  I present analytic error formulae for the energy-dependent cross-spectrum and
rms spectrum, which are Fourier statistics widely used to probe the rapid X-ray
variability observed from accreting compact objects. The new formulae cover the
modulus, phase, real and imaginary parts of the cross-spectrum, and are valid
for any value of intrinsic coherence between variability in different energy
bands. I show that existing error formulae (including that for the phase lag),
which are valid for a single cross-spectrum or power spectrum, lead to
over-fitting when applied to the energy-dependent cross-spectrum - which
consists of cross-spectra between individual energy channels and a common
reference band. I also introduce an optimal, unbiased way to define the
reference band and an accurate way to calculate the intrinsic coherence between
energy bands. I find that the traditional use of the old formulae has likely
had a rather benign impact on the literature, but recommend the use of the new
formulae in future wherever appropriate, since they are more accurate and are
no harder to implement than existing error estimates. A code to implement the
new error formulae on observational data is available online.
