Signal To Noise For Binned vs. Unbinned Images

In the first diagram, showing 1x1 binning of a stellar object, the image falls completely on one of the pixels. If x denotes the signal, then the S/N is

S/N ~ x / x^1/2 = x^1/2

In the second diagram, showing 2x2 binning of the same stellar object, the signal again is x and the S/N is again

S/N ~ x / x^1/2 = x^1/2

Thus in this case, the S/N is the same in the binned and unbinned case.

However, in the third diagram, showing 1x1 binning of the same stellar object, the image falls on four pixels. In this case, the signal for each pixel is ¼ x, so the noise on each pixel is

S/N ~ ¼ x / ( ¼ x )^1/2 = ½ x^1/2

Thus, in this unbinned case the S/N is ½ that of the binned case. Moreover, this same reasoning applies to extended objects which are spread over the entire binned pixel, so for extended objects the S/N in the unbinned imaged is ½ that in the binned image.