
    Joseph Clemmie MOULTRIE, Appellant, v. Joseph R. MARTIN, Warden, et al., Appellees.
    No. 81-6493.
    United States Court of Appeals, Fourth Circuit.
    Argued Dec. 11, 1981.
    Decided Sept. 30, 1982.
    
      Dennis N. Balske, Montgomery, Ala. (Charles B. Macloskie, Beaufort, S. C., on brief), for appellant.
    Lindy P. Funkhouser, Asst. Atty. Gen., Columbia, S. C. (Daniel R. McLeod, Atty. Gen., Columbia, S. C., on brief), for appellees.
    Before FIELD, Senior Circuit Judge, and WIDENER and HALL, Circuit Judges.
   WIDENER, Circuit Judge:

Petitioner has appealed the denial of his petition for habeas corpus relief pursuant to 28 U.S.C. § 2254, claiming that his incarceration by South Carolina authorities is illegal. He asserts that he was denied equal protection under the fourteenth amendment because blacks allegedly were underrepresented on the grand jury which indicted him. Moultrie’s claim is based wholly on statistics. He claims that he proved by way of statistics a prima facie case of discrimination which was not rebutted. We are of opinion that he did not establish a prima facie case, and affirm. We need not address the matter of rebuttal, although the district court held that if a prima facie case had been shown it was rebutted.

I

Petitioner was arrested on September 19, 1977 for the murder of a Colleton County deputy sheriff. A Colleton County grand jury composed of three blacks and fifteen whites returned an indictment against him on November 7,1977. At trial, the petitioner sought to quash the indictment on the ground that blacks were underrepresented on the grand jury. The trial court denied the motion to quash, and, on March 6, 1978, the petitioner was convicted of voluntary manslaughter. The South Carolina Supreme Court affirmed the conviction, finding no error in the composition of the grand jury. State v. Moultrie, 273 S.C. 532, 257 S.E.2d 730 (1978). Petitioner subsequently filed his petition for habeas corpus relief in the district court, making the constitutional argument that underrepresentation of blacks on the grand jury was grounds for relief. A federal magistrate recommended that the petition be denied, and the district court agreed. The court concluded that the petitioner had failed to demonstrate a prima facie case of discrimination because there was inconclusive proof of long term discrimination in grand jury selection. The court added that even if the petitioner had demonstrated a prima facie case, the state had successfully rebutted the case by offering credible testimony on the non-di'scriminatory nature of the Colleton County grand jury selection process. Furthermore, the district court held the petitioner had failed to demonstrate that the factual determinations by the state courts were erroneous under 28 U.S.C. § 2254(d), i.e., a failure to demonstrate by convincing evidence.

II

The method of selecting grand juries in Colleton County is mandated by the South Carolina Code. The county auditor, treasurer and clerk of court, who are all elected officials, serve as the jury commissioners. S.C.Code Ann. § 14-7-110 (1976). The commissioners meet annually to formulate a list of citizens able to perform jury duty. This list is taken from the county voting rolls and must include a minimum of two-thirds of the names on the voting rolls. Potential jurors may be exempted from the list for a variety of reasons. See generally S.C.Code Ann. § 14-7-810 to 900. Testimony at trial indicated that the Colleton County jury commissioners made most such exemptions on the basis of personal knowledge. The voting rolls contain information on each registrant’s race, sex and age, but the commissioners testified that they ignored this information. Prom the list of potential jurors, twelve people are randomly chosen each year to serve as grand jurors. Six of these twelve subsequently are chosen at random to serve for a second year. There are thus a total of eighteen people on the Colleton County grand jury each year. Id. §§ 14-7-1510 to 1560. This system produced grand juries with the indicated racial compositions during the years 1971-77:

# Blacks % Black % White Year

1971 6 94

1972 28 72

1973 28 72

1974 39 61

1975 39 61

1976 22 78

1977 17 83

TOTAL 31/126 25 75

The record does not disclose the number of blacks who were among the 12 new grand jurors chosen each year; neither does it disclose the number of blacks selected each year to serve a second year, although this information was in petitioner’s possession at the state court trial. In 1977 the Colleton County voting rolls were 38% black. Comparable statistics on the voting rolls are not in the record for other years, although such statistics should have been provided by the petitioner in order to determine accurately whether there was under-representation of blacks on Colleton County grand juries over the full 1971-77 period. Again, such information was available, but not presented at the state court trial.

Petitioner has placed particular emphasis on the racial composition of the 1977 grand jury. He notes that not only was he indicted by the 1977 grand jury, but 1977 was the only year in which all three of the 1977 jury commissioners served together. The petitioner places a secondary emphasis on the 1976 grand jury statistics because two of the three 1977 commissioners also served that year. His brief continues, “The years 1971-75 have little relevance,” but adds “Data for the years 1971-75, for what it is worth, is presented nevertheless.” Petitioner’s request that we isolate the 1977 statistics is not in accord with the mandate of the Supreme Court that “underrepresentation must be proved... over a significant period of time.” Castaneda v. Partida, 430 U.S. 482, 494, 97 S.Ct. 1272, 1280, 51 L.Ed.2d 498 (1977). We note here, and explain in greater detail later, that if we honor the petitioner’s request and isolate the 1977 statistics, a statistical analysis shows that the 1977 statistics do not provide even a marginal inference of discrimination. This is because of the small sample size that they represent. In this opinion, however, we will follow the mandate of the Castaneda court and consider the full 1971-77 period in determining whether petitioner has presented a prima facie case. We commence with 1971 only because that is the first year about which petitioner offered evidence.

Ill

The Supreme Court, on numerous occasions, has considered constitutional equal protection violations stemming from racial underrepresentation on grand juries. E.g., Castaneda v. Partida, 430 U.S. 482, 97 S.Ct. 1272, 51 L.Ed.2d 498 (1977); Sims v. Georgia, 389 U.S. 404, 88 S.Ct. 523, 19 L.Ed.2d 634 (1967); Jones v. Georgia, 389 U.S. 24, 88 S.Ct. 4, 19 L.Ed.2d 25 (1967); Whitus v. Georgia, 385 U.S. 545, 87 S.Ct. 643, 17 L.Ed.2d 599 (1967). The Court has provided the following guidelines for deciding when a criminal defendant has presented a prima facie case of an equal protection violation stemming from racial underrepresentation on a grand jury:

[I]n order to show that an equal protection violation has occurred in the context of grand jury selection, the defendant must show that the procedure employed resulted in substantial underrepresentation of his race or of the identifiable group to which he belongs. The first step is to establish that the group is one that is a recognizable, distinct class, singled out for different treatment under the laws, as written or as applied. Hernandez v. Texas, 347 U.S. [475], at 478-479 [74 S.Ct. 667, at 670-671, 98 L.Ed. 866]. Next, the degree of underrepresentation must be proved, by comparing the proportion of the group in the total population to the proportion called to serve as grand jurors, over a significant period of time. Id., at 480 [74 S.Ct., at 671], See Norris v. Alabama, 294 U.S. 587 [55 S.Ct. 579, 79 L.Ed. 1074] (1935). This method of proof, sometimes called the “rule of exclusion,” has been held to be available as a method of proving discrimination in jury selection against a delineated class. Hernandez v. Texas, 347 U.S., at 480 [74 S.Ct., at 671], Finally, as noted above, a selection procedure that is susceptible of abuse or is not racially neutral supports the presumption of discrimination raised by the statistical showing. Washington v. Davis, 426 U.S., at 241 [96 S.Ct., at 2048]; Alexander v. Louisiana, 405 U.S. [625], at 630 [92 S.Ct. 1221, at 1225, 31 L.Ed.2d 536], Once the defendant has shown substantial under-representation of his group, he has made out a prima facie case of discriminatory purpose, and the burden then shifts to the State to rebut the case.

Castaneda, 430 U.S. at 494-95, 97 S.Ct. at 1280 (footnote omitted).

In the instant case, the petitioner has met the first requirement of demonstrating a prima facie case because he is black and thus is a member of a recognizable, distinct class. Washington v. Davis, 426 U.S. 229, 241, 96 S.Ct. 2040, 2048, 48 L.Ed.2d 597 (1976). The second element of the test, however, underrepresentation over a significant period of time, is more difficult to prove. Petitioner seeks to prove underrepresentation by making a raw percentage comparison between the percentage of blacks on the grand jury and the percentage of blacks in the total population of Colleton County. This method produces varying underrepresentations between 8% and 41% over the 1971-77 period, with an average of 22%.

We reject the petitioner’s methodology for two reasons. First, the petitioner utilized the percentage of blacks in Colleton County (47%) as the statistic for comparison with the percentage of blacks actually on the grand juries. The use of this statistic for this purpose is inappropriate because the grand jury membership was based on the county voting rolls. See Sims, 389 U.S. at 407, 88 S.Ct. at 525; Jones, 389 U.S. at 25, 88 S.Ct. at 5. The voting rolls were 38% black in 1977, and, for purposes of this opinion, we will assume that figure is representative of the 1971-77 period. We give the petitioner the benefit of a considerable doubt in this assumption, for he, as noted, has not introduced evidence of what the true figures were although they were readily available.

More importantly, though, we reject the petitioner’s and district court’s method of evaluating discrimination through the comparison of straight racial percentages. Such methodology is mathematically incorrect, and we are of opinion that it has been rejected by the Supreme Court. Admittedly, the Court in Castaneda did compare the percentage underrepresentation of Mexican-Americans on grand juries with straight percentage underrepresentations in other grand jury cases. 430 U.S. at 495-96, 97 S.Ct. at 1280-1281. Nevertheless, in a lengthy footnote, the Court explained the methodology for determining whether the difference between two population percentages would be mathematically significant to a statistician and thus in court. Id. at 496-97 n. 17, 97 S.Ct. at 1281 n. 17. Subsequently, the Court has explained and adopted the reasoning of this footnote, saying “A precise method of measuring such statistical disparities was explained in Castaneda v. Partida, 430 U.S. 482, 496-97 n. 17, 97 S.Ct. 1272, 1281 n. 17, 51 L.Ed.2d 498. It involved the calculation of the ‘standard deviation’ as a measure of the predicted fluctuations from the sample.” Hazelwood School District v. United States, 433 U.S. 299, 309 n. 19, 97 S.Ct. 2736, 2742 n. 19, 53 L.Ed.2d 768. Courts in this circuit, from time to time, have used straight percentage comparisons without the necessary standard deviation analysis in proving and rebutting discrimination cases. Such decisions failed to observe standard mathematical procedures. Consequently, we now hold that, in all eases involving racial discrimination, the courts of this circuit must apply a standard deviation analysis such as that approved by the Supreme Court in Hazelwood before drawing conclusions from statistical comparisons. As part of this analysis, courts must, of course, compensate for the size of the sample being evaluated through methodology such as the student’s t distribution. See fn. 10 infra.

The reason for our holding is quite apparent. When a litigant seeks to prove his point exclusively through the use of statistics, he is borrowing the principles of another discipline, mathematics, and applying these principles to the law. In borrowing from another discipline, a litigant cannot be selective in which principles are applied. He must employ a standard mathematical analysis. Any other requirement defies logic to the point of being unjust. Statisticians do not simply look at two statistics, such as the actual and expected percentage of blacks on a grand jury, and make a subjective conclusion that the statistics are significantly different. Rather, statisticians compare figures through an objective process known as hypothesis testing. W. Hines & D. Montgomery, Probability & Statistics in Engineering Science, 235-37 (1972); F. Mosteller, R. Rourke & G. Thomas, Probability with Statistical Applications, 302-05 (2d ed. 1970) (hereinafter Mosteller, Rourke & Thomas); R. Winkler & W. Hays, Statistics: Probability, Inference, and Decision, 402-03 (2d ed. 1975) (hereinafter Winkler & Hays). The process of hypothesis testing is readily adapted to evaluation of the racial composition of grand juries. Finkelstein, The Application of Statistical Decision Theory to the Jury Discrimination cases, 80 Harv.L.Rev. 338, 349-53 (1967) (hereinafter Finkelstein).

One of the principal reasons for using a standard deviation analysis and hypothesis testing is that it is axiomatic in statistical analysis that the precision and dependability of statistics is directly related to the size of the sample being evaluated. K. Hammond & J. Householder, Introduction to the Statistical Method, 299 (1962) (hereinafter Hammond & Householder); Winkler & Hays, supra, at 444-46. Without the use of hypothesis testing, a court may give weight to statistical differences which are actually mathematically insignificant. Such a situation often arises when the sample sizes are relatively small. For this reason, it is particularly important that courts follow such formulae before drawing conclusions from statistical evidence, and we so require it. See Mayor v. Educational Equality League, 415 U.S. 605, 621, 94 S.Ct. 1323, 1333, 39 L.Ed.2d 630 (1974).

IV

The means of applying a standard deviation analysis to grand jury statistics is fairly straightforward and the appropriate formula is set forth in the margin. Applying the formula to the statistics at hand, we have determined that the actual and expected percentages of blacks on the 1977 Colleton County grand jury differ by 1.8 standard deviations. Applying the formula to the entire 1971-77 period, we find that actual and expected percentages differ by 2.9 standard deviations. Deleting the statistics for 1971, the earliest year in the period and a year for which the statistics are considerably different from the other years, we-find that the actual and expected percentages differ by 2.0 standard deviations. But for only 1971, considered separately, do the standard deviations exceed 3, and in no year in the period 1972-1977 do the standard deviations reach 2. In two of those years the standard deviations show slight overrepresentation by blacks. See table fn. 11, infra.

In Castaneda the Supreme Court stated the following guidelines to be used in interpreting such differences in standard deviations: “As a general rule for. . .large samples, if the difference between the expected value and the observed number is greater than two or three standard deviations, then the hypothesis that the jury drawing was random would be suspect to a social scientist.” 430 U.S. at 497 n. 17, 97 S.Ct. at 1281 n. 17. While the Supreme Court does not explain the adjustments necessary for small sample sizes, we have discussed them in the margin.

Applying the Castaneda criteria to the computations in the instant case, we do not believe that the petitioner has demonstrated underrepresentation. Isolating the year 1977, as the petitioner urges us to do, shows that the 1.8 standard deviations is below the Castaneda standard, even before using the small sample size adjustment. Applying the standard to the six and seven year totals reinforces our conclusion. While the seven year total at 2.9 standard deviations is greater than the lesser of two or three standard deviations, the six year total of 2.0 does not exceed that level, and we do not believe that the petitioner has demonstrated a prima facie case from the limited evidence he has offered.

The borderline nature of the above computations for the periods of 1971-77 and 1972-77 causes us to turn to other evidence. Additional evidence that could have supported the petitioner’s case would include statistics showing that the jury commissioners exempted a disproportionate number of blacks during their proceedings. The commissioners actually testified in state court that they exempted more whites than blacks, although there was no testimony as to whether these exemptions were proportionate to the racial composition of the voting rolls. In any case, we note that both trial courts which have previously considered petitioner’s argument have criticized the petitioner for failing to compile statistics on the percentage of blacks and whites exempted by the jury commissioners, even though such material was available to the petitioner. In light of this failure and the previously noted failures of the petitioner to provide data on the number of blacks among the new grand jurors chosen each year, the number of blacks among the holdover grand jurors each year, and the number of blacks among the registered voters and on the jury list for each year, we are of opinion that petitioner has not provided sufficient evidence of discriminatory underrepresentation of blacks. Coupled with the at best marginal inferences of discrimination from the evidence which petitioner has elected to provide, the district court’s conclusion that petitioner has not proved a prima facie case of discrimination merits affirmance, although we do so for a different reason. S.E.C. v. Chenery, 318 U.S. 80, 88, 63 S.Ct. 454, 459, 87 L.Ed. 626 (1943).

Because petitioner has failed’to demonstrate that blacks were significantly underrepresented on Colleton County grand juries, he has failed to meet one of the Supreme Court’s standards for establishing a prima facie case. Castaneda, 430 U.S. at 494, 97 S.Ct. at 1280. There is thus no basis for habeas corpus relief, and the district court’s decision to deny the petition for such relief is

AFFIRMED. 
      
      . The case was decided on motion for summary judgment, with which the state filed a copy of the transcript of the state trial. Other parts of the state record are not before us, no offer of that having been made.
     
      
      . The South Carolina Code provides:
      Grand jurors shall be drawn, summoned and returned in the same manner as jurors for trials and when drawn at the same time as jurors for trials, the persons whose names are first drawn, to the number required, shall be returned as grand jurors and those after-wards drawn, to the number required, shall be jurors for trials.
      § 14-7-1550 (1976).
     
      
      . The South Carolina Code provides:
      The jury commissioners of each county shall, in the month of December of each year, prepare from the official enrollment books of qualified electors a list of such electors of their county, qualified under the provisions of the Constitution, and of good moral character as they may deem otherwise well qualified to serve as jurors, being persons of sound judgment and free from all legal exceptions. Such list shall include not less than two from every three of such electors qualified under the provisions of the Constitution and of good moral character to be selected without regard to whether such persons live within five miles or more than five miles from the courthouse.
      § 14-7-140.’
     
      
      . The United States Supreme Court has upheld the constitutionality of the South Carolina jury selection scheme under a quite similar previous statute. Franklin v. South Carolina, 218 U.S. 161, 30 S.Ct. 640, 54 L.Ed. 980 (1910); see also Carter v. Jury Commission at Green County, 396 U.S. 320, 90 S.Ct. 518, 24 L.Ed.2d 549 (1976). This court, however, has determined that the scheme provides the opportunity for deliberate or inadvertent discrimination. Blackwell v. Thomas, 476 F.2d 443, 444 (4th Cir. 1973).
     
      
      . See footnote 9, infra.
      
     
      
      . Census figures showed that blacks comprised 40% of the voting age population in Colleton County in 1977.
     
      
      . Population samples of less than 30 to 40 are generally considered to be small samples and require special treatment in certain statistical analyses. P. Hoel & R. Jessen, Basic Statistics for Business and Economics 190 (1977); Winkler & Hays, supra, at 366; see footnote 10 infra. The effect of sample size on the precision of a statistic is easily illustrated. Assume there is a very large population of which 20% of the members are black. A random sample would also be expected to be 20% black, but suppose a certain sample is only 12% black. A statistician would state the difference between 12% and 20% was significant (i.e. not due to chance) with a certain degree of precision, depending on the size of the sample. For instance, if there were 1000 people in the sample, a statistician would say with 99.9% certainty that the 8% difference was significant. If there were 100 people in the sample, the precision falls to 95%; with 50 people the precision is 84%; with 25 it is 68%; with 10 it is 47%; and with 5 it is 14%.
      Statisticians do not usually state their conclusion in terms of precision. Rather, they state whether the difference between actual and expected values is statistically significant at a given confidence level. Statisticians usually use 95% or 99% confidence levels. Winkler & Hays, supra, at 413; see Finkelstein, supra, at 364. Thus in the above example, only when the sample is 100 or larger would a statistician find the difference between 12% and 20% statistically significant at a 95% confidence level.
     
      
      . The formula for determining the standard deviation of a binomial distribution is:
      
        
      
      where n is the sample size (i.e. total number of jurors in the sample being scrutinized) and p is the proportion of individuals possessing the characteristic which is being investigated (i.e. the percentage of blacks on the voting rolls). Winkler & Hays, supra, at 214. This formula can be adjusted so that it gives the standard deviations in terms of population percentage by dividing by n. This is called the standard error. Id. at 308.
      'A formula which can be used to determine by how much the actual and expected population percentages differ in terms of standard deviations is as follows:
      
        
      
      where a is the proportion of the minority actually present, p is the proportion that would be expected and n is the total sample size. This is the formula that will be used in the balance of this opinion.
       
      
      . In computing this statistic, we analyzed the representation of blacks on the 1977 grand jury, rather than the representation of blacks among the group selected by the 1977 jury commissioners. Especially in view of petitioner’s assertion that the problem is with the 1977 selection process, the latter statistic would be more appropriate. The racial composition of the new jurors selected in each year is analogous to the applicant flow data used in employment discrimination cases. Courts have noted that applicant flow data is highly relevant in determining the existence of discrimination. E.g., United States v. Fairfax County, 629 F.2d 932, 940 (4th Cir. 1980), cert. denied, 449 U.S. 1078, 101 S.Ct. 858, 66 L.Ed.2d 801 (1981). Such information, however, is not in the record. The transcript of the state trial discloses, though, that one of the petitioner’s own witnesses had that information at hand because the witness had the rosters of the grand juries for all years in question. The racial composition of the 12 jurors chosen in 1977 easily could have been determined by comparing the 1976 and 1977 jury rosters. Similarly, the racial composition of the jurors selected each year 1971 76 also could have been so determined.
     
      
      . The Supreme Court’s rule in Castaneda of course can be adjusted for small sample sizes through the use of the student’s t distribution. The student’s t distribution teaches that when the sample size is less than approximately thirty, the number of standard deviations must be increased in order to achieve the same significance level. See Winkler & Hays, supra, at 366; Mosteller, Rourke & Thomas, supra, at 437 38. The student’s t distribution, like the binomial distribution used in Castaneda, is represented by a bell shaped curve. When the sample size is small, the student’s t curve is flatter in the middle and plumper in the tails. As the sample size increases, the student’s t curve is approximated by the binomial distribution. Winkler & Hays, supra, at 364. While a precise mathematical formula exists for computing the student’s t distribution, in practice it is more easily computed by the use of tables found in standard books concerning statistics.
      Employing these tables, the Court’s Castaneda analysis is easily adapted to small samples. The two to three standard deviations noted in Castaneda correspond to a 95% to 99.9% significance level on the two-tailed binomial distribution. Winkler & Hays, supra, at 355-58; see footnote 7, supra. Student’s t tables are stated in terms of significance level and degrees of freedom. The number of degrees of freedom is equal to the sample size minus one. The tables thus show that when the sample size is 15 (i.e. 14 degrees'of freedom), one needs to use 2.1 standard deviations for a two-tailed 95% significance level and 3.8 standard deviations for a two-tailed 99.9% significance level. For a sample size of five, the standard deviations become 2.8 and 7.2 respectively. Winkler & Hays, supra, at xv.
     
      
      . For a sample size of 18 (i.e. the number of jurors), the Castaneda range becomes 2.1 to 3.6 standard deviations. See footnote 10 supra.
      
      Computing the difference between actual and expected number of jurors, in terms of standard deviation, for each of the years 1971-77 shows the following results:
      Year i Black Jurors No. Std. Dev.
      1971 c 6 -3.4
      1972 28 -0.9
      1973 28 -0.9
      1974 39 + 0.1
      1975 39 + 0.1
      1976 22 -1.4
      1977 17 -1.8
      We emphasize again that the computations contained in this opinion are based on the limited statistics which the petitioner has provided. They are not the statistics which are necessary for a truly accurate computation of whether the actual and expected percentage of blacks on the grand jury are significantly different. As mentioned before, these are the numbers (or percentages) of blacks on the jury lists and voting rolls, those drawn for new grand juror service and those drawn for holdover service. We would expect such a full disclosure of statistics because the petitioner’s case is based exclusively on statistical evidence.
      We repeat the two principal assumptions made in these computations because the petitioner did not give us full information. First, we assumed that the percentage of blacks on the 1977 voting rolls was constant during the entire 1971-77 period. This assumption may well be invalid. If the percentage of black voters actually was higher in 1977 than in any or all of the years 1971-76, then the petitioner’s already weak statistical case would have been even weaker. Second, the lack of complete information required that our computations be based on the total number of jurors who served each year, rather than the number of new and holdover grand jurors selected each year. This leads to an inaccurate result because the petitioner has assumed, as we must to make any computation, that the expected ratio of blacks among the holdover grand jurors is the same as those drawn from the list. Without full racial data on the new and holdover jurors for each year, as well as the voting lists and jury pool, however, we cannot make a complete statistical inquiry into the existence of discrimination.
     
      
      . The petitioner’s claim in this case is based wholly on statistical evidence. Of course we do not preclude litigants from demonstrating racial discrimination by other means than by statistical evidence.
     