The quantum mechanics of position measurement of a macroscopic object is typically inaccessible because of strong coupling to the environment and classical noise. In this work, we monitor a mechanical resonator subject to an increasingly strong continuous position measurement and observe a quantum mechanical back-action force that rises in accordance with the Heisenberg uncertainty limit. For our optically based position measurements, the back-action takes the form of a fluctuating radiation pressure from the Poisson-distributed photons in the coherent measurement field, termed radiation pressure shot noise. We demonstrate a back-action force that is comparable in magnitude to the thermal forces in our system. Additionally, we observe a temporal correlation between fluctuations in the radiation force and in the position of the resonator.