our statistical model of reasonable hours” developed in the benchmark proxy. J.A. 445. The problem, however, is that neither the Special Master nor the district court explained the extent to which the “statistical model of reasonable hours” factors in absenteeism due to injuries. Consequently, the district court had no objective basis on which to determine when a claimant’s injury-time was sufficiently excessive to render it beyond the statistical average.

It is not surprising, then, that the analysis below is an ad hoc, internally inconsistent evaluation of the back pay calculation for claimants who suffered injuries during the back pay period. For example, the Special Master reduced Eldridge Harmon’s back pay for the one month that he could not work due to a back injury, J.A. 409, and reduced Sherman Johnson’s back pay for the one month that he could not work due to an asthma condition. J.A. 144. At the same time, the Special Master ignored the evidence relating to Jessie Berger’s 12-week prostate surgery recovery in 1974, J.A. 367, ignored evidence relating to Silburn Francis’ six-week injury in 1980 after a rod struck him in the stomach, J.A. 405, and ignored evidence of Thomas Kirkland’s six-week back injury in-1976, awarding full back pay for the relevant period to each claimant. Clearly, if a one-month injury warrants a reduction, so too must injuries lasting six and twelve weeks. But we, like the district court, have no objective basis on which to resolve the inconsistency because there has been no finding as to the statistical injury average or how long an injury must last to go beyond that average. We therefore remand to the district court for a determination of the average injury-time built into the benchmark figures, and a thorough inquiry into whether each challenged award involves a claimant whose injury-time exceeds that average.

There is one back pay challenge, however, that we can partially resolve now. The district court awarded Thomas Kirkland a full back pay award for 1976, even though Kirkland stipulated that he was unable to work for the six weeks when he had a back injury and was therefore not asking for back pay for that period. J.A. 660. The award is therefore clear error; we remand for the district court to reduce Kirkland’s award to reflect the six-week injury.

7. John Thomas. The class challenges the district court’s denial of back pay to John Thomas for the years 1973 to 1975. The sole basis on which the Special Master and the district court denied back pay for this period was that Thomas’ pension records show steady and full-time employment during that period. But the class is correct that steady employment only deprives a claimant of back pay if the earnings from that employment exceed the benchmark earnings for that year. The Special Master did not conduct the necessary analysis, and as the class demonstrates, it appears that Thomas’ earnings for each year between 1972 and 1975 fell short of the benchmark amount. The unions concede the class’ general argument, but argue that the district court did not err by denying Thomas an award for 1973 because Thomas’ employer for that year reported the maximum amount ($10,800) that any single employer was required to report for FICA taxes. The class counters that the question is whether Thomas -earned less than the $13,-917 benchmark in 1973, which according to their calculations, he did.

The denial of all back pay for 1972-75 years is clear error. We remand Thomas’ award for the district court to carry out the analysis and award the appropriate back pay pursuant to the method used to resolve the other claimants’ awards. In this regard, the 1973 award is no different from the 1972, 1974, and 1975 awards that the unions concede were erroneously denied to Thomas. We leave it to the district court to determine whether, as the class contends, Thomas’ 1973 earnings result in a short-fall entitling him to an award for that year. This calculation may obviously be affected by any alteration to the benchmark proxy figures on remand.

8. Charles Dean. The class also argues that the district court’s determination that Charles Dean was admitted to Local 201 in January 1981 is clear error; the record clearly shows he was admitted in January 1982, and the unions concede the class’ argument. If the district court does not alter the benchmarks on remand, or lowers them, this error is of no consequence because Dean’s 1981 earnings exceed the current benchmark. However, if the benchmarks are raised on remand such that Dean would otherwise be entitled to an award for 1981, we instruct the district court not to apply its clearly erroneous finding to deprive Dean of an award.

V. Mitigation

The unions challenge many of the Special Master’s back pay awards on the ground that the claimants failed to mitigate their damages adequately. Under Title VII, “[i]nterim earnings or amounts earnable with reasonable diligence by the person or persons discriminated against shall operate to reduce the back pay otherwise allowable.” 42 U.S.C. § 2000e-5(g). This creates a statutory duty to minimize damages on the part of Title VII claimants, which requires them “to use reasonable diligence in finding other suitable employment.” Ford Motor Co. v. EEOC, 458 U.S. 219, 231, 102 S.Ct. 3057, 73 L.Ed.2d 721 (1982). The victim of discrimination, however, is “merely required to make ‘reasonable efforts’ to mitigate his loss of income, and only unjustified refusals to find or accept other employment are penalized under this rule.” Oil, Chem. & Atomic Workers Int’l Union v. NLRB, 547 F.2d 598, 603 (D.C.Cir.1976). “[T]he employee is held ... only to reasonable exertions in this regard, not the highest standard of diligence.” NLRB v. Madison Courier, Inc., 472 F.2d 1307, 1318 (D.C.Cir.1972) (Madison Courier I) (internal quotations and citations omitted).

A claimant “forfeits his right to back pay if he refuses a job substantially equivalent to the one he was denied.” Ford Motor, 458 U.S. at 232, 102 S.Ct. 3057. But “the unemployed or underemployed claimant need not go into another line of work, accept a demotion or take a demeaning position.” Id. at 231, 102 S.Ct. 3057. Nor is he “required to accept employment at a great distance from his home.” Oil, Chem. & Atomic Workers, 547 F.2d at 604. On the other hand, a claimant may reasonably conclude that he should lower his sights and seek other work, including work outside the industry. NLRB v. Madison Cornier, Inc., 505 F.2d 391, 396 (D.C.Cir.1974) (Madison Courier II). “The claimant,” after all, “cannot afford to stand aside while the wheels of justice grind slowly toward the ultimate resolution of the lawsuit. The claimant needs work that will feed a family and restore self-respect.” Ford Motor, 458 U.S. at 221, 102 S.Ct. 3057. Indeed, a claimant “may be required ... to ‘lower his sights’ by seeking less remunerative work after he has unsuccessfully attempted for a reasonable period of time to locate interim employment comparable with his improperly denied position.” Madison Courier I, 472 F.2d at 1321.

As the above discussion suggests, the elements of the mitigation doctrine can create a dilemma for a claimant. As we said in Madison Courier I,

If the discriminatee accepts significantly lower paying work too soon after the discrimination in question, he may be subject to a reduction in back pay on the ground that he willfully incurred a loss by accepting an unsuitably low paying position. On the other hand ... if he fails to ‘lower his sights’ after the passage of a reasonable period of unsuccessful employment searching, he may be held to have forfeited his right to reimbursement on the ground that he failed to make the requisite effort to mitigate his losses.

Id. Because of this dilemma, we held that “courts must be careful when applying” the mitigation doctrine, and that “it would not be unreasonable ... to resolve doubts in this area in favor of the innocent discriminatee.” Id. “[T]he burden of establishing facts in mitigation of the back pay liability” is therefore upon the violator. Id. at 1318; accord Oil) Chem. & Atomic Workers, 547 F.2d at 603.

In addressing the unions’ mitigation challenges, we are hampered by the Master’s failure to address the mitigation question with respect to a number of the challenged claimants. Where the Master has been silent, we can uphold an award only if the unions offer nothing to support a claim of non-mitigation other than an inadequate legal theory, and hence fail to satisfy their burden of proving non-mitigation.

A.

We begin with four claimants whom the unions contend “did not consistently seek Local 201 referrals” between 1972 and 1975, when there was a surfeit of work available through the Local. Def. Br. at 54. Because the Local had more than enough work during this time for any permit man who wanted it, the unions contend that a failure to seek referrals from the union constituted a failure to mitigate. See id. at 54-55, 59-60.

1. O.C. Brown. Surprisingly, the first claimant the unions offer as an example of one who failed to seek referrals from the Local is O.C. Brown, who the unions concede did seek and receive many referrals between 1972 and 1978. Id. at 56. The problem with Brown, the unions assert, is that he held few of those referred jobs for very long because of his “chronic, voluntary, premature quits,” and that as a result his yearly work hours were low. Id. at 57. The Special Master, however, credited Brown’s testimony on the subject and found that the reason for those “quits” was that “even when referred, Brown was fired on instructions from the business agent [for the union] solely because of his status as a permit man.... ” J.A. 382. The unions cite only one specific example of a “quit,” Brown’s decision to leave a job at Wahib Steel because of a dispute with a foreman. But the Master found that “Brown was not unemployed after quitting Wahib but appears to have immediately obtained employment” from another employer. J.A. 381. Accordingly, the Master concluded that no deduction from Brown’s back pay award was required, and we cannot find that conclusion clearly erroneous.

As noted in Part IV above, however, the unions have asserted another ground for deduction in Brown’s case: that Brown voluntarily absented himself from the workforce every winter. There is evidence in the record to support this assertion. See J.A. 772-75. Although there may be reasons why such absences do not constitute a failure to mitigate (e.g., because little rodwork was done in the winter, a point made by the unions’ own expert, see J.A. 332), the Master did not address Brown’s seasonal absences at all, and we therefore must remand his award for further consideration.

2. Silburn Francis. The second claimant the unions challenge for not seeking Local 201 referrals is, again, a claimant who the unions concede did seek and receive referrals from the union. Moreover, the unions concede that Silburn Francis, unlike O.C. Brown, “worked high numbers of hours through Local 201 between January 1, 1971 and June 30, 1974.” Def. Br. at 57. Nonetheless, the unions contend that Francis “achieved those hours only by working an unusually high number of different jobs,” and thus was “a chronically lackadaisical worker” who could not hold a job. Id. at 56-57. Once again, the Special Master drew a different conclusion from the same testimony and documentary record. The Master read Francis’ employment history not as indicating that he was “lackadaisical,” but as “demonstrating] Francis’ tenacity in seeking work as an ironworker.” J.A. 405. Francis, the Master found, “made good faith and diligent efforts to obtain employment through references from Local 201.” J.A. 405. That finding is not clearly erroneous.

3. Eldridge Harmon and James Hicks. Although Eldridge Harmon did seek referrals from the union, he mostly worked on non-union jobs. Similarly, James Hicks worked for a number of non-rodwork employers. Because there was more than enough union rodwork available during this period, and because that work presumably paid higher wages, the unions contend that claimants’ failure to seek solely Local 201 work during this period constituted a failure to reasonably mitigate. They were “not available for Local 201 referrals,” the unions contend, when they were “working elsewhere.” Def. Br. at 60.

Whether the decisions of these claimants not to seek work solely through the Local constituted a failure to mitigate depends on the reasons they had for taking other work. As we noted in Oil, Chemical & Atomic Workers, it may be reasonable for a claimant to decline an interim job from his employer (other, of course, than the very job at issue in his lawsuit) in favor of a lower-paying but more permanent job from someone else. 547 F.2d at 604-05. A fortiori, it may be reasonable to decline to leave an existing job when doing so would only make oneself available for possible referral to a better-paid one. We do not know whether these kinds of considerations explain Harmon’s or Hicks’ decisions, however, because the Special Master did not discuss mitigation with respect to Harmon or Hicks at all. Accordingly we have no choice but to remand their awards for further consideration.

B.

The unions next address the post-1975 period, which saw employment patterns in the rodwork industry fluctuate. “Even during this period,” the unions argue, “Local 201-referral jobs went unfilled for lack of applicants.” Def. Br. at 60. The unions therefore again contend that a claimant did not reasonably mitigate if he did not seek work through Local 201. In a set of further, sometimes contradictory arguments, however, the unions contend that a claimant did not reasonably mitigate if he did not also seek union rodwork in other cities, seek non-union rodwork, seek other construction work, and register with government employment agencies. We consider these individual challenges below.

1. James Brown. The unions do not dispute that James Brown sought and received referrals from Local 201. In a one-sentence challenge to Brown’s award, however, they argue that he “did not seek work through any other Iron Workers Local union or through any other union during 1975 and 1976.” Id. at 63. That argument is insufficient' to satisfy the unions’ burden. First, this kind of challenge to Brown’s 1975 award directly contradicts the unions’ argument— discussed in Part V.A above—that because Local 201 had more than enough work for permit men from 1972-75, a claimant’s failure to seek work solely through Local 201 during that period constituted a failure to mitigate. Nor do the unions offer evidence that in 1976 Brown would have had a better chance of obtaining union work in other cities than by continuing to seek referrals from Local 201. Since, as the Master noted, Brown’s strategy of seeking work through Local 201 earned him nearly as much or more than the benchmark wages in 1972-74, J.A. 375, and since the unions concede “Local 201-referral jobs went unfilled for lack of applicants” even during the post-1975 period, Def. Br. at 60, the unions’ single-sentence challenge does not meet their burden of showing that Brown was unreasonable in continuing to seek work through Local 201. Brown’s awards are affirmed.

2. Sherman Johnson. The unions contend that Johnson should not have been awarded back pay for 1975 because he sought no work from non-union companies, non-Local 201 unions, or non-rodwork employment during that year. Once again, this directly contradicts their contention that 1975 was a “full employment” year at Local 201, with plenty of work for any permit man who wanted it, and consequently that any claimant who did not seek work solely through Local 201 failed to mitigate. The award is affirmed.

3. Charles Dean. The unions argue that Charles Dean failed to mitigate because he never applied to the leading nonunion employer, Miller & Long, during the 1975-79 period. The unions, concede, however, that Dean did work for other non-union firms, id. at 64, and offer no evidence that Dean could have done better at Miller & Long. Indeed, the Master found that Dean’s actual hours during this period approached or exceeded the benchmark figures in all relevant years. Accordingly, the unions cannot meet their burden of showing a failure to reasonably. mitigate. The awards are affirmed.

4. Van Edward Lewis. The unions challenge Lewis’ awards for 1977 and 1979, claiming that he ceased seeking work through Local 201 in 1976 and did not return to the union until 1980. The unions offer no evidence, however, that supports this claim. The portion of Lewis’ testimony cited in support says that he ceased trying to “join the training program” in 1976, not that he ceased seeking permit man work through Local 201. See Def. Br. at 65 (citing J.A. 736). The Master’s report does suggest that Lewis may not have sought Local 201 work in 1979, but that is" only because it shows he worked for Miller & Long during that year—the same non-union employer from which the unions insist Charles Dean should have gotten his work. Accordingly, the awards are affirmed.

5. Thomas Kirkland. Citing a less-than-clear portion of Kirkland’s testimony, the unions contend that he voluntarily ceased looking for work during the last quarter of 1976 and hence failed to mitigate during that period. The Master did not address this issue at all, and we therefore remand this portion of Kirkland’s award for reconsideration.

6. John Offer. The unions challenge Offer’s awards for 1975 and 1976, on the ground that he did not seek in-town work from non-union rod companies, or out-of-town work from union companies. The chállenge to the 1975 award fails for the same reason it failed in the cases of James Brown and Sherman Johnson. We must remand the 1976 award, however, because the Special Master failed to discuss the mitigation issue despite claimant’s low earnings that year.

7. John Thomas. The unions challenge Thomas’ awards for 1976, 1977 and 1979 on the basis of his asserted failure to adequately mitigate. With respect to 1976, they contend that he principally collected unemployment compensation rather than working. Because Thomas earned only $338 in that year, and because the Master did not address the mitigation issue for that year at all, we remand the 1976 award. With respect to the remaining two years, however, the Master noted that Thomas earned approximately 80% of the benchmark figure in 1977 (by working for Miller & Long) and 98% of the benchmark figure in 1979. J.A. 480. These figures suggest reasonable mitigation in those years, and because the unions offer no evidence that Thomas could have done better by seeking any other kind of employment, we affirm those awards.

8. Ronald Tucker. The unions challenge the awards of back pay to Tucker for 1975 and 1977, on the ground that he did not seek work through any union other than Local 201 or register with an employment-agency. The challenge with respect to 1975 fails for the same reason it failed in the case of the other claimants’ 1975 awards. With respect to 1977, the Master awarded back pay to Tucker for only one calendar quarter and noted that during that entire quarter Tucker worked for a steel company in Baltimore. The unions have proffered no evidence that other work would have paid more, or that Tucker’s mitigation efforts were otherwise unreasonable. They are thus unable to satisfy their burden of showing a failure to mitigate. The awards are affirmed.

C.

Finally, we also consider the class’ challenges to the Special Master’s decision to truncate the awards of four claimants on the ground that after certain dates those claimants “abandoned” Local 201. We remand two of those decisions, and affirm the other two.

1. James Brown and Ronald Tucker. The Special Master cut off Brown’s and Tucker’s back pay awards after 1976 and 1977, respectively, because they obtained non-union employment and ceased to seek union referrals. J.A. 376, 483. We agree with the unions that this issue is properly evaluated by applying the mitigation doctrines described above. See Def. Reply Br. at 6, 13-14. But the Master’s decision to truncate the awards solely because the claimants chose to keep working at alternative employment, rather than constantly to seek new union referrals, misapplies those doctrines and requires a remand.

To infer a breach of the duty to mitigate solely from a claimant’s acceptance of other work implicates the dilemma noted at the beginning of this Part, and creates a Catch-22 situation for the claimant. As the Third Circuit has said:

[T]he fact that a plaintiff takes a job in an unrelated field to meet her obligation of mitigation should not be construed as a voluntary withdrawal from her former profession. Otherwise, a plaintiff would be put in the intolerable position of choosing between foregoing a source of earnings during the interim before trial or risking an adverse finding on abandonment of her profession. Such a rule would also work to the disadvantage of employers because the scope of the mitigation obligation necessarily would be relaxed. It is conceivable that a plaintiff, wronged by discrimination, would decline to take a job that would substantially mitigate damages because such employment could be construed as an abandonment of her former vocation.

Ellis v. Ringgold Sch. Dist., 832 F.2d 27, 30 (3d Cir.1987). Here, claimants did not even choose work in an unrelated field, as the plaintiff did in Ellis. Rather, they did just what the unions have asserted they were obligated to do: when unable to fill their hours with Local 201 work, they sought and successfully obtained non-union work instead. See supra Part V.B. To cut off their back pay now would truly be to apply a Catch-22: claimants would have been ineligible for back pay had they not tried to obtain non-union work, and would now be ineligible precisely because they succeeded in obtaining it.

It may be that a reasonable claimant would have known he could have done even better by constantly checking with Local 201 for referrals, although there was record evidence that such constant checking (and the tardiness it would have caused at the claimant’s current employment) would have put his nonunion work in jeopardy. See J.A. 663, 761-62, 808-11. It is also possible that having obtained non-union work, those claimants were satisfied and had no intention of ever returning to Local 201, although they contend they “would have preferred to work out of Local 201 as journeymen” and took the non-union work only because they had no choice. PI. Br. at 60. None of these points was discussed by the Special Master, however, and no findings were made on either side. Accordingly, we must remand the truncation of these awards for further consideration and appropriate application of the law relating to mitigation.

2. Sherman Johnson and John Offer. The Special Master’s decision to truncate the awards of Johnson and Offer presents a different question. The Master found that Johnson’s medical condition (chronic bronchial asthma) caused him to abandon his pursuit of union referrals in 1976. J.A. 437. Similarly, the Master found that Offer abandoned rodwork altogether after 1976 because the work was too physically demanding for him. J.A. 458. Although both claimants obtained work in other fields, the Master’s decision to truncate their awards did not rest simply on the fact that they took that work, but rather on his finding that the reason they did was because they were no longer able to do the kind of work referred by Local 201. This does not raise the Catch-22 concern noted above, and the Master’s finding was not clearly erroneous. The truncation of these two awards is affirmed.

VI. Compensatory Damages

The unions challenge the Special Master’s decision to award compensatory damages to 18 claimants. They correctly note that compensatory damages serve only to compensate injuries that result from violations of constitutional or statutory rights, and may not be “presumed to flow from every deprivation” of those rights. Carey v. Piphus, 435 U.S. 247, 263, 98 S.Ct. 1042, 55 L.Ed.2d 252 (1978). “Where no injury [is] present, no ‘compensatory damages [may] be awarded.” Memphis Community Sch. Dist. v. Stachura, 477 U.S. 299, 308, 106 S.Ct. 2537, 91 L.Ed.2d 249 (1986).

The Supreme Court, however, has distinguished the impermissible award of compensatory damages — where they are presumed merely from the violation of a right— from the “form of presumed damages [that] may possibly be appropriate ... [to] roughly approximate the harm that the plaintiffs suffered....” Id. at 311, 106 S.Ct. 2537. Similarly, in Hobson v. Wilson, this court stated that “in appropriate circumstances the infliction of emotional distress may be inferred from the circumstances of the violation.-” 737 F.2d 1, 62 n. 173 (D.C.Cir.1984). The critical distinction made by both Memphis and Hob-son is that courts may properly infer emotional distress from factual circumstances— and award damages to compensate for that distress — but may not presume damages from a bare violation of a statutory or constitutional right. See All U.S. at 311, 106 S.Ct. 2537, 737 F.2d at 62 n. 173.

The awards in the instant case are supported by the proper kind of inference. There can be little doubt that claimants, who were experienced rodmen, suffered emotional distress by having to subject themselves to an unnecessary training program for up to two years before being permitted to take the union entrance exam. Those circumstances more than adequately support the extremely modest awards granted here, which range from $2,500 to $25,000.

The unions also complain that the Special Master granted compensatory damages to three claimants who did not seek them. The district court upheld those awards on the basis of Fed. R. Civ. P. 54(c), which provides that “every final judgment shall grant the relief to which the party in whose favor it is rendered is entitled, even if the party has not demanded such relief in the party’s pleadings.” J.A. 537 (Mem. Op.) (quoting Fed. R. Civ. P. 54(c)). Although the cited rule may provide the discretionary authority necessary to make the awards to the three claimants, no explanation was offered as to why-the Special Master simultaneously failed to award compensatory damages to another claimant for no reason other than that he did “not seek compensatory damages.” J.A. 435 n.101 (Sherman Johnson); see also J.A. 451 (estate of James McGee). Without such an explanation, we are unable to determine whether this inconsistency reflects a rational distinction or an abuse of discretion, and we therefore remand the compensatory damage awards to James Brown, Paul Brown and Silburn Francis for reconsideration and explanation. With those exceptions, the compensatory damages awards are affirmed.

VII. Punitive Damages

The class alleges that the Special Master applied an incorrect standard in denying them an award of punitive damages. The Master stated that “punitive damages will be recoverable for conduct exhibiting malice, evil motive, recklessness or callous indifference to a federally protected right.” J.A. 319 (citing Smith v. Wade, 461 U.S. 30, 52, 103 S.Ct. 1625, 75 L.Ed.2d 632 (1983)). This is the same standard relied upon by both the majority and the dissent in this circuit’s leading case on the issue. See Kolstad v. American Dental Ass’n, 139 F.3d 958, 964-65 (D.C.Cir.1998) (en banc), cert. granted, — U.S. —, 119 S.Ct. 401, 142 L.Ed.2d 326 (1998); id. at 971 (Tatel, J., dissenting). Moreover, as we also noted in Kolstad, “punitive damages ‘are never awarded as of right, no matter how egregious the defendant’s conduct.’ ” 139 F.3d at 965 (quoting Smith, 461 U.S. at 52, 103 S.Ct. 1625). Rather, they are “awarded or rejected in a particular case at the discretion of the fact finder.” Id. (internal quotation omitted); accord id. at 972 (Tatel, J., dissenting). We have no basis for overturning the Master’s discretionary decision here.

VIII. Prejudgment Interest

Finally, both the unions and the class challenge the award of prejudgment interest at a rate of 6%, compounded annually, for the entire period of the litigation. The unions argue that no interest should have been awarded at all or, in the alternative, that no interest should have been awarded for certain periods of time. The class argues that 6% is too low a rate, and that interest should have been awarded at a variable rate. We reject all of these contentions and affirm the decision of the district court.

The back pay provision of Title VII “is a manifestation of Congress’ intent to make persons whole for injuries suffered through past discrimination,” and “[p]rejudgment interest, of course, is an element of complete compensation.” Loeffler v. Frank, 486 U.S. 549, 558, 108 S.Ct. 1965, 100 L.Ed.2d 549 (1988) (internal citations and quotations omitted). For that reason, we have held that “prejudgment interest ‘must be an ordinary part of any award of back pay ... under § 1981.’ ” Barbour v. Merrill, 48 F.3d 1270, 1278 (1995) (quoting Williamson v. Handy Button Mach. Co., 817 F.2d 1290, 1297 (7th Cir.1987)). The decision as to how to compute prejudgment interest is within the discretion of the district court. Forman v. Korean Air Lines Co., 84 F.3d 446, 450 (D.C.Cir.1996).

In the instant case, both sides cited a variety of circumstances that might support an exclusion of certain time periods on the' one hand, or a variable rate of interest on the other. The district court, after reviewing these arguments, chose the 6% rate for the entire period, principally on the ground that the parties had once consented to that rate. J.A. 521-22. The unions are wrong in arguing that delays for which they are not responsible mandate tolling of prejudgment interest. See Bufco Corp. v. NLRB, 147 F.3d 964, 967 (D.C.Cir.1998) (refusing to toll interest when NLRB may have been responsible for delay). The class, on the other hand, is equally wrong in contending that it was an abuse of discretion for the district court to impose a fixed rate in large part because of their earlier concession that such an interest rate would make them whole. See TI Fed. Credit Union v. DelBonis, 72 F.3d 921, 928 (1st Cir.1995) (distinguishing between binding effects of factual and legal stipulations). We thus affirm the decision of the district court awarding prejudgment interest at the rate of 6%, compounded annually, for the entire period of the litigation.

SILBERMAN, Circuit Judge,

concurring:

We have strained hard — perhaps too hard — to decide as much of this case as we could. As our background section indicates, the district court’s interminable delays are inexcusable and have caused a great hardship to the parties, particularly the class. I am terribly concerned that our remand to this district judge is equivalent to dropping the case into a well, and, therefore, we should be prepared to grant extraordinary relief if there is further unjustified delay.

It seems to me that all the district judges — the whole district court — should assume responsibility for unwarranted delays in the processing of cases. The court of appeals has a rule, the September Rule, which has been vigorously enforced, that prevents any judge from sitting on cases in the fall if he or she has more than three assigned majority opinions outstanding over six months. I see no reason why the district court could not adopt an analogous rule, more tailored to its circumstances, that would force district judges to process cases in a timely fashion or else be disqualified. The court of appeals can only act episodically as cases are brought to us; it is not our responsibility to supervise district judges.

SENTELLE, Circuit Judge,

concurring and dissenting in part:

I concur in the court’s opinion with the exception of Part III.B.7, discussing the eligibility of claimants Paul Brown, James Hicks, and James Brown. There the court determines that failing the Open Period examination, or failing to take it when offered, does not exclude a claimant from class membership. The conclusions reached in the court’s opinion do not follow from our holding in Berger I. Accordingly, I respectfully dissent.

The language quoted by the court as defining the class is correct, so far as it goes, but it is based on an incomplete examination of our opinion. As the majority notes, “[t]he union is liable, we said ‘to those class members who were experienced workers, but were delayed entry to union ranks by the particular educational prerequisites affecting them from the end of the Open Period until the filing of suit on October 21, 1975.’ ” An applicant who failed the Open Period exam, as Paul Brown and James Hicks did, or failed to take it despite being offered an opportunity to do so, as James Brown did, was kept out of the union on the basis of that failure, not on the basis of an impermissible educational prerequisite.

In Berger I, we recognized that the rod trade has historically been apprenticeable, and noted that “it stands to reason that on-the-job experience alone may not necessarily teach all that a fully qualified rodman should know.” Berger, 843 F.2d 1395, 1420. We pointed to the existence of the Open Period exam as proof that the union could devise an examination that properly tested experienced rodmen to see if they were qualified even though they had not been through an apprenticeship program. “In our view, the Open Period establishes that experience can qualify one to be a journeyman rodman, and, not incidentally, that the Union is capable of devising an exam that screens out insufficiently competent applicants for journeyman status." Id. at 1421 (emphasis in original).

Under our analysis, the Union remains free, among other things, to (1) require significant rodman experience before an applicant may be admitted to the journeyman exam, (2) offer (cured of discrimination against experienced workers) both the Apprenticeship and Training programs, and (3) devise a more exacting or thorough exam for rodmen who eschew classroom training to assure that skills (e.g., reading blueprints) learned in the classroom have been learned on the job (so long, of course, as any such “stepped-up” exam satisfies the bedrock requirements of job-relatedness).

Id. If we are pointing to the Open Period exam as proof that the union could create an acceptable exam, it does not follow that failure of that exam should not properly be deemed to preclude someone from membership as unqualified. Therefore the - proper course for the union to take with regard to someone who failed the Open Period examination was to do precisely what it did, require them to take courses in an apprenticeship program, and then administer the second test. In Berger I, we recognized that it may have been harder, but accepted the increased difficulty. However, one significant factor, overlooked by the majority opinion, that may explain the difference in passage rates between the two exams is that the rodmen taking the second examination had just finished taking a course designed specifically to help them pass that examination. Rodmen who could not pass the membership examination and were thus deemed “insufficiently competent applicants for journeyman status” cannot show that they were impermissibly discriminated against by the unions, and are not properly members of the class.

GARLAND, Circuit Judge,

concurring and dissenting in part:

I concur in the court’s opinion with the exception of Part II. In that Part, the court remands the Special Master’s benchmark determination — that is, his calculation of the hours claimants would have worked in the absence of discrimination. The questions my colleagues raise about the Master’s calculation are not unreasonable ones. But that is not the test on appeal. It does not matter that we might have made a different calculation had we been sitting as the triers of fact. See Anderson v. Bessemer City, 470 U.S. 564, 573-74, 105 S.Ct. 1504, 84 L.Ed.2d 518 (1985). Instead, to justify remand, appellants must demonstrate that the Master’s calculation was clearly erroneous. See id.; 9A Charles A. Wright & Arthur R. Miller, Federal Practice and Procedure § 2585, at 565 (2d ed.1995). Because they have not done so, I would affirm the Master’s determination rather than needlessly prolong this decades-old case.

I.

In order to determine the number of hours the claimants would have worked had they not been subject to discrimination, the Special Master consulted Local 201 pension records to calculate the average number of hours a representative group of union workers actually worked during the relevant period. The court’s first objection to the Master’s methodology is that he excluded from that group “those who for several years ... worked no hours at all.” J.A. 342-43. By not including these “zero-hour” workers in calculating the hours of an average worker, the court contends that “the Special Master remove[d] from the equation the risk of disabling injury, or of finding another more desirable job, or whatever other reason a person might not work full time.” Op. at 190.

The inclusion of zero-hour workers may be a reasonable way to account for the risk that an individual claimant would have stopped working even if he had been admitted to the union. But it is not the only, or even the most direct, way. The most direct way is simply to deny back pay to those claimants who actually did stop working, rather than build into the benchmark the statistical probability that a hypothetical claimant would have done so. Not unreasonably, the Special Master chose the direct approach.

The Master’s benchmark accounted for the risks of injury and attrition as follows. First, he included within the representative pool those workers whose hours had been reduced by short-term injuries or other absences. As the court notes in Part IV, the “Special Master’s proxy does ... factor in the ‘average number of days lost due to injury, sickness, and attrition.’ ” Op. at 200 (quoting J.A. 341). Second, the Master excluded those who had worked zero hours “for several years,” because they were not representative of union members who were actually working during the relevant period. J.A. 342-43. Finally, to ensure that a claimant who worked zero hours did not receive a windfall, the Master reduced the pay of claimants where there was “an ‘extensive period of unemployment due to injury [that] falls outside [the] statistical model of reasonable hours’ developed in the benchmark proxy.” Op. at 190 (quoting J.A 445).

I do not disagree that the Master appears to have performed this last calculus inconsistently. For that reason, I join Part IV.6 of the court’s opinion, which remands certain challenged back pay determinations for an “inquiry into whether each challenged award involves a claimant whose injury-time exceeds th[e] average.” Op. at 200. But that limited remand is sufficient to remedy the error. As long as the Master denies claimants back pay for actual absenteeism “so extreme as to be beyond the proxy’s statistical average,” Op. at 200, there is no reason to require him also to build the probability of lengthy absences into the benchmark. As the court itself notes in Part II, “[b]oth ... are means of discounting back pay awards to reflect unavailability for work during the liability period.” Op. at 200. Even if the probability-based, zero-hour approach were preferable, it cannot be clearly erroneous for the Master to have chosen the direct-reduction approach instead. See Anderson, 470 U.S. at 574, 105 S.Ct. 1504 (“Where there are two permissible views of the evidence, the factfinder’s choice between them cannot be clearly erroneous.”).

II.

The court’s second criticism of the Special Master’s calculation is its asserted failure to recognize that the number of work-hours available for union members during the relevant period was a “fixed pie.” If the claimants had been admitted into the union, the court contends, that fixed pie of hours would have been divided among a greater number of workers. Hence, each union member would have worked fewer hours than union members actually worked during the period. There are two reasons to reject this critique.

First, defendants did not make this argument in their briefs before this court. Indeed, the term “fixed pie” cannot be found anywhere therein. See Def. Br. at 24-32; Def. Reply Br. at 4-6. We routinely and for good reason refuse to consider contentions not raised in a party’s briefs. See Boggs v. Rubin, 161 F.3d 37, 42 (D.C.Cir.1998) (holding that “[w]e will not consider at this late stage an argument that the appellant failed to raise” in his briefs); Diamond Walnut Growers, Inc., v. NLRB, 113 F.3d 1259, 1263 (D.C.Cir.1997) (en banc) (“[I]t is well-established ... that we do not consider arguments not presented to us.”). Although it is not impossible to tease the recipe for a fixed-pie argument out of a single sentence in which defendants described the calculations performed by their own expert, their briefs did not argue that a fixed-pie problem rendered the Special Master’s calculations clearly erroneous. As we have said in another context, a reviewing body “need not sift pleadings and documents to identify arguments that are not stated with clarity by a petitioner.” Bartholdi v. FCC, 114 F.3d 274, 279 (D.C.Cir.1997) (internal quotations omitted); see also United States v. Gilliam, 167 F.3d 628, 640 n.10 (D.C.Cir.1999) (“[T]he court will not construe the briefs to raise an argument that is hinted at but never stated.”).

Second, there is substantial support in the record for the Master’s conclusion that “there really [was] no fixed pie” of available hours. J.A. 339. As the court explains, the Local referred work to both union members and non-union workers. The defendants’ expert assumed that had claimants become members of the union, they would have displaced non-union workers first. There would thus be no fixed-pie problem, the expert said, as long as the hours referred to non-union workers in a given year were more than the potential “claimant hours” — which he defined as the product of the number of eligible claimants and the mean hours worked by union members that year. J.A. 1383 (report of defendants’ expert).

Examination of two charts prominently displayed in defendants’ own brief reveals that in fact, the number of hours the Local referred to non-union workers did exceed the number of potential claimant hours every year through 1981. Def. Br. at 15, 28. That is because in each of those years the union referred in excess of 100,000 hours to nonunion workers, more than enough to accommodate 173 claimants without displacing any union members. Hence, even assuming that the union could not have attracted additional work for additional members, there simply was no fixed-pie problem through 1981. If there were an error in the Master’s calculation, then, it would apply only to awards for years after 1981 — and only one claimant received such an award. At most, the fixed-pie theory would necessitate a remand of the award to Eldridge Harmon, who received $2,075 for 1985-86.

But the court is also factually incorrect in stating that “[tjhere is no evidence in the record to suggest that there was additional work in the D.C. area for Local 201 rodmen” beyond that actually handled by its union members and nonunion referrals. Op. at 191. Defendants’ own briefs provide the contrary evidence, demonstrating that during the entire relevant period, Local 201 had more job requests from employers than both its union and non-union workers could absorb. Defendants state that from 1972 to 1975, “Local 201 was unable to fill 4,432 jobs due to an insufficient number of workers seeking jobs through the hiring hall.” Def. Br. at 14 (citing J.A. 274-75). And from 1976 to 1986, “Local 201 was unable to fill I,649 jobs.” Id. at 15; see also id. at 60 (“Even after the 1972-75 ‘full employment’ period, Local 201-referral jobs went unfilled for lack of applicants.”). Defendants’ evidence makes clear that this circumstance existed in every year for which there are records, see J.A. 275, despite the fact that the “union never deliberately let a job go unfilled.” Def. Br. at 14. Hence, the defendants’ own briefs provide the “prima facie showing that additional hours were available to Local 201” upon which the court insists. Op. at 192.

Finally, the fact that the union had to turn down jobs also undermines the court’s declaration that “[ejommon sense and experience suggest that a union will attempt to bring as many projects as possible under union control. ...” Op. át 191. That may be the case where a union and its non-union referrals are able to handle all the work they can bring in. But where a local is already turning down unsolicited job referrals, it has no incentive to bring still more projects under its control. Under these circumstances, neither common sense nor experience militates against the Master’s finding that there was no fixed pie. J.A. 339.

III.

The burden is on the appellants to establish that the decision below was clearly erroneous. See Bellevue Gardens, Inc. v. Hill, 297 F.2d 185, 187 (D.C.Cir.1961); 9A Charles A. Wright & Arthur R. Miller § 2585, at 565. Because they have not met that burden, there is no reason for us to prolong the final resolution of plaintiffs’ back pay awards by remanding the Special Master’s benchmark determination for further consideration. 
      
      . We also upheld liability for several individual claims of retaliation, none of which is relevant in this appeal.
     
      
      . We clarified on rehearing that the Apprenticeship Committee and the Training Program, which administered the educational prerequisites, were not jointly liable with Local 201 and the International. Liability, we said, was established only with respect to the unions' imposition of the requirements themselves, and was not based on the administration of the programs by the Apprenticeship Committee and Training Program. See Berger v. Iron Workers Reinforced Rodmen Local 201, 852 F.2d 619, 620-21 (D.C.Cir.1988) (Berger II).
      
     
      
      . One hundred and seventy-three claimants participated in the damages phase originally, but many settled their claims and others were excluded for filing untimely claims.
     
      
      . Because this portion of the panel's decision resolves an apparent conflict between two of our prior decisions and Price Waterhouse v. Hopkins, it has been separately considered and approved by the full court and thus constitutes the law of the circuit. See Irons v. Diamond, 670 F.2d 265, 268 n. 11 (D.C.Cir.1981).
     
      
      . In making his determination of the appropriate back pay period for Francis, the Special Master noted that Francis was told he could not apply for membership until he was a U.S. citizen. J.A. 404. Since discrimination on the basis of citizenship was neither alleged in the lawsuit nor made a part of the liability finding in Berger I, on remand it should play no part in determining Francis’ class membership or eligibility for back pay.
     
      
      . The class asserts that the unions' challenges to the back pay awards are numerically incorrect because the challenges rely on the Special Master’s benchmark proxy, which erroneously excluded overtime earnings. But as the unions concede, the parties stipulated in 1990 that overtime should be included, and the district court in its March 16, 1995 order concluded that the Special Master erred and ordered the parties to recalculate the back pay figures. J.A. 572. This presents some confusion since the specific dollar amounts discussed in the briefs are not technically accurate. We leave it to the district court to calculate the actual amount owed to any claimant consistent with the correct back pay figures, including overtime.
     
      
      . The unions also contend that Ernest Bellamy's back pay award for 1972 was erroneously calculated, given that his 1972 tax return listed his earnings at $13,217, whereas the Special Master used Bellamy’s social security earning record which listed his 1972 earnings as $9,178.75. The class agrees that the district court clearly erred in the 1972 award. On remand, Bellamy’s 1972 award should be recalculated using the correct 1972 earnings amount.
     
      
      . The unions also claim that, because O.C. Brown testified to his special difficulty in working in cold weather, his back pay award also should be reduced to account for the three-month period in which he was unavailable for work each year. Unlike the unions’ challenges to claimants who had unusually excessive injury-time, we think this challenge to O.C. Brown’s award is more appropriately resolved under the duty to mitigate doctrine, which we discuss below.
     
      
      . The uni.ons contend, and the class concedes, that the Special Master erroneously stated that Harmon's injury began in November 1974, when it in fact began on September 28, 1974. Since the difference between a one-month and a three-month injury might be significant in determining the amount of back pay reduction, if any, we reverse for clear error the district court’s finding that Harmon was injured in November of 1974. Also, the unions are correct that Harmon's interim earnings for 1974 were $10,299, and not $8,316, as the Special Master clearly erroneously found. J.A. 1501-03; 409.
     
      
      . The Special Master also appears to have adopted conflicting methods for reducing back pay because of excessive injury-time. In some instances, he reduced the benchmark proxy for the relevant period by the percentage of time during which the claimant was unavailable for work, and subtracted the claimant’s actual work-time from the reduced benchmark. See, e.g., J.A. 436 (Sherman Johnson). However, on other occasions he simply reduced the earnings shortfall (the final back pay award) by the percentage of time during which the claimant was unable to work. See, e.g., J.A. 409-10 (Eldridge Harmon). On remand, we instruct the district court to apply one method of injury-time reduction consistently, and to explain the basis for picking that method.
     
      
      . The back pay provisions of Title VII were modeled on those of the National Labor Relations Act (NLRA), and the Supreme Court has therefore applied principles developed in the NLRA context to Title VII remedies. See Ford Motor, 458 U.S. at 226 n. 8, 102 S.Ct. 3057.
     
      
      . On remand, the district court should also consider the unions' contention that Brown failed to mitigate during the period 1975-78. The unions cite evidence that Brown never sought employment from a specific company he believed would have hired him, see Def. Br. at 64, notwithstanding that he worked few hours during that period, see J.A. 379-80.
     
      
      . Francis’ award is subject, however, to the outcome of the remand of his class membership, as discussed in Part III.B.2 above.
     
      
      . This appears to be the unstated (and undisputed) premise of the unions' argument.
     
      
      . See Def. Br. at 54-55, 59-60. As noted above, the unions contended that a claimant failed to mitigate if he took jobs with non-union employers during this period, thus making him "not available for Local 201 referrals when he was working elsewhere." Id. at 60. The unions further contended that:
      Local 201 was unable to fulfill employer requests that it dispatch workers for 4,432 jobs during the period 1972-75.... Local 201 fell short of supplying workers only because it ex-
      hausted the rodmen who were available to be referred on all classes on its list. And, during pension years 1973 to 1976, union members were only able to work 36%, 32%, 35% and 48% respectively, of the hours worked by all workers referred by Local 201. At least during this period, then, a claimant would exercise reasonable diligence only by consistently seeking employment through Local 201, just as the union's member did.
      
        Id. at 55.
     
      
      . There was also testimony that returning to the Local after taking a non-union job would have been futile, since rodmen who worked for nonunion employers were regarded as "scabs” and not given referrals by the union hiring hall. See J.A. 903, 1095.
     
      
      . In this case, the class' punitive damages claim is based on 42 U.S.C. § 1981, since Title VII's punitive damages remedy was not added to the statute until 1991, long after this lawsuit was filed. It is nonetheless appropriate to apply the principles outlined in Kolstad, which • was brought under Title VII, and Smith, which was brought under 42 U.S.C. § 1983, because we have consistently applied the same punitive damages standard under all three statutes. See Kolstad, 139 F.3d at 962-65 (applying Smith and § 1981 standards in Title VII action); Barbour v. Merrill, 48 F.3d 1270, 1277 (D.C.Cir.1995) (applying Smith standard in § 1981 action).
     
      
      . It is important to note that the Special Master did not exclude all zero-hour workers from the proxy group — he excluded only those who had worked zero hours “for several years." J.A. 342-43. Similarly, defendants’ own expert excluded some, but not all, zero-hour workers from his preferred benchmark proxy — he excluded “zero-hour rodmen who had died, retired, were incarcerated or permanently disabled.” Op. at 190 (citing J.A. 1381). Neither the court nor defendants explain why the exclusions made by defendants’ expert were permissible, while those made by the Master were clearly erroneous.
     
      
      . To a lesser extent, the zero-hour argument discussed above suffers from the same disability. Although one of defendants' briefs did use the term "zero-hour" (once), it did so only in describing the work of defendants' expert. It did not expressly argue that the failure to include zero-hour workers rendered the Master's benchmark clearly erroneous.
     
      
      . The chart on page 15 of defendants’ brief discloses the number of hours the union referred to non-union workers in each year. The chart on page 28 shows the mean union-member hours for each year as determined by defendants’ expert. When the latter figures are multiplied by the 173 putative class members, the resulting claimant hours are less than the non-union hours for every year through 1981.
     
      
      . This roughly accords with the concession of defendants' own expert that in the 1970s there would have been sufficient hours available for the claimants had they been admitted to the union. See J.A. 339 (citing expert’s testimony).
     
      
      . The court states that the fact that jobs went unfilled does not necessarily mean that the union had enough work for the claimants, since it might be explained merely by "[fluctuations on a given day that would result in a specific job referral being listed as unfilled.” Op. at 191. But this theoretical possibility, like the year-to-year disparities to which the court also points, is hardly sufficient to justify a conclusion that the Master's determination was clearly erroneous. See Anderson, 470 U.S. at 573-74, 105 S.Ct. 1504. The court also contends that defendants "made a prima facie showing that the hours were not available” for the claimants, based on evidence of "the declining hours referred out of the hall and [on] evidence [defendants] introduced of the declining market share available to' the union.” Op. at 191. But it is the gross hours potentially available for the claimants, not the evidence of trends and percentages, that is relevant to the validity of the fixed-pie theory.
     
      
      .For the same reasons, and contrary to the court’s contention, the Special Master did not shift the burden of proof to defendants by characterizing the fixed-pie theory as "at best speculative.” J.A. 340. Indeed, the defendants' expert himself described the theory in words of speculation. See J.A. 1383 (stating that the benchmark "may require [an] adjustment ... [to] reflect[] the fixed number of union-referred hours”) (emphasis added); id. at 1384 (stating that a fixed-pie adjustment would be required "if there are no permitmen and travelers working in a given year”) (emphasis added).
     