JON 0. NEWMAN, Chief Judge:

This appeal concerns the quality of evidence that may support a finding of uncharged criminal conduct relied upon by a sentencing judge to increase a sentence beyond the punishment appropriate for the offense of conviction. The issue arises in the frequently litigated context of determining the quantity of drugs attributable to á defendant for purposes of sentencing under the “relevant conduct” provision of the Sentencing Guidelines. See U.S.S.G. § 1B1.3. Charles 0. Shonubi appeals from the sentence of the District Court for the Eastern District of New York (Jack B. Weinstein, Judge), resentencing him to a prison term of 151 months after the same sentence had been vacated by this Court and the case remanded for resentencing. See United States v. Shonubi, 998 F.2d 84 (2d Cir.1993) (Shonubi II). We conclude that the record lacks the “specific evidence,” id. at 89, required by our prior decision to support punishment for drug quantities sought to be attributed to the defendant, over and above the quantity for which he was convicted. We therefore vacate the sentence and remand with directions to impose a sentence based on the 427.4 grams of heroin that the defendant was convicted of bringing into this - country from Nigeria, secreted in his gastrointestinal tract.

Background

The basic facts of the offense were set forth in our prior opinion and need only be summarized here. Shonubi was arrested on December 10, 1991, at JFK International Airport after flying from Lagos, Nigeria, by way of Amsterdam. He aroused the suspicion of a customs inspector, who obtained Shonubi’s consent to an X-ray examination. The X-ray revealed foreign bodies in Shonubi’s digestive tract. During a two-day detention, Shonubi passed 103 balloons containing 427.4 grams of heroin.

Shonubi was charged with importing heroin, in violation of 21 U.S.C. §§ 952, 960, and possession of heroin with intent to distribute, in violation of 21 U.S.C. § 841. Both offenses were alleged to have occurred on December 10, 1991, the date of the arrest. A jury found Shonubi guilty of both offenses. At the initial sentencing, Judge Weinstein found that, in addition to the trip that occasioned Shonubi’s arrest, he had made seven other trips to Nigeria between September 1990 and December 1991. The District Judge, implicitly finding that these seven trips had been made for the purpose of importing heroin, multiplied by eight the 427.4 grams-imported on December 10, 1991, and imposed a sentence based on a quantity of 3,419.2 grams. United States v. Shonubi 802 F.Supp. 859 (E.D.N.Y.1992) (Shonubi I).

The drug quantity table of the Sentencing Guidelines establishes 28 as the base offense level for a quantity of heroin between 400 and 700 grams, U.S.S.G. § 2Dl.l(c)(6), and level 34 for a quantity between 3 and 10 kilograms, id. § 2Dl.l(c)(3). Shonubi’s base offense level calculated only with regard to the 427.4 grams brought in on the day of his arrest would have been 28. His base offense level calculated by the District Court for the extrapolated quantity of 3,419.2 grams was 34. In Criminal History Category I, level 28 would have yielded a sentencing range of 78 to 97 months; at level 34, the range was 151 to 188 months. Judge Weinstein declined to make a two-level enhancement of the base offense level for obstruction of justice, id. § 3C1.1, despite finding that Shonubi had committed perjury at his trial. The District Court imposed a sentence of 151 months, plus supervised release.

On the first appeal, we ruled that prior case law “uniformly requires specific evi dence — e.g., drug records, admissions or live testimony- — to calculate drug quantities for sentencing purposes.” Shonubi II, 998 F.2d at 89. Concluding that such evidence was not contained in the record and that multipli *1088 cation of the quantity seized on the night of the arrest by the total number of trips was an inadequate substitute for the required “specific evidence,” we vacated the sentence and remanded for resentencing. We also ruled in the Government’s favor on its cross-appeal from the denial of the two-level enhancement for obstruction of justice. Id. at 87-88.

On remand, Judge Weinstein conducted an elaborate hearing. He took testimony from a Government expert on statistics, a defense expert on statistics, and a panel of two statistics experts, appointed by the Court pursuant to Fed.R.Evid. 706. He also received reports of heroin quantities seized from 117 Nigerian heroin swallowers arrested at JFK Airport during the same time period that spanned Shonubi’s eight trips. In addition, he surveyed the federal judges of the Eastern District to obtain their opinions concerning heroin swallowers. 1 Based on the record made at an extensive hearing, Judge Weinstein then wrote an elaborate opinion of 177 typescript pages to support his finding that Shonubi had carried between 1,000 and 3,000 grams of heroin during the eight trips. United States v. Shonubi, 895 F.Supp. 460 (E.D.N.Y.1995) (Shonubi III). That quantity translated into a base offense level of 32, U.S.S.G. § 2D1.1.(c)(4) (at least one kilogram but less than three kilograms of heroin). The two-level enhancement required by our prior decision increased the offense level to 34, yielding the same sentencing range applicable at the original sentencing, 151 to 188 months. The original sentence of 151 months, plus supervised release, was re-imposed.

Discussion

I. Punishment for “Unconvicted” Conduct

One of the most significant changes effected by the Sentencing Guidelines is the prescription of precisely calibrated punishment for conduct of which the defendant has not been convicted. Prior to the Guidelines, the law was settled that a defendant’s wrongful conduct, beyond the conduct constituting the offense of conviction, was relevant to punishment, see Williams v. New York, 337 U.S. 241, 246-47, 69 S.Ct. 1079, 1082-83, 93 L.Ed. 1337 (1949), but the law established no specification of the additional punishment a defendant was to receive for such “unconvicted” conduct. Instead, the sentencing judge was entitled, but not required, to take such conduct into account and enhance the sentence to whatever extent the judge thought appropriate, up to the maximum sentence applicable to the offense of conviction.

The Guidelines introduced a new approach to sentencing for “unconvicted” conduct. Endeavoring to strike a balance between punishing only for the offense of conviction and punishing for all wrongful conduct that could be established at a sentencing hearing, the Guidelines opted for incremental punishment for conduct deemed to be “relevant” to the offense of conviction. U.S.S.G. § 1B1.3. As to such “relevant conduct,” the Guidelines then took the extraordinary and totally unprecedented step of punishing the relevant conduct at precisely the same degree of severity as if the defendant had been charged with arid convicted of the activity constituting the “relevant conduct.” No other guideline system in any of the states has instituted such an approach to punishment.

*1089 The Guidelines’ approach for “relevant conduct” can be illustrated most clearly in the context in which the pending appeal arises — drug offenses. The Guidelines contain a “drug quantity table” to be used for sentencing drug offenders. The table contains 17 gradations, of offense level, each correlated to a different range of quantity for various drugs. Id. § 2Dl.l(e). Each offense level, in turn, correlates to a different range of punishment. For example, for heroin, the quantity ranges, corresponding offense levels, and corresponding punishment ranges (for a defendant with no prior criminal record and without any aggravating or mitigating adjustments) are as follows: to punishment

heroin quantity offense level punishment

<5 grams 12 10-16 months

5 — <10 grams 14 15-21 months

10 — <20 grams 16 21-27 months

20 — <40 grams 18 27-33 months

40 — <60 grams 20 33-41 months

60 — <80 grams 22 41-51 months

80 — <100 grams 24 51-63 months

100 — <400 grams 26 63-78 months

400 — <700 grams 28 78-97 months

700 grams — <1 kilo 30 97-121 months

1 —<3 kilos 32 121-151 months

3 — <10 kilos 34 151-188 months

10 — <30 kilos 36 188-235 months

30 kilos or more 38 235-293 months

Having specified these offense levels and punishments for various heroin quantities, the Guidelines then take the crucial step of specifying that the “relevant conduct”, for a defendant convicted of a drug offense includes all of the additional drugs, beyond the quantity in the offense of conviction, that were unlawfully distributed or possessed with intent to distribute either by the defendant personally or by the reasonably foreseeT able acts of others in furtherance of a jointly undertaken criminal activity. Id, § lB1.3(a)(l), (2).

Thus, a defendant convicted of distributing 50 grams of heroin may be punished with imprisonment of up to 41 months, but if the sentencing judge finds that he (or a confederate acting foreseeably) also distributed an additional 550 grams, the defendant may be punished for the aggregate 600 grams with imprisonment of up to 97 months, precisely the same punishment he could have received if he had been charged with and convicted of distributing the entire quantity of 600 grams. 2

A guideline system that prescribes punishment for unconvicted conduct at the same level of severity as convicted conduct obviously obliges courts to proceed carefully in determining the standards for establishing whether the relevant conduct has been proven. We have recognized the need for such care with regard to the basic issue of the degree of the burden of proof. Thus, though the Sentencing Commission has favored the preponderance-of-the-evidence standard for resolving all disputed fact issues at sentencing, U.S.S.G. § 6A1.3., p.s., comment., we have ruled that a more rigorous standard should be used in determining disputed aspects of relevant conduct where such conduct, if proven, will significantly enhance a sentence. See United States v. Gigante, 94 F.3d 53, 56-57 (2d Cir.1996) (denying petition for rehearing).

A similar concern guided our decision on the prior appeal in this case. Aware of the consequences of a relevant conduct finding as to drug quantities, we invoked the rule from prior case law of our Circuit that, we observed, “uniformly requires specific evidence — e.g., drug records, admissions or live testimony — to calculate drug quantities for sentencing purposes.” Shonubi II, 998 F.2d at 89 (emphasis added) (citations omitted).

II. The “Specific Evidence” Requirement

The “specific evidence” we required to prove a relevant-conduct quantity of drugs for purposes of enhancing a sentence must *1090 be evidence that points specifically to a drug quantity for which the defendant is responsible. By mentioning “drug records” and “admissions” as examples of specific evidence we thought it reasonably clear that we were referring to the defendant — his admissions and records of his drug transactions. And by “live testimony” we were referring to testimony about his drug transactions. Judge Weinstein apparently misunderstood our prior opinion to equate “specific” evidence with “direct” evidence, a consequence that, as he pointed out, Shonubi III, 895 F.Supp. at 478, would preclude all use of circumstantial evidence. However, our identification of drug records as one example of “specific evidence” should have dispelled that misunderstanding since such records are a form of circumstantial evidence. If a defendant’s drug records reflect drug transactions of a specific quantity, that is circumstantial evidence permitting the inference that the defendant has trafficked in that quantity of drugs.

Our approach might fairly be criticized on the arguable ground that since sentencing facts need normally be established only by a preponderance of the evidence, except where Gigante is applicable, a rigorous standard concerning the quality of evidence should not be applied in a context where the degree of persuasion required is reduced.

This argument does not persuade us to abandon a safeguard adopted in response to the Sentencing Commission’s insistence that a defendant should be punished for unconvicted “relevant conduct” exactly as if he had been convicted of such conduct. In sentencing proceedings, as they existed before the Sentencing Guidelines, and still exist throughout all of the states of this Nation, we tolerate a wide ranging scope of evidence to prove related criminal conduct, even with the reduced “preponderance of the evidence” standard of proof, because such evidence is generally available only for such consideration as the sentencing judge (or in some states the sentencing jury) decides to accord to it. But under the Sentencing Guidelines, evidence tending to prove “relevant conduct” is not merely taken into consideration at sentencing, it determines sentencing (subject only to departure authority), and it does so at the same level of severity as if the defendant had been convicted of the relevant conduct. That circumstance prompted us to require “specific evidence” of a “relevant conduct” drug quantity, and we adhere to that requirement.

If some “specific evidence” of quantity is presented, we do not rule out the possibility that evidence of the sort considered by the District Court might be usefully assessed in determining whether the alleged quantity had been established by a preponderance of the evidence. However, if “specific evidence” of the quantity handled by the defendant (or others for whose acts he is responsible) is available, it is not likely that the time and effort required to conduct probability analyses of quantities carried by other drug couriers would be worthwhile.

III. Is There “Specific Evidence” of a Relevant Conduct Drug Quantity?

Though disapproving of our requirement that the relevant conduct quantity of drugs be based on “specific evidence,” see Shonubi III, 895 F.Supp. at 475-79, the District Court endeavored to apply this requirement. Judge Weinstein acknowledged that we had required “specific evidence” such as drug records, admissions, or live testimony, and identified evidence that he believed met our standard. Id. at 524. For “records” he cited “a combination of drug records (including DEA and Customs Service records) and the records of Shonubi’s trial, sentencing hearing, and presentence report.” Id. For “admissions” he cited Shonubi’s “admissions at the time of his arrest.” Id. For “live testimony” he cited “the statistical analysis introduced on remand as well as testimony on the economics of heroin swallowing.” Id. The Judge said he also relied on Shonubi’s demeanor at trial and sentencing and the Judge’s own “acquired knowledge of the drug trade.” Id.

These items of evidence are not “specific evidence” of drug quantities carried by Shonubi on his prior seven trips. We required specific evidence of what Shonubi had done. The DEA records informed Judge Weinstein *1091 of what 117 other balloon swallowers from Nigeria had done during the same time period as Shonubi’s eight trips. Those records of other defendants’ crimes arguably provided some basis for an estimate of the quantities that were carried by Shonubi on his seven prior trips, but they are not “specific evidence” of the quantities he carried. The defendant’s distinguished expert on statistics, Michael 0. Finkelstein, Esq., correctly informed the District Court that ‘“statistics relating to others would not usually be characterized as “specific evidence” relating to Shonubi.’” Id. at 505 (quoting Finkelstein affidavit at 3). The experts on the Court’s Rule 706 panel rendered the same advice. Id. at 510 (citing Rule 706 Panel report at 45). Though the records of Shonubi’s trial, sentencing hearing, and presentenee report, relate specifically to Shonubi, they do not provide “specific evidence” of the quantities carried on his prior seven trips, any more than they did when these records were before us on the prior appeal. Shonubi’s admissions likewise are “specific” as to him, but contain no “specific evidence” of the quantities carried on his prior trips. The statistical and economic analyses relate to drug trafficking generally and not to Shonubi specifically. 3 The demeanor evidence relates to Shonubi specifically, but, with all respect to the experienced District Judge’s ability to gauge character from a defendant’s demean- or, his conclusions about the defendant’s “extremely low level of risk aversion and an overconfidence in his own powers,” id. at 489, *1092 are not based on “specific evidence” of the quantities carried on the prior seven trips.

Though we conclude that the extrapolation analyses relied on by the District Court do not yield the “specific evidence” that our remand required for determination of the “relevant conduct” quantities carried by Shonubi on his seven prior trips, we are obliged to reckon with the District Court’s point that such extrapolation is defensible because it was accepted for use to estimate the quantity carried by Shonubi on his eighth trip. See id. at 470. As Judge Weinstein pointed out, a forensic chemist had selected at random four of the 103 balloons passed by Shonubi after his arrest, determined the average weight of the heroin contained in these four balloons, and then multiplied that average weight by 103 to conclude that the weight of the heroin contained in all 103 balloons was 427.4 grams. This approach rested on the assumption that the four balloons selected were representative of the entire 103 balloons, an assumption that, in turn, rested on subsidiary assumptions that each of the 103 balloons contained heroin, that the average quantity of heroin in each of the four balloons selected was the same as the average quantity in all of the 103 balloons, and that the average purity of the heroin in the four balloons selected was the same as the purity of the heroin in all of the 103 balloons because all the heroin came from the same batch of heroin.

The short answer to the District Court’s attempt to justify extrapolation to estimate the quantity carried on the seven prior trips because that technique was used to estimate the quantity carried on the eighth trip is that this Court accepted the estimate used for the eighth trip, in the absence of any objection by the appellant, but did not accept an estimate of the quantity for the seven trips, once the appellant specifically raised the issue on the prior appeal.

The further answer is that the seeming inconsistency fails to take account of the different purposes for which the two estimates were made. The estimate of the quantity carried on the eighth trip was made to determine the quantity for the counts on which Shonubi was convicted. The estimate for the prior trips was used to punish Shonubi for conduct of which he had not even been charged, much less convicted. The distinction warrants caution in the use of estimates. Furthermore, the extrapolation as to the eighth trip was based on evidence of what Shonubi had done; the extrapolation for the prior seven trips was based on what 117 other people had done. 4

Conclusion

Judge Weinstein considered this case to be “an opportunity to observe, explain, and discuss forensic decision-making,” Shonubi III, 895 F.Supp. at 464, an opportunity he seized with his customary thoroughness and erudition. Though his comprehensive opinion is a valuable addition to the legal literature on the subject of evidence in particular and judicial decision-making in general, we conclude that he relied on evidence beyond the category of “specific evidence” that our prior opinion ruled was required for determination of a “relevant conduct” drug quantity for purposes of imposing a criminal sentence. In the absence of such “specific evidence,” the finding of a “relevant conduct” quantity must again be rejected. Since the Government has now had two opportunities to present the required “specific evidence” to the sentencing court, no further opportunity is warranted, and the case must be remanded for imposition of a sentence based on the quantity of drugs Shonubi carried on. the night of his arrest, adjusted only by the *1093 previously adjudicated enhancement for obstruction of justice.

The sentence is vacated and remanded for resentencing in conformity with this opinion.

.Judge Weinstein described his inquiry to the district judges as follows:

The judges were asked to rate, "based on their trial and sentencing experience,” their confidence in five hypotheses about heroin swallowers, using a scale of 1 (unlikely) to 5 (likely).... The hypotheses, and the average "scores” received, were:

1. They start with smaller amounts and increase the amounts on later trips.

2.4

2. They start with an amount close to the maximum amount they can cárry and keep carrying that amount until caught.

4.3

3. There is no relation between the trip number and amount.

2.2

4. Those carrying more are more likely to be caught.

2.2

5. First-time smugglers are more likely to be caught.

3.8

Shonubi III, 895 F.Supp. at 511.

. The relevant conduct regime, as constructed by the Sentencing Commission and applied by this Circuit, even permits punishment of the defendant for the entire quantity of drugs, in the example given in the text, if he was convicted of distributing only the 50 grams and acquitted of distributing the additional 550 grams. See United States v. Concepcion, 983 F.2d 369, 386-89 (2d Cir.1992), rehearing in banc denied, id. at 395. But see United States v. Brady, 928 F.2d 844, 851 (9th Cir.1991) (court may not "reconsider facts during sentencing that have been rejected by a jury’s not guilty verdict’’); Concepcion, 983 F.2d at 393 (Newman, J., concurring); id. at 395 (Newman, J., dissenting from denial of rehearing in banc). This Circuit mitigates that rule by permitting the sentencing judge to make a downward departure from a sentencing range calculated on the basis of acquitted conduct. Id. at 389.

. The District Court considered several statistical analyses. The Government's expert, Dr. David Boyum, made two analyses, each based on the DEA's report of 117 balloon swallowers from Nigeria who were arrested at JFK Airport during the time period of Shonubi's eight trips. The first analysis calculated how many of the 117 balloon swallowers in the DEA report carried quantities within 13 100-gram ranges from 0 to 1,300 grams. Shonubi III, 895 F.Supp. at 500 (Table 1). From this classification, Dr. Boyum calculated that the mean net weight was 432.1 grams and the median net weight was 414.5 grams, id. at 502, figures he deemed reasonable to estimate for Shonubi's previous seven trips. Second, Dr. Boyum entered into a computer the weights carried by these 117 smugglers and asked the computer to calculate the sum carried on seven trips, selected at random from the 117. He then asked the computer to repeat this random selection and calculation 100,000 times. Id. at 503 (Chart B). From this process he determined that there was a 75 percent probability that Shonubi carried more than 2712.6 grams on his prior seven trips. Id. at 504.

The District Court’s Rule 706 panel of experts submitted two analyses of their own. As a first step, one of the Rule 706 experts, Prof. David Schum, distributed a pound of powdered sugar into 103 balloons, id. at 506, and also “reflect[ed] on the task of swallowing them,” id. at 507. There is no indication that he carried his investigation to the point of swallowing the balloons. Dr. Schum concluded that the activity of carrying heroin in swallowed balloons involves a learning curve. Next, he constructed two charts, each reflecting quantities Shonubi might have carried on his eight trips. For the first trip he assumed the amount was the smallest amount carried by any of the 117 smugglers from the DEA report. For the last trip he used the quantity Shonubi carried when arrested. The first chart estimated the intervening trips by using an arithmetic progression (increasing the quantities in equal intervals). Id. at 509 (Chart C). The second chart used the same quantities for the first and last trips, but estimated slightly smaller quantities than the first chart for the intervening trips, to reflect a slower learning curve. Id. at 510 (Chart D). The aggregate quantity for the eight trips was 1,930 grams from the first chart and 1,479 grams from the second chart. Id. at 508.

Judge Weinstein also constructed his own “non-Bayesian and non-statistical model.” Id. at 522 (Table 4). First, he estimated the probabilities that Shonubi carried heroin on his eight trips. Id. at 522. He used 99 percent for the eighth trip, 95 percent for the seventh trip, and decreased the probability by .five percentage points for each of the prior trips. Then, he estimated a range of the quantities carried on each trip. These ranges included the quantity recovered from the last trip, but estimated a bottom of the range that diminished with each earlier trip. Id. He then multiplied the estimated probability by the estimated quantity, id., using the bottom of the estimated ranges in order to "favor[] the defendant.” Id. at 523. Finally, he aggregated three different total quantities by including only those quantities from trips where the estimated probability of carrying heroin exceeded a level that Judge Weinstein associated with different burdens of proof — beyond a reasonable doubt, 95 percent + probability; clear and convincing, 70 percent + probability; and preponderance, 50 percent + probability. This yielded total quantities of 752 grams for the two trips .with at least a 95 percent probability of carrying heroin, 1,964 grams for the seven trips with at least a 70 percent probability of carrying heroin, and 2,110 grams for all eight trips (the probability for the first trip was 65 percent). Id.

. Professor Finkelstein pointed out a further distinction, which he advanced in the context of distinguishing between extrapolation from four balloons to the 103 balloons carried by Shonubi on his eighth trip and extrapolation from the eighth trip to the prior seven trips. The first extrapolation involves a statistical sample, in which the mechanism for selection is randomization, while the second involves an observational study, in which the method of selection might be correlated with biasing factors, referred to as confounders. Shonubi III, 895 F.Supp. at 520 (citing Finkelstein affidavit at 1-3). Professor Finkelstein’s distinction applies with special force to extrapolation based on the 117 couriers reported by the DEA.