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I have census block data polygons. I want to overlay them over Zip Codes (larger polygons) in order to estimate the population in the Zip Codes. Or overlay buffers around toxic waste sites over census blocks to estimate the population within the buffer. How is this done In QGIS?

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I sort of understand what you're trying to do, but I'd like to ask you to expand on what you're trying to achieve and perhaps where are you getting stuck at? – dassouki Jul 11 '12 at 1:41
You may want to read this section of the Census Geography and Maps FAQ: FAQ - Zip Codes - You may potentially want to look into the Zip Code Tabulation Area Product for your statistics instead. – Get Spatial Jul 11 '12 at 1:43
It is important to note that doing this sort of thing can introduce large error in your results especially since the census block and zip codes do not necessarily follow the same boundaries and also the population may not be evenly distributed. You should check out the Modifiable Areal Unit Problem and also here. Also see the ecological fallacy – Fezter Jul 11 '12 at 3:06
To get an estimate of say, the population of Zips, which are not census areas. One overlays cen blks and thru areal interpolation determines an estimate of the Zip population. The notion is how to determine characteristics of one geography from information you have in another. It does not matter is the boundaries are coincident. The GIS will determine what fraction of a Zip is covered by one or more census blocks. Uniform pop distribution is assumed. This is how ZCTA pops are determined. Most GIS have this. I am new to QGIS, figure it must be there. Essential feature for spatial statistics. – HealthMaps Jul 11 '12 at 4:13
TO make it clearer what I am rying to do with QGIS: – HealthMaps Jul 12 '12 at 17:22

Taking into account that all the comments (Ecological Fallacy and Mismatched Polygons inherent to combining ZipCode and Census Tract data), this would be my suggestion.

1) Estimate Population Density Using Census Tracts using Random Points.

2) Vector -> Analysis Tools -> Points in polygon

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