Chaotic behaviour due to round-off characteristics in digital compensators is called a chaotic rounding error. First, we show sufficient conditions for the existence of chaotic rounding errors in digital control systems with first-order plants and first-order finite-wordlength compensators. Next, it is shown that a chaotic rounding error, which appears as a chaotic attractor in the state space, can be described by a Markov chain with an appropriate partition of the attractor. Finally, by analysing the Markov chain we calculate several statistical properties of the chaotic rounding error.