  Uncertainties $(\Delta x)^2$ and $(\Delta p)^2$ are analytically derived in
an $N$-coupled harmonic oscillator system when spring and coupling constants
are arbitrarily time-dependent and each oscillator is in an arbitrary excited
state. When $N = 2$, those uncertainties are shown as just arithmetic average
of uncertainties of two single harmonic oscillators. We call this property as
"sum rule of quantum uncertainty". However, this arithmetic average property is
not generally maintained when $N \geq 3$, but it is recovered in $N$-coupled
oscillator systems if and only if $(N-1)$ quantum numbers are equal. The
generalization of our results to a more general quantum system is briefly
discussed.
