The documentation seems to state that MovingAverage[list, 6] would do this.
This example seems to show that this gives the correct result when r=3
In[1]:= MovingAverage[{1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4}, 3]
Out[1]= {1, 4/3, 5/3, 2, 7/3, 8/3, 3, 10/3, 11/3, 4}
