Convicted of murder and sentenced to death, Ronald Sanders appeals the district court’s denial of his federal habeas petition, challenging both his conviction and his death sentence. We hold that the district court correctly rejected Sanders’ claim that the jury that convicted him was drawn from a jury venire that unconstitutionally failed to reflect the number of Hispanics in Kern County, where he was tried. We conclude, however, that Sanders did not receive an individualized death sentence, as required by the Eighth Amendment. The California Supreme Court neither independently reweighed aggravating and mitigating sentencing factors after it had invalidated two of the aggravating factors, nor did it conduct an appropriate harmless-error analysis. We also conclude that this error was not harmless. We therefore reverse the district court’s denial of Sanders’ habeas petition as it relates to the imposition of the death penalty and remand with instructions to grant the petition if the state does not either provide a new penalty trial or replace the sentence of death with another legally appropriate punishment.

FACTUAL AND PROCEDURAL BACKGROUND

Sanders was found guilty of murdering of Janice Allen. According to the state, the murder arose out of an escalating dispute between Allen’s boyfriend, Dale Boender, who was a drug dealer, and two of Boender’s customers, Brenda Maxwell and her aunt Donna Thompson.

In 1980, Boender dealt cocaine around Bakersfield, California. Maxwell was one of his customers, as was Thompson. Boender had stopped selling cocaine to Maxwell because she owed him money. As for her aunt, she felt that Boender had “burned her” in a drug transaction. So— as Maxwell testified — she, Thompson, and Sanders hatched a plan to rob Boender and steal his cocaine. The plan was to lure Boender to Maxwell’s home, have Sanders — whom Boender did not know— attack and rob Boender, and then have Sanders bind and “rob” Maxwell to make her look innocent of the set-up. Thompson would later “discover” and free both Boender and Maxwell.

On the morning of January 21, 1981, Maxwell called Boender, asking him to come to her house with a large amount of cocaine. Boender went to Maxwell’s home accompanied by Allen. Upon arriving, Boender was attacked with a piece of a pool cue by a man whom he had never seen before, but whom Boender later identified as Ronald Sanders. The robbery, however, did not go according to plan. Boender gained the upper hand over Sanders and left with the drugs. By that time, Allen had already fled the scene.

Immediately after the botched robbery, Maxwell feared that Boender would realize he had been set up, and (according to Maxwell) Sanders also feared Boender could identify him. Maxwell, Thompson and Sanders went by the house of another person, John Cebreros, to enlist his help. The group then went to Thompson’s house, where Maxwell called mutual friends of hers and Boender’s to tell them she had been robbed and raped so as to enhance her claim that she had been victimized along with Boender.

The next Friday evening, after drinking wine and smoking marijuana with friends, Boender and Allen bought groceries and returned to their apartment. While they were preparing dinner, there was a knock at the door. Leaving Allen in the kitchen, Boender went to the front door and opened it, encountering Sanders and a man he later identified as Cebreros (whom Boender had not seen before). Sanders spun Boender around and pushed him to the floor, face down. Allen emerged from the kitchen and was also made to lie on the floor. Boender’s glasses were ripped from his face and both he and Allen were bound and blindfolded.

Boender testified that the assailants asked for his cocaine and his money. He heard the assailants rummaging around his apartment. One of the assailants dragged him to what seemed like his bedroom and left the room. He heard more footsteps, muffled talking and more banging around the apartment. Boender heard one of the assailants say that he wanted to leave, but heard the other say that he wanted to stay. Boender could not identify the speaker who wanted to leave. Boender then heard someone approach, felt a blow to the head and recalled nothing further.

Later that night, Boender’s roommates returned to the apartment and found it full of smoke from a pot left on the stove. They discovered Boender in his bedroom, lying in a pool of blood. After calling an ambulance, they noticed that the apartment was in disarray, there were spots of blood around and a bag of marijuana was missing. One roommate found Allen’s body in another bedroom and called the police.

Both Boender and Allen had been bound by lengths of electrical cord cut from a vacuum cleaner. Allen sustained a fatal head wound from a heavy, blunt object which fractured her skull and lacerated her brain. Boender suffered a skull fracture but was conscious when the police arrived. Maxwell, Sanders and Cebreros were originally arrested, but Maxwell was released and granted immunity in return for her testimony.

Sanders and Cebreros were tried jointly. As the state acknowledged at trial, there was no direct evidence that determined whether Sanders or Cebreros had killed Allen. The most important witnesses for the prosecution were Boender and Maxwell; Boender identified Sanders from the robbery and Maxwell implicated Sanders in the plot to rob or murder Boender.

The defendants challenged Boender’s identification and presented an alibi defense. Three defense witnesses testified that on the night of the murder, both Sanders and Cebreros were at the home of Cebreros’ brother, Salvador, talking, play ing chess and drinking beer. No physical evidence was found at the murder scene to link Sanders or Cebreros to the murder.

The first trial of the co-defendants resulted in a hung jury. On January 22, 1982, after a retrial, both Sanders and Cebreros were convicted of robbery, burglary, attempted murder of Boender and first degree murder for the death of Allen. For both defendants, the jury found to be true four of the “special circumstances” that are necessary under California law for the imposition of the death penalty after a murder conviction.

At the penalty phase, for reasons that are unclear from the record, the prosecution waived its right to seek the death penalty for Cebreros but decided to seek death for Sanders. Sanders instructed his counsel that he did not want to present any evidence or argument at the penalty phase, because (as he told the trial court at the time) he felt that both life in prison and death were “equally unacceptable” sentences. Apparently because of these instructions, Sanders’ counsel did not pursue a thorough investigation into potential mitigating evidence. He presented no evidence of mitigating circumstances and gave no argument whatsoever to the jury at the penalty phase.

The state argued only one aggravating circumstance to the jury at the penalty phase — namely, that Sanders had committed five armed robberies in Orange County, California, in 1970, to which several witnesses testified. Sanders had been convicted for these robberies, pled guilty, served time in state prison and, in 1973, was granted parole, from which he had been removed in 1980. After hearing the prosecution’s penalty evidence and argument, the jury returned a verdict of death after deliberating for about two hours over two days.

As discussed in more detail below, on automatic appeal, the California Supreme Court invalidated two of the four special circumstances found by the jury in convicting Sanders. The California Supreme Court upheld the conviction and sentence in all other respects, and the United States Supreme Court denied certiorari.

On December 20, 1993, Sanders filed his first federal petition for a writ of habeas corpus in the district court. The district court ordered Sanders to exhaust state remedies, which he proceeded to do. Sanders then filed an amended petition in the district court, which denied the petition in its entirety on August 24, 2001. We granted Sanders a certificate of appealability on several of his claims on July 30, 2002.

JURISDICTION AND STANDARD OF REVIEW

The district court had jurisdiction over Sanders’ habeas petition under 28 U.S.C. § 2254 . We have jurisdiction under 28 U.S.C. § 1291 . Because Sanders filed his habeas petition before the effective date of the Anti-Terrorism and Effective Death Penalty Act of 1996, AEDPA does not apply. Alcala v. Woodford, 334 F.3d 862, 868 (9th Cir.2003). The district court’s decision to deny relief is reviewed de novo. Clark v. Murphy, 331 F.3d 1062, 1067 (9th Cir.2003). Factual findings made by the district court are reviewed for clear error. Alcala, 334 F.3d at 868 .

DISCUSSION

We first address Sanders’ challenge to the imposition of the death penalty, and then discuss his challenge to his conviction.

I. Sentencing Error

In assessing whether a death sentence satisfies the Eighth Amendment’s prohibi tion on cruel and unusual punishment, a “primary concern ... has been that the sentencing decision be based on the facts and circumstances of the defendant, his background, and his crime.” Clemons v. Mississippi, 494 U.S. 738, 748 , 110 S.Ct. 1441 , 108 L.Ed.2d 725 (1990). “Given that the imposition of death by public authority is so profoundly different from all other penalties, we cannot avoid the conclusion that an individualized decision is essential in capital cases. The need for treating each defendant in a capital case with that degree of respect due the uniqueness of the individual is far more important than in noncapital cases.” Lockett v. Ohio, 438 U.S. 586, 605 , 98 S.Ct. 2954 , 57 L.Ed.2d 973 (1978).

A serious concern about whether a death sentence is truly individualized arises when a jury decides to impose a death sentence based on its own assessment of aggravating and mitigating factors, and an appellate court later declares some or all of those aggravating factors legally invalid. Later invalidation of aggravating factors may undermine a jury’s original calculus for imposing death, introducing the risk that a defendant in such cases will not receive “the individualized treatment that would result from actual reweighing of the mix of mitigating factors and aggravating circumstances.” Clemons, 494 U.S. at 752 , 110 S.Ct. 1441 ; see also Stringer v. Black, 503 U.S. 222, 230-31 , 112 S.Ct. 1130 , 117 L.Ed.2d 367 (1992). Moreover, “[ejmploy-ing an invalid aggravating factor in the weighing process creates the possibility of randomness, by placing a thumb on death’s side of the scale, thus creating the risk of treating the defendant as more deserving of the death penalty.” Sochor v. Florida, 504 U.S. 527, 532 , 112 S.Ct. 2114 , 119 L.Ed.2d 326 (1992) (internal quotation marks, citations and alterations omitted).

The Supreme Court has set out clear rules for the procedures appellate courts must follow when an aggravating factor has been held invalid. See id. at 532 , 112 S.Ct. 2114 . The procedures differ significantly between so-called “weighing” and “nonweighing” states. In nonweighing states, aggravating factors matter for determining eligibility for the death sentence, but have no specific function in the sentencing process itself. At sentencing in nonweighing states, the factfinder always takes into consideration all circumstances from both the guilt and the sentencing phases of the trial. See Stringer, 503 U.S. at 229-230 , 112 S.Ct. 1130 . Thus, in these states, as long as the jury finds that at least one aggravating factor makes the defendant death-eligible, the subsequent elimination of another aggravating factor does not pose a problem for individualized sentencing, because the aggravating circumstances are not considered as separate factors in sentencing. See Zant v. Stephens, 462 U.S. 862, 889 , 103 S.Ct. 2733 , 77 L.Ed.2d 235 (1983); Williams v. Calderon, 52 F.3d 1465, 1479 (9th Cir.1995) (holding that California’s pre-1978 death penalty system was nonweighing).

In contrast, in weighing states-including, as we explain below, California after 1978-"the finding of aggravating factors is part of the jury's sentencing determination, and the jury is required to weigh any mitigating factors against the aggravating circumstances." Clemons, 494 U.S. at 745 , 110 S.Ct. 1441 . In these states, "there is Eighth Amendment error when the sen-tencer weighs an `invalid' aggravating circumstance in reaching the ultimate decision to impose a death sentence." Sochor, 504 U.S. at 532 , 112 S.Ct. 2114 .

A remand for resentencing is not necessarily required, however, in order to correct this error. In weighing states, when a jury has made the sentencing de termination, state appellate courts that have declared an aggravating factor invalid in a capital case have three options. They may either: (1) remand for resentencing; (2) independently reweigh the remaining aggravating and mitigating circumstances under the procedure set forth in Clemons , in which the “state appellate court reweighs aggravating and mitigating circumstances that have already been found by a jury to exist,” Valerio v. Crawford, 306 F.3d 742, 757 (9th Cir.2002) (en banc); or (3) independently conclude that the sentencing body’s consideration of the invalid aggravating circumstance was “harmless beyond a reasonable doubt” under the standard elaborated in Chapman v. California, 386 U.S. 18, 23 , 87 S.Ct. 824 , 17 L.Ed.2d 705 (1967). See Morales v. Woodford, 336 F.3d 1136, 1147 (9th Cir.2003). “While federal law does not require the state appellate court ... to remand for resentencing, it must, short of remand, either itself reweigh without the invalid aggravating factor or determine that weighing the invalid factor was harmless error.” Sochor, 504 U.S. at 532 , 112 S.Ct. 2114 .

Even if a state appellate court has not adhered to these principles and thus failed to ensure constitutionally adequate sentencing, such an error does not automatically mean that a petitioner will receive habeas relief in federal court. Rather, in habeas cases, we apply a second level of harmless-error review in order to determine whether the state court’s failure to conduct the constitutionally mandated review was itself harmless. See Morales, 336 F.3d at 1148 . In doing so, we apply the standard of Brecht v. Abrahamson, 507 U.S. 619, 638 , 113 S.Ct. 1710 , 123 L.Ed.2d 353 (1993), which requires us to determine whether, in light of the record as a whole, the error had substantial and injurious effect or influence in determining the jury’s verdict. Morales, 336 F.3d at 1148 .

In Sanders’ case, the California Supreme Court invalidated two of the “special circumstances” that the jury had found at the guilt phase and was required to weigh in deciding whether to impose a death sentence. See Cal.Penal Code 190.3(a) (West 2003) (stating that a sentencing jury shall take into consideration if relevant “[t]he circumstances of the crime of which the defendant was convicted in the present proceeding and the existence of any special circumstances found to be true pursuant to [Cal.Penal Code] Section 190.1”). Sanders claims that he is entitled to relief because after invalidating these two special circumstances, the California Supreme Court neither remanded for re-sentencing nor conducted an independent reweighing or a proper harmless-error analysis. For Sanders to prevail on this claim, he must demonstrate (1) that California is a weighing state; (2) that the California Supreme Court did not conduct a proper review; and (3) that the failure to conduct such a review, in light of the record as a whole, had a substantial and injurious effect on his sentencing. We hold that Sanders has met all three of these requirements.

A. California’s Death Sentencing System

We have indicated that California’s posU-1978 death penalty law created a “weighing” system. See Allen v. Woodford, 366 F.3d 823, 857 (9th Cir.2004). We have not, however, explained precisely how the weighing of sentencing factors operates under California law. Because specific features of California law are important to our ruling here, we shall do so now.

We consider a state death penalty regime to be a weighing system when “the sentencer [is] restricted to a ‘weighing’ of aggravation against mitigation” and “the sentencer [is] prevented from considering evidence in aggravation other than discrete, statutorily-defined factors.” Williams v. Calderon, 52 F.3d 1465, 1477 (9th Cir.1995). Both of these elements are present under California’s post-1978 system. California Penal Code § 190.3 specifies that the sentencing jury in a capital case “shall take into account ... if relevant” any of 11 factors. The trier of fact “shall impose a sentence of death if [it] concludes that the aggravating circumstances outweigh the mitigating circumstances. If the trier of fact determines that the mitigating circumstances outweigh the aggravating circumstances the trier of fact shall impose a sentence of confinement in state prison for a term of life without the possibility of parole.” Cal.Penal Code § 190.3. The California Supreme Court has expressly interpreted § 109.3 as precluding the jury from considering aggravating factors other than those statutorily defined. People v. Boyd, 38 Cal.3d 762, 773 , 215 Cal.Rptr. 1 , 700 P.2d 782 (1985) (“By ... requiring the jury to decide the appropriateness of the death penalty by a process of weighing the specific factors listed in the statute, the [posH978 death penalty law] necessarily implie[s] that matters not within the statutory list are not enti-tied to any weight in the penalty determination.”). Thus, as we have previously indicated, it is clear that California’s post-1978 death penalty system is a weighing system. See Allen, 366 F.3d at 857 .

We note, however, that California’s system has features that are not present in all weighing states, and that are important for understanding the effect of the invalidation of the special circumstances in Sanders’ case. A death penalty trial in California proceeds in two stages. At the initial phase of the trial, when the trier of fact decides the issue of the defendant’s guilt or innocence, “a determination must be made as to the existence of any ‘special circumstances.’ ” People v. Bacigalupo, 6 Cal.4th 457, 467 , 24 Cal.Rptr.2d 808 , 862 P.2d 808 (1993). Special circumstances found at the guilt phase serve to make a defendant eligible for the death penalty, and are thus the “criteria in the California capital scheme that define the class of murders for which death is a potential penalty.” Id. at 467-68 , 24 Cal.Rptr.2d 808 , 862 P.2d 808 .

The weighing of factors under § 190.3 becomes relevant only at a subsequent “penalty” or sentencing phase that occurs once the defendant has been found death-eligible during the guilt phase.

Id. at 814 (internal citation, quotation marks and alterations omitted). Although the statute plainly instructs that the fact-finder “shall” at this stage impose death if it finds that the aggravating circumstances outweigh the mitigating ones, “[t]his weighing is a process that by nature is incapable of precise description.” Id. at 470, 24 Cal.Rptr.2d 808 , 862 P.2d 808 (internal quotation marks omitted); see also People v. Brown, 40 Cal.3d 512, 541-45 , 230 Cal.Rptr. 834 , 726 P.2d 516 (1983), reversed on other grounds by California v. Brown, 479 U.S. 538 , 107 S.Ct. 837 , 93 L.Ed.2d 934 (1987).

Bacigalupo, 6 Cal.4th at 470 , 24 Cal.Rptr.2d 808 , 862 P.2d 808 (internal quotation marks and citations omitted); see also Allen, 366 F.3d at 855 .

Thus, under California law, when a jury decides whether to impose the death penalty, it does more than merely count aggravating and mitigating factors, and individual jurors may ascribe varying weight to any single aggravating factor. This makes it difficult for an appellate court that later reviews the jury’s sentencing decision to surmise what weight the jury gave to a particular factor.

Nonetheless, California still qualifies as a weighing state, because the jury’s sentencing discretion is not boundless — it must consider the defined list of aggravating factors, and may not consider other aggravating factors, in making its penalty determination. Therefore, an appellate court’s invalidation of one or more of the sentencing factors may have a serious effect on individualized sentencing, because there is a real risk that the jury’s decision to impose the death penalty rather than life imprisonment may have turned on the weight it gave to an invalid aggravating factor. With this sentencing structure in mind, we turn to the California Supreme Court’s affirmance of Sanders’ death sentence in light of its invalidation of two of the aggravating factors the jury considered.

B. The California Supreme Court’s Affirmance of the Death Sentence

The jury found to be true four special circumstance allegations against Sanders: (1) that the murder was committed while he was engaged in a robbery (see Cal.Penal Code § 190.2(a)(17)(A)); (2) that it was committed while he was engaged in a burglary (§ 190.2(a)(17)(G)); (3) that Allen was killed to prevent her testimony (§ 190.2(a)(10)); and (4) that the murder was heinous, atrocious and cruel (§ 190.2(a)(14)). People v. Sanders, 51 Cal.3d 471, 515 , 273 Cal.Rptr. 537 , 797 P.2d 561 (1990). The California Supreme Court invalidated the burglary special circumstance because the jury could have found the requisite mental state for burglary based on Sanders’ intention to commit assault, not a murder, and then have impermissibly merged the burglary with the murder to create the burglary-murder special circumstance. Id. at 517 , 273 Cal.Rptr. 537 , 797 P.2d 561 ; see also People v. Wilson, 1 Cal.3d 431, 441 , 82 Cal.Rptr. 494 , 462 P.2d 22 (1969) (rejecting “bootstrapping” of burglary and felony-murder). The court also set aside the heinous-murder special circumstance because in a prior opinion it had found that special circumstance to be unconstitutionally vague. Sanders, 51 Cal.3d at 520 , 273 Cal.Rptr. 537 , 797 P.2d 561 ; see People v. Superior Court (Engert), 31 Cal.3d 797 , 183 Cal.Rptr. 800 , 647 P.2d 76 (1982).

The California Supreme Court issued its decision shortly after the United States Supreme Court’s decision in Clemons v. Mississippi, 494 U.S. 738 , 110 S.Ct. 1441 , 108 L.Ed.2d 725 (1990), and before the Court’s decision in Stringer v. Black, 503 U.S. 222, 230 , 112 S.Ct. 1130 , 117 L.Ed.2d 367 (1992). Thus, perhaps understandably, the California court did not follow the procedures constitutionally mandated for appellate review in a weighing state where an aggravating circumstance has been invalidated. The California court did not remand for resentencing. It also did not independently reweigh the aggravating and mitigating factors to ensure an individualized sentence. The state does not argue otherwise.

Although the California court did apparently conduct some type of harmless-error analysis, it did not find, as it was required to do, that the error was “harmless beyond a reasonable doubt.” See Morales, 336 F.3d at 1147-48 . The court first focused on the heinous-murder special circumstance and its effect on the jury in light of the prosecutor’s closing argument. The court observed, “Although the prosecutor mentioned the heinous-murder special circumstance in closing argument, he did not heavily rely on it.” Sanders, 51 Cal.3d at 521 , 273 Cal.Rptr. 537 , 797 P.2d 561 . It concluded, “[A] reasonable juror would not have been swayed by abstract concepts of heinous, atrocious or cruel ... but would instead have focused on the actual circumstances of the offense which formed the foundation for finding those special circumstances to be true.” Id. (internal quotation marks and alterations omitted). Then, the court turned to the burglary-murder special circumstance, noting that “the prosecutor did not focus on the bare number of special circumstance findings but urged the jury to consider the brutality of the crimes.” Id. It determined that “there was little chance defendant was prejudiced by consideration of the burglary-murder special circumstance.” Id.

We cannot uphold a state appellate court's harmless error review as adequate when we have substantial uncertainty about whether the state court actually concluded that the invalid aggravating factor was harmless beyond a reasonable doubt. In Sochor v. Florida, 504 U.S. 527 , 112 S.Ct. 2114 , 119 L.Ed.2d 326 (1992), a state trial court in Florida had found four aggravating circumstances and no circumstances in mitigation. Id. at 530 , 112 S.Ct. 2114 . The Supreme Court of Florida invalidated one of the aggravating circumstances-the "cold, calculated, and premeditated manner" circumstance-because this factor required a "heightened" degree of premeditation that was not supported by the evidence in the case. See id. at 531 , 112 S.Ct. 2114 . Despite the error, the Florida court affirmed the death sentence and said:

Sochor v. State, 580 So.2d 595, 604 (Fla.1991). The United States Supreme Court concluded that the Supreme Court of Florida had not adequately performed a harmless-error review. The Court noted that the state court failed “so much as to mention ‘harmless error.’ ” Sochor, 504 U.S. at 539-40 , 112 S.Ct. 2114 . It also pointed out that “[o]nly one of the four cases [cited by the Florida court] contains language giving an explicit indication that the State Supreme Court had performed harmless error analysis. The other three simply do not, and the result is ambiguity.” Id. at 540 , 112 S.Ct. 2114 (citation omitted). The Court thus held, “Since the Supreme Court of Florida did not explain or even ‘declare a belief that’ this error ‘was harmless beyond a reasonable doubt’ in that ‘it did not contribute to the [sentence] obtained, Chapman, [ 386 U.S. at 24 , 87 S.Ct. 824 ,] the error cannot be taken as cured by the State Supreme Court’s consideration of the case.” Sochor, 504 U.S. at 540 , 112 S.Ct. 2114 .

The California Supreme Court’s review in this case is similar to the Florida Supreme Court’s review in Sochor. The California court never used the words “harmless error” or “reasonable doubt” in analyzing the effect of removing the special circumstance. Moreover, it appears that the California court erroneously believed that it could apply the rule of Zant v. Stephens, 462 U.S. 862 , 103 S.Ct. 2733 , 77 L.Ed.2d 235 (1983)-which is applicable only to nonweighing states-and uphold the verdict despite the invalidation of two special circumstances because it was upholding other special circumstances. See Sanders, 51 Cal.3d at 520 , 273 Cal.Rptr. 537 , 797 P.2d 561 ("The United States Supreme Court has upheld a death penalty judgment despite invalidation of one of several aggravating factors [citing Zant], and this court is in accord."). In these circumstances, we cannot conclude that the California Supreme Court found that the invalidation of the special circumstance was harmless beyond a reasonable doubt. See Clemons, 494 U.S. at 754 , 110 S.Ct. 1441 ("It is perhaps possible that the Mississippi Supreme Court intended to ask whether beyond a reasonable doubt the result would have been the same.... Because we cannot be sure which course was followed in Clemons' case, however, we vacate the judgment insofar as it rested on harmless error. . . ."). We therefore hold that Sanders did not receive the individualized death sentence to which he was entitled because the California Supreme Court did not conduct an adequate, independent appellate review.

C. Substantial and Injurious Effect

Before Sanders is entitled to habe-as relief, however, we must also apply our own harmless-error analysis to determine whether the Eighth Amendment error had a substantial and injurious effect or influence on the jury’s verdict. “When a federal judge in a habeas proceeding is in grave doubt about whether a trial error of federal law had substantial and injurious effect or influence in determining the jury’s ver- diet, that error is not harmless. And, the petitioner must win.” O’Neal v. McAninch, 513 U.S. 432, 436 , 115 S.Ct. 992 , 130 L.Ed.2d 947 (1995) (internal quotation marks omitted).

Morales v. Woodford, 336 F.3d 1136 (9th Cir.2003), another federal habeas case involving California’s posW.978 death-penalty law, provides a point of contrast. In that case, applying Brecht v. Abrahamson, 507 U.S. 619, 638 , 113 S.Ct. 1710 , 123 L.Ed.2d 353 (1993), we found harmless an invalid special circumstance weighed by the jury in the penalty phase, where the jury also relied on another valid special circumstance in deciding to impose the death penalty. The invalid special circumstance required the jury to find that “the murder involved the infliction of torture” (defined as “the infliction of extreme pain”). Morales, 336 F.3d at 1145 & n. 20 (internal quotation marks omitted). However, the special circumstance did not require the jury to find that the defendant intentionally tortured the victim. Id. Applying our prior caselaw, we held that special circumstance constitutionally invalid because the unintentional infliction of extreme pain might “have nothing to do with the mental state or culpability of the defendant and would not seem to provide a principled basis for distinguishing capital murder from any other murder.” Id. at 1146 ; see also Wade v. Calderon, 29 F.3d 1312, 1320 (9th Cir.1994) (invalidating this special circumstance), overruled on other grounds, Rohan ex rel. Gates v. Woodford, 334 F.3d 803, 815 (9th Cir.2003).

Despite this legal conclusion, we noted that the facts strongly suggested that the defendant had tortured the victim and that he had done so intentionally. “There[was] no reason to doubt that after [the defendant] failed to kill [the victim] by strangling her with [a] belt, he beat her head in with a hammer, and when she still lived, dragged her out of the car, raped her, and stabbed her several times.” Morales, 336 F.3d at 1149 . Given this “overwhelming” evidence, the jury would have reached the unavoidable conclusion that the defendant intended to inflict extreme pain on the victim as part of the murder. Id. Because we did not doubt in Morales that the jury in fact did find that the defendant intended to torture the victim, we concluded that the constitutional error had no substantial or injurious effect on the defendant’s sentencing.

Here, in contrast, the jury likely considered the legally improper aspects of the invalid special circumstances. The jury could well have relied on the vague language in the heinous-murder instruction in finding the murder “heinous, atrocious, and cruel.” The facts do nothing to cure the problem with this special circumstance, because the terms of that circumstance are “ ‘so vague that men of common intelligence must necessarily guess at its meaning and differ as to its application.’ ” People v. Superior Court (Engert) 31 Cal.3d 797, 801 , 183 Cal.Rptr. 800 , 647 P.2d 76 (1982) (quoting Connolly v. Gen. Constr. Co., 269 U.S. 385, 391 , 46 S.Ct. 126 , 70 L.Ed. 322 (1926)).

The jury also easily could have considered the improper aspect of the burglary-murder special circumstance. According to the California Supreme Court, the flaw in the burglary-murder special circumstance was that the trial court's felony-murder jury instructions during the guilt phase had "improperly permitted the jury to find a burglary based on~Sanders'] intent to commit an assault." Sanders, 51 Cal.3d at 517 , 273 Cal.Rptr. 537 , 797 P.2d 561 . Because the jury could have improperly found first degree felony murder by "bootstrapping"-flnding burglary based on intent to assault and then using the finding of burglary to convict Sanders of “first degree murder without proof of malice aforethought and premeditation” — the California Supreme Court held the burglary-murder special circumstance instruction generally invalid. Id. at 509, 517 , 273 Cal.Rptr. 537 , 797 P.2d 561 . Given that it was unclear from the evidence presented at trial whether Sanders or Cebreros actually killed Allen, and that at least one of the defendants, who may have been Sanders, said that he wanted to leave before the murder began, it is realistic to conceive that a juror could have concluded that Sanders entered Boender’s apartment intending only to commit assault, not murder. If the jury had concluded that Sanders’ burglary was committed with only an intent to commit assault, the jury’s finding of the burglary-murder special circumstance was improper. Therefore, we cannot conclude that the facts of the case made the legal problems in the aggravating circumstance instructions harmless.

We may not conclude that the jury’s consideration of these aggravating circumstances did not substantially influence the jury’s assessment of Sanders’ suitability for the death penalty. As outlined above, California’s weighing process differs from that of other weighing states. Under California law, “ Veighing’ ... connotes a mental balancing process, but certainly not one which calls for a mere mechanical counting of factors ... or the arbitrary assignment of ‘weights’ to any of them.” Brown, 40 Cal.3d at 542 , 230 Cal.Rptr. 834 , 726 P.2d 516 . We cannot know as an appellate court what individual weight a juror assigned to a finding of an aggravating special circumstance. Thus we may not simply assume harmless error because of the presence of other aggravating circumstances or the absence of mitigating ones.

On the facts here, we cannot say with sufficient certainty that the jury’s consideration of the two improper special circumstances did not substantially influence its sentencing determination. There is good reason to believe that the jury may have had doubts about Sanders’ role in the murder and that it may thus have been only marginally inclined to impose the death penalty. There was no physical or overwhelming circumstantial evidence indicat ing who, as between Sanders and Cebre-ros, delivered the fatal blow to Allen’s head.

There was also considerable uncertainty about the extent to which the murder had been pre-planned by Sanders. ' On the one hand, Maxwell testified that after the first attempted robbery, Sanders had expressed concern that Boender could identify him. On the other hand, Thompson told Maxwell after the murder that Allen “wasn’t supposed to be dead” and “that wasn’t what was planned.”

Similarly, the existence of other factors did not overwhelmingly compel a death sentence to the point where we can state with confidence that these circumstances, and not the invalid heinous-murder or burglary-murder special circumstances, were decisive determinants of the death sentence. Certainly, Sanders’ prior violent robberies and felony conviction were aggravating. However, the last robbery he committed was 11 years before the robbery of Boender and Allen. As for the robbery-murder and witness-killing aggravating circumstances, it is not clear they would have made the heinous murder circumstance merely superfluous, because the heinous-murder circumstance may have particularly emphasized the brutal nature of Allen’s murder and thus have compelled the jurors to vote for death. The jury might have chosen to be more lenient because the means by which the victim was killed, beating rather than shooting or stabbing, do not necessarily imply an intention to cause death. The jury might also have chosen to be lenient with Sanders because Maxwell, despite her initiative in bringing about the murder, was not even charged. We cannot, of course, reconstruct the jury’s penalty determination, but there is enough uncertainty, in this case to cause us to believe that consideration of the improper circumstances may well have had a substantial effect or influence on the jury’s determination.

The state contends that consideration of the two invalid special circumstances was harmless because when the “title special circumstance is removed from the evidence” the substance of that evidence remains intact. Even assuming that ■ the state is correct that on the facts of this case, the jury could still have considered the “substance of the evidence” that led it to find true the invalid special circumstances, we have grave doubt as to whether it would have imposed death absent the special-circumstance label. This was a close case, for the reasons we have already discussed, including the uncertainty as to who struck the fatal blow and which assailant wanted to leave before Allen was killed. We also note that the first jury hung on guilt. In such a situation, unlike in Morales and Allen where we could easily ascertain what led the jury to impose death, in this case the jury’s improper weighing of special circumstances may well have mattered.

In sum, the jury was told to weigh two special circumstances that were improperly deemed special circumstances. In a weighing state, if the trier of fact is erroneously directed to weigh certain aggravating factors due to an invalid jury instruction, and that misdirection substantially affects the jury’s sentencing determination, then the defendant has not received a properly individualized sentence and the error is not harmless. In these circumstances, we are required to grant habeas relief. Here, the jury was erroneously instructed on two special circumstances that it may have applied in an invalid manner, and we have grave doubt about whether that error had substantial and injurious effect or influence in determining the jury’s verdict. Accordingly, applying the standards of Brecht , 507 U.S. at 638, 113 S.Ct. 1710 , and O’Neal, 513 U.S. at 440 , 115 S.Ct. 992 , we hold the error to be not harmless.

D. Remedy

When there has been a failure of adequate appellate review of an erroneous sentencing instruction in the penalty phase of a capital case, the proper remedy is to grant the writ unless there is either a new penalty trial or the death sentence is vacated and a lesser sentence imposed. Valerio, 306 F.3d at 763 (imposing this remedy in a capital case where the Nevada Supreme Court failed to conduct adequate appellate review). We therefore reverse the district court and remand with instructions that it shall grant the petition for the writ of habeas corpus as to the penalty phase, unless the state within a reasonable period of time either grants a new penalty trial or vacates the death sentence and imposes a lesser sentence consistent with law.

Because we are remanding for a new penalty trial, we do not consider Sanders’ other challenges to his sentencing or to the process of appellate re-weighing of sentencing factors in California.

II. Guilt-Phase Challenge to Jury Ve-nire

Sanders also argues that the under-representation of Hispanics on his jury violated his Sixth Amendment right to an impartial jury drawn from a representative cross-section of the community. In Taylor v. Louisiana, 419 U.S. 522 , 95 S.Ct. 692 , 42 L.Ed.2d 690 (1975), the United States Supreme Court recognized that “the selection of a petit jury from a representative cross section of the community is an essential component of the Sixth Amendment right to a jury trial.” Id. at 528 , 95 S.Ct. 692 .

To state a prima facie violation of the representative cross section requirement, a defendant must show that (1) the group alleged to have been excluded is a “distinctive” group in the community; (2) the representation of this group in venires from which juries are selected is not fair and reasonable in relation to the number of such persons in the community; and (3) this under-representation is due to systematic exclusion of the group in the jury-selection process. Duren v. Missouri, 439 U.S. 357, 364 , 99 S.Ct. 664 , 58 L.Ed.2d 579 (1979). If the petitioner makes a prima facie showing under Duren , the burden shifts to. the state to justify the under-representation “by demonstrating that attainment of a fair cross section is incompatible with a significant state interest.” Thomas v. Borg, 159 F.3d 1147, 1150 (9th Cir.1998); see also Duren, 439 U.S. at 367-68 , 99 S.Ct. 664 .

It is undisputed that Sanders has met the first prong because “Hispanics are a ‘distinctive’ group for purposes of Sixth Amendment analysis.” United States v. Nelson, 137 F.3d 1094, 1101 (9th Cir.1998). We hold, however, that he has not met the second prong.

“The second prong ... requires proof, typically statistical data, that the jury pool does not adequately represent the distinctive group in relation to the number of such persons in the community.” United States v. Esquivel, 88 F.3d 722, 726 (9th Cir.1996). In determining whether a particular group is underrepresented in a jury venire, we use an absolute disparity analysis. Borg, 159 F.3d at 1150 . “We determine absolute disparity by taking the percentage of the group at issue in the total population and subtracting from it the percentage of that group that is represented on the master jury wheel.” United States v. Sanchez-Lopez, 879 F.2d 541, 547 (9th Cir.1989).

Here, Sanders is “unable to provide the statistics necessary for this court to determine the absolute disparity in his case,” Borg, 159 F.3d at 1150 , because the statistics he offers fail to take any account of undocumented immigrants and other legitimately ineligible jurors within the total Hispanic population. Sanders’ argument for disparity comes from testimony of Dr. Terry Newell, based on Newell’s study of jury venires in Kern County, California (where Sanders’ trial was held) from 1980 to 1981. At that time, Kern County compiled its master jury list from voter registration records.

Dr. Newell used the following method to determine underrepresentatiob on the jury venire. First, he relied on the 1980 census figures for the total population and total Hispanic population of Kern County. The census listed a total population of 402,089 individuals in Kern County, including 87,-025 Hispanic individuals, from which New-ell calculated the Hispanic percentage of the total population to be 21.59 percent. Recognizing that the census numbers for total population and total Hispanic population included noncitizens-persons presumptively ineligible to serve on a jury-Newell attempted to estimate the number of Hispanic citizens in Kern county, In that calculation, however, he used a method that is insufficient on its face. Using data from the Immigration and Naturalization Service, he obtained both the total number of legal, registered aliens in Kern County (14,387) and the number of legal, registered aliens in Kern County whose country of origin was Mexico (10,230). He then subtracted the total number of legal, registered aliens from the census count of the total population of Kern County (402,-089 minus 14,387, or 387,702), and the number of legal, registered aliens from Mexico from the total number of Hispanics in Kern County (87,025 minus 10,230, or 76,795). He compared those two numbers to. produce an estimate of the percentage of the citizen population that was Hispanic, which he put at 19.81 percent. He then used a statistical technique that we need not describe here to account for the percentage of that population who were adults, ultimately estimating the total adult-and-citizen Hispanic population in Kern County at 16.3 percent of the total adult-and-citizen population. He compared this estimate of the adult-and-citizen Hispanic population with the percentage of Hispanics in the jury venire (which he estimated as 8.3 percent) to find an absolute disparity of 8 percentage points between the percentage of adult Hispanic citizens in the general population and the percentage of Hispanics in the jury venire.

The flaw in Newell’s methodology is his assumption that every adult Hispanic person in Kern County who was not a legal, registered immigrant from Mexico was a jury-eligible United States citizen. This assumption ignored the probability that some Hispanic noncitizens were either illegal immigrants or did not originally come from Mexico. We need not engage in sophisticated statistical analysis to con- elude that Newell’s assumption is highly likely to have substantially overstated the number of Hispanic jury-eligible citizens, and thus to have substantially overstated the disparity between the percentage of Hispanics in the county and the percentage of Hispanics in the jury venire. Dr. Newell made no attempt to control for the effects of illegal immigration.

In United States v. Artero, 121 F.3d 1256, 1262 (9th Cir.1997), we rejected a challenge based on the underrepresentation of Hispanics in the jury venire when the challenge was based only on a comparison between the number of Hispanics in the total population and those in the jury wheel, instead of between the population of Hispanics who were jury-eligible citizens and those in the jury wheel. Although Sanders, unlike the defendant in Artero , has made some attempt to separate out citizens from noncitizens, the methodology employed is so inadequate that it cannot answer the “right question,” which is “whether Hispanics eligible to serve on ... juries were unreasonably underrepresented because of systematic exclusion.” Id. at 1261 . Rather, it addresses a “different question, whether Hispanics, whether eligible to serve on ... juries or not, were represented in jury wheels at a lower rate than their proportion of the population as a whole.” Id.

We take no position as to what statistical methods may be more appropriate in estimating the percentage of undoeu-mented immigrants or other noncitizens within a total population, and recognize that it may be difficult to calculate such numbers with precision. However, where no attempt whatsoever has been made to account for the percentage of undocumented immigrants within a total population that is likely to contain such persons, we are unable to perform the necessary inquiry in a prima facie Sixth Amendment jury-venire challenge: discovering whether the systematic exclusion of a distinctive group has prevented the group from being fairly and reasonably represented in the jury venire. As we noted in rejecting similarly flawed statistics in Artero, “A statistical study that fails to correct for salient explanatory variables, or even to make the most elementary comparisons, has no value as causal explanation.” Id. at 1262 (internal quotation marks and alteration omitted). Therefore, we hold that Sanders has not established a prima facie Sixth Amendment violation.

CONCLUSION

We reverse the district court and remand with instructions to grant the petition for the writ of habeas corpus as to the penalty phase, unless the state within a reasonable period of time either grants a new penalty trial or vacates the death sentence and imposes a lesser sentence consistent with law. We affirm the district court’s denial of Sanders’ habeas petition with regard to the guilt phase.

AFFIRMED in part, REVERSED in part and REMANDED.

. We take our factual statements from the California Supreme Court's opinion in People v. Sanders, 51 Cal.3d 471 , 273 Cal.Rptr. 537 , 797 P.2d 561 (1990). . In Valerio we concluded that another method of appellate curing of unconstitutional error in sentencing instruction — the combination of a narrowing construction of the improper instruction and de novo appellate review described in Walton v. Arizona, 497 U.S. 639 , 110 S.Ct. 3047 , 111 L.Ed.2d 511 (1990)—is not available when the penalty phase factfinder is a jury. 306 F.3d at 758 . . We note that mitigating evidence, unlike aggravating evidence, may not be confined to discrete, statutorily defined factors, because "the sentencer may not be precluded from considering, and may not refuse to consider, any constitutionally relevant mitigating evidence.” Buchanan v. Angelone, 522 U.S. 269, 276 , 118 S.Ct. 757 , 139 L.Ed.2d 702 (1998); see also Belmontes v. Woodford, 350 F.3d 861, 898 (9th Cir.2003) (interpreting California Penal Code § 190.3(k) as satisfying this requirement within California’s death penalty scheme). . In Florida, the weighing is done by a judge with an advisory jury verdict. See Sochor, 504 U.S. at 529-30 , 112 S.Ct. 2114 . . At trial, the jury had been instructed that "it could return a verdict of first degree murder if it found the murder was committed during a burglary in which [the] defendant entered Boender’s home with the intent to (1) steal, (2) commit an assault, (3) falsely imprison the victims, or (4) dissuade the victims from testifying.” Sanders, 51 Cal.3d at 508 , 273 Cal.Rptr. 537 , 797 P.2d 561 . . In Allen , another California death penalty case, we concluded that a jury's consideration of improperly "inflated” special circumstances and its erroneous double-counting of prior crimes as sentencing factors had no substantial and injurious effect on the verdict. 366 F.3d at 855-58 . Even disregarding the errors in counting of special circumstances and evidence of prior crimes, the jury in Allen had before it "extraordinarily damaging aggravating evidence” based on a defendant "orchestrating — from jail — a conspiracy to murder seven people, and succeeding in the actual killing of three, all to retaliate for their prior testimony against him and to prevent future damaging testimony.” Id. at 828 . In our independent review of aggravating and mitigating factors in that case, we held that "[a]fter weighing the total potential mitigating evidence against the evidence in aggravation, we are compelled to conclude that every juror would have reached only one result,” and noted that "[t]he especially aggravating circumstances of Allen's triple murder and conspiracy [i.e., multiple murders orchestrated from prison with no showing of remorse] are those for which the Supreme Court envisions the harshest penally.” As explained below, the facts of Sanders' case are not comparable. Whereas in Allen we could easily ascertain what led the jury to impose death regardless of its consideration of improper aggravating factors, here we have no such certainty. .The 1980 census numbers included both legal, documented immigrants and an unknown number of undocumented immigrants. See DAVID L. WORD, NAT'L BUREAU OF THE CENSUS, THE CENSUS BUREAU APPROACH FOR ALLOCATING INTERNATIONAL MIGRATION TO STATES, COUNTIES, AND PLACES: 1981-1991 (1992) at 2.1.2, available at http://www.census.gov/pop-ulation/www/documentation/twpsOOO 1 .html. . Dividing the same numbers used by Newell (87,025 by 402,089), the correct number would appear to be 21.64 percent. . Thus, Newell divided (87,025 minus 10,230) by (402,089 minus 14,387) to get his estimate of the percentage (19.81 percent) of Hispanic citizens in Kern County. . This is true because it is highly likely that Hispanic illegal immigrants accounted for substantially more than 19.81 percent of the total number of illegal immigrants in Kern County at the time. Immigrants from Mexico alone made up a substantial majority (10,-230/14,387, or 71.1 percent) of the legal immigrant population in Kern County, and there seems good reason to believe that the percentage of Mexicans and other Hispanics among the total number of illegal immigrants would be comparable. . For example, in order to determine whether a prima facie claim of an exclusionary jury venire in violation of the Sixth Amendment has been stated, we must determine whether the absolute disparity is of sufficient extent to create a constitutional violation. See, e.g., United States v. Suttiswad, 696 F.2d 645, 649 (9th Cir.1982) (finding insubstantial an absolute disparity of 7.7 percent). . We also deny Sanders’ motion to expand the certificate of appealability and his motion for judicial notice.