Even-denominator fractional quantum Hall states (FQHSs) fall outside the standard Laughlin’s and Jain’s odd-denominator hierarchy. They are of intense current interest because they may exhibit non-Abelian topological order and therefore have applications in topological quantum computation. In this Letter, we study the FQHS at filling factor ν=1/2, observed in a two-dimensional electron system confined to a wide GaAs quantum well. Unlike other even-denominator FQHSs, this state is stabilized at the lowest Landau level and has an energy gap about 10 times larger than the widely studied ν=5/2 FQHS. Moreover, a recently observed pair of daughter FQHSs theoretically predicted the parent ν=1/2 FQHS to be a non-Abelian state. Here, we report shot-noise measurements of partitioned quasiparticles at ν=1/2 by an etch-defined quantum point contact (QPC). Our measurements were performed on two nominally identical devices in two independent experimental setups. Analysis of the shot noise in the weak-backscattering regime (of the QPC) reveals quasiparticles with charge e/4, consistent with a non-Abelian order, although it does not rule out an Abelian order. These observations provide a clear benchmark for future studies aimed at probing the topological order of the ν=1/2 FQHS and its quasiparticles’ exchange statistics.