  In ultrarelativistic heavy-ion experiments, one estimates the centrality of a
collision by using a single observable, say $n$, typically given by the
transverse energy or the number of tracks observed in a dedicated detector. The
correlation between $n$ and the impact parameter, $b$, of the collision is then
inferred by fitting a specific model of the collision dynamics, such as the
Glauber model, to experimental data. The goal of this paper is to assess
precisely which information about $b$ can be extracted from data without any
specific model of the collision. Under the sole assumption that the probability
distribution of $n$ for a fixed $b$ is Gaussian, we show that the probability
distribution of the impact parameter in a narrow centrality bin can be
accurately reconstructed up to $5\%$ centrality. We apply our methodology to
data from the Relativistic Heavy Ion Collider and the Large Hadron Collider. We
propose a simple measure of the precision of the centrality determination,
which can be used to compare different experiments.
