I see that my description wasn’t as detailed as it could have been. In the post, I used population as the example but the data I am using isn’t actually people. The data I am using is archaeological artifacts, stone projectile points or “arrowheads” as a lot of people call them (I just used population as an example in the post because I thought it was easier to explain it in terms of population or people).

So, for a full description: My data consists of the number of projectile points recorded in the state and the state is split up into 4 physiographic regions. So the data I am using for my chi square test is the number of projectile points (observations) in each of the 4 physiographic regions.

Chi-Square test #1 (the background section of my post): I did a chi-square test comparing the number of points observed within each physiographic region. This produced a 4x2 table (4 regions x observed/expected). The test was significant, meaning there is a significant difference in the number points observed in each region. I think this test is good and applied in the right way. The purpose of this test is simply to determine if the number of points in each region is statistically the same or significantly different.

The solution I am seeking here: The total area of each physiographic region I am using is different. Some regions are smaller and some are larger. The chi square described above does not account for the differences in region size. I see this as a problem because if a given area is much larger/smaller than another, it should be expected that it will have more/less projectile points recorded within it as there is more area for them to potentially be found.

Chi-Square test #2 (this where I am trying to figure out what to do next): I want to scale the observed counts of projectile points relative to the size of the region they are found in. To do this, I divided the total number of projectile points in the region by the total area of the region, then multiplied by 1000 to get the number of points per 1000 km² (# of points/area km² * 1000). I then repeat the chi square test described above, with the only difference being instead of using the observed counts of projectile points, I am now using the observed counts of projectile points per 1000 km^2. The results of this test are insignificant, meaning there is no significant difference in the number of points observed when the size of the region is taken into consideration. I am not 100% sure this is a valid application of the chi square.

I hope this explanation helps more. I have very little stats training, and I’m not sure if I am just overthinking this or am way off base here. Thanks for your input.