  The center of mass of an operator $A$ (denoted St($A$), and called in this
paper as the {\em Stampfli point} of A) was introduced by Stampfli in his
Pacific J. Math (1970) paper as the unique $\lambda\in\mathbb C$ delivering the
minimum value of the norm of $A-\lambda I$. We derive some results concerning
the location of St($A$) for several classes of operators, including 2-by-2
block operator matrices with scalar diagonal blocks and 3-by-3 matrices with
repeated eigenvalues. We also show that for almost normal $A$ its Stampfli
point lies in the convex hull of the spectrum, which is not the case in
general. Some relations between the property St($A$)=0 and Roberts
orthogonality of $A$ to the identity operator are established.
