To clarify - in the two pictures above you originally posted 1) you have the larger circumference selected. 2) that selection is the one you are scaling 3) when you scale it down it is the second picture [the top one] and it doesn't stay centered.
Rotation and scaling are relative to the pivot point. Even if you have the cursor centered but have the pivot on something else it won't work. Both the cursor and pivot point have to be set.
There is also another slight possibility. You may have inadvertently moved the objects origin relative to the geometry while in edit mode. This shouldn't be the case here, but I will explain anyway in case it helps later with something else.
Create a cube. Notice the orange dot that floats around in its center. Go into edit mode [tab], select all vertexes and move them - the whole box will move, but the origin marker [the orange dot] stays still. Now go back to object mode. Now move the box around again. Notice the orange dot isn't in the center of the cube anymore. You have re-oriented the entire objects geometry's origin.
If you tell the cursor to go to selected, and it is trying to find the origin, it will wind up somewhere you may not intend. Reset the origin to the geometry. See image. Other then that you may wish to reset rotation and scale (CTRL-A)... ?