In 1964, the Supreme Court ruled in Reynolds vs. Sims that legislative districts had to have close to the same populations. A district near Birmingham, Alabama had about 41 times the number of eligible voters than in another district. Persons in districts whose populations are below the ideal population have more voting power than those in districts with populrations above the ideal population.
The blue-red maps are color-coded to display deviations from the ideal population. A blue district is below the ideal population. A red district is above the ideal population. The intensity of the color is how extreme the deviation is. The deviations range from -5% to 5%. So a dark blue district is between -5% and -4%. A dark red district is between 4% and 5%.
The visualizations of the 2021, Linehan and Wayne legislative maps show that Wayne’s districts do not have a pattern in how deviations are assigned to rural and urban districts. In other words, Wayne’s map is fair to both rural and urban districts. In Linehan’s map, we see a clear pattern; rural districts have populations below the ideal population and urban districts have populations above. The 2021 map is less extreme than Linehan’s, but still shows more of a tendency for rural maps to be under the ideal population and urban to be over.
These visual observations are corroborated in the “Comparing Trends” tab that displays scatterplots with trend lines. The vertical axis on the scatterplot records the percent deviation from ideal population. The horizontal axis shows population density, measured in persons per square mile. In an urban district, people live closer together, so there are more persons per square mile. In a rural district the persons per square mile will be much smaller. As you move from the left to the right on the horizontal axis, you are moving from most rural to most urban district.
The scatterplot has one dot for each district. There are three colors of dots, one for each of the three maps: 2021, Linehan, Wayne. The trend lines show the overall shape or trend of the dots. A horizontal trend line tells us there is no relationship between deviation and ruralness/urbanness, as we see with the Wayne trend line. This is a desirable trait in a map. The steeper the line, the more unfair the map is. We see that the Linehan trend line is the steepest one; as you go left to right, the line goes up. This tells us that the rural districts in the Linehan map are more likely to have negative deviations and urban districts are more likely to have positive deviations. In this way, the Linehan map favors rural districts.
Compactness is one of the few requirements for redistricting stipulated in the constitution of Nebraska. Loosely speaking, a district is compact if it is held together nicely. It’s “tidy”. A district that is not compact will have a more unusual shape.
There are several different ways to measure compactness. The most common is the Polsby-Popper Score (PPS). There is a mathematical fact that says that for a fixed perimeter (distance around a shape), a circle encloses the largest area. More easily stated, of all shapes, a circle uses its boundary most efficiently.
So the PPS compares a district to a circle. Specifically, the PPS is the area of the district divided by the area of a circle that has the same perimeter as the district. Thus, the very largest a PPS can be is 1, and that is if the district is in the shape of a circle. The very smallest a PPS can be is 0, and that is if the boundary of a district encloses no area; this might be hard to imagine, and that is because you would never make a district that has an area of 0 square miles since nobody could live there!
The larger the PPS, the more compact the district; a small PPS reflects lack of compactness (a crazy shape).
To compare compactness of the maps, we first calulate the average (mean) PPS of the districts of a map.
On comparing the 2021, Linehan and Wayne congressional maps, this is the ranking from most compact to least:
For the legislative maps, the ranking, from most compact to least, is: