We study winds in 12 X-ray AGN host galaxies at z ~ 1. We find, using the low-ionization Fe II 2586 absorption in the stacked spectra, that the probability distribution function (PDF) of the centroid velocity shift in AGN has a median, 16th and 84th percentiles of (-87, -251, +86) km/s respectively. The PDF of the velocity dispersion in AGN has a median, 84th and 16th percentile of (139, 253, 52) km/s respectively. The centroid velocity and the velocity dispersions are obtained from a two

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... ent (ISM+wind) absorption line model. The equivalent width PDF of the outflow in AGN has median, 84th and 16th percentiles of (0.4, 0.8, 0.1) Angstrom. There is a strong ISM component in Fe II 2586 absorption with (1.2, 1.5, 0.8) Angstrom, implying presence of substantial amount cold gas in the host galaxies. For comparison, star-forming and X-ray undetected galaxies at a similar redshift, matched roughly in stellar mass and galaxy inclination, have a centroid velocity PDF with percentiles of (-74, -258, +90) km/s, and a velocity dispersion PDF percentiles of (150, 259, 57) km/s. Thus, winds in the AGN are similar to star-formation-driven winds, and are too weak to escape and expel substantial cool gas from galaxies. Our sample doubles the previous sample of AGN studied at z ~ 0.5 and extends the analysis to z ~ 1. A joint reanalysis of the z ~ 0.5 AGN sample and our sample yields consistent results to the measurements above.