Spin measurements of supermassive black holes (SMBHs) provide crucial constraints on the accretion processes that power active galactic nuclei (AGN), fuel outflows, and trigger black hole growth. However, spin measurements are mainly limited to a few dozen nearby sources for which high quality, high S/N spectra (e.g., from Chandra, XMM-Newton, Suzaku, NuSTAR) are available. Here we measure the average SMBH spin of $sim$1900 AGN in the Chandra COSMOS-Legacy survey using spectral stacking analysis. We find broad Fe K$alpha$ line emission in the average COSMOS spectrum (Gaussian width $sigma=0.27pm0.05$ keV), and by fitting this emission line profile with relativistic line models, we measure the average black hole spin parameter $a=0.62~substack{+0.07 -0.17}$. The sample size, availability of multiwavelength data, and spatial resolution of the COSMOS Legacy field also provide a unique environment to investigate the average SMBH spin as a function of other observables (e.g., redshift, luminosity) up to $zsim5.3$. We find that optically classified Type 1 sources have broader Fe K$alpha$ line emission than Type 2 sources. X-ray unobscured and obscured sources, as defined by their column densities, have widths that are consistent with the optically defined unobscured and obscured sources, respectively. There is some evidence for evolution of the Fe K$alpha$ width and black hole spin parameter with luminosity, but not conclusively with redshift. The results of this work provide insights into the average spins of SMBHs in AGN, shedding light on their growth mechanisms and observed co-evolution with their host galaxies.
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