SEYMOUR, Circuit Judge.
Zuni Public School District 89 (Zuni) and Gallup-McKinley County • Public School District No. 1 (Gallup-McKinley) seek review of a final decision of the Secretary of Education which addressed whether, and to what extent, the State of New Mexico can consider its receipt of federal Impact Aid when making its own statewide distributions of educational funds. Both Zuni and Gallup-McKinley filed timely petitions for review of the Secretary’s decision. We exercise jurisdiction pursuant to 20 U.S.C. § 7711(b)(1) and affirm.
I
In October 1999, Zuni filed an objection to a certification made by the Department of Education that the State of New Mexico was equalized pursuant to 20 U.S.C. § 7709(b) and the corresponding regulations at 34 C.F.R. § 222.162.1 This certification allows New Mexico .to offset its contributions to local education agencies (LEAs),2 including Zuni and Gallup-McKinley, by a proportion of the federal Impact Aid payments made to those LEAs. See 20 U.S.C. § 7702, 7703, 7709(d)(1)(B). In November 1999, Gallup-McKinley filed a similar objection with the Department, alleging that New Mexico took an inappropriate proportion of Impact Act funds into consideration when determining state aid, in violation of 20 U.S.C. § 7709(d)(1) and 34 C.F.R. § 222.163(a).3
[1162]*1162The objections of Zuni and Gallup-McKinley were consolidated for the purposes of administrative adjudication. An administrative law judge (ALJ) issued a decision sustaining the Department’s certification and rejected all other arguments presented by Zuni and Gallup-McKinley. Both school districts appealed the ALJ’s decision to the Secretary, who affirmed the ALJ’s decision. We will discuss as necessary any additional relevant facts throughout the course of this opinion.4
II
Zuni frames its challenge on appeal as “whether the United States Department of Education and State of New Mexico applied the mandatory statutory formula for determining whether a state has an equalized funding system for its schools which would entitle the state to take credit for federal Impact Aid funding.” Aplt. br. at 2. As we will detail below, the Secretary determined the State of New Mexico was equalized under 20 U.S.C. § 7709, and reached this decision by reference to and application of the implementing regulations and appendix supporting the statute. See 34 C.F.R. § 222.162(a); 34 C.F.R. pt. 222, subpt. K, app. Thus, we must determine
whether the text of the statute resolves the issue, or if not, whether the [Department’s] interpretation [as evidenced by its application of the regulatory index] is permissible in light of the deference to be accorded to the agency under the statutory scheme. If the statute speaks clearly to the precise question at issue, we must give effect to the unambiguously expressed intent of Congress. If the statute is instead silent or ambiguous with respect to the specific issue, we must sustain the agency’s interpretation if it is based on a permissible construction of the statute.
Yellow Transp., Inc. v. Michigan, 537 U.S. 36, 45, 123 S.Ct. 371, 154 L.Ed.2d 377 (2002) (quoting Chevron, U.S.A., Inc. v. Natural Res. Def. Council, Inc., 467 U.S. 837, 842-43, 104 S.Ct. 2778, 81 L.Ed.2d 694 (1984); Barnhart v. Walton, 535 U.S. 212, 217-18, 122 S.Ct. 1265, 152 L.Ed.2d 330 (2002)). In so doing, it is important for us to examine the current statute as well as its implementing regulations and appendix, and the history accompanying each.
Impact Aid is compensatory financial assistance paid by the United States to an LEA experiencing financial burdens because the LEA’s ability to raise local revenues has been limited as a result of the real property within its boundaries having tax exempt status due to the property’s acquisition by the federal government, or because the LEA educates children residing on, or whose parents are employed on federal property, including Indian lands. See 20 U.S.C. §§ 7701-7714.
[1163]*1163Generally, states are prohibited from reducing the state aid they provide to their LEAs if the LEAs receive Impact Aid. See 20 U.S.C. § 7709(a). Pursuant to 20 U.S.C. § 7709(b), however, if a state is certified by the Department as having a program of state aid “that equalizes expenditures for free public education among local educational agencies in the State,” id., then the state is permitted to factor in the receipt of Impact Aid funds when making its own distributions of educational aid to its LEAs. The statute sets out a general disparity test to determine whether a state is equalized. Under this test, LEAs are ranked in descending order by their per-pupil expenditures or revenues. If the LEA that is ranked highest in terms of its per-pupil expenditures or revenues does not exceed the per-pupil revenues or expenditures of the lowest ranked LEA by more than twenty-five percent, then the state qualifies as equalized under the statute. 20 U.S.C. § 7709(b)(2)(A).
In making this calculation, LEAs ranked at the extremes with the highest and lowest per-pupil expenditures are excluded. As directed by the statute, “local educational agencies with per-pupil expenditures or revenues above the 95th percentile or below the 5th percentile of such expenditures or revenues in the State,” 20 U.S.C. § 7709(b)(2)(B)(i), are disregarded when determining whether a twenty-five percent disparity exists between the highest and lowest LEAs.
The exception detailed in 20 U.S.C. § 7709(b) allowing “equalized” states to take into consideration Impact Aid when making their own distributions of state aid first appeared in the Impact Aid laws in 1974. That statute used general language similar to that which appears in the current statute,5 but instead of describing a disparity test, the 1974 statute expressly delegated to the Secretary the power to define the term “equalize expenditures.” See 20 U.S.C. § 240(d)(2)(B) (1982).
In 1976, after engaging in the process of notice and comment rulemaking, the Secretary promulgated regulations in which it outlined the disparity test. It directed that a state would be deemed equalized if “the disparity in the amount of current expenditures or revenue per pupil for free public education among local educational agencies having similar grade levels in the State is no more than 25 per centum, as determined according to the procedures set forth in Appendix A to this subpart.” 41 Fed.Reg. 26320, 26327 (June 25, 1976). Appendix A detailed the specific method by which to make the disparity determination. First, LEAs were ranked by expenditures or revenues per pupil, and then those LEAs which fell “at the 95th and 5th percentiles of the total numbers of pupils in attendance in the schools of those agencies,” were eliminated. Id. at 26329. The twenty-five percent disparity comparison was then made between the remaining highest and lowest ranked LEAs. Id.
In the course of the 1976 notice and comment process, the Department of Education responded in length to a question regarding whether the ninety-fifth and fifth percentiles were to be calculated based on the total number of pupils in the state or by school districts. It stated that
the referenced percentiles are based on number of pwpils. [The regulation] provides that in calculating the disparity [1164]*1164standard according to the procedures set forth in Appendix A, the districts in a State will be ranked on the basis of current expenditures or revenue per pupil, and that those districts which fall above the 95th and below the 5th percentile of those agencies in terms of the number of pupils in attendance in the schools of those agencies will be excluded for purposes of the calculation. The percentiles will be determined on the basis of numbers of pupils and not on the basis of numbers of school districts. However, it appears that the proposed regulation may not have been as clearly expressed as desired. In regard to the question of pupils versus districts for the percentages used in calculating the disparity standard, it is the Commissioner’s view that basing an exclusion on numbers of districts would act to apply the disparity standard in an unfair and inconsistent manner among States. The purpose of the exclusion is to eliminate those anomalous characteristics of a distribution of expenditures. In States with a small number of large districts, an exclusion based on percentage of school districts might exclude from the measure of disparity a substantial percentage of the pupil population in those States. Conversely, in States with large numbers of small districts, such an approach might exclude only an insignificant fraction of the pupil population and would not exclude anomalous characteristics.
Id. at 26324 (emphasis added). In 1993, the 1976 regulations and the accompanying appendix were codified with substantially the same language in the Code of Federal Regulations. See 34 C.F.R. § 222.63 (1993), 34 C.F.R. pt. 222, subpt. K, app. (1993).
In 1994, the statute at issue in this case was passed, repealing and replacing 20 U.S.C. § 240(d). As noted above, the replacement statute, 20 U.S.C. § 7709, specifically details within its text the twenty-five percent disparity test, and also includes language setting out the basic parameters of the required ninety-fifth and fifth percentile determinations. This statutory language largely reiterates the disparity test and percentile determinations laid out in the earlier regulations and appendix. However, the percentile language appearing in the 1994 statute differs slightly from that which appeared in the earlier regulations. While the former appendix directed that the ninety-fifth and fifth percentiles should be calculated by “the total number of pupils in attendance in the schools” to determine which LEAs to eliminate, 34 C.F.R. pt. 222, subpt. K, app. (1993), the 1994 statute dictates that LEAs “with per-pupil expenditures or revenues above the 95th percentile or below the 5th percentile of such expenditures or revenues in the State,” 20 U.S.C. § 7709(b)(2)(B)®, should be disregarded. Nothing in the legislative history for the 1994 changes to the law gives any indication as to what meaning, if any, should be ascribed to the language “above the 95th percentile or below the 5th percentile of such expenditures or revenues in the State,” id., when calculating whether a state is equalized. At most, the legislative history merely notes that the statute “prohibits a State from taking Impact Aid payments into account in determining the amount of State aid to be paid to LEAs that receive Impact Aid, unless that State has an equalization plan approved by the Secretary and describes the standard which State plans must meet.” H.R. No. 103 — 425, at 71 (1994), reprinted in 1994 U.S.C.C.A.N. 2807, 2877.
Following the passage of 20 U.S.C. § 7709, the Department enacted new regulations to replace the 1976 regulations. [1165]*1165The new disparity calculation regulation closely mirrors the language of the statute:
The Secretary considers that a State aid program equalizes expenditures if the disparity in the amount of current expenditures or revenues per pupil for free public education among LEAs in the State is no more than 25 percent. In determining the disparity percentage, the Secretary disregards LEAs with per pupil expenditures or revenues above the 95th or below the 5th percentile of those expenditures or revenues in the State.
34 C.F.R. § 222.162(a). But the regulation goes on to state, in a fashion similar to its predecessors, that “[t]he method for calculating the percentage disparity in a State is in the appendix to this subpart.” Id. The appendix, 34 C.F.R. pt. 222, subpt. K, app., contains the same per-pupil expenditure percentile ranking method as was outlined in the earlier appendices, and details the same specific methodology for calculating the twenty-five percent disparity test. It directs as follows:
(a) The determinations of disparity in current expenditures or revenue per pupil are made by—
(i) Ranking all LEAs having similar grade levels within the State on the basis of current expenditures or revenue per pupil for the second preceding year before the year of determination;
(ii) Identifying those LEAs in each ranking that fall at the 95th and 5th percentiles of the total number of pupils in attendance in the schools of those LEAs; and
(iii)Subtracting the lower current expenditure or revenue per pupil figure from the higher for those agencies identified in paragraph (ii) and dividing the difference by the lower figure.
Id. (emphasis added). The history accompanying the regulation notes that the regulation outlines the single statutory standard for determining whether a state is equalized and “specifies the method the Secretary will employ to measure the statutory disparity standard. Detailed examples of the application of this method to State funding programs are provided in the Appendix following subpart K.” 60 Fed.Reg. 50774, 50777 (September 29, 1995).6 In sum, the Department has adhered since 1976 to a rule which dictates that when disregarding LEAs to determine whether a state is equalized, the ninety-fifth and fifth percentiles should be calculated by the total number of students in the state, rather than by the number of LEAs.
With this history in place, we turn to an examination of the Department’s percentile calculation to determine whether the Department’s construction is permissible in light of the language in 20 U.S.C. § 7709 which directs that “local educational agencies with per-pupil expenditures or revenues above the 95th percentile or below the 5th percentile of such expenditures or revenues in the State,” should be disregarded. 20 U.S.C. § 7709(b)(2)(B)®.
In determining whether New Mexico qualified in the year in question here as an equalized state under 20 U.S.C. § 7709, the Department followed the methodology provided in. the appendix to 34 C.F.R. [1166]*1166§ 222.162 to decide which LEAs fell within the ninety-fifth and fifth percentiles of “the total number of pupils in attendance in the schools of those LEAs.” 34 C.F.R. pt 222, subpt. K, app. Applying this methodology to New Mexico, which has predominately small LEAs among its eighty-nine public LEAs, the Department eliminated the eighteen highest-ranked and the seven lowest-ranked LEAs before ascertaining that the newly-top-ranked LEA received only 14^43% more state funding than the newly-bottom-ranked LEA. The Department thus concluded that New Mexico’s public school funding is substantially equalized.
Zuni argues that this methodology directly conflicts with the statutory language of 20 U.S.C. § 7709(b)(2)(B)®, which it reads to require the Department to eliminate LEAs on the basis of LEA percentiles rather than by student population percentiles. It asserts that the statute requires one of only two methods for determining the ninety-fifth and fifth percentiles of per-pupil expenditures: (1) eliminate five percent of the LEAs at the top and bottom of the ranked chart of LEAs, or (2) figure five percent of the highest and lowest actual per-pupil expenditure figures and disregard those LEAs that are funded above and below those numbers, respectively.7 Zuni asserts the total number of students is irrelevant to determining percentiles of per-pupil expenditures and thus conflicts with the statute. We disagree.
Where a law entrusted to an agency’s administration is ambiguous, or where Congress implicitly or explicitly left a gap in the law to be filled in by the agency through the formulation of policy or rules, “we must accept the agency’s position so long as it reflects a permissible construction of the language in question.” Hill v. Ibarra, 954 F.2d 1516, 1523 (10th Cir.1992) (quotation omitted). See also Chevron, 467 U.S. at 842-43, 104 S.Ct. 2778. “Such an interpretation is given controlling weight unless it is arbitrary, capricious, or contrary to law.” McNamar v. Apfel, 172 F.3d 764, 766 (10th Cir.1999).
When Congress passed 20 U.S.C. § 7709 in 1994, it codified the disparity standard essentially as it appeared in the former regulations, albeit with an unexplained and slight alteration to the wording regarding percentile exclusions. Compare 34 C.F.R. § 222.63 (1993) and 34 C.F.R. pt. 222, subpt. K, app. (1993) with 20 U.S.C. § 7709(b). While the prior statutory scheme and regulations made clear that the percentiles were to be calculated by total student population, the new statutory language is not as explicit: the precise language directs the Secretary to “disregard local educational agencies with per-pupil expenditures or revenues above the 95th percentile of such [per pupil] expenditures in the State 20 U.S.C. § 7709(b)(2)(B)® (emphasis added). We agree with the Secretary’s determination that the statute is ambiguous: [1167]*1167Ree., vol. Ill, doc. 25 at 3. In other words, we do not know what calculations Congress intended by “such [per pupil] expenditures in the State.” Even Zuni conceded in its argument before the ALJ that the statute “may be ambiguous [as] to the precise formula that is to be used.” Id, doc. 16 at 10. The statute’s ambiguity,8 coupled with the gap left by Congress regarding the specific means by which to implement the disparity test, requires us, in accordance with the well-established rule laid out in Chevron, to give deference to the Secretary’s interpretation of the statute if we deem that interpretation to be reasonable or “based on a permissible construction of the statute.” Chevron, 467 U.S. at 843, 104 S.Ct. 2778. Based on our review of the statute, the regulations and the history accompanying both, we cannot hold that the Department’s construction of the statute and the method by which it applied the disparity test was impermissible.
[1166]*1166[a]lthough the impact aid statute sets forth the parameters for calculating state public education expenditures or revenues under the disparity test, the statute does not contain a specific implementation of the disparity test; instead, Congress left that gap to be filled by regulation, which has been duly promulgated at an appendix to subpart K of 34 CFR Part 222.
[1167]*1167The statute mandates that the expenditures or revenue for each LEA be calculated on a per pupil basis. See 20 U.S.C. § 7709(b)(2)(B)(i). The statute then directs that those LEAs which fall above the ninety-fifth percentile or below the fifth percentile of “such [per-pupil] expenditures or revenues in the state” should be disregarded for the purpose of calculating the disparity standard. Id Because the statute mandates that expenditures or revenues be examined on a per-pupil basis, we conclude the Department’s method of determining the ninety-fifth and fifth percentiles based on the total student enrollment in the state is a permissible construction. Having already ranked the LEAs in descending order by their per-pupil expenditures, it makes sense that the cut-off points for percentiles wohld also be based on total student enrollment. Such an approach supports the basic purpose of the percentile exclusion because it eliminates in a fair and effective manner any unusual or noncharacteristic per-pupil revenues or expenditures that may appear at the extremes of the range- of LEAs in the state. This is especially true for the State of New Mexico, which has predominately small LEAs, several of which rank near the top of per-pupil expenditures. See Rec., vol. I, doc. 4, attach. 4 at 1. To base the percentile calculation on LEAs, rather than student population, would not further the goal of eliminating this unusual distribution of per-pupil expenditures in New Mexico. As we previously pointed out, the Department rejected determining percentiles solely on an LEA basis for this very reason.
[B]asing an exclusion on numbers of districts would act to apply the disparity standard in an unfair and inconsistent manner among States. The purpose of the exclusion is to eliminate those anomalous characteristics of a distribution of expenditures. In States with a small number of large districts, an exclusion based on percentage of school districts might exclude from the measure of disparity a substantial percentage of the pupil population in those States. Conversely, in States with large numbers of small districts, such an approach might exclude only an insignificant fraction of the pupil population and would not exclude anomalous characteristics.
41 Fed.Reg. 26324 (June 25, 1976). Finally, as detailed above, the Department’s method of calculating percentiles by student population comports with the longstanding history surrounding such calculations.
There may exist some other, and perhaps even better, way to eliminate anoma[1168]*1168lies in school district funding in New Mexico. However, given the large number of small LEAs in the state, the Department’s disparity calculation methodology implements the statute in a manner that will give a more consistent result when applied to a variety of state school systems. We need not conclude the “agency construction was the only one it permissibly could have adopted to uphold the construction, or even the reading [we] would have reached if the question initially had risen in a judicial proceeding.” Chevron, 467 U.S. at 843 n. 11, 104 S.Ct. 2778. Rather, we are compelled to “sustain the agency’s determination if it is based on a permissible construction of the statute.” Yellow Transp., Inc., 537 U.S. at 45, 123 S.Ct. 371 (quotation omitted). Based on our analysis above, we hold that the Department’s construction of 20 U.S.C. § 7709(b)(2)(B)®, through its application of the disparity calculation methodology laid out in 34 C.F.R. § 222.162(a) and its accompanying appendix, is permissible, and therefore warrants our deference under Chevron.
Ill
We turn next to Gallup-McKinley’s argument challenging New Mexico’s offset of contributions to LEAs. We decline to devote any substantial analysis to the merits of Gallup-McKinley’s contention because we conclude it failed to raise this issue in a sufficient manner before the agency below.
Congress limits state offsets for federal Impact Aid funding to a “proportion to the share that local tax revenues covered under a State equalization program are of total local tax revenues.” 20 U.S.C. § 7709(d)(1)(B). The corresponding regulation provides, in part, that “[d]etermina-tions of proportionality must be made on a case-by-case basis for each LEA affected and not on the basis of a general rule to be applied throughout a State.” 34 C.F.R. § 222.163(a). On appeal, Gallup-McKinley complains that, in violation of the regulatory language requiring the state to determine proportionality on a “case-by-case” basis, New Mexico applies an across-the-board standard to its school districts. Gallup-McKinley contends it raised this specific issue during oral arguments before the ALJ.
In the course of the proceedings below, Gallup-McKinley outlined three different methods by which it believed the proportional Impact Aid offset should be calculated. One of these methods was admittedly presented to the ALJ for the first time during oral arguments. Rec., vol. Ill, doc. 16 at 86-87. In that argument, Gallup-McKinley asserted a method of determining proportionality that conflicted with New Mexico’s method. It was only at the very close of oral argument that Gallup-McKinley made the most general of references to the regulation’s “case-by-case” language. Id. at 111-12. But Gallup-McKinley did not present any argument to the ALJ, either during the course of oral argument or in any of its initial written submissions to the judge, to explain why the State of New Mexico’s proportionality calculation allegedly violated the “case-by-case” language appearing in 34 C.F.R. § 222.163(a) while its suggested method did not. The ALJ asked the parties to include in their supplemental briefs a discussion of whether Gallup-McKinley’s additional method fell within the regulations. Id. at 102.
In their supplemental briefs to the ALJ, the State of New Mexico and Department of Education appropriately focused their arguments on whether Gallup-McKinley’s newly proffered proportionality offset method fit within the regulations. Gallup-McKinley did the same, claiming in conclu-sory fashion and without any substantial [1169]*1169discussion, argument, or analysis, that the regulation’s “case-by-case” language was violated by New Mexico’s method. It never offered any explanation as to why its suggested proportionality formula met the “case-by-case” requirement while New Mexico’s did not, id., doc. 17 at 4, and it still offers no such explanation in its appeal brief to us.
In its opinion, the ALJ considered Gallup-McKinley’s three different interpretations of how 20 U.S.C. § 7709(d)(1) should be applied in light of the general proportionality issues timely and sufficiently presented to it, and rejected each as inconsistent with the statute and its corresponding regulation. Id., doc. 21 at 8-10. It did not mention the “case-by-case” language of the regulation. In seeking review of the ALJ’s decision to the Secretary of Education, Gallup-McKinley reframed its challenge against the State of New Mexico’s proportionality offset determination, and stated the “sole issue” on appeal was “whether determinations of proportionality must be made on a case-by-case basis for each LEA or if it can be based upon a general rule applied throughout the State.” Id., doc. 23 at 2. In addressing Gallup-McKinley’s appeal, the Secretary limited its analysis to the more general proportionality issues that were presented to the ALJ, and did not devote any discussion to Gallup-McKinley’s new and undeveloped “case-by-case” argument. Id., doc. 25 at 2 n.3.
It is well established that “in the absence of exceptional circumstances, a reviewing court will refuse to consider contentions not presented before the administrative proceeding at the appropriate time.” Micheli v. Director, OWCP, 846 F.2d 632, 635 (10th Cir.1988) (quoting Duncanson-Harrelson Co. v. Director, OWCP, 644 F.2d 827, 832 (9th Cir.1981) (further quotation omitted)). Likewise,
[v]ague, arguable references to [a] point in the [lower] court proceedings do not ... preserve the issue on appeal. [W]here a litigant changes to a new theory on appeal that falls under the same general category as an argument present at trial or presents a theory that was discussed in a vague and ambiguous way the theory will not be considered on appeal.
Bancamerica Commercial Corp. v. Mosher Steel of Kansas, Inc., 100 F.3d 792, 798-99 (10th Cir.1996) (citations and quotations omitted). Hence, when a litigant fails to raise an issue below in a timely and clear fashion and the lower court does not address the merits of the issue, the issue is not preserved for our review. FDIC v. Noel, 177 F.3d 911, 915 (10th Cir.1999).
Based on our review of the record, Gallup-McKinley’s passing and conclusory references to 34 C.F.R. § 222.163(a)’s “case-by-case” language during oral argument and in its supplemental briefs to the ALJ are, at best, ambiguous and vague. As a result, neither the ALJ nor the Secretary addressed the merits of this potential argument. We therefore conclude that Gallup-McKinley failed to preserve this issue for appeal, and we decline to address it.
The petition for review is DENIED.