
    Donald PRICE and Melvin E. Price, Minors, et al., Plaintiffs-Appellees, v. The DENISON INDEPENDENT SCHOOL DISTRICT, Defendant-Appellant.
    Nos. 81-2264, 81-2379.
    United States Court of Appeals, Fifth Circuit.
    Dec. 8, 1982.
    Opinion on Denial of Rehearing and Rehearing En Banc May 12, 1983.
    See also, 5 Cir., 348 F.2d 1010.
    
      Munson, Munson, Hynds, Watkins & Gordon, J. Don Gordon, Peter Kerr Munson, Sherman, Tex., William C. Bednar, Jr., Austin, Tex., for defendant-appellant in both cases.
    R.C. Slagle, III, Sherman, Tex., for plaintiffs-appellees.
    Eskew, Muir & Bednar, Austin, Tex., for defendant-appellant in No. 81-2379.
    Before COLEMAN, POLITZ and GAR-WOOD, Circuit Judges.
   GARWOOD, Circuit Judge:

The Denison Independent School District [DISD] appeals from certain orders of the district court entered in this school desegregation litigation.

I. Introduction.

Denison, Texas is a small city located some seventy-five miles north of Dallas. The urban portion of the school district is likewise small, measuring approximately five miles by five miles, though the district also extends several miles beyond the city limits, principally to the north and south, and includes largely rural areas. Residential patterns are such that the black population is principally concentrated in a relatively compact segment of the northeast quadrant of the city, while the western half of Denison, which contains the newer centers of population, is almost exclusively white. During the 1979-80 school year, the DISD provided education to some 5,200 student's, of whom approximately 12 percent were black.

II. Background of this Appeal.

Until the spring of 1963, the DISD operated a school system that was completely segregated by race, with a dual set of attendance zones for black and white students. On June 24, 1963, the school board, the district’s governing body, voluntarily passed a resolution instituting a “freedom of choice” plan of desegregation, to be phased in on a one-grade-a-year “stair-step” basis. That plan was challenged in a suit brought by a class of black school children and their parents who sought desegregation of all grades in the Denison public schools by the 1964-65 school year. The school board’s plan was approved by the district court, but this Court found the nine years required to fully implement the plan to be excessive in light of the command of Brown v. Board of Education (Brown I), 347 U.S. 483, 74 S.Ct. 686, 98 L.Ed. 873 (1954). Price v. Denison Independent School Distinct, 348 F.2d 1010 (5th Cir.1965). Upon remand, the district court entered an order, dated July 19, 1965, which established a “freedom of choice” plan effective the ensuing school year for all grades in the district. There was no appeal from this order. Thereafter and until the revival of this litigation in 1979, the lawsuit remained essentially quiescent. In 1968 the DISD commenced operating only one high school for all students, and has continued to do so since that time. In 1969 the DISD voluntarily and unilaterally abandoned the “freedom of choice” scheme for grade levels below high school in favor of a neighborhood attendance zone plan. The attendance zones in that plan appear to have remained in effect without substantial change until the 1979-80 school year. In the summer of 1979, the DISD promulgated, effective that fall, a new neighborhood attendance scheme involving the closing of three elementary schools, the conversion of a middle school to an elementary and the drawing of new attendance zone boundaries, principally for the elementary schools.

It was the adoption of this new scheme which prompted institution of the present action. On July 31, 1979, shortly after the announcement of this new plan, the plaintiffs and a group of plaintiff-intervenors filed a joint motion for substitution of parties, intervention, and an amended complaint claiming that the 1979-80 attendance plan continued the vestiges of historical segregation in the DISD and seeking relief from the allegedly unconstitutional elementary and junior high student assignment plans and unequal junior high school educational conditions, as well as purported deficiencies respecting teachers, assistant principals and principals. Plaintiffs specifically complained of the closing of the three elementary schools, alleging, among other things, that “white students from the less wealthy socio-economic groups have had their neighborhood schools closed and attendance lines redrawn where they can be counted to cut the minority black percentage down at Terrell [elementary]” and that some 41 black students “would be bussed from North Denison to an elementary school [Mayes] in West Denison.” A temporary restraining order was also sought (unsuccessfully) against the closing of the three elementary schools. It is evident that the contemplated closing of these schools was the motivating factor in the revival of the lawsuit.

Hearings were held on August 11-12, 1980. On December 17, 1980, the district court found that the combined differences, in educational conditions and racial composition, between Denison’s two junior high schools, Hughes and McDaniel, violated the Equal Protection Clause of the Fourteenth Amendment and ordered unitization of the seventh, eighth and ninth grades served by those schools. Pursuant to the DISD’s request, the district court, in March 1981, amended this order to include the sixth grade, placing the sixth and seventh grades at Hughes, the eighth and ninth at McDaniel. The junior high school unitization was to go into effect with the 1981-82 school year.

On June 10, 1981, the district court entered an order and memorandum opinion which (1) reiterated its findings and order as to the junior high schools; (2) found the 1979-80 elementary school student assignment plan to leave some schools racially identifiable and in violation of the Constitution; and, (3) found that the failure of the DISD to currently have a black principal in any of its schools presented a constitutional violation. The school board was required to submit to the court, within thirty days, an elementary assignment plan which would “assure that the population of each elementary school will be at least six percent black by the beginning of the 1981-1982 school year”; and, to employ a black individual in the next available principalship vacancy. On July 6, 1981, the DISD filed notice of appeal from the June 10 order.

On August 12, 1981, the DISD submitted its proposed elementary assignment plan for the 1981-82 school year. The district court rejected that plan in an order of August 14, 1981. The court found that although the DISD’s proposed plan was “in formal compliance with the minimum standards set forth in the June 10 order,” it nevertheless “violate[d] the spirit of the order” because it insufficiently reduced (from 33.8 percent to 29.3 percent) the percentage of black students at Terrell elementary. The -August 14 order required the DISD to submit an “elementary assignment plan which would assure that the population of each elementary school will be no less than six percent black and no more than seventeen percent black by the beginning of the 1981-1982 school year.”

On September 1,1981, a hearing was held on a motion for stay of the August 14 order filed by the DISD. At this hearing the parties introduced a “joint exhibit” reflecting school attendance zones that, on the basis of rough calculation, would create elementary school populations in approximate compliance with the August 14 order, both parties agreeing that these attendance zones would be submitted to the school board for approval. On September 8, the school board rejected the proposed plan.

On September 18, 1981, the district court entered its final order in this suit. The court (1) denied the motion for stay pending appeal; (2) vacated its order of August 14; and, (3) ordered that the elementary school student assignment plan reflected by the referenced joint exhibit, with some relatively minor modifications, be implemented in January 1982.

III. The Issues.

In this appeal, the DISD raises diverse questions as to every aspect of the district court’s orders. Its principal challenges are to the district court’s determinations of constitutional violations in (1) the elementary school student assignments; (2) the junior high schools; and, (3) the absence of a black principal. The DISD further contends that the court’s findings and conclusions do not support its determination of constitutional violations.

IY. The Elementary Schools.

A. Background.

For the school year 1979-80, the DISD operated seven elementary schools for approximately 2,800 students. The seven facilities were relatively evenly dispersed throughout the city, with Terrell elementary in the northeast, Lamar in the east, Golden Rule in the south, Hyde Park in the southwest, Mayes in the west, Layne in the northwest, and Houston elementary located in the center of Denison.

The previous school year, 1978-79, Denison had nine elementary schools. The plan which prompted this litigation and went into effect for the 1979-80 year closed three of these elementary schools, Central (then 55.3 or 56.9 percent black), Peabody (.9 percent black) and Raynal (9.7 percent black), and converted Terrell, then a grade six, seven and eight middle school (24.2 percent black), to an elementary. The former Peabody attendance zone was divided between Houston, Golden Rule and Lamar. Part of the Central attendance zone was assigned to Mayes, the remainder to Terrell, which is located some five blocks north and .three east of Central. Raynal and Lamar, each located somewhat east of Terrell, had previously served students in grades five and below, with sixth graders from these zones attending Terrell middle school. Under the 1979-80 scheme, all of the Raynal attendance zone was assigned to Terrell. Lamar sixth graders continued to attend Terrell. Each of the seven elementary zones under the 1979-80 plan was geographically a wholly contiguous unit, as was also the case with the previous nine-zone plan.

The evidence indicated that the principal purposes of the 1979-80 plan were to effect some economies and instructional improvement by reducing the number of elementary schools and to eliminate the then majority black elementary school, Central. The United States Department of Health, Education and Welfare had notified the DISD that the black concentration at Central should not exceed 32 percent. A competing plan advanced by some of the plaintiffs was rejected by the board because it did not reduce the number of schools and insufficiently reduced the concentration of blacks, which would have been over 40 percent at Central.

The following table shows the elementary school statistics for the 1980-81 school year (the statistics which the district court principally focused on), as well as the black student percentages for the years 1978-79 and 1979-80:

1980-81 1980-81 1979-80 1978-79
School Black All Other Total Black % Black % Black %
Golden Rule 12 318 330 3.6 1.7 2.4
Houston 35 348 383 9.1 13.1 12.2
Hyde Park 5 362 367 1.4 .8 .9
Lamar 25 322 347 7.2 10.0 9.5
Layne 2 339 341 .6 .8 .3
Mayes 38 390 428 8.9 8.4 .3
1980-81 1980-81 1979-80 1978-79
School Black All Other Total Black % Black % Black %
Terrell 211 414 625 33.8 32.6 24.2
Central - Closed Closed 56.9
Raynal - Closed Closed 9.7
Peabody - Closed Closed .9
328 2,493 2,821 11.6

During the 1980-81 year, some 59 percent of the white elementary students attended schools having more than 7 percent black students, and some 64 percent of the black students attended Terrell.

During the period of dual racial attendance zones, Central was an all-white elementary school, and Terrell was an entirely black school apparently serving all grade levels. Terrell was entirely closed at the end of the 1967-68 school year. It reopened for the 1969-70 year as, in the words of the joint pretrial order, “a desegregated middle school serving grades 6-8.” It then had a 23.9 percent minority enrollment. Terrell remained as a middle school through the 1978-79 year, its highest minority enrollment being 29.05 percent in 1975-76 and its lowest 22.5 percent in 1970-71 and 1971-72. Central’s minority enrollment was less than 2.5 percent until the 1966-67 year when it became 25.5 percent. In 1967-68, it became 53.9 percent minority. Thereafter, minority enrollment varied from a low of 49.4 percent in 1969-70 to a high of 63.5 percent in 1976-77, falling somewhat in 1977-78 to 56.7 percent.

Layne, Hyde Park, Houston and Lamar apparently were all-white schools during the years of dual racial attendance zones. In the years 1966-67 through 1977-78, Houston’s minority enrollment varied from a low of 8.4 percent in 1966-67 to a high of 16.2 percent in 1976-77. Hyde Park’s low was .75 percent in 1967-68, its high being 2.4 percent in 1971-72. Layne had no minority students until 1971-72, and thereafter the percentage varied between 1 percent and 1.4 percent. During this period Lamar’s minority percentage increased from 15.3 percent in 1966-67 to 22.6 percent in 1971-72. It was closed for renovation in 1973-74, and reopened (apparently without a sixth grade) in 1974-75 with an 11.2 percent minority enrollment, thereafter reaching a high of 12.54 percent in 1977-78.

There is no evidence concerning elementary school closing or construction prior to the 1962-63 year. Until 1979-80, the only school closings, other than Terrell, were the all-black Langston and Dalton schools, apparently elementary, which were closed just after the 1964^65 year. Apparently only two elementary schools have been added. Golden Rule, in the south central part of the district, previously served a small adjoining rural school district which merged into the DISD commencing with the 1964-65 year. Its minority enrollment that year was .3 percent. It had no minority students the following year. Thereafter, its minority enrollment varied from a low of .9 percent in 1969-70 to a high of 6.1 percent in 1976-77. Mayes, located in west Denison, was opened in 1968-69 and had no minority students until 1971-72, when the minority enrollment was 1.7 percent. Thereafter and prior to 1979-80, minority enrollment at Mayes varied from a low of 1 percent to a high of 4.11 percent. In 1979-80 it became 8.4 percent.

The DISD’s elementary schools were, and apparently always had been, essentially “neighborhood” schools, in that the great majority of students attending each school lived in its immediate vicinity. Students living more than two miles distant, principally those residing in the rural areas, were bused to school. Prior to the 1979-80 year, some 10 to 12 percent of the elementary students were transported in this manner. Apparently, these students were disproportionately white, reflecting the racial composition of the rural students. Under the 1979-80 zones, the percentage of elementary students bused increased to about 24 percent. About 9 percent of those bused were black (some 19 percent of all black elementary students). Most of the increase in busing of black elementary students was necessitated by the northeastern extension of the Mayes attendance zone to encompass a predominantly black area in the western part of the former Central zone.

Two desegregation experts, Dr. Edward Tapscott and Mr. James Williams, were appointed by the court in December 1979 to study and report on the DISD. Neither recommended any material change in the DISD’s elementary school attendance scheme put in effect in 1979-80. Williams described it as “an appropriate plan, particularly evidencing good faith,” and Tapscott said it was “a good-faith effort to try to keep neighborhood schools and yet at the same time draw an attendance zone which would move minority students into some of the other elementary attendance zones.” Integrative aspects of the 1979-80 system which the experts commended were the northeastern expansion of the Mayes zone to incorporate a mostly black area formerly in the Central elementary zone, having the Lamar sixth graders attend the new Terrell elementary, and the closing of Raynal, with the concommitant assignment of its heavily white, northeast rural students to the new Terrell elementary.

The experts refused to fault the district for not moving the Mayes boundary still farther east, since the 1979-80 move had already placed its line only a few blocks west of the Terrell facility. Because of Layne’s size and condition, Tapscott “would be hesitant of reassigning large numbers of students to it.” He also would not condemn the elementary system because its only pairing involved the Lamar zone sixth graders attending Terrell. Though certain other plans which had been suggested resulted in substantially greater racial “balance” in the elementary schools by extensive “pairing” and transportation, Tapscott and Williams were unwilling to recommend any of them in preference to the 1979-80 system or to say that they were materially superior or other than “about equal” to it.

Respecting the Peabody closing, Tapscott testified that among the elementary schools Golden Rule, Houston and Peabody, which were located relatively close to each other, it made sense to close one, and Peabody was a logical choice as it “was the closest to the downtown area and would probably be able to serve the community for the least amount of time.” The downtown area was losing population, particularly of school age. Further, Tapscott considered that the board “made a proper decision” in closing Raynal, which was “too small,” and “reassigning those students to Terrell.” None of the experts criticized the closing of Central or the conversion of Terrell to an elementary school.

The experts were concerned with the disproportionately high percentage of black students at Terrell elementary, located in the heart of an overwhelmingly black residential neighborhood. However, this appeared to be more a fear for the future than a criticism of the present. The concern principally related to the possibility of future “white flight,” which would take place if the “tip ratio” usually experienced in Texas of “approximately 35% black” were reached. As Dr. Tapscott explained the “tip ratio,” “Once [a school’s black population] reaches that 35%, then it seems to jump very quickly to 50 to 60 and move very fast to become a minority school .... The patrons find some way to another elementary zone . However, the experts felt that this did not require a present change in the Terrell attendance zone, but rather indicated that the board should carefully monitor the situation there. Superintendent Alexander testified that the board was committed to such monitoring, and would take steps if Terrell became over 35 percent black. Testimony also disclosed that the district’s honors program for grades four, five and six was located at Terrell, and intra-district elementary transfers were limited to that program. Similarly, all elementary transfers from without the district were assigned to Terrell. Dr. Tapscott, in describing the 1979-80 plan, said it placed at Terrell a larger number of white students “than would normally be there with the elementary attendance boundaries as they would normally be drawn.” The experts were high in their praise of Terrell. Dr. Tapscott described it as “probably the best educational program that I observed going on in the elementary schools of Denison” and as “the best elementary school in the district.” Williams “wholeheartedly” concurred, and testified Terrell represented “the best use of a formerly all-black school” he “had ever seen.”

While Tapscott and Williams were generally commendatory of the board’s 1979-80 elementary program, they were somewhat critical of the district’s performance in past years. The criticism, however, appears to have been essentially that the board had waited too long to do what it did in 1979. Thus, Dr. Tapscott addressed the prior situation where Central had been over 50 percent minority and most of the other elementary schools (generally five out of eight) had minority enrollments several percentage points under the minority elementary percentage of the district as a whole, stating:

“I would have to conclude that they should have responded earlier, that they didn’t and, therefore, I would have to assume that they wanted to keep it that way for some reason, and they should have responded much earlier than they did.”

Mr. Williams also was critical of the board’s prior inactivity, though he would not ascribe it to segregative intent. He answered in the affirmative when asked, “whatever their intention,” whether the building of Mayes was a cause of the Annex and Central having become “resegregated.” However, Williams seemed to regard the previous high concentrations of blacks at Central (and at McDaniel junior high) to be largely the result of demographic factors unrelated to school board actions or educational considerations. He testified in that regard:

“[W]hy and how did this city get to seem to be a town that had the east half of town and the west half of town with an awful lot of railroad yards and ponds and lakes and all sorts of things in between, but ... it really boils down to Old East Denison and New West Denison.
“One of the explanations ... was Perrin Air Force Base. A tremendous housing development in the late fifties and, primarily, the sixties all, of course, to the west, typical Texas pattern, and with the closing of Perrin Air Force Base, it seemed to me that the district and the town went under a considerable condition of economic stress, ... and, consequently, it seemed to me that even though a district may have had the best of intents at any given moment, it did not have the best tax base....
“[I]t was a poor economic base, a disruptive closing of an Air Force Base, and an apparent pattern established by realtors, none of whom, you know, are responsible to the school, of putting all the new folks out west and leaving an interior city, the old City of Denison, on pretty shaky ground when it comes to who is left there.
“Quite often you have elderly people, retired railroad people, people with no children. We found this a unique and disturbing type of condition ....”

Other evidence showed that the district suffered an approximately 20 percent enrollment decline from 1969-70 to 1979-80. While the black student percentage in the district as a whole remained in the 10 to 12 percent range during this period, the testimony of Mr. Williams suggests that the Perrin-sparked westward population movement, largely composed of whites of an age to have younger children, raised the percentage of black elementary students residing in the areas of “old” or “east” Denison. Williams also testified:

“Q. Do you see anything in the past actions of the district as it relates to school closings and attendance zones and that sort of thing that has caused a greater concentration of black people in the Terrell area?
“A. No, I did not.”

Elementary classes in the DISD were generally composed of 25 students. While some black residents testified that they preferred an even district-wide dispersion (theoretically resulting in three blacks per class), at least one of the plaintiffs’ witnesses stated there should be no less than five blacks per class of 25 or 30. Williams indicated his general concern with only 10 percent blacks in a class, and also stated, “I have never approved of having only 1 or 2 or 3 blacks in an elementary classroom.” Plaintiffs’ expert Dr. Gay testified that eight or nine blacks in a class of 25 (32 to 36 percent) was generally a better situation than two in such a class (8 percent).

B. District Court Orders.

(i) Violations found and relief granted.

In the portion of its June 10, 1981, opinion dealing with the elementary schools, the district court stated:

“The primary issue as to the new elementary assignment scheme is whether it allows any particular elementary schools to remain as racially identifiable vestiges of segregation. It is well settled that the existence of racially identifiable schools in a formerly de jure dual school system constitutes prima facie evidence of a constitutional violation, which can be overcome only by a showing that the racial composition of the schools is unrelated to the prior discriminatory educational structure. Swann v. Charlotte-Mecklenburg Board of Education, 402 U.S. 1, 18, 26 [, 91 S.Ct. 1267, 1277, 1281, 28 L.Ed.2d 554] (1971).” (Footnote omitted.)

Relying on the language in Swann v. Charlotte-Mecklenburg Board of Education, 402 U.S. 1, 26, 91 S.Ct. 1267, 1281, 28 L.Ed.2d 554 (1971), concerning “a presumption against schools that are substantially disproportionate in their racial composition,” the district court ruled that Layne, Hyde Park and Golden Rule elementary schools were racially identifiable (as white), since the proportion of black students in each varied by more than five percentage points from that of the elementary school population of the district as a whole (11.6 percent).

While in regard to Terrell certain aspects of the June 10 opinion are ambiguous, considered as a whole the only fair conclusion is that it did not find Terrell racially identifiable. The court accordingly ordered the board to submit “a plan designed to ensure that each elementary school within the district will contain a minimum Black population of six percent as of the beginning of school in the fall of 1981.”

The plan submitted by the school board met the requirements of the June 10 order, and was not objected to by the plaintiffs, but was rejected by the district court in its opinion of August 14. The court there noted that in its opinion of “June 10, 1981, Golden Rule, Hyde Park, Layne and Terrell were found to be racially identifiable.” While the plan submitted by the board reduced the percentage of blacks at Terrell from 33.8 percent to 29.3 percent, the district court held that “Terrell remains a racially identifiable school” since its proportion of black students varied by more than five percentage points from the 11.6 percent district-wide black elementary percentage. For this reason, the court rejected the plan, and ordered the board to submit a plan “which assures that the population of each elementary school will be no less than six percent black, and no more than seventeen percent black.”

At a September 1 hearing on the district’s motion for stay, a potential compromise elementary plan was developed which was later rejected by the board. On September 14, at a resumption of the stay hearing, the plaintiffs requested the court to order a modification of such plan into effect, and the court did so, effective January 1982, by its September 18 order.

Although this order vacated the August 14 order and purported to enforce that of June 10, nevertheless, under the plan ordered implemented the projected elementary attendance figures would meet the requirements of the August 14 order that no school’s proportion of black students be less than 6 percent or more than 17 percent. This was largely achieved through noncontiguous zones and increased busing.

(ii) Determination of racial identifiability-

It is in some respects difficult to determine whether the district court treated the issue of racial identifiability as essentially one of law, ultimately determined by the general statistical criterion of a variation as large as five percentage points in the school’s proportion of black students from that of the district as a whole, or as a question of fact to be determined for each individual school and district in the light of all the relevant circumstances.

The June 10 opinion in part evidences something of the latter approach. Its analysis in this regard commences by stating “an institution is racially identifiable when its racial composition differs significantly from that in the district as a whole.” It then cites authorities for the proposition that, in districts with black student proportions similar to the DISD’s, racially identifiable schools “have been held to be those whose Black population deviate by more than five percentage points” from the district-wide percentage. Nevertheless, the June opinion then sets out the following qualification;

“Of course, it must be emphasized that such a statistical measure is only a starting point, albeit a valuable one, in determining the presence of racially identifiable schools. Other factors may militate against a conclusion of racial identifiability and counteract the import of the statistics.”

However, the “only a starting point” approach seems to have been applied chiefly, if not entirely, to Terrell, with the result that, in the June 10 opinion, Terrell was not found racially identifiable. The determination in the June 10 opinion that Golden Rule, Hyde Park and Layne were racially identifiable appears to be essentially a statistical one, coupled with the fact that these schools had been all white in the time of dual racial attendance zones.

Moreover, the district court’s subsequent orders herein reflect that it treated the question of racial identifiability essentially as one of law, to be ultimately determined strictly on the basis of a generally applicable statistical formula. This approach is evident in the court’s order of July 17 (denying motion for stay) which states that any school in the district is “racially identifiable” if its black population is “greater than 17.6 percent, or less than 6.6 percent.” Similarly, in the order of August 14, in which it rejected the board’s plan submitted pursuant to the June 10 order, the court stated, “Terrell school is racially identifiable, because black students are relatively over-represented in that school.” (Emphasis added.) No new or different factual analysis is made of Terrell, nor are any different facts stated or conclusions drawn, as compared with the June 10 order, with the sole exception of the ultimate statistical determination of racial identifiability. The August 14 order states that “a racially identifiable school is one in which the racial composition of the school varies more than five percentage points, plus or minus, from the racial composition of the school district,” then recites the DISD percentages, and concludes by stating, “any school with more than 16.6 percent black population is a racially identifiable school.” (Emphasis added.)

In its final order of September 18 the district court again reviewed the methodology it had followed in its determinations of racial identifiability. This order reflects that the district court did not consider the issue of racial identifiability as, in any meaningful sense, a factual or evidentiary question, dependent on local circumstances or history, but rather viewed it as purely a question of law to be ultimately determined by a generally applicable statistical formula.

The court labeled as “mistaken” the contention that selection of an appropriate statistical variation for application in a given case was an “essentially ‘evidentiary’ ” process, and stated that among the DISD’s “many misapprehensions” was the “assumption that the determination of racial identifiability is a factual issue.” Instead, the court ruled that “a standard for determining racial identifiability is a legal matter.” It explained that its choice of a five percentage point deviation factor as the relevant standard for the DISD, though it represented an application of the statistical formula espoused by Dr. Gordon Foster as reported in the opinions in Penick and Higgins (see note 30 supra), was nevertheless a “legal conclusion” amounting to the adoption by the district court of this statistical formula as the broadly applicable legal standard by which racial identifiability in all cases of this general nature, not just in the DISD’s case, would be determined.

The September 18 opinion then addressed the analysis of racial identifiability as made in the June 10 opinion. Preliminarily, it again observed that “racial identifiability” is “a legal conclusion,” and that when such a legal determination is made respecting “a school” in a former de jure system, such “determination establishes a prima facie case that the school system violates the Constitution.” This prima facie case, however, can be rebutted by evidence that the racial identifiability of the school in question “is unrelated to the prior history of legally mandated segregation.” The court then turned to the specific language in its June 10 opinion that “a statistical measure is only a starting point, albeit a valuable one, in determining the presence of racially identifiable schools.” The court observed that such phraseology contained “an unfortunate verbal confusion,” as the term “racially identifiable” was used in the quoted sentence of the June 10 opinion to mean “a finding of constitutional dimension, rather than the prima facie conclusion.” As we understand it, then, the court’s September 18 opinion holds that a school’s “racial identifiability” is determined finally by a purely statistical analysis, and that for purposes of such determination the statistics are not “only a starting point.” Moreover, as we read this portion of the September 18 opinion, the status of racial identifiability (“the prima facie conclusion”), determined statistically and as a matter of law, gives rise, again as a matter of law, to a prima facie case of unconstitutionality. The school district cannot prevent the prima facie case from arising by presenting evidence of “other factors” which “militate against a conclusion of racial identifiability and counteract the import of the statistics.” But the case is only prima facie, and may be rebutted by showing that the racial identifiability is unrelated to past (or present) de jure segregation; however, if the prima facie case is not rebutted, then the racial identifiability is conclusively presumed to result from de jure segregation and hence constitutes a condition the school board is constitutionally required to eliminate (“a finding of constitutional dimension”).

In line with this analysis, the September opinion finds that Terrell, on the basis of the statistical formula, is “racially identifiable as black,” although it concedes that “unquestionably, Terrell school is predominantly white, since the student population is two-thirds white.” Accordingly, the court holds that the “DISD is charged with the responsibility of establishing that this characteristic [racial identifiability as black] is not traceable to a history of racial discrimination.” At this point, however, the path of the September 18 opinion becomes less plain. It states “as the June 10 order makes clear, DISD singularly failed to carry this burden.” We are unable to so construe the June 10 opinion. Not only does it conspicuously fail to find Terrell racially identifiable (see note 25 supra), it likewise contains no indication that the DISD failed to carry any burden respecting Terrell. The September opinion continues by stating, however, that other evidence “militated against the conclusion of racial identifiability” of Terrell, citing its educational quality. This seems to suggest, contrary to the above-noted implications of the September opinion’s discussion of the “unfortunate verbal confusion” concerning “racial identifiability,” that statistical racial identifiability may be rebutted by evidence, such as quality of education, not directly related to whether the statistical condition results from present or past intentionally segregative acts by the educational authorities. In any event, in the remaining portion of the September opinion, the district court does not expressly address the effect, factually or legally, of the evidence which it found to “militate against the conclusion of racial identifiability” of Terrell. Nor does it expressly address the presence or absence of any other rebuttal as to either Terrell’s racial identifiability or any prima facie case arising therefrom. The September opinion does refer to “the conclusions of the June 10 order ... that the racial identifiability of Terrell establishes a presumption of unconstitutionality.” However, as we have noted, the June 10 order makes no such determination.

The district court in its September opinion placed heavy reliance, in its racial identifiability analysis, on the following from this Court’s opinion in Lemon v. Bossier Parish School Board, 566 F.2d 985, 987 (5th Cir.1978):

“In the conversion from dual school systems based on race to unitary school systems, the continued existence of all-black or virtually all-black schools is unacceptable where reasonable alternatives exist.”

The district court then concluded that school desegregation law had become sufficiently “refined” so that it was appropriate to substitute the concept of “racially identifiable” as white or black (presumably as determined statistically under the Foster formula) for the concept of “all-black or virtually all-black” as set forth in Lemon.

Accordingly, though the matter is not entirely free from doubt, considering all the district court’s opinions herein dealing with the elementary schools, we conclude that it finally treated the issue of whether a school was racially identifiable as a question of law ultimately determined by a legally established statistical formula of broad general application, calling by its terms in this instance for a maximum deviation of five percentage points, and not as a discrete question of fact depending on, or subject to rebuttal by evidence of, the particular history or circumstances of the school or district in question.

(iii) Determination of constitutional violation.

Having determined the existence of racially identifiable elementary schools in a former de jure district, the district court noted that such schools, absent a showing that their racial composition is unrelated to racially segregative action by the school authorities, must be “found to be violative of the Constitution.”

We observe at this point that the ultimate constitutional violation, the existence of which may be presumed from the unrebutted prima facie case, is not merely racial imbalance in the schools, Milliken v. Bradley, 433 U.S. 267, 280-81, 97 S.Ct. 2749, 2757, 53 L.Ed.2d 745 (1977), but is rather “a current condition of segregation resulting from intentional state action.” Washington v. Davis, 426 U.S. 229, 240, 96 S.Ct. 2040, 2047, 48 L.Ed.2d 597 (1976). Accordingly, assessment of whether the prima facie case has been rebutted involves a determination of whether the current racial composition of the schools does not result from past or present intentionally segregative action by the educational authorities. Obviously such action includes that which is covertly, as well as avowedly, for a segregative purpose. The current racial composition of the schools will, of course, result from intentionally segregative action if the present system of student assignment is adopted or maintained with a segregative purpose. Where that is not the case, the current racial composition of schools may nonetheless reflect, in whole or in part, the present effect of past intentional educational segregation. This may be the ease where the effects of such past educational segregation are reflected in current housing patterns, or school locations, to the extent that such conditions are in turn reflected in the complained of current racial composition of the schools. See United States v. Texas Education Agency (Lubbock I.S.D.), 600 F.2d 518, 525 (5th Cir.1979).

The district court in its June 10 opinion expressly found that the “DISD has failed to fulfill its burden of proving that the current racial identifiability of Layne, Hyde Park and Golden Rule is unrelated to past discriminatory action.”

It seems apparent that this determination was not to any extent based on a conclusion that the DISD had failed to demonstrate that its actions, either in adopting the 1979-80 plan or at any time prior thereto and after the dismantling of the dual racial attendance zones some fifteen or sixteen years previously, were taken without segregative purpose or intent. There was certainly evidence reflecting both present and past good faith on the part of the DISD. In its June 10 opinion the district court stated:

“Certainly, some progressive efforts have been made in the past by DISD, and the plan instituted in 1979 represented an improvement over the previous assignment pattern.”

And in its order of July 17, denying the district’s request for stay of the June 10 order, the court further addressed the question of intent, as follows:

“[M]any of the measures taken [by the DISD] in the past fifteen years were conscientious efforts to ameliorate the disturbing legacy of segregation that was inherited from a time when separate and unequal schooling for students in the Denison school district was enshrined in law and custom.... However, the command of the law is clear — the good intentions of officials in the school system cannot compensate for actual failures of any plans ultimately implemented. If the plans do not work, then they must be rejected .... Thus the laudable past efforts of DISD to diminish the vestiges of racial discrimination may not now be used as a foil to the continuing affirmative duty, incumbent on DISD, ultimately to eliminate all vestiges and establish a truly unitary system of public education.” (Emphasis added.)

Moreover, we note that just before reconvening following the mid-afternoon recess on the first day of trial, while the plaintiffs’ case-in-chief was still being presented, the court, sua sponte, advised the parties as follows:

“THE COURT: All right. It’s my sense that we’re wasting a lot of time in this case talking about intent. The Supreme Court held in Dayton Board of Education against Brinkman in 1978 at 443 U.S. 526 [99 S.Ct. 2971, 61 L.Ed.2d 720] — the decision came down July 2, 1979 — says, ‘The measure of the post Brown I conduct of a school board under an unsatisfied duty to liquidate a school system is the effectiveness, not the purpose, of the actions in decreasing or increasing the segregation caused by a school system.’
“So, we don’t have here a question of intent; we’ve got a question of effectiveness, so I suggest that all of this testimony about intent is totally irrelevant, so let’s go forward with the evidence that is relevant.”

The above statements not only expressly reflect the court’s view that intent was irrelevant and should not be further explored, but also, by implication, its view that there was no issue for further evidentiary development, or for factual determination, as to whether the current racial compositions of the schools resulted from past illegal segregation by the educational authorities.

The district court did not expressly address the question of whether the residen-. tial concentration of blacks in a segment of the northeast quadrant of Denison was shown not to have been caused by intentionally segregative acts of the educational authorities. It merely stated in this connection:

“The school district cannot meet the presumptive burden raised by the presence of the racially identifiable schools with the contention that the present level of segregation results from the neighborhood school policy, in combination with segregated housing patterns.”

No evaluation was made of the testimony of Mr. Williams, the court-appointed expert, as to some of the causes for Denison’s racially concentrated housing.

It seems apparent, then, that the district court did not make any kind of factual determination or analysis as to the presence or extent of any relationship between the current racial compositions of the respective elementary schools and intentionally segregative conduct, past or present, on the part of the educational authorities. Rather, the court simply stated that the presumption from statistical imbalance was unrebutted, and hence the statistical imbalance was automatically equated to a constitutional violation.

C. Review of the District Court Orders.

As the district court aptly noted, Denison’s case presents “uncharted terrain” in this field of jurisprudence and “is unusual among the gamut of southern desegregation litigation, in that the city’s school system has a relatively small percentage of Black students.”

Approximately a decade and a half has elapsed since all overt segregation in the DISD ceased, and in that time it is unquestioned that considerable progress has been made without judicial intervention. There are no all-black or virtually all-black schools. There are no majority black schools. In regard to its several elementary schools, the DISD follows, in apparent good faith, a neighborhood school policy, and there is no suggestion that the attendance lines have been segregatively gerrymandered. Because of the small number and residentially concentrated nature of the black elementary students, it is plain that, even if somewhat integratively gerrymandered, normal geographically continuous neighborhood attendance zones would nevertheless still result in a disproportionately large number of blacks in one of the zones and a disproportionately large number of whites in others. The disproportionately black elementary school was nonetheless two-thirds white, and was described by impartial experts as the best in the district. Transfers to it were encouraged. It had had large white majorities for over ten years. It is described as being a “desegregated” school as long ago as 1969-70. There was evidence that the residential concentration of blacks was caused at least in part by factors unrelated to educational policies or actions. The proportion of blacks in the district’s elementary student body, 11.6 percent, was so small that a completely even dispersion would result in all schools having white percentages so large (over 88 percent, or, with a five percentage point variance, several schools over 90 percent) as to be at a level generally considered “virtually one race.” Nevertheless, there were no 100 percent white schools, and over half the white students attended schools with approximately 6 percent or more black students.

As to Layne, Hyde Park, Golden Rule and Terrell, the district court’s judgment is based on an interrelated two-step process. First, these schools were ultimately determined to be “racially identifiable,” as a matter of law, essentially on the sole basis of their former overt “de jure” status combined with the deviation of their respective minority percentages from that of the district as a whole by a number of percentage points greater than that allowed by the Foster formula. This finding of “racial identifiability” was basically arrived at purely as a matter of law, as distinguished from being the result of factual conclusions or determinations with reference to the particular conditions and circumstances of these schools and the DISD. Second, it was concluded, as a legal consequence of the presumption arising from the “racially identifiable” status of these schools, that their condition in this respect resulted from intentionally segregative conduct by the educational authorities. Similarly, this determination was in essence a legal one, and did not rest on an articulated consideration of the particular facts and circumstances of these schools and the DISD or on a drawing of factual inferences or conclusions therefrom. We hold that, under the peculiar circumstances of this case as outlined in the preceding paragraph, the referenced method employed by the district court, to determine whether the racial composition of these elementary schools represented an unconstitutional vestige of “de jure” educational segregation, was erroneous in that such determination was made essentially only as a matter of law and was not supported by any sufficient underlying factual determinations and conclusions.

Our consideration of this matter must necessarily begin with, and be guided by our understanding of, the Supreme Court’s opinion in Swann, particularly the following portions thereof:

“(1) Racial Balances or Racial Quotas. U
“Our objective in dealing with the issues presented by these eases is to see that school authorities exclude no pupil of a racial minority from any school, directly or indirectly, on account of race; ... u
“[I]n the opinion and order of the District Court of December 1, 1969, we find that court directing ‘that efforts should be made to reach a 71-29 ratio in the various schools so that there will be no basis for contending that one school is racially different from the others . .., [t]hat no school [should] be operated with an all-black or predominantly black student body, [and] [t]hat pupils of all grades [should] be assigned in such a way that as nearly as practicable the various schools at various grade levels have about the same proportion of black and white students.’
“The District Judge went on to acknowledge that variation ‘from that norm may be unavoidable.’ This contains intimations that the ‘norm’ is a fixed mathematical racial balance reflecting the pupil constituency of the system. If we were to read the holding of the District Court to require, as a matter of substantive constitutional right, any particular degree of racial balance or mixing, that approach would be disapproved and we would be obliged to reverse. The constitutional command to desegregate schools does not mean that every school in every community must always reflect the racial composition of the school system as a whole. a
“We see therefore that the use made of mathematical ratios was no more than a starting point in the process of shaping a remedy, rather than an inflexible requirement. From that starting point the District Court proceeded to frame a decree that was within its discretionary powers, as an equitable remedy for the particular circumstances.... Awareness of the ra cial composition of the whole school system is likely to be a useful starting point in shaping a remedy to correct past constitutional violations. In sum, the very limited use made of mathematical ratios was within the equitable remedial discretion of the District Court.
“(2) One-race Schools.
“... Schools all or predominantly of one race in a district of mixed population will require close scrutiny to determine that school assignments are not part of state-enforced segregation.
“In light of the above, it should be clear that the existence of some small number of one-race, or virtually one-race, schools within a district is not in and of itself the mark of a system that still practices segregation by law. The district judge or school authorities should make every effort to achieve the greatest possible degree of actual desegregation and will thus necessarily be concerned with the elimination of one-race schools. No per se rule can adequately embrace all the difficulties of reconciling the competing interests involved; but in a system with a history of segregation the need for remedial criteria of sufficient specificity to assure a school authority’s compliance with its constitutional duty warrants a presumption against schools that are substantially disproportionate in their racial composition. Where the school authority’s proposed plan for conversion from a dual to a unitary system contemplates the continued existence of some schools that are all or predominately of one race, they have the burden of showing that such school assignments are genuinely nondiscriminatory. The court should scrutinize such schools, and the burden upon the school authorities will be to satisfy the court that their racial composition is not the result of present or past discriminatory action on their part.” 402 U.S. at 22-24, 91 S.Ct. at 1279-80 (footnotes omitted; emphasis added, except in headings).

In applying the foregoing principles, we turn first to Terrell, which the district court determined to be, as a matter of law, racially identifiable as black, despite its clear white majority, solely on the basis of its prior overtly de jure status and its failure to meet the Foster formula percentage criteria. Under the circumstances of this case, such a mechanical method of determination was legally erroneous.

While, as reflected by the above quotation, Swann does mention “a presumption against schools that are substantially disproportionate in their racial composition,” the context in which this language appears strongly suggests that the intended reference is limited to schools that are all or predominantly of one race. The language in question appears in a section of the opinion entitled “One-race Schools.” Earlier in the opinion, the Court has explained that this section will deal with the question of “whether every all-Negro and all-white school must be eliminated as an indispensable part of a remedial process of desegregation.” Id. 402 U.S. at 22, 91 S.Ct. at 1279. In the portion of the section in question preceding the “substantially disproportionate” sentence, there are several references to schools that are “all or predominantly of one race,” or “one-race, or virtually one-race,” or simply “one-race,” which reinforces the view that schools which may fairly be so described constitute the subject matter of this section of the opinion. Moreover, in the sentence immediately following the “substantially disproportionate” language, the Court again refers to “schools that are all or predominantly of one race,” and says that the school authorities “have the burden of showing that such school assignments are genuinely nondiscriminatory.” And in the next sentence, it is said that the “court should scrutinize such schools,” and the school authorities have the burden to show “that their racial composition is not the result of present or past discriminatory action” (emphasis added). It seems clear that in such sentence of the Swann opinion the words “such schools” and “their” (which we have italicized) refer to the phrase “schools that are all or predominately of one race” appearing in the immediately preceding sentence in Swann.

Moreover, the sentence in Swann mentioning “a presumption against schools that are substantially disproportionate” does not explain the operation or object of such presumption. The only mention of any burden on the school authorities, or any requirement for a showing by them respecting the effects of current or past discriminatory actions, is contained in the next two sentences. These sentences make clear that what such burden and showing relates to is “schools that are all or predominately of one race.” And, it is those schools which constitute the subject matter of this section of the Swann opinion. Further, it is “in a district of mixed population” that the opinion has said such all or predominantly one-race schools “require close scrutiny.” Accordingly, to the extent the sentence in Swann concerning “substantially disproportionate” may be understood to modify the language in that opinion concerning a burden of proof arising from the continued existence of former de jure “schools that are all or predominately of one race,” we believe that such modification would likely have been conceived of as operating more in the nature of a limiting condition to, rather than an expansion of, the one-race or virtually one-race concept. Finally, we observe that the central issue before the Court in Swann related to the elementary assignment plan, and the Court observed that under the board’s plan, which it held the district court properly rejected, “[m]ore than half of the Negro elementary pupils were left in nine schools that were 86% to 100% Negro”, id., 402 U.S. at 9, 91 S.Ct. at 1272, though the minority population in the entire district was approximately 29 percent.

The Supreme Court’s discussion in Swann of “Racial Balances” in schools likewise reinforces our conclusion. In the section of the opinion in which that topic constitutes the subject matter under consideration, there is no indication that mere racial imbalance in the schools of a former de jure district gives rise as a matter of law to a presumption that the imbalance results from intentionally segregative conduct by the educational authorities. Rather, the discussion of racial imbalance is in terms of its relevance to the general nature of the remedy which the district court may frame for a condition it has already found to be the result of intentionally segregative conduct. That, of course, was the context in which the matter was presented to the Supreme Court in Swann. Moreover, even in that context the Court was obviously concerned with limiting the use and significance of mathematical ratios and statistical racial balance. The opinion counsels against using the district-wide racial percentages as a “norm” (even a norm from which variations might be permitted) by which to judge the several schools, and states that remedial decrees should not be framed to require as a substantive constitutional right “any particular degree of racial balance” and that the “command to desegregate” does not require each school to mirror the racial balance of the whole district. Mechanical or “inflexible” use of general statistical ratios is clearly discouraged. On the other hand, where mathematical ratios are used as “no more than a starting point in the process of shaping a remedy ... for the particular circumstances,” then this “very limited use” is within the district court’s “remedial discretion.” Id. 402 U.S. at 24, 25, 91 S.Ct. at 1280.

Here the district court employed mathematical ratios primarily to determine the existence of a condition of unlawful segregation. Moreover, its use of such ratios cannot fairly be described as “very limited.” The district court did not use “awareness of the racial composition of the whole school system” as only a “starting point” in the flexible and discrete exercise of its remedial jurisdiction to fashion a remedy for the particular circumstances of this case. Rather, it determined “racial identifiability” by applying the Foster formula, fixing the maximum permissible variation from the district-wide racial composition “norm,” as an essentially inflexible rule of law of general application.

Our reading of Swann appears consistent with other decisions of the Supreme Court. None has been cited to us, nor are we aware of any, presuming continued illegal segregation merely because of the existence of a disproportionately large black enrollment in a formerly de jure school having a clear majority of white students. We will mention some of the leading decisions of the Court since Brown II.

In Green v. School Board of New Kent County, 391 U.S. 430, 88 S.Ct. 1689, 20 L.Ed.2d 716 (1968), where the Court spoke of the duty to create “a unitary system in which racial discrimination would be eliminated root and branch,” id. 391 U.S. at 438, 88 S.Ct. at 1694, the opinion notes the absence of residential segregation in the district and that the district has only two schools, one of which is “all-Negro” and is attended by “85% of the Negro children in the system.” Id. 88 S.Ct. at 431, 441, 88 S.Ct. at 1691, 1696. In Davis v. Board of School Commissioners of Mobile County, 402 U.S. 33, 91 S.Ct. 1289, 28 L.Ed.2d 577 (1971), a case handed down the same day as Swann and like it a unanimous opinion authored by Chief Justice Burger, the Court states that having “found a violation” there should be “every effort to achieve the greatest possible degree of actual desegregation” and that “the measure of any desegregation plan is its effectiveness.” Id. 402 U.S. at 37, 91 S.Ct. at 1291. The opinion notes that under the plan, which the Court rejects, “64% of all the Negro elementary school pupils” in the district attended schools “over 90% Negro,” and “over half” of the “Negro junior and senior high school students” were attending “all-Negro or nearly all-Negro schools.” Id.

Wright v. Council of City of Emporia, 407 U.S. 451, 92 S.Ct. 2196, 33 L.Ed.2d 51 (1972), sustained an injunction against the division of a school district which was attempted two weeks after the issuance of a district-wide court desegregation order. The division would have exacerbated the racial imbalance in the schools, and the district court found that “ ‘in a sense, race was a factor in the city’s decision to secede’ ” (though this was apparently not the primary or dominant purpose). Id. 407 U.S. at 461, 92 S.Ct. at 2202. Justice Stewart’s opinion for the majority was at pains to point out that the Court was not holding that the racial imbalance factor was sufficient to sustain the injunction, absent “other considerations.” Chief Justice Burger, dissenting for himself and Justices Blaekmun, Powell and Rehnquist, concluded that the “other considerations” relied on by the majority were not significant, and addressed the racial balance factor at some length. His opinion states, inter alia, that “a geographic assignment pattern is prima facie” valid, id. 407 U.S. at 472, 92 S.Ct. at 2208, that “racial balance” is neither “the norm to be sought” nor “the condition requiring a judicial remedy,” id. 407 U.S. at 473, 92 S.Ct. at 2208, and, regarding the six percentage point increase in racial imbalance which the division would cause, that:

“Obsession with such minor statistical differences reflects the gravely mistaken view that a plan providing more consistent racial ratios is somehow more unitary than one which tolerates a lack of racial balance.” Id. 407 U.S. at 474, 92 S.Ct. 2209.

Citing his opinion in Swann, the Chief Justice also called attention to the absence in the proposed systems of “ ‘one-race, or virtually one-race schools.’ ” Id. 407 U.S. at 477, 92 S.Ct. at 2210.

In Keyes v. School District No. 1, 413 U.S. 189, 93 S.Ct. 2686, 37 L.Ed.2d 548 (1973), the Court stated that where “in a meaningful portion of a school system” segregated schools exist as a result of intentionally segregative conduct of the educational authorities, the latter have “the burden of proving that other segregated schools within the system are not also the result of intentionally segregative actions.” Id. 413 U.S. at 208, 93 S.Ct. at 2697. The opinion reflects the trial court’s factual finding, made in 1969, that the “School Board had engaged over almost a decade after 1960 in an unconstitutional policy of deliberate racial segregation with respect to” schools in the Park Hill area, which embraced over 37 percent of the district’s Negro students, by such devices as “gerrymandering of student attendance zones” and the like. Id. 413 U.S. at 192, 93 S.Ct. at 2689. The minority percentage in the entire district was 34 percent (14 percent Negro and 20 percent Hispano). Id. 413 U.S. at 195, 93 S.Ct. at 2690. Of the five elementary schools in the Park Hill area, two were over 96 percent minority and another was over 89 percent; of the three junior high schools there, one was over 96 percent minority and the other was over 75 percent. Id. 413 U.S. at 199 n. 10, 93 S.Ct. at 2692 n. 10. Adjoining the Park Hill area was the “core city” area, where, the opinion observes, there were 22 schools over 70 percent minority, 11 of which were over 90 percent minority. The Supreme Court held that the district court, by proceeding on the premise that its finding respecting the Park Hill schools was “irrelevant” to the core city schools without stating any reason therefor except its view that the areas were “different,” id. 413 U.S. at 204, 206, 93 S.Ct. at 2695, 2696, “applied an incorrect legal standard in addressing ... [the] contention that ... [the] Board engaged in an unconstitutional policy of deliberate segregation in the core city schools.” Id. 413 U.S. at 198, 93 S.Ct. at 2692. The Court also stated:

“What is or is not a segregated school will necessarily depend on the facts of each particular case. In addition to the racial and ethnic composition of a school’s student body, other factors, such as the racial and ethnic composition of faculty and staff and the community and administration attitudes toward the school, must be taken into consideration.” Id. 413 U.S. at 196, 93 S.Ct. at 2691

In two 1977 decisions, the Supreme Court reemphasized that the mere racial imbalance in public schools, not caused by intentionally segregative actions of educational authorities, was not violative of the Constitution, and that where such authorities had taken purposefully discriminatory actions the remedy should be designed to redress the segregation resulting therefrom, or as nearly as possible to restore the victims to the position they would have had absent such conduct, rather than to achieve any particular degree of racial balance.' Millikan v. Bradley, 433 U.S. 267, 280, 97 S.Ct. 2749, 2757, 53 L.Ed.2d 745 (1977); Dayton Board of Education v. Brinkman (Dayton I), 433 U.S. 406, 413, 417, 420, 97 S.Ct. 2766, 2772, 2774, 2775, 53 L.Ed.2d 851 (1977).

Dayton Board of Education v. Brinkman (Dayton II), 443 U.S. 526, 99 S.Ct. 2971, 61 L.Ed.2d 720 (1979), and Columbus Board of Education v. Penick, 443 U.S. 449, 99 S.Ct. 2982, 61 L.Ed.2d 666 (1979), decided the same day, were each in many respects similar to Keyes. Dayton II, citing Wright and Keyes, stressed that a school board which, by actions taken with discriminatory purpose, had established a dual system, as had Dayton at the time of Brown I, was thereafter under a duty to take effective action to “liquidate” “the segregation caused by the dual system.” 443 U.S. at 538, 99 S.Ct. at 2979. The Court pointed out that in the 1950’s over half the Dayton black students attended “100% black schools,” that all these institutions (or their successors) remained “black schools” at the time of trial, that every school which was “90 percent or more black” in the 1950’s or 1960’s or early 1970’s and which was still in use at the time of trial (1972-73) remained “90 percent or more black,” and that when the complaint was filed (1971-72) there were 21 schools 90 percent or more black (22 the following year) attended by over 75 percent of the black students. Despite a 43 percent district-wide black student population, approximately three-fourths of the schools were “virtually all” (90 percent or more) one race. Id. 443 U.S. at 529 and n. 1, 535, 537, 99 S.Ct. at 2974 and n. 1, 2977, 2978. Citing this and other evidence, including numerous school board actions extending into the 1970’s having segregative effect and in several instances taken with segregative purpose, the Court refused to disturb the Sixth Circuit’s reversal of the district court’s ruling that the present segregated condition of Dayton’s schools was not shown to have resulted from purposeful school board action.

In Columbus the Court affirmed the determinations of the district court and the Sixth Circuit that the “Columbus Public Schools are in fact unlawfully segregated.” The Supreme Court, noting the extensive evidence introduced in the lengthy trial, and the district court’s finding that the current racial segregation in the system “directly resulted from [the board’s] intentional segregative acts and omissions,” 443 U.S. at 452-53, 99 S.Ct. at 2943-44, held:

“We have discovered no reason, however, to disturb the judgment of the Court of Appeals, based on the findings and conclusions of the District Court, that the Board’s conduct at the time of trial and before not only was animated by an unconstitutional, segregative purpose, but also had current, segregative impact that was sufficiently systemwide to warrant the remedy ordered.... ” Id. 443 U.S. at 455, 99 S.Ct. at 2945.

Justice White’s opinion for the Court notes that at the time of trial some 32 percent of the Columbus students were black, about 70 percent of all students attended schools 80 percent or more of one race, and half of all schools “were 90% black or 90% white.” Id. 443 U.S. at 452, 99 S.Ct. at 2943. As the Sixth Circuit’s opinion reflects, these schools included 22 over 90 percent black, of which 14 were over 96 percent black. 583 F.2d 787 at 800. Calling attention to the district court’s finding that the school board intentionally maintained “ ‘an enclave of separate, black schools on the near east side of Columbus,’ ” Justice White states:

“Proof of purposeful and effective maintenance of a body of separate black schools in a substantial part of the system itself is prima facie proof of a dual school system and supports a finding to this effect absent sufficient contrary proof . ..Id. 443 U.S. at 458, 99 S.Ct. at 2946.
Justice White thereafter notes the district court’s finding that these same schools continued to the present to be “ ‘almost completely segregated.’ ” Id. 443 U.S. at 466 n. 16, 99 S.Ct. at 2951 n. 16. The opinion also refers to the district court’s finding that since 1954 the board engaged in a series of “actions and practices” that “ ‘intentionally aggravated, rather than alleviated,’ racial separation in the schools,” instances of which continued until well after the suit was filed, including “ ‘intentionally segregative use of optional attendance zones, dis-contiguous attendance areas, and boundary changes,’ ” as well as school site selection. Id. 433 U.S. at 461-62, 99 S.Ct. at 2948-49. “Against this background,” Justice White states, the Court “cannot fault” the lower courts’ “conclusion ... that at the time of trial there was systemwide segregation in the Columbus schools that was the result of recent and remote intentionally segregative actions of the Columbus Board,” and that “on this record ... there is no apparent reason to disturb the factual findings and conclusions entered by the District Court and strongly affirmed by the Court of Appeals .... ” Id. 443 U.S. at 463-64,99 S.Ct. at 2949. Attention is called to the district court having “repeatedly emphasized that it had found purposefully segregative practices with current, systemwide impact,” id. 443 U.S. at 466, 99 S.Ct. at 2951. The opinion concludes by stating:

“There was no undue reliance here on the inferences permitted by Keyes, or upon those recognized by Swann. Furthermore, the Board was given ample opportunity to counter the evidence of segregative purpose and current, systemwide impact, and the findings of the courts below were against it in both respects.” Id. 443 U.S. at 468, 99 S.Ct. at 2952.

As heretofore noted, in the case at bar, the court below, in support of its holding that a five percentage point (or greater) variation between the proportion of minority enrollment in a given school and that in the whole district constituted the school “racially identifiable,” relied on the district court opinion in Columbus. 429 F.Supp. 229 (S.D. Ohio 1977). Dr. Foster testified in Columbus, and the limited use made in that case of his formula was based on the factual determination that the evidence there supported its reasonableness in that context. Here Dr. Foster did not testify, and there was no evidence supporting the validity of his formula or the propriety of its application to the DISD. Moreover, the use made of this formula in Columbus was distinctly limited and factual in nature. The Columbus district court describes the Foster formula as:

“. .. a rough gauge which is a useful reference point when examining particular schools. Standing alone, such statis tics are of little evidentiary value.” 429 F.Supp. at 268-69 (emphasis added).

Indeed, no determinative importance seems to have been attached to the Foster formula in the actual decisional process of the district court in Columbus, particularly not in respect to schools which though “racially identifiable” as black under the formula were nevertheless in fact clearly majority white. Considering the Columbus district court opinion as a whole, it gives no talismanic or presumptive effect to the Foster formula, and certainly does not accord it crucial or legal significance. Rather, in Columbus, the district court appears to have considered deviations from the formula’s permissible ranges as but one, and by no means the most important, of many circumstances to be taken into account and weighed in the overall process of drawing factual inferences and reaching factual conclusions, on the basis of the evidence as a ¡ whole, concerning the presence, extent and! causes of segregation in the Columbus s school system. Nor do we read the Sixth Circuit opinion in Columbus as giving any significance to Dr. Foster’s formula different from or greater than that accorded it by the district court in that case. Plainly, the Supreme Court opinion in Columbus, as can be seen from our above analysis of it, in no sense rests on an application of Dr. Foster’s formula for racial identifiability. Indeed, the closest the opinion comes to making any reference to it is in a footnote answering a complaint as to the remedy ordered, where the Court states:

“We see no misuse of mathematical ratios under our decision in Swann ... especially in light of the Board’s failure to justify the continued existence of ‘some schools that are all or predominantly of one race’ [citing Swann ].” 443 U.S. at 455 n. 3, 99 S.Ct. at 2945 n. 3 (emphasis added).

The court below also relied on Higgins v. Board of Education of City of Grand Rapids, 508 F.2d 779 (6th Cir.1974), in respect to the Foster formula. In Higgins the Sixth Circuit adverted to Dr. Foster’s testimony there, describing him as “plaintiff’s major witness,” and to his use of a maximum permissible deviation factor of five percentage points where the system-wide minority percentage was small. Id. at 787 n. 12. Nevertheless, it affirmed the district court’s judgment for the defendant, holding that the Grand Rapids schools were not unlawfully segregated, despite the fact that approximately three-fourths of the elementary schools were “racially identifiable” under the formula testified to there by Dr. Foster. Id. at 786-87. The Sixth Circuit in this connection relied heavily on its en banc decision in Goss v. Board of Education, 482 F.2d 1044 (6th Cir.1973), cert. denied, 414 U.S. 1171, 94 S.Ct. 933, 39 L.Ed.2d 120 (1974), which sustained a district court determination (340 F.Supp. 711; E.D.Tenn. 1972) that the formerly “de jure” segregated Knoxville, Tennessee schools had become “unitary” despite seven-eights of such schools being “racially identifiable” under a formula essentially the same as Dr. Foster’s. Higgins, 508 F.2d at 787. Higgins does not support the use made by the district court here of the Foster formula.

We are aware of no decisions of this Court approving a mathematical formula for determining racial identifiability such as Dr. Foster’s. Of particular relevance is Carr v. Montgomery County Board of Education, 377 F.Supp. 1123 (M.D.Ala.1974), aff’d, 511 F.2d 1374, rehearing and rehearing en banc denied, 511 F.2d 1390 (5th Cir.), cert. denied, 423 U.S. 986, 96 S.Ct. 394, 46 L.Ed.2d 303 (1975). There then District, now Circuit, Judge Frank Johnson reviewed the applicable authorities, including Swann, and held that:

“... it appears that a balance must be reached, one unquestionably subtle in its implications: while school system segregation must be actively disestablished, racial quotas for student population are not to be instituted.” 377 F.Supp. at 1133.

In Carr, Judge Johnson described the testimony of Doctors Foster and Winecoff, witnesses for the plaintiffs against the defendant Board of Education, concerning their methods or formulas for determining that various schools were “racially identifiable,” as follows:

“In doing this, Dr. Foster uses a 15 percent variation or tolerance and Dr. Winecoff uses a 10 to 15 percent variation or tolerance to determine the racial identifiability of elementary and junior high schools in the Montgomery school system. These variations are determined on the elementary and junior high school level on the basis that 48.5 percent of the total elementary and junior high school students enrolled in the system are black. This means that, pursuant to Dr. Foster’s computation (a similar procedure is followed by Dr. Winecoff), any elementary or junior high school with an enrollment less than 33.5 percent black is ‘racially identifiable’ as white. If the enrollment is more than 63.5 percent black, the school, according to Dr. Foster, is ‘racially identifiable’ as black.” Id. at 1141.

Commenting that “the use of such variances or tolerances is highly artificial,” id. at 1141, Judge Johnson refused to follow the Foster formula and stated:

“The application of such formulas must of necessity proceed on the theory that a racial balance is to be achieved and is required under the law. Furthermore, the formalistic and mechanical application of the 15 percent tolerance of Dr. Foster or the 10 to 15 percent tolerance or deviation of Dr. Winecoff gives no consideration whatsoever to the other indicia in school desegregation cases such as faculty, transportation, facilities and extra-curricular activities. To label schools that do not fall within these tolerances or deviations as ‘racially identifiable’ means that, in order not to be ‘racially identifiable,’ each school within any school system must meet certain predetermined ratios. As this Court has stated through the years time and time again, racial balance is not constitutionally required.” Id. at 1141.

In sustaining Judge Johnson on appeal this Court said “[w]e affirm the judgment of the district court for the reasons set forth in its opinion, 377 F.Supp. 1123 (M.D.Ala.1974).” 511 F.2d 1374. A petition for

rehearing en banc was rejected by an eleven-to-four vote, 511 F.2d at 1390, and the Supreme Court denied certiorari. 423 U.S. 986, 96 S.Ct. 394, 46 L.Ed.2d 303. Judge

Johnson’s opinion in Carr has never subsequently been questioned by this Court, and we have on several occasions cited it with

approval. See e.g., Calhoun v. Cook, 522 F.2d 717, rehearing and rehearing en banc denied, 525 F.2d 1203 (5th Cir.1975); Stout

v. Jefferson County Board of Education, 537 F.2d 800, 803 (5th Cir.1976). See also Tasby v. Wright, 520 F.Supp. 683, 709-12

(N.D.Tex.1981), 542 F.Supp. 134 (N.D.Tex. 1982) (520 F.Supp. at 711 and n. 63, rejecting the “sliding tolerances” formula ap-

proach testified to there by Dr. Foster). We regard Carr as fully authoritative here.

Moreover, the emphasis given in Lemon v. Bossier Parish School Board, 566 F.2d 985, 987 (5th Cir.1978), and in a host of other decisions of this Court, to the presumptions against the continued existence, in a former de jure system, of “all-bláck or virtually all-black schools,” strongly suggests that schools whose student bodies are clearly majority white are not to be automatically held racially identifiable as black, and hence presumptively unlawfully segregated, merely because their fraction of black students is disproportionately large compared to that of the district as a whole.

For these reasons, we hold that, under the circumstances of this case, the method by which the district court determined Terrell to be racially identifiable as black, and hence presumptively a vestige of de jure educational segregation, was erroneous in that the determination was made essentially as a matter of law merely on the basis of Terrell’s deviation from the ratios permitted by the Foster formula, and was not supported by any sufficient underlying factual determinations and conclusions.

We now turn to Golden Rule, Layne and Hyde Park. Just as it did with Terrell, the district court determined these schools to be racially identifiable, and hence presumptively vestiges of de jure segregation, essentially on the legal principle that they were such merely because their black student ratios varied from the district-wide ratio by an amount in excess of that permitted by the Foster formula. We hold, as we have in regard to Terrell, that this constitutes a legally incorrect method of determining whether the complained of current conditions at these schools result, directly or indirectly, from past or present purposefully segregative or otherwise racially discriminatory acts or policies, covert or avowed, of the educational authorities.

We recognize, of course, that these three schools are “virtually all” white, and are hence literally within the language of Swann and many other decisions concerning one-race or virtually one-race schools. However, neither Swann nor any other decision of the Supreme Court of which we are aware, nor any of this Court, has applied such language to disproportionately white schools in a district where the overall white enrollment is so large as to essentially constitute the system as a whole “virtually all” white, the disproportionately “white” schools are not superior to any others, and there are no schools which do not have a clear white majority. Such a situation, if there are no schools properly determined to be racially identifiable as black, is not one in which any black child appears to be denied either an integrated or an otherwise equal education. This is of particular significance in light of the historic aim of the Fourteenth Amendment in general and, more specifically, the major purpose behind the school desegregation decisions. As Justice Blackmun observed in Washington v. Seattle School District No. 1, - U.S. -, ---, 102 S.Ct. 3187, 3195-96, 73 L.Ed.2d 896, 908-09 (1982):

“... [0]ur cases suggest that desegregation of the public schools, like the Akron open housing ordinance, at bottom inures primarily to the benefit of the minority, and is designed for that purpose.... [Minority children can achieve their full measure of success only if they learn to function in — and are fully accepted by— the larger community. Attending an ethnically diverse school may help accomplish this goal....”

Recognition of the foregoing is implicit in the numerous desegregation opinions of this Court where the primary emphasis or thrust of the opinion has been in regard to the continued existence of all-, or virtually all-, black schools. In Allen v. Board of Public Instruction of Broward County, Florida, 432 F.2d 362, 367 (5th Cir.1970), cert. denied, 402 U.S. 952, 91 S.Ct. 1609, 29 L.Ed.2d 123 (1971), Judge Goldberg stated for this Court:

“In the conversion from dual school systems based on race to unitary school systems, the continued existence of all-black or virtually all-black schools is unacceptable where reasonable alternatives exist.... The tenor of our decisions is unmistakable: where all-black or virtually all-black schools remain under a zoning plan, but it is practicable to desegregate some or all of the black schools by using the tool of pairing, the tool must be used.”

This language has often been quoted with approval in subsequent decisions of this Court. See, e.g, Ross v. Eckels, 434 F.2d 1140, 1147 (5th Cir.1970); Boykins v. Fairfield Board of Education, 457 F.2d 1091, 1095 (5th Cir.1972); Flax v. Potts, 464 F.2d 865, 868 (5th Cir.), cert. denied, 409 U.S. 1007, 93 S.Ct. 433, 34 L.Ed.2d 295 (1972); Lemon v. Bossier Parish School Board, 566 F.2d 985, 987-88 (5th Cir.1978); United States v. DeSoto Parish School Board, 574 F.2d 804, 816, 819 (5th Cir.), cert. denied, 439 U.S. 982, 99 S.Ct. 571, 58 L.Ed.2d 653 (1978). In several of these decisions, as well as in others, this Court has not required the termination of all-white or virtually all-white schools in the circumstances of those particular cases. See e.g., Bradley v. Board of Public Instruction of Pinellas County, Florida, 431 F.2d 1377, 1383-85 (5th Cir.1970), cert. denied, 402 U.S. 943, 91 S.Ct. 1608, 29 L.Ed.2d 111 (1971) (court-ordered plan eliminates a large number of all-black or virtually all-black schools, leaving many all-white schools); Allen v. Board of Public Instruction of Broward County, Florida, supra (comparison of the opinion in this case, 432 F.2d at 367-70, with that of the district court, 312 F.Supp. 1127, 1138-49 (S.D.Fla.1970), reflects that this Court’s order eliminated each of the remaining all-black or virtually all-black elementary schools, but left a large number of all-white schools); Ross v. Eckels, supra (this Court’s order eliminates most all-black or virtually all-black schools, leaving many all-white and virtually all-white schools); Boykins v. Fairfield Board of Education, supra (in a district with four elementary schools and 57.4 percent overall black enrollment, this Court disapproves the continued existence of one 100 percent black elementary school, but does not require change at an elementary with over 88 percent white enrollment); Flax v. Potts, supra (this Court’s opinion reflects the existence of 16 schools over 88 percent black and 40 schools similarly “all white,” and requires action to eliminate the former); Calhoun v. Cook, 522 F.2d 717 (5th Cir.), rehearing en banc denied, 525 F.2d 1203 (1975) (permitting schools up to 70 percent white in a district 85 percent black); Lemon v. Bossier Parish School Board supra; United States v. Board of Education of Valdosta, Georgia, 576 F.2d 37 (5th Cir.1978) (indicating that in a 56 percent black district, schools over 90 percent black were racially identifiable, but not expressly so indicating as to a 92 percent white school); Horton v. Lawrence County Board of Education, 578 F.2d 147 (5th Cir.1978) (three all-white schools, predicated on residential patterns, in a 13 school district with 24 percent black enrollment and no all-black or virtually all-black schools); Alvarado v. El Paso Independent School District, 593 F.2d 577, 581 (5th Cir.1979) (in a district 38 percent “Anglo,” six-sevenths of the “Anglo” students attend “predominantly Anglo” schools); United States v. Texas Education Agency (Lubbock I.S.D.), 600 F.2d 518 (5th Cir.1979) (opinion’s entire focus is on predominantly minority schools, though the Appendix there discloses 10 over 90 percent white schools in a 58 percent white district).

Our holding in regard to the elementary schools is, of course, limited in several respects. Plainly we are dealing with adjective, not substantive, law. We naturally recognize that the mere presence of some integrated elementary schools in a district affords no legal justification for the continuing existence, at other elementary schools in the district, of the effects of purposefully segregative or otherwise racially discriminatory conduct of the educational authorities. And a black child’s constitutional rights are certainly violated by a purposefully segregative school assignment even if the school to which the assignment is made is not majority black. “White schools” are not exempted from the Constitution. Schools with disproportionately large black enrollments are not necessarily free from the effects of purposefully segregative or otherwise racially discriminatory acts or policies merely by reason of having white majority enrollment. We deal here, however, with the method to be employed in determining whether or not existing conditions at given schools are in fact a direct or indirect result of purposefully segregative or otherwise racially discriminatory acts or policies of the educational authorities. A finding of such an existing constitutional violation may, of course, be based on circumstantial evidence and the drawing of inferences. Moreover, we do not hold that a disproportionately large black enrollment is an irrelevant circumstance in any school which has a white majority, or that disproportionately low black enrollment is not to be considered unless the district has a certain minimum percentage of black students overall or a majority black school. Certainly, these factors can be relevant circumstances from which, together with other evidence, it may be proper to draw the factual inference, on a consideration of all the evidence, that current conditions at the school or schools in question directly or indirectly result from past or present purposefully segregative or otherwise racially discriminatory actions of the educational authorities. Cf. United States v. Texas Education Agency (Lubbock I.S.D.), supra at 528 (“Disproportionate impact is, however, not irrelevant to determining whether a governmental entity has acted with an intent to discriminate. It is among the factors in the calculus.”); Washington v. Davis, 426 U.S. 229, 241, 242, 96 S.Ct. 2040, 2048, 48 L.Ed.2d 597 (1976) (“[Necessarily, an invidious discriminatory purpose may often be inferred from the totality of the * relevant facts, including the fact, if it is true, that the law bears more heavily on one race than another.”); Rogers v. Lodge, - U.S. -, -, 102 S.Ct. 3272, 3276, 73 L.Ed.2d 1012, 1018 (1982). But, in circumstances such as those at bar, the district court may not substantially bypass the inference drawing, fact-finding, evidentiary process, and simply conclude as a matter of legal principle that schools are racially identifiable and hence presumptively vestiges of illegal segregation merely because their black percentages vary from the district-wide percentage by an amount in excess of that “permitted” by a general formula testified to by a witness in another case, when the district as a whole is overwhelmingly white and has no school which does not have a clear white majority. Finally, we write here only in the context of the facts of this case and others exhibiting essentially similar characteristics.

In cases of this nature we, of course, attach great weight to the district court’s special sensitivity and insight respecting local conditions. Here, however, such considerations seem to have played no material part in the ultimate decision respecting the elementary schools, which rather seems to have been made on the basis of a legal principle grounded in the general formula of an expert not familiar with Denison or testifying in this case, and despite the discrete mix of circumstances and history peculiar to this particular school district. We also recognize the deference properly due by us to a district court’s fact-findings, ultimate and otherwise, as well as to its exercise of discretion. Where, however, an erroneous view of the law appears to have materially influenced or limited the fact-finding process or the exercise of discretion, then we are normally obliged to remand. See eg., Pullman-Standard v. Swint, - U.S. -, ---, 102 S.Ct. 1781, 1795-97, 72 L.Ed.2d 66, 79-82 (1982); Dayton I, supra; Keyes, 413 U.S. at 198, 93 S.Ct. at 2692; United States v. Texas Education Agency (Lubbock I.S.D.), supra. Such being the case here, we remand the district court’s orders respecting the elementary schools.

Because the case appears to have been tried at least in part on an incorrect theory, and during trial the parties were erroneously directed not to present further evidence respecting the board’s intent, on remand the district court shall afford the parties an opportunity to introduce such proper additional relevant evidence concerning the elementary schools as they may desire, including but not necessarily limited to that bearing on the district’s intent, subject, of course, to the usual rules and the district court’s normal discretion in governing the conduct of proceedings before it. The district court shall thereafter make appropriate specific underlying, as well as ultimate, factual findings and conclusions, including those concerning whether, and if so to what extent, the DISD’s elementary schools are currently segregated as a direct or indirect result of past or present purposefully segregative or otherwise racially discriminatory actions or policies (whether avowed or covert) of the educational authorities. If appropriate, the findings shall also cover the proper remedy to correct any such thus caused existing conditions so that the schools will cease to thereby infringe the constitutional rights of students and their parents.

VT~ The Junior High Schools.

A. Background.

The DISD had for many years operated two junior high schools, Hughes, located in west Central Denison, and McDaniel, located in the center of east Denison. The attendance boundary line for the junior high schools ran essentially through the center of the school district in a north-south direction, splitting the community into east and west segments. As of September 1979 these schools enrolled 1,218 students, of whom 171, or 14 percent, were black. With the attendance line placed in the center of the district, the racial composition of the individual schools reflected the alignment of residential housing. Hughes junior high school, which had 498 students, was then attended by 11 black pupils, who represented 2.2 percent of the student body; McDaniel junior high school housed a student population of 720, of whom 160, or 22.2 percent, were black. The only change made in these boundaries under the plan enacted for the 1979-80 school year was the addition of two rather narrow “optional” attendance zones, which adjoined the east side of the basic attendance line and ran roughly parallel to it almost the entire length of the urban part of the district except for a relatively short gap in the center of Denison. Junior high students residing in these optional zones were permitted the choice of attending either Hughes or McDaniel. However, few elected to attend a school they would not have been assigned to absent the “option.”

McDaniel, located in the middle of downtown Denison, was built in 1913, served as a high school, and then, sometime well prior to 1962-63, was converted to a junior high school. Apparently it remained 100 percent white until the 1966-67 year, when it had a 9 percent minority enrollment. In the 1968-69 year minority enrollment was 11.9 percent, and from then through 1974-75 it varied from a low of 12.8 percent in 1970-71 to a high of 17.8 percent in 1973-74 and 17.7 percent in 1974-75. In 1975-76 minority enrollment at McDaniel was 20.1 percent and the following year it reached its high of 25.1 percent, reducing in 1977-78 and 1978-79 to 20.7 percent and 21.1 percent respectively.

Hughes was built in 1963 in central west Denison as a junior high. It remained all white until the 1966-67 year, when its minority enrollment was .2 percent. It had no minority enrollment the following year. Thereafter, its minority enrollment varied from a low of. .15 percent in 1970-71 to a high of 4.8 percent in 1976-77. Hughes apparently had no black students in 1978-79.

The evidence reflected significant differences between the physical facilities and educational programs at McDaniel and Hughes.

McDaniel is located on one square block in downtown Denison, is housed in a building over sixty years old which is in poor condition, and has limited athletic facilities. Hughes, by contrast, is situated on a 36-acre site, has tennis courts and ample playing fields, and is located in a building in good condition which is less than a third as old as McDaniel. The curriculum at Hughes offered “advanced placement” courses, nothing comparable to which was offered at McDaniel, its only specialization being some courses in vocational work.. Although transfers were allowed from McDaniel to Hughes for “qualified” students to enroll in the “advanced placement” courses, in practice there were few such transfers.

The court-appointed experts were strong in their condemnation of McDaniel. Dr. Tapscott testified that its condition was “very poor” and described McDaniel as “an educational facility that had outlived its location and its usefulness.” Mr. Williams, testifying concerning their recommendation that McDaniel be closed, stated:

“... — [W]e said, ‘This structure is no longer suitable for appropriate instruction in a Texas free public school district comparable to Denison Independent School District,’ and I just don’t know how more strongly it could be put.”

And on another occasion, Williams remarked concerning McDaniel: “There is just no way that [the] Texas public school system should condone the continuation of a school in that shape.”

Even the DISD made no serious effort to defend McDaniel. The superintendent, Dr. Alexander, admitted that it should be closed in the near future due to its antiquated condition. Indeed, the bond issue which was defeated in early 1980 contemplated McDaniel’s retirement, use of the current high school as a grades six through eight middle school serving the entire district, and construction of a new high school for grades nine through twelve.

Williams and Tapscott accordingly recommended closing McDaniel and unitizing grades seven, eight and nine, with grades seven and eight at Hughes and grade nine at another facility. The Texas Education Agency likewise recommended closing McDaniel and unitizing the junior high grades.

While racial considerations were not the principal basis for the experts’ junior high school recommendations, they recognized as of some significance the fact that a disproportionately large fraction (94 percent) of all black junior high students attended McDaniel. Dr. Tapscott testified:

“A. My major concern is I feel that they are maintaining an inadequate educational facility in McDaniel.
“Q. Whether they’re white children or black?
“A. Whether they’re white or black. I think the intent in the past has been to maintain it as a school predominantly — or a place for black students to attend. I think that’s beginning to change, but I feel that they need to go ahead and get on with it.”

Williams gave similar testimony. He also described McDaniel and Hughes as follows:

“If you just rode into Denison and asked somebody where’s the black junior high and where’s the white junior high, people in Denison could quickly tell you that.”

B. District Court Order.

In the district court’s order of December 17, 1980 it found that Hughes was “in excellent physical shape” and “offers several advanced placement courses,” while McDaniel “is in poor, even deplorable condition” and offers “no advanced placement courses.” The court also stated:

“Unanimity and agreement characterized the testimony by the witnesses ... that McDaniel’s facilities and curriculum are exceedingly inferior to those of Hughes. Indeed, it was the opinion not only of the two court-appointed experts, but also of the DISD superintendent, Dr. Warren Alexander, that McDaniel school should be closed in the near future, because of its decrepit condition.”

The court found that for the last ten years McDaniel “has housed a highly disproportionate number of the school district’s Black junior high students,” and observed that “[e]ven if the educational resources at McDaniel were not decidedly inferior, the clustering of ninety-four percent of Black junior high students in one school would raise serious questions of a fourteenth amendment violation.” However, the court did not purport to resolve such “serious questions” and did not base its decision on such ground. Rather, it held that:

“... [Tjhere is no questioning that equal protection was transgressed, when a highly disproportionate number of Black students, were relegated to the worst school facilities in the city. As the Supreme Court noted in the landmark case of Swann v. Charlotte-Mecklenburg Board of Education, 402 U.S. 1, 18 [91 S.Ct. 1267, 1277, 28 L.Ed.2d 554] (1971):
‘ “Independent of student assignment, where it is possible to identify a ‘white school’ or a ‘Negro school’ simply by reference to the racial composition of teachers and staff, the quality of school buildings and equipment or the organization of sports activities, a prima facie case of violation of substantive constitutional rights under the Equal Protection clause is shown.’
“(emphasis added). That prima facie case has not been successfully rebutted here. The present racial composition of the Denison junior high schools is, therefore, violative of the Equal Protection clause of the fourteenth amendment.”

The court then noted that in August the school board had resolved, and formally advised the court, that if it found a constitutional violation respecting the junior high schools, the ninth grade be unitized at McDaniel and the seventh and eighth at Hughes. The court accordingly ordered such relief, and in March 1981, again at the formal request of the DISD following resolution of its board, amended the order to provide for unitized eighth and ninth grades at McDaniel and sixth and seventh grades at Hughes, effective for the 1981-82 school year.

C. Review of the District Court Order.

The district court’s ruling respecting the junior high schools is not based on the “Foster” (or any similar) formula, which is not even mentioned in the December order, or on a lack of racial balance in the schools as such, but rather on the long-existing significantly inferior condition of McDaniel, the school currently and for many years past attended by almost all the district’s black junior high students.

Undoubtedly, the burden of ■ inadequate junior high school facilities in the DISD fell with significant" disproportionate severity on the district’s black students. This was not a recent condition, but one which had obtained for at least the last five years, if not longer. There was no suggestion of efforts to correct it. As is frequently, though not inevitably, the case in a two-zone plan such as this, it appears that slight shifts in the angle of the line dividing the Hughes and McDaniel zones would have materially reduced this racially unequal sharing of substandard conditions. And, the above-recited testimony of the experts likewise furnishes the basis for an inference that race was a material factor in the district’s failure to upgrade McDaniel. Of course, with respect to governmental conduct not avowedly based on race, decisions construing the Fourteenth Amendment:

“... have not embraced the proposition that a law or other official act, without regard to whether it reflects a racially discriminatory purpose, is unconstitutional solely because it has a racially disproportionate impact.” Washington v. Davis, 426 U.S. 229 at 239, 96 S.Ct. at 2047 (emphasis supplied).

Nevertheless, an invidiously discriminatory purpose may frequently be inferred where racial minorities suffer a disproportionately adverse impact. Washington v. Davis, supra 426 U.S. at 265, 96 S.Ct. at 2059; United States v. Texas Education Agency (Lubbock I.S.D.), supra at 528. Moreover, it is established that this rule applies not only to instances where the official conduct initially bringing about unequal conditions is called into question, but also to the failure of responsible officials to thereafter take action to terminate such inequality in respect to matters within their continuing charge. Rogers v. Lodge, supra at - U.S. -, -, -, 102 S.Ct. 3272, 3276, 3280, 73 L.Ed.2d 1012, 1018, 1024. Thus, despite the fact that the disparity between McDaniel and Hughes appears to have come about totally indepen- ' dent of any racially discriminatory purpose on the part of the governmental authorities, the DISD’s long-continued retention of such disparity when it so unequally impacted black students could, under the facts of this case, be inferred to have resulted in material part from racial considerations on the part of the district officials and to hence violate the Fourteenth Amendment. Rogers v. Lodge, supra. In such a situation, remedial orders requiring equalization of facilities may be proper. Swann, supra; Alvarado v. El Paso Independent School District, supra (order to fully air condition “predominantly Mexican-American schools” until their number equaled the number of fully air conditioned “predominantly Anglo-American schools”).

Accordingly, we reject the DISD’s contention that the evidence establishes no violation respecting the junior high schools. Nevertheless, we are concerned because, as noted in our discussion of the elementary schools, the district court during the trial in effect instructed the attorneys for both parties that intent was irrelevant and further evidence concerning it should not be presented. While the district court’s remarks in this regard may have been intended to relate primarily to the elementary schools, that is by no means clear. We note that both Hughes and McDaniel were established, in the days of complete and avowed segregation, as all-“white” schools. It seems plain that the disparity between them existed in those times, as well as in more recent years. Thus, to trace the offensive disparity in their respective conditions to purposefully discriminatory or segregative school authority conduct during the period of open and avowed “de jure” segregation might be problematical. However, we do not resolve this question, and it may be unnecessary to do so. Other than the stay until August 1981, which the district court granted shortly after its December 1980 order, the DISD has never sought a stay of the junior high school order, and, unlike the elementary school order, that respecting the junior high schools has gone into effect. There is no suggestion that this order is in any way disruptive, and, indeed, its format is in accordance with the district’s desires. The junior high school order obviously remedies an inequity.

The main focus of the DISD’s appeal has been in regard to the elementary schools, and, to a lesser extent, the principalship. We have found it necessary to reverse and remand this case particularly with respect to those matters. Under the circumstances, we believe that the rights of all parties respecting the junior high schools will be adequately protected if the district court’s order respecting them is left in effect, subject to the following modification. If the DISD desires to do so, and gives the plaintiffs and the district court such reasonable notice as that court may direct, the DISD shall be allowed to seek a redetermination by the district court of its ruling regarding the junior high schools. In the event the DISD does so, then: the parties shall be afforded an opportunity to present additional relevant evidence concerning the junior high schools, including that respecting intent, subject to the usual rules and the district court’s normal discretion in governing the conduct of its proceedings; and, the district court, whether or not additional evidence is presented, shall make further factual findings respecting the junior high schools, including findings as to whether the continued disparity between Hughes and McDaniel to any material extent results from racial motivations on the part of the district, and whether the unequal conditions borne by the black junior high students otherwise result from racially purposeful action of the educational authorities. The district court may then reconfirm its junior high school order or modify or rescind it, subject in any event to further appeal by either party to this Court.

VI. The Principals.

A. Background.

The plaintiffs’ only allegations concerning faculty and staff were that “role models ... are being denied minority students” in that:

“... [1] [N]o black teachers have been advanced to the positions of principal or vice principal in any of the elementary or junior high schools of D.I.S.D. [2] No minority instructor has been granted the opportunity to qualify while emergency certificates for administration have been requested and granted for white teachers.... [3] [T]he D.I.S.D. has not had an effective minority recruitment program to hire black teachers, and some of the black teachers are under-utilized.”

The amended complaint does not charge any intentional discrimination respecting any of the foregoing. There are no allegations that whites, either generally or in specific instances, were promoted or hired in preference to equally or better qualified black applicants. Nor are any terminations or demotions of blacks alleged, generally or specifically. No assignment of teachers to schools on the basis of race is charged.

While the evidence is not entirely clear, it appears that in 1980-81 approximately 8.5 percent of the teaching staff was black, none of the ten principals was black and two of the four assistant principals were black. This apparently was also the approximate situation in 1979-80. Respecting employment of minorities other than blacks, the only evidence was that one of the principals was Mexican-American. One of the black assistant principals was at McDaniel. There is no indication where the other was assigned. There was no evidence as to the relative duties, responsibilities and compensation of assistant principals and principals, nor as to the duties, responsibilities and compensation of other professional positions within the DISD.

The historical evidence showed a decline in the number of black professionals employed by the DISD during the period from about 1964 to 1974. The district then adopted a faculty and staff affirmative action plan which was approved by the United States Department of Health, Education and Welfare, Office of Civil Rights, in 1974. The goals of this plan, in terms of the number of black professionals in the system, were approximately 90 percent met, and the level of black teachers was increased to something in excess of what it had been prior to the commencement of the decline which had started about 1964. Student enrollment declined some 20 percent from 1969-70 to 1979-80 (and declined again by about 7 percent from 1979-80 to 1981-82). However, the number of teachers had not declined in the four or five years preceding trial. During that period of time annual teacher turnover was about 5 percent. There was no historical evidence as to whether, and if so when, any black had served as principal, nor as to how many principal vacancies, if any, had occurred in any prior period of time or when the last such vacancy had occurred. There is no evidence as to any black having applied, or been available for, any principalship vacancy, or any assistant principalship vacancy which was filled by a non-black. There was no evidence as to how the DISD filled principalships or assistant principalships, whether by promoting from within or by outside hiring or both. The record reflects nothing concerning the minimum qualifications for these positions. There was no support for the allegations of failure to grant minority instructors the “opportunity to qualify for emergency certificates” or of black teachers being “under-utilized” compared to whites. There was no evidence of any position being filled by a white in preference to a qualified black applicant, or of any black personnel being discharged or demoted from any position.

B. District Court Order.

In its June 10, 1981 order, the district court ruled that plaintiffs had neither alleged nor proved a violation of Singleton, either by discriminatory dismissals or demotions during desegregation or by the failure of faculty at individual schools to approximate the district-wide racial composition.

It also refused to find any other violation respecting teachers or assistant principals, stating:

“Approximately twelve percent of the overall student body in the Denison schools in Black, while the teaching staff is 9.3 percent Black. Although this underrepresentation occasions some concern, it cannot be said to amount to a prima facie constitutional violation .... Moreover, the superintendent of DISD testified that the district devised an affirmative action hiring plan, in an effort to recruit Black professionals, which was submitted to the Office of Civil Rights for the United States Department of Health, Education and Welfare in 1974. While the goals of the plan have not been fully implemented, enough intermittent progress has been made in hiring Black teachers since 1974 to shield DISD from the necessity of injunctive relief as to the teaching staff at this time. U
“Presently, of the four assistant principals employed in the Denison school system, two are Black. Thus, no statistical case supports the contention that the DISD is presently guilty of a constitutional violation with respect to the assistant principals.”

However, the district court did find a violation as to the principals. It considered that the plaintiffs had alleged “an under-representation in the number of Blacks occupying positions as principals.” The standard by which the court proceeded to gauge this question of underrepresentation was a comparison of the “racial population ratios for” principals “with those of the student body.” The court further noted:

“... [0]f the ten [principals] who serve the Denison schools, all are white. This zero percentage, combined with the history of discrimination involved here, makes out a prima facie case in support of the plaintiffs’ contentions.... Yet, DISD has come forward with no evidence to rebut the prima facie showing, and no demonstration has been made that the all-white principalships are the result of something other than past or present racial discrimination.
ii
“Nor does the fact that integration has been achieved in the ranks of the assistant principals and, to a much lesser degree, the teachers, carry DISD’s burden of proving clearly and convincingly that racial discrimination did not play a part in the present all-white composition of the district’s principalships.”

No express finding of purposeful racial discrimination was made, the court merely stating that “it can only be concluded that the total lack of Black principals in the DISD signals a failure of constitutional compliance.” The court then observed:

“Only one Black educator need be hired or promoted to the position of principal, in order to bring the racial composition of principalships sufficiently in line with that of the Denison student body.”

It ordered that the DISD “fill the first available principalship with a Black person,” provided that if “a reasonably short period of time reveals that no vacancy has occurred, more drastic injunctive relief will be necessary.”

C. Review of District Court Order.

Plaintiffs did not bring their suit, individually or as a class, in the capacity of former, present or prospective DISD professional employees or applicants for employment. Nevertheless, it is well settled that plaintiffs, as black students and parents, had standing to urge their complaint that their Fourteenth Amendment rights were violated because minority students were denied role models by the DISD’s failure to promote black teachers to elementary or junior high school principalships or vice principalships. See e.g., Castaneda v. Pickard, 648 F.2d 989, 999-1001 (5th Cir.1981). However, it is likewise clear that no such failure to promote is actionable under the Fourteenth Amendment unless it was motivated by purposefully racial considerations, and mere disproportionality in school district employment matters does not amount to an infringement of constitutional rights, absent a violation of Singleton v. Jackson Municipal Separate School District, 419 F.2d 1211 (5th Cir.1969), cert. denied, 396 U.S. 1032, 90 S.Ct. 611, 24 L.Ed.2d 698 (1970). See Castaneda, 648 F.2d at 1001.

The general rule, of course, is that disparate impact is relevant to, and may permit an inference of, purposeful discrimination, Washington v. Davis, supra; United States v. Texas Educational Agency (Lubbock I.S.D.), supra. And, where purposeful discrimination by a school district is shown in a meaningful portion of its actions, the district has the burden of proving that other actions on its part having significant disparate or segregative impact are not also the product of a discriminatory intent. Keyes, 413 U.S. at 207, 208, 93 S.Ct. at 2696, 2697. And see United States v. Gregory-Portland ISD, 654 F.2d 989, 995, 996 (5th Cir.1981). In the specific area of public school employment practices, our decisions have stated the rule that:

“... [When] a pattern or practice of discrimination in the employment of faculty and staff ... claim is asserted against a school district having a relatively recent history of discrimination, the burden placed on the defendant school board to rebut a plaintiff’s prima facie case is heavier than ... in the usual employment discrimination case ... the defendant must rebut the plaintiff’s prima facie case by clear and convincing evidence that the challenged employment decisions were motivated by legitimate nondiscriminatory reasons.” Castaneda, 648 F.2d at 994 (emphasis added).

We think it is plain from the foregoing in Castaneda, as well as being implicit in our other decisions in this regard, that in the public school employment practices area plaintiffs must make a prima facie case before the burden will shift to the defendant school board, and that the necessary prima facie ease does not consist of merely showing a “relatively recent history of discrimination.”

In concluding that the plaintiffs had established a prima facie equal protection violation here, the district court used a statistical measure that compared the racial population of the principals with that of the student body. This was not the relevant comparison. In Castaneda, we held in reference to a claim of discrimination in hiring and promotion to administrative levels, that where a promote-from-within policy was followed, the relevant comparison was between the respective minority percentages at the administrative level and at the faculty level, assuming there was not discriminatory hiring at the faculty level. Id. at 1003. A similar approach was taken in Hereford v. Huntsville Board of Education, 574 F.2d 268 (5th Cir.1978). Castaneda specifically rejects comparison of faculty to student body, and points out that “the relevant labor market must be separately and distinctly defined.” Castaneda was expressly reaffirmed in this regard in Fort Bend Independent School District v. City of Stafford, 651 F.2d 1133, 1138 (5th Cir.1981).

Here, plaintiffs made absolutely no effort to define the relevant labor market. There was no showing whether the DISD filled principalship vacancies by outside hire, promotion from within or some combination of the two. There was no showing of the minimum qualifications for the position of principal, and we certainly cannot assume that all teachers possessed the minimum qualifications to be considered for principal-ships. As there were only ten principal-ships, and blacks were but one-twelfth of the teaching staff, this is of potential significance.

We also note the extremely small sample involved. The district court based its decision on simply a one-year “snapshot” of a single, and small, classification of professional employees. The measure of violation was the irreducible minimum of one person. There was no evidence concerning past vacancies or whether there were any black applicants, or the like. For all the record shows, in the last eight years there may have been a total of nine principalship vacancies, of which one was filled by a black. We do not suggest that any one of these deficiencies is fatal in and of itself. In combination, however, they deprive the asserted “prima facie” case of significant force. Had the district court combined the principals and assistant principals a larger sample would have been obtained, and even if double weight were assigned the principalships the black proportion of the combined total would have been one-twelfth. Relevant in this regard is the statement in United States v. South Park Independent School District, 566 F.2d 1221, 1226 (5th Cir.), cert. denied, 439 U.S. 1007, 99 S.Ct. 622, 58 L.Ed.2d 684 (1978):

“... [W]e recognize that the assignment of principals alone is not necessarily the important factor but rather one must look at the racial composition of a school district’s entire staff. We are not ready to hold that each particular level of employment in a school system must have a particular racial composition.”

See also Hereford, 574 F.2d at 273 (“[discrimination in promotions is not proved merely by the absence of blacks in particular positions in particular schools”). Moreover, our decisions have cautioned against creating a prima facie employment discrimination case on too narrow a statistical sample. See e.g., Turner v. Texas Instruments, Inc., 555 F.2d 1251, 1254, 1257 (5th Cir.1977); Robinson v. City of Dallas, 514 F.2d 1271, 1273 (5th Cir.1975).

Under these circumstances, we cannot approve the district court’s method of establishing a prima facie case here.

Finally, we observe again that the ultimate issue of fact is whether the complained of failure to promote was racially motivated. Castaneda, 648 F.2d at 1001. Here there is neither express allegation nor express finding that the DISD was racially motivated in its decisions concerning promotions to principal positions (or hirings at that level). This becomes significant when considered in context with the district court’s above-referenced ruling during trial that intent was irrelevant and further evidence on that subject should not be presented.

We accordingly conclude that the district court’s order respecting the principals must be remanded. On remand, the district court shall allow the parties to introduce such proper additional relevant evidence concerning the filling of principalship vacancies as they may desire, including but not necessarily limited to that bearing on intent, subject, of course, to the usual rules and the district court’s normal discretion in governing the conduct of proceedings before it. The district court should factually determine, on consideration of all the evidence, whether any of the DISD’s decisions in filling elementary and junior high school principalship vacancies were racially motivated and if so, whether the present lack of one or more such black principals is attributable thereto. In determining the existence of a prima facie case, the comparison shall not be to the student population, but rather to the pool of those eligible for promotion (or hiring) to the position of principal. It is suggested that a broader basis of comparison be used than merely the existing principalships in one or two years without evidence of vacancies or applications.

VII. Conclusion.

In sum, we reverse and remand the district court’s orders respecting the elementary schools and the filling of the first principalship vacancy. We affirm respecting the junior high schools, subject to the modifications and instructions noted in the last paragraph of part V(C) hereof above.

Affirmed in part with modification and instructions. Remanded in part.

ON SUGGESTION FOR REHEARING EN BANC

PER CURIAM:

Treating the Suggestion for Rehearing En Banc as a Petition for Panel Rehearing, it is ordered that the Petition for Panel Rehearing is DENIED. No member of the panel nor Judge in regular active service of this Court having requested that the Court be polled- on rehearing en banc (Rule 35, Federal Rules of Appellate Procedure; Local Fifth Circuit Rule 16), the Suggestion for Rehearing En Banc is DENIED.

Contrary to the suggestion of the appellees, we did not and do not hold that in a former de jure school system, proof and finding of discriminatory intent on the part of the educational authorities is necessarily required to establish an existing constitutional violation. When a trial court has duly found, after proper consideration of all the relevant factors, that some current condition of racial segregation or inequality exists within a school system, it will then be determined whether the existence of such condition amounts to a constitutional violation. Where the referenced current condition results from the prior de jure segregation in the school system, a present constitutional violation is established without proof or finding that the former de jure segregation was intentional, for it is so as a matter of law, and the fact that the school authorities may more recently have been in good faith and had no segregative or discriminatory intent is not dispositive of whether an existing constitutional violation is present. On the other hand, where the referenced current condition does not result from the prior de jure segregation, proof and finding of subsequent discriminatory or segregative intent on the part of the educational authorities is necessary to establish a present constitutional violation. 
      
      . The joint pretrial order filed August 4, 1980, describes the revision of attendance zones which went into effect with the 1979-80 school year as being “the subject of the current controversy.” There is some uncertainty as to the precise figures for that year. One paragraph of the pretrial order states that the correct figures for the 1979-80 school year are set out on an attachment which reflects that in September 1979 there were 5,247 students, of whom 636, or 12.1 percent, were black. The next paragraph reflects that in the 1979-80 year there was a total of 5,156 students, of whom 653 (which would be 12.6 percent) were black.
     
      
      . At the time of Brown I, the Constitution of the State of Texas required the segregation of black children in all public schools.
     
      
      . This Court, in its opinion on the prior appeal, complimented the DISD on its actions: “A word is certainly due for Denison’s having initiated its plan voluntarily.... [I]t is clear that [the district judge] was convinced — as were we on argument — of the School Board’s good faith desire to do what the law requires.” 348 F.2d at 1014.
     
      
      . The plaintiffs are a group of black students and parents who are representatives of a class of black students and parents in the DISD. The plaintiff-intervenors include a group of white students and their parents and a group of adult residents of Denison who are the elected officers of a voluntary organization known as the Save Our Schools Association. Both the plaintiffs and the plaintiff-intervenors made the same allegations and seek the same relief. They are represented by the same counsel. We will hereafter refer to them collectively as plaintiffs.
     
      
      . Attached to the amended complaint was a petition purportedly bearing the signatures of approximately 1,000 residents volunteering to be plaintiffs in the lawsuit and protesting the closing of the three elementary schools, the contemplated future closing of three more elementary schools and the school board’s decision to float a bond issue.
      The referenced bond issue was defeated in early 1980. It had been intended to finance a new high school, with the existing high school being used to replace McDaniel junior high, the older of the two junior high schools, which would be “retired”. This scheme was designed to place grades six through eight in one middle school, grades nine through twelve in one high school, and grades five and below in several elementaries.
     
      
      . For example, plaintiffs’ counsel stated at the September 1, 1981, hearing that the DISD had “really by their closing of three neighborhood schools prompt[ed] this lawsuit” and that:
      “[Tjhis lawsuit was brought by a mixed group of white and blacks, citizens and parents, who were offended that their neighborhood schools, Raynal in the northeast sector, Peabody in the south central sector, and Central in the north central sector, had been closed....”
      In 1978-79, the last year these elementary schools were operated, their percentages of black students were: Raynal, 9.7 percent; Peabody, .9 percent; Central, 56.9 (or 55.3) percent.
     
      
      . On July 10, 1981, the original due date for submission of its elementary assignment plan, the DISD filed a motion for stay of the June 10 order (other than its provisions respecting the junior high schools) during its pending appeal. The district court denied the stay on July 17, 1981. On August 5, 1981, this Court also refused to stay the district court’s order.
     
      
      . The DISD filed a notice of appeal from the August 14 order and requested a stay or extension of time. These requests were denied by the district court in an order of August 24, 1981, which also required the DISD to submit by August 28 an elementary assignment plan in compliance with the August 14 order, failing which the court would appoint a special master to prepare a plan. Thereafter, the DISD, acting through new counsel, filed in this Court a motion for stay of the August 14 order, raising different grounds than those asserted in the motion previously filed in the district court. On August 28, this Court remanded the DISD’s motion to the district court for hearing on an expedited basis, which the district court then held on September 1 and 14, 1981.
     
      
      . The DISD again gave notice of appeal and thereafter all its appeals in this matter were consolidated. Following oral argument before this panel, we issued a partial stay keeping in force the elementary school attendance zones in effect in December 1981 pending our decision in this case.
     
      
      . Complaint is additionally made of the scope of the elementary order. Procedurally, the DISD asserts that the district court lacked jurisdiction to enter the orders of August 14 and September 18 because the appeal filed by the DISD on July 6, 1981, ousted the district court of its jurisdiction to make substantive changes in its June 10 order. We do not reach this contention.
     
      
      As a middle school (grades 6-8) in 1978-79; 1979-80 and 1980-81 figures include the “Annex.”
     
      
      Or 55.3%, it being unclear which.
     
      
      . Peabody and Raynal were also all-white schools under the dual racial attendance zone system, Raynal first having minority students in 1968-69 (.3 percent), Peabody in 1966-67 (.27 percent). Thereafter and prior to 1978-79, the minority percentages at Peabody varied from .54 percent to 5.5 percent, and at Raynal from 1.38 percent to 13.3 percent. During this time Raynal apparently had no sixth grade. Raynal’s (and Lamar’s, at least since 1974-75) sixth graders attended Terrell middle school.
      Some confusion in the racial percentages arises from the presence of the “Annex,” a facility located about four blocks west of Terrell. Apparently this had been the all-black Wims school, serving the first and second grades, until the 1966-67 year when it was “paired” with Central (all white in 1965-66), so both the second grades attended Central and both the first grades attended Wims. This arrangement seems to have lasted through the 1969-70 or 1970-71 school years. Thereafter, this facility appears to have served only kindergarten. For the years 1966-67 through 1971-72 minority enrollment at the Annex varied from a low of 43.7 percent in 1966-67 to a high of 63.8 percent in 1968-69. In 1971-72, it was 51.6 percent. In the years 1973-74 through 1976-77, it varied from 12.9 percent to 16.5 percent. In 1977-78, when full-day kindergarten was first established, it went to 70 percent minority and was 67.9 percent in 1978-79. Under the 1979-80 plan this “Annex” became a kindergarten administratively attached to Terrell. The Annex’s minority enroHment was 37.3 percent in 1979-80. Separate figures for it are not available for 1980-81.
      The Terrell figures given in the text for 1979-80 and 1980-81 include the Annex. Without the Annex, the Terrell black enrollment in 1979-80 was 31.5 percent. For the 1981-82 school year, the Annex was closed, and its kindergarten students attended at the Terrell campus. For the 1981-82 year all the sixth graders attended Hughes middle school serving grades six and seven, pursuant to the court’s order.
      Other difficulties exist with the racial percentages. Figures for years before 1978-79 are denominated “minority” and apparently lump together Mexican-Americans, Blacks and Indians. Strangely, however, the Black percentage for Central in 1978-79 is shown as 56.9 percent, which is higher than its “minority” percentage (55.3 percent) that year. There are no figures at all for the 1972-73 school year.
     
      
      . This may somewhat overstate the increase in elementary school busing, as it appears that prior to 1979-80 sixth graders bused to Terrell when it was a middle school were not included in the elementary totals, but were so included in 1979-80 when Terrell became an elementary.
     
      
      . Each of these witnesses had extensive experience and qualifications, both as educators generally and in school desegregation matters in particular, primarily in Texas. Additionally, Williams is an attorney.
     
      
      . In connection with their recommendations for changing the junior high school scheme, the experts recommended closing McDaniel junior high, converting Golden Rule elementary to a ninth grade facility, and that Peabody, one of the elementaries closed in the 1979-80 zone revision, “be reopened as an elementary school to take the place of Golden Rule.” The experts’ recommendation in respect to Golden Rule and Peabody was not in any way intended to affect racial dispersion in the elementary schools, and there is no evidence that it would have done so. In the 1978-79 year, Peabody’s black enrollment was .9 percent; Golden Rule’s was 2.4 percent that year and 1.7 percent in 1979-80.
     
      
      . And in answer to a question on deposition inquiring whether “given the geographical location of the minority populations ... the line drawing of the attendance district ... [is] a good faith effort to achieve integration,” Tapscott testified that, “I feel that they’ve made a good-faith effort to draw ... the elementary school attendance lines, as carefully as they can to provide a good mix, given the limitations of the population of Denison. I consider it a good-faith effort.”
      Similarly, Gilbert Conoley, for the past twelve years the Texas Education Agency Director of the Division for Technical Assistance for School Desegregation, answered in the affirmative when asked whether the elementary system “represents a good faith effort on the part of the Denison Independent School District to cure the vestiges of segregation and attempt to establish a unitary school system in the elementary schools.” The Texas Education Agency reported to the court that under the 1979-80 plan the DISD would be “a unitary system.”
      The president of the Denison NAACP chapter testified he would prefer that each elementary school have a 12 percent black student body. However, in 1978 he had proposed to the board an elementary plan which he described as “close to the plan” adopted for the 1979-80 year.
     
      
      . The school superintendent, Dr. Alexander, corroborated this. He also testified that the board had expected a 10.8 percent black student body at Mayes for 1979-80, but that it turned out to be 8.8 percent. He said, “we had some movement.” Apparently, there was some resistance among blacks in the part of Central added to Mayes to attending the latter.
     
      
      . Similarly, Conoley testified he did not “find any evidence of attempts by the School Board to perpetuate discrimination” by rejecting these other suggested plans, some of which, by extensive “pairing” and the like, resulted in more even racial “balance” in the elementary schools.
     
      
      . There was no “tip ratio” evidence particularly applicable to the DISD.
      While some concern was expressed as to higher black percentages in the lower grades at Terrell, particularly 44.9 percent in the second grade in 1979-80, it was the percentage of the entire facility, rather than particular grades, that Dr. Tapscott deemed significant. Superintendent Alexander explained that higher black percentages in the lower elementary grades tended to occur because in those grades more children (presumably disproportionately white) attended private schools (apparently available at those, but not higher, grade levels). Moreover, some of the variation between grades appears random. Thus, for 1979-80, the black percentages by grade at Terrell were: kindergarten (Annex), 37.3 percent; first, 34.2 percent; second, 44.9 percent, third, 26.2 percent; fourth, 27.5 percent; fifth, 30.3 percent, sixth, 23.6 percent.
      At the beginning of the 1981-82 year, the black percentage at Terrell had risen to 41.2 percent (percentages at the other elementaries were: Golden Rule, 4.3; Houston, 8.5; Hyde Park, 1.7; Lamar, 8.4; Layne, .7; Mayes, 5.8). However, in that year, as required by the court order, all sixth graders were attending the Hughes middle school (grades six and seven). Though “per grade” figures are not shown for 1980-81 or 1981-82, the evidence indicates the sixth graders who would have attended Terrell numbered about 150 in September 1981 (though as a fourth grade in 1979-80 this class only numbered 91 in December 1979). While the black percentage at Terrell was increased at the beginning of the 1981-82 year over what it was at the end of the 1980-81 year, the same period also witnessed small increases in the respective black percentages at Lamar, Golden Rule, Hyde Park and Layne. The plan submitted by the board in response to the court’s June 10, 1981 order, which the court rejected in August, would have reduced the Terrell black percentage to 29.3 percent.
     
      
      . And the unidentified “CBAE program.” Dr. Jacobs, successor superintendent to Dr. Alexander, testified in September 1981 that the board committee monitoring Terrell had not recommended further changes for the 1981-82 year because of the pending litigation.
     
      
      . This is apparently what Tapscott was referring to in his earlier testimony that previous to 1979 “there had been segregative intent,” though “an effort had been made with the last revision to correct some of that” and “a change in direction seemed to have been made.” This testimony apparently was further qualified when the witness said, on subsequently being again examined as to this subject, “I responded only to the [last] two years. I have not tried to respond to what went on back a long time ago as far as the history is concerned.”
      
        Tapscott’s testimony quoted in the text actually responds to a question assuming Central at or over 50 percent “minority” and “nine of the ten schools” with substantially disproportionately low “minority” enrollment. In fact, such a situation never existed in the DISD. Central first became over 26 percent minority in 1967-68. Counting Central and the Annex as two schools, there were ten elementaries. In neither of the years 1967-68 and 1968-69 were there more than six elementaries with minority percentages below 8.4 percent; in no year after 1968-69 have there been more than six elementaries with minority percentages below 11.6 percent; in none of the years 1973-74 through 1978-79 were there more than five elementaries with minority percentages below 9.1 percent, except in 1975-76 when a sixth elementary had 7.19 percent; in 1976-77 only three elementaries had minority percentages less than 5.5 percent.
     
      
      . The simple “yes” answer to the rather involved question is unexplained. The record shows the average of the minority percentages at Central (formerly an all-white school) for the year Mayes opened (1968-69) and the next three years, 51.58 percent, is less than Central’s minority percentage for the year just prior to Mayes’ opening, which was 53.9 percent. The year before Mayes opened the minority percentage at the Annex was 54.7 percent; though in the next three years it was higher on the average, by the third year following the year Mayes opened it was 51.6 percent. The question did not disclose these figures, and the witness did not then have them before him. The trial court did not find Mayes to be a racially identifiable school in 1979-80 or 1980-81, although it indicated that it may have become so, by .2 percent, in September 1981.
     
      
      . Dr. Alexander testified Perrin closed in “about” 1971.
     
      
      . The district experienced a further attendance decline of some 7 percent from 1979-80 to 1981-82.
     
      
      . And Williams “had no difficulty with” the suggestion by the board’s counsel that the closing of some black schools in the early or mid-1960’s (apparently Dalton and Langston, and perhaps Terrell) reflected an intention to “pursue a more unified system.”
     
      
      . The June 10 opinion mentions that Terrell’s racial composition exceeds the crucial deviation factor of five percentage points, and notes that if the percentage of blacks were increased at other schools, Terrell’s would be decreased. However, elsewhere the opinion clearly shows that the concern respecting Terrell was not with the school itself, but rather with its effect on Golden Rule, Layne and Hyde Park. Thus, the opinion states:
      “Clearly, the racial composition at Terrell Elementary is vastly disproportionate to the district-wide average. As previously noted, however, statistics are only a starting point in determining racial identifiability. When Terrell is viewed by itself, the disproportion occasions no great concern, for the school has significant numbers of Black and white students together in attendance. While Terrell was an all-Black institution in the days of strict segregation, it was reopened in 1979 as a predominantly white school and remains predominantly white. The evidence, on the whole, did not indicate that Terrell students are suffering from inferior physical facilities or a limited educational curriculum. In isolation, then, Terrell appears to fulfill the spirit of the Brown v. Board of Education mandate.
      “Despite such an appearance, student assignment in Terrell is not totally innocuous in terms of its effect on the rest of the district. The housing of two-thirds of Denison’s Black students within Terrell allows the maintenance of Golden Rule, Layne and Hyde Park as practically all-white schools. Thus, while not necessarily racially identifiable per se, the racial composition of the Terrell student body obviously contributes to the racially isolated character of other schools in the district.”
      Other aspects of the June 10 opinion confirm the indication that Terrell was not found racially identifiable. The other three schools were expressly so found, but there is no such express finding as to Terrell. The 1979-80 plan was faulted because it “failed to erase the racial identifiability of” the three “white” schools, not because of any similar failure as to Terrell. Further, it is recited that “three out of the district’s seven elementary schools are racially identifiable, and the composition of a fourth contributes to the level of segregation in the three.” The June 10 opinion recites that the district failed to prove the racial identifiability of Layne, Golden Rule and Hyde Park was unrelated to past discriminatory actions, but contains no such determination respecting Terrell. It states, “some relief must be ordered to eliminate the segregated character of” the three schools, but makes no such statement as to Terrell. Finally, the June 10 opinion also states: “It is not significant that only the three elementary schools within the DISD are racially identifiable.” (Emphasis added.)
     
      
      . For the reasons stated in note 25 supra, we cannot accept this characterization of the June 10, opinion as respects Terrell. Nor can we accept the similar characterization of the June 10 opinion which is implicit in the district court’s September 18 opinion.
     
      
      . The plan submitted by the board, which the court rejected, contemplated the following black percentages for the 1981-82 year in the other elementaries: Golden Rule, 11.9 percent; Houston, 7.1 percent; Hyde Park, 10.6 percent; Lamar, 6.8 percent; Layne, 7.2 percent; and, Mayes, 6.9 percent. These were found acceptable. The major features of the board’s plan involved moving the east boundary of the Layne attendance zone east across state highway 75 to encompass an area formerly in the Terrell zone, and assigning one group of Terrell zone students to Hyde Park and another to Golden Rule. Those assigned to Hyde Park were all students within a compact area in the Terrell zone. The Houston school, and some of the remaining Terrell zone, lay between those students and Hyde Park school. Those assigned to Golden Rule were the black students residing in the eastern part of the Terrell zone (whites there continued to Terrell); a portion of the Terrell zone separated these students from Golden Rule, and it was more distant than either Terrell itself, Lamar or Houston. This plan was apparently the board’s first use of “noncontiguous” zoning. All students zoned out of Terrell would be bused to their new schools.
     
      
      . The projected black ratios were: Terrell, 16.6 percent; Mayes, 8.9 percent; Layne, 14.9 percent; Lamar, 8.4 percent; Hyde Park, 13.9 percent; Houston, 7.7 percent; and, Golden Rule, 13.1 percent.
     
      
      . The essential features of the plan contained in the September 18 order were: (1) moving the east line of the Layne zone east (across state highway 75) into the former Terrell Zone; (2) assigning a group of Terrell zone students to Hyde Park, across the intervening Houston zone; (3) assigning to Golden Rule the black students residing east of Terrell (whites there continuing to attend Terrell); (4) moving the northeast line of Mayes slightly to the east farther into the former Terrell zone; and, (5) reassigning all northwest rural students from Layne to Terrell. The features referenced in (1), (2) and (3) above were similar to those contained in the plan which the court rejected in August (see note 27 supra), except that the September plan moved the Layne line about twice as far east, the group assigned to Hyde Park came from slightly closer to Terrell school, and the group assigned to Golden Rule was expanded slightly. The features referenced in (4) and (5) were new. Virtually all reassignments resulted in students being bused to schools other than those closest to them, and in case of the reassignments to Hyde Park and Golden Rule, the busing would be past intervening schools (Houston or Lamar).
      The plan resulted in an increase in busing of approximately 114 students, 56 black and 58 white. Under the plan, 42.9 percent of the black elementary students would be bused, and 24 percent of the white would be. Under the 1979-80 configuration, 19.2 percent of the black elementary students, and 21.9 percent of the white, were bused.
     
      
      . The authorities cited are Penick v. Columbus Board of Education, 429 F.Supp. 229, 240, 268 (S.D.Ohio 1977), aff'd, 583 F.2d 787, 799-800 (6th Cir.1978), aff'd, 443 U.S. 449, 99 S.Ct. 2982, 61 L.Ed.2d 666 (1979), and Higgins v. Board of Education, 508 F.2d 779, 787 n. 12 (6th Cir.1974).
     
      
      . The June 10 opinion also refers in this connection (but apparently attaches little significance) to the lack of black faculty at Layne and Hyde Park. However, no such lack was noted respecting Golden Rule, and it was deemed no less racially identifiable. Neither the presence nor absence of such a factor was noted in the June 10 opinion, or subsequently, respecting Terrell. Also, the June 10 opinion, though it notes the requirement that “the faculty be truly integrated, not only physically but numerically” and that there be no discrimination in the “assignment of teachers,” expressly declines to find a violation by the DISD in regard to its teaching staff. Absent a violation of Singleton v. Jackson Municipal Separate School District, 419 F.2d 1211 (5th Cir.1969), cert. denied, 396 U.S. 1032, 90 S.Ct. 612, 24 L.Ed.2d 530 (1970), which was neither charged nor found, or segregative intent or actions currently or in relatively recent years on the part of the board or administration, which the district court apparently refused to find, the mere current lack of black faculty at Layne and Hyde Park may, under the circumstances, have little significant relevance to their racial identifiability. More importantly, it appears that this factor played no material part in the district court’s determinations of racial identifiability.
     
      
      . Dr. Foster’s method for determining whether any given school is racially identifiable turns on whether the school’s percentage of black students differs from the percentage of black students in the entire district by a greater number of percentage points than the maximum permissible deviation fixed by his formula. The particular maximum number of percentage points of permissible deviation is set according to which of several ranges of percentages the overall black percentage falls into, with more deviation being allowed at higher district-wide black percentages. Thus, where the black district-wide percentage is as low as 15 percent, the maximum permissible deviation in any school is only five percentage points; where the district-wide percentage is 20 percent, the maximum permissible deviation is ten percentage points; where the overall percentage is as high as 25 percent, the maximum deviation is fifteen percentage points. Penick, 429 F.Supp. at 268. Apparently, in no case is more than fifteen percentage points acceptable. See Carr v. Montgomery County Board of Education, 377 F.Supp. 1123, 1129 n. 21 (M.D.Ala.1974), aff’d, 511 F.2d 1374 (5th Cir.), cert. denied, 423 U.S. 986, 96 S.Ct. 394, 46 L.Ed.2d 303 (1975). We are aware of no recorded specification of the various precise levels fixing the respective ranges of district-wide percentages within which the various respective maximum numbers of percentage points of permissible deviation in individual schools are applicable.
     
      
      . The district court notes that it uses the word “predominantly” only in the sense of more than half. This contrasts with the Penick district court opinion, where “predominantly” is defined as referring to a school “substantially outside the range of statistical deviation” (presumably as determined by the Foster formula) and which “the average ... resident might describe as black or white.” 429 F.Supp. at 269.
     
      
      . The district court also states that Terrell was formerly all black, a factor specifically adverted to in the June opinion. The September opinion notes, however, that the inference in the prior opinion that Terrell was not reopened until 1979 is erroneous, as Terrell was closed at the end of the 1967-68 year, and reopened for the 1969-70 year as a grade six through eight middle school, which it remained for ten years (until converted to an elementary in 1979-80) with a student body of from 70.9 percent to 77.5 percent white (described in the agreed pretrial order as “desegregated”). The district court further observes that during these years Central, serving much of the same area as Terrell, had black percentages ranging from 49.7 percent to 63.8 percent. During the years of dual racial school zones, however, Central was an all-white school. Later in its September opinion the district court also observes that by the fall of 1981 Terrell (at that point without a sixth grade) had become 41.2 percent black. See note 18 supra.
      
     
      
      . In contrast, the June 10 opinion, at a point subsequent to its discussion of the situation at Layne, Hyde Park, Golden Rule and Terrell (including the paragraphs quoted in note 25 supra), states: “DISD has failed to fulfill its burden of proving that the current racial identifiability of Layne, Hyde Park, and Golden Rule is unrelated to past discriminatory action.” No comparable ruling is anywhere made, or fairly inferable, as to Terrell in the June 10 opinion.
     
      
      . The September opinion does not here mention another factor which was specifically noted in this connection in the June opinion, namely “the school has significant numbers of Black and white students together” and “remains predominantly white.” See note 25 supra.
      
     
      
      . The September opinion does refer to the June opinion’s recital that Terrell’s racial composition was “not totally innocuous.” This, however, was only “in terms of its effect on the rest of the district,” namely, Hyde Park, Layne and Golden Rule, while Terrell itself was found “not necessarily racially identifiable per se.” The court found that the board’s plan submitted in response to the June order, and rejected by the court in August, “did succeed in achieving the level of desegregation required in the other elementary schools” (as well as reducing Terrell’s black percentage from 33.8 percent to 29.3 percent). Accordingly, the “not totally innocuous” finding in the June opinion is not related to the August and September conclusions of Terrell’s racial identifiability.
     
      
      . As noted in the discussion of racial identifiability, no such finding was made respecting Terrell, nor was it found racially identifiable. Concerning Layne, Hyde Park and Golden Rule, the court observed that “very little evidence was offered by the defendants in explanation of the racial compositions of these schools.”
     
      
      . And it observed that “the efforts of DISD, however much in good faith, did not prove totally effective.” The district court expressly declined to find anything wrong with the closing of Central, Raynal or Peabody.
     
      
      . When, in December 1969, the district court in Swann found the school district in violation and ordered it to submit another remedial plan, over half of the black students in the district attended schools which were 98 percent to 100 percent black, and the district court observed “[t]he black school problem has not been solved.” Swann v. Charlotte-Mecklenburg Board of Education, 306 F.Supp. 1299, 1304, 1308 (W.D.N.C.1969). This opinion likewise states:
      “On the facts in this record and with this background of de jure segregation extending full fifteen years since Brown I, this court is of the opinion that all the black and predominantly black schools in the system are illegally segregated, ...” Id. at 1312.
      We interpret this statement to mean that the district court had in its previous unreported orders found that the school board had, at least until the April 1969 order, administered the school system with the intent of promoting and maintaining segregation. Similarly, Judge Sobeloff s dissenting opinion in the Court of Appeals in Swann recognizes that residentially the district is divided “along racial lines” and states, without contradiction by the majority, that:
      “This is not a fortuity. It is the result, as the majority has recognized, of government fostered residential patterns, school planning, placement, and, as the District Court found, gerrymandering.” 431 F.2d 138, 149 (4th Cir. 1970) (emphasis added).
      The foregoing also provides instructive context for the Supreme Court’s statements in Swann concerning how actions of school authorities may affect housing patterns and promote segregation:
      “In addition to the classic pattern of building schools specifically intended for Negro or white students, school authorities have sometimes, since Brown, closed schools which appeared likely to become racially mixed ... sometimes accompanied by building new schools in the areas ... farthest from Negro population centers in order to maintain the separation of the races.... Such a policy ... may well promote segregated residential patterns....
      “... ‘Racially neutral’ assignment plans ... may fail to counteract the continuing effects of past school segregation resulting from discriminatory location of school sites or distortion of school size in order to achieve or maintain an artificial racial separation.” 402 U.S. at 21, 28, 91 S.Ct. at 1278, 1282 (emphasis added).
      In the case at bar, there are no findings of any such action on the part of the DISD. Indeed, the district court seems to have been of the view that for the some fifteen years prior to trial the DISD’s actions were not motivated by segregative intent.
     
      
      . Companion cases to Green were Raney v. Board of Education, 391 U.S. 443, 88 S.Ct. 1697, 20 L.Ed.2d 727 (1968), and Monroe v. Board of Commissioners, 391 U.S. 450, 88 S.Ct. 1700, 20 L.Ed.2d 733 (1968). Raney involved a district without residential segregation in which over “85% of the Negro children” attended “all-Negro” schools. In Monroe, the minority percentage was about 40 percent; Merry Junior High was “completely a ‘Negro’ school” attended by “over 80% of the system’s Negro junior high school students,” and three of the eight elementary schools were “attended only by Negroes”; the “transfer” system was admittedly designed to foster racial separation.
      In Alexander v. Holmes County Board of Education, 396 U.S. 19, 90 S.Ct. 29, 24 L.Ed.2d 19 (1969), the Court decreed that the continued operation of segregated schools under the “all deliberate speed” standard must end and called for immediate operation “as unitary school systems within which no person is to be effectively excluded from'any school because of race or color.” Id. 396 U.S. at 20, 90 S.Ct. at 29.
     
      
      . The opinion states:
      “As already noted, our holding today does not rest upon a conclusion that the disparity in racial balance ... resulting from separate systems would, absent any other considerations, be unacceptable.... We hold only that a new school district may not be created where its effect would be to impede the process of dismantling a dual system. And in making that essentially factual determination in any particular case, ... [the Court will rely on the discretion of the district judge]....” 407 U.S. at 470, 92 S.Ct. at 2207.
     
      
      . A companion case to Wright was United States v. Scotland Neck City Board of Education, 407 U.S. 484, 92 S.Ct. 2214, 33 L.Ed.2d 75 (1972), which involved an attempted district division following Justice Department demand for implementation of desegregation. The Justices dissenting in Wright concurred in the result in Scotland Neck largely because of clearer evidence and findings of substantial segregative motivation in the attempted division coupled with the resulting large number of students who would attend “virtually all-Negro schools.” Id. 407 U.S. at 491-92, 92 S.Ct. at 2218.
     
      
      . The Court also noted that there was “evidence tending to show that the Board, ... over a period of years, intentionally created and maintained the segregated character of the core city schools,” 413 U.S. at 206, 93 S.Ct. at 2696, that “uncontroverted evidence” showed teacher and staff assignments “throughout the system” had been racially based “for years,” id 413 U.S. at 200, 93 S.Ct. at 2693, that “there was cogent evidence that the ultimate effect of the Board’s actions in Park Hill was not limited to that area,” id. 413 U.S. at 204, 93 S.Ct. at 2695, and that the district court found “the segregated core city schools were educationally inferior to the predominantly ‘white’ or ‘Anglo’ schools in other parts of the district.” Id. 413 U.S. at 193, 93 S.Ct. at 2689.
     
      
      . However, the Court held “the District Court erred in separating Negroes and Hispanos for purposes of defining a ‘segregated’ school.” 413 U.S. at 197, 93 S.Ct. at 2691. The Court noted that the district court had said a school would be considered educationally “inferior” only if it had “ ‘a concentration of either Negro or Hispano students in the general area of 70% to 75%.’ ” Id. 413 U.S. at 196, 93 S.Ct. at 2691. The Court would “intimate no opinion” as to whether the “70 to 75% requirement was correct,” and observed that the record did not suggest the figures were “used to define a ‘segregated’ school in the de jure context.” Id. 413 U.S. at 196, 93 S.Ct. at 2691.
     
      
      . The Supreme Court noted the Court of Appeals’ determination that the board had “ ‘intentionally maintained a segregated school system down to the time the complaint was filed in the present case,’ ” 443 U.S. at 537, 99 S.Ct. at 2978, and observed that it saw “no reason to disturb” the “factual determinations” by the Court of Appeals (in some instances, by the district court) that since 1954, despite frequent complaint and being on notice of the effects of its acts and omissions, the board made no attempt to reduce racial separation in the schools (with one counterproductive exception) and instead engaged in activities which increased it, that “intentional faculty segregation ... continued into the 1970s” as a “systemwide practice” constituting “strong evidence that the Board was continuing its efforts to segregate students,” that the board used “ ‘optional attendance zones for racially discriminatory purposes,’ ” and that from 1950 to trial 22 of the 24 schools which opened did so as 90 percent (or over) one race and 78 of the 86 space additions made at existing schools were to schools 90 percent or more of one race. Id. 443 U.S. at 539-40, 99 S.Ct. at 2980. The Court held it was proper for the Court of Appeals under this evidence and the principles of Keyes, to trace the existing segregation “back to the purposefully dual system of the 1950s and to the subsequent acts of intentional discrimination.” Id. 443 U.S. at 541, 99 S.Ct. at 2981.
     
      
      . Penick v. Columbus Board of Education, 429 F.Supp. 229, 264 (E.D.Ohio 1977), aff’d, 583 F.2d 787 (6th Cir.1978).
     
      
      . The Sixth Circuit opinion states:
      “The Columbus record, as found by the District Judge, presented a situation where a segregated school system existed in 1954, when Brown I was decided, and has been intentionally maintained as such by the Columbus School Board down to the date of trial of this case.” 583 F.2d 787 at 815.
     
      
      . Answering the contention that by 1888 Ohio had abandoned any statutory requirement or authorization for school segregation, the Court, relying on Keyes, states that:
      “... in the educational context ... there is no magical difference between segregated schools mandated by statute and those that result from local segregative acts and policies.” 443 U.S. at 457 n. 5, 99 S.Ct. at 2946 n. 5.
     
      
      . The opinion also observes that the district court’s findings of purposeful segregation are not limited to the specific instances recited in its opinion. 443 U.S. at 466 n. 14, 99 S.Ct. at 2951 n. 14.
     
      
      . In footnotes 6 and 15, 443 U.S. at 457, 466, 99 S.Ct. at 2946, 2951, the majority criticizes the dissent for attempting to substitute its factual judgment for the “factual determinations” of the lower courts (n. 6) which, in the majority view, have “a much more knowledgeable and reliable view of the facts and of the record than do our dissenting Brethren” (n. 15).
     
      
      . In Columbus the district court said: “Plaintiffs’ expert witness, Dr. Gordon Foster, used statistical criteria in terming a school ‘racially identifiable.’ ... There is ample evidence to support the use of such ranges and the evidence indicates that Dr. Foster’s estimates are reasonable.” 429 F.Supp. at 240. The court also stated “. .. this case is unique; .. . this decision is based on those facts brought out in this trial and no others.” Id. at 232.
     
      
      . It should be noted that “Racially Identifiable” is only one of several terms defined in the district court’s “Glossary” in Columbus. It also defines “One Race School” as one in which 90 percent or more of the students are of a single race. Further, “Predominantly,” in reference to the racial composition of a school, though stated to be “not subject to precise definition in terms of percentages,” is used to denote a school whose racial composition is “substantially outside the range of statistical deviation from the system-wide ... percentage” or “which the average Columbus resident might describe as black or white.” 429 F.Supp. at 269.' This seems to indicate the district court’s recognition that Dr. Foster’s formula might label a school “racially identifiable” which would not be so considered by “the average Columbus resident.”
     
      
      . Apart from various specific instances of purposefully segregative actions, two matters appear to have had particular prominence in the factual reasoning process of the district court in Columbus. First, that although the district had a black student population of some 32.5 percent, over 70 percent of all students “attended schools which were 80-100% populated by either black or white students.” 429 F.Supp. at 240. Second, special attention was paid to the five schools (Champion, Mt. Vernon, Garfield, Felton and Pilgrim) in the east area of the district. These are said to have been “predominantly black schools” in 1941, “attended almost exclusively by black children” in 1943, to have constituted “an enclave of separate, black schools on the near east side” and have been “overwhelmingly black schools” in 1954, id. at 236, and to be and have been “almost completely segregated in 1954, 1964, 1973 and today.” Id. at 260. The Sixth Circuit opinion indicates Champion and Garfield were 96 percent or over black. 583 F.2d at 800. While figures are not available for the other schools, it seems a fair inference from the language in the district court opinion (e.g., “almost exclusively ... black,” “overwhelmingly black,” or “almost completely segregated”) that they, too, were at least well in excess of 50 percent, and probably not less than 90 percent, black. Moreover, the reference in the court’s “conclusion” to “those ... schools in ... Columbus ... which presently have a predominantly black student enrollment,” id. at 266, clearly implies schools having a black enrollment “substantially” over 50 percent (see note 53 supra).
      
      Likewise, the Foster formula does not appear to have played a significant role in the district court’s analysis of the various specific instances of purposefully segregative actions. Thus, in its analysis of the “Near-Bexley Option,” it deals with an attendance zone scheme allowing white children to avoid their logical assignment to nearby schools over 90 percent black in favor of more distant “white (or whiter)” schools, some of which latter schools had black student proportions higher (in one case by several percentage points) than the district average. Id. at 244. See also the “Sixth Avenue Elementary” discussion. Id. at 242. Similarly, in its discussion of faculty policies the district court notes that 63.3 percent of all black elementary teachers were “assigned to schools in which the student body was 80%-100% black.” Id. at 238.
     
      
      . Much of the Sixth Circuit opinion consists of an analysis and approval of the district court opinion. While the Sixth Circuit quotes the district court’s definition of “racially identifiable” (under the Foster formula) and “one race” and says that “the ratios” were “properly employed” by the district court under Swann, it also specifically calls attention to the numerous schools 80 percent or more, 90 percent or more, and 96 percent or more, of one race, so many of which were black. 583 F.2d 787 at 799-800.
     
      
      . It may be noted that Judge Goldberg’s forceful panel dissent, 511 F.2d at 1375, on which the judges dissenting to the denial of rehearing en banc relied, 511 F.2d at 1390, does not to any extent find fault with Judge Johnson’s rejection of the Foster formula. Rather, the concern expressed by Judge Goldberg relates almost exclusively to “the concentration of black students in virtually all-black schools,” 511 F.2d at 1379, evidenced by the fact that, under the projections, of the black elementary students 55 percent would attend “schools 87% or more black” and 44 percent would attend “schools 93% or more black.” Id at 1377.
     
      
      . Cases in which Carr has been distinguished have all considered crucial the existence of a relatively high percentage of black students attending all- or virtually all-black schools. See e.g., United States v. Seminole City School Dist., 553 F.2d 992, 993, 995 (5th Cir.1977) (“... at issue here is an elementary school that has remained identifiably black .... In a county of 15% blacks, Midway’s black enrollment has never been less than 92% and is currently more than 98%.” Id. at 993); Lee v. Demopolis City School System, 557 F.2d 1053, 1054 (5th Cir.1977), cert. denied, 434 U.S. 1014, 98 S.Ct. 729, 54 L.Ed.2d 758 (1978) (an elementary system having only two schools, one was 98 percent black); United States v. DeSoto Parish Sch. Bd., 574 F.2d 804, 815-16 (5th Cir.), cert. denied, 439 U.S. 982, 99 S.Ct. 571, 58 L.Ed.2d 653 (1978) (“In DeSoto, by contrast, over 83 percent of the black pupils attend all-black schools, and, under the current plan, will never be exposed to a desegregated school. There is no entrenched residential segregation----”); Lee v. Macon Cty. Bd. of Ed., 616 F.2d 805, 809, 811 (5th Cir.1980) (“... about 67% of Tuscalousa’s black children would attend elementary schools more than 95% black,” id. at 809, and “... constitutional boundaries in school desegregation cases necessarily remain nebulous ... [but] four predominantly black schools ... can no longer be tolerated.” Id. at 811).
     
      
      . We do not consider Carr’s rejection of the Foster, Winecoff and similar formulas, and their formalistic and mechanical application of mathematical ratios, to be in any way undercut by our statement in Tasby v. Estes, 517 F.2d 92, 104 (5th Cir.), cert. denied, 423 U.S. 939, 96 S.Ct. 299, 46 L.Ed.2d 271 (1975), that:
      “The objective of reducing the proportionate share of a racial group’s composition of the student population of a particular school to just below the 90% mark is short of the Supreme Court’s standard of conversion from a dual to a unitary system. The 90% figure ... was never intended ... to represent the magic level below which a school would no longer be categorized as ‘one race.’ ” (Emphasis added.)
      This language (in a case involving numerous schools over 90 percent black, id. at 97) is plainly directed against both the rigid use of mathematical ratios and the manipulative practice of rigging attendance in order to do no more than barely meet preset arbitrary outward indicia of substantive constitutional standards. We do not consider these obviously correct principles, nor their subsequent reaffirmation in Tasby v. Estes, 572 F.2d 1010, 1012 n. 3 (5th Cir.1978), cert. granted, 440 U.S. 906, 99 S.Ct. 1212, 59 L.Ed.2d 453 (1979), cert. dismissed, 444 U.S. 437, 100 S.Ct. 716, 62 L.Ed.2d 626 (1980), to in any way endorse the magic of the Foster or similar formulas, or to suggest that a school which is clearly majority white should be automatically considered racially identifiable as black merely because its black enrollment is disproportionately large.
     
      
      . We also note in this connection several decisions of this Court where schools with disproportionately large black enrollments (including some with black majorities) that were not, however, such as to constitute the school “one race or virtually one race,” have been either approved or accorded a distinctly different and lesser significance than schools that were all black or virtually all black. In Bradley v. Board of Public Instruction of Pinellas Co., Fla., 431 F.2d 1377 (5th Cir.1970), cert. denied, 402 U.S. 943, 91 S.Ct. 1608, 29 L.Ed.2d 111 (1971), the district-wide black student proportion was 16 percent (or perhaps 12.5 percent, it is unclear which), and this Court ordered the elimination of a large number of “all-Negro or virtually all-Negro” schools. The opinion describes eight remedial zones or pairings ordered; three will be above 50 percent black, six over 42 percent and seven over 38 percent. Id. at 1381-83. Bradley was cited with approval by this Court in Lemon. 566 F.2d at 988. In Allen v. Board of Public Instruction of Broward Co., Fla., 432 F.2d 362 (5th Cir.1970), cert. denied, 402 U.S. 952, 91 S.Ct. 1609, 29 L.Ed.2d 123 (1971), this Court, to eliminate a large number of 90 to 100 percent black schools in a district where the overall black ratio was 22 percent, ordered numerous pairings and clusterings; about half of those ordered would have black percentages in excess of 42 percent, two of which would be over 50 percent. Id. at 370-71. Conley v. Lake Charles School Board, 434 F.2d 35 (5th Cir.1970), dealt with the “Ward 3” schools, where the black enrollment ratio was 36 percent, disapproved four schools over 97 percent black (and a fifth over 86 percent, because of its crowded condition) and ordered adjustments to reduce their enrollments to between 78 percent and 83 percent black, which is described as “desegregated.” Id. at 36, 38, 40-41. Ellis v. Board of Public Instruction of Orange Cty., Fla., 465 F.2d 878 (5th Cir.1972), cert. denied, 410 U.S. 966, 93 S.Ct. 1438, 35 L.Ed.2d 700 (1973), held that in a school district with an 18 percent black enrollment, two 79 percent black schools were desegregated, while three other schools, 100 percent, 99.8 percent and 96.1 percent black, were not. Id. at 880. In Horton v. Lawrence County Bd. of Ed., 578 F.2d 147 (5th Cir.1978), it is indicated that in a district having a 24 percent black enrollment two over 60 percent black schools were desegregated, their black predominance being “attributable solely to the residential patterns.” Id. at 149-50.
     
      
      . We recognize that the Fourteenth Amendment in general and its school desegregation law in particular is not for the exclusive benefit of minorities, and that the benefits from compliance with the constitutional command to liquidate the remnants of officially imposed school segregation are not limited, for black or white children, to affording the former an integrated education. Nevertheless, such is certainly a predominant benefit to be achieved. Recognition of this is appropriate in the application of procedural rules and methods to the process of making the relevant factual determinations in these cases. Cf. Sumner v. Mata, 449 U.S. 539, 551, 101 S.Ct. 764, 771, 66 L.Ed.2d 722 (1981).
     
      
      . This Court’s opinion in Lemon reflects that there were three “virtually all-white” elementary schools within two miles of Butler, the “all-black” elementary school. The opinion observes that the only question presented is whether Butler can continue “all black” in these circumstances. One of the three “white” schools, Bossier, was within a mile of Butler and had much more than enough unused capacity to absorb all the Butler students. Butler likewise had capacity to handle many more students. This Court outlined three alternatives. The first, and preferred, was to pair Butler and Bossier; the next was to bus some of the students from the referenced three white schools to Butler; the third was to close Butler and assign its students either all to Bossier or part to Bossier and part to one of the other “all-white” schools. Each alternative would have left two virtually “all-white” schools (except one variant of the third alternative, which would have left one virtually “all-white” school).
     
      
      . See also Goss v. Board of Education, 340 F.Supp. 711, 717, 727-29 (E.D.Tenn.1972), aff’d, 482 F.2d 1044 (6th Cir.1973; en banc), cert. denied, 414 U.S. 1171, 94 S.Ct. 933, 39 L.Ed.2d 120 (1974), cited with approval in Higgins v. Board of Education of City of Grand Rapids, 508 F.2d 779, 787 (6th Cir.1974).
     
      
      . See also Horton v. Lawrence County Bd. of Ed., supra at 149 (“These statistics, standing alone, would ordinarily create a fairly strong inference that a racially segregated public school system exists in Lawrence County, at least with respect to the named schools.”).
     
      
      . We do not here address extremes. Cf. Lee v. Demopolis City School System, 557 F.2d 1053 (5th Cir.1977), cert. denied, 434 U.S. 1013, 98 S.Ct. 729, 54 L.Ed.2d 758 (1978) (two elementary school system, one 98 percent black, the other virtually all white). But see United States v. Gregory-Portland I.S.D., 654 F.2d 989 (5th Cir.1981) (two school elementary system, one 94.5 percent Mexican-American, the other 81 percent Anglo). See also Judge Gee’s concurring opinion in United States v. DeSoto Parish School Board, 574 F.2d at 824.
      In the DISD, residential patterns are racially concentrated; there are seven elementary schools, each attended by blacks, at several in significant numbers; though there is no school without a clear white majority, the school with the largest concentration of blacks is judged by the experts to be the best, and has had large white majorities for over ten years; there is no appearance of segregative gerrymandering or assignment patterns or the like, and apparently these, as well as avowedly segregationist policies, have been absent for over fifteen years, during which time there is evidence that noneducational factors have increased racial residential concentrations; court-appointed experts do not recommend a change in the elementary zones. These are all matters which, together with others, need to be included as factors in the factual calculus to be made by the district court.
     
      
      . We do not suggest that Denison, or any other district, is so “special” as to be above the law. Denison is not; no district is. Our point, rather, is that in this case the determination that Denison’s elementary schools are in violation of law may not find support in the notion that such determination rests on the district court’s special sensitivity and insight respecting local conditions. It does not rest on such matters.
     
      
      . Other evidence reflects that in December 1979 the black enrollment percentage at Hughes was 1.8 percent and at McDaniel was 20.8 percent.
     
      
      . Terrell, located north and east of McDaniel, served as an all-black junior high school through the year 1967-68, at the end of which it was closed (during this time it was also an all-black elementary, and an all-black high school until the 1966-67 year). Since 1967-68 Hughes and McDaniel have been the only junior highs in the DISD.
     
      
      . They suggested using Golden Rule as the ninth grade facility, and substituting Peabody (which had been closed under the 1979-80 elementary plan) for Golden Rule as the seventh elementary.
     
      
      . He stated McDaniels’ inferior condition was “possibly a racial matter,” and referred to the fact that nearly all black junior high students had long attended McDaniel as being “either an accident of neglect or ... an unfortunate historical circumstance.” He also testified:
      “Q. ... [Djon’t you think the fact that the black population being concentrated on the east side of that line [dividing the Hughes and McDaniel zones] has played a role in the district’s decision about how to handle that school [McDaniel]?
      “A. I have said so. I do think it has....” There were currently no blacks among the DISD’s seven-person board of trustees.
     
      
      . In its December order, the court specifically rejected the request of the plaintiffs and the recommendation of the experts that McDaniel be closed, “because it is a substandard school,” noting that the Constitution does not forbid such schools per se, but rather “prohibit[s] the consignment of practically all of the community’s Black citizens to those substandard schools, which would effectively deny them equal educational opportunity and retain the vestiges of years of racial discrimination.” As plaintiffs do not appeal the failure to close McDaniel, and the DISD may not (and does not) dispute the form of relief which it itself requested, we have no occasion to consider whether relief directly related to the condition of the facilities is to be preferred. See Swann, 402 U.S. at 18, 19, 91 S.Ct. at 1277 (paragraph just following that set out in the above quotation from the district court’s opinion here).
     
      
      . This was initially to go into effect January 15, 1981, but by its January 8, 1981 order the court postponed the effective date to August 15, 1981. On January 13, the board resolved that its attorneys “submit to the court that the 6th, 7th, 8th and 9th grades be unitized ... the 6th and 7th housed at Hughes ... and the 8th and 9th grades housed at McDaniel .... ” The district court granted this relief by its March order. Thereafter, no changes were made in the orders respecting the junior high schools. The court’s June 10, 1981 memorandum opinion, so far as it deals with the junior high schools, simply reincorporates the December 1980 order, as amended.
     
      
      . We assume any such proceedings would be combined with the proceedings on remand respecting the other facets of this case.
      In the event the DISD does not formally advise the district court within such reasonable time as that court may direct, and in any event prior to the expiration of ninety days after the issuance of our mandate herein, that it seeks redetermination hereunder of the junior high school order, then such order shall become in all respects final, and shall not be subject to further appeal, to the same extent as if it had been in all matters unqualifiedly affirmed herein.
     
      
      . These allegations are neither expanded on nor particularized in the pretrial order. In delineating the plaintiffs’ contentions the pretrial order either simply repeats these same contentions, in language almost identical to that of the amended complaint, or states that “role models are being denied to minority students as more' fully set out in” plaintiffs’ amended complaint.
     
      
      . If the “administrative staff’ is considered as consisting of positions appearing at and above the “assistant principal” level on the May 1981 stipulated list of DISD professional employees, with the teaching staff being all others, then 10 percent of the DISD “administrative staff’ was black, and 8 percent of the “teaching staff’ was black, at that time. The district court apparently considered “counselors” and the like as not a part of the teaching staff, and noted the absence of black counselors, but observed that the plaintiffs’ complaint was “limited strictly to principals, assistant principals and teachers.” Despite the court’s observing that the plaintiffs were “free to return” with a claim as to counselors, no such claim was made. The superintendent testified there had been two black counselors, of whom one died and the other moved away. There was no indication when either event occurred.
     
      
      . The district court stated:
      “There is no claim that the district has violated Singleton v. Jackson Municipal Separate School District, 419 F.2d 1211 (5th Cir.1969), cert. denied, 396 U.S. 1032 [90 S.Ct. 611, 24 L.Ed.2d 698] (1970), which prohibits discriminatory dismissals or demotions as part of the desegregation process and requires the faculty of each school to approximate the district-wide racial composition. Neither have the plaintiffs presented the type of historical evidence necessary to support such a claim.” (Footnote omitted.)
      The court, by a footnote, in effect invited plaintiffs to file a Singleton -type complaint by amended pleadings, if they felt it justified. They did not attempt to do so.
     
      
      . The district court’s use of the 9.3 percent teacher figure apparently results from dividing the number of black teachers in May 1981, 27, by the number of “all other” teachers then, 289, instead of by the total of the two, 316. Using this data, the correct figure for black teachers is 8.54 percent.
     
      
      . The court noted that if a “Black educator” were to intervene with sufficient proof that he or she had been “discriminatorily passed over for the job of principal in favor of a white person,” then the court would grant “immediate relief by placing the particular Black person in the principal’s position to which he or she is rightfully entitled.” Despite this invitation, it appears that no such intervention was sought.
     
      
      . Castaneda also states, 648 F.2d at 994 n. 2, that this rule is unaffected by Texas Department of Community Affairs v. Burdine, 450 U.S. 248, 101 S.Ct. 1089 (1981). Neither Castaneda, nor apparently any of our other decisions, addresses in this particular connection the language in Dayton II cautioning against proof of purposefully racial faculty assignments being used to constitute a prima facie case that racial imbalance in student assignments likewise results from similar official motivation. 443 U.S. at 536 n. 9, 99 S.Ct. at 2978 n. 9.
      Our decisions in this area appear to stem from Roper v. Effingham County Board of Education, 528 F.2d 1024, 1025 (5th Cir.1976), where, in the course of affirming denial of relief to a black assistant principal refused promotion to principalship vacancies apparently filled by whites, we stated, citing United States v. Jefferson County Board of Education, 380 F.2d 385 (5th Cir. en banc), cert. denied, 389 U.S. 840, 88 S.Ct. 77, 19 L.Ed.2d 104 (1967), that “[t]he School Board has the burden of proving that its personnel decisions were free from racial considerations.” The only seemingly relevant matter in Jefferson County is the panel opinion statement, made in reference to the required annual court reports of each school’s faculty, that:
      “... school authorities have an affirmative duty to break up the historical pattern of segregated faculties, the hall-mark of the dual system. To aid the court[ ] in its task, the decree requires the school authorities to report to the district courts the progress made toward faculty integration. The school authorities bear the burden of justifying an apparent lack of progress.” 372 F.2d 836 at 895 (5th Cir.1966) (footnote omitted).
      In Barnes v. James County School Dist., 544 F.2d 804, 807 (5th Cir.1977), involving a demotion and firing claim by a former black teacher which was remanded for determination other than on a Singleton basis, the “clear and convincing evidence” wording was added, citing, inter alia, Keyes “dictum.” Subsequent decisions to the same effect, each involving complaint of specific instances where apparently qualified blacks were refused promotions to vacancies filled by whites, are Hereford v. Huntsville Bd. of Ed., 574 F.2d 268 (5th Cir.1978); Davis v. Board of School Com’rs of Mobile County, 600 F.2d 470 (5th Cir.1979); Hardy v. Porter, 613 F.2d 112 (5th Cir.1980), affirming and approving, 443 F.Supp. 1164 (N.D.Miss.1977); and Lee v. Conecuh County Bd. of Ed., 634 F.2d 959 (5th Cir.1981). Castaneda was apparently the first case to apply these decisions to a general student-type class suit. Castaneda was followed in the somewhat analogous case of Fort Bend Indep. School Dist. v. City of Stafford, 651 F.2d 1133 (5th Cir.1981).
     
      
      . “Because only 11.2% of the teachers are black, however, only 11.2% of the persons in the initial pool of potential applicants for promotions are black.” 574 F.2d at 272-73.
     
      
      . “In this case, plaintiffs have relied heavily on the disparity between the percentage of the Raymondville school population consisting of Mexican-Americans (approximately 85%) and the percentage of the faculty in the Raymond-ville schools who are Mexican-American (27%), in support of their contention that RISD discriminates in its employment decisions. Plaintiffs urge that this statistical disparity coupled with the evidence of a past history of segregation in the Raymondville schools sufficed to make out a prima facie case of discrimination which shifted to the defendants a heavy burden of rebuttal which they failed to meet.
      “We think the plaintiffs’ suggested comparison is not the relevant one. Where, as here, the nature of the employment involved suggests that the pool of people qualified to fill the positions is not likely to be substantially congruent with the general population, the relevant labor market must be separately and distinctly defined.” 648 F.2d at 1002.
     
      
      . In Singleton cases, we have not adopted a rigid “per se” or “title” rule for determining whether “demotions” took place and, depending on the actual duties, responsibilities and compensation of the two particular jobs involved in a given case, a transfer from principal to assistant principal may or may not be a demotion. See Hereford, 574 F.2d at 273-74; United States v. Gadsen County School Dist., 539 F.2d 1369, 1376 (5th Cir.1976) (“... [T]he focus in a case like this must remain on the responsibilities actually borne by the particular person in the particular position.”); Roper v. Effingham County Board of Education, 528 F.2d 1024, 1025 (5th Cir.1976). See also the district court opinion in Hardy v. Porter, 443 F.Supp. 1164, 1172 (N.D.Miss.1977), affirmed and opinion approved, 613 F.2d 112, 114 (5th Cir.1980). Here there is no evidence as to the relative duties, responsibilities and compensation of principals and assistant principals. Of course, we do not suggest that a proper essentially statistical showing can under no circumstances ever suffice to establish a prima facie case as to principals or any other single classification of employees.
     
      
      . The case at bar is substantially different from Lee v. Washington County Bd. of Ed., 625 F.2d 1235 (5th Cir.1980). In Lee, both individual and class claims were presented on behalf of school district employees. Over a six-year period, out of more than 32 vacancies filled in head coaching, assistant coaching and office staff positions, only two (assistant coaches) were filled with blacks. Id. at 1237. All six high school principalship vacancies that occurred in the eight years since the original desegregation order were filled with whites “despite the existence of qualified black applicants,” and there had been no black principals in any of the five high schools over the past four years, and none ever at four of these. Id. at 1239. The district court opinion in Lee reflects that the school board conceded Singleton violations, and was also found to have purposefully racially discriminated against two black teachers. 456 F.Supp. 1175, 1178, 1187-88 (S.D.Ala.1978). See also the district court opinion in the companion case, 456 F.Supp. 1164, 1167-68, 1172-74 (S.D.Ala.1978) (finding student and faculty and staff assignment violations). The district court found no violation in the coaching and central office area, noting that several blacks in the system prior to the original 1970 desegregation order had thereafter remained in coaching, in football only one black had applied since 1970 and he was hired for the year following trial, there was “no evidence with respect to the number of applicants in other sports” nor of qualified blacks being passed over in favor of whites, and only one black had applied for central office staff. 456 F.Supp. at 1186-87. This Court affirmed the denial of relief as to the coaching and central office area, though we observed that the plaintiffs’ “statistical evidence ... considered in light of the historical background of this litigation, proved a prima facie case of purposeful discrimination on the basis of race that, if unrebutted, would have supported a district court’s grant of the requested relief.” 625 F.2d at 1238. This Court also reversed and remanded for retrial the denial of relief as to two admittedly qualified black applicants for principal-ship vacancies filled by whites because the district court erroneously placed on the plaintiffs the burden to prove “they were the most qualified applicants.” Id. at 1239-40. The opinion reflects that over the years in question the average fraction of black teachers in the system was 35 percent. Id. at 1241.
     
      
      . When this Court previously denied a stay of that part of the district court’s order requiring the DISD to hire a black for the first occurring principal vacancy, we observed “there is no indication that a vacancy exists at this time, and there is no need to issue any stay” of such part of the order. We assume this is still the case, and request the parties to promptly notify us if it is not. On this assumption, we further direct that, if a principalship vacancy is hereafter filled prior to the final order of the district court following remand and after such position is filled there are no black principals, then any non-black so filling such vacancy shall hold the position subject to the final order of the district court, or of this Court, following remand.
     