Suppose a half-white, half-black school system suspends 1 percent of its students for disruptive behavior. What is the most probable racial makeup of the suspended group?

   The question began the section, "Games with Fuzzy Variables," but December's game is June's analytical scalpel. With it we will peel back layers of misinformation to reveal race and school discipline in a greater social context.

   The year 1975 witnessed the first widely-disseminated report on racial disparities in school discipline. Issued by the Children's Defense Fund, the report found that black students at all levels were suspended at 2 to 3 times the rate of whites. The bias hunters had arrived. Hundreds of studies and millions of dollars later they would learn little more.

   Powerful statistical techniques - analysis of variance, χ², regression, correlation analysis and more - have been brought to bear on school discipline and race. Each new study adds to the annals of redundant findings. The essential result is now well established: Referrals, suspensions and expulsions are distributed asymmetrically with respect to race and ethnicity. So what new can La Griffe bring to an area so thoroughly picked over? Surprisingly, quite a bit.

   The groundwork for our analysis was laid in December in the essay, "Aggressiveness, Criminality and Sex Drive by Race, Gender and Ethnicity." There we developed the method of thresholds, a procedure that takes rates at which different groups cross behavioral thresholds (like committing assault), and from them constructs yardsticks for "fuzzy" variables that are otherwise hard to define (like aggressiveness). The method lends itself well to the analysis of race and school discipline.

   In December, using assault as a threshold for aggressiveness, we found a black-white aggressiveness gap of 0.37 standard deviations (SD), blacks being the more aggressive. Data came from two sources. From INTERPOL we obtained rates of "serious assault" in 64 African and European countries. From the Justice Department's National Crime Victimization Survey (NCVS) we obtained US assault data. Each source independently produced the same 0.37 SD aggressiveness gap. Even allowing for some degree of fortuity, the black-white aggressiveness gap was found to be invariant to cultural shifts across continents, compelling evidence that aggressiveness distributions differ among races and are intrinsic to them.

   The threshold for "serious assault" falls way out on the aggressiveness scale, 2.86 SD from the black mean. The NCVS survey identified lower levels of assault from simple on up. Its threshold was much lower, 1.64 SD from the black mean. That aside, both data sets yielded the same black-white gap. From the standpoint of the method of thresholds, expulsion and suspension from school are simply new thresholds to be placed on the aggressiveness axis.

Discipline and Group Differences
   Bias hunters who look for racial disparities usually find them. Their papers, though sullied by subjectivity, are valuable for the data they include. Such is the case of a much-referenced paper that appeared a few years ago. The paper, "Office referrals and suspension: Disciplinary intervention in middle schools," Skiba, R. J., Peterson, R. L., and Williams, T., Education and Treatment of Children, 20, 295-315, (1997), examined disciplinary records in a large Midwestern urban school district for the 1994-95 school year. The large-scale study included all 11,001 middle-school students in the district, 98 percent of whom were either black (56%) or white (42%). By analysis of variance Skiba et al. convincingly demonstrated that African American students were suspended and expelled at rates disproportionate to their population in the schools. More important, the paper included raw data uncompromised by expectation or prejudice.

   Skiba et al. found suspension rates of 17.1 percent for whites and 27.0 percent for blacks. We applied the method of thresholds (see Appendix) to these rates. It revealed a black-white aggressiveness gap of 0.34 SD, a value differing insignificantly from the assault data value of 0.37 SD. On the aggressiveness axis, suspension fell at 0.61 SD from the black mean, much lower than assault (1.64 SD), and lower still than "serious assault" (2.86 SD).

   Figure 1 shows distributions of aggressiveness for blacks and whites. Drawn to scale, the curves are separated by a mean difference, Δ, of 0.37 SD. The thresholds for suspension (Skiba data), assault and "serious assault" are marked on the aggressiveness axis.

Figure 1.  Aggressiveness distributions. The curves for blacks and whites are separated by a mean difference, Δ = 0.37 SD. Thresholds for suspension, assault and "serious assault" are designated λ₁, λ₂ and λ₃, respectively.

 


where P is the normalized aggressiveness distribution for blacks, Δ is the black-white mean difference (0.37 SD) and λ is the threshold of aggressiveness beyond which suspension (or expulsion) kicks in. (The quantity λ is school or system dependent.)

   Equation 1 describes how discipline would break down by race if students received punishment in rank order of aggressiveness. The first term on the left side of (1) represents the number of blacks who would be disciplined, the second term the number of whites. Solution of (1) yields the threshold, λ, from which the individual terms may be evaluated.

   In the Skiba study, N_B = 6161, N_W = 4620, and N_S = 2461 suspended or 42 expelled. Solution of (1) yielded the results shown in Table 1.

 

When the number of suspended or expelled students from a given (racial) group differs from that predicted by Equation 1 by two or more standard errors (computed from Equation 2), there is reason to suspect extrinsic factors including (racial) bias as a cause. 

   In the Skiba example, n* = 1692, and the total number of suspended students, N_S = 2461, giving a standard error of 23. The difference between the predicted and actual numbers of suspended blacks is 0.17 σ. For expulsion, the difference is 0.48 σ. Plainly, bias was not a factor.

Closing the Behavior Gap
   What human attribute is most responsible for the advancement of civilization? What capability has it to build an internal combustion engine, what capacity to produce Chateau Margaux 1982? If forced to choose but one, it is the ability to solve problems. Unacceptable group behavior is just another problem to be solved.

   Recently, a remarkable paper appeared in the Journal of the American Medical Association that bears upon this issue. The article, "Long-term Effects of an Early Childhood Intervention on Educational Achievement and Juvenile Arrest," A. J. Reynolds, J. A. Temple, D. L. Robertson and E. A, Mann, JAMA, 285, 2339-2346 (2001), is a 15-year follow-up of urban, mostly black, low-income children who matriculated through an early intervention program in the Chicago public schools. The authors describe impressive long-term gains made by the children. We put their data through the Griffian engine and thus obtained an unexpected and extraordinary result.

   The intervention strategy described in the paper is that of the Chicago Child-Parent Center Program (CCPCP). According to Reynolds et al., it incorporates "comprehensive education, family, and health services and includes half-day preschool at ages 3 to 4 years, half- or full-day kindergarten, and school-age services in linked elementary schools at ages 6 to 9 years."

   Outcomes for the 989 children who went through the program were compared with those of a 550-child matched cohort. The comparison group participated in alternative programs such as kindergarten with or without additional services, but without early preschool. Both group outcomes are summarized in Table 2.

where P is the normalized distribution of aggressiveness for blacks, and Δ is the black-white mean difference in aggressiveness.

   In Skiba et al., the fractions F_B and F_W are 0.270 and 0.171 respectively. Using these values (A.1) and (A.2) may be solved simultaneously for λ and Δ. Taking P to be Gaussian, the result is Δ = 0.34 and λ = 0.61.