Many experiments detect discrete events rather than continuous values: radioactive decays, photons, particle hits or rare transitions. If events occur independently at a constant mean rate, the number counted in a fixed interval is modeled by a Poisson distribution.

If $N$ events are counted, the standard deviation is approximately $$\sigma_N=\sqrt N$$ for sufficiently large $N$. Relative counting uncertainty therefore scales as $$\frac{\sigma_N}{N}=\frac{1}{\sqrt N}.$$ Collecting four times as many events halves the relative statistical uncertainty.

Real detectors also record background events. Suppose the signal-plus-background run produces $N_{SB}$ counts over time $t_{SB}$ and a separate background run produces $N_B$ counts over time $t_B$. The estimated signal count in the first exposure is $$N_S=N_{SB}-\frac{t_{SB}}{t_B}N_B.$$ If the two counts are independent Poisson variables, their variances add: $$\sigma_{N_S}^2=N_{SB}+\left(\frac{t_{SB}}{t_B}\right)^2N_B.$$

For equal exposure times, $N_{SB}=1200$ and $N_B=800$ give an estimated signal of $400$ counts but uncertainty $$\sqrt{1200+800}\approx44.7,$$ not $\sqrt{400}=20$. Background subtraction removes an estimated mean background; it does not remove the background's statistical fluctuation.

Counting experiments therefore require both rate normalization and correct uncertainty propagation. This logic underlies radiation measurements, photon counting, particle detectors and many low-signal experiments.