  Despite spatial and temporal fluctuations in turbulent astrophysical systems,
mean-field theories can be used to describe their secular evolution. However,
observations taken over time scales much shorter than dynamical time scales
capture a system in a single state of its turbulence ensemble. Comparing with
mean-field theory can falsify the latter only if the theory is additionally
supplied with a quantified precision. The central limit theorem provides
appropriate estimates to the precision only when fluctuations contribute
linearly to an observable and with constant coherent scales. Here we introduce
an error propagation formula that relaxes both limitations, allowing for
nonlinear functional forms of observables and inhomogeneous coherent scales and
amplitudes of fluctuations. The method is exemplified in the context of
accretion disc theories, where inhomogeneous fluctuations in the surface
temperature are propagated to the disc emission spectrum--the latter being a
nonlinear and non-local function of the former. The derived precision depends
non-monotonically on emission frequency. Using the same method, we investigate
how binned spectral fluctuations in telescope data change with the spectral
resolving power. We discuss the broader implications for falsifiability of a
mean-field theory.
