  We investigate the crystal structure in multiferroic tetragonal perovskite
Sr$_{1/2}$Ba$_{1/2}$MnO$_3$ with high accuracy of the order of 10$^{-3}$
Angstrom for an atomic displacement. The large atomic displacement for Mn ion
from the centerosymmetric position, comparable with the off-centering
distortion in the tetragonal ferroelectric BaTiO$_3$, is observed in the
ferroelectric phase ($T_\mathrm{N}$ $\leq$ $T$ $\leq$ $T_\mathrm{C}$). In stark
contrast, in the multiferroic phase ($T$ $\leq$ $T_\mathrm{N}$), the atomic
displacement for Mn ion is suppressed, but those for O ions are enlarged. The
atomic displacements in the polar crystal structures are also analyzed in terms
of the ferroelectric modes. In the ferroelectric phase, the atomic
displacements are decomposed into dominant positive Slater, negative Last, and
small positive Axe modes. The suppression of Slater and Last modes, the sign
change of Last mode, and the enlargement of Axe mode are found in the
multiferroic phase. The ferroelectric distortion is well reproduced by a
first-principles calculation based on Berry phase method, providing an
additional information on competing mechanisms to induce the polarization,
electronic $p$-$d$ hybridization vs. magnetic exchange-striction.
