# calculating "center of population"

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### #1 Dennis McClendon Posted 02 November 2007 - 06:29 PM

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Has anyone seen a recipe for how to do this?

I'm trying to figure out the population center of Chicago. One way of describing this is the pinpoint on which the city polygon would "balance." Here's the Wikipedia entry explaining this concept.

Since I happen to already have a dot density map in FreeHand, it's simple for me to find an east-west line with an equal number of dots above and below, and the similar north-south line. That's the "median population center." But if I rotate the whole map 45 degrees, that center point (half the dots east and half west, etc.) changes, making it clear that's insufficient to determine the mean, or weighted population center.
Dennis McClendon, Chicago CartoGraphics
chicagocarto.com

### #2 jokestaaah Posted 02 November 2007 - 07:19 PM

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Has anyone seen a recipe for how to do this?

I'm trying to figure out the population center of Chicago. One way of describing this is the pinpoint on which the city polygon would "balance." Here's the Wikipedia entry explaining this concept.

Since I happen to already have a dot density map in FreeHand, it's simple for me to find an east-west line with an equal number of dots above and below, and the similar north-south line. That's the "median population center." But if I rotate the whole map 45 degrees, that center point (half the dots east and half west, etc.) changes, making it clear that's insufficient to determine the mean, or weighted population center.

There are two ways to do this:
First, if you want an unweighted centroid (just solely based on location), you take the average "X" coordinate and the average "Y" Coordinate, which gives you your unweighted centroid.

The second way, for each coordinate you assign a weight (# of whatever at that location of the dot) with the above equation. I am not too sure as to the forumula, but that is the concept.

### #3 Charlie Frye Posted 05 November 2007 - 02:47 PM

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Has anyone seen a recipe for how to do this?

I'm trying to figure out the population center of Chicago. One way of describing this is the pinpoint on which the city polygon would "balance." Here's the Wikipedia entry explaining this concept.

This left me thinking, just because you can generalize something down to one point, does it mean you should?

What if (averaging the Xs & Ys) that point is not in Chicago? What if it's in a rail yard or a warehouse district?

Is there just one point? If you created a surface of population by interpolating the population for census block points, and then analyzed that "terrain" would you have just one peak?

You could run a cluster analysis on your dots; though in theory it should work just as well to do so on census block points weighting by population. The number and location of clusters should tell you something.
Charlie Frye
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### #4 p-dub Posted 07 November 2007 - 11:02 PM

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Hi Dennis,

The mean center formula is pretty simple, I think this is the 'recipe':

Xbar=sum(PiXi)/N
Ybar=sum(PiYi)/N

P=population of area
X=x coord
Y=y coord
N=sum(Pi)

How were your points created? I think most GIS apps apply dot density points in a somewhat random fashion, to fill an area.
Also, each dot in a dot density should have the same population value, so, you can't really weight the points, they will each have the same influence.
Can you create centroids from the points?

Charlie, you have a point. Some areas have odd shapes, and spatial patterns are functions of area.
Sometimes an average doesn't say much... Creating a surface from dot density points does make sense.
However, a mean center is a measure of concentration best observed in relation to other concentrations.
i.e. Where is the poor center of town?(very possibly near the freight or stock yards or even 'outside') in relation to the wealthy center?

Does this make sense to anyone? Any geo-statisticians out there?

PW

### #5 p-dub Posted 08 November 2007 - 02:12 AM

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Dennis,
Your method of counting points and dividing them into halves is flawed. It removes all spatial objectivity from the points. Without an objective spatial reference the ‘median’ as you described is meaningless. You rotate the map, and see a different result because you are assuming that your perspective is the origin. What happens if you look at the map on a vertical plane? I guess they are in a line… I’m sitting to your right, so I say the center is here…

There needs to be an origin, a datum… I think using lat/long would work but, wouldn’t make much sense and the calculations would be difficult. Try plotting the dots in a local rectangular coordinate system, maybe SPCS. Even make up your own rectangular system for Chicago; I don’t see why this wouldn't work.

So yah, we covered mean center in my cart class today, then I found this thread…

PW

(edits -> time for bed)

### #6 Dennis McClendon Posted 08 November 2007 - 02:58 PM

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In Chicago, it's not entirely meaningless to calculate the median based on cardinal directions because the city is so orthogonal and natives think of the world that way. So the median center (half the population east and half west; half the population north and half south) is pretty easy to calculate. It falls near Leavitt and Monroe:

Each dot represents 200 persons in 2000

But if you rotate the map 45 degrees (half the population to the northwest, half southeast, etc.), you get a different median point, near Roosevelt & California, pointing out the problem with this approach, which I think you and I agree on, but in different phrasing.

You also point out the problem with random placement of dots by the GIS to begin with, but these were done by small areas (census tracts, which within the city are comprised of only a few blocks) and with many dots representing a large population, any error should be quite tiny.

Later today I will try averaging the Xs and Ys.
Dennis McClendon, Chicago CartoGraphics
chicagocarto.com

### #7 Dennis McClendon Posted 08 November 2007 - 04:32 PM

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A quick export to Adobe Illustrator 3, a text editor, and Excel made quick work of calculating the mean x value and mean y value for the purple dots. I was surprised that it moved south from the median, to near 16th/Claremont:

Dennis McClendon, Chicago CartoGraphics
chicagocarto.com

### #8 dsl Posted 22 November 2007 - 05:49 PM

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You've probably moved past this, but you could use the Census' methodology for mean population center. Here is a pdf:
http://www.census.go...calculate2k.pdf

You could also use some of the one's they've calculated:
http://www.census.go.../bg_cenpop.html
http://www.census.go.../coucntr17.html

Just curious, is your dot density map actual locations of people or are they randomly generated? If they are randomly generated, you might consider getting the centroids of census blocks, and weighting them based on population to get a better sense of the population center. If I'm understanding what your goal is correctly.

Hope that helps.
David

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