Exponential inequality

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Exponential inequality treats the equal changes in the multiple of the range as an equal change in the score.

An average state with 72 counties and 7 apportionment regions has 10.3 counties per region on average. For each county that is chopped to reduce the range it decreases the number of regions by one. With one chop the average state would have 6 regions with 12 counties per region. With two chops the average state would have 14.4 counties per region. For this average state each chop corresponds to a change of approximately two in the measure of counties per region. Based on the slope factor described above, there should be a decrease in the range by a factor of 10 for each 5 chops.

This factor has a weakness for small ranges. Consider the same average state in the preceding paragraph using the exponential formula based on the correlation between the range and the number of geographic units. The predicted range for 7 regions is 2343. If it was only divided into 2 regions with 5 chops there would be 36 counties per region and the formula would predict a range of 6. A purely exponential table like the one below would assign the 7 region plan a likely score of 17, but the 2 region plan would likely get a score of 4. The inequality score would improve 13 points at the expense of 5 chops. This shows a lack of balance between the two factors.

Exponential
Inequality
Factor
Range
0 0 - 1
1 2
2 3
3 4
4 5 - 6
5 7 - 10
6 11 - 16
7 17 - 25
8 26 - 40
9 41 - 63
10 64 - 100
11 101 - 160
12 161 - 250
13 251 - 400
14 401 - 630
15 631 - 1000
16 1001 - 1600
17 1601 - 2500
18 2501 - 4000
19 4001 - 6300
20 6301 - 10000
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