Full opinion text
AMENDED MEMORANDUM AND JUDGMENT WEINSTEIN, Senior District Judge. CONTENTS I. Introduction.464 II. Facts.465 III. Procedural History.466 A. Aborted plea bargain.466 B. Trial.466 C. Sentencing.466 D. Appeal.467 E. Proceedings on remand utilizing Rule 706 of the Federal Rules of F. Summary of arguments on remand-OO O G. Statement of the issue before the court 05 <£> IV. Sentencing Law.470 A. Burden of proof.470 1. Generally.470 2. Burden of proof at sentencing.471 B. Operation of the Guidelines.472 1. Drug cases.473 2. Non-drug cases.474 C. Caselaw on estimation and specific evidence.475 1. Estimation.475 2. Estimates based on extrapolation.477 3. Specific evidence.478 V. Available Sources of Information.479 A. Background knowledge .479 B. Knowledge of the drug trade.480 C. Demeanor.480 1. How judges use demeanor.481 2. Demeanor and appellate review.481 D. Assumptions about criminal behavior.481 VI. How Decision-Makers Learn and Decide. 00 CO A. Inferences based on prior information and training £8. 00 CO B. Methods of reaching conclusions. 00 CO 1. Classical step-by-step analysis. 00 CO 2. Bayesian and statistical analysis. 00 3. Biases. 00 05 4. Storytelling. 00 VIL Application of Law to Facts on Original Sentence.488 A. Evidence from trial .488 B. Demeanor and character.489 C. Knowledge of the drug trade: the trip effect .490 D. Storytelling analysis.491 VIII. Desirability of Further Analysis.492 IX. The General Federal Rule Favoring Admissibility Of and Reliance on All Helpful Evidence.492 A. Mechanistic rules versus flexible general principles.493 B. Development of twentieth century conceptions of evidence codes.494 C. Model Code of Evidence.496 D. Federal Rules of Evidence.497 E. Recent developments.499 X. Additional Material Available to Sentencing Judge On Remand A.Experts’ reports . 1. Government expert. 2. Defense expert. 3. Rule 706 Panel. a. Use of fictions. b. Problems with government’s assumptions . c. Comments on defendant’s report. d. Non-statistical analysis. ©> LO e. Statistical analysis. ©■ LO f. Simulations accounting for trip effect.... <© LO g. Conclusion . T — l LO B. Survey of the Eastern District bench. tH 1C C. Testimony on economics of heroin smuggling_ 1 — I )lO XI. Law Applicable to Statistical and Other Information Supplied After Remand CO i — i U5 A. Admissibility of probabilistic evidence. CO i“H B. Use of bare statistics. © tH 1. Generally. © t*H 2. Criminal cases. 00 tH lO XII. Application of Law to Facts After Remand. co H iO A. Conclusions about experts’ reports.-.. ca H lO B. Random versus non-random sampling. © CJ © C. Use of statistics to illustrate non-statistical decision-making. h (M © D. Conclusions on proper role of statistics in this ease. co (M © E. Cross-checking. co (M © F. Conclusion in light of statistics and other information provided on remand. Cn 52 XIII. Obstruction of Justice Enhancement. cn to iN A. Purposes of sentencing and of § 3C1.1. cn to B. Double-counting. cn to cn C. Discretion to enhance under Dunnigan. cn to © D. Unconstitutionally of automatic enhancement. cn to -q E. Particularized finding of perjury. cn to F. Conclusion. cn to OO XIV. Additional Sentencing Considerations. cn to oo A. “Penalty” for going to trial. Cn to oo B. Sentencing within prescribed Guidelines range. cn to © C. Added time in prison required by Guidelines system cn co © XV. Conclusion. © CO no I. Introduction The defendant was caught entering the country with 427.4 grams of heroin in his digestive tract. It is believed that he made seven prior drug smuggling trips, but not how much he carried on those trips. The court is required, under the Sentencing Guidelines, to estimate the total quantity of heroin imported. At the original sentence, the court multiplied 427.4 by eight, arriving at a total of 3419.2 grams. The sentence was 151 months in prison, the low end of the Guidelines range for importation of 3,000 grams of heroin or more. See United States v. Shonubi, 802 F.Supp. 859 (E.D.N.Y.1992) [Shonubi 7], conviction affirmed, sentence reversed, 998 F.2d 84 (2d Cir.1993) [Shonubi III The court of appeals rejected this solution and remanded. On reconsideration, the trial court now concludes that the defendant should be sentenced for importing between 1,000 and 3,000 grams of heroin. For reasons indicated in Parts XIII, XIV, and XV, infra, the term of imprisonment is unchanged. This memorandum is largely devoted to explaining how a sentencing judge — and a trier of fact generally — reaches a decision. The case presents an opportunity to observe, explain, and discuss forensic decision-making. The absence of exclusionary rules of evidence at sentencing, the availability of statistics against which to “check” non-statistical proof, and the assistance of skilled experts have permitted the court to examine the decision-making process more fully than circumstances usually permit. It should come as no surprise that in addition to rational analysis, the forensic factfin-der depends upon assumptions and methods of thinking that may introduce biases and errors. Articulating the assumptions used in this case may provide valuable perspectives on decision-making practice and theory. Because the sentencing judge found the problems raised by this case difficult, he issued a draft memorandum before resen-tencing and circulated it for comment. The parties and a number of scholars responded. These responses are on file with the court. The sentencing judge also posed questions to his colleagues on the court; their responses are described below. The final memorandum benefits from all of this assistance. While it generally follows the earlier draft, it includes a number of revisions. The memorandum is written with great diffidence since those of us involved in the daily quest for the truth in the courtroom are out of close touch with those academics endeavoring to understand and explain decisionmaking. See, e.g., Ronald J. Allen, David J. Balding, Peter Donnelly, Richard Friedman, David H. Kaye, Lewis Henry LaRue, Roger C. Park, Bernard Robertson & Alexander Stein, Probability and Proof in State v. Skipper: An Internet Exchange, 35 Jurimetrics Journal 277 (1995) [hereinafter, Ronald J. Allen et al., An Internet Exchange ]. The separation between courts on the one hand and scientists, including statisticians and students of the human mind and emotions, on the other needs to be reduced by such projects as those of the Carnegie Foundation and the Federal Judicial Center described below. How to make available to triers, in a way they can understand, the work of scientists remains a central task of science and the law. II. Facts Charles 0. Shonubi, a 34-year-old Nigerian citizen, lived in New Jersey while studying architecture and working as a toll collector at the George Washington Bridge. His salary was $12,000 a year. On December 10, 1991, Shonubi flew from Lagos, Nigeria, to Amsterdam, and then on to New York. A customs service officer at John F. Kennedy Airport noticed Shonubi near a baggage carousel; he was “turning rapidly” and “scanning the customs area” rather than looking for his luggage. The officer examined Shonubi’s passport and then questioned him about the frequent trips to Nigeria indicated on the passport. The defendant’s answers were contradictory and confusing. For example, Shonubi said the purpose of his last trip was to get married. Minutes later, he said the object was to visit his ailing mother. Reminded of his previous statement, he told the officer that he had forgotten that he had just said that he had gone to Nigeria to get married. Shonubi was “visibly nervous, avoided eye contact, and was wringing his hands”; he was “trembling and sweating.” (The interesting question of what “avoiding eye contact” means in different cultures is not at this stage of the case a matter of concern.) In response to the agent’s request, Shonu-bi consented to an X-ray. Taken to a “search room,” he was patted down, read his Miranda rights, and handcuffed. A search of his person turned up a slip of paper bearing the name of a Nigerian customs official. An X-ray revealed a number of foreign bodies in his intestine. Shonubi was escorted to a trailer designed for the observation of passengers suspected of carrying drugs internally. The trailer has two levels. Material expelled into a toilet on the upper level drops into a holding tank, where agents on the lower level can observe it. Shonubi passed a total of 103 balloons over two days. A forensic chemist found that four of the balloons, selected at random, contained a heroin mixture. The mixture represented 60.49 percent of the gross weight of the four balloons. Cf. United States v. Esieke, 940 F.2d 29, 32 (2d Cir.1991) (weight of heroin mixture “is usually several hundred grams lower than the gross weight [of heroin plus balloons]”). Multiplying the average weight of the heroin mixture in the four tested balloons by 103, the chemist arrived at an aggregate weight of 427.4 grams. Chemical testing established that the purity of the heroin mixture in the four balloons was 53 percent. A customs service agent interviewed Sho-nubi while he was in the tráiler. When she asked him where he had obtained the narcotics, Shonubi answered that he had bought them from “an ordinary man on the street.” She asked him what he was supposed to do with the drugs, and he answered that they were his, and that he “wasn’t giving them to anyone else.” III. Procedural History A. Aborted plea bargain The defendant attempted to plea bargain. However, during a plea allocution before the chief magistrate judge, he alternately admitted and denied knowledge of the drugs found in his system. The allocution was aborted. Had Shonubi accepted the plea agreement, it is likely — given the disposition of similar cases in this district — that he would have been sentenced to a maximum of thirty-six months in prison. Instead, he elected to go to trial. B. Trial At trial, two customs agents described Shonubi’s behavior at Kennedy Airport. Another customs agent testified that 427.4 grams of heroin, at 53 percent purity, would have a wholesale value of $44,000 and, if cut following the normal procedures of heroin distributors in New York, would produce 20,000 “hits.” That the heroin had to be diluted was, according to the agent, certain: ingestion of 53 percent heroin can be fatal. The heroin, he concluded, could not have been for Shonubi’s personal use. Cf. United States v. Martinez, 54 F.3d 1040, 1043 (2d Cir.1995) (jury properly inferred, “viewing the evidence in its totality,” that cocaine found on defendant was not for defendant’s own use). The date stamps on Shonubi’s passport conflicted with his accounts of when he had traveled, as well as with employment records at the George Washington Bridge. Those records showed eight absences from work. Entries on Shonubi’s passport strongly supported the inference that eight trips had, in fact, been made, but that Shonubi had made some legs of the trips using a second passport. Warned by his attorney not to testify, Sho-nubi — after visibly remonstrating with his attorney — chose to take the stand. He proved an effective witness for the prosecution. Confronted with his passport and work records proving that he had made at least eight trips to Nigeria in the previous fifteen months, Shonubi nonetheless denied making more than four trips to Nigeria. He also denied traveling abroad except to see his family. He was unable to explain how he could afford eight round-trip tickets, costing a minimum of $900 each, while earning $12,-000 a year and paying college expenses. He also denied defecating in the medical van. Shonubi was convicted by the jury of heroin importation, 21 U.S.C. §§ 952(a) and 960(b)(2)(A), and possession of a controlled substance with intent to distribute, 21 U.S.C. § 841(b)(1)(B). C.Sentencing At his sentencing hearing, Shonubi reiterated, under oath, that he had made only four trips to Nigeria and had not imported heroin on any of those trips. The sentencing judge found that Shonubi had lied throughout the trial and was lying at the sentencing proceeding. His fabrications included his statements about the number of trips he had made, about the purposes of those trips, about his employment history, about his use of multiple passports, and about events inside the customs service trailer. Based on evidence at the trial and at sentencing, the judge found that the defendant had made a total of eight smuggling trips to Nigeria between September 1, 1990 and December 10, 1991. Multiplying the amount of heroin found on Shonubi by eight produced a total of 3,419.2 grams. Under the Sentencing Guidelines, possession of at least 3,000 but less than 10,000 grams of heroin is a “level 34” offense. See U.S.S.G. § 2D1.1 (1994) (drug quantity table). The sentencing court rejected the defendant’s argument that he deserved a reduction for his “minor” role in the offense. See U.S.S.G. § 3B1.2. It held that a preponderance of the evidence at his trial established that defendant imported large quantities of heroin on eight occasions, using two passports. Defendant was not the foolish, occasionally even unwitting, drag mule who frequently comes before this court. He regularly and systematically engaged in the business of narcotics trafficking between Nigeria and the United States. He deserves no leniency for being a “minor” or “minimal” criminal. Shonubi I, 802 F.Supp. at 864. The government contended that the defendant’s perjury at trial and sentencing constituted per se obstruction of justice, requiring enhancement under § 3C1.1 of the Sentencing Guidelines. The sentencing court held the enhancement provision should not be imposed whenever a defendant lies in court, but only in furtherance of the goals of sentencing enumerated by Congress at 18 U.S.C. § 3551 et seq. In this case, the defendant’s lies had led to the discovery of seven previously unknown smuggling trips. The sentencing judge observed: The Guidelines’ predilection for incarceration should be satisfied by adding ten years for defendant’s exercising his right to a trial and to testify. Penalizing defendant additionally for what amounts to the same conduct (the maintenance of his own defense) would be inappropriate and cruel. 802 F.Supp. at 863. The sentencing range for a level 34 offense, for an offender in defendant’s criminal history category—“I”—is 151-188 months. Shonubi was sentenced to 151 months, plus five years’ supervised release and a $100 assessment. D. Appeal Shonubi appealed from his sentence on two grounds. First, he argued that the government had failed to prove how much, if any, heroin he had imported on trips prior to the offense of conviction. Second, he contended that the court should have reduced his offense level by four points for his “minimal role in the offense.” U.S.S.G. § 3B1.2. The government cross-appealed. It argued that the court was obligated, in light of Shonubi’s perjury, to impose a two-point obstruction of justice enhancement under § 3C1.1. The court of appeals found that the record amply supported the trial court’s determination that Shonubi had made a total of eight related smuggling trips in 1990 and 1991. The conflicting accounts in Shonubi’s passport and his work attendance records amply prove that he made eight trips to Nigeria on more than one passport. It may also be inferred that appellant imported heroin during each of these journeys because he used two passports, traveled frequently, avoided using direct flights ... and could not have afforded the air fare on his toll collector’s salary. Shonubi II, 998 F.2d at 89. The court of appeals also found that, for sentencing purposes, the eight trips should be considered a single course of conduct, which the court defined as “an identifiable pattern of criminal conduct.” Id. In making this finding, the court considered “such factors as the nature of defendant’s acts and how frequently the ... acts have been repeated.” Id. (citing United States v. Perdomo, 927 F.2d 111, 115 (2d Cir.1991)). It also found that Shonubi was not a minor participant in the offense. In reaching this conclusion, the cpurt noted that “appellant. regularly engaged in drag smuggling_[and that his] pattern of travel ... suggests broad knowledge and culpability in the heroin trade.” Shonubi II, 998 F.2d at 90. The court of appeals was not satisfied with the sentencing judge’s finding that Shonubi had imported more than 3,400 grams of heroin on his eight trips. It pointed to a lack of “specific evidence,” which it defined as “e.g., drag records, admissions or live testimony.” Id. at 89. It found that on the issue of quantity the evidence “does not constitute proof by the requisite preponderance of the evidence,” id. at 86, and that the sentencing court had relied on “surmise and conjecture.” Id. at 90. Finally, the court of appeals held that the sentencing judge was required to impose the obstruction of justice enhancement once it had determined that Shonubi had lied under oath. In light of United States v. Dunnigan, — U.S. -, 113 S.Ct. 1111, 122 L.Ed.2d 445 (1993), decided after Shonubi’s initial sentence, the court held that enhancement is mandatory when a defendant lies about a material matter on the stand. Id. at 88. The court of appeals vacated and remanded for resentencing. E. Proceedings on remand utilizing Rule 706 of the Federal Rules of Evidence On remand the government indicated that it would rely on statistical evidence to establish the total quantity of heroin imported by Shonubi. The defendant was authorized to retain an expert at government expense to respond. To help the court evaluate the statistical evidence proffered by the parties’ experts, the court appointed a panel of experts. Federal Rule of Evidence 706 provides that “the court may appoint expert witnesses of its own selection.” Judges are encouraged to avail themselves of this procedure. See, e.g., Daubert v. Merrell Dow Pharmaceuticals, — U.S. -, -, 113 S.Ct. 2786, 2797-98, 125 L.Ed.2d 469 (1993) (Judges should be “mindful” of the Rule 706 power); see also Kenneth R. Foster et. al., Phantom Risk: Scientific Inference and the Law 437 (1994) (encouraging judges to appoint experts); The Evolving Role of Statistical Assessments as Evidence in the Courts 171 (Report of the Panel on Statistical Assessments as Evidence in the Courts) (Stephen E. Fienberg ed., 1989) [hereinafter Evolving Role ] (recommending “the increased use of court-appointment procedures”); cf. Science and Technology in Judicial Decision Making: Creating Opportunities and Meeting Challenges 16 (Report of the Carnegie Commission on Science, Technology, and Government, March 1993) (urging judges to “take an active role” in evaluating scientific proof); Federal Judicial Center, Reference Manual on Scientific Evidence 531-42 (1994) (describing uses of court-appointed experts) [hereinafter Reference Manual ]. Pursuant to Rule 706, both the defense and the government were given opportunities to respond to the report of the Rule 706 Panel. See generally Joe S. Cecil & Thomas E. Willging, “Court-Appointed Experts,” in Reference Manual, supra, at 527-73 (describing Rule 706 procedures). Cf. Edwin J. Wesely, Practical Workings of the Special Master, N.Y.L.J., Mar. 22, 1994, at 1 (recommending circulation of reports of special masters). The parties were also permitted to respond to the draft memorandum. Based on responses to that draft, the court requested the parties to provide additional information on the economics of heroin smuggling. This occurred at a final Fatico hearing described in Part X.C, infra. F. Summary of arguments on remand The government statistician, Dr. David Bo-yum, obtained customs service data on the amount of heroin recovered from 117 Nigerian heroin swallowers apprehended at John F. Kennedy Airport between September 1, 1990 and December 10,1991, the dates of the first and last trips indicated on Shonubi’s passport. Using this data, statistical methods, and a computer program, Dr. Boyum determined that it was “99 percent probable” that Shonubi had imported more than 2090.2 grams of heroin. This analysis, the government argued, satisfied its burden of proving that Shonubi had smuggled enough heroin to constitute a level 32 offense. It did not attempt to prove importation of at least 3,000 grams of heroin, the amount found by the court at the first sentence. The defense, relying on an analysis by Professor Michael O. Finkelstein, argued that no statistics about other smugglers could satisfy the government’s burden of proof vis-a-vis Shonubi. It also disputed Dr. Boyum’s methodology. His calculations, it argued, failed to account for the likelihood that smugglers carry more heroin with each successive trip, a putative learning curve it termed the “trip effect.” Finally, it argued that smugglers of large amounts of heroin are more likely than smugglers of small amounts to be caught, suggesting that the DEA data overstate average quantities smuggled. The Rule 706 Panel agreed with the defense that statistics about other smugglers, taken alone, could not satisfy the government’s burden of proof. The Panel, however, suggested a number of possible uses for the statistical evidence in conjunction with evidence already available to the court. The combination of statistical and non-statistical proof, the Panel suggested, would support a finding that Shonubi had imported, at a minimum, 1,000 grams of heroin on his eight trips. The experts’ reports will be examined in detail in Part X.A, infra. G. Statement of the issue before the court As the Second Circuit Court of Appeals recently observed “direct evidence” is described as evidence tending to show, without resort to inference, the existence of a fact in question... But ... “all knowledge is inferential.” Tyler v. Bethlehem Steel Corp., 958 F.2d 1176, 1183 (2d Cir.1992) (quoting Visser v. Packer Engineering Assoc., 924 F.2d 655, 659 (7th Cir.1991) (en banc) (emphasis added)). The relevance of any evidentiary fact depends on the premises adopted by the trier. Whenever an item of evidence is offered as tending circumstantially — that is, inferentially — to establish a proposition the truth of which is at issue in a case, it is essential to articulate honestly and fully the inference or series of inferences invited. Each specific step of reasoning must invariably match a premise, usually unarticulated, which the judge judicially notices. “Relevancy and Related Problems” in John H. Mansfield, Norman Abrams, Margaret E. Berger, et al., Cases and Materials on Evidence 1, 3 (8th ed. 1988) (emphasis added). No rule prevents a judge or jury from “drawing an inference upon an inference.” Fenner v. General Motors Corp., 657 F.2d 647, 650 (5th Cir.1981). Without this ability hardly a single trial could be adequately prosecuted. For example, on a charge of murder, the defendant’s gun is found discharged; from this we infer that he discharged it; and from this we infer that it was his bullet which struck and killed the deceased.... All departments of reasoning, all scientific work, every day’s life and every day’s trial, proceed upon such data. 1 Wigmore on Evidence § 41, at 435-36 (Peter Tillers rev. ed. 1983) (footnote omitted). Estimation is a type of inference. As Professor Alan Wolf observed in his comparison of science and the law, both require “ ‘[a]p-proximations of many types.... Often it is necessary to make assumptions that are not testable ... simply to be able to proceed with a computation.’” See Eva H. Hanks, Michael E. Herz & Steven S. Nemerson, Elements of Law 505 (1994) (quoting Wolf). Under the Sentencing Guidelines, approximations are not only permitted but are frequently required. See infra Part IV.B. One variety of estimation is extrapolation — broadly defined as the application of knowledge about an event to a series of similar events, using “inferences based on an assumed continuity, correspondence, or other parallelism.” Webster’s Third New Internay tional Dictionary 807 (1967). Extrapolation requires circumstances that permit the inference that known conduct will recur, or has recurred, under similar conditions. See, e.g., United States v. McMillen, 8 F.3d 1246 (7th Cir.1993) (using extrapolation to determine drug quantity under the Guidelines), cert. denied, — U.S. -, 114 S.Ct. 1649, 128 L.Ed.2d 368 (1994); United States v. Sklar, 920 F.2d 107 (1st Cir.1990) (same), described in Part IV.C, infra. Extrapolation has been used by courts in a variety of contexts. In mass torts cases, extrapolation is required to predict the quantity and quality of future claims. See, e.g., In re Joint E. & S. Dist. Asbestos Litig. (Findley v. Falise), 878 F.Supp. 473 (E. & S.D.N.Y.1995) (“Courts cannot wait for the near certainty that may come with a post hoc perspective.”). For other examples of extrapolation in forensic decision-making, see, e.g., Integrated Circuits Unlimited v. E.F. Johnson Co., 691 F.Supp. 630 (E.D.N.Y.1988) (purchaser’s actions in rejecting shipment of radio parts, based on sampling of eight percent of shipped parts, was proper use of statistical methods) (citing James, Relevancy, Probability and the Law, 29 Calif.L.Rev. 689, 692 (1941)), rev’d on other grounds, 875 F.2d 1040 (2d Cir.1989); United States v. 449 Cases Containing Tomato Paste, 212 F.2d 567 (2d Cir.1954) (approving sampling of defective shipment as meeting burden of proof); In re Estate of Ferdinand E. Marcos Human Rights Litigation, MDL No. 840, Report of Special Master Sol Schreiber (Dec. 30, 1994) (depositions of 137 randomly selected claimants used to predict validity rates of 10,059 claims). Extrapolation has already played a role in the instant case. Only four of the 103 balloons expelled by Shonubi were tested for heroin. Yet it is highly probable that all 103 balloons contained heroin mixture at a purity of roughly 53 percent. See People v. Argro, 37 N.Y.2d 929, 930, 379 N.Y.S.2d 840, 841, 342 N.E.2d 601, 602 (1975) (approving toxicologist’s procedure of choosing envelopes of heroin at random from a large batch to estimate total weight seized), approved, People v. Hill, 85 N.Y.2d 256, 624 N.Y.S.2d 79, 648 N.E.2d 455 (1995) (statistical sampling may be used to determine quantity of drugs). Sampling, as practiced in Hill, is common in drug cases. In the Second Circuit, where sentences are not reduced in accordance with reasonable belief, see, e.g., Fred A. Bernstein, The Denigration of Mens Rea in Drug Sentencing, 7 Fed.Sent.Rep. 121 (1994), sampling is often the basis of drug quantity determinations. Conclusions based on extrapolation require assumptions about human behavior (i.e., no one carries sugar in balloons from Nigeria to New York) and about statistics (i.e., 4 balloons chosen at random out of 103, when there is no ostensible reason for variation, is a sufficient sample). Moreover, the trier’s sense of “story,” see infra Part VLB.4, suggests that the balloons were all filled from the same batch of heroin, using the same filling technique. This combination of thought processes permits the rejection as highly unlikely of any scenario in which the 103 balloons did not contain heroin in roughly similar amounts. The same combination of beliefs about how people act in the real world and assessments of probability may permit a finding that each of Shonubi’s eight trips involved similar quantities of heroin. This is true despite the differences between a random sample — the choice of four out of 103 balloons — and the court’s non-random reliance on the eighth trip as a basis for conclusions about earlier offenses. This matter is discussed at length in Part XII.B, infra. Understood this way, the question before the court is how likely it is, based on all available information, that the seven smuggling trips about which little is known were similar to the eighth, about which a great deal is known. The sentencing judge had previously concluded that the defendant’s first seven trips were substantially like the trip leading to his arrest. This memorandum explores the bases for that conclusion. See infra Parts V through XII. IV. Sentencing Law A. Burden of proof 1. Generally In a judicial proceeding, “all the factfinder can acquire is a belief of what probably happened.” In re Winship, 397 U.S. 358, 370, 90 S.Ct. 1068, 1075-76, 25 L.Ed.2d 368 (1970) (Harlan, J., concurring) (emphasis in original). For that belief to become a decision, the court must utilize a burden of proof. [A] standard of proof represents an attempt to instruct the factfinder concerning the degree of confidence our society thinks he should have in the correctness of factual conclusions. Id.; see also, e.g., Ronald J. Allen, Burdens of Proof, Uncertainty, and Ambiguity in Modern Legal Discourse, 17 Harv.J.L. & Pub.Pol’y 627, 633 (1994) (burden of proof “provides the decision rule under uncertainty”). Courts do not generally quantify burdens of proof as a matter of law. See John M. Maguire, James H. Chadbourn, et al., Cases and Materials on Evidence 871-73 (6th ed. 1973) (collecting literature on quantification of burdens of proof); see also Richard H. Gaskins, Burdens of Proof in Modern Discourse 20 (1992) (observing that implications of burdens of proof have been “neglected” in legal discourse). But see Terence Anderson & William Twining, Analysis of Evidence 338 (1991) (correlating verbal and mathematical measures of certainty and doubt). Despite the absence of explicit quantification, the phrases ‘preponderance of the evidence’ and ‘proof beyond a reasonable doubt’ ... communicate to the finder of fact different notions concerning the degree of confidence he or she is expected to have in the correctness of [his or her] conclusions. In re Winship, 397 U.S. at 370, 90 S.Ct. at 1076. Judges — who deal with burdens of proof on a daily basis — are inclined to think of those burdens in probabilistic terms. A survey of judges in the Eastern District of New York found general; agreement that “a preponderance of the evidence” translates into 50 + percent probability. Eight judges estimated “clear and convincing” as between 60 and 70 percent probable (while two found this standard unquantifiable). Estimates for “beyond a reasonable doubt” ranged from 76 to 90 percent, with 85 percent the modal response. See United States v. Fatico, 458 F.Supp. 388, 410 (E.D.N.Y.1978) (chart), aff'd on other grounds, 603 F.2d 1053 (2d Cir.1979), cert. denied, 444 U.S. 1073, 100 S.Ct. 1018, 62 L.Ed.2d 755 (1980); C.M.A. McCauliff, Burdens of Proof: Degrees of Belief, Quanta of Evidence, or Constitutional Guarantees?, 35 Vand.L.Rev. 1293 (1982); Vern R. Walker, Direct Inference in the Lost Chance Cases: Factfinding Constraints Under Minimal Fairness to Parties, 23 Hofstra L.Rev. 247, 257 (1994) (“A number of courts have ... explained the meaning of preponderance ... using the quantitative terminology of mathematical probability.”). As Fatico demonstrated, there is a consensus among judges that burdens of proof can be stated in numerical terms. Moreover, with some variation, there is agreement as to what those numbers are. 2. Burden of proof at sentencing At sentencing, a defendant is entitled to a full evidentiary hearing. See United States v. Fatico, 458 F.Supp. 388, 409-11 (E.D.N.Y.1978), aff'd, 603 F.2d 1053 (2d Cir.1979), cert. denied, 444 U.S. 1073, 100 S.Ct. 1018, 62 L.Ed.2d 755 (1980); United States v. Naugle, 879 F.Supp. 262 (E.D.N.Y.1995) (Appendix) (listing defendant’s rights at sentencing proceeding). The government must prove disputed facts by a preponderance of the evidence. See, e.g., United States v. Concepcion, 983 F.2d 369, 388 (2d Cir.1992), cert. denied, — U.S. -, 114 S.Ct. 163, 126 L.Ed.2d 124 (1993); see also U.S.S.G. § 6A1.3 (1994) (commentary) (endorsing preponderance standard). A fact has been proven by a preponderance of the evidence, according to the court of appeals, if “ ‘the scales tip, however slightly, in favor of the party with th[e] burden of proof.’” Ostrowski v. Atlantic Mut. Ins. Cos., 968 F.2d 171, 187 (2d Cir.1992) (quoting Leonard B. Sand et al., Modern Federal Jury Instructions ¶ 73.01, at 73-4 (1992)). See also Herman & MacLean v. Huddleston, 459 U.S. 375, 390, 103 S.Ct. 683, 691-92, 74 L.Ed.2d 548 (1983) (preponderance standard allows “both parties to share the risk of error in roughly equal fashion”); McCormick, Evidence § 339, at 794 (2d ed. 1972) (preponderance standard requires proof that “the existence of the contested fact is more probable than its nonexistence”). Although these definitions derive from civil cases, the courts have not defined the standard differently in criminal litigation. See, e.g., United States v. Gigante, 39 F.3d 42, 47 (2d Cir.1994) (in sentencing, “preponderance standard is ... a tie-breaker dictating that when the evidence on an issue is evenly balanced, the party with the burden of proof loses”) (citing Sand’s civil instruction). Commentators have argued that a preponderance standard provides the defendant with inadequate protection at sentencing under the Guidelines. See, e.g., Joseph P. Sargent, Comment, The Standard of Proof Under the Federal Sentencing Guidelines: Raising the Standard to Beyond a Reasonable Doubt, 28 Wake Forest L.Rev. 463 (1993); W. Crews Lott, Balancing the Burden of Proof and Relevant Conduct: At What Point Is Due Process Violated?, 45 Baylor L.Rev. 877 (1993); Richard Husseini, Note, The Federal Sentencing Guidelines: Adopting Clear and Convincing Evidence as the Burden of Proof, 57 U.Chi.L.Rev. 1387 (1990); Susan N. Herman, The Tail That Wagged the Dog: Bifurcated Fact-Finding Under the Federal Sentencing Guidelines and the Limits of Due Process, 66 S.Cal.L.Rev. 289, 309 (1966) (advocating burden higher than preponderance). Several appellate courts have expressed concern about the burden of proof at sentencing. See, e.g., United States v. Restrepo, 946 F.2d 654, 659-60 (9th Cir.1991) (en banc) (suggesting that a clear and convincing standard be applied to facts that will dramatically increase a sentence), cert. denied, 503 U.S. 961, 112 S.Ct. 1564, 118 L.Ed.2d 211 (1992); United States v. Townley, 929 F.2d 365, 370 (8th Cir.1991) (same); see also United States v. Sepulveda, 15 F.3d 1161, 1198 (1st Cir.1993) (that “drug quantity has a dramatic leveraging effect [under the Guidelines] ... informs the preponderance standard, requiring that ... where uncertainty reigns, [district courts] must ‘err on the side of caution’ ”) (quoting United States v. Sklar, 920 F.2d 107, 113 (1st Cir.1990)), cert. denied, — U.S. -, 114 S.Ct. 2714, 129 L.Ed.2d 840 (1994). Cf. O’Neal v. McAninch, — U.S. -, -, 115 S.Ct. 992, 993, 130 L.Ed.2d 947 (1995) (rejecting the application of a civil burden of proof to a habeas action, although it is “technically” a civil proceeding, because “the errors being considered by a habeas court occurred in a criminal proceeding [where] ... someone’s custody, rather than mere civil liability, is at stake”) (emphasis in original). In the Second Circuit, the Chief Judge has stated that “a strong argument can be made that the ‘clear and convincing evidence’ standard should be used, at least for substantial enhancements [under the Guidelines].” See United States v. Concepcion, 983 F.2d 369, 394 (2d Cir.1992) (Newman, J., concurring), cert. denied, — U.S. -, 114 S.Ct. 163, 126 L.Ed.2d 124 (1993). Another panel of the court of appeals has suggested that, at the very least, “there is a constitutional requirement of some rough proportionality between the weight of the evidence of ... uncharged conduct and the degree of adjustment or departure [to which that evidence leads].” United States v. Gigante, 39 F.3d 42, 47 (2d Cir.1994). Although it noted that “the preponderance test continues to govern,” id. at 48, the Gigante court suggested that some correspondence between the evidentiary standard and the consequence of the fact in question is warranted: We find it difficult to believe that a district judge who had to resort to a tie-breaking rule to find facts justifying an upward departure would actually depart in any substantial way. We also believe that an appellate court would and should take the weight of the evidence into account in reviewing the reasonableness of a departure. Id. The court further warned of “the danger of substantial error [that] would permeate a substantial upward departure based on a finding of, say, six uncharged crimes ... proven by a bare preponderance,” id. at 47, and noted that this problem is particularly acute under the Guidelines, which “mandat[e] that [judges] take uncharged conduct into account.” Id. at 48. In light of these concerns, many sentencing judges employ a sliding scale, adjusting burdens of proof in proportion to the significance of the fact at issue. See United States v. Fatico, 458 F.Supp. 388, 409-11 (E.D.N.Y.1978) (surveying sentencing practices of judges in the Eastern District of New York), aff'd on other grounds, 603 F.2d 1053 (2d Cir.1979), cert. denied, 444 U.S. 1073, 100 S.Ct. 1018, 62 L.Ed.2d 755 (1980); see also A Trial Judge’s Second Impression of the Federal Sentencing Guidelines, 66 S.Cal.L.Rev. 357, 360 (1992) (surveying judges). This court follows the Second Circuit practice— described most recently in Gigante — of assessing the weight of evidence in light of its impact on the sentence. B. Operation of the Guidelines Sentencing for nearly all federal crimes is controlled by the Guidelines. United States v. DeRiggi 45 F.3d 713 (2d Cir.1995); see also United States Sentencing Commission, Guidelines Manual 10 (Nov. 1, 1994) (Guidelines apply to 90 percent of federal crimes) [hereinafter Guidelines Manual ]. In creating the Guidelines, the Sentencing Commission relied on “statistical analyses” of “40,000 convictions [and] ... 10,000 augmented pre-sentence reports.” Guidelines Manual at 10. See also Stephen G. Breyer, The Federal Sentencing Guidelines and the Key Compromises Upon Which They Rest, 17 Hofstra L.Rev. 1, 7 n. 50 (1988) (describing computer program used by the Commission to analyze data about 100,000 “criminal dispositions”); Reena Raggi, Foreword to Tony Garoppolo, The Sentencing Reform Act: A Guide for Defense Counsel at v (2d ed. 1994) (describing Guidelines as a “statistically based” system). The goal was to tie the Guidelines ranges to “typical, or average ... past practice.” Breyer, supra, 17 Hofstra L.Rev. at 17. In this respect, the Guidelines are at heart probabilistic. The centerpiece of the Guidelines system is a one-page “Sentencing Table.” The table resembles a tax chart. Its vertical axis lists “offense levels,” numbered from 1 to 43; its horizontal axis lists “criminal history categories,” from I to VI. The meeting point of offense level and history category determines the applicable sentencing range. Both horizontally and vertically, each range “overlap[s] ... the ranges in the preceding and succeeding levels.” Guidelines Manual at 9. The objective of overlapping ranges, according to the Sentencing Commission, was to “discourage unnecessary litigation.” Id. For example, the Commission explained, “little purpose will be served in ... trying to determine ... whether $10,000 or $11,000 was obtained as a result of fraud.” Id. at 9-10. Instead of discouraging litigation, the Guidelines have, of course, generated tens of thousands of appeals, required resentencing in a large number of cases, and caused unacceptable distortions, disparities, and irrationalities in sentencing. See, e.g., Steven L. Chanenson, Consistently Inconsistent: Circuits Rulings on the Guidelines in 1991, 7 Fed.Sent.Rep. 224, 225 (1995) (in 1994, 8,400 cases—some 59.5% of all federal criminal appeals — presented Guidelines issues). The proliferation of appeals and reversals illustrates the dangers of using statistical data for substantive law-making purposes. This use is quite different from the function of statistics in the instant and like cases as a means of improving fact-finding in litigation. 1. Drug cases In drug cases, offense level is based on quantity, possessed, sold, or imported. U.S.S.G. § 2D1.1(c) (1994) (drug quantity table). For example, possession of more than 1,000, but less than 3,000, grams of heroin is a Level 32 offense. Within such ranges, estimation is encouraged. However, at the margins, a difference of a few grams can make a large difference in prison time. See, e.g., United States v. Rivera, 821 F.Supp. 868, 869 (E.D.N.Y.1993) (describing effects on sentence of possible errors in weighing); cf. United States v. Ekwunoh, 12 F.3d 368 (2d Cir.1993) (applying mandatory minimum sentencing statute requiring 10 years in prison for possession of more than 1,000 grams to offense involving 1,013 grams); modified on remand, 888 F.Supp. 369 (E.D.N.Y.1994). When the amounts are close to a critical step-up, at, for example, 1,000 or 3,000 grams, the possibility of a substantial increase in sentence suggests the need for extreme care in estimation. Under the “modified real offense” system adopted by the Sentencing Commission, relevant conduct, in drug cases, is not limited to the offense of conviction. See Guidelines Manual at 4-5. Instead, the quantity of drugs includes any amounts “that were part of the same course of conduct” or “common scheme or plan.” U.S.S.G. § 1B1.3(a)(2) (1994); United States v. Madkour, 930 F.2d 234, 237 (2d Cir.), cert. denied, 502 U.S. 911, 112 S.Ct. 308, 116 L.Ed.2d 251 (1991); United States v. Cousineau, 929 F.2d 64, 67 (2d Cir.1991). In the Second Circuit, inclusion of such quantities is mandatory. See, e.g., United States v. Schaper, 903 F.2d 891, 898 (2d Cir.1990) (reversing trial judge’s decision to base sentence solely on quantity seized); United States v. Telesco, 962 F.2d 165, 168 (2d Cir.1992) (such quantities “must be considered in determining the base offense level”) (emphasis in original); United States v. Vazzano, 906 F.2d 879, 882 (2d Cir.1990) (same; citing, inter aha, William J. Wilkins, Jr. & John R. Steer, Relevant Conduct: The Cornerstone of the Federal Sentencing Guidelines, 41 S.C.L.Rev. 495, 513-17 (1990)). Given their obligation to consider drugs from transactions that predate the offense of conviction — in some cases by over a year— judges must sometimes estimate the weight of drugs that neither the court nor the gov-eminent has seen. See, e.g., United States v. Santiago, 906 F.2d 867, 872 (2d Cir.1990) (offenses 14 months apart included in quantity computation). In such cases, the Guidelines offer this instruction: Where there is no drug seizure or the amount seized does not reflect the scale of the offense, the court shall approximate the quantity of the controlled substance. U.S.S.G. § 2D1.1, application note 12 (1994) (emphasis added). The Guidelines give examples of the kinds of evidence on which approximations may be based: [T]he court may consider, for example, the price generally obtained for the controlled substance, financial or other records, similar transactions in controlled substances by the defendant, and the size or capability of any laboratory involved. Id., application note 12 (emphasis added). This non-exclusive list is consistent with the rule that sentencing judges may consider any evidence that has “sufficient indicia of reliability to support its probable accuracy.” U.S.S.G. § 6A1.3 (1994); see also 18 U.S.C. § 3661 (“[N]o limitation shall be placed on the information concerning the background, character, and conduct” of the defendant which the court may consider in sentencing). For many drugs, including marijuana, mescaline, and PCP, the Guidelines provide farther instructions for estimation. If the number of doses, pills, or capsules but not the weight of the controlled substance is known, multiply the number of doses, pills or capsules by the typical weight per dose in the table below.... The Typical Weight Per Unit Table, prepared from information provided by the Drug Enforcement Administration, displays the typical weight per dose, pill or capsule for certain controlled substances. Do not use this table if any more reliable estimate of the total weight is available from case-specific information. U.S.S.G. § 2D1.1, application note 11 (1994) (emphasis added). The Guidelines thus provide for estimation based on the activities of typical offenders (as determined by the Drug Enforcement Administration) when “ease-specific” evidence is lacking. Similarly, in cases involving clandestine laboratories “in which the manufacture of a controlled substance has not been completed ... the court must estimate the amount of controlled substance that would have been manufactured....” 60 Fed.Reg. 2430, 2456 (Jan. 9, 1995) (construing U.S.S.G. § 2D1.1 (1994)). For this purpose, the Drug Enforcement Administration provides a formula for estimating theoretical yields from the quantities of precursor chemicals seized. Because actual yield depends on a multiplicity of factors, according to the Sentencing Commission, “[t]he use of theoretical yield frequently [results] in a higher offense level for someone who ... does not produce any controlled substance than for someone who [does].” 60 Fed.Reg. at 2456-57. The Sentencing Commission has proposed adding a rebuttable presumption that actual yield is half of theoretical yield. Id. This amendment, even if it improves the result, will still require courts to impose sentences based on data culled from other cases. 2. Non-drug cases The use of approximations under the Guidelines is not limited to drug cases. Such provisions can be found throughout the Guidelines. For example, in tax evasion cases, the “tax loss shall be treated as equal to 28% of the unreported gross income ... unless a more accurate determination ... can be made.” U.S.S.G. § 2T1.1(c)(1)(A) (1994) (emphasis added). Similarly, in sentences for fraud, the loss need not be determined with precision. The court need only make a reasonable estimate of the loss, given the available information. This estimate, for example, may be based on the approximate number of victims and an estimate of the average loss to each victim, or on more general factors, such as ... the revenues generated by similar operations. U.S.S.G. § 2F1.1, application note 8 (1994) (emphasis added); see also U.S.S.G. § 2B1.1, application note 3 (1994) (permitting estimation of loss in theft and embezzlement cases). See United States v. Stanley, 54 F.3d 103, 106 (2d Cir.1995) (approving “reasonable” calculation under § 2F1.1 because “[t]he Guidelines require no more”); see also, e.g., United States v. Whitlow, 979 F.2d 1008 (5th Cir.1992) (estimating loss in odometer rollback case); United States v. Hernandez, 952 F.2d 1110 (9th Cir.1991), cert. denied, — U.S. -, 113 S.Ct. 334, 121 L.Ed.2d 252 (1992) (estimating loss in audiotape counterfeiting ease). In Hernandez, the court rejected a method of loss calculation, suggested by the defendant, which would have “required detailed information and complex ... calculations.” Id. at 1119. The court held that “the Guidelines do not require such precision.” Id. In short, the Guidelines recognize that precise quantification required may be impossible in many cases. In such instances the trier, relying on practical good sense and experience, must estimate from the best available information, including information culled from similar cases. C. Caselaw on estimation and specific evidence In reversing the sentencing judge’s finding in the instant ease, the court of appeals stated that estimates of drug quantity must be based on “specific evidence.” Shonubi II, 998 F.2d at 89. As examples of such evidence it listed “drug records, admissions or live testimony” — that is, written or spoken statements. Id. This rule, it asserted, is a “careful practice” necessitated by a system in which drug quantity determinations can significantly affect terms of incarceration. 998 F.2d at 89-90. 1. Estimation The court of appeals pointed to six cases that illustrate this “careful practice”: United States v. Lasanta, 978 F.2d 1300, 1309 (2d Cir.1992); United States v. Rivera, 971 F.2d 876, 892-93 (2d Cir.1992); United States v. Burnett, 968 F.2d 278, 280 (2d Cir.1992); United States v. Colon, 961 F.2d 41, 43 (2d Cir.1992); United States v. Santiago, 906 F.2d 867, 871-72 (2d Cir.1990); and United States v. Schaper, 903 F.2d 891, 896-97 (2d Cir.1990). Examination of the evidence and inferences relied on in these cases is instructive. In Lasanta, a government analysis of wiretapped conversations led to a finding that the defendant had sold 14.7205 kilograms of cocaine and .336 kilograms of heroin. The court adopted these amounts after reviewing the conversations. By “converting” heroin to cocaine (using the “drug equivalency table” of U.S.S.G. § 2D1.1 and the conversion practices described in application notes 6 and 10 of that section), the court determined that Lasanta had sold 16.4005 kilos of “cocaine equivalent.” This, it determined, was a “level 34” offense, which requires possession of “[a]t least 15 KG but less than 50 KG of Cocaine.” See U.S.S.G. § 2D1.1 (1994). The court of appeals affirmed. 978 F.2d at 1309. In Rivera, the sentencing court found that defendants who worked at one of a conspiracy’s five “distribution points” were responsible for quantities estimated to have been sold at the other four points. Finding that the defendants’ familiarity with the “management team to which they reported” and knowledge of the sophisticated wrapping of the' heroin packets, along with the fact that they had attended a party with eo-conspira-tors, indicated awareness of the scope of the conspiracy, the court affirmed the sentence. 971 F.2d at 893. In Burnett, a government witness testified that he had sold the defendant “[t]wo kilos, three kilos” of cocaine. 968 F.2d at 280. He later “clarified” his testimony, indicating that he had sold three kilos to the defendant. Id. The court sentenced the defendant to a term based on possession of more than three kilograms of cocaine. Deferring “to [the] sentencing judge’s credibility determinations,” the court of appeals affirmed. 968 F.2d at 280. In Colon, the defendant told a probation officer that he had sold 80 glassine envelopes of heroin every two or three days for a “few years.” The court estimated, “conservatively,” that the defendant must have sold 8,000 envelopes. It multiplied this figure by the average estimated amount of heroin (.05 grams) in the 149 bags that had been seized, arriving at a gross weight of 400 grams. The court of appeals affirmed. 961 F.2d at 43. In Santiago, the defendant admitted that he had made five heroin sales to an informant named Shattuck. Shattuck, in turn, testified that he bought heroin from Santiago on “approximately a dozen occasions.” 906 F.2d at 869 (emphasis added). The court multiplied the amount found in one bag of heroin by five and twelve to arrive at low and high estimates — either of which, it noted, would translate into a level 26 offense. The court of appeals held that a finding “that Santiago had in fact made a dozen prior sales to Shattuck was amply supported.” Id. at 872. In Schaper, the sentencing court considered only amounts seized. The appeals court remanded for estimation of drug quantity above the amount known to have been possessed. 903 F.2d at 899. A few “rules” can be derived from this series of cases. The court of appeals will affirm a sentence based on an estimated quantity 50 times greater than the amount actually seized. Colon. It will accept an estimate that pushes a defendant just over the “line” into a higher offense level. Lasan-ta. It will accept figures that, in their apparent precision, mask the inexactitude of human interactions. Lasanta (analysis of phone conversations translates into finding that 14.7205 kilograms of cocaine was sold). It will permit estimates based on judges’ inferences about other people’s inferences about what third parties are doing. Rivera. It will base sentences on quantity estimates proffered by witnesses whose statements are inconsistent, Burnett, or equivocal. Santiago. Finally, it will accept, without discounting, estimates of drug quantity based on statements of witnesses whose credibility is doubtful. Burnett. Referring to the six cases, the court of appeals stated that “[c]ase law uniformly requires” such “specific evidence” for quantity determinations. Shonubi II, 998 F.2d at 89. The practice, however, is less “uniform” than the court believes. In United States v. Mickens, 926 F.2d 1323 (2d Cir.1991), cert. denied, 502 U.S. 1060, 112 S.Ct. 940, 117 L.Ed.2d 111 (1992), the sentencing court “approximated that [a] conspiracy distributed in excess of fifty kilograms of cocaine,” a finding it “based on [the defendant’s] unexplained income of over $2,000,000 during the operation of the conspiracy.” Id. at 1331. The court of appeals held that this estimate — based upon inferences from financial records — was “proper.” Id. at 1332; see also United States v. Jacobs, 955 F.2d 7, 9 (2d Cir.1992) (endorsing estimation procedure used in Miekens); cf. United States v. Perrone, 936 F.2d 1403, 1419 (2d Cir.), clarified, 949 F.2d 36 (2d Cir.1991) (disapproving estimation because “[t]here is no evidence here of any conversations about the amount of drugs, ... no records of past sales, no money”) (emphasis added). In drug manufacturing cases, the court has permitted estimation based on seized chemical bottles whose contents were discarded before testing. See, e.g., United States v. Macklin, 927 F.2d 1272 (2d Cir.), cert. denied, 502 U.S. 847, 112 S.Ct. 146, 116 L.Ed.2d 112 (1991). In Macklin, the court found an estimate based primarily on the bottles’ labels to be “thoroughly reasonable,” id. at 1282, despite significant testimony that the government’s expert had overstated the laboratory’s capability. By contrast, in Perrone, 936 F.2d at 1403, the court rejected an estimate of the amount of cocaine a laboratory could produce. Distinguishing Macklin, the Perrone court noted that neither cocaine base nor hydrochloric acid — two crucial ingredients — had been found in the laboratory. 936 F.2d at 1419. The court was careful to note, however, that “we are not holding that a court can never reach a reliable estimate as to quantity on the basis of some drug ingredients if other drug ingredients are missing,” especially “[i]f the missing ingredients are demonstrated to be readily available to the defendant.” Id.; see also United States v. Pirre, 927 F.2d 694, 696-97 (2d Cir.1991) (upholding sentence based on possession of more than 15 kilograms of cocaine, when chemist estimated that the offense involved 15.09 kilograms of cocaine, then discarded much of the evidence, in order to save space, prior to trial). In another estimation case, the court relied heavily on the defendant’s phone bill. See United States v. Prescott, 920 F.2d 139, 142 (2d Cir.1990) (in finding that defendant distributed 150 kilograms of cocaine, “[pjerhaps most telling was the fact that in the two and one-half months that [he] owned a cellular telephone ... telephone company records revealed a total of 12,611 incoming and outgoing calls in 79 days or nearly 160 calls a day on average”). Other circuits have followed the Mickens approach, translating sums of money into quantities of drugs. See, e.g., United States v. Ortiz-Martinez, 1 F.3d 662, 675 (8th Cir.1993) (approving estimate of cocaine quantity by dividing sum of money by cost of cocaine), cert. denied, — U.S. -, 114 S.Ct. 355, 126 L.Ed.2d 319 (1993); United States v. Duarte, 950 F.2d 1255, 1265 (7th Cir.1991) (same), cert. denied, — U.S. -, 113 S.Ct. 174, 121 L.Ed.2d 120 (1992); United States v. Jackson, 3 F.3d 506, 511 (1st Cir.1993) (same; large amounts of cash in proximity to drug supply can be assumed to represent drug profits). They have also utilized the Macklin technique for estimating the capacity of laboratories. See, e.g., United States v. Evans, 891 F.2d 686, 687-88 (8th Cir.1989) (sentence may be based on amount defendant is capable of producing), cert. denied, 495 U.S. 931, 110 S.Ct. 2170, 109 L.Ed.2d 499 (1990); United States v. Smallwood, 920 F.2d 1231, 1236-37 (5th Cir.) (same, despite findings that facility was not operational and that several “necessary precursor [chemicals]” were missing), cert. denied, 501 U.S. 1238, 111 S.Ct. 2870, 115 L.Ed.2d 1035 (1991). 2. Estimates based on extrapolation Estimates based on extrapolation from known events have been approved in a number of eases. For example, in United States v. McMillen, 8 F.3d 1246, 1249-51 (7th Cir.1993), cert. denied, — U.S. -, 114 S.Ct. 1649, 128 L.Ed.2d 368 (1994), the defendant was a member of a conspiracy for eight weeks, during which approximately 34 heroin sales took place. An average of 39 grams of heroin changed hands in each transaction. The trial court, with some rounding, multiplied the number of transactions by the average size of the transactions to arrive at a total quantity of 1,280 grams. The court of appeals approved the extrapolation: Because these transactions were similar in nature, the court inferred that these transactions involved a similar quantity of heroin. It was the district judge’s function to ... make a finding supported by a preponderance of the evidence.... 8 F.3d at 1250-51. Similarly in United States v. Thomas, 12 F.3d 1350, 1369 (5th Cir.), cert. denied, — U.S. -, -, 114 S.Ct. 1861, 2119, 128 L.Ed.2d 483, 676 (1994), ledgers showed that 56 kilograms of cocaine were distributed during one third of the duration of a conspiracy. The court multiplied by three to arrive at a sentence based on “more than 150 kilograms” of cocaine. In United States v. Sklar, 920 F.2d 107 (1st Cir.1990), the defendant was arrested after mailing a package of cocaine. His relevant conduct included sending eleven other packages, none of which was seized. The government knew the weights of the eleven packages. It multiplied those weights by the percentage of cocaine (by weight) of the seized package, rounded down. The court of appeals affirmed “this conservative approach,” noting that, by rounding down, the court had “insulated the challenged finding from clear-error attack.” 920 F.2d at 113. United States v. Hilton, 894 F.2d 485 (1st Cir.1990), involved a sailboat which caught fire during a mid-ocean interception by government agents. A member of the boarding party saw ten packages in the bilge and retrieved one of them; he observed twelve additional packages floating away. The retrieved bundle had a gross weight of 14 pounds and was found to contain marijuana. The sentencing court “concluded that it was reasonable to assume that the remaining twelve packages each contained at least 10 pounds of marijuana, notwithstanding that they were never seized.” 894 F.2d at 486. The court of appeals affirmed, “agree[ing] with the district court that the evidence offered by the government ... satisfies the sufficiency test, even beyond a reasonable doubt.” Id. at 488 (emphasis added). Such cases follow Guidelines policy by basing estimates on “similar transactions in controlled substances by the defendant.” U.S.S.G. § 2D1.1, application note 12 (1994) (emphasis added). Such estimates have been disapproved where the court overlooks evidence that the defendant’s prior transactions were not similar to the offense of conviction. In United States v. Zimmer, 14 F.3d 286 (6th Cir.1994), the defendant possessed 802 marijuana plants at the time of arrest; he also admitted to having grown marijuana for seven or eight years. Based on this evidence, the district court found that the defendant had produced an “absolute minimum” of 200 additional plants over the last eight years — for a total of 1,000 plants. Id. at 289. The court of appeals reversed, finding that “the court’s determination that the defendant grew an additional 200 plants is not