The Estate of Paul Mitchell (the “Estate”) petitions for review of the United States Tax Court’s decision allowing the Commissioner of the Internal Revenue Service (the “Commissioner”) to assess an additional $2,404,571 in federal estate taxes. The Estate claims that the Tax Court erred by: (1) finding that the Commissioner timely notified the Estate of the additional taxes due; (2) failing to shift the burden of proof to the Commissioner and failing to require the government to justify its calculation of the additional taxes; (3) failing to provide a detailed explanation of the methodology used to calculate the fair market value of the John Paul Mitchell Systems (“JPMS”) stock held by the Estate; and (4) miscalculating the value of JPMS stock held by the Estate in a manner inconsistent with its own holding. Although we find that the Commissioner’s notice was timely, we nevertheless vacate the Tax Court’s judgment and remand because the Tax Court failed to shift the burden of proving the accuracy of the ad*698ditional estate tax to the Commissioner and failed to provide an adequate explanation for its valuation of the JPMS stock at the time of Paul Mitchell’s death.
I. BACKGROUND
Paul Mitchell, co-founder of the highly successful hair-care products company of the same name, died on April 21, 1989.1 Pursuant to 26 U.S.C. § 6075(a), the Estate filed for, and received, a six month extension of time to file its estate tax return, delaying the filing deadline from January 21, 1990, to July 21, 1990. Because July 21, 1990, fell on a Saturday, the Estate mailed its return on Friday, July 20, 1990. The IRS received the Estate’s return on Monday, July 23,1990.
On July 21, 1993, the IRS mailed to the Estate a notice of deficiency (the “Notice”), determining a deficiency in the federal estate tax in the amount of $45,117,089, and a total of $8,543,643 in penalties.2 The IRS asserted that the Estate had undervalued its 1,226 shares of JPMS stock. The Estate had reported the stock was worth $28.5 million based on a valuation conducted by a private accounting firm. The IRS, however, calculated the stock’s value at $105 million and assessed additional taxes based on the $76.5 million discrepancy. .
In October 1993, the Estate petitioned the United States Tax Court for review of the Commissioner’s additional assessment. The Estate first disputed the timeliness of the Notice. In moving for summary judgment, the Estate maintained that, under 26 U.S.C. § 7502, its return was filed on July 20, 1990 — the date on which the return was mailed to the IRS. Therefore, the Estate argued that the Notice was untimely under 26 U.S.C. § 6501 because it was mailed on July 21, 1993 — a day after the three year statute of limitations ran.
In October 1994, the Tax Court ruled that the Notice was timely. Estate of Mitchell v. Commissioner, 103 T.C. 520, 1994 WL 585226 (1994). The Tax Court reasoned that 26 U.S.C. § 7502 was not relevant because it only applies in situations where a return is untimely filed. Because the deadline, July 21, 1990, fell on a Saturday, 26 U.S.C. § 7503 applied, which considers as timely filed a return due on a weekend or holiday that is received by the IRS on the first business day following that weekend or holiday. The Estate’s return was timely because the IRS received it on Monday, July 23, 1990. Therefore, § 7502 did not apply.
On June 11, 1996, the Estate filed a motion with the Tax Court disputing that it bore the burden of persuasion to show the Commissioner’s assessment was inaccurate. The Estate argued that the evidence established that it owned 49.04 percent of the outstanding stock in JPMS on the valuation date and thus its interest in JPMS was a minority interest, not a controlling interest. Therefore, the Commissioner’s appraisal, determining that the Estate’s 49.04 percent interest was a controlling interest, was erroneous, and any additional estate taxes were excessive. *699The Estate contended that pursuant to Herbert v. Commissioner, 377 F.2d 65 (9th Cir.1966), the burden should be placed on the Commissioner to justify the government’s original assessment or to submit a more accurate figure. On July 8, 1996, the Tax Court denied the Estate’s motion to shift the burden of persuasion without explanation.
The dispute over the value of the stock proceeded to trial. In addition to a substantial amount of documentary evidence, both the Estate and the Commissioner offered the testimony of expert witnesses as to the value of the 1,226 shares of stock JPMS stock. The experts’ testimony offered a wide variety of estimates and methods for calculating the stock’s value. As may be expected, the experts for the Estate minimized the stock’s value, testifying that its value on the date of Paul Mitchell’s death ranged from approximately $20 to $29 million, while the experts for the Commissioner maximized the stock’s value in a range from $57 to $165 million. The methodology each expert used was equally varied, with some producing estimates based on the stock prices of similar companies and others using elaborate economic formulae. The experts generally agreed that the most significant factors included the impact of Paul Mitchell’s death on the reputation of the company, the costs of litigation between the Estate and John Paul “Jones” DeJoria, (Mitchell’s co-founder and business partner),3 cash-flow patterns, the marketability of the Estate’s minority (ie. non-controlling) interest of stock in the company,4 and the overall competition in the hair care industry.
In 1997, the Tax Court issued its opinion as to the stock’s value. Estate of Mitchell v. Commissioner, 74 T.C.M. (CCH) 872 (1997). The Tax Court found that the stock’s fair market value was $41,532,600. The court began by assigning a $150 million value to JPMS based on the testimony of Robert Taylor, the president of Minne-tonka Corporations, who testified that his company had offered $125 million for JPMS, but was rebuffed by DeJoria who informed Taylor that he had received a $150 million offer from Gillette, Co. The court then discounted the company’s value by ten percent to account for the.loss of Mitchell’s public presence and creativity. From the now $135 million total company value, the Tax Court calculated the value of the Estate’s 49.04 percent share in the company at $66,204,000. Finally, the court granted a total 35 percent discount to reflect the combined discounts of lack of marketability and minority interest and a $1.5 million discount to reflect the possibility of a lawsuit over Mr. DeJoria’s compensation.
In November 1997, the Estate filed a motion for reconsideration and to correct the Tax Court’s opinion, claiming the Tax Court had overvalued the stock by imposing a 35 percent combined discount to reflect the minority interest and lack of marketability, which it argued was unsupported, by and contrary to the record. Specifically, the Estate argued that the Tax Court misstated the testimony of expert witness George Weiksner, erroneous*700ly reciting that he had adjusted the public value of the shares by a 45 percent discount to reflect a combined discount for the minority interest and the lack of marketability, when in fact Weiksner’s uncon-troverted testimony suggested a combined discount of 61.5 percent. On December 19, 1997, the Tax Court granted the Estate’s motion in part, modifying it to correctly reflect Weiksner’s testimony, but denied the Estate’s request for reconsideration.
On June 10, 1998, the Estate filed a second motion for reconsideration, pursuant to Tax Court Rules of Practice and Procedure 161, renewing its objection to the Tax Court’s valuation of the stock. The Estate also argued that our holding in Leonard Pipeline Contractors v. Commissioner, 142 F.3d 1133 (9th Cir.1998), which had been filed but not published at the time the Estate filed its first motion for reconsideration,5 required the Tax Court to more adequately explain how it arrived at the combined discount for minority interest and lack of marketability. On July 8, 1998, the Tax Court denied the Estate’s second motion for reconsideration, ruling that the motion was untimely and that Estate had failed to “show unusual circumstances or substantial error.” The court further explained that its determination of the 35 percent combined discount rate was' appropriate and “fell within the ranges suggested by the parties’ experts.” *lt further stated that “valuation is necessarily an approximation and a matter of judgment, rather than one of mathematics, Hamm v. Commissioner, 325 F.2d [934,] 940 [(8th Cir.1963)], on which petitioner has the burden of proof.”
The Estate filed a timely appeal. We have jurisdiction under 28 U.S.C. § 1291 and review the merits of each of the Estate’s claims in turn.
II. DISCUSSION
A. Timeliness of the Notice of Deficiency
The Estate claims that the Notice of Deficiency was untimely, and therefore the statute of limitations bars the assessment of additional Federal estate tax and penalties. This presents a mixed question of law and fact, which is reviewed de novo. Mayors v. Comm’r, 785 F.2d 757, 759 (9th Cir.1986). The Estate contends that, under 26 U.S.C. § 7502(a)(1), its return was filed on July 20, 1990 — the postmarked date. Section 7502(a)(1) provides:
If any return, claim, statement, or other document required -to be filed, or any payment required to be made, within a prescribed period or on or before a prescribed date under authority of any provision of the internal revenue laws is, after such period or such date, delivered by the United States mail to the agency, officer, or office with which such return, claim, statement, or other document is required to be filed, or to which such payment is required to be made, the date of the United States postmark stamped on the cover in which such return, claim, statement, or other document, or payment, is mailed shall be deemed to be the date of delivery or the date of payment, as the case may be.
26 U.S.C. § 7502(a)(1) (emphasis added). Therefore, the Estate argues, the Notice was late because it was mailed on July 21, 1993 — one day after the three year statute *701of limitations ran. See 26 U.S.C. § 6501(a) (setting a three year statute of limitations for the Commissioner to send a notice of deficiency).
The Tax Court rejected the Estate’s argument, reasoning that 26 U.S.C. § 7502 was not relevant because the Estate’s return was timely filed. The Tax Court held that because the deadline, July 21, 1990, fell on a Saturday, the provisions of 26 U.S.C. § 7503 were applicable:
When the last day prescribed under authority of the internal revenue laws for performing any act falls on Saturday, Sunday, or a legal holiday, the performance of such act shall be considered timely if it is performed on the next succeeding day which is not a Saturday, Sunday, or a legal holiday.
26 U.S.C. § 7503. Given that the prescribed deadline for the Estate’s return was July 21, 1990, a Saturday, the Tax Court held that § 7503 automatically extended the timely filing period to Monday, July 23, 1990. Accordingly, the Tax Court concluded that because the IRS received the Estate’s return on Monday, July 23, 1990, the return was timely filed that day, and therefore § 7502 did not apply.
We agree with the Tax Court. As we previously held in First Charter Financial Corp. v. United States, 669 F.2d 1342 (9th Cir.1982), “Section 7502(a) intended to make the date of mailing the date of delivery only where a document would otherwise be considered untimely filed.” Id. at 1346; see also Miller v. United States, 784 F.2d 728, 730 (6th Cir.1986) (holding that § 7502 “applies only in cases where the document is actually received by the I.R.S. after the statutory period”). Therefore, the Tax Court correctly concluded that the Estate’s return was timely filed under § 7503, not § 7502.
Even if we were to assume that the Estate is correct and § 7502 applies, the Notice would be timely. Under § 7502, the return would be considered delivered, and therefore filed, when it was mailed on July 20, 1990 — one day before the July 21, 1990 deadline. It also would be an early return. 26 U.S.C. § 6501(b)(1) states: “For purposes of this section, a return of tax imposed by this title ... filed before the last day prescribed by law or by regulations promulgated pursuant to law for the filing thereof, shall be considered as filed on such last day.” Id. (emphasis added); see also Hotel Equities Corp. v. Comm’r, 546 F.2d 725, 727 n. 3 (7th Cir.1976) (discussing the effect of § 6501(b)(1) on the three year statute of limitations). As an early filed return, the Estate’s return would be considered filed on July 21, 1990; therefore, the statute of limitations would begin to run on that day, and not July 20, 1990, as the Estate contends. Consequently, the Notice would be timely because it was sent on the last day before the statute of limitations ran: July 21, 1993. Thus, in light of either section 7502 or 7503, the Notice was timely.
B. Shifting the Burden of Proof
The Estate argues that the Tax Court erred by denying its motion to shift the burden of persuasion, leaving the burden of proof on the Estate. See Herbert v. Comm’r, 377 F.2d 65, 69 (9th Cir.1966). The Tax Court denied the Estate’s motion without explanation, and we will refrain from speculating as to the reasons for its decision. We nevertheless review de novo the Tax Court’s decision to deny the Estate’s motion to shift the burden. See Moss v. Comm’r, 831 F.2d 833, 837 (9th Cir.1987).
In Cohen v. Commissioner, 266 F.2d 5 (9th Cir.1959), we stated:
At the outset of a Tax Court proceeding to redetermine a tax deficiency, the Commissioner’s determination is pre*702sumed to be correct. The burden of proof is thus placed upon the taxpayer to show that the Commissioner’s determination is invalid.
When the Commissioner’s determination has been shown to be invalid, the Tax Court must redetermine the deficiency. The presumption as to the correctness of the Commissioner’s determination is then out of the case. The Commissioner and not the taxpayer then has the burden of proving whether any deficiency exists and if so the amount. It is not incumbent upon the taxpayer under these circumstances to prove that he owed no tax or the amount of the tax which he did owe.
Id. at 11 (citations omitted).
According to the Notice, the Commissioner concluded the value of the JPMS stock at the time of Paul Mitchell’s death was $105 million. The Estate had reported the value at $28.5 million in its tax return. Due to the $76.5 million difference in value, the Commissioner asserted that the Estate owed an additional $45,117,089 in estate taxes, not including a total of $8,543,643 in penalties. At trial, Martin Hanan, a witness for the Commissioner, valued the stock at $81 million — $34 million less than the Commissioner’s original valuation. Furthermore, a letter written by the Commissioner’s appraiser, AIBE Valuation, dated March 18, 1993, indicates that AIBE Valuation originally appraised Mitchell’s interest at $85 million as a minority interest, but increased it to $105 million, at the request of the IRS, to reflect the Estate’s interest as a controlling interest. We find that Hanan’s testimony and the AIBE letter support the conclusion that the Commissioner’s assessment was arbitrary and excessive. United States v. Stonehill, 702 F.2d 1288, 1294 (9th Cir.1983) (holding that “where the assessment has separable items, ... error which demonstrates a pattern of arbitrariness or carelessness will destroy the presumption for the entire assessment”); Cohen, 266 F.2d at 11 (holding that when the taxpayer has shown the determination to be arbitrary and excessive, the burden of persuasion shifts to the Commissioner to prove the correct amount of tax owed and the presumption as to the correctness of the Commissioner’s determination is out of the case); see also Helvering v. Taylor, 293 U.S. 507, 513-15, 55 S.Ct. 287, 79 L.Ed. 623 (1935).
We conclude that the Tax Court erred in denying the Estate’s Motion to Shift the Burden of Persuasion. Consistent with Cohen, because the Commissioner’s determination was demonstrated, by its own experts, to be invalid, the Commissioner — and not the Estate — had the “burden of proving whether any deficiency exists and if so the amount.” Cohen, 266 F.2d at 11. The Tax Court treated the case as one where the burden of proof made no difference; it did not find that one party failed to carry its burden, but proceeded with its own valuation, “weighing the evidence and choosing from among conflicting inferences and conclusions those which it considers most reasonable.” Tax Court Order, Docket No. 21805-93 (July 8, 1998) (citing Comm’r v. Scottish Am. Inv. Co., 323 U.S. 119, 123-24, 65 S.Ct. 169, 89 L.Ed. 113 (1944)). However, in responding to the petitioner’s second motion for reconsideration, the Tax Court erroneously stated that valuation was a matter of approximation and judgment “on which the petitioner has the burden of proof.” (emphasis added). Because the burden of proving the evaluation of the Estate and the commensurate deficiency shifted to the Commissioner, it was error not to put the Commissioner to its proof.
*703C. Explanation of Valuation
The Estate argues that the Tax Court also erred by failing to provide a more detailed explanation of its stock valuation pursuant to our holding in Leonard Pipeline Contractors v. Commissioner, 142 F.3d 1133 (9th Cir.1998). It further contends that the Tax Court’s decision is internally inconsistent, noting that the Tax Court erroneously stated that it selected a combined discount value for the stock that was within the ranges established by the expert witnesses testimony. Finally, the Estate argues that the Tax Court’s error in identifying the correct range follows from its failure adequately to explain its holding.
In Leonard Pipeline, the Tax Court had concluded that it was “impossible” to calculate the values at issue with any “mathematical precision” and instead substituted its “best judgment” to arrive at the appropriate figures. Id. at 1135. On appeal, we held that “it is the obligation of the Tax Court to spell out its reasoning and to do more than enumerate the factors and leap to a figure intermediate between petitioner’s and the Commissioner’s.” Id. In reversing and remanding that case to the Tax Court, we further held: “A reasoned decision as to what is reasonable in this context must bring together the disparate elements and give some account of how the judge has reached his conclusion.” Id.
We next considered this issue in Estate of Magnin v. Commissioner, 184 F.3d 1074 (9th Cir.1999), where we applied Leonard Pipeline to hold that the Tax Court had again failed to provide a sufficient explanation of its conclusions. Id. at 1081. There we reasoned that “the Tax Court only provided the ‘pieces of the puzzle’ but did not divulge how it put them together,” a task necessary for our review. Id.
In denying the Estate’s second motion for reconsideration, the Tax Court explained that its “determination of a 35-percent combined minority and marketability discount fell within the ranges suggested by the parties’ experts.”6 After determining that the Estate’s 49.04 percent share in the company was appropriately valued at $66,204,000, it then attempted to calculate the discounts described by the experts. The Tax Court relied upon the testimony of three experts to establish the ranges. First, it relied upon testimony of Martin Hanan, a witness for the Commissioner, to set the floor of the range at 30 percent for lack of marketability. The court noted that Hanan “determined a value for the JPMS shares using a comparable companies analysis on a publicly traded, minority interest basis (i.e., which took into account any applicable minority discount).” The Tax Court then turned to the Estate’s expert witness, George Weiksner, to set the ceiling of the range at a 45 percent discount to reflect lack of marketability. The court noted that Weiksner “used a comparable companies analysis and determined petitioner’s proportionate public value of the shares (which took into account a 30 percent minority discount).” Weiksner’s discount was consistent with the testimony of Kenneth W. McGraw, another Estate ex*704pert witness, who also applied a 45 percent marketability discount. From the testimony of these three expert witnesses, the Tax Court concluded that a 30 to 45 percent range for the applicable discounts was established. It ruled that “based upon a thorough review of the entire record before us, we believe that we correctly arrived at a 85 percent combined discount rate.”
We review this analysis de novo, and conclude that the Tax Court’s explanations are insufficient. See Ann Jackson Family Found, v. Comm’r, 15 F.3d 917, 920 (9th Cir.1994); Walt Disney, Inc. v. Comm’r, 4 F.3d 735, 738 (9th Cir.1993). Even following the explanation offered by the Tax Court, we cannot determine just how it arrived at the 35 percent rate.
First, the Tax Court appears to be comparing “apples and oranges.” The expert witnesses used various starting values for JPMS and calculated them discounts differently. The experts relied upon by the Tax Court — Hanan, Weiksner, and McGraw — conducted their analyses based on a hypothetical or estimated publicly traded value of the minority interest of JPMS stock to determine the initial value of the company before applying discounts for lack of marketability. The Tax Court, on the other hand, started with an acquisition value, the $150 million bid by Gillette Co., and began discounting from there. Acquisition value and publicly traded value are different because acquisition prices involve a premium for the purchase of the entire company in one deal. Such a lump-sum 'valuation was not taken into account when the minority interest value of the stock was calculated by the experts. In general, the acquisition price is higher, resulting in an inflated tax consequence for the Estate.
Second, the court does not explain how it reached a combined discount of 30-45 percent. Hanan’s 30 percent discount was for marketability only, having already accounted for minority interest in his starting point. McGraw and Weiksner likewise testified to a marketability discount based on public trading value that already included a minority discount. In addition, the Tax Court’s initial misstatement of Weiks-ner’s testimony compounded the problem because it failed to recognize that he had actually testified to a 61.5 percent combined discount. In examining the expert’s testimony, we conclude that Hanan did not set the floor of the range for a combined discount at 30 percent, nor did Weiksner’s testimony establish a ceiling of 45 percent. Because the range is unsupported by the testimony of any of the experts, singularly or together, it is unclear whether the Tax Court’s combined discount actually falls within any particular range that might be supported in the record.
Finally, the Commissioner offers us a multitude of avenues through which one might arrive at a 35 percent combined discount. This strained effort, in and of itself, is the most telling evidence of the inadequacy of the Tax Court’s explanation. We are left to speculate, like the Commissioner, as to the basis for the final valuation of the Estate’s JPMS stock. We therefore vacate the Tax Court’s decision and remand so that it may provide an explanation of its conclusion consistent with the standards we established in Leonard Pipeline.
CONCLUSION
Based on the foregoing, we deny the Estate’s petition in part and affirm the Tax Court’s decision with respect to the timeliness of the notice of deficiency. We grant the Estate’s petition in part, vacate, and remand for the Tax Court to shift the *705burden of proof to the Commissioner regarding the determination of additional taxes and explain its valuation of the stock consistent with Leonard Pipeline. In light of the foregoing, we do not reach the question whether the Tax Court correctly valued the Estate, as we are unable to conduct a meaningful review.
AFFIRMED IN PART, VACATED AND REMANDED IN PART.
. Because of the limited issues before us on this appeal, we do not find it necessary to detail the rather colorful background of the development of JPMS and its phenomenal rise through the hair care industry. For a thorough and highly detailed exposition of the development of JPMS as well as- the specific testimony regarding its value at Mitchell’s death, see the Tax Court’s Opinion, reprinted at 74 T.C.M. (CCH) 872 (1997).
. The Commissioner determined penalties of $8,396,020 pursuant to I.R.C. § 6662(g) and $147,623 pursuant to I.R.C. § 6662(h). The combined total of the penalties is $8,543,643. Sea Estate of Mitchell v. Comm'r, 103 T.C. 520, 521, 1994 WL 585226 (1994).
. In 1993, Patrick T. Fujieki, the Estate’s executor, had brought suit on behalf of the Estate against DeJoria, alleging that DeJoria's compensation from the company was excessive. Although the parties settled in 1995, the litigation remained pending and could have affected the company’s value at the time of the Tax Court trial.
. Of the 2,500 total shares of JPMS stock, the Estate held 1,226. DeJoria held 1,250, which represented the controlling interest in the company. DeJoria and Mitchell originally owned 1,250 shares a piece, but Mitchell assigned a total of 24 shares to two other people in 1987.
. We decided Leonard Pipeline on April 24, 1998, which chronologically fell between the Estate's first and second motions for reconsideration. Accordingly, the Estate cited Leonard Pipeline as a new legal development in filing its second motion for reconsideration.
. The Commissioner also argues that a sufficient explanation had been provided, relying upon an older body of case law that purports to grant great deference to the Tax Court. See Ebben v. Comm’r, 783 F.2d 906, 909 (9th Cir.1986); Estate of O’Connell v. Comm'r, 640 F.2d 249, 251-52 (9th Cir.1981). In these cases, however, the Tax Court provided some justification for its conclusions in a manner that allowed us to understand and reconstruct the Tax Court’s rationale. In the case at hand, the Tax Court merely announced the discount it applied to the Estate’s stock without any explanation.