  Squared deviations from the equilibrium positions of one-dimensional coupled
harmonic oscillators with fixed and free endpoints are calculated, and the time
averages are expressed as a function of the initial displacements and
velocities. Furthermore, we consider the averages of squared deviations over an
ensemble of initial displacements and velocities, which distribute based on a
product of the same distribution functions with variances of the initial
coordinates $\sigma_{x}^2$ and velocities $\sigma_{v}^2$, respectively. We
demonstrate that the mean squared deviation linearly increases as the
oscillator separates further from the fixed endpoint because of the
asymmetrical boundary conditions, and that the increase rate depends only on
$\sigma_{v}^2$ and not on $\sigma_{x}^2$. This simple statistical property of
harmonic oscillation is similarly observed in the oscillations of a graphene
sheet and carbon nanotube in molecular dynamics simulations, in which the
interacting forces are nonlinear.
