  Single crystalline YBa$_{\text{2}}$Cu$_{\text{3}}$O$_{\text{7}-\delta}$
(YBCO) thin films were grown by pulsed laser deposition (PLD) in order to probe
the oxygen deficiency $\delta$ using a mono-energetic positron beam. The sample
set covered a large range of $\delta$ (0.191<$\delta$<0.791) yielding a
variation of the critical temperature $T_{\text{c}}$ between 25 and 90\,K. We
found a linear correlation between the Doppler broadening of the positron
electron annihilation line and $\delta$ determined by X-ray diffraction (XRD).
Both, the origin of the found correlation and the influence of metallic
vacancies, were examined with the aid of ab-initio calculations that allowed us
(i) to exclude the presence of Y vacancies and (ii) to ensure that positrons
still probe $\delta$ despite the potential presence of Ba or Cu vacancies. In
addition, by scanning with the positron beam the spatial variation of $\delta$
could be analyzed. It was found to fluctuate with a standard deviation of up to
$0.079(5)$ within a single YBCO film.
