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May 23, 2015 at 21:47 comment added Chris W Good point. If they are randomly located within the zip code or just at the centroid (or either if they're just in tabular form) then there's some question over how to figure out which county they're in if even possible. Even without an address, if the points were in a specific location they could be geocoded to get one. Which wouldn't be necessary anyway, because at that point it's a simple point-in-polygon operation as your answer outlines and is discussed at a number of other questions here.
May 23, 2015 at 3:10 comment added evv_gis The OP should clarify how the point data was created. If the point data is simply a centroid of the zip code area, then yes, adding information on how to handle more than one potential county should (and will) be added.
May 23, 2015 at 2:27 comment added Chris W Since zip and county boundaries aren't coincident, I think it would improve this answer if you address that. How will the spatial join handle a zip with two counties? Mention of the potential need for apportionment prior to aggregation might be important, since without it you skew results (and even with you're only estimating).
May 22, 2015 at 13:33 comment added evv_gis @Vesanto - really that is a different question than what the OP asked, but you can reference this post.
May 22, 2015 at 13:30 history edited evv_gis CC BY-SA 3.0
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May 22, 2015 at 13:27 comment added HeikkiVesanto How would you get x and y's onto your postcodes layer?
May 22, 2015 at 13:26 history answered evv_gis CC BY-SA 3.0