Given base $b$, what is the expected value of round off error (relative, not absolute) when we round $x$ to $d$ digits, where $x$ is a random variable? Assume $x$ is drawn from a uniform distribution over an interval.
It's easy to determine maximum error: e.g. for $b = 10, d = 1$, it's $1/3$, which occurs at $15$. The expected value requires integrating over the floor function and seems to require using the harmonic series.