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Zipf's law is a law that states that the frequency of something will be equal to the most frequent thing divided by it's rank. So something that is rank 2 shows up half as much as rank 1, and rank 3 shows up a third as often as rank 1. Vsauce does a good job of explaining it.
So for any particular category does your country follow Zipf's law?
I will do the US as an example.
Rank. Metro city: Population | Predicted population according to Zipf's law
1. New York: 20,153,634 | 20,153,634
2. Los Angeles: 13,310,447 | 10,076,817
3. Chicago: 9,512,999 | 6,717,878
4. Dallas: 7,233,323 | 5,038,409
5. Houston: 6,772,470 | 4,030,727
6. Washington: 6,131,977 | 3,358,939
7. Philadelphia: 6,070,500 | 2,879,091
8. Miami: 6,066,387 | 2,519,204
9. Atlanta: 5,789,700 | 2,239,293
10. Boston: 4,794,447 | 2,015,363
So Zipf's law does a pretty terrible job at predicting the metro area's population, but city limit population's follow Zipf's law much more closely.
1. New York: 8,537,673 | 8,537,673
2. Los Angeles: 3,976,322 | 4,268,837
3. Chicago: 2,704,958 | 2,845,891
4. Houston: 2,303,482 | 2,134,418
5. Phoenix: 1,615,017 | 1,707,535
6. Philadelphia: 1,567,872 | 1,422,946
7. San Antonio: 1,492,510 | 1,219,668
8. San Diego: 1,406,630 | 1,067,209
9. Dallas: 1,317,929 | 948,630
10. San Jose: 1,025,350 | 853,767
You don't only have to do city populations, you could also do any other category, for instance state area, which by the way is a bit off in the case of the US.
Rank. State: Area | Predicted area according to Zipf's law
1. Alaska: 665,384.04 sq mi (1,723,337 km2) | 665,384.04 sq mi (1,723,337 km2)
2. Texas: 268,596.46 sq mi (695,662 km2) | 332,692 sq mi (861,668.5 km2)
3. California: 163,694.74 sq mi (423,967 km2) | 221794.67 sq mi (574,445.67 km2)
4. Montana: 147,039.71 sq mi (380,831 km2) | 166,346 sq mi (430,834.25 km2)
5. New Mexico: 121,590.30 sq mi (314,917 km2) | 133,076.8 sq mi (344,667.4 km2)
6. Arizona: 113,990.30 sq mi (295,234 km2) | 110,897.33 sq mi (287,222.83 km2)
7. Nevada: 110,571.82 sq mi (286,380 km2) | 95,054.86 sq mi (246,191 km2)
8. Colorado: 104,093.67 sq mi (269,601 km2) | 83,173 sq mi (215,417.13 km2)
9. Oregon: 98,378.54 sq mi (254,799 km2) | 73,931.56 sq mi (191,481.89 km2)
10. Wyoming: 97,813.01 sq mi (253,335 km2) | 66,538.4 sq mi (172,333.7 km2)
Whoa, a 21 minute video of I don't know what the heck?
What does it tell us?
You could watch the video and find out? But you don't have to watch the entire video to get a handle of the subject. You only really need to watch the first 4 minutes. If you like what you see you can watch the rest.
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