Figure 1. Loci of points in the diversity space of hypothetical Groups A and B. The curves were generated from Gaussian criminality distributions using a different set of parameter values for each curve. The dashed line through the origin with unit slope corresponds to A and B having identical criminality distributions.

   Justice changes with the times. The tough-on-crime political climate of the past few decades caused prison populations to swell, but at different rates. Minority imprisonment grew fastest. As the gap between whites and minorities widened, tons of interpretive commentary were generated, and more than a few doctoral dissertations were spawned. The fact is, however, that incarceration rates precisely followed trajectories in diversity space predicted by Equations 1.

   Incarceration rates are a window on criminality. When plotted in diversity space, the rates trace out a path belonging to the family of curves illustrated in Figure 1, and determined by underlying criminality distributions. Finding the distributions, then, is simply a matter of identifying which of the infinity of curves in the family has been traced out.

Incarceration statistics
   The Bureau of Justice Statistics (BJS) assembles and makes public national incarceration data disaggregated by race and ethnicity. But raw incarceration rates are not suitable axes for diversity space. The problem with raw data is that they do not include ex-prisoners. Living ex-prisoners, however, are as important to the assessment of group criminality as the current crop. In 2001 former prisoners accounted for 77% of all adult U.S. residents who had ever been incarcerated. Omitting them from the analysis would seriously distort its outcome.

   Another problem surfaces in periods of rapid demographic flux. Many prisoners are in the middle and terminal stages of long sentences. Fast growing groups like Hispanics have fewer such prisoners because their population was appreciably smaller when these sentences began. As a result, criminality in these groups may be underestimated.

   Both difficulties can be overcome. For selected years between 1974 and 2001, BJS statisticians have estimated the number of people ever sentenced to State or Federal prison. The estimates, broken down by gender, race, ethnicity and age are reported in the BJS publication "Prevalence of Imprisonment in the U.S. Population, 1974-2001." (Regrettably, no estimates were made for Asians.) Figure 2 is a graphical representation of the BJS data for adult black, Hispanic and white males.

Figure 2. Percentages of adult black, Hispanic and white males ever sentenced to State or Federal prison. Source: Bureau of Justice Statistics; "Prevalence of Imprisonment in the U.S. Population, 1974-2001."

   A second BJS tabulation is also useful. From mortality and incarceration data, BJS demographers have estimated the lifetime likelihood of being confined at least once in a State or Federal prison. The estimates, for selected birth years between 1974 and 2001, disaggregated by race, ethnicity, age and sex can also be found in "Prevalence of Imprisonment in the U.S. Population, 1974-2001." 

   Some extraordinary results emerged from the lifetime-likelihood calculations such as the conclusion that a black male born in 2001 has a 32 percent chance of being imprisoned at least once in his lifetime. Figure 3 displays, by birth year, lifetime likelihoods of incarceration for black, Hispanic and white males.

Figure 3. Lifetime likelihood of being sentenced to State or Federal prison for black, Hispanic and white males. Source: Bureau of Justice Statistics; "Prevalence of Imprisonment in the U.S. Population, 1974-2001"

Criminality distributions
   Parameters in the criminality distributions of white, Hispanic and black males were adjusted to find the trajectories in diversity space that best fit the BJS ever-incarcerated and lifetime-likelihood data. The procedure is described in the Appendix to Intelligence, Gender and Race. (Substitute "criminality" for "g," and "incarceration rate" for "cognitive threshold" wherever these terms appear.)

   Figure 4 shows the trajectory in black-white diversity space traced out by the BJS data together with the path predicted by Gaussian distributions of criminality. As is evident, the fit is excellent.

Figure 4. Least-squares fit of BJS incarceration data for black and white males to the trajectory in diversity space derived from Gaussian criminality distributions.

   Figure 5 shows a closeup of the fit.

Figure 5. Closeup of the fit of BJS incarceration data for black and white males to the trajectory in diversity space derived from Gaussian criminality distributions.

   Figures 6 and 7 are the corresponding graphs for Hispanic and white males.

Figure 6. Least-squares fit of BJS incarceration data for Hispanic and white males to the trajectory in diversity space derived from Gaussian criminality distributions.

Figure 7. Closeup of the fit of BJS incarceration data for Hispanic and white males to the trajectory in diversity space derived from Gaussian criminality distributions.

   As seen above, in Figures 4 - 7, twenty-seven years and several kinds of incarceration data fall on a single curve in diversity space -- one of the family of curves generated from Gaussian distributions of criminality. We infer from this that criminality distributions for white, black and Hispanic men have persisted unchanged for more than a quarter century.

   Parameters of the distributions are listed in Table 1.

	Standard Deviation	Mean Criminality	Variance Ratio (x/white)
black	0.676	1.269	0.458
Hispanic	0.630	1.018	0.397
white	1.0	0	1.0
Table 1. Gaussian criminality-distribution parameters for black, Hispanic and white males. The white distribution is taken as the reference distribution. Its mean is set to zero, and its standard deviation to unity.

   Normalized criminality distributions for adult white, Hispanic and black males are shown in Figure 8. They are accurately drawn using the parameters in Table 1.

Figure 8. Criminality distributions of white, Hispanic and black men. The abscissa is marked in standard deviation units of the white distribution.

   Profound differences between white and minority criminality distributions are evident in Figure 8. There is a white-black gap of 1.27 SD and a white-Hispanic gap of 1.02 SD (both in units of the white standard deviation). Also conspicuous is how narrow the minority distributions are compared to the white. The black-to-white variance ratio is just 0.458, and the Hispanic-to-white ratio is even smaller at 0.397. Black and Hispanic men form remarkably homogeneous groups, each displaying relatively modest variation within their ranks.

   The contrast between white and minority variance is better visualized when viewed with a common origin (Figure 9).

Figure 9. Criminality distributions for Hispanic and white men plotted with a common origin.

Reducing the black/white disparity ratio
   A standard way of gauging racial disparities in the criminal justice system is via the disparity ratio -- the ratio of minority to white incarceration rates. Numerical values of the ratio vary depending on how incarceration rates are defined. Because we are concerned with criminality, meaningful rates must include former as well as current prisoners.

   What happens to disparity ratios when general levels of incarceration change?  Consider the black-to-white ratio. We have already established a relationship between white and black incarceration rates (Equations 1). Consequently, finding how a ratio of these two quantities varies with either rate it is a straightforward affair. Figure 10 shows how the disparity ratio is predicted to vary with white incarceration. The curve in Figure 10 was generated using the criminality-distribution parameters in Table 1.

Figure 10.  Predicted variation of the black/white disparity ratio with white incarceration.

   Incarceration rates skyrocketed in the years for which we have suitable BJS data (1974 - 2001). But despite substantial increases in the prison population -- white incarceration increased by 86% -- the black/white disparity ratio held fast at 6.2 + 0.18.

   Though the entire range of white incarceration, 0% to 100%, is formally possible, only a tiny part of this range is ever observed. Between 1974 and 2001, for example, the percentage of white men ever serving time in State or Federal prison varied from a low of 1.4% in 1974 to a high of 2.6% in 2001. Coincidentally, this range is where the black-to-white disparity ratio changes most slowly (Figure 11). At the levels of white incarceration seen between 1974 and 2001, the predicted dependence of the disparity ratio on white incarceration is almost flat. But the near constancy of the ratio is accidental. It is the result of the particular balance of crime and punishment extant in the U.S. between 1974 and 2001 that kept white incarceration within the bounds shown in the figure. 

Figure 11.  Observed black/white disparity ratios shown with the curve of predicted ratios.

   Like the man who looks around him and concludes the earth is flat, the criminologist is blind to what will happen if incarceration rates spread beyond the limited domain of his experience. He is unaware that the black-to-white disparity ratio peaks at about 6, and that either growing or shrinking the prison population should cause the ratio to drop. He cannot know these things because he has not had the benefit of a coherent theory that puts incarceration into statistical perspective.

   To lower the disparity ratio, La Griffe offers two courses of action: Set the bar for incarceration so low that even jaywalkers are jailed. Or, set it so high that only murder would warrant imprisonment. Either way, dramatic reductions would ensue. If you are willing to settle for only a 50% drop in the disparity ratio, that could be achieved simply by adjusting the crime and punishment balance so that 28% of adult white males would be sentenced to prison. And, if that should prove too costly, 0.06% will achieve the same result.

We began this essay with three questions from Professors Oliver and Yocom.

Q. Why would the black-white disparity in imprisonment grow over time?

Q. Why would the prison system become increasingly targeted on Black people?

A. Nothing suggesting targeting of black people matches the facts. When general levels of incarceration increase, black and white incarceration rates follow a path precisely accounted for by group criminality distributions. In the tough-on-crime environment of the past 30 years, incarceration grew across the board. As levels of imprisonment rose, black and white incarceration rates traced out a predictable trajectory in diversity space. Black imprisonment grew as predicted -- approximately six times as fast as the white. (See Figure 4 or Figure 5.) None of this had anything to do with bias, racism, repressive injustice, racial profiling, political repression or the economy. It was an inevitable consequence of underlying group criminality distributions.

Q. What has been going on in United States race relations to make racial differences in imprisonment so much worse in the years after the Civil Rights Movement?

A. Since criminality distributions account quantitatively for racial disparities in imprisonment, I suppose one could argue that race relations have shaped the distributions, but La Griffe leaves that to those better schooled in the casuistic arts.

Serial killers
  Serial murder lends itself nicely to the analytical methods developed here provided suitable data are available. Only a few systematic accounts of the race and ethnicity of serial killers exist. There is an exhaustive study of African American serial killers (Homicide Studies 2005; 9; 271) by criminologist Anthony Walsh, and also Eric W. Hickey's book, Serial killers and their victims, Brooks/Cole, 1997. That's about it.

   Walsh identified 90 African American and 323 white American serial murderers, all men operating between 1945 and 2004. (Though Asians, Hispanics and Native Americans were not included in the study, their addition would not alter the fact that most serial killers are white men.) African American serial killers made up 22% of Walsh's sample, a figure in close agreement Hickey's enumeration. But, as blacks number well below 22% of the U.S. population, they are actually overrepresented among the ranks of serial killers -- roughly by a factor of 2.

   So, why are most serial killers white men? The short answer is that serial murder is primarily a male enterprise, and in the U.S. white men outnumber black men seven to one. The long answer, however, is instructive**.

   The ratio of black to white criminal offenders generally increases with the progression from passive white collar crime to pernicious violent crime. In 2003, for example, blacks were imprisoned for violent crimes at approximately 7 times the per capita rate of whites, but only 4 times the white rate for property crimes. Serial murder, by any reckoning, ranks at the top of this progression. Consequently, black serial murderers should, per capita, outnumber their white counterparts by at least a factor of seven. In actuality, the factor is closer to two. The appropriate question, then, is not why most serial killers are white men, but rather why black men are only twice as likely as white men to be serial murderers. Criminality distributions provide the answer. The mean criminality of black males is almost 1.3 SD greater than the white. But, with a black-to-white variance ratio of 0.46, white men display much greater variability. A symmetric way to view this is in terms of black homogeneity rather than white variability. Either way, at the extraordinarily high level of criminality peculiar to the serial murderer, white representation increases. We explore this in more detail in the next section.

Hispanic serial killers
   Though, data on Hispanic serial killers is scant, we have learned enough to make some predictions concerning them. We ask two questions: What is the minimum value of criminality in the population of serial killers? And, what percentage of white, black and Hispanic men achieve this criminality?

   Consistent with Walsh's black to white serial-killer ratio of 90/323, the minimum criminality, C_o, of serial murderers will satisfy the following relation: