  The power spectral density in Fourier frequency domain, and the different
variants of the Allan deviation (ADEV) in dependence on the averaging time are
well established tools to analyse the fluctuation properties and frequency
instability of an oscillatory signal. It is often supposed that the statistical
uncertainty of a measured average frequency is given by the ADEV at a well
considered averaging time. However, this approach requires further mathematical
justification and refinement, which has already been done regarding the
original ADEV for certain noise types. Here we provide the necessary background
to use the modified Allan deviation (modADEV) and other two-sample deviations
to determine the uncertainty of weighted frequency averages. The type of
two-sample deviation used to determine the uncertainty depends on the method
used for determination of the average. We find that the modADEV, which is
connected with $\Lambda$-weighted averaging, and the two sample deviation
associated to a linear phase regression weighting (parADEV) are in particular
advantageous for measurements, in which white phase noise is dominating.
Furthermore, we derive a procedure how to minimize the uncertainty of a
measurement for a typical combination of white phase and frequency noise by
adaptive averaging of the data set with different weighting functions. Finally,
some aspects of the theoretical considerations for real-world frequency
measurement equipment are discussed.
