We study the noise of the intensity variance and of the intensity correlation and structure functions measured in light scattering from a random medium in the case when these quantities are obtained by averaging over a finite number N of pixels of a digital camera. We show that the noise scales as 1/N in all cases and that it is sensitive to correlations of signals corresponding to adjacent pixels as well as to the effective time averaging (due to the finite integration time) and spatial averaging (due to the finite pixel size). Our results provide a guide to estimation of noise levels in such applications as multi-speckle dynamic light scattering, time-resolved correlation spectroscopy, speckle visibility spectroscopy, laser speckle imaging etc.