Scaled statistical error

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Everyone — and by “everyone” I mean anyone who analyzes discrete random processes on a regular basis (quite an inclusive group, I know) — knows that the statistical error in a random count of events is the square root of the count. (For those of you not in the know, when you’re examining some process in which events occur at random intervals, e.g. the decay of radioactive atoms, if you count the number of events that occur in a minute over and over again for a large number of minutes, you’ll have some average number of decays per minute, and the distribution of counts will have a standard deviation of the square root of that average.)

But what happens when the quantity you’re really interested in is not the count itself, but something proportional to the count? What’s the statistical uncertainty in that? It’s actually fairly simple to figure out based on the usual formula for error propagation. If the quantity you’re measuring is denoted \(I\) and it’s related to the count by

$$I = A N$$

where \(A\) is some constant (or anything independent of the count), then

$$\delta I = A \delta N = A \sqrt{N} = A \sqrt{\frac{I}{A}} = \sqrt{A I}$$

A prime example is light: when you’re using a photodetector to measure the amount of light emitted by something, the detector counts photons per unit time, but what you usually care about is power. The relationship is simple

$$P = \frac{h c}{\lambda t}N$$

so according to the formula above, the uncertainty in power is just

$$\delta P = \sqrt{\frac{h c}{\lambda t}P}$$

(of course, this is only valid at a single wavelength).