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I am a new user to ArcMap.

I have two layers of polygon features and I want to calculate the minimum and mean distance between polygons of one layer and polygons of the other layer.

I am guessing that Euclidean Distance is a tool I would use, but I'm not positive.

How do I approach this problem?

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Generate Near Table using the "closest" parameter set to ALL will build you a table showing the distance between every feature in one layer with every feature in another layer. You can then run summary statistics on this table (in Excel or some other table software) to identify the min and mean distance values for each input polygon. The more polygons you have, the longer (exponentially) this tool will take to run, so be aware that with a large dataset it will take a long time to run.

The Euclidean Distance tool is designed to be run only on raster datasets, not vector datasets (i.e. a grid not polygons).

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    No need in Excel. Summarise statistics on output will do. – FelixIP Feb 1 at 22:20

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