Criminality Distributions

Via iSteve, "The Growth, Scope, and Spatial Distribution of People with Felony Records in the United States":

"We estimate that 3 % of the total U.S. adult population and 15 % of the African American adult male population has ever been to prison; people with felony convictions account for 8 % of all adults and 33 % of the African American adult male population."

Note that given these numbers, and assuming blacks constitute 12% of the U.S. population, the non-black felony conviction rate is 4.6% (arrived at by this formula:(.08 - .33*.12)/(1 - .12)); Likewise, the non-black incarceration rate is 1.4%.

This implies that blacks are 7.2x overrepresented among felony convicts and 11x overrepresented among prisoners.

These data can be explained by a model in which anti-social behavior is normally distributed, with the black mean higher than the non-black mean and a higher threshold for incarceration than felony conviction. We are all familiar with the bell curve:

A few words of explanation: Shown are two normal distributions, each with variance one. The "Group A" mean is zero, as shown on the x-axis; the "Group B" mean is one standard deviation above the Group A mean. The left Y-axis is the probability distribution function (PDF) value. It is meaningless in itself; obtaining percentages requires integration (i.e. the area under the curve). On the right axis is the value of the ratio of the percentages of the two distributions lying above the threshold indicated on the x-axis. For fellow stats geeks, the ratio is (1 - F_B(x))/(1 - F_A(x)), where F(x) is the cumulative density function (CDF) corresponding to each PDF.

For instance, Exactly 50% of the Group A distribution falls above zero, while 84% of the group B distribution does; their ratio is .84/.5 = 1.7. 16% of the Group A distribution falls above 1, while 50% of the Group B distribution does; their ratio is 3.1. 2% of the Group A distribution falls above 2, while 16 % of the Group B distribution does; their ratio is 7. These ratios continue to climb as shown.

The ratios given by the study can be accounted for by a model of anti-social behavior where the black mean of anti-social behavior is 1.3 standard deviations of the non-black distribution above the non-black mean, whereas the non-black standard deviation is .87 of that of the non-black standard deviation:

That the black criminality distribution is narrower than the non-black distribution should not surprise us; the non-black population is by definition more heterogeneous, including Hispanics with higher rates of criminality than the white majority and Asians with lower rates. A more rigrous model would attempt to break these out separately. In the meantime, one of the implications of the narrower black distribution is that at sufficiently high thresholds we see from the graph that the ratio begins (just barely) to level off; however, this is an artifact of combining disparate populations into a non-black category.

This model is useful for predicting outcomes as reported in this Columbus Dispatch article about racial disparities in the juvenile system:

Delinquency filings in county Juvenile Court fell from 6,247 in 2010 to 2,457 in 2020, a 61% drop. The number of youths admitted to the county's juvenile detention center fell from 2,196 in 2010 to 602 in 2020, a 73% decline. . . .

In 2020, Black teens in Franklin County were two-and-a-half times more likely to be charged with a delinquency offense than white teens — in a county where less than 25% of the population is Black. . . .

In Franklin County, the detention numbers are particularly stark. Black juveniles represented 84% of the youths admitted to the county detention center in 2020. That number has grown worse since 2010, when Blacks accounted for 71% of the juveniles detained.

On average, the daily population in the county detention center in 2020 included 47 Blacks and four whites. And the median length of stay in detention — where youths are held before their cases are resolved — was 12 days for Black juveniles and three days for white juveniles. . . .

The 65 Black juveniles who were given an out-of-home placement in 2020 represented a 70% drop from 2014. But the 13 whites youths placed out-of-home last year represented an 89% drop during the same period. . . .

Based on raw numbers, Balis said, Franklin County's Juvenile Court is to be commended for the drop in out-of-home placements for Black teens. The number of such placements stood at 215 in 2014.

Here is the demographic information for Franklin County, Ohio, of which 22.9% is black and 62% is white. As is common in mainstream reporting, the article combines multiple measures; I count "delinquency filings", "charged and detained", "charged", "admitted to county detention", "detained", "daily population", and "out-of-home placement". I kind of get that journalists want to mix up their formulations for stylistic reasons, but it means that I can only guess which terms are supposed to be equivalent. But youths are "charged", where the article claims is a 2.5x black/white rate ratio, far more often than they are "detained", where there is a (47/.229)/(4/.62) = 32x black/white rate ratio, assuming that the ratios of the juvenile population is the same as the overall population, which assumption is almost certainly incorrect, but is the best data we have.

However, it is apparent that Franklin County raised the thresholds at which juveniles are both charged and detained. So it should not suprise us to see that of the two measures for which black numbers are provided in the same context for two different years, we see that the percent black of detentions climbed from 71% to 84% between 2010 and 2020, implying an increase in the black/non-black rate ratio from 8x to 18x, and the black/white ratio of out-of-home placements climbed from 1.8x (i.e. 215//(13/(1-.89)) = 215/118) in 2014 to 5x (65/13); given their percentages of the population, this implies an increase in the black/white rate ratio from 5x to 14x.

That said, these numbers don't necessarily fit the parameters developed from the nationwide conviction and incarceration data shared earlier. For instance, as you can see from the graph, the black/non-black anti-social behavior ratio of 5x occurs right around the median of the black distribution, and obviously somewhat less than half of all black youths in Franklin County received out-of-home placements in 2014. But otherwise we should expect the black/white and black/non-black rate ratios to be at least this high, and higher still as the threshold for punishment increases.