  The well established procedure of constructing phenomenological ensemble from
a single long time series is investigated. It is determined that a time series
generated by a simple Uhlenbeck-Ornstein Langevin equation is mean ergodic.
However the probability ensemble average yields a variance that is different
from that determined using the phenomenological ensemble (time average). We
conclude that the latter ensemble is often neither stationary nor ergodic and
consequently the probability ensemble averages can misrepresent the underlying
dynamic process.
