PER CURIAM:

I. FACTS AND PROCEDURAL HISTORY

Shawn Daniels, Tyrone Scott, Paul George, Sadie Elizabeth Green, Johnny Morris Anderson, and Kenneth Bruce Hicks (“defendants”) were among sixteen *453 people indicted on federal money laundering and drug charges. One count charged all the defendants, with the exception of Sadie Green, with conspiring to distribute cocaine in violation of 21 U.S.C. § 846. The evidence presented at trial shows that the defendants, in and around Savannah, Georgia, were involved together in buying and selling multi-kilo quantities of cocaine and multi-ounce quantities of cocaine base (“crack”), obtained primarily from southern Florida. Zachary Scott was a leader in this conspiracy. He was a witness for the prosecution at trial pursuant to a plea agreement in which he pleaded guilty to Count 2 of the indictment, engaging in a continuing criminal enterprise. 21 U.S.C. § 848. 1 Several other indicted individuals entered guilty pleas and agreed to cooperate with the Government by testifying at trial. The Government also called as witnesses several unindicted co-conspirators. In addition to the testimony of various people who know the defendants, the Government introduced telephone records and charts that summarize the records.

The jury returned not guilty verdicts on some counts, including the conspiracy count as to one defendant. Shawn Daniels, Tyrone Scott, Paul George, Johnny Morris Anderson, and Kenneth Bruce Hicks were convicted on the conspiracy count. Several defendants were convicted on substantive counts, which they do not challenge on this appeal. The jury found Kenneth Bruce Hicks guilty on Count 12, possessing with the intent to distribute approximately seventeen kilograms of cocaine around December 1989. The jury also returned a guilty verdict on Count 22, which charged Sadie Green with making false and material statements before the grand jury. The defendants appeal.

II. ISSUES ON APPEAL

The defendants raise a number of issues on appeal that merit discussion. Each of the defendants argues that the district court erred in refusing to allow individual voir dire or to strike the panel after one potential juror, in the presence of the other jurors, said, “Zachary Scott killed my brother.” (R. 9 at 21.)

The defendants also contend that the trial court erred in allowing the introduction of a Government exhibit. This exhibit is a set of three posters showing completed calls between telephone numbers associated with various defendants.

The defendants question the appropriateness of two of the jury instructions. The first challenged instruction describes reasonable doubt as “real doubt” such that one “would be willing to rely and act upon it without hesitation in the most important of [one’s] affairs.” The other instruction describes the court’s discretion regarding Government requests for sentencing leniency toward its witnesses. 2

III. DISCUSSION

A. PREJUDICE CAUSED BY THE JUROR’S REMARK

During voir dire, the widespread awareness of the case among the jurors became *454 apparent. When the district judge asked any potential jurors who had heard or read anything about the case in the news media to stand and give their names, so many responded that he later remarked that it would have been easier to ask for the identity of those who had not heard or read about the case.

When the judge asked one juror whether he had heard or read about the case, the juror responded, “Been following it pretty closely because Zachary Scott killed my brother.” (R. 9 at 21.) Following this remark, the judge asked whether any of the jurors had formed an opinion about the guilt or innocence of any of the defendants. The judge then dismissed nine jurors who answered the question affirmatively, including the juror who made the statement at issue. Defense counsel later moved for individual voir dire or to dismiss the panel. The district court denied the motions.

In later questioning of the panel, the judge explained that every defendant, “including these defendants is presumed by the law to be innocent ... until you as an impartial juror, if you are selected as a juror, are convinced beyond a reasonable doubt, with proper evidence, of that person’s guilt.” (R. 9 at 25.) He then asked the jurors whether any would not be able to give force to this rule of law if selected as a juror. None of the jurors indicated that they could not impartially decide the case consistent with the judge’s instructions.

The defendants contend that the court erred in denying their motions for individual voir dire or dismissal of the panel. We, however, find no abuse of discretion.

“The purpose of voir dire is to ascertain whether potential jurors can render a verdict solely on the basis of evidence presented and the charge of the trial court.” Wilcox v. Ford, 813 F.2d 1140 (11th Cir.), cert. denied, 484 U.S. 925, 108 S.Ct. 287, 98 L.Ed.2d 247 (1987). A district judge has wide latitude in determining which questions will be asked during voir dire. Id.; United States v. Nash, 910 F.2d 749 (11th Cir.1990). The trial court also has broad discretion about when to allow individual voir dire of the jurors. See Wilcox, 813 F.2d at 1150. This court will review such decisions for abuse of discretion and reverse only if the questioning as a whole did not comply with “the essential demands of fairness” and did not give “reasonable assurance to parties that any prejudice of potential jurors would be discovered.” Nash, 910 F.2d at 753.

The district court also has discretion regarding motions to strike the panel. United States v. Khoury, 901 F.2d 948, 955 (11th Cir.1990). To demonstrate error in the denial of a motion to strike the panel, the defendant must overcome the presumption of juror impartiality. Id.

The defendants attempt to distinguish this case from Khoury and United States v. Tegzes, 715 F.2d 505 (11th Cir.1983). In both cases, a juror made a statement that the defendants alleged was so likely to cause prejudice that dismissal of the panel was necessary. The defendants in Tegzes also sought individual voir dire of the panel members. We upheld the Khoury and Tegzes courts’ refusals to strike the panel or to question the jurors individually. In both cases we noted that the remark by the juror “did not constitute an opinion concerning the guilt or innocence of the defendants, nor did it relate to knowledge about the facts, parties, or witness involved.” Khoury, 901 F.2d at 955; Tegzes, 715 F.2d at 508. The defendants appear to argue that when a juror’s statement involves these elements, which we noted were not present in the juror statements at issue in Khoury and Tegzes, then a judge’s denial of motions for dismissal of the panel or individual voir dire is erroneous. Our opinions in Khoury and Tegzes, however, do not set out what characteristics of a juror’s remark would require the court to strike the panel or to question the jurors individually.

In this case, the juror’s remark was not clearly prejudicial. His statement that he had been following the case asserted that he had read or heard about it; however, it is undisputed that many of the jurors had also read and heard about the case. *455 His statement, “Zachary Scott killed my brother,” did not constitute an opinion regarding the guilt or innocence of any of the defendants on trial. The juror did not assert that he knew any of the parties or witnesses, nor did he make any allegations regarding any of the defendants on trial. Therefore, the defendants fail in their attempt to distinguish the statement at issue in this case from those at issue in Tegzes and Khoury.

Zachary Scott was a witness for the Government, not the defendants. Each of the defendants disclaimed any criminal connection with Zachary Scott. Further, no witness alleged that any of the defendants committed any acts of violence. Under these circumstances, the statement about Zachary Scott would not necessarily prejudice the defendants.

The defendants’ citation of United States v. Nell, 526 F.2d 1223 (5th Cir.1976), in support of their argument that they were denied a fair and impartial jury is inapposite. Nell involved the district court’s improper refusal to question individually or to dismiss the juror whose remarks indicated prejudice. This case does not involve that issue; the juror who accused Zachary Scott of killing his brother was dismissed from the panel.

The defendants have not shown evidence of jury bias, which would require dismissal of the panel, nor have they demonstrated that individual voir dire was necessary to discover any bias among the jurors resulting from the dismissed juror’s remark. The district judge was in the best position to discern any need for additional questions and to decide whether to strike the panel. See Tegzes, 715 F.2d at 508. We conclude that the district court did not abuse its discretion in denying the defendants’ motions for individual voir dire and for dismissal of the panel of prospective jurors.

B. ADMISSION OF THE TELEPHONE CHARTS

The defendants contend that they were denied a fair trial because the district court improperly admitted three telephone association posters as exhibits. The posters were diagrams showing completed calls between telephone numbers associated with the defendants. Each defendant’s name appeared in a circle with the telephone numbers the Government had associated with that defendant. Lines connected the circles to signify that there had been a completed telephone call. A number on each line indicated the number of phone calls the Government had reported between the two numbers.

At trial, the defendants objected to the charts on various grounds; they asserted that the charts did not include a time frame during which the calls were made, that the evidence did not show that the people whose names were connected actually spoke to each other during that completed call, and that the Government’s associations were either false or not substantiated by the evidence in the record. They also argued that the prejudicial impact of the posters outweighed their probative value. The defendants make these same arguments on appeal.

Federal Rule of Evidence 1006 provides that “[t]he contents of voluminous writings, recordings, or photographs which cannot conveniently be examined in court may be presented in the form of a chart, summary, or calculation.” The decision to admit or exclude such summary exhibits is within the discretion of the trial court, and we will review that decision for abuse of discretion only. United States v. Harmas, 974 F.2d 1262, 1269 (11th Cir.1992) (citing United States v. Duff, 707 F.2d 1315, 1319 (11th Cir.1983)).

The contention made by some appellants that the associations made by the Government are not supported by evidence in the record is without merit. We have fully examined the record and find evidence supporting the association of each defendant with each of the telephone numbers the chart associates with that defendant.

The defendants complain that the charts could not with certainty show that a particular defendant spoke with any other defendant on the telephone concerning drugs or any other matter and yet the charts have *456 the effect of putting all of the defendants together in an exhibit to give the impression of a conspiracy. The probative value of the charts, the defendants conclude, is outweighed by their prejudicial effect.

The defendants have presented no indications that the jury was in any way misled as to the information conveyed by the charts. The Government’s witness, Madeleine Pinckney, clearly explained the different elements of the charts and their meanings. Each of the defendants’ attorneys who cross-examined Ms. Pinckney asked her about the charts’ limitations, which she also explained. The court also cautioned the jury that the charts presented what the Government contended the evidence in the record shows and instructed the jury to evaluate the evidence independently.

The charts show a probability that certain defendants charged in the conspiracy spoke together a specified number of times on the telephone. This evidence alone does not show that a conspiracy existed, but it is relevant in that it tends to make that fact more likely than not. See Fed.R.Evid. 401 (defining relevant evidence). We hold that the district court did not abuse its discretion in admitting the charts as summary exhibits.

C. JURY INSTRUCTION REGARDING REASONABLE DOUBT

The defendants contend that they were denied a fair trial because the district judge incorrectly instructed the jury as to the meaning of “reasonable doubt.” The challenged instruction reads as follows:

It is the Government that has the burden of proving a defendant guilty beyond a reasonable doubt. If it fails to do so, you must, under your oath, find the defendant not guilty.

But while the Government’s burden of proof is a strict or heavy burden, it is not necessary that the defendant’s guilt be proved beyond all possible doubt. The law does not require mathematical certainty, because that is generally impossible. What is required is that the Government’s proof exclude any reasonable doubt concerning the defendant’s guilt.

A reasonable doubt is just precisely what it says. It is a real doubt based upon reason and common sense after a careful and impartial consideration of all of the evidence in the case. Proof beyond a reasonable doubt, stated a little bit differently, is proof of such a convincing character that you would be willing to rely and act upon it without hesitation in the most important of your own affairs.

(R. 13 at 4-5.) This instruction closely tracks the pattern instruction for the Eleventh Circuit. United States Eleventh Circuit District Judges Ass’n, Pattern Jury Instructions: Criminal Cases, basic instruction 3 (1985).

In reviewing “reasonable doubt” charges we consider the whole instruction rather than isolate any particular sentence. United States v. Rodriguez, 585 F.2d 1234, 1241 (5th Cir.1978). We recognize the broad discretion of the trial judge in formulating a jury instruction, and we review the instruction to determine whether it correctly states the substance of the law and the facts. United States v. Hines, 955 F.2d 1449, 1453 (11th Cir.1992); United States v. Elkins, 885 F.2d 775, 788 (11th Cir.1989), cert. denied, 494 U.S. 1005, 110 S.Ct. 1300, 108 L.Ed.2d 477 (1990).

The defendants first contend that this jury instruction was improper in light of the Supreme Court decision in Cage v. Louisiana, 498 U.S. 39, 39, 111 S.Ct. 328, 329, 112 L.Ed.2d 339 (1990), which found that a jury instruction improperly defining reasonable doubt violated the standard set forth in In re Winship, 397 U.S. 358, 364, 90 S.Ct. 1068, 1073, 25 L.Ed.2d 368 (1970). The instruction rejected by the Supreme Court in Cage defined reasonable doubt as:

founded upon a real tangible substantial basis and not upon caprice or conjecture. It must be such doubt as would give rise to a grave uncertainty, raised in your mind by reasons of the unsatisfactory character of the evidence or lack thereof. A reasonable doubt is not a mere possible doubt. It is an actual substantial doubt. It is a doubt that a reasonable man can seriously entertain. *457 What is required is not an absolute or mathematical certainty, but a moral certainty.

Cage, 498 U.S. at 40, 111 S.Ct. at 329 (emphasis in original). The Supreme Court was concerned that the words “grave” and “substantial” would cause the jurors to think that the degree of doubt about guilt that would require a not-guilty verdict was greater than a reasonable doubt. Id. at 40-41, 111 S.Ct. at 329-30. The Court also concluded that' “[w]hen those statements are then considered with the reference to ‘moral certainty,’ rather than evidentiary certainty, it becomes clear that a reasonable juror could have interpreted the instruction to allow a finding of guilt based on a degree of proof below that required by the Due Process Clause.” Id. at 41, 111 S.Ct. at 330.

The “real doubt” instruction is distinguishable from the instruction the Supreme Court rejected in Cage. The terms that the Court focused on as problematic in Cage do not appear anywhere in the instruction the defendants challenge. We disagree with the plaintiff’s contention that “real” carries the same connotation as “substantial.” “Substantial” necessarily indicates weight or degree; “real” does not refer to the degree of doubt, only its existence. The phrase “real doubt” differentiates “reasonable doubt” from possible doubt or the potential for doubt. Cf. United States v. Polan, 970 F.2d 1280 (3d Cir. 1992) (upholding a trial court’s use of an instruction that reasonable doubt is an “honest doubt” and “not a fanciful doubt”). In this case, the trial judge’s use of “real doubt” in the context of the entire charge did not violate the defendants’ due process rights. 3

The defendants’ second contention of error regarding this instruction is that the description of proof beyond a reasonable doubt as “proof of such a convincing character that you would be willing to rely and act upon it without hesitation in the most important of your own affairs” is inaccurate and prejudicial. The defendants did not object at trial to this latter portion of the charge; therefore, we review it for plain error only. United States v. Hope, 901 F.2d 1013, 1020 (11th Cir.1990), cert. denied, 498 U.S. 1041, 111 S.Ct. 713, 112 L.Ed.2d 702 (1991); United States v. Soloman, 856 F.2d 1572, 1575 (11th Cir.1988), cert. denied, 489 U.S. 1070, 109 S.Ct. 1352, 103 L.Ed.2d 820 (1989). A plain error is an obvious and substantial error that seriously affects the fairness, integrity, or public reputation of judicial proceedings. Hope, 901 F.2d at 1020; Soloman, 856 F.2d at 1575.

This court has expressly upheld this portion of the challenged charge as proper. United States v. Jones, 663 F.2d 567, 571 (5th Cir. Unit B Dec. 1981). 4 The court recognized in Jones, that descriptions of proof beyond a reasonable doubt as proof upon which one would be “willing to act” are sometimes disfavored. The court explained, however, that this version was acceptable because it contained the limiting words “without hesitation” and in the conduct of “the most important of [one’s] own affairs.” Jones, 663 F.2d at 571.

We also reject the defendants' contention that the combination of “real doubt” and the “willing to act” language of the instruction cumulatively and impermissibly lowered the Government’s burden of proof. The Tenth Circuit decision in Monk v. Zelez, 901 F.2d 885 (10th Cir.1990), which the defendants cite in support of this final contention, is clearly distinguishable from the circumstances of this case. First, the jury instruction rejected by the Tenth Circuit in Monk, like the Cage instruction, contained the phrase “substantial doubt,” not “real doubt.” Id. at 889. Furthermore, the “willingness to act” language in Monk described the necessary proof as proof such as one “would be willing to act upon in the *458 more weighty and important matters relating to [one’s] own affairs.” Id. Thus, it did not express the high level of certainty that is communicated in the willingness to rely and act upon that proof “without hesitation in the most important” of one’s own affairs. We find that the instruction as a whole adequately expresses the meaning of reasonable doubt; thus, the use of it did not violate the defendants’ rights secured by the Due Process Clause of the Fifth Amendment.

D. JURY INSTRUCTION REGARDING JUDGE’S DISCRETION ABOUT PROSECUTORIAL RECOMMENDATIONS

A number of persons who were originally named as defendants in the indictment and others who were already serving prison sentences for earlier convictions testified for the Government in this case. These people had made agreements with the Government to testify in exchange for the dismissal of some charges and a recommendation of favorable sentencing treatment. The judge instructed the jury accordingly and informed the jury that such persons’ testimony ought to be treated with more caution than the testimony of others.

The judge also instructed the jury in a later portion of the charge:

[A]ny motions to reduce sentences or recommendations to limit sentences of those not yet sentenced are to be decided by the sentencing judge, and they will be decided by that sentencing judge.

The Court is not bound by any agreement the Government may have made with any such witness nor limited in any way by the Government’s promises or recommendations. The only limitations on the Court’s power are the law, the facts, our judicial discretion and our consciences.

(R. 13 at 35-36.)

This instruction, the defendants allege, deprived them of due process by confusing the jury as to the importance of plea agreements to the credibility of witnesses. We disagree. This instruction, although arguably irrelevant to the issues before the jury, is an accurate statement of the law. It is difficult to see how the instruction could have prejudiced the defendants.

During the cross-examination of witnesses who had made such agreements with the Government, the defendants asked the witnesses about what they expected or hoped to gain by their testimony. The defendants do not challenge the accuracy of any of the witnesses’ responses. 5

During direct examination, Mr. Newman, the prosecutor, asked Ernest Moore if he was hoping to have his sentence reduced. Mr. Newman also asked him if he knew that any reduction of the sentence was “entirely up to the judge.” To both questions Moore answered, “yes.” The defendants did not object. Thus, the jury was fully aware that the Government had promised to recommend to the court beneficial sentencing treatment for some witnesses and that this recommendation was what the witnesses hoped to gain by testifying.

The existence and contents of these agreements were not hidden from the jury. The judge’s charge as a whole adequately explained to the jury the significance of the agreements as motivations for testifying, affecting the credibility of the witnesses who had made such agreements. The judge did not violate the defendants’ rights to due process by explaining the discretion of the sentencing judge.

IY. CONCLUSION

We have reviewed each of the defendants’ contentions of error. We find them *459 all to be without merit. For the reasons discussed, we affirm the convictions and sentences of Shawn Daniels, Tyrone Scott, Paul George, Sadie Elizabeth Green, Johnny Morris Anderson, and Kenneth Bruce Hicks in all respects.

AFFIRMED.

Notes

. Johnny Morris Anderson was tried on this count, but the jury found him not guilty.

. The defendants also raise certain issues that are meritless and do not warrant discussion. The defendants have not demonstrated error in the district court’s denial of their motions for severance or in its admission of testimony the defendants claim to be hearsay. Also, the evidence presented at trial is sufficient to support the jury’s verdicts, finding Shawn Daniels, Tyrone Scott, Kenneth Bruce Hicks, and Paul George guilty on the conspiracy count. The district court, therefore, did not err in denying motions for acquittal or for a new trial on this count as to these defendants.

Finally, the district court did not err in denying Kenneth Bruce Hicks’s motion for acquittal or for a new trial on Count 12, which charged him with possessing with the intent to distribute approximately seventeen kilograms of cocaine hydrochloride around December 1989 and aiding and abetting Terry Scanes to do the same, in violation of 21 U.S.C. § 841(a)(1). The primary argument Hicks makes on this issue concerns the Government’s lack of proof of venue. He waived this argument, however, by failing to object to venue prior to trial. United States v. Dryden, 423 F.2d 1175 (5th Cir.), cert. denied, 398 U.S. 950, 90 S.Ct. 1869, 26 L.Ed.2d 290 (1970). The evidence relevant to the charge in Count 12 is clearly sufficient to support his conviction.

. In United States v. Schaffer, 600 F.2d 1120, 1123 (5th Cir.1979), prior to the Supreme Court’s decision in Cage, this court held that the use of the term "real doubt” to describe "reasonable doubt” was not plain error.

. The Eleventh Circuit, in Stein v. Reynolds Securities, Inc., 667 F.2d 33 (11th Cir.1982), adopted as precedent decisions of the former Fifth Circuit, Unit B, rendered after September 30, 1981.

. The defendants cite Giglio v. United States, 405 U.S. 150, 92 S.Ct. 763, 31 L.Ed.2d 104 (1972), in support of their argument that they were prejudiced by the court's instruction. Giglio involved the presentation of misleading evidence concerning the agreement made between the Government and its primary witness. The defendants in this case do not contend that any evidence offered by the Government deceived the jury regarding the true nature of any agreement made between the Government and witnesses.