Photon shot noise

Independent photon arrivals have a Poisson distribution: a mean count $\mu$ has variance $\mu$ and standard deviation $\sqrt{\mu }$. Subtraction of independent counting measurements adds their variances.

For a known-background Poisson measurement with mean $N$, the standard deviation is $\sqrt{N}$. A small added mean $\Delta N$ has approximate signal-to-noise ratio $\Delta N/\sqrt{N}$ when background counting noise dominates. A fixed criterion $z$ therefore gives $\Delta N\approx z\sqrt{N}$ and fractional contrast $z/\sqrt{N}$. This is a stated observation model, not a universal human threshold or a two-alternative comparison formula.

Subtracting a reproducible sky, dark or bias mean removes its offset but does not undo random arrivals or electronic fluctuations already present in a science exposure. An independently estimated calibration adds its own uncertainty. Longer integration reduces relative independent counting noise; correlated backgrounds and systematic residuals need separate models and may set an accuracy floor.