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I'm performing a radioactivity experiment where I measure a specific number of counts in some time period t.

Later on I take the total count rate. (Number of counts/time: $N/t$)

I'm supposed to find the error in total count rate. My initial impression was to do some sort of standard error measurement and go from there. I've been told, however, that the error in total counts measured is simply the square root of $N$

Why is this the case? I've got the impression it's to do with the data following some sort of distribution or something, but is there an obvious answer that I'm missing?

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    $\begingroup$ This comes from the central limit theorem. There is a nice if somewhat detailed discussion of this on the Stats SE. $\endgroup$
    – John Rennie
    Commented Oct 8, 2015 at 10:34
  • $\begingroup$ Thanks @John Rennie. I do now also have a simpler answer to my question but I'm not sure about posting it as I've been making a habit of answering my own questions lately! Better to delete the question or answer it myself? $\endgroup$
    – Matt
    Commented Oct 8, 2015 at 11:49
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    $\begingroup$ Answer it yourself. If it's a good answer then it will help future visitors to the site and earn you some rep :-) $\endgroup$
    – John Rennie
    Commented Oct 8, 2015 at 11:56

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The answer to my question turned out to be in a fantastic book called Radiation Detection and Measurement by Glenn Knoll, which has been my nuclear physics bible so far and has come up trumps again!

The key is the fact that my data were distributed in a Gaussian distribution. If you're given a Gaussian or Poisson distribution, you can approximate the standard deviation to the square root of the mean.

So, for each individual count, the standard error is the square root of the mean for that measurement. The mean for that count, given that it's only one measurement, is the count itself.

If you sum a lot of individual counts, to then get the variance you sum the variances for each count. The variances of each count are the squares of their individual standard deviations. In this case, it's the square of the square root of the mean for an individual count, i.e., the count itself.

Therefore, the variance of the sum of measurements is the sum of the measurements themselves.

The error is the square root of the variance, i.e., the square root of the sum of the counts.

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