1. Statistical Measures and Methods for Analyzing Data on the Representativeness of Jury Pools
For the Workshop on Statistical Evidence at the Newton Institute
Presenter: Prof. Joseph L. Gastwirth

2. Outline 1. Legal Background
2. Measures of Underrepresentation or Disparity
3. Application to data from Duren v. Missouri
4. Application to data from People v. Bryant and Ambrose v. Booker
5. The potential impact of peremptory challenges on minority representation on juries.

3. Statistical Measures Used to Evaluate the Representativeness of Jury Pools
Duren v. Missouri: 54% (π) of the adult inhabitants of Jackson County were women. Data on the jury pools during June-October 1975 and January-March 1976 indicated that 11,197 persons were summoned, 2992 (p=26.7% were women).
absolute disparity (AD) =p- π= = -.273
comparative disparity (CD) =1-p/π=.273/.54=.506
SR=p(1-π )/(π(1-p))={(.267)/(.733)}/{(.54/.46)}=.3103
The Duren opinion emphasized that in order to establish a prima facie case the defendant needs to show that the representation of this group is not fair relative to their proportion in the community over a reasonable time period (Duren examined 8 months of data).

4. Formal Statistical Tests
The probability of observing x minorities in a sample of size n is P [B=x] = 𝑛 𝑥 𝜋 𝑥 (1−𝜋) 𝑛−𝑥
Test Statistic 𝑧= 𝑥−𝑛𝜋 𝑛𝜋(1−𝜋)
For the data in Duren Z is
𝑍= 2992−11197∗ ∗.54∗.46 = 2992− = − =−57.9.
P-value < 10 −9 i.e. a highly significant result.

5. The Disparity of the Risk measure
Detre(1994): statistical testing is not appropriate for a fair cross section analysis because the probability the composition of a jury wheel arose by random selection from the community is not directly related to the defendant’s chances of drawing a jury of a certain composition.
Notice the mistake in interpreting the p-value.
k*: the value of k for which the difference in the two binomial probabilities is greatest.

6. Table 1: The Disparity of the Risk and the Number of Minorities on the Jury when π=.50 and p=.30
F(k)
G(k)
|F(k)-G(k)| =D(k)
.0138
.0002
.0136 1
.0850
.0032
.0818 2
.2528
.0193
.2335 3
.4925
.0730
.4195 4
.7236
.1938
.5298 5
.8821
.3872
.4949 6
.9614
.6128
.3486
[1] F(k) and G(k) are the c.d.f.s of Binomial(n,p) and Binomial(n,π) random variables, respectively.

7. Comments on DR
the value k* of k, where this maximum difference occurs is not directly related to the issue of whether minority members have the same chance of being called for jury service as individuals from the majority group.
When n=12 and k*= 0, minority defendants would benefit by having a jury with one minority member.
race and ethnicity of jurors does influence jury decisions (King, 1993)and whites are influenced by he presence of Blacks on the jury (Sommers, 2006)

9. The Probability a Jury Has Fewer Minority Jurors than One Chosen from the Jury-Eligible Population
Minority under-representation is reflected by the probability (PL) a jury of 12 randomly selected from the jury pool contains fewer minorities than a jury chosen from the age-eligible population
ties are excluded in calculating PL.
The choice of threshold value of PL should be made by the courts and might depend on whether one is considering a jury of 12, a grand jury of 23 or the entire venire
Can adjust PL by subtracting the small probability that a jury chosen from a pool with minority fraction p has more minorities than one drawn form the area

10. Table 2: Statistical Measures of Minority Under-representation when n=12, π=0.2, p=p
.16
.14
.1333
.12
.10 AD
.04
.06
.0667
.08 CD
.20
.30
.3333
.40
.50 SR
.7619
.6512
.6154
.5455
.4444 DR
.1427
.2113
.2362
.2937
.3841 k* 2 1 PL
.4949
.5470
.5649
.6012
.6569

11. Figure 1: The density function of the N(0,1) and N(1
Figure 1: The density function of the N(0,1) and N(1.349,1) distributions corresponding D=.50

12. Analysis of the data submitted in the Kent County cases
Formal Statistical Test
𝑍= 163−3898∗(.0825) ∗.0825∗ =−9.23. (p< 10 −6 )
AD= , CD= and SR=.4839
The threshold of .10 for AD is inapplicable as the minority forms less than 10% of the age-eligible community.
n=12, DR=0.2439, K*=0, PL= (Mich. S. Ct. considered n=12, required DR >=.50).
n=45, DR=0.44, K*=2, PL=0.7212
N=132, DR=0.6736, K*=7

13. Questioning unfairness in the peremptory challenges
DiPrima, 1995; Gastwirth, 2005 and 2016: Recommend Fisher’s exact test
The Code of Criminal Procedure in Michigan allows 5 or 12 peremptory challenges depending on severity of the crime.
Now we focus on the reduction in the probability Fisher’s exact test has to detect an unfair system of peremptory challenges as a consequence of the underrepresentation of African-Americans in the jury pool.

14. Table 3: The probability that a venire of 45 has k African-Americans and whether (1) or not (0) Fisher’s exact test would be statistically significant if the prosecutor removes all or all but one of the k minority members from a venire of 45 in 12 peremptory challenges.
proportion k
prob.
cum prob.
remove all
remove all pvalue
0.0825 1
0.0840
0.1048
0.267 2
0.1662
0.2710
0.067 3
0.2142
0.4852
0.016 4
0.2022
0.6874
0.003 5
0.1491
0.8365
0.001 6
0.0894
0.9259
0.000 7
0.0448
0.9706 8
0.0191
0.9898 9
0.0071
0.9968
0.0417
0.2880
0.4351
0.2757
0.7108
0.1720
0.8828
0.0786
0.9614
0.0280
0.9894
0.0081
0.9975
0.0020
0.9995
0.0004
0.9999
0.0001
1.0000

15. “Undetectable” biased peremptory challenges
in Table 7 Fisher’s test will not be significant if there are no more than two minorities on the venire that are peremptorily challenged.
This occurs with probability (.2710) when minorities form 4.17% (8.25%) of the source pool.
When the venire of 45 has three minority members, the prosecutor can remove two without Fisher’s test reaching significance but not all three.
Let’s assume the prosecution will remove as many minorities as possible without triggering a stat. signif. Fisher test while the defendant will not challenge any minority member of the venire.

16. Effect of undetected challenges
Minority Share of Source Pool of Remaining Venire
8.25% %
4.1% %
1.5% (would pass DR >.50 criteria) %
Note: A jury of 12 randomly selected from a remaining venire with a minority representation of only two-tenths of 1% has probability .976 of
including NO minorities.
Thus, by accepting the DR >.50 criteria, the Mich.
Court allows prosecutors to exclude a minority group composing 8% of the eligible community from 97%
of the juries.

17. Summary
DR does not measure what claims, rather PL tells us the probability a jury will have fewer minority members than one drawn from the community.
Michigan Supreme Court was wrong to accept a DR > .50.
It is far too stringent and allows prosecutors to practically eliminate minority participation on juries in Kent County.
if a statistical analysis shows underrepresentation, but the underrepresentation does not substantially affect the representation of the group in the actual jury pool, then the under-representation does not have legal significance in the fair cross-section context (U.S. v. Hernandez-Estrada). Our analysis indicates that the effect of peremptory challenges on minority representation on the final jury, not just the venire, should be considered in assessing “legal significance”
The reanalysis of the jury composition data in Kent County is consistent with the conclusion of the 6th Circuit in Ambrose that the statistical evidence should suffice to establish a prima facie case of minority underrepresentation.