Why The War On Drugs Matters In Mass Incarceration, Part 2: The Two Dimensions Of Prison Populations.


This is the second part of a (hopefully) five-part series on the effects of “The War On Drugs” on “mass incarceration”.

In the first part, I pointed out that drug offenses account for a large share (perhaps 40% +/- a bit depending on the baseline year) of the growth in prison terms issued over the “mass incarceration” period, while violent offenses only account for 21% or so:

So why do people get different answers if they look at “the incarceration rate” instead of “the prison sentence rate”? John Pfaff, for example, made this table noting that drug offenses only account for 21% of the growth in state prison populations, while violent offenses account for 52%:

Which is more important, then? Drug offenses or violent offenses? And again, where does the difference come from?

Frankly it took me some time to understand how to think about the difference between “prison admissions” and “prison populations”. I had to come up with a particular visualization approach that I call “prison population rectangles” before I really saw it clearly.

Since this approach helped me, maybe it can help other people as well, and perhaps lead to more productive discussions on this topic.

The Math Of Prison Populations:

The math of the distribution of prison terms can get pretty complicated, but in some ways the math of “prison populations” is pretty simple. Basically, if nothing is changing, then over time the equilibrium is:

Prison population = (Annual number of prison terms) * (Average number of years served in prison)

For example imagine a small county that sends one person to prison a month (12 a year) for six months each (0.5 years). Right now the “prison population” would consist of the prisoners who entered in April, May, June, July, August, and September, so there’d be six prisoners — that is, (12 prison terms a year)* (0.5 years)= 6 prisoners.

This isn’t math unique to me, of course. In fact “prison population/annual number of prison terms” is often used in studies as a proxy measure for average prison time. This is the approach I follow below as well.

Visualizing The Two Dimensions Of The Incarceration Rate:

I eventually realized that prison populations lend themselves to a natural visualization format: If, as I said above,

Prison population = (Annual number of prison terms) * (Average number of years served in prison),

then you can think of “the prison population” as the area of a rectangle, with the annual number of prison terms as the height, and the average number of years in prison as the width. Area = (height)*(width), prison populations = (terms)*(length).

On its own this kind of graphic isn’t very interesting, since the horizontal and vertical scales are arbitrary. Where it can be helpful, I think, is in visualizing comparisons.

Using “Prison Population Rectangles” For Comparisons:

Here is the “rectangle” of the 1980 incarceration rate on top of the “rectangle” of the 2011 incarceration rate:

The 2011 rectangle is more than twice is high as the 1980 rectangle — that is, the new state prison term rate went from 58 in 1980 to 128 in 2011. The average prison term length increased by about 49%, from 2.2 years to 3.3 years, based on the incarceration rate increasing from 130 in 1980 to 431 in 201 ((431/130)/(128/58) = 1.50, although it’s 1.49 without rounding).

Visually, it certainly seems like a large majority of the growth in “the incarceration rate” is vertical (from more prison terms) rather than horizontal (from longer terms). However as Neal and Rick also noticed (see sources and notes), this is somewhat contradicted when you break the statistics down by offense.

Prison Rectangles By Offense Categories:

As it turns out, breaking this visualization down by offense helps to clarify the answer to our original question: Why do different offenses seem important when you look at “prison admissions” and “prison populations”?

Here are the 1980 and 2011 “prison population rectangles” by offense category, along with how much each category contributed to the growth in incarceration rate/area:

Most of the vertical growth — the growth in admissions — was in “public order/other” and “drug” offenses, as I discussed in my last article. But those offenses apparently had short prison terms that didn’t get much longer, so they didn’t contribute as much to area growth — the growth in the incarceration rate.

Instead, there was more horizontal growth — more growth in prison time served — than expected, much of it in violent offenses.

So if you look at the growth in height, then most of it was in drugs and public order/other offenses. This does not at all contradict that most of the growth in area was in violent offenses.

It sounds like a tricky paradox that most of the population growth was in violent offenses, and most of the population growth was from admissions, even as rather little of the population growth was from violent admissions. But it’s not a paradox at all — it’s just what a two-dimensional process can look like.

Breaking the rectangles down into narrower offense categories shows this as well:

The horizontal growth, the growth from longer prison terms, is mostly concentrated in more “serious” offenses like murder/non-negligent manslaughter, rape/sexual assault, robbery, and burglary. The people who are sent to prison on those charges are staying longer, but there aren’t that many more of them in a given year (perhaps except for rape/sexual assault charges).

The vertical growth, the growth from more prison terms, is mostly concentrated in lower-level offenses: “Other property”, drugs, “public order/other”. But people sent to prison on these charges don’t seem to be staying longer, on average (perhaps except for “other violent” charges which I think is mostly assault/aggravated assault).

Again, this analysis is not unique to me —for example, the National Research Council report “The Growth Of Incarceration In The United States” said that “data point clearly to the increased rate of prison admission (particularly marked for drug crimes) and the increase in time served (especially for violent offenses) as sources of increased incarceration rates.”

You can also see the distinction in these line charts. (Although I don’t encourage it! I’m trying to promote an alternative visualization approach here.) The longer terms/more terms division doesn’t perfectly map onto violent/non-violent offenses, but that was our original question. Here’s new court commitments and incarceration rates for violent offenses, 1980–2011:

New prison terms for violent offenses were relatively flat, but the population grew quickly anyway. If that isn’t because of longer time served, what could explain it?

Here’s new court commitments and incarceration rates for non-violent offenses, 1980–2011:

New prison terms for non-violent offenses weren’t at all flat — and the prison population for such offenses only grew a little faster than the number of offenses. There isn’t as much for changes in time served to explain here.


Why do different offenses seem important when looking at “prison sentences” as when looking at “prison populations”? To try to understand that, visualize “prison populations” as two-dimensional figures. Different parts of the figure might grow in different ways— and looking at height might tell you something different than looking at area.

According to these visualizations, the 2011 state prison system had more prison terms for drugs, “public order/other”, and some violent and property offenses than the 1980 state prison system, but these were mostly short. Some prison terms did grow longer, but on average mostly for murder/non-negligent manslaughter, rape/sexual assault, robbery, and burglary. (EDIT: This originally said “lower-level violent and property offenses” but I don’t know if that’s exactly true — I think a lot of the growth in “other violent” terms was in assault/aggravated assault and I don’t know the composition of “other property”.)

These two changes might not have happened in the same places, for the same reasons, or at the same time. And the rectangle visualizations are only approximations of the actual changes in distribution of prison terms. We might see something like longer time served on some drug charges, or more prison sentences on some murder charges, if we could look at the full distributions or if we could break them down by different . As with many questions, precise answers might have to wait for long-term, individual-level administrative data.

Decomposing prison population growth into admissions and time served isn’t just an intellectual or visualization exercise. As I keep saying in this series, focusing on one statistic glosses over real human consequences. Violent offenders serving longer prison terms, along with additional prison terms for “rape/sexual assault” and “other violent” offenses, really did contribute more to “the incarceration rate” per se than the War on Drugs did.

That doesn’t mean the War on Drugs didn’t happen, or that all those extra prison terms for drugs and other lower-level offenses had no effects. By placing admissions and time served in different dimensions, we might make that distinction clearer, and more fully understand what mass incarceration has really meant.

Sources And Notes:

Click-through versions of these graphics are available here: http://xenocrypt.github.io/PrisonRectangleGraphic.html

I honestly don’t remember if I read about “population over admissions equals time served” before I made these graphics, but it’s a standard approximation. EG again from that National Research Council book: “Blumstein and Beck (1999; Beck and Blumstein, 2012) base estimates of time served on the ratio of the stock population — the number of people in prison on the day of the annual population count — to new court commitments in that year.”

Similarly, while I did some previous visualizations based on the Neal/Rick paper “The Prison Boom & Sentencing Policy”, I didn’t really understand arguments like this until I made the graphics shown here: “The relative stability of the distribution of time-served among admitted prisoners during the 1980s and 1990s resulted from off-setting increases in various types of admissions. The number of persons serving short terms increased as the number of admissions associated with parole revocations and convictions for minor crimes increased, but the number serving medium and long terms also increased as offenders charged with more serious crimes began serving more time in prison.” Or, in my language, the seemingly small horizontal growth in the “total” rectangles hides the significant horizontal growth visible in the “offense category” rectangles.

1980–1989 new court commitments taken from “Prisoners in 1992” Appendix Tables 1 and 2 on Page 10. 1980–1992 prison populations are BJS estimates taken from “Correctional Populations in the United States, 1993”, Table 1.9 on page 11. 1993–1995 populations and 1990–1995 new court commitments taken from https://books.google.com/books?id=0CgnfzWO8HEC&lpg=PA14&pg=PA9#v=onepage&q&f=false, Table 1.11 on Page 9 and Table 1.20 on Page 16 respectively. 2001, 2006, 2011 populations and new court commitments taken from “Prisoners in 2012” Table 3 on Page 5 and Table 5 on Page 7 respectively. 2011 populations cover all jurisdictional prisoners sentenced to at least one year; 1980 populations cover all in custody including unsentenced. Population definitions and offense percents are slightly different in years both are available. I believe “Assault” and “Aggravated Assault” are equivalent but the different phrasing makes me uncertain whether to include them as a category.