  The ability to measure metallicities and {\alpha}-element abundances in
individual red giant branch (RGB) stars using medium-resolution spectra ($R
\approx 6000$) is a valuable tool for deciphering the nature of Milky Way dwarf
satellites and the history of the Galactic halo. Extending such studies to more
distant systems like Andromeda is beyond the ability of the current generation
of telescopes, but by co-adding the spectra of similar stars, we can attain the
necessary signal-to-noise ratio to make detailed abundance measurements. In
this paper, we present a method to determine metallicities and {\alpha}-element
abundances using the co-addition of medium resolution spectra. We test the
method of spectral co-addition using high-S/N spectra of more than 1300 RGB
stars from Milky Way globular clusters and dwarf spheroidal galaxies obtained
with the Keck II telescope/DEIMOS spectrograph. We group similar stars using
photometric criteria and compare the weighted ensemble average abundances
([Fe/H], [Mg/Fe], [Si/Fe], [Ca/Fe] and [Ti/Fe]) of individual stars in each
group with the measurements made on the corresponding co-added spectrum. We
find a high level of agreement between the two methods, which permits us to
apply this co-added spectra technique to more distant RGB stars, like stars in
the M31 satellite galaxies. This paper outlines our spectral co-addition and
abundance measurement methodology and describes the potential biases in making
these measurements.
