Thinking about the possible determinants of the shape of school attendance zones, Richards and Stroub ask: are they drawn to increase or decrease racial segregation?
They call this "attendance zone gerrymandering."
Their findings are presented in two recent papers: Richards (2014) and Richards and Stroub (2015). Roughly, the answer is yes, attendance zones do seem to be gerrymandered, possibly to increase racial segregation.
There are some problems that make it hard to draw the kinds of conclusions we would like from their research, but I'll save my snarky remarks until the end. First, let's talk about the concrete things we can learn from these papers.
Results in brief
- In terms of the same geometric statistics political scientists use to measure how "gerrymandered" congressional districts are, school attendance zones are less gerrymandered on average. Attendance zones are about -0.3 to -0.5 standard deviations less gerrymandered (2015, table 2).
- Relative to "ideal non-gerrymandered fake attendance zones" (I talk about how they draw these below), the true zones are about 1.8% of a SD less "diverse", and about 2.5% of a SD more "segregated" (2014, tables 2 and 4). (Differences are significant at p<0.001.)
- The differences are smaller when restricting to measures of black-white diversity or segregation, and are larger when restricting to measures of black-hispanic-asian diversity/segregation. So maybe attendance zones are drawn partly to separate blacks from hispanics? District boundaries are probably the dividing lines between blacks and whites.
- School districts under an active court desegregation order have less segregation in actual zones than in fake zones; districts whose court order has been dismissed have more segregation in real zones than fake zones; and those that have never been under court order are in the middle. I find this an important justification for their method.
Methods
Measuring gerrymandering
To measure gerrymandering in the 2015 paper, they focus on two geometric properties of school attendance zones: dispersion and indentation.
Dispersion
First approach is to measure "dispersion." More disperse districts are likely more gerrymandered.
The most intuitive way to describe this is as follows. Take a shape (school attendance zone). Wrap the shape with a rubber band and take the resulting "rubber-band shape"--in math that's called taking its convex hull. Then consider the area in the rubber-band shape that isn't in the original school attendance zone (i.e., take the difference between the hull and the shape). The area of this difference measures how "non-convex" the original shape was. We might call this "degree of non-convexity," "dispersion."
Another way to do this is to circumscribe the original shape with the circle (rather than doing the convex hull thing); then the difference measures how "non-circular" the attendance zone is. Either way, an attendance zone that departs from an ideal convex shape is considered more gerrymandered.
Indentation
Second approach is to measure "indentation."
Here, the idea is to compare how much perimeter is actually used to form the attendance zone, with what the bare minimum "needed" to be used to form the attendance zone. An attendance zone that is more wiggly is considered more gerrymandered.
Creating counterfactual attendance zones
In her 2014 paper, Richards applies the "Voronoi approach" to create ideal counterfactual attendance zones for comparison. This means:
(a) use school locations to construct fake "Voronoi" attendance zones
(b) compare the levels of diversity or segregation observed among the Voronoi attendance zones with those observed among the real attendance zones
(c) if the Voronoi (fake) attendances zones are more segregated than the true attendance zones, conclude that attendance zones are drawn to limit segregation; if the opposite, conclude that attendance zones are drawn to exacerbate segregation.
What is this Voronoi macaroni?
The Voronoi approach to construct fake attendance zones uses the school location to make up fake attendance zone boundaries.
Each attendance zone is carved out so that all land area inside of that attendance zone is closest to the school for that zone, relative to any other school in the district. That is, if uniformly distributed students walked to school in a flat desert, this would be the ideal attendance zoning for transportation purposes.
That sounds complicated, so I had to try it myself. It turns out, if you know what you are doing, it is only a few clicks in QGIS (tutorial). I tried it for New York City. Here are the maps of the elementary schools in New York City (from the Common Core of Data), along with NYC's 3rd grade attendance zones obtained from SABINS. (All data is for the 2009-10 school year. For the record, this is the same school attendance zone dataset that was used in the papers I'm discussing.)
If I hit the dots with QGIS' Voronoi command, I get the following result.
Maybe an easier way to phrase this process: to construct the Voronoi polygons, you take each point on the map and assign it to its nearest school.
There are some problems: the Voronoi method in QGIS doesn't seem to care about water, and it looks like it wants to be square. I suspect the way around these problems is to do a clipping of the Voronoi polygon using the school district area... but I couldn't figure out how to get that to work in QGIS.
Comparing Voronoi-fake to true attendance zones using census blocks
In the 2014 paper, Richards matches attendances zones, both real and fake, to census blocks. The census block is apparently the smallest geographic unit for which race and age composition is publicly provided by the Census. She compares the youth racial composition in fake versus real attendance zones by aggregating up Census blocks. This is how she gets her results.
Criticisms and conclusions
Obviously they cannot say which came first: segregation, or these attendance zones? That's the biggest problem.
Moreover, from my reading of their work, all of their results are consistent with attendance zones being drawn wildly because of transportation. They don't do anything to counter this idea in their "tabulations of gerrymandering." And because their Voronoi estimates are small overall, I think that gerrymandering of school attendance zones happens, but it is not that common, and certainly not that big of a deal.
References
Richards, Meredith P. and Kori J. Stroub (2015). "An Accident of Geography? Assessing the Gerrymandering of School Attendance Zones." Teachers College Record
Richards, Meredith P. (2014). "The Gerrymandering of School Attendance Zones and the Segregation of Public Schools: A Geospatial Analysis." American Educational Research Journal
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