The capacity regions of Gaussian broadcast channels depends on the knowledge of channel state information (CSI). When there is only statistical CSI at the transmitter and full CSI at the receiver, the ergodic capacity region is unknown in general. In this paper we investigate the relation between the degradedness and stochastic orders among channels from the transmitter to different receivers. We derive criteria to identify the degradedness for single and multiple-antenna cases when the channels belong to the usual stochastic order or the increasing convex order. Examples illustrate the usage of the derived criteria. We also show a case in which the channel enhancement technique can be applied even when there is only statistical CSIT.