  We measure abundances of 12 elements (Na, Mg, Si, Ca, Sc, Ti, V, Cr, Mn, Fe,
Co, Ni) in a sample of 86 metal-poor ($-2 \lesssim \text{[Fe/H]} \lesssim -1$)
subgiant stars in the solar neighborhood. Abundances are derived from
high-resolution spectra taken with the Potsdam Echelle Polarimetric and
Spectroscopic Instrument on the Large Binocular Telescope, modeled using iSpec
and MOOG. By carefully quantifying the impact of photon-noise ($< 0.05$ dex for
all elements) we robustly measure the intrinsic scatter of abundance ratios. At
fixed [Fe/H] the RMS intrinsic scatter in [X/Fe] ranges from 0.04 dex (Cr) to
0.16 dex (Na), with a median of 0.08 dex. Scatter in [X/Mg] is similar, and
accounting for [$\alpha$/Fe] only reduces the overall scatter moderately. We
consider several possible origins of the intrinsic scatter with particular
attention to fluctuations in the relative enrichment by core-collapse
supernovae (CCSN) and Type Ia supernovae (SNIa) and stochastic sampling of the
CCSN progenitor mass distribution. The stochastic sampling scenario provides a
good quantitative explanation of our data if the effective number of CCSN
contributing to the enrichment of a typical sample star is $N \sim 50$. At the
median metallicity of our sample, this interpretation implies that the CCSN
ejecta are mixed over a gas mass $\sim 10^5 M_{\odot}$ before forming stars.
The scatter of elemental abundance ratios is a powerful diagnostic test for
simulations of star formation, feedback, and gas mixing in the early phases of
the Galaxy.
