Perhaps rephrase your question and clearly say what it is you want to visualise. I think the visualisation problem is probably independent of your application so perhaps avoid jargon from your field.
What is it that you want to visualise? Can you explain what you mean by "I have one code that creates a sphere but I don't know how to add in my measurements."? Do you mean that you want to colour the surface of the sphere according to some angular (spherical coordinate system) measurements? Or do you want to alter the sphere radius depending on measurements? What do you mean by "add in"? The colour/shape mappings can be done.
What do you mean by: "They are all taken on a horizontal plane with the router in the middle but the router was rotated every 22.5 degrees and 16 measurements were taken each rotation."? That sounds like 2D angular (in the plane) measurements but earlier you talked about a 3D sphere.