Using the Westerbork Synthesis Radio Telescope , we carried out shallow H I absorption observations of a flux-selected ( S _ { 1.4 GHz } > 50 mJy ) sample of 93 radio active galactic nuclei ( AGN ) which have available SDSS ( Sloan Digital Sky Survey ) redshifts between 0.02 < z < 0.23 . Our main goal is to study the gas properties of radio sources down to S _ { 1.4 GHz } flux densities not systematically explored before using , for the first time , stacking of absorption spectra of extragalactic H I . Despite the shallow observations , we obtained a direct detection rate of \sim 29 \% , comparable with deeper studies of radio galaxies . Furthermore , detections are found at every S _ { 1.4 GHz } flux level , showing that H I absorption detections are not biased toward brighter sources . The stacked profiles of detections and non-detections reveal a clear dichotomy in the presence of H I , with the 27 detections showing an average peak \tau = 0.02 corresponding to N ( H I ) \sim ( 7.4 \pm 0.2 ) \times 10 ^ { 18 } ( T _ { spin } /c _ { f } ) cm ^ { -2 } , while the 66 non-detections remain undetected upon stacking with a peak optical depth upper limit \tau < 0.002 corresponding to N ( H I ) < ( 2.26 \pm 0.06 ) \times 10 ^ { 17 } ( T _ { spin } /c _ { f } ) cm ^ { -2 } ( using a FWHM of 62 km s ^ { -1 } , derived from the mean width of the detections ) . Separating the sample into compact and extended radio sources increases the detection rate , optical depth , and FWHM for the compact sample . The dichotomy for the stacked profiles of detections and non-detections still holds between these two groups of objects . We argue that orientation effects connected to a disk-like distribution of the H I can be partly responsible for the dichotomy that we see in our sample . However , orientation effects alone can not explain all the observational results , and some of our galaxies must be genuinely depleted of cold gas . A fraction of the compact sources in the sample are confirmed by previous studies as likely young radio sources ( compact steep spectrum and gigahertz peaked spectrum sources ) . These show an even higher detection rate of 55 % . Along with their high integrated optical depth and wider profile , this reinforces the idea that young radio AGN are embedded in a medium that is rich in atomic gas . Part of our motivation is to probe for the presence of faint H I outflows at low optical depth using stacking . However , the stacked profiles do not reveal any significant blueshifted wing . We are currently collecting more data to investigate the presence of outflows . The results presented in this paper are particularly relevant for future surveys in two ways . The lack of bias toward bright sources is encouraging for the search for H I in sources with even lower radio fluxes planned by such surveys . The results also represent a reference point when searching for H I absorption at higher redshifts .