Measurements of oxygen abundance throughout galaxies provide insight to the formation histories and ongoing processes. Here we present a study of the gas-phase oxygen abundance in the H ii regions and diffuse gas of the nearby starburst dwarf galaxy, IC 10. Using the Keck Cosmic Web Imager at W. M. Keck Observatory, we map the central region of IC 10 from 3500 to 5500 Å. The auroral [O III] 4363 Å line is detected with a high signal-to-noise ratio in 12 of 46 H II regions observed, allowing for direct measurement of the oxygen abundance, yielding a median and standard deviation of 12+log(O/H)=8.37±0.25 . We investigate trends between these directly measured oxygen abundances and other H II region properties, finding weak negative correlations with the radius, velocity dispersion, and luminosity. We also find weak negative correlations between the oxygen abundance and the derived quantities of turbulent pressure and ionized gas mass, and a moderate correlation with the derived dynamical mass. Strong line, R 23 abundance estimates are used in the remainder of the H II regions and on a resolved spaxel-by-spaxel basis. There is a large offset between the abundances measured with R 23 and the auroral line method. We find that the R 23 method is unable to capture the large range of abundances observed via the auroral line measurements. The extent of this variation in the measured abundances further indicates a poorly mixed interstellar medium in IC 10, which is not typical of dwarf galaxies and may be partly due to the ongoing starburst, accretion of pristine gas, or a late stage merger.
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