
    UNITED STATES of America, Plaintiff-Appellee, v. Obinna Felix UKWU, Defendant-Appellant.
    No. 12-4866.
    United States Court of Appeals, Fourth Circuit.
    Submitted: Aug. 21, 2013.
    Decided: Nov. 22, 2013.
    Bruce Fein, Bruce Fein & Associates, Inc., Washington, D.C., for Appellant. Rod J. Rosenstein, United States Attorney, Kathleen O. Gavin, Assistant United States Attorney, Office of the United States Attorney, Baltimore, Maryland, for Appellee.
    Before NIEMEYER, GREGORY, and DUNCAN, Circuit Judges.
   Affirmed by unpublished PER CURIAM opinion.

Unpublished opinions are not binding precedent in this circuit.

PER CURIAM:

Appellant Obinna Ukwu was convicted of twelve counts of aiding and assisting in the preparation of false income tax returns. 26 U.S.C. ■ § 7206(2). Mr. Ukwu was sentenced to 51 months in prison. He now challenges this sentence, arguing that the district court erred when it estimated the amount of tax loss Mr. Ukwu caused. Because a preponderance of the evidence supports the district court’s estimate, we affirm the sentence.

I.

Mr. Ukwu was an officer with the Maryland State Division of Corrections, but in 2006, he started an accounting business as side employment. The business offered tax return preparation services, and Mr. Ukwu operated the business until midway through 2010, when his legal problems began. In the intervening years, business boomed: in 2006, his revenue was roughly $8,000, but by 2009, it soared to $175,000.

A criminal investigation in 2010 revealed that Mr. Ukwu’s business was less criminally successful than successfully criminal. On many of his clients’ returns, Mr. Ukwu would claim fictional business losses in order to garner tax benefits. At trial, the vast majority of witnesses testified that these losses were entirely false and that they were not aware that Mr. Ukwu had invented these numbers on their returns.

Mr. Ukwu’s malfeasance went beyond false business losses. Mr. Ukwu claimed false charitable deductions on his clients’ forms. He also committed tax fraud on his own income taxes, filing a joint return for his wife and himself, but also filing a separate individual return for his wife under a different name. Finally, Mr. Ukwu took fees from his clients’ bank accounts and refund checks without notification.

After Mr. Ukwu’s jury conviction, the government estimated how much money Mr. Ukwu took from federal and state coffers. It concluded that Mr. Ukwu’s criminal behavior created tax losses of $2.1 million, which corresponds to a base offense level of 22 under § 2T4.1 of the United States Sentencing Guidelines Manual.

On appeal, Mr. Ukwu takes issue with the $2.1 million estimate, arguing that a preponderance of the evidence shows that his ill-gotten gains amounted to less than $1 million. Specifically, he argues that the district court’s method of estimating the tax shortfall was unsound because it used a small, flawed sample of tax returns to make inferences about another 1000 returns that he prepared. Based in part on its estimate, the district court sentenced Mr. Ukwu to 51 months in prison. Mr. Ukwu filed a timely appeal.

II.

We have jurisdiction to review Mr. Ukwu’s sentence under 28 U.S.C. § 1291 and 18 U.S.C. § 3742. The government has the burden of establishing the amount of tax loss by a preponderance of the evidence. United States v. Mehta, 594 F.3d 277, 281 (4th Cir.2010). The district court need not calculate the amount with a pharmacist’s precision: the sentencing guidelines require only a reasonable estimate. Id. Further, the district court may consider any relevant information regardless of its admissibility, provided that the information is sufficiently reliable. Id.

While we generally review for clear error, Mr. Ukwu did not challenge the district court’s tax loss estimate at sentencing. Therefore, we will apply a plain error standard of review. United, States v. Slade, 631 F.3d 185, 188 (4th Cir.2011). Mr. Ukwu must demonstrate that an error was made, that the error was plain, and that the error affected his substantial rights. Id. at 190. In the sentencing context, an error affects substantial rights if a different sentence would have been imposed absent the error. Id. In addition, even if these three elements are met, we retain discretion over whether to correct the forfeited error and do not exercise this discretion “unless the error seriously affects the fairness, integrity or public reputation of judicial proceedings.” United States v. Olano, 507 U.S. 725, 732, 118 S.Ct. 1770, 123 L.Ed.2d 508 (1993) (internal quotations and citations omitted).

Mr. Ukwu takes issue with how the district court reached its conclusion that his crimes caused over $1 million in tax losses. The sentencing court faced a difficult problem because of the sheer size of Mr. Ukwu’s potential fraud. Mr. Ukwu prepared roughly 1,000 tax returns that reported business losses, but the sentencing court and the IRS do not have time to audit each return, interview each taxpayer, and identify the extent of Mr. Ukwu’s crimes. As a result, the government had to rely on sampling techniques to make inferences about the universe of 1,000 tax returns. Essentially, the government had to take a spoonful of sauce out of the pot to assess whether the whole batch was spoiled.

The government used two samples of Mr. Ukwu’s 1,000 prepared tax returns to answer the following question: how often did Mr. Ukwu invent Schedule C losses from whole cloth? First, the government relied on a sample of 18 returns that were used at Mr. Ukwu’s criminal trial. These returns all reported Schedule C losses and contained loss descriptions that were vague, undocumented, and suspicious. Based on the testimony from the taxpayers involved, the government concluded that 16 out of 18 returns had Schedule C losses that were entirely false. The two remaining returns were disputed. Using these numbers, the government found that 88.88% of the returns in this sample used entirely false Schedule C losses. Note, however, that the returns investigated at trial were chosen for investigation specifically because they contained very high tax loss amounts. Thus, this was not a random sample of returns.

To solve this problem, the government then collected a random sample of returns to confirm its initial findings. The government drew 24 returns from the universe of 1,000 returns that contained Schedule C losses. Then, investigators analyzed these returns and found that every single one had large Schedule C losses that were vague, undocumented, and suspicious. That is, these returns exhibited the same pattern questionable Schedule C descriptions as the non-random sample of returns that were investigated at trial.

In sum, the government analyzed a nonrandom sample of returns at trial and found that 90% of the Schedule C losses were entirely false. Then, investigators used a random sample to confirm this estimate, reasoning that since the random sample bore the same patterns as the nonrandom sample, the two samples likely contained similar levels of fraud. That is, since the random sample looked like the non-random one, and since 90% of returns in the non-random sample were completely false, then 90% of the random sample was also likely to be completely false.

Finally, the government used this 90% number to calculate Mr. Ukwu’s tax loss estimate. The investigators could establish that among the 1000 returns where a Schedule C loss was claimed, Mr. Ukwu claimed roughly $16.4 million in Schedule C losses. If 90% of these losses were entirely fabricated, then this means that roughly $14.6 million of false losses were claimed. Assuming the lowest marginal tax rate of 10%, and factoring in state tax losses, the estimated tax loss was roughly $2.1 million. Because this estimate is between $1 million and $2.5 million, the district court concluded that Mr. Ukwu merited a base offense level of 22. U.S.S.G. § 2T4.1.

Mr. Ukwu takes issue with several methodological moves made by the government in reaching its $2.1 million estimate. First, he argues that the samples used were too small. Second, he argues that it was error to rely on the non-random sample of returns. Third, he argues that the government never established that the $14.6 million in Schedule C losses were totally fraudulent, rather than partially fraudulent.

A.

As a preliminary matter, we can reject with ease Mr. Ukwu’s argument that the government’s samples were too small to make a robust inference about the universe as a whole. His argument has intuitive appeal — how can 24 cases tell us about 1000? But Mr. Ukwu’s claim that small sample sizes render estimates useless is statistically incorrect. See David H. Kaye & David A. Freedman, Reference Guide on Statistics, in Reference Manual on Scientific Evidence 83, 126 n. 145 (2d ed. 2000) (“Analyzing data from small samples may require more stringent assumptions, but there is no fundamental difference in” how we make statistical inferences in small versus large samples). Certainly, a larger sample size is preferable, since it decreases the odds that one’s sample will be misleading. See Joseph Sanders, The Bendectin Litigation: A Case Study in the Life Cycle of Mass Torts, 43 Hastings L.J. 301, 342-43 (1992). However, even very small samples can be useful, as any political polling agency can attest: in many elections, a sample of 1,000 Americans can show, with enough certainty to satisfy the preponderance of the evidence standard, what is likely to happen in an election involving over 100 million voters. See Nate Silver, The Signal and the Noise 63 fig.2-4 (2012). While 24 is a relatively small sample, it amounts to 2% of the entire universe. This sample size does not paralyze us in our attempts to make inferences about the universe of all cases. See United States v. Littrice, 666 F.3d 1053, 1061 (“[Requiring the government to go through all the needles in the haystack of materially fraudulent and false returns ... would place a burden on the government beyond what the preponderance standard requires.”). As any chef or statistician can attest, even a small spoonful of sauce can indicate how much salt to add.

Mr. Ukwu’s next argument is that the government’s estimate was erroneous because it relied on a non-random sample, but this argument is similarly unavailing. He cites to Mehta, in which we questioned a district court’s use of a non-random sample to estimate the amount of tax loss among a broader universe of returns. 594 F.3d 277 (4th Cir.2010). In Mehta, the government analyzed a sample of returns that were chosen because they had been audited by the IRS. Id. at 282-83. It calculated the average tax loss among these returns to be $1,531 and then concluded that the entire universe of returns would have a similar average tax loss. Id. This was problematic because the returns in the sample were flagged by the IRS specifically because they were more likely to contain tax losses. Id. As such, the average amount of tax loss among this sample was misleading: the broader universe of returns was likely to have a lower average tax loss. Id. The sentencing court’s tax estimate was like using a group of NBA players to estimate the average height of all Americans.

Mr. Ukwu is correct that the initial, nonrandom sample used in this case is a problematic tool to make inferences about the amount of tax loss for the broader universe of returns. The returns chosen for the non-random sample were chosen specifically because they had higher tax losses. It could be that the amount of fraud in these returns was higher than for the entire universe of returns, so relying on the non-random sample alone would be problematic. However, the government’s tax loss estimate was based on more than a non-random sample. The government went out of its way to collect a random sample of returns to bolster its initial estimate. It compared this random sample to the original, non-random sample, and the government concluded that both groups of returns contained the same pattern of suspicious, unexplained tax losses. Though the government’s original estimate is based on a non-random sample, the government cleansed this error with the use of a random sample. Thus, the district court did not make the sort of mistake identified in Mehta, and as such, it did not commit plain error. See Olano, 507 U.S. at 734, 113 S.Ct. 1770 (1993) (“‘Plain’ is synonymous with ‘clear’ or, equivalently, ‘obvious.’ ”).

Mr. Ukwu’s final argument is most challenging. He admits that the non-random sample contains 90% falsehoods. He admits that the random sample looks similar to the non-random sample. However, he argues that this similarity alone fails to prove that in the random sample, all of the unexplained Schedule C losses were due to criminality. Instead, these losses might have been exaggerated instead of false, or due to negligence instead of fraud. Mr. Ukwu points to a Seventh Circuit case in which that court expressed skepticism of a similar methodology. United States v. Schroeder, 536 F.3d 746, 754-55 (7th Cir.2008).

Mr. Ukwu’s argument fails because the government need only make a reasonable estimate of the tax loss, and the methodology here, though imperfect, meets that standard. U.S.S.G. § 2T1.1 cmt. 1; Mehta, 594 F.3d at 282. In the eighteen tax returns investigated at trial, the Schedule C forms Mr. Ukwu prepared exhibited a suspicious pattern. Many returns claimed that the taxpayer worked as a contractor for Mary Kay or worked in “Nursing Services,” but at trial, the taxpayers testified that they never worked for Mary Kay and never owned such health care businesses. These returns also contained a suspicious pattern of receipts and expenses. The invented businesses often had revenues that were low or non-existent. Nearly all expenses were low or non-existent. Labor costs, meanwhile, were enormous.

The government’s random sample of tax returns exhibited a similar or identical pattern. Many of the returns listed Mary Kay as a profession; many more listed nursing services. One return even listed “General Services” as the profession. In the random sample, as in the non-random sample, the businesses almost always claimed to have zero sales, zero expenses, but enormous labor costs. Given these similarities, the sentencing court made no plain error when it concluded that, just like the returns analyzed at trial, the random sample of returns contained business losses that were entirely fabricated. See Olano, 507 U.S. at 734, 113 S.Ct. 1770 (“ ‘Plain’ is synonymous with ‘clear’ or, equivalently, ‘obvious.’ ”).

Further, Mr. Ukwu’s reliance on Schroeder is misguided. In that case, the government used a similar argument to make a tax estimate: it found strong evidence of fraud in sample A, found a similar pattern of losses in sample B, and concluded that sample B was therefore likely to contain fraud. 536 F.3d at 754-55. The Seventh Circuit expressed skepticism of this methodology. Id. at 755. However, the court’s reversal in that case was based not on the sampling methodology but rather on fundamental legal errors made by the sentencing court. Id. at 755. The district court in that case applied the wrong burden of proof, apparently concluding “that if evidence is admissible it proves the truth of the proposition for which it is being offered.” Id. Instead of requiring the government to prove a tax loss by a preponderance of the evidence, the sentencing court accepted the government’s estimate without any analysis, concluding that as long as the evidence was reliable, the tax loss had been proven. Id. Here, meanwhile, the sentencing court conducted a careful analysis of the evidence. It noted potential shortcomings in the methodology but concluded that the estimate was more likely than not to be accurate or significantly lower than the true tax loss. Thus, Schroeder is inapposite. Though the government’s methods were not perfect, its tax loss estimate was reasonable. Further, unlike in Schroeder, the district court’s analysis was careful and legally sound. This is all that is required under the Sentencing Guidelines. U.S.S.G. § 2T1.1 cmt. 1; Mehta, 594 F.3d at 282.

B.

Finally, even if Mr. Ukwu is correct that the tax loss estimate has methodological shortcomings, these errors were harmless and therefore did not affect his substantial rights. Slade, 631 F.3d at 190. The government estimated a tax loss of $2.1 million. Mr. Ukwu argues that it is possible that most of the claimed Schedule C losses were not criminal, but instead were legitimate losses, or at least negligent ones. For example, a client might have had $1,000 in legitimate business losses, but Mr. Ukwu might have pumped the number up to $2,000.

Mr. Ukwu might be correct, but the $2.1 million estimate is so conservative that even if he is right, the total tax losses are still likely to be above $1 million, which is the level of loss that is necessary for his sentencing range. U.S.S.G. § 2T4.1. First, in addition to false Schedule C losses, Mr. Ukwu used false charitable deductions on his clients’ returns, and none of these deductions were counted towards the $2.1 million figure. In one case, Mr. Ukwu claimed a $10,000 charitable gift that was entirely fabricated, suggesting that his Schedule A fraud might be significant. Similarly, the $2.1 million figure also excludes the fraud Mr. Ukwu committed on his own tax returns, which amount to roughly $100,000.

Further, the court’s estimate only looked at Mr. Ukwu’s returns from 2006 to 2008. He continued to prepare tax returns in 2009 and for part of 2010, and none of these returns were factored in to the tax loss estimate. Factoring in Mr. Ukwu’s 2009 returns increases the estimated loss to roughly $3 million.

Most importantly, the $2.1 million figure was calculated by applying a 10% marginal tax rate to the entire universe of returns. This is likely a gross underestimate of the true tax liability, since many of the returns were likely to have been subject to a 25% marginal tax rate or higher. This alone could increase the estimated tax loss by more than two-fold. In sum, even if Mr. Ukwu’s arguments are valid, his estimated tax losses are more likely than not to be well over $1 million. As such, the district court’s alleged error did not affect his substantial rights.

For the foregoing reasons, we affirm the judgment of the district court. We dispense with oral argument because the facts and legal contentions are adequately presented in the materials before this court and argument would not aid the decisional process.

AFFIRMED. 
      
      . Specifically, the investigators alphabetized the returns by the first name of the taxpayer, then drew one out of every fifty returns. This technique passes muster, though it is not perfect. Mr. Ukwu is Nigerian, and many of his clients were Nigerian immigrants. If these immigrants were more likely to have the same first name, or the same first letter of their first name, and if Mr. Ukwu was more likely to file false returns on immigrants' forms, as the district court suggested, then the sampling technique would be problematic. However, given the burden of proof — simply a preponderance of the evidence — it is more likely than not that this issue was not so grave that it affected the outcome of the sentencing calculation. Thus, while this technique does not warrant reversal here, future sentencing courts should be wary of accepting at face value that a randomization technique is truly random.
     
      
      . Specifically, statisticians teach that larger sample sizes can cut down on two types of error. First, there is the possibility that Mr. Ukwu committed rampant corruption, but by chance, we end up with a sample of cases where he did nothing wrong. Sanders, Bendectin, supra, at 342-43. Second, there is the possibility that Mr. Ukwu committed almost no corruption, but we happen to end up with a sample of cases in which he appears to fudge numbers constantly. Id. A larger sample size decreases the chance of both false negatives and false positives. Id.
      
     