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Assume that there are photons arriving at a detector according to a Poission distribution. Let's say we detect 100 photons over 10 seconds.

Now I see two ways to calculate the statistical error $s$, which one of these is correct?

  1. First, the statistical error for the 100 counts: $s_1=\sqrt{100}=10$, and then via Gaussian error propagation the error for the counts/second: $s=\frac{s_1}{10 s}=1 \frac{1}{s}$.

  2. Alternatively, we just apply the square too immediately to counts/s: $s=\sqrt(100/10)=\sqrt{10}\approx 3$

Which one of these two ways is correct?

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The first is correct and the second is diabolically wrong.

People get the idea that Poisson errors mean you just take the square root of everything. No! The thing you take the square root of has to be an actual number of events. Any scaling factors are applied afterwards.

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