Funky Math

An obscure (to me) blog wrote Monday of a “mysterious law that predicts the size of the world’s largest cities,” called “Zipf’s Law.” Apparently, this guy Zipf was a linguist who discovered this “law” with respect to word usage, and has since discovered that it is pretty much universal. With respect to cities, it means that a nation’s largest city is twice as large as the second largest, the second is twice as large as the third largest, and so on.

Just take a look at the top ranked cities in the United States by population. In the 2010 census, the biggest city in the U.S., New York, had a population of 8,175,133. Los Angeles, ranked number 2, had a population of 3,792,621. And the cities in the next three ranks, Chicago, Houston and Philadelphia, clock in at 2,695,598, 2,100,263 and 1,526,006 respectively. You can see that obviously the numbers aren't exact, but looked at statistically, they are remarkably consistent with Zipf's predictions.

To say that “the numbers aren’t exact” is more than a little bit of an understatement, and to say that “they are remarkably consistent with Zipf's predictions” is nothing short of delusional. At my first casual glance I saw two cities which are essentially the same size.

The first one works reasonably okay; Los Angeles at 3,792,621 is at least within hand grenade distance of the 4,087,566 which would be half of New York’s 8,175,133.

But half of Los Angeles is 1,896,310 and that is a lot closer to #4 Houston than it is #3 Chicago, smaller than either one of them, and it’s getting pretty close to #5 Philadelphia. We only had one step in Mr. Zipf’s putative “law” before things went to shit, and the author of this piece didn’t notice.

And it gets worse; much worse. Half of #3 Chicago would be 1,347,799, which is nowhere even close to #4 Houston. In fact, Chicago and Houston are for all practical purposes about the same size. Half of #3 Chicago is significantly smaller than #5 Philadelphia.

For Zipf’s law to be true, Los Angeles would have a population of 4,087,566 rather than 3,792,621, and we will quickly see even larger divergence between Mr. Zipf and reality. Chicago would lose almost a quarter of its population, dropping to 2,043,783 from 2,695,598. Houston would shrink by half to 1,021,891 from 2,100,263 and Philadelphia would diminish to 510,945, one-third of it’s present 1,526,006.

China, it turns out, has one city at 44 million, one at 27 million, one at 19 million, and then a whole bunch at around 10 million. Brazil has one city at 11 million, one at 6 million and then a whole bunch are around 2 million. Argentina kind of breaks the series with one city at 3 million and several at right around 1 million. France really screws it up with one city at 12 million, the next largest at 2 million and the third at 1.7 million.

Hard do decide which is nuttier; Zipf or the writer of this piece.