# How to find "Center of Population"

I need to find the “center of population” for my State to support a political argument. Is there a way to calculate this? Or, a website “calculator” that would help me? - Jinx

Here’s a map for it for all states.. That is not something you can throw into a calculator and crunch.

That’s an interesting map. Could one say, that for any given state, if you laid a grid on the center of population with one axis going exactly East-West and the other going exactly North-South, there would be an equal number of people in each of the four quadrants?

Not at all. Although I admit I didn’t look at the site and don’t know how they define population center, but it should not be based on geographic orientation but rather population density vs. area. I’d guess that in the population center the average population in any given direction would be the same, but that does not mean that you can partition from the center into any equal size partitions (it could be impossible)

Here is the definition

Groman already noted that’s not true. Nevada provides a good counter-example. Consulting Shagnasty’s map, you’ll note that hardly anyone in Nevada lives in the north-east “quadrant”. And the south-west “quadrant” is so tiny, it can’t have too many people either.

Illinois seems to be another good counter-example. The majority of the state’s whole population is in the northeast.

I believe it’s mathematically guaranteed that there is a line, through any point in a state’s shape (and not just the center of population point) that will divide the population into two equal halves on either side. But in general that line is not going to be aligned north-south or east-west.

OK- so if the population center is the perfect balance point in terms of people, that means that if you take ANY line segment through said point, there will be the same number (or “weight”) of people on either side of the line sement, right?

One would expect the Center of Population to be analogous to Center of Mass: a weighted spatial average.

One approach is to compute the geographic center and population of each county or parish within the state. Then an estimate of the Center of Population would be:

C.P. (x-coordinate) = {Sum Over All Counties of County Pop Estimate*x-coordinate of County}/{Sum Over All Counties of County Pop Estimate},

C.P. (y-coordinate) = {Sum Over All Counties of County Pop Estimate*y-coordinate of County}/{Sum Over All Counties of County Pop Estimate},

C.P. (z-coordinate) = {Sum Over All Counties of County Pop Estimate*z-coordinate of County}/{Sum Over All Counties of County Pop Estimate}.

The nit-picky answer from Hell is that the State CP is not well defined, since the people keep moving around, and strictly speaking, everything is time-dependent.

The other issue is the choice of state divisions in the sample CP calculation. The simplest approach is to use a “residential C.P.”, where the population is indexed by residence. In practice, one could actually compute the average of locations over every single address in the state, provided that such data were available, and provided that one had a residence/address resident total count.

Then an estimate of residential CP would be:

No, that would be true only if the population distribution were radially symmetric about that particular point — which isn’t going to happen in general.

Scratch that. It doesn’t have to be radially symmetric. But it would have to have rotational symmetry in all directions around that point, which still won’t be the case in general for an arbitrary distribution of people.

If you look at New York State on the map cited by Shagnasty, you will see that the centre of population is almost on the state border, and there is almost no population in the SW quadrant. (New York has an unusual population distribution, with a big population concentration in the SE quadrant, and several major concentrations in the southern half of the NW quadrant, with most of the rest of the state pretty empty.)

You can’t guarantee that the population of all four quadrants are equal. Consider, for instance, a hypothetical state with almost all of the population in two equally-large cities, one in the southeast corner, and one in the northwest corner. By any definition, the center of population will be directly between those two cities. So almost all of the population will be in the SE and NW quadrants, and almost none in the SW or NE quadrants.

There does exist a (almost unique) point such that there is equal population to the east and west, and equal to the north and south, and there may even be a point such that there is equal population on either side of any line through that point. But this point will not, in general, be the same as the center of population, under the standard definition.

One more thing:

It is possible that the centre of population is outside the state. Indeed, only if the state is convex can you guarantee that, for any population distribution, the centre be will inside the state.

Proof:
If the state is not convex then there exists at least one pair of points within the state (A and B) such that there is a point C on the line segment between A and B with C outside the state. So you can put all the population on those points A and B, distributed so that the centre is on point C.

And apparently, Hawaii’s centre of population is outside the state.

The center of population is as NOAA stated above: the point at which a weightless plane in the shape of the geographic area would balance if an equal weight was placed at the location (residence) of each person.

As others have said, “No”.

What it does mean is that if you multiplied the number of people in the north section by the area of the north section, that will equal the number of people in the south section multiplied by the area of the south section.

But, that’s just a description of the calculation really. It doesn’t work quadrant-wise either. Just “hemisphere”-wise.

That doesn’t sound right. Imagine a perfectly square state, 100 miles on a side.

Everybody in this state is living in four cities - 3 million people in each city. Two of the cities are extremely close to the southwest and southeast corners of the state.
The other two cities are along the west and east borders, 4 miles north of the midline of the state.

According to your statement, trunk, the center of population would have to be in the geographic center of the square.

However, if I made a light square and added four equal weights placed so, I wouldn’t be able to balance it on it’s middle; it would be nearly halfway from the midpoint to the south border of the state.

It isn’t the area of the north and south sections that matter - it is how far the population is from the center that must be adjusted for to balance out. (Just like how far the weight is from the center of balance adjusts the force it exerts.)

You’re right.

It’s actually each person multiplied by their distance to the dividing line, not the area of their side. I actually work with those things. :eek:

oops.

Yes, there is a way How accurately do you need to know it? It is just a moment problem and as a first approximation I would do it by counties. The brute force approximation method is to draw a pair of arbitrary, orthoganal axes on the map. Multiply the population of each county by the signed distance of the geographic center of the county from the horizontal axis. Add these numbers and divide by the total population of the state. This gives you the location of a new horizontal axis on your map. Do the same thing for the vertical axis to get a new vertical axis. The approximate population center is where these new axes cross.