  We introduce randomness into a class of integrable models and study the
spectral form factor as a diagnostic to distinguish between randomness and
chaos. Spectral form factors exhibit a characteristic dip-ramp-plateau behavior
in the $N>2$ SYK$_2$ model at high temperatures that is absent in the $N=2$
SYK$_2$ model. Our results suggest that this dip-ramp-plateau behavior implies
the existence of random eigenvectors in a quantum many-body system. To further
support this observation, we examine the Gaussian random transverse Ising model
and obtain consistent results without suffering from small $N$ issues. Finally,
we demonstrate numerically that expectation values of observables computed in a
random quantum state at late times are equivalent to the expectation values
computed in the thermal ensemble in a Gaussian random one-qubit model.
