LUMBARD, Circuit Judge:
Defendants New York City Board of Education and New York State Commissioner of Education appeal from the decision in the Eastern District of New York, Constantino, /., invalidating the Board’s plan for reassigning students of Andrew Jackson High School to other City high schools, on the ground that defendants failed to justify as necessary at least one of the racially restrictive provisions of the plan. We reverse and remand for further proceedings.
I.
This case originated in June, 1976, as a class action filed on behalf of students and parents in the Andrew Jackson High School District in Queens, New York, challenging the constitutionality of a proposed school desegregation plan voluntarily adopted by defendant New York City Board of Education, and approved by defendant State Commissioner of Education Gordon Ambach. The genesis of the plan, known as the “1976 Controlled Rate of Change Plan,” is explained in detail in two opinions in an earlier stage of this litigation, Parent Ass’n of Andrew Jackson High School v. Ambach, 598 F.2d 705, 710-711 (2d Cir.1979), reversing in part, 451 F.Supp. 1056, 1063-69 (E.D.N.Y.1978).
In brief, the minority enrollment at Andrew Jackson High School, which had been 18% in 1957, steadily increased over the next two decades until, by 1976, it exceeded 99%. Beginning in 1963, the New York City Board of Education (“the Board”) adopted a succession of plans to stem Jackson’s acceleration toward an exclusively minority student population by transferring white students in from other zones. As the statistics show, none of these plans was successful. In 1973, the Board finally abandoned the hope of integrating Jackson.1 Instead, it developed a “Choice of Admissions” scheme, under which minority students in the Jackson zone had the choice of attending any of a number of receiving schools at which the minority population was lower than at Jackson, and the influx of additional minority students would not destroy whatever racial balance currently existed.
With revisions in 1975 and 1976, the scheme became the “1976 Controlled Rate of Change Plan” (“the Plan”) at issue here. The Plan was intended to balance the goal of placing Jackson’s minority students in integrated schools of their choice, against the perceived reality that if the minority enrollment in an individual receiving school increased too rapidly or reached a critical absolute level (the so-called “tipping point”), white students would leave the receiving school at an increasing rate. By controlling both the rate and extent of change in racial composition in the receiving schools, the Board hoped gradually to transfer most or all of the student population out of segregated Jackson, without at the same time triggering resegregation of currently integrated receiving schools, thereby giving the largest number of children an integrated education over the longest period of time.
The Plan provided that Black and Hispanic students residing within the Jackson school zone could elect to attend any New [577] York City high school not already overenrolled, in which (1) the percentage of white students in the school exceeds either 50% of the school population or the borough-wide average in the borough in which the receiving school is located (whichever is higher); provided further that (2) the admission of such students, coupled with the admission of minority students from other integration programs and/or through demographic changes in the attendance area servicing the school, will not (a) decrease the receiving facility’s white/minority ethnic balance by 4% or more in any one school year, or (b) produce a rate of change in any one year that exceeds one-fourth of the difference between the school’s current white enrollment and a 50% white enrollment, whichever is less. At the same time, it provided that white students in the Jackson zone could elect to attend any City high school in which the percentage of white students is lower than either the borough’s white percentage or 50% of the school population, whichever is less.
In June 1976, students and parents at Jackson filed this class action against the Board of Education and the State Commissioner of Education, charging (in relevant portion here) that the Plan’s imposition of a racial quota against the admission of minority students to potential receiving schools violated the equal protection clause of the Fourteenth Amendment. The district court, agreeing with plaintiffs, invalidated the Plan as unconstitutional. Although acknowledging that the Board’s goals in adopting the Plan were benign, the court held that by “limitpng] minority pupils’ access to schools because of their minority status in order to provide integrated schooling for as long as possible to a progressively limited number of minority and other students,” the Plan impermissibly recreated “a dual school system” of integrated and segregated schools. 451 F.Supp. at 1080-81.
On appeal, we reversed, holding that the Plan’s aim “to promote a more lasting integration is a sufficiently compelling purpose to justify as a matter of law excluding some minority students from schools of their choice under the obviously race-conscious Rate of Change Plan.” 598 F.2d at 719. However, we remanded for several factual determinations concerning specific provisions of the Plan, including the two relevant here: (1) whether the Board had shown a factual justification for its choice of a 50% tipping point; and (2) whether demographic projections supported the Board’s determination that the maximum allowable rate of change in racial composition of a school should be set at the lesser of 4% or lh of the différence between the current white enrollment and 50% white enrollment.
On remand before Judge Dooling in February 1980, the Board provided extensive statistical data to support both determinations. In respect to the 50% tipping point, the Board submitted two exhibits to substantiate its claim that white students left a school at an increasing rate when the percentage of white students dropped below 50%. The first, Exhibit 3C, compared the rates of change for each City high school when and if it was between two 30-point intervals of ethnic composition— 80%-50% white, and 50%-20% white — during the years 1957-1978. As explained by Philip Tama, the Associate Director of Statistics and Demographics in the Board’s Office of Zoning and Integration, the Exhibit revealed that most individual schools experienced a sharp increase in the annual rate of change when they moved from the 80%-50% range to the 50%-20% range. The city-wide interval average (equal to the mean of the schools’ averages within each interval) showed that the percentage of white pupils per year declined at an average rate of 2.94% per year during the period that schools were in the interval from 80%-50% white; and declined at an accelerated average rate of 5.54% during the period they were in the interval from 50%-20% white. The second, Exhibit 3E, broke down the two 30-percentage point intervals in Exhibit 3C into six 10-percentage point intervals. The city-wide average figures revealed a steady increase in the decline in the percentage of white pupils in the first [578] three intervals (3.67% from 80%-70%; 4.42% from 70%-60%; and 4.80% from 60%-50%), and a marked increase in such decline in the fourth interval (6.06% from 50%-40%).2
In respect to the 4% maximum allowable annual rate of change, Joseph Elias, Director of the Board’s Office of Zoning and Integration, testified that the intent of the Board in limiting the rate of change was to approximate the 3% natural rate of change (through attrition) that the schools were experiencing at the time. He noted further that the 4% figure referred to the school as a whole (i.e., all four classes), not simply the entering class that contained the new minority transfer students. As a result, the entering class could actually absorb up to a 16% decrease in white enrollment annually without exceeding the 4% ceiling.
In addition, in support of both the 50% tipping point and the 4% rate of change, the Board submitted Exhibit 4, comparing the actual experience in Queens receiving schools since the Controlled Rate of Change Plan went into effect in 1976,3 with projections of what would have happened had all minority students been granted their first choice assignments without regard to extent or rate of change. (Computations were done only for Queens high schools because virtually all the students who otherwise would have gone to Jackson designated another Queens high school as their first choice.) As explained by Elias, whose office prepared the Exhibit, the limitations on extent and rate of change had two desirable effects on integration. They greatly slowed the rate of decrease in white enrollment for those receiving schools that were below the 70% point prior to 1976, in several cases avoiding a projected precipitous drop to well below the 50% mark; and they nearly equalized the racial balance in all receiving schools, by decreasing more sharply the white enrollment in the few schools that had been well above 70% white in 1975.4 As a result, it was anticipated that by 1980, all Queens receiving high schools would have between 52% and 63% white enrollment if the Plan remained in effect, compared to a projected spread of 39.5% to 75.7% white enrollment if Jackson students were given their first preferences without limitation.
Finally, Tama testified that, even with the Plan in effect, in 1980 92.5% of the applicants in the Queens choice of admissions area were assigned to one of their top three, and 98% to one of their top four, choices of schools.
Plaintiffs produced no witnesses of their own on remand, and introduced no evidence directly contesting the Board’s maximum allowable rate of change figures. How[579] ever, following the February 1980 hearing, they submitted a report prepared by Dr. Steven Cohen, an Associate Professor of Sociology at Queens College, taking issue with the statistical methodology employed by defendants in the two exhibits offered to substantiate the 50% tipping point. When Dr. Cohen recomputed the average city-wide rates of change by his preferred methodology, his figures showed that the greatest decline occurred at the 60%, not the 50%, level.
A second hearing was held before Judge Dooling in October 1980, at which Elias and Tama were requestioned, but no new witnesses or evidence were introduced. Before he could render a decision, Judge Dooling passed away. The case was reassigned to Judge Costantino on January 22, 1981. With the agreement of all the parties, Judge Constantino indicated he would decide the matter on the basis of the record before Judge Dooling and any written evidence submitted by the parties subsequent to the first hearing.
On July 5, 1983, Judge Constantino issued his decision, finding that the Board had failed to provide factual justification for its choice of a 50% tipping point. As that finding alone was grounds for declaring the Plan invalid under the terms of this court’s remand, Judge Constantino never reached the maximum rate of change issue. Defendants appeal from that decision.
II.
In an earlier stage in these proceedings, we held that the Board’s goal of ensuring the continuation of relatively integrated schools for the maximum number of students, even at the cost of limiting freedom of choice for some minority students, survived strict scrutiny as a matter of law. 598 F.2d at 717-720. However, in remanding to the district court for further factual determinations, we stated that “the Board and the Commissioner must bear the burden of proving that the Plan, in each of its components, is necessary to achieve the goal set forth.” Id. at 721. The question before us now is whether defendants have met that heavy burden of proof as to the two elements of the Plan still in dispute: the choice of a 50% tipping point; and the choice of (a) 4% or (b) lk of the difference between the current white enrollment and 50%, whichever is less, as the maximum allowable rate of change per year.5
We begin with the 50% tipping point, the only issue addressed by the district court. As a matter of normal appellate procedure, this court may overturn the district court’s finding that defendants failed to meet their burden of proof with respect to the 50% tipping point only if it was clearly erroneous. See Fed.R.Civ.P. 52(a). We conclude that it was, at least on the grounds asserted by the court.
The district court based its finding that defendants had failed to provide factual justification for the 50% tipping point on two observations the court drew from its own analysis of defendants’ school-by-school raw statistical data: (1) that three City high schools (Erasmus Hall, Springfield Gardens, and Tilden), two of which defendants' expert had cited as typical illustrations of the 50% tipping point phenomenon, in fact showed almost as great a drop in white enrollment at the 60% or 70% level as they did at the 50% level; and (2) that of nine public high schools in Brooklyn and another nine in Queens, only two experienced their greatest decline in white enrollment at the 50% level.6 The court made no reference to the two exhibits submitted by defendants showing the average interval drops on a city-wide basis, or to Dr. Cohen’s critique of those exhibits.
Appellants argue that both observations on which the court relied are based on an [580] erroneous or misleading reading of their statistical data. We agree.
On the court’s first point, in defense of its conclusion that “a similar rate of decline occurred [for each of the three schools] at figures where white student population was well above the 50% mark,” the court noted that Erasmus Hall, which had a 10.1% decline at the 50% point, had a 8.5% decline at approximately the 60% point; Springfield Gardens, which had a 10.1% decline at the 50% point, had a 7.5% decline at the 60% point; and Tilden, which had a 9.7% decline at the 50% point, had a 10.1% decline at approximately the 70% mark.
As to Tilden, the district court’s statistical inference is incorrect. The 9.7% decline at the 50% level took place over one year; the 10.1% decline at the 70% level over two years. An accurate comparison of the average annual declines at the two levels would thus require halving the 10.1%, with resulting figures of 9.7% at the 50% level and 5.05% at the 70% level — figures that amply support defendants’ choice of 50% as the critical tipping point.
As to Erasmus Hall and Springfield Gardens, we are unsure what the district court’s observation proves. Defendants contend not that the 50% level triggers the only significant drop in white enrollment, but that it triggers (on average) the most significant drop. The experience of these two schools bears that observation out. Finally, although defendants cited to these two schools as illustrative of their argument, their case for the 50% tipping point rests not on the individual experiences of the two schools, but on the average experience of schools on a city-wide basis.
On the court’s second point, of the eighteen schools the court cited, during the relevant period eleven never went below 50% white enrollment, one more went only barely below 50%, and one was never above 50%.7 As a result, for thirteen of the eighteen schools, there were no, or virtually no, statistics available at the 50% level. The remaining five schools did have experience on both sides of the 50% level. But two of the five (Erasmus Hall and Newtown) had their greatest decline at around 50%, thus bearing out defendants’ theory. The other three showed higher declines at levels other than 50%. But two of these three (Bay Ridge and Hillcrest) had only one year’s experience (their first year of operation) substantially above the 50% level, casting doubt on the meaningfulness of their statistics for that year, leaving only one (Bowne) as a reliable example of a tipping point other than 50%.
Thus, of the eighteen schools cited by the district court, only one — Bowne—unequivocally diverges from defendants' predictions of a 50% tipping point. Bowne may not be the exception that proves defendants’ rule, but surely it is also not the exception that disproves it.
We therefore cannot agree with the district court in holding, on the basis of the evidence on which it purported to rely, that defendants failed to justify the choice of a 50% tipping figure.
That leaves two questions: whether a proper analysis of the evidence submitted on the issue of a tipping point would show that defendants justified their choice of the 50% figure; and whether the evidence submitted but not considered by the district court on the rate of change issue justifies as necessary defendants’ selection of a 4% ceiling. In the hope that both questions could be resolved by us on the basis of statistical evidence submitted below, see Shackelton v. J. Kaufman Iron Works, Inc., 689 F.2d 334, 337 (2d Cir.1982); Philip v. Mayer, Rothkopf Industries, Inc., 635 F.2d 1056, 1061 (2d Cir.1980), we have reviewed that evidence with some care. Reluctantly — given the already lengthy histo[581] ry of this litigation — we conclude that we have no choice but to remand both questions to the district court to be resolved as expeditiously as possible.
50% ceiling on minority enrollment. In support of their claim of a 50% tipping point, defendants rely chiefly on Exhibits 3C and 3E, showing the average annual percentage decline in white enrollment in the City’s high schools from 1957-1978. Although we have doubts how much can be concluded from Exhibit 3C,8 which groups the data into two 30-percentage point intervals, Exhibit 3E, which breaks the data down into six 10-percentage point intervals, appears on its face to support defendants’ contention that a significant increase in the rate of departure of white students occurs between the 60%-50% and 50%-40% intervals. See n. 3 supra. However, several questions concerning the signifieance of that statistical finding prevent us from concluding outright, on the strength of that alone, that defendants have met their burden of proof concerning the 50% tipping point.9
The first question, raised below by plaintiffs but never addressed by the district court, concerns whether, in computing the city-wide mean averages in Exhibit 3E, defendants properly classified and weighted each school’s experience within a given 10-percentage point interval. Dr. Steven Cohen, plaintiffs’ expert, raised two objections to defendants’ methodology: (1) that defendants determined the interval within which a school fell for any given year based on its average white enrollment for the year, rather than on its white enrollment at the beginning of the year; and (2) that defendants weighted each school’s statistical experience within a ten-point inter[582] val by school rather than by school year.10 Reanalyzing the Board’s raw data under his preferred methodology, Dr. Cohen came up with city-wide averages that show a sharper drop at both the 60% and 30% levels than at the 50% level. As those figures, if the more accurate, would seriously undercut defendants’ case for a 50% tipping point, an informed choice between the two would seem essential. Lacking adequate information to make that choice ourselves,11 we leave it to the district court to resolve on remand.
Second, our own independent review of the data presented in Exhibit 3E raises an additional question not dealt with in the district court. The city-wide figure in Exhibit 3E for each ten-point interval is equal to the mean average of the individual rates of change for each of the City's 62 nonspecialized high schools with any experience within that interval. However, during the 1957-1968 period covered in Exhibit 3E, only 25 of the 62 schools had any significant experience on both sides of 50%.12 The other 37 schools either remained above 50% white throughout that entire period, or had dropped below 50% white before it began. If the time period is further limited to 1968-1978 — which the Board elsewhere claims to be the only period on which it is relying to prove the 50% tipping point — the number of schools with experience on both sides of 50% drops down to 8.13
What this means is that the statistical experience of the City’s school system on the two sides of 50% is based on largely, or almost entirely, nonoverlapping samples. Whether it is nonetheless legitimate to infer that the same statistical disparity observed between those two samples would hold true for the average individual school were it to cross the 50% point seems to us a crucial methodological question we cannot resolve. We therefore leave it for the district court to resolve on remand, with the aid of any additional data accumulated by the City since 1978.14 We note for the [583] record, however, that the experiences of those few schools that did straddle 50%, while too small a sample to be conclusive, do not allay our concerns. Of the 8 such schools for the period 1968-1978, only 4— Tilden, Erasmus, Springfield Gardens and Newtown — showed the highest percentage decline in white enrollment at the 50% level. Of the additional 17 schools for the period 1957-1978, only 5 — Eastern District, Bushwick, Seward Park, Washington and Hillcrest — showed their highest percentage decline at 50%, and 3 of those 5 — Eastern District, Washington and Hillcrest — were never above 62% white, casting some doubt on the significance of their comparative experience.
Finally, although we conclude that close judicial scrutiny of the 50% ceiling is required because of the unequal burdens that ceiling imposes on Jackson’s minority students, see n. 9 supra, we do not mean thereby to trigger a rigid and mechanical application of a strict scrutiny standard. The extent of proof required of defendants should take some realistic account- of the extent of the burdens imposed. To that end, the parties should provide on remand a more precise and updated picture of the burdens actually imposed on Jackson’s minority students by the Plan, particularly the numbers of plaintiffs who have been and will be forced to remain in a largely or exclusively minority school (presumably Jackson) because of a shortage of potential receiving schools that meet the Plan’s restrictions.
4% ceiling on annual rate of change. As this issue was never reached below, and as we are remanding in any case for reconsideration of the 50% ceiling on minority enrollment, we leave it for factual resolution by the district court. However, we note that some limitation on the rate of change seems to us a necessary corollary to a limitation on extent of change. That is, if the Board is allowed to put a 50% ceiling on minority enrollment but not allowed to control the rate of influx of additional minority students, the result will simply be that first-choice schools will reach the 50% ceiling .more rapidly — at which point no additional minority students at all will be allowed in. As we fail to see how that result would in any way lessen the burden on minority students, we are inclined to think that the case for imposing some limitation on rate of change must stand or fall with the case for a 50% ceiling on minority enrollment. However, whether the particular figure chosen by defendants as the maximum permissible rate of change has a factual justification is a matter we leave to the district court.
Reversed and remanded for further proceedings not inconsistent with this opinion. We retain jurisdiction.