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Which is Denser: New York or Los Angeles?

Your intuitions are correct. New Yorkers live in neighborhoods with much higher density than do Angelenos. But its not obvious how this conclusion is reached, and there's plenty of confusion going around about measuring density. Where you draw the lines on the map can have a significant impact on the results you get.

If you measure density purely regionally, Los Angeles comes out ahead. I've used the Census Bureau's MSA to show the two metro area's population densities in 2010. Sometimes the geography of Urbanized Area is used to capture the region, but that hasn't been determined for 2010 yet. So the MSA will do ...

However, this measurement misses the important story. As Ryan Avent explains,

"Simple population density measures the average density across a particular area. If you have a metro that covers a large area but which features a very dense core, however, you can easily have a situation in which the vast majority of the metro's population lives at densities above the average population density. I think it's more informative to focus on weighted-average population density."

  So here's the weighted-average density (by census tract) for the two metro areas:

New York metro goes from being about 30% behind LA in regional density to more than doubling LA in average neighborhood density.

If you were to drop from a parachute flying over the center of a city on a very windy day, the first regional density figure would tell you how many people to expect to see in the square mile around the random place in the region you land. However, if you currently live in the New York or Los Angeles metro areas (or are considering moving there), the latter figure would tell you how many people you would expect to see in the square mile around you. It's tethered to human experience, which is usual what we are asking about when we talk about density.