How to analyze spatial distribution of points based on common attribute value

RonTheSurveyor

I have a file with about 5000 points that can be grouped into about 200 groups based on an attribute value. For each unique attribute value, I want to:

1 - compute the average coordinate values (X,Y) for the group.

2 - compute the distance from the average point to each point in the group

3 - compute the average distance and standard deviation

4 - compare the average distance to each individual distance

The results would be used to find outliers that can be removed.

I looked at heat maps and other tools, but haven't been able to find anything that gives me this kind of analysis. I could export the coordinates to excel and do it there, but this seems like basic core GIS functionality.