Talk:Polsby–Popper test

What measures?

What are the "other measures that use dispersion"? Sollupulo (talk) 22:49, 21 February 2018 (UTC)

@Sollupulo: The ones mentioned in the source are Reock and the convex hull ratio. The Polsby-Popper test is perimeter-based, as opposed to dispersion-based, and the authors contend that perimeter-based tests like P-P are sensitive to convoluted geography and harder to correct for such errors than dispersion-based ones. PohranicniStraze (talk) 02:09, 22 February 2018 (UTC)

@PohranicniStraze: Thank you. I didn't realize I had found the answer to my own question until just now. I did find those other methods in my rooting around in this whole topic, but never made the connection. So what you are saying is that some geographical feature -- say the Delaware River being the border of Pennsylvania -- would unjustly skew the Polsby-Popper score for the districts that are bounded by it. This is less true for measures that use dispersion methods. Interesting... Sollupulo (talk) 01:13, 24 February 2018 (UTC)

Some Measures of Interest

Since the southwestern corner of Pennsylvania is (or nearly is) a right angle, the four least gerrymandered districts that could hypothetically be drawn would be 1. a quarter circle, 2. a rectangle, 3. a square and 4. a triangle. Since none of these are a perfect circle, they would never obtain a score of 1. So how do these shapes score?

The best is the perfect square shape, coming in at about 0.785. Next we get the quarter circle scoring at about 0.774. To figure for the triangle, I used an isosceles right triangle, as the best fit for any circle, given that the corner of the state forms two of the sides of the triangle. It comes in about 0.637. Lastly, I figured for a rectangle that scored the best with an area of π and a perimeter of 2+2π, it scores about 0.575.

Another consideration would be districts in the middle of the state that border the straight northern or southern boundaries of the state. These would also have the possibilities of a square or rectangular district with the same scores as above. Since they include a straight line, any triangle would be a possibility. So the only real special case they present would be the district in the shape of a semicircle. Score? About 0.747.

All this being said, these scores are most likely magnitudes above any gerrymandered district, but they are based on simple geometry with no consideration for any practical considerations that district drawing must take into account. Also, the only shape that would garner a perfect score of 1 is a circle. Most boundaries do not lend themselves well to using circles for districts and since circle do not fit well next to each other they would leave districts in between themselves that would score no better than a triangle (but probably still much better than most gerrymandered districts.) Sollupulo (talk) 02:38, 24 February 2018 (UTC)