We experimentally investigate the dynamics of backscattered light due to stimulated Brillouin scattering (SBS) in a narrow-linewidth continuous wave (CW) fiber amplifier. We observe the onset of sharp intensity variations in the backscattered radiation as we increase the pump power, which when analyzed using Karl-Pearson's correlation coefficient reveals a distinct structure. We find that such structure is associated with the onset of stimulated Brillouin scattering (SBS) in the fiber amplifier. Moreover, at higher pump power levels we observe a periodic signature in the Karl-Pearson correlation trace that precedes an observation of kW pulses in the backscattered radiation. Based on controlled experiments, we conclude that the formation of the above kW pulses in our system is preceded by the onset of SBS.