  The first generations of stars left their chemical fingerprints on metal-poor
stars in the Milky Way and its surrounding dwarf galaxies. While instantaneous
and homogeneous enrichment implies that groups of co-natal stars should have
the same element abundances, small amplitudes of abundance scatter are seen at
fixed [Fe/H]. Measurements of intrinsic abundance scatter have been made with
small, high-resolution spectroscopic datasets where measurement uncertainty is
small compared to this scatter. In this work, we present a method to use
mid-resolution survey data, which has larger errors, to make this measurement.
Using APOGEE DR17, we calculate the intrinsic scatter of Al, O, Mg, Si, Ti, Ni,
and Mn relative to Fe for 333 metal-poor stars across 6 classical dwarf
galaxies around the Milky Way, and 1604 stars across 19 globular clusters. We
first calibrate the reported abundance errors in bins of signal-to-noise and
[Fe/H] using a high-fidelity halo dataset. We then apply these calibrated
errors to the APOGEE data, and find small amplitudes of average intrinsic
abundance scatter in dwarf galaxies ranging from 0.032 $-$ 0.14 dex with a
median value of 0.043 dex. For the globular clusters, we find intrinsic
scatters ranging from 0.018 $-$ 0.21 dex, with particularly high scatter for Al
and O. Our measurements of intrinsic abundance scatter place important upper
limits on the intrinsic scatter in these systems, as well as constraints on
their underlying star formation history and mixing, that we can look to
simulations to interpret.
