Are school district boundaries the dividing lines between the poor and nonpoor in New York State? Let’s check by using the NCES Common Core of Data.
Mixed-income school districts
This post is motivated by the work of Saporito and Sohoni (discussed in a previous post), by a question from a friend regarding the value of mixed-income school districts, and by a pretty map from EdBuild.
Are mixed-income school districts good? Sounds plausible. A mixed-income school district would have high income parents to support the district with taxes, volunteer time, and as role-models, while also having low-income parents whose kids are in need of this support. So the case sounds clear from an equity standpoint.
On the other hand, the efficiency case for mixed-income school districts rests on a dollar worth of educational resources for a poor student being more useful than that same dollar for a rich student. Research on whether this “diminishing marginal returns” idea is true is still hazy.
For output maximization, resources should be sent to where they are most productive, and this might mean that students from wealthy backgrounds should get allocated more educational resources than students from poor backgrounds. A special case worth debating are the substantial educational funds directed toward students with disabilities. This is worth a future exploration. I am sure the reader will not find it atrocious to think that the US may be overallocating educational resources to disabled students.
Debating the merit of mixed-income school districts is something worth doing, but I suggest that we should first ask how many school districts are “mixed-income,” and if they exist, where are they? I hypothesize that there will be very few such districts: if education is redistribution, poor voters in mixed districts will vote for very high taxes, and the rich will move out.
Using free or reduced-price lunch as a proxy for poverty
EdBuild points out that new federal policies make the free or reduced-price lunch (FRPL) measure a possibly unreasonable proxy for poverty going forward, since districts are now able to elect for “Community Eligibility,” thereby making the FRPL rate for the district artificially 100%. In this blog post I focus on the 2012-13 CCD, which does not appear to be contaminated by this policy change. Moreover, FRPL continues to have the advantage of being a school-level measure of poverty. Schools are a much finer locational measure than districts, which is something I exploit later in this blog post.
So I ignore this criticism, and use the school-level FRPL rate for all calculations in this blog post, simply calling it the “poverty rate.” The FRPL poverty rate is easily calculated using the CCD, a very useful dataset I’ve previously discussed.
This poverty rate sets a high bar for poverty: around 40k annual income for a family of four. With this measure, in New York State, an average school has a poverty rate of about 52.7%, and 47.9% of children are considered poor.
How many mixed-income districts are there?
Let’s start cutting the data.
How might I answer the question: “what proportion of school districts in New York State are mixed income”?
Below I provide a histogram of district-level poverty rates for New York State school districts.
If I weight by district total enrollments, New York City stands out:
If I exclude New York City, I get something more reasonable to look at:
In this last histogram, there is a bit of a U-shape near about 60%, but there is also another U-shape near 30%, suggesting sorting into three school district types: not poor (about 10% FRPL rate), mixed (about 45% FRPL rate), and poor (about 75% FRPL rate). The large mass of students in this potentially “income mixed” type school district is contrary to my hypothesis of strong sorting out of that range.
(Note: in all analysis I exclude charter schools and magnet schools. Excluding charter schools is very important, since in the CCD, each charter school is counted as a separate district. There are a bit fewer than 700 traditional public school districts in New York, and over 200 charter schools, so this exclusion is important.)
Where are the mixed-income districts and schools?
Let’s take a look.
New York

● poverty greater than 80%
● poverty between 60% and 80%
● poverty between 40% and 60%
● poverty between 20% and 40%
● poverty less than 20%
(same key applies to districts)
Generally, each city has a ring of wealthy suburbs around it, and an impoverished urban core, while the more distant country is more mixed up. There are some exceptions.
You will also notice in the map that, if we assume all schools are neighborhood schools (not a great assumption, but worth starting with), district boundaries tend to be the “dividing lines” between impoverished and not impoverished neighborhoods. This is what EdBuild says on their website, although they don’t dig deeper in the data to show that, in fact, school districts per se are “what divides the poor from the nonpoor.” It could be that school districts are just loose containers for already strongly income-segregated neighborhoods. I explore this next.
Introducing the Theil index
The Theil index is a measure of segregation, like the dissimilarity index, that has the particular advantage of being decomposable. I won't go into the mathematical details here, but briefly, using this index, I can write this:
NY State School Theil Index for FRPL = 19 = 3 (within districts) + 15 (across districts within cities) + 1 (across cities)
This says that income segregation in New York State schools, as measured by the Theil index, equals “19” (this particular number doesn’t have an immediate interpretation), 3 of which occurs across schools within districts, 15 of which occurs across districts within cities, and 1 of which occurs across cities. From this decomposition, we can say that sorting across school districts within cities alone constitutes nearly 80% of income segregation.
Note that the dissimilarity index discussed in a previous blog post cannot be decomposed in this way.
To give you a better idea for this Theil index decomposition, I created the following simulation. Here, there are two districts in a single city, and you control the income composition of the schools. Play around with it, and you’ll quickly get the intuition of how the index works.
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(formula: city theil index = within district segregation + across district segregation)
Theil index decomposition for New York schools
I’ve already reported that, overall,
NY State School Theil Index for FRPL = 19 = 3 (within districts) + 15 (across districts within cities) + 1 (across cities)
But I can also do this decomposition for particular cities. The following clickable equation provides the decomposition for each city in New York State. Start by clicking the number. Press the reset button anytime to start over.
So, one thing that Rochester and New York City have in common is their total income segregation (as measured by the Theil index), though Rochester’s segregation occurs a bit more across district boundaries rather than within.
Are district boundaries “the dividing lines”?
If you play with the Theil index simulation, you’ll discover that a city with four schools in a line, going from a poorer neighborhood to a richer neighborhood, would yield a lot of across-district segregation (and very little within-district).
So it could just be that school districts are wrapping neighborhood segregation “loosely,” so these boundaries are not the sharp dividing lines. The maps displayed earlier tell a different story, but it is worth exploring this some more.
I conduct analysis as follows. For each school, I find the nearest neighboring school in a different district. I then plot the absolute value of the difference in poverty rates between the two “neighbors-in-different-districts” schools, against the distance between the two schools. The behavior of this plot near zero gives the size of a “typical boundary discontinuity.” While the average boundary discontinuity cannot be zero since the absolute value of any variable is censored at zero, this magnitude could on average be very small.
Overall, there is about a 15 percentage point (pp) jump near the boundary, on average–that’s a large jump. When one district is poor, the gap is about 10 to 15pp; when one district is rich, the gap is much larger: about 25pp. (The definitions of “poor” and “rich” school district here, are, respectively, a district above the 75th percentile in FRPL rate, and a district below the 25th percentile in FRPL rate.)
For the sake of my own interest, I also restricted attention to four cities in Upstate New York. For these cities, the boundary jump seems to be around 20pp.
The magnitudes of these boundary jumps do seem to suggest that school districts are the “dividing lines” between the rich and the poor. This of course doesn’t mean the schools are the ones doing the dividing–the poorest school districts’ boundaries tend to be coterminous with city boundaries, and so probably most of the “dividing” is being done by zoning boards, not school boards. However, establishing that the school district boundaries are a good way to explain sorting is a logical first step toward asking whether school choice is in fact an important reason for income segregation.
Future research
In future work, we will have to determine how much a “typical” boundary jump would be in the case of no purposeful income segregation, so as to have a baseline to compare the boundary jump estimates from the previous section. This, I think, would require a simulation.
In the maps, town or rural districts are more likely to be mixed income. This is a hypothesis worth exploring more carefully. Commuting may have something to do with the high levels of income segregation in cities.
Since the CCD is available annually from around 1990 to 2012-13, it can provide us with a high-frequency panel on school segregation. The three Census snapshots available over this time period cannot do this. I’m interested in how the time path of the spatial distribution of poor students in high growth cities differs from no-growth or negative growth cities.
The analysis done here can and will be generalized to the whole nation in a future blog post.
Hi Data Buddha,
ReplyDeleteI am still confused about the Theil Index. It has little to do with your writing above, but I figured you might be able to answer my questions anyway:
1. When I saw the formula for the Theil Index on Wikipedia (https://en.wikipedia.org/wiki/Theil_index#cite_note-Formulas-3), I immediately noticed that the Index is indeterminate as soon as any of the x_i's becomes zero (in your case, an x_i would be the poverty rate at a given school). Indeed, when I set the poverty rate to be zero in any of the schools on your Theil Index slider, the resultant Index is NaN. So in calculating the Index, how exactly do we accommodate situations where some schools are virtually poverty-free?
2. In the Wikipedia article cited above, there is no discussions about how to decompose an aggregate Theil Index into, say, within-district vs. across-district shares. Could you explain how this works exactly?
3. It seems to me that there is a lot of parallel between the Theil Index and the Dissimilarity Index. Granted, the latter primarily measures racial segregation. But if we treat students as binaries--either rich or poor--in theory we should be able to calculate some measure of income segregation in school districts using the Dissimilarity Index. Is what I just said correct? And if so, what are the advantages and drawbacks of using the Dissimilarity Index rather than the Theil Index for this endeavor?
Thank you,
ITryTenThousandTimes
Good questions!
Delete1. The answer is the index doesn't really behave very well when some schools have very few poor students... in particular it seems to easily get very large in that case.
Try 1%,1%,1%,...,5%. You'll get an index of 28%! By contrast, try 50%, 50%, 50%, ..., 100%, or even 50%, 50%, 50%, ..., 1%.
Another thought--I don't think the Theil index is invariant who which type we define as "the minority group." For example, using the Theil index, we could talk about Asian segregation, or non-Asian segregation. Asian segregation is probably very large, while non-Asian segregation is probably very small. The index of dissimilarity has the advantage that it is obviously symmetric.
2. In the wikipedia article, there is a section on "decomposibility": https://en.wikipedia.org/wiki/Theil_index#Decomposability .
The way I read it, the total Theil index is equal to a weighted sum of the individual subgroup, i.e. school district, Theil indices, plus an additional factor that measures the segregation across subgroups, i.e., across districts. The reason I can "keep decomposing" (across cities, across districts, within districts) is because I can apply the same equation to the Theil indices inside of the equation (the Theil index is "recursive").
Also note that the second term in the equation on Wikipedia (the one with log(xbar_i / xbar)) would be 0 exactly when xbar_i = xbar, for all i; that is, when every district has the same mean poverty rate the average across all districts. In that case, it wouldn't matter if schools within districts were completely segregated, so long as things look "mixed up" at the district-level. Then the rest of school-level segregation would be in the first term, which is a weighted sum of within-district segregation. (Btw, I haven't been able to prove that the across-district segregation term is nonnegative...).
3. The dissimilarity index could have been applied here, no problem. There are only two groups--FRPL and non-FRPL. My focus on the Theil index was just because I think the Theil index can be decomposed as I did above. From my understanding the dissimilarity index doesn't have any natural decomposibility properties.