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I have a point feature class with 1725 records (spread out over NYC) and a polygon feature class with 13 records (also spread out over NYC). What tools or processes can I use to figure out what percentage of those points are within a 1/2 mile radius of any of the 13 polygons?

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Use the Buffer tool to buffer your polygons with a .5 mile buffer. Use Select by Location, select the point feature class that fall within the .5 mile buffer. The resultant selection can be divided by the total, and then you will have the percentage.

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You could use the "Near" tool (under Proximity in Analysis Tools) for this. It will assign distance to the nearest polygon to the points. Then you can simply select those with a distance of less than 1/2 mile. It does require an Advanced License.

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