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I'm pretty new to GIS and I'm using ArcGIS 10.4

I've gathered point information about two different variables, X and Y. I have created the buffer zone around the X points and used a spatial join to join the Y points within a .25 mile buffer around the X points.

How can I show the count of Y points inside this buffer using graduated symbols preferably?

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    Where is all the information stored? From your question I guess it is in the buffers. Do you have a picture of the attribute table? – PythonStudent Apr 30 '17 at 20:03
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You can use Spatial Join to Join the points with the buffer polygon. Spatial Join can be found from ArcToolbox -> Analysis Tools -> Overlay -. Spatial Join.

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Use Buffer feature as Target Feature and Y points as Join Feature, and in the Field Map of Join Features, right-click the target field and select Merge rule, then choose Count, and save the output shapefile as a new file.

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You will get a new field in the attribute table with a name Join_Count, as you can see below:

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Use the Join_Count field to symbolize the result based on graduated color not symbol, because graduated symbol works only with point feature not polygon feature.

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