Assessing Under-Representation
Courts have relied on three statistical approaches to assess under-representation. All compare the minority group’s proportion, p, of venire members to their fraction, π, in the eligible population.

The first approach, absolute disparity (AD), considers the difference p-π, expressed in percentages. For the two periods, AD = -1.28% and AD = -1.11%, respectively.

The second measure, comparative disparity (CD), is AD/π, or the relative shortfall. The CDs are -18% and -15% for the periods.

The third approach is based on a standardized test statistic z, the ratio of the difference between the observed and expected numbers of minority members and its standard deviation. Courts call this method “standard deviation analysis,” and shortfalls of two to three standard deviations, corresponding roughly to statistical significance at the .05 or .01 level, support the defendant’s claim of under-representation.

To conduct a formal statistical test employing the difference between the estimated number of African-American individuals in the jury pool and its expected number, based on the census data for the county, we calculated its standard deviation. It differs from the standard deviation of the binomial model, which is typically used in jury discrimination cases. Applying this test to the data for the two periods showed that the shortfalls in African-American jurors in the first and second periods were found to be equivalent to 2.45 and 3.17 standard deviations, respectively. These statistically significant results should satisfy the second Duren v. Missouri criterion.

To assess whether the defendant’s claim that differences in juror assignment in the two periods created the shortfall, one would calculate CD1 – CD2 and find that the standardized form of statistic-T is .3, with a one-sided p-value of .382. This substantiates the court’s finding that the difference between the 18% CD and the 15.1% CD in the two periods was insufficient to meet the third Duren v. Missouri criterion (i.e., the original assignment system had a significantly adverse impact on the representation of African Americans on the circuit court venires).

Another statistical point should be noted. Many locales exclude individuals who don’t understand English, or felons still on parole from serving on juries; the minority fraction, π, obtained from the census data is not quite accurate. This could be accounted for by analyzing the minority fraction of jury questionnaires before individuals are excused from service.

Our experience with this case also suggests the judiciary would benefit from a better understanding of fundamental concepts of hypothesis testing. Both the U.S. and Michigan Supreme Court opinions state, “Standard deviation analysis seeks to determine the probability that the disparity between a group’s jury-eligible population and the group’s percentage in the qualified jury pool is attributable to random chance.” The probability referred to, of course, is the p-value, which is calculated assuming random (chance) selection of the jury pool from the eligible population.

We also believe the concept of power would help inform courts when the size of the available sample is too small to detect a legally meaningful shortfall or is so large that a minor difference would be classified as significant. Consider the “four-fifths” rule used in disparate impact employment discrimination cases in which the ratio of the probability that a jury-eligible minority member is called for service to the corresponding probability of a white should be at least .80. Suppose the court wished to have a high power, say 90%, of detecting a shortfall for this rule. Applying this to Berghuis v. Smith for the first period, the power of the usual test—based on the binomial model with π = .0728—for a sample of 929 is less than 40%. In such situations, a finding of nonsignificance should not be given much weight by the courts. Had the court been made aware of these considerations, it might have required jurisdictions to preserve data for several years, or at least a sufficient time that the demographic characteristics of an adequate number of potential jurors would be available to check the representativeness of the jury pool.

At the hearing, the sample size issue was actually noted by Justice Sonia Sotomayor. She observed that the sizes of the jury pools in the two periods differed substantially and questioned the soundness of comparing the minority proportions in the two periods.

Editor’s Note: This commentary is based on a paper submitted for publication in Law, Probability, and Risk. The paper contains citations to the articles mentioned here and can be found on the Social Science Research Network.

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