We present measurements of Fe, Mg, Si, Ca, and Ti abundances for 388 radial velocity member stars in the Sculptor dwarf spheroidal galaxy (dSph), a satellite of the Milky Way. This is the largest sample of individual alpha element (Mg, Si, Ca, Ti) abundance measurements in any single dSph. The measurements are made from Keck/DEIMOS medium-resolution spectra (6400-9000 A, R ~ 6500). Based on comparisons to published high-resolution (R >~ 20000) spectroscopic measurements, our measurements have uncertainties of sigma([Fe/H]) = 0.14 and sigma([alpha/Fe]) = 0.13. The Sculptor [Fe/H] distribution has a mean <[Fe/H]> = -1.58 and is asymmetric with a long, metal-poor tail, indicative of a history of extended star formation. Sculptor has a larger fraction of stars with [Fe/H] < -2 than the Milky Way halo. We have discovered one star with [Fe/H] = -3.80 +/- 0.28, which is the most metal-poor star known anywhere except the Milky Way halo, but high-resolution spectroscopy is needed to measure this stars detailed abundances. As has been previously reported based on high-resolution spectroscopy, [alpha/Fe] in Sculptor falls as [Fe/H] increases. The metal-rich stars ([Fe/H] ~ -1.5) have lower [alpha/Fe] than Galactic halo field stars of comparable metallicity. This indicates that star formation proceeded more gradually in Sculptor than in the Galactic halo. We also observe radial abundance gradients of -0.030 +/- 0.003 dex per arcmin in [Fe/H] and +0.013 +/- 0.003 dex per arcmin in [alpha/Fe] out to 11 arcmin (275 pc). Together, these measurements cast Sculptor and possibly other surviving dSphs as representative of the dwarf galaxies from which the metal-poor tail of the Galactic halo formed.
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