Gerrymandering Part 3: I’ve been everywhere, man

Welcome to part three of our continuing series on gerrymandering: or, the subtle art of screwing over your political opponents through the power of MAPS!

Yes, this is actually what the map of Ohio’s Congressional districts looks like. Like srsly?

Gerrymandering happens when the boundaries of legislative districts are planned with the goal of benefiting some group, often a political party. It’s one of the most infamous dirty tricks in politicians’ dirty tricks playbook. It can make the difference between a legislature that reflects the true intentions of its voters and a legislature that will pass policies that a majority of voters find abhorrent.

And remember, it’s always the same voters – the only thing that has changed is how you draw lines on the map.

In Part 1, we looked at what gerrymandering is, learned how it got its name, and looked at some of its most ridiculous results.

In Part 2, I introduced a measure of how gerrymandered a legislative district is: the Polsby-Popper (PP) score, essentially a measure of how much the district resembles a circle. The lower the PP score, the more gerrymandered the district. (The post also features a math mistake that I haven’t corrected yet – see if you can spot it!)

Today, let’s combine both approaches for a nationwide look at how legislative districts look in the United States. Gerrymandering is by no means an exclusively American phenomenon, but it was invented here, and its effects have been better studied in the U.S. than in other countries.

First, what does it look like? What happens when you look not just at a single district (like we did with Texas’s 33rd district in Part 1), and look at all U.S. House districts at once?

Here’s the map:

The current U.S. House Districts

This map will make more sense when we look at it state-by-state, which we will do over the next few months. But for now, take a look at the map (click on it to see a larger view). You should be able to get a sense of which districts are more or less gerrymandered.

What happens when we Do the Math and look at the PP score for each district?

It would be easy enough to show a table of districts and their PP values, maybe sorted from lowest (most gerrymandered) to highest (least gerrymandered). But there are 435 districts to consider – a table with 435 rows won’t be useful for getting the story of what’s going on. It’s simply too many numbers for your brain to construct a story. We need a picture.

The traditional way to show this kind of data – data showing the same numerical measurement for many different units of analysis – is with a histogram. A histogram shows the range of values in the horizontal direction (x-axis), and the number of times that value appears in the vertical direction (y-axis). When the measurements can only take on discrete values (like 1, 2, 3…), this kind of graph is easy to make. When the measurements are continuous and can take on any value within the range, histograms require some judgment calls by the person presenting the data.

The judgment comes in how to bin the data – how many data points to show together in a single bar. With 435 districts, there are many possible options.

Here is what the histogram of PP scores in all US House districts looks like when dividing the data into 17 bins. The PP score decreases to the right, indicating increased gerrymandering.

A histogram of PP scores of U.S. House Districts, divided into 17 bins

and here is a histogram of the same PP scores for all the same districts, dividing into 85 bins:

A histogram of PP scores of U.S. House Districts, divided into 85 bins

Notice how the story looks similar across the two histograms, but the two graphs emphasize different parts of the story. The first histogram (blue) shows the overall picture clearly, while the second histogram (red) shows some of the variation in district shapes from state to state.

These maps and graphs give you a good overview of what gerrymandering looks like across the entire United States at once. But it leaves a lot of important questions unanswered. How did the districts get that way? Who made them, and what strategies did they use? And maybe most importantly, how can we do better?

Gerrymandering Part 2: The Shape of You

Maryland-3, the most gerrymandered district in the USA

On Monday, we began a deep dive into gerrymandering – the practice of designing electoral districts to provide an advantage of some political party or other group. We looked at a few districts with particularly crazy shapes, including the salamanderiest district in the entire United States – Maryland’s 3rd Congressional District. But how can we say which districts are more or less gerrymandered?

There are multiple ways to measure how gerrymandered a district is, and we’ll explore several of them over the course of this series. But we’ll start here with one of the simplest, a method that requires only the shape of the district: the Polsby-Popper (PP) score.

The Polsby-Popper score was invented in 1991 by American law professors Daniel Polsby and Robert Popper, although it is closely related to concepts known to mathematicians for decades (the coastline paradox) or millennia (the isoperimetric inequality). The PP score works not just for electoral districts, but for any shape. The formula is simply the area of the shape divided by the perimeter (the distance around) squared, all multiplied by 4π.

Essentially, the PP score measures how circle-like a shape is. Some examples will help make sense of what the formula measures – and only the most high-tech mathematical figures will do here.

Calculating the PP score for a circle

The first figure shows an imaginary district that is a perfect circle. The area of a circle is π times the radius; the perimeter (circumference) is 2π times the radius. Going through the formula, the radius cancels out, leaving 4π / 4π, which is 1. There are two extremely important points here. First, the PP score for a circle is 1, which is the highest possible value. The PP score will always be a number between 0 and 1. The more gerrymandered a district is, the lower the PP score.

Second, since the radius cancels, the size of the circle doesn’t matter. The circle in question could be a fly’s egg, or the Sun, and it wouldn’t matter, the PP score is the same. And it turns out that is the size of the district will always cancel.

Calculating the PP score for a square

What about other district shapes? What is the PP score for a square? Take a look at this illustration plus equations. If x is the side length of a square, the perimeter of the square is 4x; the area is x2. And so the PP score for a square is π/4, approximately equal to 0.78. That makes sense – a square is not as round as a circle.

Calculating the PP score for a 3×1 triangle

Let’s try one more shape – a triangle that is three times higher than it is wide. Before I show the math, make a prediction: will the 3-by-1 triangle have a higher or lower PP score than the square?

The math is in the image, but the answer is about 0.51. That is less than either the square or the circle.

So what are the PP scores for the actual shapes of actual U.S. Congressional districts? This was the time I was going to tell you…. but I realized I made a mistake with the calculations. I calculated the PP scores in geographic latitude/longitude coordinates, which assume that the shapes are on a flat plane. But the Earth is not a flat plane; the districts sit on the surface of a sphere. It might not make a difference, but it might. More on Friday.

Gerrymandering Part 1: WTF?

What the hell is this?

Impractical cardboard earmuffs? Rhode Island and Anti-Rhode Island about to disappear in a flash of cosmic rays?

…or how about this?

A top-heavy steam shovel falling off a cliff? The Big Dipper viewed from inside a black hole?

…or this?

Did someone build a bridge to from Norway to Panama, and did Spain grow antlers?

No. These are United States Congressional districts under the influence of gerrymandering (from top to bottom, Illinois-4, Texas-2, and Texas-33).

What is gerrymandering, and how does it work?

Gerrymandering is the art and science of drawing the borders of Congressional districts to give an advantage to a political party or some other segment of the population.

Article 1 Section 2 of the U.S. Constitution says that the House of Representatives is elected by voters every two years. The number of representatives that each state gets is proportional to the population of the state, recalculated every ten years from the results of the decennial U.S. census. And that’s it.

The Constitution gives no details on how the representatives are distributed within the state, and different states had different practices. It took an act of Congress, the Apportionment Act of 1842, to standardize the process. Ever since, each state has been divided into districts, with one representative per district. This makes good sense – the citizens of Watertown, New York have very different needs from the citizens of Lower Manhattan, and they deserve to elect a representative who they believe will best meet their needs.

So how do you go from this general principle of local representatives to specific districts to be represented? In general, it’s up to each state legislature to divide the state into congressional districts. Districts must be contiguous (covering a single area, with no holes or outlying areas), and must have approximately equal populations. Other than that, and a few other legal requirements and guidelines we’ll look at later, it’s entirely up to the state legislature.

Leaving something so fundamental to the political process in the hands of a potentially partisan state legislature is a recipe for parties using the process to their advantage – and indeed, history has shown that creative assignment of congressional districts is one of democracy’s most effective cheat codes.

The "Gerrymander" political cartoon from 1812, showing the salamander-shaped district created by Elbridge Gerry as an actual salamander
The Gerrymander illustration by Elkanah Tisdale, 1812

The name comes from the first famous proponent of the practice, former Massachusetts Governor Elbridge Gerry. In 1812, Gerry was in charge of designing Massachusetts’s 20 electoral districts (Massachusetts elected representatives by district even before the Apportionment Act of 1842 required it).

Gerry was a member of the Democratic-Republican Party in a state with a majority of voters from the opposing Federalist Party. He figured out the One Weird Trick to guarantee his party a House seat by connecting multiple Democratic-Republican strongholds west of Boston into a single long, narrow district. Federalists looked at a map of the district and noticed that it looked like a salamander, so they named it the “Gerrymander.” The carton shown here helped popularize the name, and solidified the opposition to Gerry’s proposal. The name stuck, and has been adopted worldwide – as a verb, adjective, and noun to describe the practice and its effects. (Interesting aside: Gerry’s name is pronounced with a hard G, as in gay, while the eponymous practice of gerrymandering is pronounced with a soft G, as in “Genesis Device.” I have no idea how that change happened).

An illustration of how gerrymandering can lead to different representation for the same voters
Image by Wikipedia user M.Boli

This diagram from Wikipedia is a great simple illustration of how gerrymandering works in practice. The rectangle shows an imaginary state where 60 percent of voters vote for the Blue Party and 40 percent for the Yellow Party. Where the state legislature draws the lines around districts will have a massive effect on how the state is represented.

It is easily possible to draw lines resulting in a majority of representatives for the Yellow Party, despite a clear voter preference for the Blue Party – or alternatively, to elect only Blue representatives, ignoring the opinions of 40 percent of voters. Or, hopefully, to design a legislature that really represents the will of the people.

Stay tuned, because over the next several weeks (or months?), we’ll be taking a deep, deep dive into gerrymandering, with lots of maps and datasets to guide us. Along the way, we’ll explore the reasons for gerrymandering, the strategies used to enact it, and how it is employed (or not) in countries around the world. And, maybe most importantly, we’ll go state by state to look for ways to draw electoral district maps more fairly.

Coming up on Wednesday: a simple measure of how gerrymandered a state is, based solely on how weird the shape of the district is. This will make more sense once I tell you about the metric, but an obvious question is: what is the most weirdly-shaped district in the entire United States?

Right here. It’s the district I lived in from 2003 to 2021: Maryland’s 3rd district. Behold:

Maryland’s 3rd Congressional District (?!!)

Shall… perish from the Earth

man in yellow dress shirt sitting on chair
Who is this guy? Why does he want government of the people to perish? Why does he hate America? WHAT IS HE HIDING?

Photo by Ramaz Bluashvili on

When people share quotes from scientific articles but refuse to link to the full article, you should immediately be suspicious.

It’s a more extreme example than you’ll encounter in real life, but absolutely captures the spirit of the argument style: let me share this quote from the Gettysburg Address.

“Now we are engaged in a great civil war. It is entirely fitting and proper that we should do this. The world will little note, nor long remember, what we say here. It is for us the living, rather, to be dedicated here to the unfinished work… that government of the people, by the people, for the people, shall… perish from the Earth.”

Lincoln (1863)

So what should you do if you encounter this in real life? Ask to see the full article – politely but firmly, and repeatedly if necessary.

Sometimes the full article is hidden behind a paywall, but my friend can get you access through his employer. Send me the reference and he will send you the article.

Guess the outlier!

Here’s a quick and timely data science post.

I made a graph (histogram) of the ages of quarterbacks currently playing in the National (American) Football League. The graph is below. Age labels are along the bottom, increasing to the right. Along the left are labels of the number of quarterbacks at each age, increasing going up.

Look at that bar waaaaaaaaaaay oooooooover theeeeeeeeere to the right. Who do you think that is?

A graph (histogram) of NFL quarterback ages (n = 108). Click for a larger view.

Some stats

This guy, WTF?

Number of quarterbacks: 120

Average (mean) age: 27.6 years

Standard deviation (a measure of how spread out the data is): 4.5 years


Love him or hate him, Tom Brady is a freak of nature.

Want to see it for yourself?

Download my Excel spreadsheet!

It’s the Most Wonderful Time of the Year!

Welcome back to what is inexplicably my most popular blog topic ever: the announcement of this year’s long-awaited Best Sexy [Thing That Is Inherently Not Sexy] Halloween Costume Contest™!

What is the Best Sexy [Thing That Is Inherently Not Sexy] Halloween Costume Contest™, you ask? It’s an annual event on my social media, now entering its sixth year (although we skipped 2020), celebrating (?) the weirdness that is Halloween as celebrated on Earth, and especially as celebrated in the United States. Specifically, the weirdness of Halloween costumes.

What is so weird about Halloween Costumes?

Imagine that you are a woman (easy for about 50% of you) and that your favorite animal is a moose (easy for me). You want nothing more than to go out with your friends and celebrate the majestic moose.

Behold, the ONLY women’s moose costume I could find on the Internet:

And there is the problem: for just about any costume idea you can imagine, there are no costumes available in women’s sizes for that idea – instead, there are just SEXY costumes. And Sexy Moose is not even in the Top 100 weirdest.

And so in 2015, I decided to take the moose by the antlers and sponsor a contest. I invite you to suggest the best, weirdest, most WTF examples of sexy Halloween costumes. In particular, I invite you to suggest costumes that bring sexy to things that are totally, completely, Inherently Not Sexy.

Presenting the winners from previous years, and the people who suggested them:

2015: Sexy Orca

Suggested by Jeremy Berg

2016: Sexy Scrabble

Suggested by Kelly Simms

2017: Sexy Green Poo

Suggested by Aimee Shoff

2018: Sexy Marcel Duchamp Art Gallery Urinal

Suggested by Christina Rawls

2019: Sexy Mr. Rogers

Suggested by Elliot Kresmer

I’ve already gotten several great suggestions for costumes this year, which I will review on Friday. In the meantime, keep those suggestions coming!