  Peculiar phenomena have been observed in analyses of anisotropic flow ($v_n$)
fluctuations in ultracentral nucleus-nucleus collisions: The fourth-order
cumulant of the elliptic flow ($v_2$) distribution changes sign. In addition,
the ATLAS collaboration has shown that cumulants of $v_n$ fluctuations of all
orders depend significantly on the centrality estimator. We show that these
peculiarities are due to the fact that the impact parameter $b$ always spans a
finite range for a fixed value of the centrality estimator. We provide a
quantitative determination of this range through a simple Bayesian analysis. We
obtain excellent fits of STAR and ATLAS data, with a few parameters, by
assuming that the probability distribution of $v_n$ solely depends on $b$ at a
given centrality. This probability distribution is almost Gaussian, and its
parameters depend smoothly on $b$, in a way that is constrained by symmetry and
scaling laws. We reconstruct, thus, the impact parameter dependence of the mean
elliptic flow in the reaction plane in a model-independent manner, and assess
the robustness of the extraction using Monte Carlo simulations of the
collisions where the impact parameter is known. We argue that the
non-Gaussianity of $v_n$ fluctuations gives direct information on the
hydrodynamic response to initial anisotropies, ATLAS data being consistent with
a smaller response for $n=4$ than for $n=2$ and $n=3$, in agreement with
hydrodynamic calculations.
