A random telegraph signal is a time series whose value S (t) at time t is either one of only two possible values. Many processes including chemical reactions, cell membrane ion channels, and electronic noise generate such signals. Usually, Markov models have been used to model and analyze such data. Instead, we present a new fractal random telegraph signal that is statistically self-similar in time. We show how to analyze such signals and apply those techniques to study burst noise in a defective operational amplifier and ion currents recorded through individual ion channels in a cell membrane.