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I have the following data and have been asked to create a map that shows the same data set but displayed as contours.

Each polygon has a value in it's table called "count". The "count" field is used to show the graduated colors.

How would I go about converting the polygon data set into "contour" lines. I understand that these are not true contours and I'm not sure why "they" want to see it displayed this way, but I need to do it if I can.

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    You might want to look at a heatmap as a way to visualize this information. If you don't have a Spatial Analyst license it's going to be very difficult to generate contours. – Vince Apr 18 '16 at 16:31
  • I was afraid of that. I think I might try to use QGIS, though I don't have much experience with it. – GeoJohn Apr 18 '16 at 18:55
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You can convert the polygon to a raster using the count field, then using the contour tool to create contours. You will need either 3d or spatial for that though.

I believe there's a couple of free add-on's that can create contours if you don't have either license. GME is my favorite and I use that a lot. (http://www.spatialecology.com/gme/contour.htm)

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