Center of Population

What’s the world’s center of population?

The center of population for a region is, roughly, the center of mass of the inhabitants. The Census bureau defines the center of population of the US (currently in Missouri) as

the point at which an imaginary, flat, weightless, and rigid map of the United States would balance perfectly if weights of identical value were placed on it so that each weight represented the location of one person on the date of the census.

This definition breaks down for populations on curved surfaces. For the earth as a whole, the center of mass obviously falls deep inside the planet.

This problem is easy to fix. I figure a better definition would be the point at which the sum of straight-line surface distances to each person is minimized. This is equivalent to the standard definition for a flat region, but it has the advantage that you can use it to define the center of population for a sphere.

I’ve never seen anyone who’s calculated the earth’s center of population so defined, but it doesn’t seem like it would be hard. Does anyone have the answer?

Bonus: find the center of population for other groups. What is the center of population of native English speakers? internet users? … bloggers?

Edit: I was standing the shower just now when I realized that the generalization I was using had to be wrong. I got it from this page on Wolfram Mathworld,

The centroid of n point masses also gives the location at which a school should be built in order to minimize the distance travelled by children from n cities, located at the positions of the masses, and with m_i equal to the number of students from city i (Steinhaus 1999, pp. 113-116).

and did try to check out the citation while writing, but it was to a book and I was much too lazy for that. However, I think the Wolfram paraphrasing is wrong — it’s not the distances that are minimized; it must be some other quantity. You can see that this is wrong for center-of-mass of two people at A and one person at B. It’s probably sums of squares that are minimized (as suggested in a comment, and which works for the three-person example) but I don’t see an obvious proof of this.

239 thoughts on “Center of Population

  1. Take the center of population in the three dimensional space. Take a line passing through the mass center of earth surface(considering it very thin and uniform)and the center of population. The intersections of the this line with the surface should be near the center of population on a map you are searching for. This solution seems to be less acurate if the center of population is near the center of the earth.

    One other solution would be to take a arbitrary point and minimize the sum of the distances (following the geodesics) between this point and each persons. This would work for any planet shape and dimensions. But I don’t know if it’s really equivalent to the center of mass.

  2. Until earlier today, I had been crushed and disheartened by the failure to grasp the Physickal Sciences demonstrated by the Tappet Bros (Tom & Ray, Car Talk) on their most recent show. The discussion here has greatly improved my spirits. So, a big thank you to all, and special thanks to Kevin Price for sharing the link to the paper: I am now bewildered by centroids at a much higher level than ever before.

    OT — Lockheed-Martin, JPL, UofA and supporting cast of thousands, you and your beautiful Phoenix all kicked planetary ass today and I want to have your (organic) babies. W00T!

  3. The center of mass is the average of the torques, no?

    If we assume that all people have the same mass, then it just becomes the average of each person’s location.

    I propose that we express each person’s location as a vector in a 3-dimensional Cartesian coordinate system, and find the average of each component of the vector individually. Then put the averages of the components together to make a new vector, and this becomes your center of mass, eh?

  4. @myncknm you think YOU are late!

    My 2c

    Would this not depend on the map used?
    where there is only one country a flat model of the country limits makes sense. There is a definable ‘west most’ point.
    On a spheroid – the earth – a globe – a map of the world – where is the west-most point?

    * GMT Meridian

    * Swatch headquarters (Biel Mean Time (BMT) ) http://en.wikipedia.org/wiki/Swatch_Internet_Time

    * International date line

    * Perpetuate the imperialist view that Britain is both the center and the top of the world. ( actually this reflects the international date line – but pointing this our kinda spoils the rant)

  5. This is called a “Fréchet Mean”, and is a fairly standard part of Riemannian geometry. There’s a standard gradient descent algorithm for computing it, which is basically the method described by CJ above, where the solution from the previous iteration becomes the new “north pole” for the next. See, for example, Karcher’s classic article on the algorithm http://doi.wiley.com/10.1002/cpa.3160300502 or Kendall’s proofs on existence and uniqueness http://scholar.google.com/scholar?cluster=16807413806928015094
    Probably the best general introduction to the concepts for computer scientists I know about is Tom Fletcher’s PhD thesis: http://midag.cs.unc.edu/pubs/phd-thesis/PTFletcher04.pdf
    Read chapters 2 and 4.

  6. I’ve been reading the site for a while and this thread a while ago and an idea suddenly just now came to me.
    For the sake of the argument, assume the earth is a perfect sphere (or transmorgrify it into a sphere), and say it was a thin hollow and massless shell and the people point-masses positioned on that shell, and placed in a uniform gravitational field on a surface so that due to uneven balance from the point-masses it would reposition itself (rotate due to the frictionless nature), wouldn’t the contact-point between the sphere and the surface then be the center of mass of the population.. so to speak?
    Well.. cheers.

  7. This is called a “Fréchet Mean”, and is a fairly standard part of Riemannian geometry. There’s a gradient descent algorithm for computing it, which is basically the method described by CJ above, where the solution from the previous iteration becomes the new “north pole” for the next. See, for example, Karcher’s classic article on the algorithm http://doi.wiley.com/10.1002/cpa.3160300502 or Kendall’s proofs on existence and uniqueness http://scholar.google.com/scholar?cluster=16807413806928015094
    Probably the best general introduction to the concepts for computer scientists I know about is Tom Fletcher’s PhD thesis: http://midag.cs.unc.edu/pubs/phd-thesis/PTFletcher04.pdf
    Read chapters 2 and 4.

  8. Noticing that there is a copious number of comments already, I’ll just hope that I’m not repeating someone else here ^.^

    if you calculated the sentre of mass of the earth as a sphere, then projected it on to the surface by moving it away from the core, would that not be the population desnsity?

    If this is wild and preposterously untrue, I blame lack of sleep and/or caffeine ¬.¬

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  10. There’s a problem with just laying out the skin of a sphere and finding the center of mass on that flat plane. With a sphere (or any 3-D object for that matter), The center will adjust depending on how you unfold the object.

    To find the center of mass, we still need to treat the earth as a 3-D object. I think we should just find the population of individual latitudes (up to whatever level of precision you prefer), and then longitudes. The spot with the most populated latitude AND longitude is the center of mass.

  11. I don’t think the center of mass is the correct stuff to use for the location of school problem.
    Simple reason being that all the heavier kids would get more weight in the equation.
    other reason being that I wasn’t particularly weighty as a kid.
    of course that equation might help in reducing the sum of work done to get to school by the kids.

  12. 200 comments for an idea which is stupid for the following reason: with your definition, the point is not neccesarily unique on a periodic surface. For example, imagine a world uniformly distributed with people. All points fit your definition. Certainly, in our current world, a lot of points probably do to a decent approximation. It’s a dumb idea, certainly far dumber than 200 comments would warrant. Let’s move on, people.

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  14. I found the center of population on the surface of the earth to be at 30.89N,76.53E. I found this by calculating the weighted average distance a given point was from all the people on the earth until I found the smallest average distance. I did this at a resolution of 1 degree latitude and 1 degree longitude around the entire earth. Then I increased the resolution and decreased the search area to get the values I got. It turns out that the average person is 5032 km from this point. I also recorded the average distance for all 65,000ish points, so I could make a map showing the average distance everyone is from a given point, but I’m not sure how to make the map. I also found the center of population based on a center of mass calculation and found that, assuming everyone weighs the same, the center of mass is 637 km under 15.68N,75.60E.

  15. Since I got in on this thread late, I have not read all the comments, but I have a theory that might be able to be expanded for finding the “center of population” inside a sphere. Instead of imagining every person on the planet having a uniform mass, they can each have a uniform tension on the center of population. Picture you are inside a giant beach ball with a number of similar rubber bands connected to the shell of the ball at one end and each other at the other. This central knot will be pulled to a central point where the tension to each point on the shell is the least. A similar method idea could work where a person is at each point on the shell where a band is connected. Can anyone think of a way to find where the knot would be?

  16. My intuitive assessment is that the centre of everything is located approximately a meter and a half to the left of me, +- a couple of meters, although i can think of no obvious proof of this.

  17. I think Johan is right! I’ve thought of a number of tricky scenarios to try and fool it, but it succeeds every time. This is of course assuming that you want to find a point on the surface of the Earth which minimises distance across the surface to every person. The way to solve it would be to minimise the sum of all the (mgh)s. Or just plain (mh)s, given that g is just a constant. And come to think of it, since we’re assigning equal value to each person, we could just minimise h. Granted, h is still a function of both theta and phi.

  18. I don’t think the center of mass is the correct stuff to use for the location of school problem.
    Simple reason being that all the heavier kids would get more weight in the equation.
    other reason being that I wasn’t particularly weighty as a kid.
    of course that equation might help in reducing the sum of work done to get to school by the kids.

  19. I think it’s the average deviation from the average length of the path from each person to the center that’s being minimized i.e. if the school is at the center of population then every pupil will have roughly the same distance to travel to school.

    Example in a one-dimensional world: There are two people at point A and one person at point B, 30 km away. The best place for the school to optimize for distance would obviously be point A. The two persons there would be living inside the school and the person at B would have to travel 30 km, thus the average would be (0km+0km+30km)/3=10km. The average deviation from that would be ((10+10+20)/3)km = 13,3km.

    If the school is at the center of population C, between A and B, 10km away from A, then the average way would be ((10+10+20)/3)km = 13,3km > 10km, but the average deviation would only be ((3,3+3,3+6,7)/3)km = 4,43 km < 13,3km.

    So this would be the fairest point to build a school, but not the best.

  20. I’ve not read any of the comments, so my apologies if this has been suggested before.

    Why not do it the old-fashioned 3-dimensional way, find the CM_humanity point which, as noted, lies within the interior of the Earth, and then project a line from the center of the earth through that point, to a point on the surface?

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